## EN224: Linear Elasticity

##### Homework 6: Energy methods, anisotropic elasticity

##### Due Friday May 6, 2005

**Division of Engineering**

**Brown University**

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**1. **Let denote the solution to a linear elastostatic boundary value
problem. Let and denote the potential and complementary
energy, respectively.

Show that

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**2. Torsion**

2.1 Reconsider the
torsion problem discussed in class, with boundary conditions illustrated
above. Using a kinematically admissible
displacement field given by

obtain an
expression for the potential energy of the solid.

2.2 By minimizing
the potential energy, show that the best approximation to the displacements is
obtained by selecting a function *w* that satisfies

2.3 With *w *given in 2.2, deduce that the stiffness
of the shaft (defined as in class) must satisfy

**3. Interface** **crack between two anisotropic solids
subjected to anti-plane shear loading.** The problem to be solved is
illustrated in the figure below. Two
anisotropic elastic solids (with elastic constants such that the solid can
sustain anti-plane shear deformations) are bonded across the real axis. The
interface contains a crack, which is loaded by distributions of tractions
acting on its upper and lower surfaces.
Assume that stresses vanish at infinity. The solution is to be generated (using the standard complex
variable formulation for anisotropic solids) from two analytic functions, ,
where is analytic in and is analytic in . We will find that these functions depend
only on the generalized shear moduli for the two solids

3.1 Write down the conditions for traction and
displacement continuity across the real axis outside the crack, in terms of and

3.2 Express the traction and displacement continuity
conditions in terms of functions and ,
which are analytic in and . Hence, deduce that

are continuous across the real axis outside the crack, and
are analytic everywhere.

3.3 Deduce that

provide analytic continuations of and across the real axis

3.4 Show that the traction boundary conditions on the
upper and lower faces of the crack may be expressed as

3.5 Deduce that these boundary conditions may be expressed
in terms of as

3.6 Finally, deduce the potentials that generate the anti-plane shear solution
to a bi-material interface crack between two anisotropic solids subjected to
point forces on the crack faces as illustrated below.