Problems for Chapter 5

Analytical Techniques and Solutions for Linear Elastic Solids

5.3.  Complex Variable Solutions to Static Linear Elasticity Problems

5.3.1.      A long cylinder is made from an isotropic, linear elastic solid with shear modulus $\mu$.  The solid is loaded so

(i) the resultant forces and moments acting on the ends of the cylinder are zero;

(ii) the body force $b=b\left({x}_{1},{x}_{2}\right){e}_{3}$ in the interior of the solid acts parallel to the axis of the cylinder; and (iii) Any tractions $t=t\left({x}_{1},{x}_{2}\right){e}_{3}$ or displacements ${u}^{*}={u}^{*}\left({x}_{1},{x}_{2}\right){e}_{3}$ 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 $u=u\left({x}_{1},{x}_{2}\right){e}_{3}$, and the solid is said to deform in a state of anti-plane shear.

5.3.1.1.            Calculate the strain field in the solid in terms of u.

5.3.1.2.            Find an expression for the nonzero stress components in the solid, in terms of u and material properties.

5.3.1.3.            Find the equations of equilibrium for the nonzero stress components.

5.3.1.4.            Write down boundary conditions for stress and displacement on the side of the cylinder

5.3.1.5.            Hence, show that the governing equations for u reduce to

$\frac{{\partial }^{2}u}{\partial {x}_{1}^{2}}+\frac{{\partial }^{2}u}{\partial {x}_{2}^{2}}+b=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left\{u\left(x\right)={u}^{*}\left(x\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in {S}_{1}\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left\{\mu \left(\frac{\partial u\left(x\right)}{\partial {x}_{1}}{n}_{1}\left(x\right)+\frac{\partial u\left(x\right)}{\partial {x}_{2}}{n}_{2}\left(x\right)\right)=t\left(x\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in {S}_{1}\right\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$

5.3.2.      Let $\Theta \left(z\right)$ be an analytic function of a complex number $z={x}_{1}+i{x}_{2}$.  Let $v\left({x}_{1},{x}_{2}\right)$, $w\left({x}_{1},{x}_{2}\right)$ denote the real and imaginary parts of $\Theta \left(z\right)$.

5.3.2.1.            Since $\Theta \left(z\right)$ is analytic, the real and imaginary parts must satisfy the Cauchy Riemann conditions

$\frac{\partial v}{\partial {x}_{1}}=\frac{\partial w}{\partial {x}_{2}}\begin{array}{ccc}& & \end{array}\frac{\partial w}{\partial {x}_{1}}=-\frac{\partial v}{\partial {x}_{2}}$

Show that

$\frac{{\partial }^{2}v}{\partial {x}_{1}^{2}}+\frac{{\partial }^{2}v}{\partial {x}_{2}^{2}}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\partial }^{2}w}{\partial {x}_{1}^{2}}+\frac{{\partial }^{2}w}{\partial {x}_{2}^{2}}=0$

5.3.2.2.            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 $\Theta \left(z\right)$, using the representation

$\begin{array}{l}2\mu u\left({x}_{1},{x}_{2}\right)=\Theta \left(z\right)+\overline{\Theta \left(z\right)}\\ {\sigma }_{31}-i{\sigma }_{32}=\Theta \text{'}\left(z\right)\end{array}$

5.3.3.      Calculate the displacements and stresses generated when the complex potential

$\Theta \left(z\right)=-\frac{F}{2\pi }\mathrm{log}\left(z\right)$

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 ${e}_{3}$ direction at the origin.

5.3.4.      Calculate the displacements and stresses generated when the complex potential

$\Theta \left(z\right)=-\frac{i\mu b}{2\pi }\mathrm{log}\left(z\right)$

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 ${e}_{3}$. (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)

5.3.5.      Calculate the displacements and stresses generated when the complex potential

$\Theta \left(z\right)={\tau }_{0}\left(z-\frac{{a}^{2}}{z}\right)$

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.

5.3.6.      Calculate the displacements and stresses generated when the complex potential

$\Theta \left(z\right)=i{\tau }_{0}\sqrt{{z}^{2}-{a}^{2}}$

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 $\sqrt{{z}^{2}-{a}^{2}}$

5.3.7.      Consider complex potentials $\Omega \left(z\right)=az+b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\omega \left(z\right)=cz+d$, where a, b, c, d are complex numbers.  Let

$\frac{E}{\left(1+\nu \right)}D=\left(3-4\nu \right)\Omega \left(z\right)-z\text{\hspace{0.17em}}\text{\hspace{0.17em}}\overline{{\Omega }^{\prime }\left(z\right)}⥄-\text{\hspace{0.17em}}\overline{\omega \left(z\right)}$

${\sigma }_{11}+{\sigma }_{22}=2\left({\Omega }^{\prime }\left(z\right)+\overline{{\Omega }^{\prime }\left(z\right)}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{11}-{\sigma }_{22}+2i{\sigma }_{12}=-2\left(z\text{\hspace{0.17em}}\overline{{\Omega }^{″}\left(z\right)}+\overline{{\omega }^{\prime }\left(z\right)}\right)$

be a displacement and stress field derived from these potentials.

5.3.7.1.            Find values of a,b,c,d that represent a rigid displacement ${u}_{1}={w}_{1}+\alpha {x}_{2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{2}={w}_{2}-\alpha {x}_{1}$ where ${w}_{1},{w}_{2}$ are (real) constants representing a translation, and $\alpha$ is a real constant representing an infinitesimal rotation.

5.3.7.2.            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.

5.3.8.      Show that the complex potentials

$\Omega \left(z\right)=\frac{{p}_{a}{a}^{2}-{p}_{b}{b}^{2}}{4\left({b}^{2}-{a}^{2}\right)}z\begin{array}{cccc}& & & \end{array}\omega \left(z\right)=-\frac{\left({p}_{a}-{p}_{b}\right)}{2\left({b}^{2}-{a}^{2}\right)z}$

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)

5.3.9.      The complex potentials

$\Omega \left(z\right)=-i\frac{E\left({b}_{1}+i{b}_{2}\right)}{8\pi \left(1-{\nu }^{2}\right)}\mathrm{log}\left(z\right)\begin{array}{cccc}& & & \end{array}\omega \left(z\right)=i\frac{E\left({b}_{1}-i{b}_{2}\right)}{8\pi \left(1-{\nu }^{2}\right)}\mathrm{log}\left(z\right)$

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.

5.3.10.  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 ${b}_{i}$, the force can be calculated as follows:

Let ${\sigma }_{ij}^{D\infty }$ denote the stress field in an infinite solid containing the dislocation (calculated using the formulas in Section 5.3.4

Let ${\sigma }_{ij}$ 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 $\Delta {\sigma }_{ij}={\sigma }_{ij}-{\sigma }_{ij}^{D\infty }$ denote the difference between these quantities.

The force can then be calculated as ${F}_{i}={\in }_{ij3}\Delta {\sigma }_{jk}{b}_{k}$.

Consider two edge dislocations in an infinite solid, each with burgers vector ${b}_{1}=b,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{b}_{2}=0$.  One dislocation is located at the origin, the other is at position $\left({x}_{1},{x}_{2}\right)$.  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 $8\pi \left(1-{\nu }^{2}\right){F}_{1}\left(\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}\right)/Eb$ ).

5.3.11.  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.

5.3.12.  The figure shows an edge dislocation with burgers vector ${b}_{1}=-b,{b}_{2}=0$ 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 ${\sigma }_{11}={\sigma }_{0}$.  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.

5.3.13.  The figure shows a dislocation in an elastic solid with Young’s modulus E and Poisson’s ratio $\nu$, which is bonded to a rigid solid.   The solution can be generated from complex potentials

$\Omega \left(z\right)={\Omega }_{0}\left(z\right)+{\Omega }_{1}\left(z\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\omega \left(z\right)={\omega }_{0}\left(z\right)+{\omega }_{1}\left(z\right)$

where

$\begin{array}{l}{\Omega }_{0}\left(z\right)=-i\frac{E\left({b}_{1}+i{b}_{2}\right)}{8\pi \left(1-{\nu }^{2}\right)}\mathrm{log}\left(z-ih\right)\\ {\omega }_{0}\left(z\right)=i\frac{E\left({b}_{1}-i{b}_{2}\right)}{8\pi \left(1-{\nu }^{2}\right)}\mathrm{log}\left(z-ih\right)+\frac{E\left({b}_{1}+i{b}_{2}\right)}{8\pi \left(1-{\nu }^{2}\right)}\frac{h}{z-ih}\end{array}$

is the solution for a dislocation at position ${z}_{0}=ih$ in an infinite solid, and

$\begin{array}{l}{\Omega }_{1}\left(z\right)=\left(z\overline{{{\Omega }^{\prime }}_{0}\left(\overline{z}\right)}+\overline{{\omega }_{0}\left(\overline{z}\right)}\right)/\left(3-4\nu \right)\\ {\omega }_{1}\left(z\right)=\left(3-4\nu \right)\overline{{\Omega }_{0}\left(\overline{z}\right)}-z\left(\overline{{{\Omega }^{\prime }}_{0}\left(\overline{z}\right)}+z\overline{{{\Omega }^{″}}_{0}\left(\overline{z}\right)}+\overline{{{\omega }^{\prime }}_{0}\left(\overline{z}\right)}\right)/\left(3-4\nu \right)\end{array}$

corrects the solution to satisfy the zero displacement boundary condition at the interface.

5.3.13.1.        Show that the solution satisfies the zero displacement boundary condition

5.3.13.2.        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.

5.3.13.3.        Calculate the distribution of stress along the interface between the elastic and rigid solids, in terms of h and ${x}_{1}$

5.3.14.  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 ${\sigma }_{11}={\sigma }_{0}$ at infinity.  The goal of this problem is to calculate the stress fields in the matrix.

5.3.14.1.        Write down the boundary conditions on the displacement field at r=a

5.3.14.2.        Show that the boundary conditions can be satisfied by complex potentials of the form

$\Omega \left(z\right)=\frac{{\sigma }_{0}}{4}\left(z+\frac{\alpha {a}^{2}}{z}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\omega \left(z\right)=-\frac{{\sigma }_{0}}{2}\left(z+\frac{\beta {a}^{2}}{z}+\frac{\lambda {a}^{4}}{{z}^{3}}\right)$

where $\alpha ,\beta ,\lambda$ are three real valued coefficients whose values you will need to determine.  The algebra in this problem can be simplified by noting that $\overline{z}={a}^{2}/z$ on the boundary of the inclusion.

5.3.14.3.        Find an expression for the stresses acting at the inclusion/matrix boundary

5.3.14.4.        The interface between inclusion and matrix fails when the normal stress acting on the interface reaches a critical stress ${\sigma }_{crit}$.  Find an expression for the maximum tensile stress that can be applied to the material without causing failure.

5.3.15.  The figure shows a cylindrical inclusion with radius a and Young’s modulus and Poisson’s ratio ${E}_{p},{\nu }_{p}$  embedded in an isotropic elastic matrix with elastic constants $E,\nu$.  The solid is subjected to a uniform uniaxial stress ${\sigma }_{11}={\sigma }_{0}$ 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 ${\sigma }_{11}={p}_{11},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{22}={p}_{22}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{s}_{12}=0$, where ${p}_{11},{p}_{22}$ are to be determined.   The solution inside the particle can therefore be derived from complex potentials

${\Omega }_{p}\left(z\right)=\frac{{p}_{11}+{p}_{22}}{4}z\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\omega }_{p}\left(z\right)=\frac{{p}_{22}-{p}_{11}}{2}z$

In addition, assume that the solution in the matrix can be derived from complex potentials of the form

$\Omega \left(z\right)=\frac{{\sigma }_{0}}{4}\left(z+\frac{\alpha {a}^{2}}{z}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\omega \left(z\right)=-\frac{{\sigma }_{0}}{2}\left(z+\frac{\beta {a}^{2}}{z}+\frac{\lambda {a}^{4}}{{z}^{3}}\right)$

where $\alpha ,\beta ,\lambda$ are three real valued coefficients whose values you will need to determine.

5.3.15.1.        Write down the boundary conditions on the displacement field at r=a.  Express this boundary condition as an equation relating ${\Omega }_{p},{\omega }_{p}$ to $\Omega ,\omega$ in terms of material properties.

5.3.15.2.        Write down boundary conditions on the stress components ${\sigma }_{rr},{\sigma }_{r\theta }$ at r=a. Express this boundary condition as an equation relating ${\Omega }_{p},{\omega }_{p}$ to $\Omega ,\omega$.

5.3.15.3.        Calculate expressions for ${p}_{11},{p}_{22}$, $\alpha ,\beta ,\lambda$ in terms of ${\sigma }_{0}$ and material properties.

5.3.15.4.        The inclusion fractures when the maximum principal stress acting in the particle reaches a critical stress ${\sigma }_{crit}$.  Find an expression for the maximum tensile stress that can be applied to the material without causing failure.

5.3.16.  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 ${r}_{1}/a\to 0$ ).  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 ${\sigma }_{22}^{\infty },{\sigma }_{12}^{\infty }$ and a.

5.3.17.  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.

5.3.18.  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 $Q$ that can be transmitted through the gears.  The gears can be idealized as isotropic, linear elastic solids with Young’s modulus E and Poisson’s ratio $\nu$.

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 $r={R}_{p}\mathrm{sin}\varphi$, where ${R}_{p}$ is the pitch circle radius of the gears and $\varphi$ is the pressure angle.  The cylinders are pressed into contact by a force $P=Q/\left({R}_{p}\mathrm{cos}\varphi \right)$.

5.3.18.1.        Find a formula for the area of contact between the two gear teeth, in terms of $Q$, ${R}_{p}$, $\varphi$, b and representative material properties.

5.3.18.2.        Find a formula for the maximum contact pressure acting on the contact area, in terms of $Q$, ${R}_{p}$, $\varphi$, b and representative material properties.

5.3.18.3.        Suppose that the gears have uniaxial tensile yield stress Y.  Find a formula for the critical  value of $Q$ required to initiate yield in the gears.