Problems for Chapter 5

Analytical Techniques and Solutions for Linear Elastic Solids

5.4.  Solutions to 3D Static Problems

5.4.1.      Consider the Papkovich-Neuber potentials

${\Psi }_{i}=\frac{\left(1-\nu \right){\sigma }_{0}}{\left(1+\nu \right)}{x}_{3}{\delta }_{i3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =\frac{\nu \left(1-\nu \right){\sigma }_{0}}{\left(1+\nu \right)}\left(3{x}_{3}^{2}-{R}^{2}\right)$

5.4.1.1.            Verify that the potentials satisfy the equilibrium equations

5.4.1.2.            Show that the fields generated from the potentials correspond to a state of uniaxial stress, with magnitude ${\sigma }_{0}$ acting parallel to the ${e}_{3}$ direction of an infinite solid

5.4.2.      Consider the fields derived from the Papkovich-Neuber potentials

${\Psi }_{i}=\frac{\left(1-\nu \right)p}{\left(1+\nu \right)}{x}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =\frac{2\nu \left(1-\nu \right)p}{\left(1+\nu \right)}{R}^{2}$

5.4.2.1.            Verify that the potentials satisfy the equilibrium equations

5.4.2.2.            Show that the fields generated from the potentials correspond to a state of  hydrostatic tension ${\sigma }_{ij}=p{\delta }_{ij}$

5.4.3.      Consider the Papkovich-Neuber potentials

${\Psi }_{i}=\alpha {x}_{i}+\beta \frac{{x}_{i}}{{R}^{3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =\alpha {R}^{2}-\frac{3\beta }{R}$

5.4.3.1.            Verify that the potentials satisfy the governing equations

5.4.3.2.            Show that the potentials generate a spherically symmetric displacement field

5.4.3.3.            Calculate values of $\alpha$ and $\beta$ 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.

5.4.4.      Verify that the Papkovich-Neuber potential

${\Psi }_{i}=\frac{{P}_{i}}{4\pi R}⥄⥄⥄⥄⥄⥄⥄⥄⥄⥄⥄⥄⥄⥄\varphi =0$

generates the fields for a point force $P={P}_{1}{e}_{1}+{P}_{2}{e}_{2}+{P}_{3}{e}_{3}$ acting at the origin of a large (infinite) elastic solid with Young’s modulus E and Poisson’s ratio $\nu$.  To this end:

5.4.4.1.            Verify that the potentials satisfy the governing equation

5.4.4.2.            Calculate the stresses

5.4.4.3.            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.

5.4.5.      Consider an infinite, isotropic, linear elastic solid with Young’s modulus E and Poisson’s ratio $\nu$.  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 ${u}_{i}=0$ at the particle/matrix interface.  The solid is subjected to a uniaxial tensile stress ${\sigma }_{33}={\sigma }_{0}$ at infinity.  Calculate the stress field in the elastic solid.  To proceed, note that the potentials

${\Psi }_{i}=\frac{\left(1-\nu \right){\sigma }_{0}}{\left(1+\nu \right)}{x}_{3}{\delta }_{i3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =\frac{\nu \left(1-\nu \right){\sigma }_{0}}{\left(1+\nu \right)}\left(3{x}_{3}^{2}-{R}^{2}\right)$

generate a uniform, uniaxial stress ${\sigma }_{33}={\sigma }_{0}$ (see problem 1).  The potentials

${\Psi }_{i}=\frac{{a}^{3}{p}_{ik}^{T}{x}_{k}}{3{R}^{3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =\frac{{a}^{3}{p}_{ij}^{T}}{15{R}^{3}}\left(\left(5{R}^{2}-{a}^{2}\right){\delta }_{ij}+3{a}^{2}\frac{{x}_{i}{x}_{j}}{{R}^{2}}\right)$

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 ${p}_{ij}^{T}=A{\delta }_{ij}+B{\delta }_{i3}{\delta }_{j3}$, where A and B are constants to be determined.  The two pairs of potentials can be superposed to generate the required solution.

5.4.6.      Consider an infinite, isotropic, linear elastic solid with Young’s modulus E and Poisson’s ratio $\nu$.  Suppose that the solid contains a spherical particle (an inclusion) with radius a and center at the origin.  The particle has Young’s modulus ${E}_{p}$ and Poisson’s ratio ${\nu }_{p}$, 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 ${\sigma }_{33}={\sigma }_{0}$ at infinity.  The objective of this problem is to calculate the stress field in the elastic inclusion.

5.4.6.1.            Assume that the stress field inside the inclusion is given by ${\sigma }_{ij}=A{\sigma }_{0}{\delta }_{ij}+B{\sigma }_{0}{\delta }_{i3}{\delta }_{j3}$.  Calculate the displacement field in the inclusion (assume that the displacement and rotation of the solid vanish at the origin).

5.4.6.2.            The stress field outside the inclusion can be generated from Papkovich-Neuber potentials

${\Psi }_{i}=\frac{\left(1-\nu \right){\sigma }_{0}}{\left(1+\nu \right)}{x}_{3}{\delta }_{i3}+\frac{{a}^{3}{p}_{ik}^{T}{x}_{k}}{3{R}^{3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =\frac{\nu \left(1-\nu \right){\sigma }_{0}}{\left(1+\nu \right)}\left(3{x}_{3}^{2}-{R}^{2}\right)+\frac{{a}^{3}{p}_{ij}^{T}}{15{R}^{3}}\left(\left(5{R}^{2}-{a}^{2}\right){\delta }_{ij}+3{a}^{2}\frac{{x}_{i}{x}_{j}}{{R}^{2}}\right)$

where ${p}_{ij}^{T}=C{\sigma }_{0}{\delta }_{ij}+D{\sigma }_{0}{\delta }_{i3}{\delta }_{j3}$, and C and D are constants to be determined.

5.4.6.3.            Use the conditions at r=a to find expressions for A,B,C,D in terms of geometric and material properties.

5.4.6.4.            Hence, find the stress field inside the inclusion.

5.4.7.      Consider the Eshelby inclusion problem described in Section 5.4.6. An infinite homogeneous, stress free, linear elastic solid has Young’s modulus E and Poisson’s ratio $\nu$. The solid is initially stress free. An inelastic strain distribution ${\epsilon }_{ij}^{T}$  is introduced into an ellipsoidal region of the solid B (e.g. due to thermal expansion, or a phase transformation).  Let ${u}_{i}$ denote the displacement field, ${\epsilon }_{ij}^{}={\epsilon }_{ij}^{e}+{\epsilon }_{ij}^{T}$  denote the total strain distribution, and let ${\sigma }_{ij}$ denote the stress field in the solid.

5.4.7.1.            Write down an expression for the total strain energy ${\Phi }_{I}$ within the ellipsoidal region, in terms of ${\sigma }_{ij}$, ${\epsilon }_{ij}$ and ${\epsilon }_{ij}^{T}$.

5.4.7.2.            Write down an expression for the total strain energy outside the ellipsoidal region, expressing your answer as a volume integral in terms of ${\epsilon }_{ij}$ and ${\sigma }_{ij}$.  Using the divergence theorem, show that the result can also be expressed as

${\Phi }_{O}=-\frac{1}{2}\underset{S}{\int }{\sigma }_{ij}{n}_{j}{u}_{i}dA$

where S denotes the surface of the ellipsoid, and ${n}_{j}$ 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.

5.4.7.3.            The Eshelby solution shows that the strain ${\epsilon }_{ij}^{}={\epsilon }_{ij}^{e}+{\epsilon }_{ij}^{T}$ inside B is uniform.  Write down the displacement field inside the ellipsoidal region, in terms of ${\epsilon }_{ij}$ (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

${\Phi }_{O}=-\frac{1}{2}\underset{S}{\int }{\sigma }_{ij}{\epsilon }_{ik}{x}_{k}{n}_{j}dA$

5.4.7.4.            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

$\Phi ={\Phi }_{O}+{\Phi }_{I}=-\frac{1}{2}\underset{B}{\int }{\sigma }_{ij}{\epsilon }_{ij}^{T}⥄dV$

5.4.8.      Using the solution to Problem 7, calculate the total strain energy of an  initially stress-free isotropic, linear elastic solid with Young’s modulus E and Poisson’s ratio $\nu$, after an inelastic strain ${\epsilon }_{ij}^{T}$ is introduced into a spherical region with radius a in the solid.

5.4.9.      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.

5.4.10.  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 Young’s modulus E, Poisson’s ratio $\nu$ 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.

5.4.11.  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 Young’s modulus $E$ and Poisson’s ratio $\nu$

5.4.11.1.        Assume that the load causes the center of the inner raceway to move vertically upwards by a distance $\Delta$, 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 $\Delta$

5.4.11.2.        Hence, calculate the resultant contact forces between the balls at A, B, C and the raceways, in terms of $\Delta$ and relevant geometrical and material properties.

5.4.11.3.        Finally, calculate the contact forces in terms of P

5.4.11.4.        If the materials have uniaxial tensile yield stress Y, find an expression for the maximum force P that the bearing can withstand before yielding.

5.4.12.  A rigid, conical indenter with apex angle $2\beta$ is pressed into the surface of an isotropic, linear elastic solid with Young’s modulus $E$ and Poisson’s ratio $\nu$

5.4.12.1.        Write down the initial gap between the two surfaces $g\left(r\right)$

5.4.12.2.        Find the relationship between the depth of penetration h of the indenter and the radius of contact a

5.4.12.3.        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 $\frac{dP}{dh}=2{E}^{*}a$

5.4.12.4.        Calculate the distribution of contact pressure that acts between the contacting surfaces.

5.4.13.  A sphere, which has radius R, is dropped from height h onto the flat surface of a large solid.  The sphere has mass density $\rho$, and both the sphere and the surface can be idealized as linear elastic solids, with Young’s modulus $E$ and Poisson’s ratio $\nu$.  As a rough approximation, the impact can be idealized as a quasi-static elastic indentation.

5.4.13.1.        Write down the relationship between the force P acting on the sphere and the displacement of the center of the sphere below ${x}_{2}=R$

5.4.13.2.        Calculate the maximum vertical displacement of the sphere below the point of initial contact.

5.4.13.3.        Deduce the maximum force and contact pressure acting on the sphere

5.4.13.4.        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

5.4.13.5.        Calculate a value of h if the materials are steel, and the sphere has a 1 cm radius.