ࡱ> K  i             { U bnv__hww ` e!"""}##{$$S@ bjbj `&[ %%%%CCC8C Ot%2hDSW>"``LV@jl$R%B V`LWgPgPgPB`%LgPgPgP{Pv ' J%L8S `uB!Cs/U\ 02yB %%%vrgP\D%%$5)N>%%5 Chapter 1 Objectives and Methods of Solid Mechanics Defining a problem in solid mechanics For each of the following applications, outline briefly: What would you calculate if you were asked to model the component for a design application? What level of detail is required in modeling the geometry of the solid? How would you model loading applied to the solid? Would you conduct a static or dynamic analysis? Is it necessary to account for thermal stresses? Is it necessary to account for temperature variation as a function of time? What constitutive law would you use to model the material behavior? A load cell intended to model forces applied to a specimen in a tensile testing machine The seat-belt assembly in a vehicle The solar panels on a communications satellite. A compressor blade in a gas turbine engine A MEMS optical switch An artificial knee joint A solder joint on a printed circuit board An entire printed circuit board assembly The metal interconnects inside a microelectronic circuit What is the difference between a linear elastic stress-strain law and a hyperelastic stress-strain law? Give examples of representative applications for both material models. What is the difference between a rate-dependent (viscoplastic) and rate independent plastic constitutive law? Give examples of representative applications for both material models. Choose a recent publication describing an application of theoretical or computational solid mechanics from one of the following journals: Journal of the Mechanics and Physics of Solids; International Journal of Solids and Structures; Modeling and Simulation in Materials Science and Engineering; European Journal of Mechanics A; Computer methods in Applied Mechanics and Engineering. Write a short summary of the paper stating: (i) the goal of the paper; (ii) the problem that was solved, including idealizations and assumptions involved in the analysis; (iii) the method of analysis; and (iv) the main results; and (v) the conclusions of the study Chapter 2 Governing Equations Mathematical Description of Shape Changes in Solids  A thin film of material is deformed in simple shear during a plate impact experiment, as shown in the figure. Write down expressions for the displacement field in the film, in terms of  EMBED Equation  d and h, expressing your answer as components in the basis shown. Calculate the Lagrange strain tensor associated with the deformation, expressing your answer as components in the basis shown. Calculate the infinitesimal strain tensor for the deformation, expressing your answer as components in the basis shown. Find the principal values of the infinitesimal strain tensor, in terms of d and h Find a displacement field corresponding to a uniform infinitesimal strain field  EMBED Equation.DSMT4 . (Dont make this hard in particular do not use the complicated approach described in Section 2.1.20. Instead, think about what kind of function, when differentiated, gives a constant). Is the displacement unique? Find a formula for the displacement field that generates zero infinitesimal strain. Find a displacement field that corresponds to a uniform Lagrange strain tensor  EMBED Equation.DSMT4 . Is the displacement unique? Find a formula for the most general displacement field that generates a uniform Lagrange strain. The displacement field in a homogeneous, isotropic circular shaft twisted through angle  EMBED Equation  at one end is given by  EMBED Equation  Calculate the matrix of components of the deformation gradient tensor Calculate the matrix of components of the Lagrange strain tensor. Is the strain tensor a function of  EMBED Equation ? Why? Find an expression for the increase in length of a material fiber of initial length dl, which is on the outer surface of the cylinder and initially oriented in the  EMBED Equation direction. Show that material fibers initially oriented in the EMBED Equation  and EMBED Equation  directions do not change their length. Calculate the principal values and directions of the Lagrange strain tensor at the point  EMBED Equation . Hence, deduce the orientations of the material fibers that have the greatest and smallest increase in length. Calculate the components of the infinitesimal strain tensor. Show that, for small values of  EMBED Equation , the infinitesimal strain tensor is identical to the Lagrange strain tensor, but for finite rotations the two measures of deformation differ. Use the infinitesimal strain tensor to obtain estimates for the lengths of material fibers initially oriented with the three basis vectors. Where is the error in this estimate greatest? How large can  EMBED Equation  be before the error in this estimate reaches 10%? To measure the in-plane deformation of a sheet of metal during a forming process, your managers place three small hardness indentations on the sheet. Using a travelling microscope, they determine that the initial lengths of the sides of the triangle formed by the three indents are 1cm, 1cm, 1.414cm, as shown in the picture below. After deformation, the sides have lengths 1.5cm, 2.0cm and 2.8cm. Your managers would like to use this information to determine the inplane components of the Lagrange strain tensor. Unfortunately, being business economics graduates, they are unable to do this. Explain how the measurements can be used to determine  EMBED Equation  and do the calculation. Is it possible to determine the deformation gradient from the measurements provided? Why? If not, what additional measurements would be required to determine the deformation gradient? To track the deformation in a slowly moving glacier, three survey stations are installed in the shape of an equilateral triangle, spaced 100m apart, as shown in the picture. After a suitable period of time, the spacing between the three stations is measured again, and found to be 90m, 110m and 120m, as shown in the figure. Assuming that the deformation of the glacier is homogeneous over the region spanned by the survey stations, please compute the components of the Lagrange strain tensor associated with this deformation, expressing your answer as components in the basis shown. Compose a limerick that will help you to remember the distinction between engineering shear strains and the formal (mathematical) definition of shear strain. A rigid body motion is a nonzero displacement field that does not distort any infinitesimal volume element within a solid. Thus, a rigid body displacement induces no strain, and hence no stress, in the solid. The deformation corresponding to a 3D rigid rotation about an axis through the origin is  EMBED Equation.DSMT4  where R must satisfy  EMBED Equation.DSMT4 , det(R)>0. Show that the Lagrange strain associated with this deformation is zero. As a specific example, consider the deformation  EMBED Equation  This is the displacement field caused by rotating a solid through an angle  EMBED Equation  about the  EMBED Equation  axis. Find the deformation gradient for this displacement field, and show that the deformation gradient tensor is orthogonal, as predicted above. Show also that the infinitesimal strain tensor for this displacement field is not generally zero, but is of order  EMBED Equation  if  EMBED Equation  is small. If the displacements are small, we can find a simpler representation for a rigid body displacement. Consider a deformation of the form  EMBED Equation  Here  EMBED Equation.DSMT4  is a vector with magnitude <<1, which represents an infinitesimal rotation about an axis parallel to  EMBED Equation.DSMT4 . Show that the infinitesimal strain tensor associated with this displacement is always zero. Show further that the Lagrange strain associated with this displacement field is  EMBED Equation  This is not, in general, zero. It is small if all  EMBED Equation . The formula for the deformation due to a rotation through an angle  EMBED Equation.DSMT4  about an axis parallel to a unit vector n that passes through the origin is  EMBED Equation.DSMT4  Calculate the components of corresponding deformation gradient Verify that the deformation gradient satisfies  EMBED Equation.DSMT4  Find the components of the inverse of the deformation gradient Verify that both the Lagrange strain tensor and the Eulerian strain tensor are zero for this deformation. What does this tell you about the distorsion of the material? Calculate the Jacobian of the deformation gradient. What does this tell you about volume changes associated with the deformation?  In Section 2.1.6 it was stated that the Eulerian strain tensor  EMBED Equation.DSMT4  can be used to relate the length of a material fiber in a deformable solid before and after deformation, using the formula  EMBED Equation  where  EMBED Equation.DSMT4  are the components of a unit vector parallel to the material fiber after deformation. Derive this result. Suppose that you have measured the Lagrange strain tensor for a deformation, and wish to calculate the Eulerian strain tensor. On purely physical grounds, do you think this is possible, without calculating the deformation gradient? If so, find a formula relating Lagrange strain  EMBED Equation.DSMT4  to Eulerian strain  EMBED Equation.DSMT4 . Repeat problem 2.1.6, but instead of calculating the Lagrange strain tensor, find the components of the Eulerian strain tensor  EMBED Equation.DSMT4  (you can do this directly, or use the results of problem 2.1.12, or both) Repeat problem 2.1.7, but instead of calculating the Lagrange strain tensor, find the components of the Eulerian strain tensor  EMBED Equation.DSMT4  (you can do this directly, or use the results for problem 2.1.12, or both) The Lagrange strain tensor can be used to calculate the change in angle between any two material fibers in a solid as the solid is deformed. In this problem you will calculate the formula that can be used to do this. To this end, consider two infinitesimal material fibers in the undeformed solid, which are characterized by vectors with components  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are two unit vectors. Recall that the angle  EMBED Equation.DSMT4  between  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  before deformation can be calculated from  EMBED Equation.DSMT4 . Let  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  represent the two material fibers after deformation. Show that the angle between  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  can be calculated from the formula  EMBED Equation.DSMT4   EMBED Designer.Drawing.8  Suppose that a solid is subjected to a sequence of two homogeneous deformations (i) a rigid rotation R, followed by (ii) an arbitrary homogeneous deformation F. Taking the original configuration as reference, find formulas for the following deformation measures for the final configuration of the solid, in terms of F and R: The deformation gradient The Left and Right Cauchy-Green deformation tensors The Lagrange strain The Eulerian strain. Repeat problem 2.1.16, but this time assume that the sequence of the two deformations is reversed, i.e. the solid is first subjected to an arbitrary homogeneous deformation F, and is subsequently subjected to a rigid rotation R. A spherical shell (see the figure) is made from an incompressible material. In its undeformed state, the inner and outer radii of the shell are  EMBED Equation.DSMT4 . After deformation, the new values are  EMBED Equation.DSMT4 . The deformation in the shell can be described (in Cartesian components) by the equation  EMBED Equation.DSMT4  Calculate the components of the deformation gradient tensor Verify that the deformation is volume preserving Find the deformed length of an infinitesimal radial line that has initial length  EMBED Equation.DSMT4 , expressed as a function of R Find the deformed length of an infinitesimal circumferential line that has initial length  EMBED Equation.DSMT4 , expressed as a function of R Using the results of 2.1.18.3, 2.1.18.4, find the principal stretches for the deformation. Find the inverse of the deformation gradient, expressed as a function of  EMBED Equation.DSMT4 . It is best to do this by working out a formula that enables you to calculate  EMBED Equation.DSMT4  in terms of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  and differentiate the result rather than to attempt to invert the result of 10.1. Suppose that the spherical shell described in Problem 2.1.18 is continuously expanding (visualize a balloon being inflated). The rate of expansion can be characterized by the velocity  EMBED Equation.DSMT4  of the surface that lies at R=A in the undeformed cylinder. Calculate the velocity field  EMBED Equation.DSMT4  in the sphere as a function of  EMBED Equation.DSMT4  Calculate the velocity field as a function of  EMBED Equation.DSMT4  (there is a long, obvious way to do this and a quick, subtle way) Calculate the time derivative of the deformation gradient tensor calculated in 2.1.18.1. Calculate the components of the velocity gradient  EMBED Equation.DSMT4  by differentiating the result of 2.1.19.1 Calculate the components of the velocity gradient using the results of 2.1.19.3 and 2.1.18.6 Calculate the stretch rate tensor  EMBED Equation.DSMT4 . Verify that the result represents a volume preserving stretch rate field. Repeat Problem 2.1.18.1, 2.1.18.6 and all of 2.1.19, but this time solve the problem using spherical-polar coordinates, using the various formulas for vector and tensor operations given in Appendix E. In this case, you may assume that a point with position  EMBED Equation.DSMT4  in the undeformed solid has position vector  EMBED Equation.DSMT4  after deformation. An initially straight beam is bent into a circle with radius R as shown in the figure. Material fibers that are perpendicular to the axis of the undeformed beam are assumed to remain perpendicular to the axis after deformation, and the beams thickness and the length of its axis are assumed to be unchanged. Under these conditions the deformation can be described as  EMBED Equation.DSMT4  where, as usual x is the position of a material particle in the undeformed beam, and y is the position of the same particle after deformation. Calculate the deformation gradient field in the beam, expressing your answer as a function of  EMBED Equation.DSMT4 , and as components in the basis  EMBED Equation.DSMT4  shown. Calculate the Lagrange strain field in the beam. Calculate the infinitesimal strain field in the beam. Compare the values of Lagrange strain and infinitesimal strain for two points that lie at  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Explain briefly the physical origin of the difference between the two strain measures at each point. Recommend maximum allowable values of h/R and L/R for use of the infinitesimal strain measure in modeling beam deflections. Calculate the deformed length of an infinitesimal material fiber that has length  EMBED Equation.DSMT4  and orientation  EMBED Equation.DSMT4  in the undeformed beam. Express your answer as a function of  EMBED Equation.DSMT4 . Calculate the change in length of an infinitesimal material fiber that has length  EMBED Equation.DSMT4  and orientation  EMBED Equation.DSMT4  in the undeformed beam. Show that the two material fibers described in 2.1.21.5 and 2.1.21.6 remain mutually perpendicular after deformation. Is this true for all material fibers that are mutually perpendicular in the undeformed solid? Find the components in the basis  EMBED Equation.DSMT4  of the Left and Right stretch tensors  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  as well as the rotation tensor  EMBED Equation.DSMT4  for this deformation. You should be able to write down  EMBED Equation.DSMT4  and R by inspection, without needing to wade through the laborious general process outlined in Section 2.1.13. The results can then be used to calculate  EMBED Equation.DSMT4 . Find the principal directions of  EMBED Equation.DSMT4  as well as the principal stretches. You should be able to write these down using your physical intuition without doing any tedious calculations. Let  EMBED Equation.DSMT4  be a basis in which  EMBED Equation.DSMT4  is parallel to the axis of the deformed beam, as shown in the figure. Write down the components of each of the unit vectors  EMBED Equation.DSMT4  in the basis  EMBED Equation.DSMT4 . Hence, compute the transformation matrix  EMBED Equation.DSMT4  that is used to transform tensor components from  EMBED Equation.DSMT4  to  EMBED Equation.DSMT4 . Find the components of the deformation gradient tensor, Lagrange strain tensor, as well as  EMBED Equation.DSMT4   EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  in the basis  EMBED Equation.DSMT4 . Find the principal directions of  EMBED Equation.DSMT4  expressed as components in the basis  EMBED Equation.DSMT4 . Again, you should be able to simply write down this result. A sheet of material is subjected to a two dimensional homogeneous deformation of the form  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  are constants. Suppose that a circle of unit radius is drawn on the undeformed sheet. This circle is distorted to a smooth curve on the deformed sheet. Show that the distorted circle is an ellipse, with semi-axes that are parallel to the principal directions of the left stretch tensor V, and that the lengths of the semi-axes of the ellipse are equal to the principal stretches for the deformation. There are many different ways to approach this calculation some are very involved. The simplest way is probably to assume that the principal directions of V subtend an angle  EMBED Equation.DSMT4  to the  EMBED Equation.DSMT4  basis as shown in the figure, write the polar decomposition  EMBED Equation.DSMT4  in terms of principal stretches  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , and then show that  EMBED Equation.DSMT4  (where  EMBED Equation.DSMT4  is on the unit circle) describes an ellipse. A solid is subjected to a rigid rotation so that a unit vector a in the undeformed solid is rotated to a new orientation b. Find a rotation tensor R that is consistent with this deformation, in terms of the components of a and b. Is the rotation tensor unique? If not, find the most general formula for the rotation tensor. In a plate impact experiment, a thin film of material with thickness h is subjected to a homogeneous shear deformation by displacing the upper surface of the film horizontally with a speed v. Write down the velocity field in the film Calculate the velocity gradient, the stretch rate and the spin rate Calculate the instantaneous angular velocity of a material fiber parallel to the  EMBED Equation.DSMT4  direction in the film Calculate the instantaneous angular velocity of a material fiber parallel to  EMBED Equation.DSMT4  Calculate the stretch rates for the material fibers in 22.3 and 22.4 What is the direction of the material fiber with the greatest angular velocity? What is the direction of the material fiber with the greatest stretch rate? The velocity field  EMBED Equation.DSMT4  due to a rigid rotation about an axis through the origin can be characterized by a skew tensor  EMBED Equation.DSMT4  or an angular velocity vector  EMBED Equation.DSMT4  defined so that  EMBED Equation.DSMT4  Find a formula relating the components of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . (One way to approach this problem is to calculate a formula for W by taking the time derivative of Rodriguez formula see Sect 2.1.1). A single crystal deforms by shearing on a single active slip system as illustrated in the figure. The crystal is loaded so that the slip direction  EMBED Equation.DSMT4  and normal to the slip plane  EMBED Equation.DSMT4  maintain a constant direction during the deformation Show that the deformation gradient can be expressed in terms of the components of the slip direction  EMBED Equation.DSMT4  and the normal to the slip plane m as  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  denotes the shear, as illustrated in the figure. Suppose shearing proceeds at some rate  EMBED Equation.DSMT4 . At the instant when  EMBED Equation.DSMT4 , calculate (i) the velocity gradient tensor; (ii) the stretch rate tensor and (iii) the spin tensor associated with the deformation. The properties of many rubbers and foams are specified by functions of the following invariants of the left Cauchy-Green deformation tensor  EMBED Equation .  EMBED Equation.DSMT4  Invariants of a tensor are defined in Appendix B they are functions of the components of a tensor that are independent of the choice of basis. Verify that  EMBED Equation.DSMT4  are invariants. The simplest way to do this is to show that  EMBED Equation.DSMT4  are unchanged during a change of basis. In order to calculate stress-strain relations for these materials, it is necessary to evaluate derivatives of the invariants. Show that  EMBED Equation.DSMT4  The infinitesimal strain field in a long cylinder containing a hole at its center is given by  EMBED Equation.DSMT4  Show that the strain field satisfies the equations of compatibility. Show that the strain field is consistent with a displacement field of the form  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 . Note that although the strain field is compatible, the displacement field is multiple valued i.e. the displacements are not equal at  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , which supposedly represent the same point in the solid. Surprisingly, displacement fields like this do exist in solids they are caused by dislocations in a crystal. These are discussed in more detail in Sections 5.3.4 The figure shows a test designed to measure the response of a polymer to large shear strains. The sample is a hollow cylinder with internal radius  EMBED Equation.DSMT4  and external radius  EMBED Equation.DSMT4 . The inside diameter is bonded to a fixed rigid cylinder. The external diameter is bonded inside a rigid tube, which is rotated through an angle  EMBED Equation.DSMT4 . Assume that the specimen deforms as indicated in the figure, i.e. (a) cylindrical sections remain cylindrical; (b) no point in the specimen moves in the axial or radial directions; (c) that a cylindrical element of material at radius  EMBED Equation.DSMT4  rotates through angle  EMBED Equation.DSMT4  about the axis of the specimen. Take the undeformed configuration as reference. Let  EMBED Equation.DSMT4 denote the cylindrical-polar coordinates of a material point in the reference configuration, and let  EMBED Equation.DSMT4 be cylindrical-polar basis vectors at  EMBED Equation.DSMT4 . Let  EMBED Equation.DSMT4  denote the coordinates of this point in the deformed configuration, and let  EMBED Equation.DSMT4 by cylindrical-polar basis vectors located at  EMBED Equation.DSMT4 . Write down expressions for  EMBED Equation.DSMT4  in terms of  EMBED Equation.DSMT4 (this constitutes the deformation mapping) Let P denote the material point at at  EMBED Equation.DSMT4  in the reference configuration. Write down the reference position vector X of P, expressing your answer as components in the basis  EMBED Equation.DSMT4 . Write down the deformed position vector x of P, expressing your answer in terms of  EMBED Equation.DSMT4 and basis vectors  EMBED Equation.DSMT4 . Find the components of the deformation gradient tensor F in  EMBED Equation.DSMT4 . (Recall that the gradient operator in cylindrical-polar coordinates is  EMBED Equation.DSMT4 ; recall also that  EMBED Equation.DSMT4 ) Show that the deformation gradient can be decomposed into a sequence  EMBED Equation.DSMT4  of a simple shear  EMBED Equation.DSMT4  followed by a rigid rotation through angle  EMBED Equation.DSMT4  about the  EMBED Equation.DSMT4  direction R. In this case the simple shear deformation will have the form  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is to be determined. Find the components of F in  EMBED Equation.DSMT4 . Verify that the deformation is volume preserving (i.e. check the value of J=det(F)) Find the components of the right Cauchy-Green deformation tensors in  EMBED Equation.DSMT4  Find the components of the left Cauchy-Green deformation tensor in  EMBED Equation.DSMT4  Find  EMBED Equation.DSMT4  in  EMBED Equation.DSMT4 . Find the principal values of the stretch tensor U Write down the velocity field v in terms of  EMBED Equation.DSMT4  in the basis  EMBED Equation.DSMT4  Calculate the spatial velocity gradient L in the basis  EMBED Equation.DSMT4  Mathematical Description of Internal Forces in Solids A rectangular bar is loaded in a state of uniaxial tension, as shown in the figure. Write down the components of the stress tensor in the bar, using the basis vectors shown. Calculate the components of the normal vector to the plane ABCD shown, and hence deduce the components of the traction vector acting on this plane, expressing your answer as components in the basis shown, in terms of  EMBED Equation  Compute the normal and tangential tractions acting on the plane shown. Consider a state of hydrostatic stress  EMBED Equation . Show that the traction vector acting on any internal plane in the solid (or, more likely, fluid!) has magnitude p and direction normal to the plane.  A cylinder of radius R is partially immersed in a static fluid. Recall that the pressure at a depth d in a fluid has magnitude  EMBED Equation . Write down an expression for the horizontal and vertical components of traction acting on the surface of the cylinder in terms of  EMBED Equation . Hence compute the resultant force exerted by the fluid on the cylinder.  The figure shows two designs for a glue joint. The glue will fail if the stress acting normal to the joint exceeds 60 MPa, or if the shear stress acting parallel to the plane of the joint exceeds 300 MPa. Calculate the normal and shear stress acting on each joint, in terms of the applied stress  EMBED Equation  Hence, calculate the value of  EMBED Equation  that will cause each joint to fail. For the Cauchy stress tensor with components  EMBED Equation.DSMT4  compute The traction vector acting on an internal material plane with normal  EMBED Equation.DSMT4  The principal stresses The hydrostatic stress The deviatoric stress tensor The Von-Mises equivalent stress Show that the hydrostatic stress  EMBED Equation.DSMT4  is invariant under a change of basis i.e. if  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  denote the components of stress in bases  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , respectively, show that  EMBED Equation.DSMT4 . A rigid, cubic solid is immersed in a fluid with mass density  EMBED Equation . Recall that a stationary fluid exerts a compressive pressure of magnitude  EMBED Equation  at depth h. Write down expressions for the traction vector exerted by the fluid on each face of the cube. You might find it convenient to take the origin for your coordinate system at the center of the cube, and take basis vectors  EMBED Equation.DSMT4  perpendicular to the cube faces. Calculate the resultant force due to the tractions acting on the cube, and show that the vertical force is equal and opposite to the weight of fluid displaced by the cube. Show that the result of problem 2.2.7 applies to any arbitrarily shaped solid immersed below the surface of a fluid, i.e. prove that the resultant force acting on an immersed solid with volume V is  EMBED Equation.DSMT4 , where it is assumed that  EMBED Equation.DSMT4  is vertical. To do this Let  EMBED Equation.DSMT4  denote the components of a unit vector normal to the surface of the immersed solid Write down a formula for the traction (as a vector) exerted by the fluid on the immersed solid Integrate the traction to calculate the resultant force, and manipulate the result obtain the required formula.  A component contains a feature with a 90 degree corner as shown in the picture. The surfaces that meet at the corner are not subjected to any loading. List all the stress components that must be zero at the corner In this problem we consider further the beam bending calculation discussed in Problem 2.1.21. Suppose that the beam is made from a material in which the Material Stress tensor is related to the Lagrange strain tensor by  EMBED Equation.DSMT4  (this can be regarded as representing an elastic material with zero Poissons ratio and shear modulus  EMBED Equation.DSMT4 ) Calculate the distribution of material stress in the bar, expressing your answer as components in the  EMBED Equation.DSMT4  basis Calculate the distribution of nominal stress in the bar expressing your answer as components in the  EMBED Equation.DSMT4  basis Calculate the distribution of Cauchy stress in the bar expressing your answer as components in the  EMBED Equation.DSMT4  basis Repeat 15.1-15.3 but express the stresses as components in the  EMBED Equation.DSMT4  basis Calculate the distribution of traction on a surface in the beam that has normal  EMBED Equation.DSMT4  in the undeformed beam. Give expressions for the tractions in both  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Show that the surfaces of the beam that have positions  EMBED Equation.DSMT4  in the undeformed beam are traction free after deformation Calculate the resultant moment acting on the ends of the beam. A solid is subjected to some loading that induces a Cauchy stress  EMBED Equation.DSMT4  at some point in the solid. The solid and the loading frame are then rotated together so that the entire solid (as well as the loading frame) is subjected to a rigid rotation  EMBED Equation.DSMT4 . This causes the components of the Cauchy stress tensor to change to new values  EMBED Equation.DSMT4 . The goal of this problem is to calculate a formula relating  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Let  EMBED Equation.DSMT4  be a unit vector normal to an internal material plane in the solid before rotation. After rotation, this vector (which rotates with the solid) is  EMBED Equation.DSMT4 . Write down the formula relating  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Let  EMBED Equation.DSMT4  be the internal traction vector that acts on a material plane with normal  EMBED Equation.DSMT4  in the solid before application of the rigid rotation. Let  EMBED Equation.DSMT4  be the traction acting on the same material plane after rotation. Write down the formula relating  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Finally, using the definition of Cauchy stress, find the relationship between  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Repeat problem 2.2.11, but instead, calculate a relationship between the components of Nominal stress  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  before and after the rigid rotation. Repeat problem 2.2.11, but instead, calculate a relationship between the components of material stress  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  before and after the rigid rotation. Equations of motion and equilibrium for deformable solids A prismatic concrete column of mass density  EMBED Equation  supports its own weight, as shown in the figure. (Assume that the solid is subjected to a uniform gravitational body force of magnitude g per unit mass). Show that the stress distribution  EMBED Equation  satisfies the equations of static equilibrium  EMBED Equation  and also satisfies the boundary conditions  EMBED Equation  on all free boundaries. Show that the traction vector acting on a plane with normal  EMBED Equation  at a height  EMBED Equation  is given by  EMBED Equation  Deduce that the normal component of traction acting on the plane is  EMBED Equation  show also that the tangential component of traction acting on the plane is  EMBED Equation  (the easiest way to do this is to note that  EMBED Equation  and solve for the tangential traction). Suppose that the concrete contains a large number of randomly oriented microcracks. A crack which lies at an angle  EMBED Equation  to the horizontal will propagate if  EMBED Equation  where  EMBED Equation  is the friction coefficient between the faces of the crack and  EMBED Equation  is a critical shear stress that is related to the size of the microcracks and the fracture toughness of the concrete, and is therefore a material property. Assume that  EMBED Equation . Find the orientation of the microcrack that is most likely to propagate. Hence, find an expression for the maximum possible height of the column. Is the stress field given below in static equilibrium? If not, find the acceleration or body force density required to satisfy linear momentum balance  EMBED Equation  Let  EMBED Equation  be a twice differentiable, scalar function of position. Derive a plane stress field from  EMBED Equation  by setting  EMBED Equation  Show that this stress field satisfies the equations of stress equilibrium with zero body force. The stress field  EMBED Equation.DSMT4  represents the stress in an infinite, incompressible elastic solid that is subjected to a point force with components  EMBED Equation.DSMT4  acting at the origin (you can visualize a point force as a very large body force which is concentrated in a very small region around the origin). Verify that the stress field is in static equilibrium Consider a spherical region of material centered at the origin. This region is subjected to (1) the body force acting at the origin; and (2) a force exerted by the stress field on the outer surface of the sphere. Calculate the resultant force exerted on the outer surface of the sphere by the stress, and show that it is equal in magnitude and opposite in direction to the body force. In this problem, we consider the internal forces in the polymer specimen described in Problem 2.1.29 (you will need to solve 2.1.29 before you can attempt this one). Suppose that the specimen is homogeneous, has mass density  EMBED Equation.DSMT4  in the reference configuration, and may be idealized as a viscous fluid, in which the Kirchhoff stress is related to stretch rate by  EMBED Equation.DSMT4  where p is an indeterminate hydrostatic pressure and  EMBED Equation.DSMT4  is the viscosity. Find expressions for the Cauchy stress tensor, expressing your answer as components in  EMBED Equation.DSMT4  Assume steady, quasi-static deformation (neglect accelerations). Express the equations of equilibrium in terms of  EMBED Equation.DSMT4  Solve the equilibrium equation, together with appropriate boundary conditions, to calculate  EMBED Equation.DSMT4  Find the torque necessary to rotate the external cylinder Calculate the acceleration of a material particle in the fluid Estimate the rotation rate  EMBED Equation.DSMT4  where inertia begins to play a significant role in determining the state of stress in the fluid Work done by stresses; the principle of virtual work A solid with volume V is subjected to a distribution of traction  EMBED Equation.DSMT4  on its surface. Assume that the solid is in static equilibrium. By considering a virtual velocity of the form  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is a constant symmetric tensor, use the principle of virtual work to show that the average stress in a solid can be computed from the shape of the solid and the tractions acting on its surface using the expression  EMBED Equation.DSMT4  The figure shows a cantilever beam that is subjected to surface loading  EMBED Equation.DSMT4  per unit length. The state of stress in the beam can be approximated by  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is the area moment of inertia of the beams cross section and  EMBED Equation.DSMT4  is an arbitrary function (all other stress components are zero). By considering a virtual velocity field of the form  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is an arbitrary function satisfying  EMBED Equation.DSMT4  at  EMBED Equation.DSMT4 , show that the beam is in static equilibrium if  EMBED Equation.DSMT4  By integrating the first integral expression by parts twice, show that the equilibrium equation and boundary conditions for  EMBED Equation.DSMT4  are  EMBED Equation.DSMT4   The figure shows a plate with a clamped edge that is subjected to a pressure  EMBED Equation.DSMT4  on its surface. The state of stress in the plate can be approximated by  EMBED Equation.DSMT4  where the subscripts  EMBED Equation.DSMT4  can have values 1 or 2, and  EMBED Equation.DSMT4  is a tensor valued function. By considering a virtual velocity of the form  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is an arbitrary function satisfying  EMBED Equation.DSMT4  on the edge of the plate, show that the beam is in static equilibrium if  EMBED Equation.DSMT4  By applying the divergence theorem appropriately, show that the governing equation for  EMBED Equation.DSMT4  is  EMBED Equation.DSMT4   The shell shown in the figure is subjected to a radial body force  EMBED Equation.DSMT4 , and a radial pressure  EMBED Equation.DSMT4  acting on the surfaces at  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . The loading induces a spherically symmetric state of stress in the shell, which can be expressed in terms of its components in a spherical-polar coordinate system as  EMBED Equation.DSMT4 . By considering a virtual velocity of the form  EMBED Equation.DSMT4 , show that the stress state is in static equilibrium if  EMBED Equation.DSMT4  for all w(R). Hence, show that the stress state must satisfy  EMBED Equation.DSMT4  In this problem, we consider the internal dissipation in the polymer specimen described in Problem 2.1.29 and 2.3.5 (you will need to solve 2.1.29 and 2.3.5 before you can attempt this one). Suppose that the specimen is homogeneous, has mass density  EMBED Equation.DSMT4  in the reference configuration, and may be idealized as a viscous fluid, in which the Kirchhoff stress is related to stretch rate by  EMBED Equation.DSMT4  where p is an indeterminate hydrostatic pressure and  EMBED Equation.DSMT4  is the viscosity. Calculate the rate of external work done by the torque acting on the rotating exterior cyclinder Calculate the rate of internal dissipation in the solid as a function of r. Show that the total internal dissipation is equal to the external work done on the specimen. Chapter 3 Constitutive Models: Relations between Stress and Strain General Requirements for Constitutive Equations The following problems illustrate the consequences of general restrictions on constitutive equations: Linear elastic materials: 3.2.6, 3.2.7 Hyperelastic materials: 3.3.2 Generalized Hookes law: 3.4.3 Hyperelastic materials: 3.5.8, 3.5.9. Linear Elastic Constitutive Equations Using the table of values given in Section 3.2.4, find values of bulk modulus, Lame modulus, and shear modulus for steel, aluminum and rubber. A specimen of an isotropic, linear elastic material with Youngs modulus E and is placed inside a rigid box that prevents the material from stretching in any direction. This means that the strains in the specimen are zero. The specimen is then heated to increase its temperature by  EMBED Equation.DSMT4 . Find a formula for the stress in the specimen. Find a formula for the strain energy density. How much strain energy would be stored in a  EMBED Equation.DSMT4  sample of steel if its temperature were increased by 100C? Compare the strain energy with the heat required to change the temperature by 100C the specific heat capacity of steel is about 470 J/(kg-C) A specimen of an isotropic, linear elastic solid is free of stress, and is heated to increase its temperature by  EMBED Equation.DSMT4 . Find expressions for the strain and displacement fields in the solid. A thin isotropic, linear elastic thin film with Youngs modulus E, Poissons ratio  EMBED Equation.DSMT4  and thermal expansion coefficient  EMBED Equation.DSMT4  is bonded to a stiff substrate. The film is stress free at some initial temperature, and then heated to increase its temperature by  EMBED Equation.DSMT4 . The substrate prevents the film from stretching in its own plane, so that  EMBED Equation.DSMT4 , while the surface is traction free, so that the film deforms in a state of plane stress. Calculate the stresses in the film in terms of material properties and temperature, and deduce an expression for the strain energy density in the film. A cubic material may be characterized either by its moduli as  EMBED Equation.DSMT4  or by the engineering constants  EMBED Equation.DSMT4  Calculate formulas relating  EMBED Equation.DSMT4  to  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , and deduce an expression for the anisotropy factor  EMBED Equation.DSMT4  Suppose that the stress-strain relation for a linear elastic solid is expressed in matrix form as  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  represent the stress and strain vectors and the matrix of elastic constants defined in Section 3.2.8. Show that the material has a positive definite strain energy density ( EMBED Equation.DSMT4 ) if and only if the eigenvalues of  EMBED Equation.DSMT4  are all positive. Write down an expression for the increment in stress resulting from an increment in strain applied to a linear elastic material, in terms of the matrix of elastic constants  EMBED Equation.DSMT4 . Hence show that, for a linear elastic material to be stable in the sense of Drucker, the eigenvalues of the matrix of elastic constants  EMBED Equation.DSMT4  must all be positive or zero. Let  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  represent the stress and strain vectors and the matrix of elastic constants in the isotropic linear elastic constitutive equation  EMBED Equation  Calculate the eigenvalues of the stiffness matrix  EMBED Equation.DSMT4  for an isotropic solid in terms of Youngs modulus and Poissons ratio. Hence, show that the eigenvalues are positive (a necessary requirement for the material to be stable see problem ??) if and only if  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Find the eigenvectors of  EMBED Equation.DSMT4  and sketch the deformations associated with these eigenvectors. Let  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  represent the stress and strain vectors and the matrix of elastic constants in the isotropic linear elastic constitutive equation for a cubic crystal  EMBED Equation.DSMT4  Calculate the eigenvalues of the stiffness matrix  EMBED Equation.DSMT4  and hence find expressions for the admissible ranges of  EMBED Equation.DSMT4  for the eigenvalues to be positive. Let  EMBED Equation.DSMT4  denote the components of the elasticity tensor in a basis  EMBED Equation.DSMT4 . Let  EMBED Equation.DSMT4  be a second basis, and define  EMBED Equation.DSMT4 . Recall that the components of the stress and strain tensor in  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are related by  EMBED Equation.DSMT4 . Use this result, together with the elastic constitutive equation, to show that the components of the elasticity tensor in  EMBED Equation.DSMT4  can be calculated from  EMBED Equation.DSMT4  Consider a cube-shaped specimen of an anisotropic, linear elastic material. The tensor of elastic moduli and the thermal expansion coefficient for the solid (expressed as components in an arbitrary basis) are  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . The solid is placed inside a rigid box that prevents the material from stretching in any direction. This means that the strains in the specimen are zero. The specimen is then heated to increase its temperature by  EMBED Equation.DSMT4 . Find a formula for the strain energy density, and show that the result is independent of the orientation of the material with respect to the box.  The figure shows a cubic crystal. Basis vectors  EMBED Equation.DSMT4  are aligned perpendicular to the faces of the cubic unit cell. A tensile specimen is cut from the cube the axis of the specimen lies in the  EMBED Equation.DSMT4  plane and is oriented at an angle  EMBED Equation.DSMT4  to the  EMBED Equation.DSMT4  direction. The specimen is then loaded in uniaxial tension  EMBED Equation.DSMT4  parallel to its axis. This means that the stress components in the basis  EMBED Equation.DSMT4  shown in the picture are  EMBED Equation.DSMT4  Use the basis change formulas for tensors to calculate the components of stress in the  EMBED Equation.DSMT4  basis in terms of  EMBED Equation.DSMT4 . Use the stress-strain equations in Section 3.1.16 to find the strain components in the  EMBED Equation.DSMT4  basis, in terms of the engineering constants  EMBED Equation.DSMT4  for the cubic crystal. You need only calculate  EMBED Equation.DSMT4 . Use the basis change formulas again to calculate the strain components in the  EMBED Equation.DSMT4  basis oriented with the specimen. Again, you need only calculate  EMBED Equation.DSMT4 . Check your answer by setting  EMBED Equation.DSMT4  - this makes the crystal isotropic, and you should recover the isotropic solution. Define the effective axial Youngs modulus of the tensile specimen as  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is the strain component parallel to the  EMBED Equation.DSMT4  direction. Find a formula for  EMBED Equation.DSMT4  in terms of  EMBED Equation.DSMT4 . Using data for copper, plot a graph of  EMBED Equation.DSMT4  against  EMBED Equation.DSMT4 . For copper, what is the orientation that maximizes the longitudinal stiffness of the specimen? Which orientation minimizes the stiffness? Hypoelasticity  A thin-walled tube can be idealized using the hypoelastic constitutive equation described in Section 3.3. You may assume that the axial load induces a uniaxial stress  EMBED Equation.DSMT4  while the torque induces a shear stress  EMBED Equation.DSMT4 . The shear strains are related to the twist per unit length of the tube by  EMBED Equation.DSMT4 , while the axial strains are related to the extension of the tube by  EMBED Equation.DSMT4 . Calculate a relationship between the axial load P and the extension  EMBED Equation.DSMT4  for a tube subjected only to axial loading Calculate a relationship between the torque Q and the twist  EMBED Equation.DSMT4  for a tube subjected only to torsional loading Calculate a relationship between P, Q and  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  for a tube subjected to combined axial and torsional loading. Consider a material with the hypoelastic constitutive equation described in Section 3.3. Calculate an expression for the tangent stiffness  EMBED Equation.DSMT4 . Express your answer in matrix form by finding the matrix  EMBED Equation.DSMT4  such that stress increment  EMBED Equation.DSMT4  and strain increment  EMBED Equation.DSMT4  are related by  EMBED Equation.DSMT4 . Find the eigenvalues of  EMBED Equation.DSMT4  for the particular case  EMBED Equation.DSMT4 . Hence, show that the material is stable in the sense of Drucker as long as  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Generalized Hookes law: Materials subjected to small strains and large rotations A uniaxial tensile specimen with length L and cross-sectional area A is idealized with a constitutive law that relates the material stress  EMBED Equation.DSMT4  to the Lagrange strain  EMBED Equation.DSMT4  by  EMBED Equation  where E and  EMBED Equation.DSMT4  are elastic constants. The specimen is subjected to a uniaxial force P which induces an extension  EMBED Equation.DSMT4 . Calculate the relationship between  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , and compare the results with the predictions of a linear elastic constitutive equation.  A thin walled tube with length L, radius a and wall thickness t is subjected to a torque Q . The tube can be idealized using the constitutive equation described in the preceding problem. Assume that during deformation plane sections of the tube remain plane, and that cross sections of the tube rotate through and angle  EMBED Equation.DSMT4 . Calculate an expression for the Lagrange strain in the specimen Hence deduce an expression for the material stress in the tube Compute the Cauchy stress distribution Hence, deduce an expression relating the torque Q to the tubes twist  EMBED Equation.DSMT4 . Compare the result with the predictions of a simple linear elastic constitutive equation. Check whether the constitutive equation given in problem 3.4.1 satisfies the test for objectivity described in Section 3.1.  Hyperelasticity Derive the stress-strain relations for an incompressible, Neo-Hookean material subjected to Uniaxial tension Equibiaxial tension Pure shear Derive expressions for the Cauchy stress, the Nominal stress, and the Material stress tensors (the solutions for nominal stress are listed in the table in Section 3.5.6). You should use the following procedure: (i) assume that the specimen experiences the length changes listed in 3.5.6; (ii) use the formulas in Section 3.5.5 to compute the Cauchy stress, leaving the hydrostatic part of the stress p as an unknown; (iii) Determine the hydrostatic stress from the boundary conditions (e.g. for uniaxial tensile parallel to  EMBED Equation.DSMT4  you know  EMBED Equation.DSMT4 ; for equibiaxial tension or pure shear in the  EMBED Equation.DSMT4  plane you know that  EMBED Equation.DSMT4 ) Repeat problem 3.5.1 for an incompressible Mooney-Rivlin material. Repeat problem 3.5.1 for an incompressible Arruda-Boyce material Repeat problem 3.5.1 for an incompressible Ogden material. Using the results listed in the table in Section 3.5.6, and the material properties listed in Section 3.5.7, plot graphs showing the nominal stress as a function of stretch ratio  EMBED Equation.DSMT4  for each of (a) a Neo-Hookean material; (b) a Mooney-Rivlin material; (c) the Arruda-Boyce material and (c) the Ogden material when subjected to uniaxial tension, biaxial tension, and pure shear (for the latter case, plot the largest tensile stress  EMBED Equation.DSMT4 ). A foam specimen is idealized as an Ogden-Storakers foam with strain energy density  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are material properties. Calculate: The Cauchy stress in a specimen subjected to a pure volume change with principal stretches  EMBED Equation.DSMT4  The Cauchy stress in a specimen subjected to volume preserving uniaxial extension The Cauchy stress in a specimen subjected to uniaxial tension, as a function of the tensile stretch ratio  EMBED Equation.DSMT4 . (To solve this problem you will need to assume that the solid is subjected to principal stretches  EMBED Equation.DSMT4  parallel to  EMBED Equation.DSMT4 , and stretches  EMBED Equation.DSMT4  parallel to  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . You will need to determine  EMBED Equation.DSMT4  from the condition that  EMBED Equation.DSMT4  in a uniaxial tensile test. Suppose that a hyperelastic solid is characterized by a strain energy density  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  are invariants of the Left Cauchy-Green deformation tensor  EMBED Equation . Suppose that the solid is subjected to an infinitesimal strain, so that B can be approximated as  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is a symmetric infinitesimal strain tensor. Linearize the constitutive equations for  EMBED Equation.DSMT4 , and show that the relationship between Cauchy stress  EMBED Equation.DSMT4  and infinitesimal strain  EMBED Equation.DSMT4  is equivalent to the isotropic linear elastic constitutive equation. Give formulas for the bulk modulus and shear modulus for the equivalent solid in terms of the derivatives of  EMBED Equation.DSMT4 . The constitutive law for a hyperelastic solid is derived from a strain energy potential  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  are the invariants of the Left Cauchy-Green deformation tensor  EMBED Equation . Calculate the Cauchy stress induced in the solid when it is subjected to a rigid rotation, followed by an arbitrary homogeneous deformation. Hence, demonstrate that the constitutive law is isotropic. Apply the simple check described in Section 3.1 to test whether the constitutive law is objective. The strain energy density of a hyperelastic solid is sometimes specified as a function of the right Cauchy-Green deformation tensor  EMBED Equation.DSMT4 , instead of  EMBED Equation.DSMT4  as described in Section 3.5. (This procedure must be used if the material is anisotropic, for example) Suppose that the strain energy density has the general form  EMBED Equation.DSMT4 . Derive formulas for the Material stress, Nominal stress and Cauchy stress in the solid as functions of  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Apply the simple check described in Section 3.1 to demonstrate that the resulting stress-strain relation is objective. Calculate the Cauchy stress induced in the solid when it is subjected to a rigid rotation, followed by an arbitrary homogeneous deformation. Hence, demonstrate that the constitutive law is not, in general, isotropic. Suppose that the constitutive law is simplified further by writing the strain energy density as a function of the invariants of C, i.e.  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  Derive expressions relating the Cauchy stress components to  EMBED Equation.DSMT4 . Demonstrate that the simplified constitutive law described in 3.5.9.4 characterizes an isotropic solid. Viscoelasticity  The uniaxial tensile stress-strain behavior of a viscoelastic material is idealized using the spring-damper systems illustrated in the figure, as discussed in Section 3.6.3. Derive the differential equations relating stress to strain for each system. Calculate expressions for the relaxation modulus for the Maxwell material and the 3 parameter model. Calculate expressions for the creep compliance of all three materials Calculate expressions for the complex modulus for all three materials. Calculate expressions for the complex compliance for all three materials. The shear modulus of a viscoelastic material can be approximated by a Prony series given by  EMBED Equation.DSMT4 . Find the creep shear compliance of the material Find the complex shear modulus of the material Find the complex shear compliance of the material A uniaxial tensile specimen is made from a viscoelastic material with time independent bulk modulus K, and has a shear modulus that can be approximated by the Prony series  EMBED Equation.DSMT4 . The specimen is subjected to step increase in uniaxial stress, so that  EMBED Equation.DSMT4   EMBED Equation.DSMT4  with all other stress components zero. Find an expression for the history of strain the specimen. It is easiest to solve this problem by first calculating the creep shear compliance for the material.  A floor is covered with a pad with thickness h of viscoelastic material, as shown in the figure. The pad is perfectly bonded to the floor, so that  EMBED Equation.DSMT4 . The pad can be idealized as a viscoelastic solid with time independent bulk modulus K, and has a shear modulus that can be approximated by the Prony series  EMBED Equation.DSMT4 . The surface of the pad is subjected to a history of displacement  EMBED Equation.DSMT4 . Calculate the history of stress induced in the pad by  EMBED Equation.DSMT4  Calculate the history of stress induced in the pad by  EMBED Equation.DSMT4  Assume that the pad is subjected to a displacement  EMBED Equation.DSMT4  for long enough for the cycles of stress and strain to settle to steady state. Calculate the total energy dissipated per unit area of the pad during a cycle of loading. 27 0C35 0C35 0C55 0C65 0CTime (Sec)Modulus (GPa)Time (Sec)Modulus (GPa)Time (Sec)Modulus (GPa)Time (Sec)Modulus GPaTime (Sec)Modulus GPa923.951318.89178.81155.67174.921723.912318.46318.61275.16304.862923.844118.04558.34485.00554.765223.727317.36987.91854.97974.599323.7213116.271747.271524.951744.3216623.5123314.673106.462714.923093.9229523.1641512.635525.674824.865503.4152522.6173810.619825.168574.769792.9293321.7913129.2917465.0015244.5917422.60166020.7323348.8131064.9727114.3230972.50295219.6641518.6155234.9548223.9255082.50525018.8973818.3498224.9285743.4180002.50933718.46131267.91174674.86152482.92175002.501660018.04 The table above lists measured relaxation (shear) modulus for a (fictitious) polymer at various temperatures. The polymer has a glass transition temperature of 30 0C. Plot a graph of the modulus as a function of time for each temperature, using a log scale for both axes Hence, show that the data for various temperatures can be collapsed onto a single master-curve by scaling the times in each experiment by a temperature dependent factor  EMBED Equation.DSMT4 , as described in Section 3.6.1. Plot the master-curve corresponding to relaxation at 27 0C. Plot a graph of  EMBED Equation.DSMT4  as a function of temperature, and show that the data can be fit by a function of the form  EMBED Equation.DSMT4 . Determine the values of  EMBED Equation.DSMT4  that best fit the data Hence, determine the constants  EMBED Equation.DSMT4  for the material, as discussed in Section 3.6.1 Hence, scale the times in the experimental data to plot the relaxation modulus at the glass transition temperature  EMBED Equation.DSMT4 . Find a Prony series fit to  EMBED Equation.DSMT4 . Use four terms in the series, together with an appropriate value for  EMBED Equation.DSMT4  An instrument with mass m=10kg is mounted a set of rubber pads with combined cross sectional area A=5 EMBED Equation.DSMT4  and height h=3cm as shown in the figure. The pads are made from polyisobutylene, with properties listed in Section 3.6.6. The base vibrates harmonically with amplitude  EMBED Equation.DSMT4  and (angular) frequency  EMBED Equation.DSMT4 , causing the instrument to vibrate (also harmonically) with amplitude  EMBED Equation.DSMT4 . Find an expression relating  EMBED Equation.DSMT4  to the harmonic modulus  EMBED Equation.DSMT4  of the material and m, A, h and  EMBED Equation.DSMT4 . Assume that the pads are all subjected to a uniaxial state of stress. Find an expression for the harmonic modulus  EMBED Equation.DSMT4  in terms of the material properties  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Hence, plot a graph showing the variation of  EMBED Equation.DSMT4  as a function of frequency, for temperatures  EMBED Equation.DSMT4  Small-strain metal plasticity The stress state induced by stretching a large plate containing a cylindrical hole of radius a at the origin is given by  EMBED Equation.DSMT4  Here,  EMBED Equation.DSMT4  is the stress in the plate far from the hole. (Stress components not listed are all zero) Plot contours of von-Mises equivalent stress (normalized by  EMBED Equation.DSMT4 ) as a function of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , for a material with  EMBED Equation.DSMT4  Hence identify the point in the solid that first reaches yield. Assume that the material has a yield stress Y .Calculate the critical value of  EMBED Equation.DSMT4  that will just cause the plate to reach yield. The stress state (expressed in cylindrical-polar coordinates) in a thin disk with mass density  EMBED Equation.DSMT4  that spins with angular velocity  EMBED Equation.DSMT4  can be shown to be  EMBED Equation.DSMT4  Assume that the disk is made from an elastic-plastic material with yield stress Y and  EMBED Equation.DSMT4 . Find a formula for the critical angular velocity that will cause the disk to yield, assuming Von-Mises yield criterion. Where is the critical point in the disk where plastic flow first starts? Find a formula for the critical angular velocity that will cause the disk to yield, using the Tresca yield criterion. Where is the critical point in the disk where plastic flow first starts? Using parameters representative of steel, estimate how much kinetic energy can be stored in a disk with a 0.5m radius and 0.1m thickness. Recommend the best choice of material for the flywheel in a flywheel energy storage system.  An isotropic, elastic-perfectly plastic thin film with Youngs Modulus  EMBED Equation.DSMT4 , Poissons ratio  EMBED Equation.DSMT4  , yield stress in uniaxial tension Y and thermal expansion coefficient  EMBED Equation.DSMT4  is bonded to a stiff substrate. It is stress free at some initial temperature and then heated. The substrate prevents the film from stretching in its own plane, so that  EMBED Equation.DSMT4 , while the surface is traction free, so that the film deforms in a state of plane stress. Calculate the critical temperature change  EMBED Equation.DSMT4  that will cause the film to yield, using (a) the Von Mises yield criterion and (b) the Tresca yield criterion. Assume that the thin film described in the preceding problem shows so little strain hardening behavior that it can be idealized as an elastic-perfectly plastic solid, with uniaxial tensile yield stress Y. Suppose the film is stress free at some initial temperature, and then heated to a temperature  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is the yield temperature calculated in the preceding problem, and  EMBED Equation.DSMT4 . Find the stress in the film at this temperature. The film is then cooled back to its original temperature. Find the stress in the film after cooling. Suppose that the thin film described in the preceding problem is made from an elastic, isotropically hardening plastic material with a Mises yield surface, and yield stress-v-plastic strain as shown in the figure. The film is initially stress free, and then heated to a temperature  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is the yield temperature calculated in problem 1, and  EMBED Equation.DSMT4 . Find a formula for the stress in the film at this temperature. The film is then cooled back to its original temperature. Find the stress in the film after cooling. The film is cooled further by a temperature change  EMBED Equation.DSMT4 . Calculate the critical value of  EMBED Equation.DSMT4  that will cause the film to reach yield again.  Suppose that the thin film described in the preceding problem is made from an elastic, linear kinematically hardening plastic material with a Mises yield surface, and yield stress-v-plastic strain as shown in the figure. The stress is initially stress free, and then heated to a temperature  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is the yield temperature calculated in problem 1, and  EMBED Equation.DSMT4 . Find a formula for the stress in the film at this temperature. The film is then cooled back to its original temperature. Find the stress in the film after cooling. The film is cooled further by a temperature change  EMBED Equation.DSMT4 . Calculate the critical value of  EMBED Equation.DSMT4  that will cause the film to reach yield again. A thin-walled tube of mean radius a and wall thickness t<A, and a>b. To approximate the deformation, assume that  planes that lie perpendicular to  EMBED Equation.DSMT4  in the undeformed solid remain perpendicular to  EMBED Equation.DSMT4  after deformation  The axial stretch  EMBED Equation.DSMT4  in the tube is constant It is straightforward to show that the deformation mapping can be described as  EMBED Equation.DSMT4  Calculate the deformation gradient, expressing your answer in terms of R, as components in the  EMBED Equation.DSMT4  basis. Verify that the deformation preserves volume. Calculate the components of the left Cauchy-Green deformation tensor  EMBED Equation  Find an expression for the Cauchy stress components  EMBED Equation.DSMT4  in the cylinder in terms of  EMBED Equation  and an indeterminate hydrostatic stress p. Use 4.3 and the equilibrium equation and boundary conditions to calculate an expression for the Cauchy stress components. Finally, use the condition that the resultant force acting on any cross-section of the tube must vanish to obtain an equation for the axial stretch  EMBED Equation.DSMT4 . Does the tube get longer or shorter when it is inverted?  Two spherical, hyperelastic shells are connected by a thin tube, as shown in the picture. When stress free, both spheres have internal radius A and external radius B. The material in each sphere can be idealized as an incompressible, neo-Hookean solid, with material constant  EMBED Equation.DSMT4 . Suppose that the two spheres together contain a volume  EMBED Equation.DSMT4  of an incompressible fluid. As a result, the two spheres have deformed internal and external radii  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  as shown in the picture. Investigate the possible equilibrium configurations for the system, as functions of the dimensionless fluid volume  EMBED Equation.DSMT4  and B/A. To display your results, plot a graph showing the equilibrium values of  EMBED Equation.DSMT4  as a function of  EMBED Equation.DSMT4 , for various values of B/A. You should find that for small values of  EMBED Equation.DSMT4  there is only a single stable equilibrium configuration. For  EMBED Equation.DSMT4  exceeding a critical value, there are three possible equilibrium configurations: two in which one sphere is larger than the other (these are stable), and a third in which the two spheres have the same size (this is unstable). In a model experiment intended to duplicate the propulsion mechanism of the lysteria bacterium, a spherical bead with radius a is coated with an enzyme known as an Arp2/3 activator. When suspended in a solution of actin, the enzyme causes the actin to polymerize at the surface of the bead. The polymerization reaction causes a spherical gel of a dense actin network to form around the bead. New gel is continuously formed at the bead/gel interface, forcing the rest of the gel to expand radially around the bead. The actin gel is a long-chain polymer and consequently can be idealized as a rubber-like incompressible neo-Hookean material. Experiments show that after reaching a critical radius the actin gel loses spherical symmetry and occasionally will fracture. Stresses in the actin network are believed to drive both processes. In this problem you will calculate the stress state in the growing, spherical, actin gel. Note that this is an unusual boundary value problem in solid mechanics, because a compatible reference configuration cannot be identified for the solid. Nevertheless, it is possible to write down a deformation gradient field that characterizes the change in shape of infinitesimal volume elements in the gel. To this end: (a) write down the length of a circumferential line at the surface of the bead; (b) write down the length of a circumferential line at radius r in the gel; (c) use these results, together with the incompressibility condition, to write down the deformation gradient characterizing the shape change of a material element that has been displaced from r=a to a general position r. Assume that the bead is rigid, and that the deformation is spherically symmetric. Suppose that new actin polymer is generated at volumetric rate  EMBED Equation.DSMT4 . Use the incompressibility condition to write down the velocity field in the actin gel in terms of  EMBED Equation.DSMT4 , a and r (think about the volume of material crossing a radial line per unit time) Calculate the velocity gradient  EMBED Equation.DSMT4  in the gel (a) by direct differentiation of 5.2 and (b) by using the results of 5.1. Show that the results are consistent. Calculate the components of the left Cauchy-Green deformation tensor field and hence write down an expression for the Cauchy stress field in the solid, in terms of an indeterminate hydrostatic pressure. Use the equilibrium equations and boundary condition to calculate the full Cauchy stress distribution in the bead. Assume that the outer surface of the gel (at r=b) is traction free.  A rubber sheet is wrapped around a rigid cylindrical shaft with radius a. The sheet has thickness t, and can be idealized as an incompressible neo-Hookean solid. A constant tension  EMBED Equation.DSMT4  per unit out-of-plane distance is applied to the sheet during the wrapping process. Calculate the full stress field in the solid rubber, and find an expression for the radial pressure acting on the shaft. Assume that  EMBED Equation.DSMT4  and neglect the shear stress component  EMBED Equation.DSMT4 . Solutions to simple dynamic problems involving linear elastic solids Calculate longitudinal and shear wave speeds in (a) Aluminum nitride; (b) Steel; (d) Aluminum and (e) Rubber. A linear elastic half-space with Youngs modulus E and Poissons ratio  EMBED Equation.DSMT4  is stress free and stationary at time t=0 is then subjected to a constant pressure  EMBED Equation.DSMT4  on its surface for t>0. Calculate the stress, displacement and velocity in the solid as a function of time Calculate the total kinetic energy of the half-space as a function of time Calculate the total potential energy of the half-space as a function of time Verify that the sum of the potential and kinetic energy is equal to the work done by the tractions acting on the surface of the half-space. The surface of an infinite linear elastic half-space with Youngs modulus E and Poissons ratio  EMBED Equation.DSMT4  is subjected to a harmonic pressure on its surface, given by  EMBED Equation.DSMT4  t>0, with p=0 for t<0. Calculate the distribution of stress, velocity and displacement in the solid. What is the phase difference between the displacement and pressure at the surface? Calculate the total work done by the applied pressure in one cycle of loading. A linear elastic solid with Youngs modulus E Poissons ratio  EMBED Equation.DSMT4  and density  EMBED Equation.DSMT4  is bonded to a rigid solid at  EMBED Equation.DSMT4 . Suppose that a plane wave with displacement and stress field  EMBED Equation.DSMT4   EMBED Equation.DSMT4  is induced in the solid, and at time  EMBED Equation.DSMT4  is reflected off the interface. Find the reflected wave, and sketch the variation of stress and velocity in the elastic solid just before and just after the reflection occurs. Consider the plate impact experiment described in Section 4.4.8 Draw graphs showing the stress and velocity at the impact face of the flyer plate as a function of time. Draw graphs showing the stress and velocity at the rear face of the flyer plate as a function of time Draw graphs showing the stress and velocity at the mid-plane of the flyer plate as a function of time Draw a graph showing the total strain energy and kinetic energy of the system as a function of time. Verify that total energy is conserved. Draw a graph showing the total momentum of the flyer plate and the target plate as a function of time. Verify that momentum is conserved. In a plate impact experiment, two identical elastic plates with thickness h, Youngs modulus E, Poissons ratio  EMBED Equation.DSMT4 , density  EMBED Equation.DSMT4  and longitudinal wave speed  EMBED Equation.DSMT4  are caused to collide, as shown in the picture. Just prior to impact, the projectile has a uniform velocity  EMBED Equation.DSMT4 . Draw the (x,t) diagram for the two solids after impact. Show that the collision is perfectly elastic, in the terminology of rigid body collisions, in the sense that all the energy in the flyer is transferred to the target.  In a plate impact experiment, an elastic plates with thickness h, Youngs modulus E, Poissons ratio  EMBED Equation.DSMT4 , density  EMBED Equation.DSMT4  and longitudinal wave speed  EMBED Equation.DSMT4  impacts a second plate with identical elastic properties, but thickness 2h, as shown in the picture. Just prior to impact, the projectile has a uniform velocity  EMBED Equation.DSMT4 . Draw the (x,t) diagram for the two solids after impact. A Split-Hopkinson bar or Kolsky bar is an apparatus that is used to measure plastic flow in materials at high rates of strain (of order 1000/s). The apparatus is sketched in the figure. A small specimen of the material of interest, with length a<0 a constant pressure p is applied to the end of the incident bar, sending a plane wave down the bar. This wave eventually reaches the specimen. At this point part of the wave is reflected back up the incident bar, and part of it travels through the specimen and into the second bar (known as the transmission bar). The history of stress and strain in the specimen can be deduced from the history of strain measured by the two strain gages. For example, if the specimen behaves as an elastic-perfectly plastic solid, the incindent and reflected gages would record the data shown in the figure. The goal of this problem is to calculate a relationship between the measured strains and the stress and strain rate in the specimen. Assume that the bars are linear elastic with Youngs modulus E and density  EMBED Equation.DSMT4 , and wave speed  EMBED Equation.DSMT4 , and that the bars deform in uniaxial compression. Write down the stress, strain and velocity field in the incident bar as a function of time and distance down the bar in terms of the applied pressure p and relevant material and geometric parameters, for  EMBED Equation.DSMT4 . Assume that the waves reflected from, and transmitted through, the specimen are both plane waves. Let  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  denote the compressive strains in the regions behind the reflected and transmitted wave fronts, respectively. Write down expressions for the stress and velocity behind the wave fronts in both incident and transmitted bars in terms of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , for  EMBED Equation.DSMT4  The stress behind the reflected and transmitted waves must equal the stress in the specimen. In addition, the strain rate in the specimen can be calculated from the relative velocity of the incident and transmitted bars where they touch the specimen. Show that the strain rate in the specimen can be calculated from the measured strains as  EMBED Equation.DSMT4 , while the stress in the specimen can be calculated from  EMBED Equation.DSMT4 .  In a plate impact experiment, two plates with identical thickness h, Youngs modulus E, Poissons ratio  EMBED Equation.DSMT4  are caused to collide, as shown in the picture. The target plate has twice the mass density of the flyer plate. Find the stress and velocity behind the waves generated by the impact in both target and flyer plate. Hence, draw the (x,t) diagram for the two solids after impact.  In a plate impact experiment, two plates with identical thickness h, Youngs modulus E, Poissons ratio  EMBED Equation.DSMT4 , and density  EMBED Equation.DSMT4  are caused to collide, as shown in the picture. The flyer plate has twice the mass density of the target plate. Find the stress and velocity behind the waves generated by the impact in both target and flyer plate. Hence, draw the (x,t) diagram for the two solids after impact. The figure shows a pressure-shear plate impact experiment. A flyer plate with speed  EMBED Equation.DSMT4  impacts a stationary target. Both solids have identical thickness h, Youngs modulus E, Poissons ratio  EMBED Equation.DSMT4 , density  EMBED Equation.DSMT4  and longitudinal and shear wave speeds  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . The faces of the plates are inclined at an angle  EMBED Equation.DSMT4  to the initial velocity, as shown in the figure. Both pressure and shear waves are generated by the impact. Let  EMBED Equation.DSMT4  denote unit vectors Let  EMBED Equation.DSMT4  denote the (uniform) stress behind the propagating pressure wave in both solids just after impact, and  EMBED Equation.DSMT4  denote the shear stress behind the shear wave-front. Similarly, let  EMBED Equation.DSMT4  denote the change in longitudinal and transverse velocity in the flier across the pressure and shear wave fronts, and let  EMBED Equation.DSMT4  denote the corresponding velocity changes in the target plate. Assume that the interface does not slip after impact, so that both velocity and stress must be equal in both flier and target plate at the interface just after impact. Find expressions for  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  in terms of  EMBED Equation.DSMT4  and relevant material properties. Draw the full (x,t) diagram for the pressure-shear configuration described in problem 11. Assume that the interface remains perfectly bonded until it separates under the application of a tensile stress. Note that you will have to show (x,t) diagrams associated with both shear and pressure waves. Consider an isotropic, linear elastic solid with Youngs modulus E, Poissons ratio  EMBED Equation.DSMT4 , density  EMBED Equation.DSMT4  and shear wave speed  EMBED Equation.DSMT4 . Suppose that a plane, constant stress shear wave propagates through the solid, which is initially at rest. The wave propagates in a direction  EMBED Equation.DSMT4 , and the material has particle velocity  EMBED Equation.DSMT4  behind the wave-front. Calculate the components of stress in the solid behind the wave front. Suppose that the wave front is incident on a flat, stress free surface. Take the origin for the coodinate system at some arbitrary time t at the point where the propagating wave front just intersects the surface, as shown in the picture. Write down the velocity of this intersection point (relative to a stationary observer) in terms of V and  EMBED Equation.DSMT4 . The surface must be free of traction both ahead and behind the wave front. Show that the boundary condition can be satisfied by superposing a second constant stress wave front, which intersects the free surface at the origin of the coodinate system defined in 13.2, and propagates in a direction  EMBED Equation.DSMT4 . Hence, write down the stress and particle velocity in each of the three sectors A, B, C shown in the figure. Draw the displacement of the free surface of the half-space.  Suppose that a plane, constant stress pressure wave propagates through an isotropic, linear elastic solid that is initially at rest. The wave propagates in a direction  EMBED Equation.DSMT4 , and the material has particle velocity  EMBED Equation.DSMT4  behind the wave-front. Calculate the components of stress in the solid behind the wave front. Suppose that the wave front is incident on a flat, stress free surface. Take the origin for the coodinate system at some arbitrary time t at the point where the propagating wave front just intersects the surface, as shown in the picture. Write down the velocity of this intersection point (relative to a stationary observer) in terms of V and  EMBED Equation.DSMT4 . The pressure wave is reflected as two waves a reflected pressure wave, which propagates in direction  EMBED Equation.DSMT4  and has particle velocity  EMBED Equation.DSMT4  and a reflected shear wave, which propagates in direction  EMBED Equation.DSMT4  and has particle velocity  EMBED Equation.DSMT4 . Use the condition that the incident wave and the two reflected waves must always intersect at the same point on the surface to write down an equation for  EMBED Equation.DSMT4  in terms of  EMBED Equation.DSMT4  and Poissons ratio. The surface must be free of traction. Find equations for  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  in terms of V,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and Poissons ratio. Find the special angles for which the incident wave is reflected only as a shear wave (this is called mode conversion) Chapter 5 Analytical Techniques and Solutions for Linear Elastic Solids  General Principles A spherical shell is simultaneously subjected to internal pressure, and is heated internally to raise its temperature at  EMBED Equation.DSMT4  to a temperature  EMBED Equation.DSMT4 , while at  EMBED Equation.DSMT4  its surface is traction free, and temperature is  EMBED Equation.DSMT4 . Use the principle of superposition, together with the solutions given in Chapter 4.1, to determine the stress field in the sphere.  The stress field around a cylindrical hole in an infinite solid, which is subjected to uniaxial tension  EMBED Equation.DSMT4  far from the hole, is given by  EMBED Equation.DSMT4  Using the principle of superposition, calculate the stresses near a hole in a solid which is subjected to shear stress  EMBED Equation.DSMT4  at infinity. The stress field due to a concentrated line load, with force per unit out-of-plane distance P acting on the surface of a large flat elastic solid are given by  EMBED Equation.DSMT4  The stress field due to a uniform pressure distribution acting on a strip with width 2a is  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Show that, for  EMBED Equation.DSMT4  the stresses due to the uniform pressure become equal to the stresses induced by the line force (you can do this graphically, or analytically). The stress field in an infinite solid that contains a spherical cavity with radius a at the origin, and is subjected to a uniform uniaxial stress  EMBED Equation.DSMT4  far from the sphere is given by  EMBED Equation.DSMT4  Show that the hole only influences the stress field in a region close to the hole. Airy Function Solution to Plane Stress and Strain Static Linear Elastic Problems A rectangular dam is subjected to pressure  EMBED Equation.DSMT4  on one face, where  EMBED Equation.DSMT4  is the weight density of water. The dam is made from concrete, with weight density  EMBED Equation.DSMT4  (and is therefore subjected to a body force  EMBED Equation.DSMT4  per unit volume). The goal is to calculate formulas for a and L to avoid failure. Write down the boundary conditions on all four sides of the dam. Consider the following approximate state of stress in the dam  EMBED Equation.DSMT4  Show that (i) The stress state satisfies the equilibrium equations (ii) the stress state exactly satisfies boundary conditions on the sides  EMBED Equation.DSMT4 , (iii) The stress does not satisfy the boundary condition on  EMBED Equation.DSMT4  exactly. Show, however, that the resultant force acting on  EMBED Equation.DSMT4  is zero, so by Saint Venants principle the stress state will be accurate away from the top of the dam. The concrete cannot withstand any tension. Assuming that the greatest principal tensile stress is located at point A  EMBED Equation.DSMT4 , show that the dam width must satisfy The concrete fails by crushing when the minimum principal stress reaches  EMBED Equation.DSMT4 . Assuming the greatest principal compressive stress is located at point B,  EMBED Equation.DSMT4  show that the height of the dam cannot exceed  The stress due to a line load magnitude P per unit out-of-plane length acting tangent to the surface of a homogeneous, isotropic half-space can be generated from the Airy function  EMBED Equation.DSMT4  Calculate the displacement field in the solid, following the procedure in Section 5.2.6 The figure shows a simple design for a dam.  Write down an expression for the hydrostatic pressure in the fluid at a depth  EMBED Equation  below the surface Hence, write down an expression for the traction vector acting on face OA of the dam. Write down an expression for the traction acting on face OB Write down the components of the unit vector normal to face OB in the basis shown Hence write down the boundary conditions for the stress state in the dam on faces OA and OB Consider the candidate Airy function  EMBED Equation  Is this a valid Airy function? Why? Calculate the stresses generated by the Airy function given in 5.2.2.6 Use 5.2.2.5 and 5.2.2.7 to find values for the coefficients in the Airy function, and hence show that the stress field in the dam is  EMBED Equation  Consider the Airy function  EMBED Equation.DSMT4  Verify that the Airy function satisfies the appropriate governing equation. Show that this stress state represents the solution to a large plate containing a circular hole with radius a at the origin, which is loaded by a tensile stress  EMBED Equation.DSMT4  acting parallel to the  EMBED Equation.DSMT4  direction. To do this, Show that the surface of the hole is traction free i.e.  EMBED Equation.DSMT4  on r=a Show that the stress at  EMBED Equation.DSMT4  is  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  EMBED Equation.DSMT4 . Show that the stresses in 5.2.3.2 are equivalent to a stress  EMBED Equation.DSMT4 . It is easiest to work backwards start with the stress components in the  EMBED Equation.DSMT4  basis and use the basis change formulas to find the stresses in the  EMBED Equation.DSMT4  basis Plot a graph showing the variation of hoop stress  EMBED Equation.DSMT4  with  EMBED Equation.DSMT4  at  EMBED Equation.DSMT4  (the surface of the hole). What is the value of the maximum stress, and where does it occur? Find an expression for the vertical displacement  EMBED Equation.DSMT4  of the surface of a half-space that is subjected to a distribution of pressure p(s) as shown in the picture. Show that the slope of the surface can be calculated as  EMBED Equation.DSMT4  Complex variable Solutions to Static Linear Elasticity Problems A long cylinder is made from an isotropic, linear elastic solid with shear modulus  EMBED Equation.DSMT4 . The solid is loaded so (i) the resultant forces and moments acting on the ends of the cylinder are zero; (ii) the body force  EMBED Equation.DSMT4  in the interior of the solid acts parallel to the axis of the cylinder; and (iii) Any tractions  EMBED Equation.DSMT4  or displacements  EMBED Equation.DSMT4  imposed on the sides of the cylinder are parallel to the axis of the cylinder. Under these conditions, the displacement field at a point far from the ends of the cylinder has the form  EMBED Equation.DSMT4 , and the solid is said to deform in a state of anti-plane shear. Calculate the strain field in the solid in terms of u. Find an expression for the nonzero stress components in the solid, in terms of u and material properties. Find the equations of equilibrium for the nonzero stress components. Write down boundary conditions for stress and displacement on the side of the cylinder Hence, show that the governing equations for u reduce to  EMBED Equation.DSMT4  Let  EMBED Equation.DSMT4  be an analytic function of a complex number  EMBED Equation.DSMT4 . Let  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  denote the real and imaginary parts of  EMBED Equation.DSMT4 . Since  EMBED Equation.DSMT4  is analytic, the real and imaginary parts must satisfy the Cauchy Riemann conditions  EMBED Equation.DSMT4  Show that  EMBED Equation.DSMT4  Deduce that the displacement and stress in a solid that is free of body force, and loaded on its boundary so as to induce a state anti-plane shear (see problem 1) can be derived an analytic function  EMBED Equation.DSMT4 , using the representation  EMBED Equation.DSMT4  Calculate the displacements and stresses generated when the complex potential  EMBED Equation.DSMT4  is substituted into the representation described in Problem 2. Show that the solution represents the displacement and stress in an infinite solid due to a line force acting in the  EMBED Equation.DSMT4  direction at the origin. Calculate the displacements and stresses generated when the complex potential  EMBED Equation.DSMT4  is substituted into the representation described in Problem 2. Show that the solution represents the displacement and stress due to a screw dislocation in an infinite solid, with burgers vector and line direction parallel to  EMBED Equation.DSMT4 . (To do this, you need to show that (a) the displacement field has the correct character; and (b) the resultant force acting on a circular arc surrounding the dislocation is zero) Calculate the displacements and stresses generated when the complex potential  EMBED Equation.DSMT4  is substituted into the representation described in Problem 2. Show that the solution represents the displacement and stress in an infinite solid, which contains a hole with radius a at the origin, and is subjected to anti-plane shear at infinity. Calculate the displacements and stresses generated when the complex potential  EMBED Equation.DSMT4  is substituted into the representation described in Problem 2. Show that the solution represents the displacement and stress in an infinite solid, which contains a crack with length a at the origin, and is subjected to a prescribed anti-plane shear stress at infinity. Use the procedure given in Section 5.3.6 to calculate  EMBED Equation.DSMT4  Consider complex potentials  EMBED Equation.DSMT4 , where a, b, c, d are complex numbers. Let  EMBED Equation.DSMT4   EMBED Equation.DSMT4  be a displacement and stress field derived from these potentials. Find values of a,b,c,d that represent a rigid displacement  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  are (real) constants representing a translation, and  EMBED Equation.DSMT4  is a real constant representing an infinitesimal rotation. Find values of a,b,c,d that correspond to a state of uniform stress Note that the solutions to 7.1 and 7.2 are not unique.  Show that the complex potentials  EMBED Equation.DSMT4  give the stress and displacement field in a pressurized circular cylinder which deforms in plane strain (it is best to solve this problem using polar coordinates) The complex potentials  EMBED Equation.DSMT4  generate the plane strain solution to an edge dislocation at the origin of an infinite solid. Work through the algebra necessary to determine the stresses (you can check your answer using the solution given in Section 5.3.4). Verify that the resultant force exerted by the internal tractions on a circular surface surrounding the dislocation is zero. When a stress field acts on a dislocation, the dislocation tends to move through the solid. Formulas for these forces are derived in Section 5.9.5. For the particular case of a straight edge dislocation, with burgers vector  EMBED Equation.DSMT4 , the force can be calculated as follows: Let  EMBED Equation.DSMT4  denote the stress field in an infinite solid containing the dislocation (calculated using the formulas in Section 5.3.4 Let  EMBED Equation.DSMT4  denote the actual stress field in the solid (including the effects of the dislocation itself, as well as corrections due to boundaries in the solid, or externally applied fields) Define  EMBED Equation.DSMT4  denote the difference between these quantities. The force can then be calculated as  EMBED Equation.DSMT4 . Consider two edge dislocations in an infinite solid, each with burgers vector  EMBED Equation.DSMT4 . One dislocation is located at the origin, the other is at position  EMBED Equation.DSMT4 . Plot contours of the horizontal component of force acting on the second dislocation due to the stress field of the dislocation at the origin. (normalize the force as  EMBED Equation.DSMT4 ). The figure shows an edge dislocation below the surface of an elastic solid. Use the solution given in Section 5.3.12, together with the formula in Problem 5.3.9 to calculate an expression for the force acting on the dislocation. The figure shows an edge dislocation with burgers vector  EMBED Equation.DSMT4  that lies in a strained elastic film with thickness h. The film and substrate have the same elastic moduli. The stress in the film consists of the stress due to the dislocation, together with a tensile stress  EMBED Equation.DSMT4 . Calculate the force acting on the dislocation, and hence find the film thickness for which the dislocation will be attracted to the free surface and escape from the film. You will need to use the formula given in problem 5.3.10 to calculate the force on the dislocation. The figure shows a dislocation in an elastic solid with Youngs modulus E and Poissons ratio  EMBED Equation.DSMT4 , which is bonded to a rigid solid. The solution can be generated from complex potentials  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  is the solution for a dislocation at position  EMBED Equation.DSMT4  in an infinite solid, and  EMBED Equation.DSMT4  corrects the solution to satisfy the zero displacement boundary condition at the interface. Show that the solution satisfies the zero displacement boundary condition Calculate the force acting on the dislocation, in terms of h and relevant material properties. You will need to use the formula from 5.3.10 to calculate the force on the dislocation. Calculate the distribution of stress along the interface between the elastic and rigid solids, in terms of h and  EMBED Equation.DSMT4  The figure shows a rigid cylindrical inclusion with radius a embedded in an isotropic elastic matrix. The solid is subjected to a uniform uniaxial stress  EMBED Equation.DSMT4  at infinity. The goal of this problem is to calculate the stress fields in the matrix. Write down the boundary conditions on the displacement field at r=a Show that the boundary conditions can be satisfied by complex potentials of the form  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  are three real valued coefficients whose values you will need to determine. The algebra in this problem can be simplified by noting that  EMBED Equation.DSMT4  on the boundary of the inclusion. Find an expression for the stresses acting at the inclusion/matrix boundary The interface between inclusion and matrix fails when the normal stress acting on the interface reaches a critical stress  EMBED Equation.DSMT4 . Find an expression for the maximum tensile stress that can be applied to the material without causing failure.  The figure shows a cylindrical inclusion with radius a and Youngs modulus and Poissons ratio  EMBED Equation.DSMT4  embedded in an isotropic elastic matrix with elastic constants  EMBED Equation.DSMT4 . The solid is subjected to a uniform uniaxial stress  EMBED Equation.DSMT4  at infinity. The goal of this problem is to calculate the stress fields in both the particle and the matrix. The analylsis can be simplified greatly by assuming a priori that the stress in the particle is uniform (this is not obvious, but can be checked after the full solution has been obtained). Assume, therefore, that the stress in the inclusion is  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  are to be determined. The solution inside the particle can therefore be derived from complex potentials  EMBED Equation.DSMT4  In addition, assume that the solution in the matrix can be derived from complex potentials of the form  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  are three real valued coefficients whose values you will need to determine. Write down the boundary conditions on the displacement field at r=a. Express this boundary condition as an equation relating  EMBED Equation.DSMT4  to  EMBED Equation.DSMT4  in terms of material properties. Write down boundary conditions on the stress components  EMBED Equation.DSMT4  at r=a. Express this boundary condition as an equation relating  EMBED Equation.DSMT4  to  EMBED Equation.DSMT4 . Calculate expressions for  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  in terms of  EMBED Equation.DSMT4  and material properties. The inclusion fractures when the maximum principal stress acting in the particle reaches a critical stress  EMBED Equation.DSMT4 . Find an expression for the maximum tensile stress that can be applied to the material without causing failure.  The figure shows a slit crack in an infinite solid. Using the solution given in Section 5.3.6, calculate the stress field very near the right hand crack tip (i.e. find the stresses in the limit as  EMBED Equation.DSMT4 ). Show that the results are consistent with the asymptotic crack tip field given in Section 5.2.9, and deduce an expression for the crack tip stress intensity factors in terms of  EMBED Equation.DSMT4  and a.  Two identical cylindrical roller bearings with radius 1cm are pressed into contact by a force P per unit out of plane length as indicated in the figure. The bearings are made from 52100 steel with a uniaxial tensile yield stress of 2.8GPa. Calculate the force (per unit length) that will just initiate yield in the bearings, and calculate the width of the contact strip between the bearings at this load.  The figure shows a pair of identical involute spur gears. The contact between the two gears can be idealized as a line contact between two cylindrical surfaces. The goal of this problem is to find an expression for the maximum torque  EMBED Equation.DSMT4  that can be transmitted through the gears. The gears can be idealized as isotropic, linear elastic solids with Youngs modulus E and Poissons ratio  EMBED Equation.DSMT4 . As a representative configuration, consider the instant when a single pair of gear teeth make contact exactly at the pitch point. At this time, the geometry can be idealized as contact between two cylinders, with radius  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is the pitch circle radius of the gears and  EMBED Equation.DSMT4  is the pressure angle. The cylinders are pressed into contact by a force  EMBED Equation.DSMT4 . Find a formula for the area of contact between the two gear teeth, in terms of  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , b and representative material properties. Find a formula for the maximum contact pressure acting on the contact area, in terms of  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , b and representative material properties. Suppose that the gears have uniaxial tensile yield stress Y. Find a formula for the critical value of  EMBED Equation.DSMT4  required to initiate yield in the gears. Solutions to 3D static problems in linear elasticity Consider the Papkovich-Neuber potentials  EMBED Equation.DSMT4  Verify that the potentials satisfy the equilibrium equations Show that the fields generated from the potentials correspond to a state of uniaxial stress, with magnitude  EMBED Equation.DSMT4  acting parallel to the  EMBED Equation.DSMT4  direction of an infinite solid Consider the fields derived from the Papkovich-Neuber potentials  EMBED Equation.DSMT4  Verify that the potentials satisfy the equilibrium equations Show that the fields generated from the potentials correspond to a state of hydrostatic tension  EMBED Equation.DSMT4  Consider the Papkovich-Neuber potentials  EMBED Equation.DSMT4  Verify that the potentials satisfy the governing equations Show that the potentials generate a spherically symmetric displacement field Calculate values of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  that generate the solution to an internally pressurized spherical shell, with pressure p acting at R=a and with surface at R=b traction free. Verify that the Papkovich-Neuber potential  EMBED Equation.DSMT4  generates the fields for a point force  EMBED Equation.DSMT4  acting at the origin of a large (infinite) elastic solid with Youngs modulus E and Poissons ratio  EMBED Equation.DSMT4 . To this end: Verify that the potentials satisfy the governing equation Calculate the stresses Consider a spherical region with radius R surrounding the origin. Calculate the resultant force exerted by the stress on the outer surface of this sphere, and show that they are in equilibrium with a force P. Consider an infinite, isotropic, linear elastic solid with Youngs modulus E and Poissons ratio  EMBED Equation.DSMT4 . Suppose that the solid contains a rigid spherical particle (an inclusion) with radius a and center at the origin. The particle is perfectly bonded to the elastic matrix, so that  EMBED Equation.DSMT4  at the particle/matrix interface. The solid is subjected to a uniaxial tensile stress  EMBED Equation.DSMT4  at infinity. Calculate the stress field in the elastic solid. To proceed, note that the potentials  EMBED Equation.DSMT4  generate a uniform, uniaxial stress  EMBED Equation.DSMT4  (see problem 1). The potentials  EMBED Equation.DSMT4  are a special case of the Eshelby problem described in Section 5.4.6, and generate the stresses outside a spherical inclusion, which is subjected to a uniform transformation strain. Let  EMBED Equation.DSMT4 , where A and B are constants to be determined. The two pairs of potentials can be superposed to generate the required solution. Consider an infinite, isotropic, linear elastic solid with Youngs modulus E and Poissons ratio  EMBED Equation.DSMT4 . Suppose that the solid contains a spherical particle (an inclusion) with radius a and center at the origin. The particle has Youngs modulus  EMBED Equation.DSMT4  and Poissons ratio  EMBED Equation.DSMT4 , and is perfectly bonded to the matrix, so that the displacement and radial stress are equal in both particle and matrix at the particle/matrix interface. The solid is subjected to a uniaxial tensile stress  EMBED Equation.DSMT4  at infinity. The objective of this problem is to calculate the stress field in the elastic inclusion. Assume that the stress field inside the inclusion is given by  EMBED Equation.DSMT4 . Calculate the displacement field in the inclusion (assume that the displacement and rotation of the solid vanish at the origin). The stress field outside the inclusion can be generated from Papkovich-Neuber potentials  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4 , and C and D are constants to be determined. Use the conditions at r=a to find expressions for A,B,C,D in terms of geometric and material properties. Hence, find the stress field inside the inclusion. Consider the Eshelby inclusion problem described in Section 5.4.6. An infinite homogeneous, stress free, linear elastic solid has Youngs modulus E and Poissons ratio  EMBED Equation.DSMT4 . The solid is initially stress free. An inelastic strain distribution  EMBED Equation.DSMT4  is introduced into an ellipsoidal region of the solid B (e.g. due to thermal expansion, or a phase transformation). Let  EMBED Equation.DSMT4  denote the displacement field,  EMBED Equation.DSMT4  denote the total strain distribution, and let  EMBED Equation.DSMT4  denote the stress field in the solid. Write down an expression for the total strain energy  EMBED Equation.DSMT4  within the ellipsoidal region, in terms of  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Write down an expression for the total strain energy outside the ellipsoidal region, expressing your answer as a volume integral in terms of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Using the divergence theorem, show that the result can also be expressed as  EMBED Equation.DSMT4  where S denotes the surface of the ellipsoid, and  EMBED Equation.DSMT4  are the components of an outward unit vector normal to B. Note that, when applying the divergence theorem, you need to show that the integral taken over the (arbitrary) boundary of the solid at infinity does not contribute to the energy you can do this by using the asymptotic formula given in Section 5.4.6 for the displacements far from an Eshelby inclusion. The Eshelby solution shows that the strain  EMBED Equation.DSMT4  inside B is uniform. Write down the displacement field inside the ellipsoidal region, in terms of  EMBED Equation.DSMT4  (take the displacement and rotation of the solid at the origin to be zero). Hence, show that the result of 7.2 can be re-written as  EMBED Equation.DSMT4  Finally, use the results of 7.1 and 7.3, together with the divergence theorem, to show that the total strain energy of the solid can be calculated as  EMBED Equation.DSMT4  Using the solution to Problem 7, calculate the total strain energy of an initially stress-free isotropic, linear elastic solid with Youngs modulus E and Poissons ratio  EMBED Equation.DSMT4 , after an inelastic strain  EMBED Equation.DSMT4  is introduced into a spherical region with radius a in the solid. A steel ball-bearing with radius 1cm is pushed into a flat steel surface by a force P. Neglect friction between the contacting surfaces. Typical ball-bearing steels have uniaxial tensile yield stress of order 2.8 GPa. Calculate the maximum load that the ball-bearing can withstand without causing yield, and calculate the radius of contact and maximum contact pressure at this load.  The contact between the wheel of a locomotive and the head of a rail may be approximated as the (frictionless) contact between two cylinders, with identical radius R as illustrated in the figure. The rail and wheel can be idealized as elastic-perfectly plastic solids with identical Youngs modulus E, Poissons ratio  EMBED Equation.DSMT4  and yield stress Y. Find expressions for the radius of the contact patch, the contact area, and the contact pressure as a function of the load acting on the wheel and relevant geometric and material properties. By estimating values for relevant quantities, calculate the maximum load that can be applied to the wheel without causing the rail to yield.  The figure shows a rolling element bearing. The inner raceway has radius R, and the balls have radius r, and both inner and outer raceways are designed so that the area of contact between the ball and the raceway is circular. The balls are equally spaced circumferentially around the ring. The bearing is free of stress when unloaded. The bearing is then subjected to a force P as shown. This load is transmitted through the bearings at the contacts between the raceways and the balls marked A, B, C in the figure (the remaining balls lose contact with the raceways but are held in place by a cage, which is not shown). Assume that the entire assembly is made from an elastic material with Youngs modulus  EMBED Equation.DSMT4  and Poissons ratio  EMBED Equation.DSMT4  Assume that the load causes the center of the inner raceway to move vertically upwards by a distance  EMBED Equation.DSMT4 , while the outer raceway remains fixed. Write down the change in the gap between inner and outer raceway at A, B, C, in terms of  EMBED Equation.DSMT4  Hence, calculate the resultant contact forces between the balls at A, B, C and the raceways, in terms of  EMBED Equation.DSMT4  and relevant geometrical and material properties. Finally, calculate the contact forces in terms of P If the materials have uniaxial tensile yield stress Y, find an expression for the maximum force P that the bearing can withstand before yielding.  A rigid, conical indenter with apex angle  EMBED Equation.DSMT4  is pressed into the surface of an isotropic, linear elastic solid with Youngs modulus  EMBED Equation.DSMT4  and Poissons ratio  EMBED Equation.DSMT4 . Write down the initial gap between the two surfaces  EMBED Equation.DSMT4  Find the relationship between the depth of penetration h of the indenter and the radius of contact a Find the relationship between the force applied to the contact and the radius of contact, and hence deduce the relationship between penetration depth and force. Verify that the contact stiffness is given by  EMBED Equation.DSMT4  Calculate the distribution of contact pressure that acts between the contacting surfaces. A sphere, which has radius R, is dropped from height h onto the flat surface of a large solid. The sphere has mass density  EMBED Equation.DSMT4 , and both the sphere and the surface can be idealized as linear elastic solids, with Youngs modulus  EMBED Equation.DSMT4  and Poissons ratio  EMBED Equation.DSMT4 . As a rough approximation, the impact can be idealized as a quasi-static elastic indentation. Write down the relationship between the force P acting on the sphere and the displacement of the center of the sphere below  EMBED Equation.DSMT4  Calculate the maximum vertical displacement of the sphere below the point of initial contact. Deduce the maximum force and contact pressure acting on the sphere Suppose that the two solids have yield stress in uniaxial tension Y. Find an expression for the critical value of h which will cause the solids to yield Calculate a value of h if the materials are steel, and the sphere has a 1 cm radius. Solutions to generalized plane problems for anisotropic linear elastic solids  Consider a plane, anisotropic elastic solid, which is loaded so as to induce an anti-plane shear deformation field of the form  EMBED Equation.DSMT4 . Find the three equations of equilibrium in terms of  EMBED Equation.DSMT4  and relevant elastic constants. Show that, since the elastic constants must be positive, an anti-plane shear deformation field can only satisfy the equilibrium equations if the elastic constants satisfy  EMBED Equation.DSMT4  and that under these conditions the equilibrium equation reduces to  EMBED Equation.DSMT4  Consider a displacement field of the form  EMBED Equation.DSMT4 , where f(z) is an analytic function, and  EMBED Equation.DSMT4 . Show that  EMBED Equation.DSMT4  Hence, show that the governing equation of problem 5.5.1 can be satisfied by setting  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , with p a complex number given by  EMBED Equation.DSMT4  Show that the nonzero stresses can be computed from the expression  EMBED Equation.DSMT4   Show that the analytic function  EMBED Equation.DSMT4 , substituted in the representation of problem 5.5.1 and 5.5.2 generates the solution to a point force acting in the  EMBED Equation.DSMT4  direction at the origin of an infinite solid. To do this, you need to show (1) that the solution generates a single valued displacement field, and (2) that the resultant force exerted by the tractions acting on a circular arc surrounding the origin is in equilibrium with the point force. Guided by 5.5.3, construct the solution to a straight screw dislocation in an anisotropic elastic solid, and calculate the stress field. Calculate numerical values for the Stroh matrices A and B for (a) Cu, and (b) Al, assuming that <100> directions are parallel to the coordinate axes. Calculate numerical values for the Barnett-Lothe tensors S, H and L for copper with <100> directions parallel to the coordinate axes. Find the tensors using two approaches: (i) by substituting into the formulas in terms of A and B, and (ii) by evaluating the integral formulas in 5.5.11. Calculate an expression for the inverse of the impedance tensor for a cubic material with <100> directions parallel to the coordinate axes (the expression is simpler than the formula for the impedance tensor itself) Find an expression for the fundamental elasticity matrix N for a cubic material with <100> directions parallel to the coordinate axes. Verify that  EMBED Equation.DSMT4  are the eigenvalues of N, and that  EMBED Equation.DSMT4  are its eigenvectors. Let  EMBED Equation.DSMT4  be the sub-matrices of the fundamental elasticity tensor defined in Section 5.5.6, let A and B denote the matrices of Stroh eigenvectors, and let  EMBED Equation.DSMT4  be the diagonal matrix of Stroh eighevalues. Show that  EMBED Equation.DSMT4  The Stroh representation for a uniform state of generalized plane strain in an anisotropic solid is  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  Use the properties of the Stroh matrices listed in Section 5.5.9 to show that  EMBED Equation.DSMT4  Show also that  EMBED Equation.DSMT4  Hence, deduce that  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  Use the solution given in Section 5.5.13 to calculate the stress distribution due to an edge dislocation with burgers vector  EMBED Equation.DSMT4  in a Cu crystal, with <100> directions parallel to the coordinate directions. Plot contours of the radial and hoop stresses around the dislocation, and compare the results with those given in Section 5.3.4. Use the solution given in Section 5.5.14 to calculate the force acting on a dislocation near the free surface of a single crystal of Cu. Assume that the dislocation has burgers vector  EMBED Equation.DSMT4 , and that the <100> directions of the crystal are parallel to the coordinate axes. Solutions to dynamic problems for isotropic linear elastic solids Consider the Love potentials  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is a constant unit vector and A is a constant. Verify that the potentials satisfy the appropriate governing equations Calculate the stresses and displacements generated from these potentials. Briefly, interpret the wave motion represented by this solution. Consider the Love potentials  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is a constant unit vector and  EMBED Equation.DSMT4  is a constant unit vector. Find a condition relating  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  that must be satisfied for this to be a solution to the governing equations Calculate the stresses and displacements generated from these potentials. Briefly, interpret the wave motion represented by this solution. Show that  EMBED Equation.DSMT4  satisfy the governing equations for Love potentials. Find expressions for the corresponding displacement and stress fields. Calculate the radial distribution of Von-Mises effective stress surrounding a spherical cavity of radius a, which has pressure  EMBED Equation.DSMT4  suddenly applied to its surface at time t=0. Hence, find the location in the solid that is subjected to the largest Von-Mises stress, and the time at which the maximum occurs. Calculate the displacement and stress fields generated by the Love potentials  EMBED Equation.DSMT4  Calculate the traction acting on the surface at r=a. Hence, find the Love potential that generates the fields around a spherical cavity with radius a, which is subjected to a harmonic pressure  EMBED Equation.DSMT4 . Plot the amplitude of the surface displacement at r=a (normalized by a) as a function of  EMBED Equation.DSMT4 . Calculate the distribution of kinetic and potential energy near the surface of a half-space that contains a Rayleigh wave with displacement amplitude  EMBED Equation.DSMT4  and wave number k. Take Poissons ratio  EMBED Equation.DSMT4 . Calculate the total energy per unit area of the wave (find the total energy in one wavelength, then divide by the wavelength). Estimate the energy per unit area in Rayleigh waves associated with earthquakes. The figure shows a surface-acoustic-wave device that is intended to act as a narrow band-pass filter. A piezoelectric substrate has two transducers attached to its surface one acts as an input transducer and the other as output. The transducers are electrodes: a charge can be applied to the input transducer; or detected on the output. Applying a charge to the input transducer induces a strain on the surface of the substrate: at an appropriate frequency, this will excite a Rayleigh wave in the solid. The wave propagates to the output electrodes, and the resulting deformation of the substrate induces a charge that can be detected. If the electrodes have spacing d, calculate the frequency at which the surface will be excited. Estimate the spacing required for a 1GHz filter made from AlN with Youngs modulus 345 GPa and Poissons ratio 0.3 The figure shows a thin elastic strip, which is bonded to rigid solids on both its surfaces. The strip has shear modulus  EMBED Equation.DSMT4  and wave speed  EMBED Equation.DSMT4 , and acts as a wave-guide. The goal of this problem is to calculate the displacement field associated with transverse wave propagation down the strip. Assume that the displacement has the form  EMBED Equation.DSMT4  By substituting into the Cauchy-Navier equation, show that  EMBED Equation.DSMT4  Hence, write down the general solution for  EMBED Equation.DSMT4  Show that the boundary conditions admit solutions of the form  EMBED Equation.DSMT4  where n is an integer, so that  EMBED Equation.DSMT4  Find an expression for the phase velocity of the wave, and plot the phase velocity as a function of kH/n. Calculate the dispersion relation for the wave and hence deduce an expression for the group velocity. Plot the group velocity as a function of kH/n. Consider the Love wave described in Section 5.6.4. Consider first a system with  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 , as discussed at the end of 5.6.4. Find an expression for the group velocity of the wave, and plot a graph showing the group velocity (normalized by shear wave speed in the layer) as a function of kH. Consider a system with  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Plot graphs showing both the phase velocity and the group velocity in the layer as a function of kH. In this problem, you will investigate the energy associated with wave propagation down a simple wave guide. Consider an isotropic, linear elastic strip, with thickness 2H, shear modulus  EMBED Equation.DSMT4  and wave speed  EMBED Equation.DSMT4  as indicated in the figure. The solution for a wave propagating in the  EMBED Equation.DSMT4  direction, with particle velocity  EMBED Equation.DSMT4  is given in Section 5.6.6 of the text. The flux of energy associated with wave propagation along a wave-guide can be computed from the work done by the tractions acting on an internal material surface. The work done per cycle is given by  EMBED Equation.DSMT4  where T is the period of oscillation and  EMBED Equation.DSMT4  is a unit vector normal to an internal plane perpendicular to the direction of wave propagation. Calculate  EMBED Equation.DSMT4  for the nth wave propagation mode. The average kinetic energy of a generic cross-section of the wave-guide can be calculated from  EMBED Equation.DSMT4  Find  EMBED Equation.DSMT4  for nth wave propagation mode. The average potential energy of a generic cross-section of the wave-guide can be calculated from  EMBED Equation.DSMT4  Find  EMBED Equation.DSMT4  for nth wave propagation mode. Check that  EMBED Equation.DSMT4 = EMBED Equation.DSMT4  The speed of energy flux down the wave-guide is defined as  EMBED Equation.DSMT4 . Find  EMBED Equation.DSMT4  for the nth propagation mode, and compare the solution with the expression for the group velocity of the wave  EMBED Equation.DSMT4  Energy methods for solving static linear elasticity problems A shaft with length L and square cross section is fixed at one end, and subjected to a twisting moment T at the other. The shaft is made from a linear elastic solid with Youngs modulus E and Poissons ratio  EMBED Equation . The torque causes the top end of the shaft to rotate through an angle  EMBED Equation . Consider the following displacement field  EMBED Equation  Show that this is a kinematically admissible displacement field for the twisted shaft. Calculate the strains associated with this kinematically admissible displacement field Hence, show that the potential energy of the shaft is You may assume that the potential energy of the torsional load is  EMBED Equation  Find the value of  EMBED Equation  that minimizes the potential energy, and hence estimate the torsional stiffness of the shaft.  In this problem you will use the principle of minimum potential energy to find an approximate solution to the displacement in a pressurized cylinder. Assume that the cylinder is an isotropic, linear elastic solid with Youngs modulus  EMBED Equation.DSMT4  and Poissons ratio  EMBED Equation.DSMT4 , and subjected to internal pressure p at r=a. Approximate the radial displacement field as  EMBED Equation , where  EMBED Equation.DSMT4  are constants to be determined. Assume all other components of displacement are zero. Calculate the strains in the solid  EMBED Equation.DSMT4  Find an expression for the total strain energy of the cylinder per unit length, in terms of  EMBED Equation.DSMT4  and relevant geometric and material parameters Hence, write down the potential energy (per unit length) of the cylinder. Find the values of  EMBED Equation.DSMT4  that minimize the potential energy Plot a graph showing the normalized radial displacement field  EMBED Equation.DSMT4  as a function of the normalized position  EMBED Equation.DSMT4  in the cylinder, for  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . On the same graph, plot the exact solution, given in Section 4.1.9. A bi-metallic strip is made by welding together two materials with identical Youngs modulus and Poissons ratio  EMBED Equation.DSMT4 , but with different thermal expansion coefficients  EMBED Equation.DSMT4 , as shown in the picture. At some arbitrary temperature  EMBED Equation.DSMT4  the strip is straight and free of stress. The temperature is then increased to a new value  EMBED Equation.DSMT4 , causing the strip to bend. Assume that, after heating, the displacement field in the strip can be approximated as  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  are constants to be determined. Briefly describe the physical significance of the shape changes associated with  EMBED Equation.DSMT4 . Calculate the distribution of (infinitesimal) strain associated with the kinematically admissible displacement field Hence, calculate the strain energy density distribution in the solid. Dont forget to account for the effects of thermal expansion Minimize the potential energy to determine values for  EMBED Equation.DSMT4  in terms of relevant geometric and material parameters. By guessing the deflected shape, estimate the stiffness of a clampedclamped beam subjected to a point force at mid-span. Note that your guess for the deflected shape must satisfy  EMBED Equation , so you cant assume that it bends into a circular shape as done in class. Instead, try a deflection of the form  EMBED Equation , or a similar function of your choice (you could try a suitable polynomial, for example). If you try more than one guess and want to know which one gives the best result, remember that energy minimization always overestimates stiffness. The best guess is the one that gives the lowest stiffness. A slender rod with length L and cross sectional area A is subjected to an axial body force  EMBED Equation . Our objective is to determine an approximate solution to the displacement field in the rod. Assume that the displacement field has the form  EMBED Equation  where the function w is to be determined. Find an expression for the strains in terms of w and hence deduce the strain energy density. Show that the potential energy of the rod is  EMBED Equation  To minimize the potential energy, suppose that w is perturbed from the value the minimizes V to a value  EMBED Equation . Assume that  EMBED Equation  is kinematically admissible, which requires that  EMBED Equation  at any point on the bar where the value of w is prescribed. Calculate the potential energy  EMBED Equation and show that it can be expressed in the form  EMBED Equation  where  EMBED Equation  is a function of w only,  EMBED Equation  is a function of w and  EMBED Equation , and  EMBED Equation  is a function of  EMBED Equation  only. As discussed in Section 8 of the online notes (or in class), if  EMBED Equation  is stationary at  EMBED Equation , then  EMBED Equation . Show that, to satisfy  EMBED Equation , we must choose w to satisfy  EMBED Equation  Integrate the first term by parts to deduce that, to minimize, V, w must satisfy  EMBED Equation  Show that this is equivalent to the equilibrium condition  EMBED Equation  Furthermore, deduce that if  EMBED Equation  is not prescribed at either  EMBED Equation  or both, then the boundary conditions on the end(s) of the rod must be  EMBED Equation  Show that this corresponds to the condition that  EMBED Equation  at a free end. Use your results in (2.5) to estimate the displacement field in a bar with mass density  EMBED Equation , which is attached to a rigid wall at  EMBED Equation , is free at  EMBED Equation , and subjected to the force of gravity (acting vertically downwards) The Reciprocal Theorem and applications  A planet that deforms under its own gravitational force can be idealized as a linear elastic sphere with radius a, Youngs modulus E and Poissons ratio  EMBED Equation.DSMT4  that is subjected to a radial gravitational force  EMBED Equation.DSMT4 , where g is the acceleration due to gravity at the surface of the sphere, and R is the radial coordinate. Use the reciprocal theorem, together with a hydrostatic stress distribution  EMBED Equation.DSMT4  as the reference solution, to calculate the change in volume of the sphere, and hence deduce the radial displacement of its surface. Consider an isotropic, linear elastic solid with Youngs modulus E, mass density  EMBED Equation.DSMT4 , and Poissons ratio  EMBED Equation.DSMT4 , which is subjected to a body force distribution  EMBED Equation.DSMT4  per unit mass, and tractions  EMBED Equation.DSMT4  on its exterior surface. By using the reciprocal theorem, together with a state of uniform stress  EMBED Equation.DSMT4  as the reference solution, show that the average strains in the solid can be calculated from  EMBED Equation.DSMT4  A cylinder with arbitrary cross-section rests on a flat surface, and is subjected to a vertical gravitational body force  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is a unit vector normal to the surface. The cylinder is a linear elastic solid with Youngs modulus E, mass density  EMBED Equation.DSMT4 , and Poissons ratio  EMBED Equation.DSMT4 . Define the change in length of the cylinder as  EMBED Equation.DSMT4  where  EMBED Equation.DSMT4  denotes the displacement of the end  EMBED Equation.DSMT4  of the cylinder. Show that  EMBED Equation.DSMT4 , where W is the weight of the cylinder, and A its cross-sectional area. In this problem, you will calculate an expression for the change in potential energy that occurs when an inelastic strain  EMBED Equation.DSMT4  is introduced into some part B of an elastic solid. The inelastic strain can be visualized as a generalized version of the Eshelby inclusion problem it could occur as a result of thermal expansion, a phase transformation in the solid, or plastic flow. Note that B need not be ellipsoidal. The figure illustrates the solid of interest. Assume that: The solid has elastic constants  EMBED Equation.DSMT4  No body forces act on the solid (for simplicity) Part of the surface of the solid  EMBED Equation.DSMT4  is subjected to a prescribed displacement  EMBED Equation.DSMT4  The remainder of the surface of the solid  EMBED Equation.DSMT4  is subjected to a prescribed traction  EMBED Equation.DSMT4  Let  EMBED Equation.DSMT4  denote the displacement, strain, and stress in the solid before the inelastic strain is introduced. Let  EMBED Equation.DSMT4  denote the potential energy of the solid in this state. Next, suppose that some external process introduces an inelastic strain  EMBED Equation.DSMT4  into part of the solid. Let  EMBED Equation.DSMT4  denote the change in stress in the solid resulting from the inelastic strain. Note that these fields satisfy The strain-displacement relation  EMBED Equation.DSMT4  The stress-strain law  EMBED Equation.DSMT4  in B, and  EMBED Equation.DSMT4  outside B Boundary conditions  EMBED Equation.DSMT4  on  EMBED Equation.DSMT4 , and  EMBED Equation.DSMT4  on  EMBED Equation.DSMT4 . Write down an expressions for  EMBED Equation.DSMT4  in terms of  EMBED Equation.DSMT4  Suppose that  EMBED Equation.DSMT4  are all zero (i.e. the solid is initially stress free). Write down the potential energy  EMBED Equation.DSMT4  due to  EMBED Equation.DSMT4 . This is called the self energy of the eigenstrain the energy cost of introducing the eigenstrain  EMBED Equation.DSMT4  into a stress-free solid. Show that the expression for the self-energy can be simplified to  EMBED Equation.DSMT4  Now suppose that  EMBED Equation.DSMT4  are all nonzero. Write down the total potential energy of the system  EMBED Equation.DSMT4 , in terms of  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Finally, show that the total potential energy of the system can be expressed as  EMBED Equation.DSMT4  Here, the last term is called the interaction energy of the eigenstrain with the applied load. The steps in this derivation are very similar to the derivation of the reciprocal theorem. An infinite, isotropic, linear elastic solid with Youngs modulus E and Poissons ratio  EMBED Equation.DSMT4  is subjected to a uniaxial tensile stress  EMBED Equation.DSMT4 . As a result of a phase transformation, a uniform dilatational strain  EMBED Equation.DSMT4  is then induced in a spherical region of the solid with radius a. Using the solution to problem 1, and the Eshelby solution, find an expression for the change in potential energy of the solid, in terms of  EMBED Equation.DSMT4  and relevant geometric and material parameters. Assume that the interface between the transformed material an the matrix has an energy per unit area  EMBED Equation.DSMT4 . Find an expression for the critical stress at which the total energy of the system (elastic potential energy + interface energy) is decreased as a result of the transformation Energetics of Dislocations Calculate the stress induced by a straight screw dislocation in an infinite solid using the formula in Section 5.8.4. Compare the solution with the result of the calculation in Problem 5.3.4.  The figure shows two nearby straight screw dislocations in an infinite solid, with line direction perpendicular to the plane of the figure. The screw dislocations can be introduced into the solid by cutting the plane between the dislocations and displacing the upper of the surfaces created by the cut ( EMBED Equation.DSMT4 ) by  EMBED Equation.DSMT4 , and the lower ( EMBED Equation.DSMT4 ) by  EMBED Equation.DSMT4 , and re-connecting the surfaces. The solid deforms in anti-plane shear, with a displacement field of the form  EMBED Equation.DSMT4  Write down nonzero components of stress and strain in the solid Show that the total strain energy of the solid (per unit out of plane distance) can be expressed as  EMBED Equation.DSMT4  Show that the potential energy can be re-written as  EMBED Equation.DSMT4  where the integral is taken along the line  EMBED Equation.DSMT4 . Use the solution for a screw dislocation given in Problem 5.3.4 (or 5.9.1) to show that the energy can be calculated as  EMBED Equation.DSMT4  Note that the integral is unbounded, as expected. Calculate a bounded expression by truncating the integral at  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Calculate the force exerted on one dislocation by the other by differentiating the expression for the energy. Is the force attractive or repulsive? Check your answer using the Peach-Koehler formula. Calculate the stress induced by an edge dislocation in an infinite solid using the formula in Section 5.8.4. Compare the solution with the result given in 5.3.4 Calculate the nonsingular stress  EMBED Equation.DSMT4  induced by a screw dislocation in an infinite solid using the formula in Section 5.9.2. Compare the solution with the result of the calculation in Problem 5.3.4. Calculate the nonsingular self- energy per unit length of a straight dislocation, using the approach discussed in Section 5.9.2. (To do this, you have to calculate the energy of a dislocation segment with finite length, then take the limit of the energy per unit length as the dislocation length goes to infinity). Calculate the self-energy of a square prismatic dislocation loop with side length L. Use nonsingular dislocation theory, and give your answer to zeroth order in the parameter  EMBED Equation.DSMT4  Suppose that the dislocation loop described in the preceding problem is subjected to a uniaxial tensile stress  EMBED Equation.DSMT4 . Calculate the total potential energy of the system. Display your result as a graph of normalized potential energy  EMBED Equation.DSMT4  as a function of L/b, for various values of  EMBED Equation.DSMT4 . Take  EMBED Equation.DSMT4  as a representative value. Hence, estimate (i) an expression for the activation energy required for homogeneous nucleation of a prismatic dislocation loop, as a function of  EMBED Equation.DSMT4 ; and (ii) the critical size required for a pre-existing dislocation loop to grow, as a function of  EMBED Equation.DSMT4 . Calculate the self-energy of a rectangular glide dislocation loop with burgers vector  EMBED Equation.DSMT4  and side lengths a,d. Use nonsingular dislocation theory, and give your answer to zeroth order in the parameter  EMBED Equation.DSMT4 .  A composite material is made by sandwiching thin layers of a ductile metal between layers of a hard ceramic. Both the metal and the ceramic have identical Youngs modulus  EMBED Equation.DSMT4  and Poissons ratio  EMBED Equation.DSMT4 . The figure shows one of the metal layers, which contains a glide dislocation loop on an inclined slip-plane. The solid is subjected to a uniaxial tensile stress  EMBED Equation.DSMT4  perpendicular to the layers. Calculate the total energy of the dislocation loop, in terms of the applied stress and relevant geometric and material parameters. Use non-singular dislocation theory to calculate the self-energy of the loop. Suppose that the layer contains a large number of dislocation loops with initial width  EMBED Equation.DSMT4 . The layer starts to deform plastically if the stress is large enough to cause the loops to expand in the plane of the film (by increasing the loop dimension d). Calculate the yield stress of the composite. How does the yield stress scale with film thickness? Rayleigh-Ritz Method Use the Rayleigh-Ritz method to obtain the natural frequency of vibration of the spring-mass system shown (the displacement associated with the vibration mode is trivial) Use the Rayleigh-Ritz method to estimate the fundamental frequency of the spring-mass system shown. You should be able to obtain an exact result, by describing the mode shape in terms of a single parameter, and minimizing the frequency appropriately. Reconsider problem 5.10.2. Try to find the second frequency of vibration for the system by selecting another approximation to the mode shape, which is (by construction) orthogonal to the first.  Use the Rayleigh-Ritz method to estimate the fundamental frequency of the clamped-pinned beam illustrated in the figure. Assume that the beam has Youngs modulus  EMBED Equation.DSMT4  and mass density  EMBED Equation.DSMT4 , and its cross-section has area A and moment of area  EMBED Equation.DSMT4 .  Use the Rayleigh-Ritz method to estimate the fundamental frequency of the pinned-pinned beam illustrated in the figure. Assume that the beam has Youngs modulus  EMBED Equation.DSMT4  and mass density  EMBED Equation.DSMT4 , and its cross-section has area A and moment of area  EMBED Equation.DSMT4 .  A beam with length L Youngs modulus  EMBED Equation.DSMT4  and mass density  EMBED Equation.DSMT4 , and its cross-section has area A and moment of area  EMBED Equation.DSMT4  is bonded to an elastic foundation, which exerts a restoring force per unit length  EMBED Equation.DSMT4  on the beam. The beam is pinned at both ends. Use the Rayleigh-Ritz method to estimate the natural frequency of vibration of the beam.      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