Material functions¶

The group of material functions comprises functions for constitutive models. The available models can treat linear elastic and isotropic hardening von Mises material. These material models are defined by the functions:

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

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

mises¶

Purpose:

Compute stresses and plastic strains for an elasto-plastic isotropic hardening von Mises material.

Syntax:

[es, deps, st] = mises(ptype, mp, est, st)

Description:

The mises function computes updated stresses, es, plastic strain increments deps, and state variables st for an elasto-plastic isotropic hardening von Mises material.

The input variable ptype defines the type of analysis, see also hooke. The vector mp contains the material constants:

mp = [\(\small{\, E\;\nu\;h\,}\)]

where \(E\) is the modulus of elasticity, \(\nu\) is the Poisson’s ratio, and \(h\) is the plastic modulus.

The input matrix est contains trial stresses obtained by using the elastic material matrix D in plants or a similar s-function. The input vector st contains the state parameters:

st = [\(\small{\, yi\;\sigma_y\;\varepsilon_{eff}^p\,}\)]

at the beginning of the step. The scalar \(yi\) indicates whether the material behaviour is elasto-plastic (\(yi = 1\)) or elastic (\(yi = 0\)). The current yield stress is denoted by \(\sigma_y\) and the effective plastic strain by \(\varepsilon_{eff}^p\).

The output variables es and st contain updated values obtained by integration of the constitutive equations over the actual displacement step. The increments of the plastic strains are stored in the vector deps.

If es and st contain more than one row, then every row will be treated by the command.

Note:

It is not necessary to check whether the material behaviour is elastic or elasto-plastic; this test is performed by the function. The computation is based on an Euler-Backward method, i.e., the radial return method.

Only the cases ptype = 2, 3, 4 are implemented.

dmises¶

Purpose:

Form the elasto-plastic continuum matrix for an isotropic hardening von Mises material.

Syntax:

D = dmises(ptype, mp, es, st)

Description:

dmises forms the elasto-plastic continuum matrix for an isotropic hardening von Mises material.

The input variable ptype is used to define the type of analysis, cf. hooke.

The vector mp contains the material constants:

mp = [\(\small{\, E\;\nu\;h\,}\)]

where \(E\) is the modulus of elasticity, \(\nu\) is the Poisson’s ratio, and \(h\) is the plastic modulus.

The matrix es contains current stresses obtained from plants or some similar s-function, and the vector st contains the current state parameters:

st = [\(\small{\, yi\;\sigma_y\;\varepsilon_{eff}^p\,}\)]

where \(yi = 1\) if the material behaviour is elasto-plastic, and \(yi = 0\) if the material behaviour is elastic. The current yield stress is denoted by \(\sigma_y\), and the current effective plastic strain by \(\varepsilon_{eff}^p\).

Note:

Only the case ptype = 2 is implemented.