## EN224: Linear Elasticity

##### Homework 1: Linearized Field Equations and Constitutive Law

##### Due Friday Feb 18, 2005

#

**Division of Engineering**

**Brown University**

**1.
**Consider an
elastic material with reference mass density . Suppose that, instead of characterizing the
material through the specific Helmholtz free energy in terms of Lagrange strain
**E** and temperature, we choose to express the specific Helmholtz free
energy in terms of deformation gradient **F, **as .

1.1
Follow the argument used in class to show that nominal stress (defined as in
class – may be the transpose of what you are used to seeing) is related to
Helmholtz free energy by

1.2
As a specific example, consider an elastic solid with Helmholtz free energy
given by

. Find an expression for the nominal stress in
terms of **F**.

Linearize
the constitutive law for infinitesimal motions. Show that the linearized constitutive law characterizes an isotropic
solid, and find formulas for the shear modulus and Poisson’s ratio for the
linearized material in terms of *A* and *B*.

1.3
Consider a solid whose constitutive response can be characterized using the
free energy in 1.2. Suppose that the
solid is first subjected to loading that induces a (time independent)
homogeneous deformation . A small additional load is then applied to
the solid that induces an additional displacement field ,
with . Obtain a set of linearized field equations
for the displacement field and suitable strain and stress fields. You should include (i) An appropriate
linearized strain measure; (ii) an appropriate linearized stress measure, with
momentum balance equations, and (iii) a linearized constitutive equation.

**2. **Express
the constitutive law for a linear elastic solid in matrix form as ,
where and represent column vectors of stress and strain
and represents a 6x6 elasticity matrix.

2.1 Show that the material has a positive definite
strain energy density ( ) if and only if the eigenvalues of are all positive.

2.2 Hence, show that is invertible for a material with positive
definite strain energy density.

2.3 Calculate the eigenvalues of the compliance
matrix (where ) for an isotropic solid in terms of Young’s
modulus and Poisson’s ratio. Hence,
re-derive the restrictions on values of Poisson’s ratio and Young’s modulus
that were found in class. Find the
eigenvectors of and sketch the deformations associated with
these eigenvectors.

2.4 Suppose that a cubic material is characterize by
the values of ,
and in the elasticity matrix. Use the same approach to derive restrictions
on the admissible ranges of ,
and .