# The StandardElasticity brick

The StandardElasticity brick describes the linear elastic part of the behaviour of an isotropic or orthotropic material.

## Evolution of the elastic strain

This bricks relies on the fact that the behaviour is based on the strain split hypothesis.

The elastic strain must be defined as the first integration variable. The associated variable must be called eel and its glossary name must be ElasticStrain. This is automatically the case with the @Implicit dsl.

The total strain increment deto is automatically substracted to the equation associated with the elastic (feel), which is equivalent to the following statement: $feel -= deto$

## Computation of the stress

If the elastic behaviour is orthotropic, the stiffness tensor must be available available (using the keyword @RequireStiffnessTensor) or computed by the behaviour (using the keyword @ComputeStiffnessTensor). If those keywords are not explicitly used, the brick will automatically sets the attribute requireStiffnessTensor to true which has the same effect than the @RequireStiffnessTensor keyword.

Thus, two cases arise:

• the stiffness tensor is available (using keyword @RequireStiffnessTensor) or computed by the behaviour (@ComputeStiffnessTensor).
• the behaviour has is an isotropic elastic behaviour and the stiffness tensor is not available.

### First case: the stiffness tensor is available

#### Computation of the stress at $$t+\theta\,dt$$

At $$t+\theta\,dt$$, the stress are computed using:

${\left.\sigma\right|_{t+\theta\,\Delta\,t}}=\underline{\mathbf{D}}\,\colon\,{\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t+\theta\,\Delta\,t}}$

#### Computation of the final stress at $$t+dt$$

If the stiffness tensor is avaible using the @RequireStiffnessTensor, the final stress $${\left.\sigma\right|_{t+\Delta\,t}}$$ is computed using the following formula :

${\left.\sigma\right|_{t+\Delta\,t}}=\underline{\mathbf{D}}\,\colon\,{\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t+\Delta\,t}}$

If the stiffness tensor is computed using @ComputeStiffnessTensor, the final $${\left.\sigma\right|_{t+\Delta\,t}}$$ stress is computed using:

${\left.\sigma\right|_{t+\Delta\,t}}={\left.\underline{\mathbf{D}}\right|_{t+\Delta\,t}}\,\colon\,{\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t+\Delta\,t}}$

### Second case: the stiffness tensor is not available

In this case, the elastic behaviour of the material is isotropic. The computation of the stress requires the definition of the first Lamé coefficient and the shear modulus (second Lamé coefficient).

The Lamé coefficients are derived from the Young modulus and Poisson ratio. They can be defined using:

• the @ElasticMaterialProperties keyword. In this case, the Implicit dsl already automatically computes the following variables (See the documentation of the @ElasticMaterialProperties keyword):
• young: the Young modulus at $$t+\theta\,dt$$
• nu: the Poisson ratio modulus at $$t+\theta\,dt$$
• lambda: the first Lamé coefficient at $$t+\theta\,dt$$
• mu: the second Lamé coefficient at $$t+\theta\,dt$$
• young_tdt: the Young modulus at $$t+dt$$
• nu_tdt: the Poisson ratio modulus at $$t+dt$$
• lambda_tdt: the first Lamé coefficient at $$t+dt$$
• mu_tdt: the second Lamé coefficient at $$t+dt$$
• the Young modulus and Poisson ratio has been defined as material properties or parameters. In this case, the names of those variables must be young and the nu and the glossary names associated with those variables must be respectively YoungModulus and PoissonRatio. The Lamé coefficients will be computed and stored in a data structure used internally by the brick.

If the material properties are not defined using one of those two ways, the appropriate material properties will be automatically defined by the brick.

#### Computation of the stress at $$t+\theta\,dt$$

At $$t+\theta\,dt$$, the stress are computed using the following formula: ${\left.\sigma\right|_{t+\theta\,\Delta\,t}}=\lambda\,{\mathrm{tr}{\left({\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t+\theta\,\Delta\,t}}\right)}}+2\,\mu\,{\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t+\theta\,\Delta\,t}}$ where $$\lambda$$ and $$\mu$$ are respectively the values of the first and second Lamé coefficients at $$t+\theta\,dt$$

#### Computation of the final stress at $$t+dt$$

The final stress $${\left.\sigma\right|_{t+\Delta\,t}}$$ is computed using the following formula :

${\left.\sigma\right|_{t+\Delta\,t}}={\left.\lambda\right|_{t+\Delta\,t}}\,{\mathrm{tr}{\left({\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t+\Delta\,t}}\right)}}+2\,{\left.\mu\right|_{t+\Delta\,t}}\,{\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t+\Delta\,t}}$

## Enforcement of the plane stress conditions: computation of the axial strain

If the user has explicitly specified that the axisymmetric generalised plane stress modelling hypothesis must be supported by the behaviour using the @ModellingHypothesis keyword or the @ModellingHypotheses keyword, this support is performed by automatically introducing an additional state variable: the axial strain. The associated variable is etozz, although this variable shall not be used by the end user. The glossary name of this variable is AxialStrain.

The introduction of the variable modify the strain split equation like this: $feel(2) += detozz;$ where $$detozz$$ is the increment of the axial strain. The associated jacobian term is added if necessary.

The plane stress condition is enforced by adding an additional equation to the implicit system ensuring that: ${\left.\sigma_{zz}\right|_{t+\Delta\,t}}=0$

This equation is appropriately normalised using one of the elastic properties. The associated jacobian term are added if necessary.

## Enforcement of the generalised plane stress conditions: computation of the axial strain

If the user has explicitly specified that the axisymmetric generalised plane stress modelling hypothesis must be supported by the behaviour using the @ModellingHypothesis keyword or the @ModellingHypotheses keyword, this support is performed by automatically introducing an additional state variable, the axial strain and an additional external state variable, the axial stress.

The variable associated to the axial strain is etozz, although this variable shall not be used by the end user. The glossary name of this variable is AxialStrain.

The variable associated to the axial stress is sigzz, although this variable shall not be used by the end user. The glossary name of this variable is AxialStress.

The introduction of the variable modify the strain split equation like this: $feel(1) += detozz;$ where $$detozz$$ is the increment of the axial strain. The associated jacobian term is added if necessary.

The plane stress condition is enforced by adding an additional equation to the implicit system ensuring that: ${\left.\sigma_{zz}\right|_{t+\Delta\,t}}-\sigma^{zz}-d\sigma^{zz}=0$

where $$\sigma^{zz}$$ is the value of the axial stress at the beginning of the time step and $$d\sigma^{zz}$$ is the value of the increment of the axial stress.

This equation is appropriately normalised using one of the elastic properties. The associated jacobian terms are added if necessary.

## Generic computation of the tangent operator

The elastic and secant operator are equal to the elastic stiffness matrix at the end of the time step. How this elastic stiffness matrix is obtained depends on the many cases described before.

The consistent tangent operator is computed by multiplying the elastic stiffness matrix at the end of the time step by a partial invert of the jacobian matrix. This procedure is discussed in depth in the MFront manuals.

## Options of the brick

The StandardElasticity brick supports the following options:

• Isotropic: the elastic part of the behaviour will be assumed isotropic. This is the default for an isotropic material. This is usefull if the material is declared orthotropic.
• Orthotropic: the elastic part of the behaviour will be assumed isotropic. This is the default for an isotropic material.
• NoPlaneStressSupport: plane stress and axisymmetrical generalised plane stress hypotheses support will not be added.
• NoGenericTangentOperator: generic computation of the consistent tangent operator will not be added.
• NoGenericPredictionOperator: generic computation of the prediction operator will not be added.

## Example

@Brick "StandardElasticity";