- Evolution of the elastic strain
- Computation of the stress
- Enforcement of the plane stress conditions: computation of the axial strain
- Enforcement of the generalised plane stress conditions: computation of the axial strain
- Computation of the elastic prediction of the stress
- Options of the stress potential

The `Hooke`

stress potential describes the linear elastic
part of the behaviour of an isotropic or orthotropic material.

This stress potential 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 \(\Delta\,\underline{\epsilon}^{\mathrm{to}}\) is automatically substracted to the equation associated with the elastic (\(f_{\underline{\epsilon}^{\mathrm{el}}}\)), which is equivalent to the following statement:

`-= deto feel `

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 stress potential 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.

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

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

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:

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

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 stress potential.

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

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 \(t+\theta\,dt\)

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

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

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.

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 as follows:

`(1) += detozz; feel`

where \(\epsilon^{\mathrm{to}}_{zz}\) 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.

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.

The `Hooke`

stress potential automatically defines the
`computeElasticPrediction`

method which computes a prediction
of the stress under the assumption that all states variables are equal
to their values at the beginning of the time step except the elastic
strain which are assumed to be equal to \({\left.\underline{\epsilon}^{\mathrm{el}}\right|_{t}}+\Delta\,\underline{\epsilon}^{\mathrm{to}}\).

The `Hooke`

stress potential supports the following
options:

`young_modulus`

`young_modulus1`

`young_modulus2`

`young_modulus3`

`poisson_ratio`

`poisson_ratio12`

`poisson_ratio23`

`poisson_ratio13`

`shear_modulus12`

`shear_modulus23`

`shear_modulus13`

`thermal_expansion`

`thermal_expansion1`

`thermal_expansion2`

`thermal_expansion3`

`thermal_expansion_reference_temperature`

`plane_stress_support`

`generic_tangent_operator`

`generic_prediction_operator`