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{\LARGE \ Physics 861 -- Fall 01}

{\bf Problem set 10 - }Due Thursday, Nov 29

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\noindent {\bf 1. }The stress-strain relation in an isotropic solid can be
written 
\begin{equation}
{\cal E}_{ij}=A(S_{ij}+p\delta _{ij})+A^{\prime }p\delta _{ij}
\end{equation}

where $p=-(S_{11}+S_{22}+S_{33})/3$ is the pressure.

(a) Relate the constants $A$ and $A^{\prime }$ to the bulk modulus $B$ and
shear modulus $\mu $.

(b) Relate the constants $A$ and $A^{\prime }$ to Young's modulus $E$ and
Poisson's ratio $\nu .$

\ \quad [The next questions can be answered independently of (a) and (b]

(c) In the limit (near melting, perhaps) when an isotropic solid becomes
like a liquid with bulk modulus $B$, what are the values of $E$, $\mu $ and $%
\nu $?

(d) In a cubic crystal, the Cauchy relations reduce to $C_{12}=C_{44}$. When 
$C_{11}-C_{12}=2C_{44}$ the crystal becomes an isotropic solid. Knowing
this, find the values of $E$, $\mu $ and $\nu $ for an isotropic solid of
bulk modulus $B$ when Cauchy's relations are satisfied.

(e) You will see from the above that a solid that obeys Cauchy's relations
cannot transform continuously into a liquid. Why not? 

$\medskip $

\noindent {\bf 2. }

Problem 2, page 486, of Ashcroft - Mermin

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\noindent {\bf 3. }

Problem 7, page 640, of Ashcroft - Mermin.

(a) Answer the questions posed by Ashcroft- Mermin.

(b) In eq. (30.36), what is the value of $G$ for an isotropic crystal in
terms of the standard elastic constants $B,\,E,\,\mu ,\,\nu $? Try to get
the simplest answer, rather than an awful combination of constants.

(c) What is the displacement $u_{z}$ as a function of $x$ and $y$? Here $z$
is the dislocation axis and $r^{2}=x^{2}+y^{2}$. Assume, again, an isotropic
crystal.

(d) Assume that the dislocation is parallel to one of the axes in a cubic
crystal. Recall that the equations of elastic equilibrium are, quite
generally, 
\begin{equation}
\sum_{j}\partial S_{ij}/\partial x_{j}=0
\end{equation}
Is $u_{z}$ of the same form as in part (c), for the appropriate value of $G$%
, and how is $G$ related to $C_{11},\,C_{12}$ and $C_{44}$?

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