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APPENDIX F Exercises

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APPENDIX F Exercises 540 APPENDIX F Exercises 1. Chapter 1: Objectives and Methods of Solid Mechanics 1.1. Defining a problem in solid mechanics 1.1.1. 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? 1.1.1.1. A load cell intended to model forces applied to a specimen in a tensile testing machine 1.1.1.2. The seat-belt assembly in a vehicle 1.1.1.3. The solar panels on a communications satellite. 1.1.1.4. A compressor blade in a gas turbine engine 1.1.1.5. A MEMS optical switch 1.1.1.6. An artificial knee joint 1.1.1.7. A solder joint on a printed circuit board 1.1.1.8. An entire printed circuit board assembly 1.1.1.9. The metal interconnects inside a microelectronic circuit 1.2. 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. 1.3. What is the difference between a rate-dependent (viscoplastic) and rate independent plastic constitutive law? Give examples of representative applications for both material models. 1.4. 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 541 2. Chapter 2: Governing Equations 2.1. Mathematical Description of Shape Changes in Solids e 2.1.1. A thin film of material is deformed in simple shear during a plate 2 impact experiment, as shown in the figure. Undeformed 2.1.1.1. Write down expressions for the displacement field in the h e film, in terms of xx12, d and h, expressing your answer as 1 components in the basis shown. e2 2.1.1.2. Calculate the Lagrange strain tensor associated with the d deformation, expressing your answer as components in the h Deformed basis shown. e1 2.1.1.3. Calculate the infinitesimal strain tensor for the deformation, expressing your answer as components in the basis shown. 2.1.1.4. Find the principal values of the infinitesimal strain tensor, in terms of d and h 2.1.2. Find a displacement field corresponding to a uniform infinitesimal strain field εij . (Don’t 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? 2.1.3. Find a formula for the displacement field that generates zero infinitesimal strain. 2.1.4. Find a displacement field that corresponds to a uniform Lagrange strain tensor Eij . Is the displacement unique? Find a formula for the most general displacement field that generates a uniform Lagrange strain. 2.1.5. The displacement field in a homogeneous, isotropic circular shaft twisted through angle α at one end is e2 given by ⎛⎞ααx33 ⎛⎞x e1 ux11=−−[cos⎜⎟ 1] x 2 sin ⎜⎟ ⎝⎠LL ⎝⎠ e3 α ⎛⎞ααxx33 ⎛⎞ ux21=+sin⎜⎟ x 2 [cos ⎜⎟ − 1] ⎝⎠LL ⎝⎠ L u3 = 0 2.1.5.1. Calculate the matrix of components of the deformation gradient tensor 2.1.5.2. Calculate the matrix of components of the Lagrange strain tensor. Is the strain tensor a function of x3 ? Why? 2.1.5.3. 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 e3 direction. 2.1.5.4. Show that material fibers initially oriented in the e2 and e2 directions do not change their length. 2.1.5.5. Calculate the principal values and directions of the Lagrange strain tensor at the point xax12===,,00 x 3. Hence, deduce the orientations of the material fibers that have the greatest and smallest increase in length. 542 2.1.5.6. Calculate the components of the infinitesimal strain tensor. Show that, for small values of α , the infinitesimal strain tensor is identical to the Lagrange strain tensor, but for finite rotations the two measures of deformation differ. 2.1.5.7. 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 α be before the error in this estimate reaches 10%? 2.1.6. To measure the in-plane deformation of a e2 sheet of metal during a forming process, your e2 (c) managers place three small hardness indentations on the sheet. Using a travelling microscope, they (c) 2cm 2.8cm determine that the initial lengths of the sides of 1cm 1.414cm the triangle formed by the three indents are 1cm, (a) (a) (b) e1 1cm, 1.414cm, as shown in the picture below. e1 After deformation, the sides have lengths 1.5cm, 1cm 1.5cm (b) 2.0cm and 2.8cm. Your managers would like to use this information to determine the in—plane components of the Lagrange strain tensor. Unfortunately, being business economics graduates, they are unable to do this. 2.1.6.1. Explain how the measurements can be used to determine EE11,, 22 E 12 and do the calculation. 2.1.6.2. 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? 2.1.7. To track the deformation in a slowly moving glacier, e2 e three survey stations are installed in the shape of an 2 equilateral triangle, spaced 100m apart, as shown in the 120m picture. After a suitable period of time, the spacing 100m 100m between the three stations is measured again, and found 110m e e to be 90m, 110m and 120m, as shown in the figure. 1 1 Assuming that the deformation of the glacier is 100m 90m homogeneous over the region spanned by the survey Before deformation stations, please compute the components of the Lagrange After deformation strain tensor associated with this deformation, expressing your answer as components in the basis shown. 2.1.8. Compose a limerick that will help you to remember the distinction between engineering shear strains and the formal (mathematical) definition of shear strain 2.1.9. 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 yRx= ⋅=or yRxiijj where R must satisfy RR⋅ TT=⋅= R R I, det(R)>0. 2.1.9.1. Show that the Lagrange strain associated with this deformation is zero. 2.1.9.2. As a specific example, consider the deformation 543 ⎡⎤yx11⎡⎤cosθθ− sin 0 ⎡⎤ ⎢⎥⎢⎥ ⎢⎥ ⎢⎥yx22= ⎢⎥sinθθ cos 0 ⎢⎥ ⎣⎦⎢⎥yx33⎣⎦⎢⎥001 ⎣⎦⎢⎥ This is the displacement field caused by rotating a solid through an angle θ about the e3 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 θ 2 if θ is small. 2.1.9.3. If the displacements are small, we can find a simpler representation for a rigid body displacement. Consider a deformation of the form yxi=∈ ijkω j k Here ω is a vector with magnitude <<1, which represents an infinitesimal rotation about an axis parallel to ω . 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 1 E =−δ ωω ωω ij2 () ij k k i j This is not, in general, zero. It is small if all ωk <<1 . 2.1.10. The formula for the deformation due to a rotation through an angle θ about an axis parallel to a unit vector n that passes through the origin is ⎡⎤ ynnnxiijijikjkj=+−+∈⎣⎦cosθδ (1 cos θ ) sin θ 2.1.10.1. Calculate the components of corresponding deformation gradient 2.1.10.2. Verify that the deformation gradient satisfies FFik jk= FF ki kj= δ ij 2.1.10.3. Find the components of the inverse of the deformation gradient 2.1.10.4. 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? 2.1.10.5. Calculate the Jacobian of the deformation gradient. What does this tell you about volume changes associated with the deformation? 2.1.11. In Section 2.1.6 it was stated that the Eulerian * strain tensor Eij can be used to relate the length n of a material fiber in a deformable solid before and after deformation, using the formula l0 l 22 ll− 0 * e = Eijnn i j 2 2l2 where ni are the components of a unit vector Deformed Original Configuration parallel to the material fiber after deformation.
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