soli8s¶

Purpose:

Compute stresses and strains in an 8 node isoparametric solid element.

Syntax:

[es,et,eci]=soli8s(ex,ey,ez,ep,D,ed)

Description:

soli8s computes stresses es and the strains et in an 8 node isoparametric solid element.

The input variables \(\mathbf{ex}\), \(\mathbf{ey}\), \(\mathbf{ez}\), \(\mathbf{ep}\) and the matrix \(\mathbf{D}\) are defined in soli8e. The vector \(\mathbf{ed}\) contains the nodal displacements \(\mathbf{a}^e\) of the element and is obtained by the function extract as

\[\mathbf{ed} = (\mathbf{a}^e)^T = [\;u_1\;\; u_2\;\; \dots \;\; u_{24}\;]\]

The output variables

\[\begin{split}\mathrm{es} = \boldsymbol{\sigma}^T = \begin{bmatrix} \sigma^1_{xx} & \sigma^1_{yy} & \sigma^1_{zz} & \sigma^1_{xy} & \sigma^1_{xz} & \sigma^1_{yz} \\ \sigma^2_{xx} & \sigma^2_{yy} & \sigma^2_{zz} & \sigma^2_{xy} & \sigma^2_{xz} & \sigma^2_{yz} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \sigma^{n^3}_{xx} & \sigma^{n^3}_{yy} & \sigma^{n^3}_{zz} & \sigma^{n^3}_{xy} & \sigma^{n^3}_{xz} & \sigma^{n^3}_{yz} \end{bmatrix}\end{split}\]

\[\begin{split}\mathrm{et} = \boldsymbol{\varepsilon}^T = \begin{bmatrix} \varepsilon^1_{xx} & \varepsilon^1_{yy} & \varepsilon^1_{zz} & \gamma^1_{xy} & \gamma^1_{xz} & \gamma^1_{yz} \\ \varepsilon^2_{xx} & \varepsilon^2_{yy} & \varepsilon^2_{zz} & \gamma^2_{xy} & \gamma^2_{xz} & \gamma^2_{yz} \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \varepsilon^{n^3}_{xx} & \varepsilon^{n^3}_{yy} & \varepsilon^{n^3}_{zz} & \gamma^{n^3}_{xy} & \gamma^{n^3}_{xz} & \gamma^{n^3}_{yz} \end{bmatrix} \qquad \mathbf{eci} = \begin{bmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ \vdots & \vdots & \vdots \\ x_{n^3} & y_{n^3} & z_{n^3} \end{bmatrix}\end{split}\]

contain the stress and strain components, and the coordinates of the integration points. The index \(n\) denotes the number of integration points used within the element, cf. soli8e.

Theory:

The strains and stresses are computed according to

\[\boldsymbol{\varepsilon} = \mathbf{B}^e \mathbf{a}^e\]

\[\boldsymbol{\sigma} = \mathbf{D} \boldsymbol{\varepsilon}\]

where the matrices \(\mathbf{D}\), \(\mathbf{B}^e\), and \(\mathbf{a}^e\) are described in soli8e, and where the integration points are chosen as evaluation points.