Strain

Strain Definitions

Simcenter STAR-CCM+ allows you to model linear geometry applications, where both displacements and strains are small, and nonlinear geometry applications with large displacements but small strains. In linear geometry applications, the strain can be described using the infinitesimal strain approximation, whereas nonlinear geometry applications require a finite strain (nonlinear) approximation to describe the state of strain.

The infinitesimal strain assumption is often used in structural engineering to describe the elastic behavior of materials such as steel or concrete, for which the deformations are usually small. The large-displacement, small-strain assumption is useful to describe the deformation of thin structures, that are often subject to large rotations with relatively small strains.

Infinitesimal Strain

The infinitesimal strain is defined as:

$$\epsilon =\frac{1}{2}\left(\frac{\partial u}{\partial X}+{[\frac{\partial u}{\partial X}]}^T\right)$$

(4444)

The infinitesimal strain is also called linear strain, since the strain depends linearly on the displacement.

Green-Lagrange Finite Strain

The Green-Lagrange strain tensor defines the strain in the undeformed configuration as:

$$E=\frac{1}{2}(F^{T}F-I)$$

(4445)

where $F$ is the deformation gradient (Eqn. (4428)) and $I$ is the 3 x 3 identity matrix.

The Cauchy Stress and Green-Lagrange Finite strain are not energy conjugate stress-strain pairs.

Right Cauchy-Green Deformation Tensor

The right Cauchy-Green deformation tensor defines the deformation as:

$$C=F^{T}F=U^{T}U$$

(4446)

where $F$ is the deformation gradient and $U$ is the right stretch tensor (Eqn. (4429)). By writing the deformation gradient in terms of its deviatoric and volumetric parts (see Eqn. (4430)), the modified right Cauchy-Green deformation tensor and right stretch tensor can be written as:

$$C^d=F^{dT}{F}^d=J^{-2/3}{F}^{T}F=J^{-2/3}C$$

(4447)

$$U^d=J^{-1/3}U$$

(4448)

Spatial Logarithmic Strain

The spatial logarithmic strain tensor is defined as:

$$h=\sum _{\alpha =1}^3(\ln {\lambda }_{\alpha })n_{\alpha }\otimes n_{\alpha }=\ln V$$

(4449)

where the natural log of the left or spatial stretch tensor ( $V$ ) is expressed as the sum of the natural log of the 3 principal stretches ( ${\lambda }_{\alpha }$ ) multiplied by the dyadic product of the principal directions ( $n_{\alpha }$ ) .

The spatial stretch tensor follows from a polar decomposition of the deformation gradient ( $F$ ):

$$F=VR$$

(4450)

where $R$ is a rotational tensor [871].

The Cauchy Stress and Spatial Logarithmic Strain are not energy conjugate stress-strain pairs.

Volumetric Strain

The volumetric strain is the change of volume relative to the undeformed volume. In the infinitesimal strain approximation, the volumetric strain is:

$$\frac{\Delta V}{V}={\epsilon }_{xx}+{\epsilon }_{yy}+{\epsilon }_{zz}$$

(4451)

Thermal Strain

Thermal strain is a measure of the deformation of a body due to changes in the body temperature.

If the solid is in an unconstrained state, a change in the temperature does not induce internal stresses and the material is free to expand due to a temperature increase, or to shrink due to a temperature decrease. If a solid body is constrained, a change in the temperature induces thermal stresses in the body.

The thermal strain ${\epsilon }_T$ is defined as:

$${\epsilon }_T=\alpha (T-T_{ref})$$

(4452)

where the reference temperature $T_{ref}$ is the temperature at which the thermal strain is assumed to be zero and $\alpha$ is the vector of thermal expansion coefficients ${\alpha }_i$.

Simcenter STAR-CCM+ supports linear elastic materials with isotropic, orthotropic, or anisotropic thermal strain, and hyperelastic materials with isotropic thermal strain (see Linear Elastic Materials and Hyperelastic Materials). Isotropic materials have the same thermal properties in all directions, and therefore, $\alpha$ is defined using a single coefficient. Orthotropic materials require three coefficients (see Eqn. (4510)), whereas anisotropic materials require six coefficients (see Eqn. (4504)).

Energy-Conjugate Stress-Strain Pairs

The variation of the strain energy per unit volume, due to stress, can be expressed in either the initial or current configuration, by using the correct pairing of stress and strain definitions:

$\sigma$ is the Cauchy stress, $\delta e$ is the variation of the Euler-Almansi strain ([eqnlink]), $\delta E$ is the variation of the Green-Lagrange strain (Eqn. (4445)), and $S$ is the 2nd Piola-Kirchhoff stress tensor (Eqn. (4435)). $\sigma$ and $\delta e$ are called a conjugate stress-strain pair, and so are called $S$ and $\delta E$ . Both $\delta E$ and $S$ are computed internally by Simcenter STAR-CCM+ and so are not visible outputs.

$\delta e$ and $\delta E$ can be written as: