 Problems for Chapter 6

Analytical Techniques and Solutions for Plastic Solids

6.1.  Slip-line Fields 6.1.1.      The figure shows the slip-line field for a rigid plastic double-notched bar deforming under uniaxial tensile loading.  The material has yield stress in shear k

6.1.1.1.            Draw the Mohr’s circle representing the state of stress at A.  Write down (i) the value of $\varphi$ at this point, and (ii) the magnitude of the hydrostatic stress $\overline{\sigma }$ at this point.

6.1.1.2.            Calculate the value of $\varphi$ at point B, and deduce the magnitude of $\overline{\sigma }$.  Draw the Mohr’s circle of stress at point B, and calculate the horizontal and vertical components of stress

6.1.1.3.            Repeat 1.1 and 1.2, but trace the $\beta$ slip-line from point C to point B.

6.1.1.4.            Find an expression for the force P that causes plastic collapse in the bar. 6.1.2.      The figure shows a slip-line field for oblique indentation of a rigid-plastic surface by a flat punch

6.1.2.1.            Draw the Mohr’s circle representing the state of stress at A.  Write down (i) the value of $\varphi$ at this point, and (ii) the magnitude of the hydrostatic stress $\overline{\sigma }$ at this point.

6.1.2.2.            Calculate the value of $\varphi$ at point B, and deduce the magnitude of $\overline{\sigma }$.

6.1.2.3.            Draw the Mohr’s circle representation for the stress state at B, and hence calculate the tractions acting on the contacting surface, as a function of k and $\theta$.

6.1.2.4.            Calculate expressions for P and Q in terms of k, a and $\theta$, and find an expression for Q/P

6.1.2.5.            What is the maximum possible value of friction coefficient Q/P?  What does the slip-line field look like in this limit? 6.1.3.      The figure shows the slip-line field for a rigid plastic double-notched bar under uniaxial tension. The material has yield stress in shear k. The slip-lines are logarithmic spirals, as discussed in Section 6.1.3.

6.1.3.1.            Write down a relationship between the angle $\psi$, the notch radius a and the bar width b.

6.1.3.2.            Draw the Mohr’s circle representing the state of stress at A.  Write down (i) the value of $\varphi$ at this point, and (ii) the magnitude of the hydrostatic stress $\overline{\sigma }$ at this point.

6.1.3.3.            Determine the value of $\varphi$ and the hydrostatic stress at point B, and draw the Mohr’s circle representing the stress state at this point.

6.1.3.4.            Hence, deduce the distribution of vertical stress along the line BC, and calculate the force P in terms of k, a and b. 6.1.4.      The figure shows the slip-line field for a notched, rigid plastic bar deforming under pure bending (the solution is valid for $\psi <\pi /2-1$, for reasons discussed in Section 6.1.3).  The solid has yield stress in shear k.

6.1.4.1.            Write down the distribution of stress in the triangular region OBD

6.1.4.2.            Using the solution to problem 2, write down the stress distribution along the line OA

6.1.4.3.            Calculate the resultant force exerted by tractions on the line AOC.  Find the ratio of d/b for the resultant force to vanish, in terms if $\psi$, and hence find an equation relating a/(b+d) and $\psi$.

6.1.4.4.            Finally, calculate the resultant moment of the tractions about O, and hence find a relationship between Ma, b+d and $\psi$.

6.1.4.5.            Show that the slip-line field is valid only for b+d less than a critical value, and determine an expression for the maximum allowable value for b+d.

6.1.5.      Consider the problem in 6.1.4.  Propose a slip-line field solution that is valid for $\psi >\pi /2-1$, and use it to calculate the collapse moment in terms of relevant material and geometric parameters. 6.1.6.      The figure shows the slip-line field for a rigid plastic double-notched bar subjected to a bending moment.  The slip-lines are logarithmic spirals.

6.1.6.1.            Write down a relationship between the angle $\psi$, the notch radius a and the bar width b.

6.1.6.2.            Draw the Mohr’s circle representing the state of stress at A.  Write down (i) the value of $\varphi$ at this point, and (ii) the magnitude of the hydrostatic stress $\overline{\sigma }$ at this point.

6.1.6.3.            Determine the value of $\varphi$ and the hydrostatic stress just to the right of point B

6.1.6.4.            Hence, deduce the distribution of vertical stress along the line BC

6.1.6.5.            Without calculations, write down the variation of stress along the line BD.  What happens to the stress at point B?

6.1.6.6.            Hence, calculate the value of the bending moment M in terms of b, a, and k.

6.1.6.7.            Show that the slip-line field is valid only for b less than a critical value, and determine an expression for the maximum allowable value for b.

6.1.7.      Consider the problem in 6.1.6.  Propose a slip-line field solution that is valid for large b, and use it to calculate the collapse moment in terms of relevant material and geometric parameters. 6.1.8.      A rigid flat punch is pressed into the surface of an elastic-perfectly plastic half-space, with Young’s modulus $E$, Poisson’s ratio $\nu$ and shear yield stress k.  The punch is then withdrawn.

6.1.8.1.            At maximum load the stress state under the punch can be estimated using the rigid-plastic slip-line field solution (the solution is accurate as long as plastic strains are much greater than elastic strains).   Calculate the stress state in this condition (i) just under the contact, and (ii) at the surface just outside the contact.

6.1.8.2.            The unloading process can be assumed to be elastic $–$ this means that the change in stress during unloading can be calculated using the solution to an elastic half-space subjected to uniform pressure on its surface.   Calculate the change in stress (i) just under the contact, and (ii) just outside the contact, using the solution given in Section 5.2.8.

6.1.8.3.            Calculate the residual stress (i.e. the state of stress that remains in the solid after unloading) at points A and B on the surface.