Applied Mechanics of Solids



1. Chapter 1

Objectives and Methods of Solid Mechanics

1. Defining a problem in solid mechanics

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. A load cell intended to model forces applied to a specimen in a tensile testing machine

2. The seat-belt assembly in a vehicle

3. The solar panels on a communications satellite.

4. A compressor blade in a gas turbine engine

5. A MEMS optical switch

6. An artificial knee joint

7. A solder joint on a printed circuit board

8. An entire printed circuit board assembly

9. The metal interconnects inside a microelectronic circuit

1. 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.

2. What is the difference between a rate-dependent (viscoplastic) and rate independent plastic constitutive law? Give examples of representative applications for both material models.

3. 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

1.

Governing Equations

1. Mathematical Description of Shape Changes in Solids

1. A thin film of material is deformed in simple shear during a plate impact experiment, as shown in the figure.

1. Write down expressions for the displacement field in the film, in terms of [pic] d and h, expressing your answer as components in the basis shown.

2. Calculate the Lagrange strain tensor associated with the deformation, expressing your answer as components in the basis shown.

3. Calculate the infinitesimal strain tensor for the deformation, expressing your answer as components in the basis shown.

4. Find the principal values of the infinitesimal strain tensor, in terms of d and h

2. Find a displacement field corresponding to a uniform infinitesimal strain field [pic]. (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?

3. Find a formula for the displacement field that generates zero infinitesimal strain.

4. Find a displacement field that corresponds to a uniform Lagrange strain tensor [pic]. Is the displacement unique? Find a formula for the most general displacement field that generates a uniform Lagrange strain.

5. The displacement field in a homogeneous, isotropic circular shaft twisted through angle [pic] at one end is given by

[pic]

1. Calculate the matrix of components of the deformation gradient tensor

2. Calculate the matrix of components of the Lagrange strain tensor. Is the strain tensor a function of [pic]? Why?

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 [pic]direction.

4. Show that material fibers initially oriented in the[pic] and[pic] directions do not change their length.

5. Calculate the principal values and directions of the Lagrange strain tensor at the point [pic]. Hence, deduce the orientations of the material fibers that have the greatest and smallest increase in length.

6. Calculate the components of the infinitesimal strain tensor. Show that, for small values of [pic], the infinitesimal strain tensor is identical to the Lagrange strain tensor, but for finite rotations the two measures of deformation differ.

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 [pic] be before the error in this estimate reaches 10%?

6. 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 in—plane components of the Lagrange strain tensor. Unfortunately, being business economics graduates, they are unable to do this.

1. Explain how the measurements can be used to determine [pic] and do the calculation.

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?

7. 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.

8. Compose a limerick that will help you to remember the distinction between engineering shear strains and the formal (mathematical) definition of shear strain.

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

[pic]

where R must satisfy [pic], det(R)>0.

1. Show that the Lagrange strain associated with this deformation is zero.

2. As a specific example, consider the deformation

[pic]

This is the displacement field caused by rotating a solid through an angle [pic] about the [pic] 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 [pic] if [pic] is small.

3. If the displacements are small, we can find a simpler representation for a rigid body displacement. Consider a deformation of the form

[pic]

Here [pic] is a vector with magnitude ................
................

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