 Problems for Chapter 2

Governing Equations

2.3.  Equations of motion and equilibrium for deformable solids 2.3.1.      A prismatic concrete column of mass density $\rho$ 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).

2.3.1.1.            Show that the stress distribution

${\sigma }_{22}=-\rho g\left(H-{x}_{2}\right)$

satisfies the equations of static equilibrium

$\frac{\partial {\sigma }_{ij}}{\partial {x}_{i}}+\rho {b}_{j}=0$

and also satisfies the boundary conditions ${\sigma }_{ij}{n}_{i}=0$

on all free boundaries.

2.3.1.2.            Show that the traction vector acting on a plane with normal $n=\mathrm{sin}\theta \text{\hspace{0.17em}}\text{\hspace{0.17em}}{e}_{1}+\mathrm{cos}\theta \text{\hspace{0.17em}}\text{\hspace{0.17em}}{e}_{2}$ at a height ${x}_{2}$ is given by

$T=-\rho g\left(H-{x}_{2}\right)\mathrm{cos}\theta \text{\hspace{0.17em}}{e}_{2}$

2.3.1.3.            Deduce that the normal component of traction acting on the plane is

${T}_{n}=-\rho g\left(H-{x}_{2}\right){\mathrm{cos}}^{2}\theta$

2.3.1.4.            show also that the tangential component of traction acting on the plane is

${T}_{t}=\rho g\left(H-{x}_{2}\right)\mathrm{sin}\theta \text{\hspace{0.17em}}\mathrm{cos}\theta \text{\hspace{0.17em}}\left(\mathrm{cos}\theta \text{\hspace{0.17em}}{e}_{1}-\mathrm{sin}\theta \text{\hspace{0.17em}}{e}_{2}\right)$

(the easiest way to do this is to note that $T={T}_{n}n+{T}_{t}$ and solve for the tangential traction).

### 2.3.1.5.            Suppose that the concrete contains a large number of randomly oriented microcracks.  A crack which lies at an angle $\theta$ to the horizontal will propagate if

$|{T}_{t}|+\mu {T}_{n}>{\tau }_{0}$

where $\mu$ is the friction coefficient between the faces of the crack and ${\tau }_{0}$ 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.

2.3.1.6.            Assume that $\mu =1$.  Find the orientation of the microcrack that is most likely to propagate.  Hence, find an expression for the maximum possible height of the column.

2.3.2.      Is the stress field given below in static equilibrium?  If not, find the acceleration or body force density required to satisfy linear momentum balance

$\begin{array}{l}{\sigma }_{11}=C{x}_{1}{x}_{2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{12}={\sigma }_{21}=C\left({a}^{2}-{x}_{2}^{2}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ {\sigma }_{33}={\sigma }_{23}={\sigma }_{13}=0\end{array}$

2.3.3.      Let $\varphi$ be a twice differentiable, scalar function of position.  Derive a plane stress field from $\varphi$ by setting

${\sigma }_{11}=\frac{{\partial }^{2}\varphi }{\partial {x}_{2}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{22}=\frac{{\partial }^{2}\varphi }{\partial {x}_{1}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{12}={\sigma }_{21}=-\frac{{\partial }^{2}\varphi }{\partial {x}_{1}\partial {x}_{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$

Show that this stress field satisfies the equations of stress equilibrium with zero body force.

2.3.4.      The stress field

${\sigma }_{ij}=\frac{-3{P}_{k}{x}_{k}{x}_{i}{x}_{j}}{4\pi {R}^{5}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}R=\sqrt{{x}_{k}{x}_{k}}$

represents the stress in an infinite, incompressible elastic solid that is subjected to a point force with components ${P}_{k}$ 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).

2.3.4.1.            Verify that the stress field is in static equilibrium

2.3.4.2.            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. 2.3.5.      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 $\rho$ in the reference configuration, and may be idealized as a viscous fluid, in which the Kirchhoff stress is related to stretch rate by

$\tau =\mu D+pI$

where p is an indeterminate hydrostatic pressure and $\mu$ is the viscosity.

2.3.5.1.            Find expressions for the Cauchy stress tensor, expressing your answer as components in $\left\{{e}_{r},{e}_{\theta },{e}_{z}\right\}$

2.3.5.2.            Assume steady, quasi-static deformation (neglect accelerations).  Express the equations of equilibrium in terms of the angle $\varphi \left(r,t\right)$

2.3.5.3.            Solve the equilibrium equation, together with appropriate boundary conditions, to calculate $\varphi \left(r,t\right)$

2.3.5.4.            Find the torque necessary to rotate the external cylinder

2.3.5.5.            Calculate the acceleration of a material particle in the fluid

2.3.5.6.            Estimate the rotation rate $\stackrel{˙}{\alpha }$ where inertia begins to play a significant role in determining the state of stress in the fluid