Problems for Chapter 3

Constitutive Models: Relations between Stress and Strain

3.12.    Crystal Plasticity

3.12.1.  Draw an inverse pole figure for an fcc single crystal, with [100], [010] and [001] directions parallel to the {i,j,k} directions, showing the following:

3.12.1.1.        The trace of the (010), (100), (110) and $\left(\overline{1}10\right)$ planes

3.12.1.2.        The trace of the {111} planes

3.12.1.3.        The twelve slip directions, labeled according to the convention given in Section 3.12.2 (i.e. ${a}_{1},{a}_{2},{a}_{3}$, etc)

3.12.2.  Consider the inverse pole figure for an fcc crystal with  [100], [010] and [001] directions parallel to the {i,j,k} directions, as shown in the figure.  Show that the circle corresponding to the trace of the $\left(\overline{1}01\right)$ plane has radius $R=\sqrt{2}$, and is centered at the point corresponding to the $\left[\overline{1}01\right]$ as shown in the figure.  The simplest approach to this problem is to note that the [010] direction and the [101] direction both lie on the circle.

3.12.3.  Plot a contour map on the standard triangle of the inverse pole figure for an fcc single crystal, showing the magnitude of the resolved shear stress induced by uniaxial tensile stress on the critical (d1) slip system.  Find the orientation of the tensile axis that gives the largest resolved shear stress

3.12.4.  An fcc single crystal deforms by shearing at rate $\stackrel{˙}{\gamma }$ on the d1 $\left(1\overline{1}1\right)$ $\left[011\right]$ and $-\stackrel{˙}{\gamma }$ on the c2 $\left(\overline{1}11\right)$ $\left[\overline{1}0\overline{1}\right]$ slip systems.

3.12.4.1.        Show that the material fiber parallel to the [112] direction has zero angular velocity;

3.12.4.2.        Calculate the rate of stretching of the material fiber parallel to the [112] direction.

3.12.5.  A single crystal is loaded in uniaxial tension.  The direction of the loading axis, specified by a unit vector p remains fixed during straining.  The crystal deforms by slip on a single system, with slip direction ${s}^{\alpha }$ and slip plane normal ${m}^{\alpha }$The deformation gradient resulting from a shear strain ${\gamma }^{\alpha }$ is

${F}_{ij}={R}_{ik}\left({\delta }_{kj}+{\gamma }^{\alpha }{s}_{k}^{\alpha }{m}_{j}^{\alpha }\right)$

where ${R}_{ij}$ is a proper orthogonal tensor (i.e. det(R)=1, ${R}_{ik}{R}_{jk}={\delta }_{ij}$ ), representing a rigid rotation.  Assume that that the material fiber parallel to the loading axis does not rotate during deformation.  Show that:

${R}_{ij}={\delta }_{ij}\mathrm{cos}\theta +\left(1-\mathrm{cos}\theta \right){n}_{i}{n}_{j}+\mathrm{sin}\theta {\in }_{ikj}{n}_{k}$

where ${n}_{i}={\in }_{ijk}{s}_{j}{p}_{k}$ $\mathrm{cos}\theta =\left(1+{\gamma }^{\alpha }{p}_{i}{s}_{i}^{\alpha }{p}_{k}{m}_{k}^{\alpha }\right)/C$, $\mathrm{sin}\theta ={\gamma }^{\alpha }\left({p}_{i}{m}_{i}^{\alpha }\right)\sqrt{\left(1-{\left({p}_{i}{s}_{i}^{\alpha }\right)}^{2}\right)}/C$, and $C=\sqrt{1+{\gamma }^{\alpha 2}{\left({p}_{i}{m}_{i}^{\alpha }\right)}^{2}+2{\gamma }^{\alpha }{p}_{i}{s}_{i}^{\alpha }{p}_{k}{m}_{k}^{\alpha }}$.

3.12.6.  Consider the single crystal loaded in uniaxial tension described in the preceding problem.  Calculate

3.12.6.1.        The angular velocity vector that describes the angular velocity of the slip direction and slip plane normal (it is easiest to do this by first assuming that the slip direction and slip plane normal are fixed, and calculating the angular velocity of the material fiber parallel to p, and hence deducing the angular velocity that must be imposed on the crystal to keep p pointing in the same direction)

3.12.6.2.        Calculate the spin tensor W associated with the angular velocity calculated in 3.10.6.1

3.12.7.  The resolved shear stress on a slip system in a crystal is related to the Kirchhoff stress $\tau$ by ${\tau }^{\alpha }={s}^{*\alpha }\cdot \tau \cdot {m}^{*\alpha }$.  Show that the rate of change of resolved shear stress is related to the elastic Jaumann rate of Kirchhoff stress by

$\frac{d{\tau }^{\alpha }}{dt}={s}_{i}^{*\alpha }\left(\stackrel{\nabla e}{{\tau }_{ij}}-{D}_{ik}^{e}{\tau }_{kj}+{D}_{ik}^{e}{\tau }_{kj}\right){m}_{j}^{*\alpha }$

3.12.8.  A rigid $–$perfectly plastic single crystal contains two slip systems, oriented at angles ${\varphi }_{1}$ and ${\varphi }_{2}$ as illustrated in the figure.  The solid is deformed in simple shear as indicated

3.12.8.1.        Suppose that ${\varphi }_{1}={60}^{0}$, ${\varphi }_{2}={120}^{0}$.  Sketch the yield locus (in ${\sigma }_{11},{\sigma }_{22}$ space) for the crystal.

3.12.8.2.        Write down the velocity gradient L in the strip, and compute an expression the deformation rate D.  Hence, show that (at the instant shown) the slip rates on the two slip systems are given by

$\begin{array}{l}{\stackrel{˙}{\gamma }}^{\left(1\right)}=\stackrel{˙}{\gamma }\mathrm{sin}2{\varphi }_{2}/\mathrm{sin}2\left({\varphi }_{2}-{\varphi }_{1}\right)\\ {\stackrel{˙}{\gamma }}^{\left(2\right)}=\stackrel{˙}{\gamma }\mathrm{sin}2{\varphi }_{1}/\mathrm{sin}2\left({\varphi }_{1}-{\varphi }_{2}\right)\end{array}$

and give an expression for $\stackrel{˙}{\gamma }$ in terms of ${v}_{0}$ and h.

3.12.8.3.        Assume that ${\sigma }_{11}+{\sigma }_{22}=0$, and that the slip systems have critical resolved shear stress ${\tau }_{c}$.  Show that the stress in the crystal is

$\begin{array}{l}{\sigma }_{11}=-{\sigma }_{22}={\tau }_{c}\left(\mathrm{sin}2{\varphi }_{2}-\mathrm{sin}2{\varphi }_{1}\right)/\mathrm{sin}2\left({\varphi }_{2}-{\varphi }_{1}\right)\\ {\sigma }_{12}={\tau }_{c}\left(\mathrm{cos}2{\varphi }_{2}-\mathrm{cos}2{\varphi }_{1}\right)/\mathrm{sin}2\left({\varphi }_{2}-{\varphi }_{1}\right)\end{array}$

3.12.9.  Consider a single crystal of copper, with constitutive equation given in section 3.12.5. and properties listed in 3.12.6.

3.12.9.1.        Plot a graph showing the uniaxial true stress-v-true strain curve for a crystal that is loaded parallel to the [112] direction

3.12.9.2.        Plot a graph showing the uniaxial  true stress-v-true strain curve for a crystal loaded parallel to the [111] direction

3.12.9.3.        Plot a graph showing the uniaxial true stress-v-true strain curve for a crystal loaded parallel to the [001] direction.

3.12.10.                      A rate independent, rigid perfectly plastic fcc single crystal is loaded in uniaxial tension, with tensile axis parallel to the [102] crystallographic direction.  Assume that the crystal rotates to maintain the material fiber parallel to the [102] direction aligned with the tensile axis.

3.12.10.1.    Assuming the magnitude of the shearing rate is $\stackrel{˙}{\gamma }$ on all active slip systems, calculate the velocity gradient in the crystal, expressing your answer as components in a basis aligned with the {100} directions.

3.12.10.2.    Hence, show that the [102] loading direction is not a stable orientation $–$ i.e. the tensile axis rotates with respect to the crystallographic directions.

3.12.10.3.    Calculate the instantaneous stretching rate of the tensile axis as a function of the magnitude of the shearing rate.

3.12.10.4.    Deduce the angular velocity vector that characterizes the instantaneous rotation of the crystal relative to the tensile axis.

3.12.10.5.    Show the motion of the tensile axis on an inverse pole figure.  Without calculations, predict the eventual steady-state orientation of the tensile axis with respect to the loading axis.