Problems for Chapter 2

Governing Equations

2.2.  Mathematical Description of internal forces in solids in Solids

2.2.1.      A rectangular bar is loaded in a state of uniaxial tension, as shown in the figure.

2.2.1.1.            Write down the components of the stress tensor in the bar, using the basis vectors shown.

2.2.1.2.            Calculate the components of the normal vector to the plane ABCD shown, and hence deduce the components of the traction vector acting on this plane, expressing your answer as components in the basis shown, in terms of $\theta$

2.2.1.3.            Compute the normal and tangential tractions acting on the plane shown.

2.2.2.      Consider a state of hydrostatic stress ${\sigma }_{ij}=p{\delta }_{ij}$.  Show that the traction vector acting on any internal plane in the solid (or, more likely, fluid!) has magnitude p and direction normal to the plane.

2.2.3.      A cylinder of radius R is partially immersed in a static fluid.

2.2.3.1.            Recall that the pressure at a depth d in a fluid has magnitude $\rho gd$.  Write down an expression for the horizontal and vertical components of traction acting on the surface of the cylinder in terms of $\theta$

2.2.3.2.            Hence compute the resultant force exerted by the fluid on the cylinder.

2.2.4.      The figure  shows two designs for a glue joint.  The glue will fail if the stress acting normal to the joint exceeds 60 MPa, or if the shear stress acting parallel to the plane of the joint exceeds 300 MPa.

2.2.4.1.            Calculate the normal and shear stress acting on each joint, in terms of the applied stress $\sigma$

2.2.4.2.            Hence, calculate the value of $\sigma$ that will cause each joint to fail.

2.2.5.      For the Cauchy stress tensor with components

$\left[\begin{array}{ccc}100& 250& 0\\ 250& 200& 0\\ 0& 0& 300\end{array}\right]$

compute

2.2.5.1.            The traction vector acting on an internal material plane with normal $n=\frac{1}{\sqrt{2}}{e}_{1}-\frac{1}{\sqrt{2}}{e}_{2}\text{\hspace{0.17em}}$

2.2.5.2.            The principal stresses

2.2.5.3.            The hydrostatic stress

2.2.5.4.            The deviatoric stress tensor

2.2.5.5.            The Von-Mises equivalent stress

2.2.6.      Show that the hydrostatic stress ${\sigma }_{kk}$ is invariant under a change of basis $–$ i.e. if ${\sigma }_{ij}^{e}$ and ${\sigma }_{ij}^{m}$ denote the components of stress in bases $\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$ and $\left\{{m}_{1},{m}_{2},{m}_{3}\right\}$, respectively, show that ${\sigma }_{kk}^{e}={\sigma }_{kk}^{m}$.

2.2.7.      A rigid, cubic solid is immersed in a fluid with mass density $\rho$. Recall that a stationary fluid exerts a compressive pressure of magnitude $\rho gh$ at depth h.

2.2.7.1.             Write down expressions for the traction vector exerted by the fluid on each face of the cube.  You might find it convenient to take the origin for your coordinate system at the center of the cube, and take basis vectors $\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$ perpendicular to the cube faces.

2.2.7.2.            Calculate the resultant force due to the tractions acting on the cube, and show that the vertical force is equal and opposite to the weight of fluid displaced by the cube.

2.2.8.      Show that the result of problem 2.2.7 applies to any arbitrarily shaped solid immersed below the surface of a fluid, i.e. prove that the resultant force acting on an immersed solid with volume V is ${P}_{i}=\rho gV{\delta }_{i3}$, where it is assumed that ${e}_{3}$ is vertical.   To do this

2.2.8.1.            Let ${n}_{j}$ denote the components of a unit vector normal to the surface of the immersed solid

2.2.8.2.            Write down a formula for the traction (as a vector) exerted by the fluid on the immersed solid

2.2.8.3.            Integrate the traction to calculate the resultant force, and manipulate the result obtain the required formula.

2.2.9.      A component contains a feature with a 90 degree corner as shown in the picture.   The surfaces that meet at the corner are not subjected to any loading.  List all the stress components that must be zero at the corner

2.2.10.  In this problem we consider further the beam bending calculation discussed in Problem 2.1.21. Suppose that the beam is made from a material in which the Material Stress tensor is related to the Lagrange strain tensor by

${\Sigma }_{ij}=2\mu {E}_{ij}$

(this can be regarded as representing an elastic material with zero Poisson’s ratio and shear modulus $\mu$ )

2.2.10.1.        Calculate the distribution of material stress in the bar, expressing your answer as components in the $\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$ basis

2.2.10.2.        Calculate the distribution of nominal stress in the bar expressing your answer as components in the $\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$ basis

2.2.10.3.        Calculate the distribution of Cauchy stress in the bar expressing your answer as components in the $\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$ basis

2.2.10.4.        Repeat 15.1-15.3 but express the stresses as components in the $\left\{{m}_{1},{m}_{2},{m}_{3}\right\}$ basis

2.2.10.5.        Calculate the distribution of traction on a surface in the beam that has normal ${e}_{1}$ in the undeformed beam.  Give expressions for the tractions in both $\left\{{e}_{1},{e}_{2},{e}_{3}\right\}$ and $\left\{{m}_{1},{m}_{2},{m}_{3}\right\}$

2.2.10.6.        Show that the surfaces of the beam that have positions ${x}_{2}=±h/2$ in the undeformed beam are traction free after deformation

2.2.10.7.        Calculate the resultant moment acting on the ends of the beam.

2.2.11.  A solid is subjected to some loading that induces a Cauchy stress ${\sigma }_{ij}^{\left(0\right)}$ at some point in the solid.  The solid and the loading frame are then rotated together so that the entire solid (as well as the loading frame) is subjected to a rigid rotation ${R}_{ij}$This causes the components of the Cauchy stress tensor to change to new values ${\sigma }_{ij}^{\left(1\right)}$.  The goal of this problem is to calculate a formula relating  ${\sigma }_{ij}^{\left(0\right)}$, ${\sigma }_{ij}^{\left(1\right)}$ and ${R}_{ij}$.

2.2.11.1.        Let ${n}_{i}^{\left(0\right)}$ be a unit vector normal to an internal material plane in the solid before rotation.  After rotation, this vector (which rotates with the solid) is ${n}_{i}^{\left(1\right)}$.   Write down the formula relating ${n}_{i}^{\left(0\right)}$ and ${n}_{i}^{\left(1\right)}$

2.2.11.2.        Let ${T}_{i}^{\left(0\right)}$ be the internal traction vector that acts on a material plane with normal ${n}_{i}^{\left(0\right)}$ in the solid before application of the rigid rotation.  Let ${T}_{i}^{\left(1\right)}$ be the traction acting on the same material plane after rotation.  Write down the formula relating ${T}_{i}^{\left(0\right)}$ and ${T}_{i}^{\left(1\right)}$

2.2.11.3.        Finally, using the definition of Cauchy stress, find the relationship between ${\sigma }_{ij}^{\left(0\right)}$, ${\sigma }_{ij}^{\left(1\right)}$ and ${R}_{ij}$.

2.2.12.  Repeat problem 2.2.11, but instead, calculate a relationship between the components of Nominal stress ${S}_{ij}^{\left(0\right)}$ and ${S}_{ij}^{\left(1\right)}$ before and after the rigid rotation.

2.2.13.  Repeat problem 2.2.11, but instead, calculate a relationship between the components of material stress ${\Sigma }_{ij}^{\left(0\right)}$ and ${\Sigma }_{ij}^{\left(1\right)}$ before and after the rigid rotation.