hooke¶

Purpose:

Compute material matrix for a linear elastic and isotropic material.

Syntax:

D = hooke(ptype, E, v)

Description:

The function hooke computes the material matrix \(\mathbf{D}\) for a linear elastic and isotropic material.

The variable ptype is used to define the type of analysis:

• ptype = 1 - plane stress

• ptype = 2 - plane strain

• ptype = 3 - axisymmetry

• ptype = 4 - three dimensional analysis

The material parameters \(E\) and \(\nu\) define the modulus of elasticity and the Poisson’s ratio, respectively.

Theory:

For plane stress (ptype=1), \(\mathbf{D}\) is formed as

\[\begin{split}\mathbf{D} = \frac{E}{1-\nu^2} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac{1-\nu}{2} \end{bmatrix}\end{split}\]

For plane strain (ptype=2) and axisymmetry (ptype=3), \(\mathbf{D}\) is formed as

\[\begin{split}\mathbf{D} = \frac{E}{(1+\nu)(1-2\nu)} \begin{bmatrix} 1-\nu & \nu & \nu & 0 \\ \nu & 1-\nu & \nu & 0 \\ \nu & \nu & 1-\nu & 0 \\ 0 & 0 & 0 & \frac{1}{2}(1-2\nu) \end{bmatrix}\end{split}\]

For the three dimensional case (ptype=4), \(\mathbf{D}\) is formed as

\[\begin{split}\mathbf{D} = \frac{E}{(1+\nu)(1-2\nu)} \begin{bmatrix} 1-\nu & \nu & \nu & 0 & 0 & 0 \\ \nu & 1-\nu & \nu & 0 & 0 & 0 \\ \nu & \nu & 1-\nu & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2}(1-2\nu) & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{2}(1-2\nu) & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{2}(1-2\nu) \end{bmatrix}\end{split}\]