Elasticipy.Plasticity module

class Elasticipy.Plasticity.DruckerPrager(alpha)

Bases: PlasticityCriterion

Create a Drucker-Prager (DG) plasticity criterion.

Parameters:

alpha (float) – Pressure dependence parameters (see notes for details)

Notes

The pressure-dependent DG plasticity criterion assumes that the equivalent stress is defined as:

eq_stress(stress, **kwargs)

Return the equivalent stress, with respect to the plasticity criterion.

Parameters:

• stress (StressTensor) – Stress to compute the equivalent stress from

• kwargs (dict) – keyword arguments passed to the function

Return type:

float or numpy.ndarray

normal(stress, **kwargs)

Apply the normality rule

Parameters:

• stress (StressTensor) – Stress tensor to apply the normality rule

• kwargs (dict) – Keyword arguments passed to the function

Returns:

Normalized direction of plastic flow

Return type:

StrainTensor

class Elasticipy.Plasticity.IsotropicHardening(criterion='von Mises')

Bases: object

Template class for isotropic hardening plasticity models

Create an instance of a plastic model, assuming isotropic hardening

Parameters:

criterion (str or PlasticityCriterion) – Plasticity criterion to use. Can be ‘von Mises’, ‘Tresca’ or ‘J2’. J2 is the same as von Mises.

apply_strain(strain, **kwargs)

Apply strain to the current JC model.

This function updates the internal variable to store hardening state.

Parameters:

• strain (float or StrainTensor)

• kwargs (dict) – Keyword arguments passed to flow_stress()

Returns:

Associated flow stress (positive)

Return type:

float

See also

flow_stress

compute the flow stress, given a cumulative equivalent strain

compute_strain_increment(stress, **kwargs)

flow_stress(strain, **kwargs)

reset_strain()

class Elasticipy.Plasticity.JohnsonCook(A, B, n, C=None, eps_dot_ref=1.0, m=None, T0=25, Tm=None, criterion='von Mises')

Bases: IsotropicHardening

Constructor for a Jonhson-Cook (JC) model.

The JC model is an exponential-law strain hardening model, which can take into account strain-rate sensibility and temperature-dependence (although they are not mandatory). See notes for details.

Parameters:

• A (float) – Yield stress

• B (float) – Work hardening coefficient

• n (float) – Work hardening exponent

• C (float, optional) – Strain-rate sensitivity coefficient

• eps_dot_ref (float, optional) – Reference strain-rate

• m (float, optional) – Temperature sensitivity exponent

• T0 (float, optional) – Reference temperature

• Tm (float, optional) – Melting temperature (at which the flow stress is zero)

• criterion (str or PlasticityCriterion, optional) – Plasticity criterion to use. It can be ‘von Mises’ or ‘Tresca’.

Notes

The flow stress (\(\sigma\)) depends on the strain (\(\varepsilon\)), the strain rate \(\dot{\varepsilon}\) and the temperature (\(T\)) so that:

\[\sigma = \left(A + B\varepsilon^n\right) \left(1 + C\log\left(\frac{\varepsilon}{\dot{\varepsilon}_0}\right)\right) \left(1-\theta^m\right)\]

with

\[\begin{split}\theta = \begin{cases} \frac{T-T_0}{T_m-T_0} & \text{if } T<T_m\\ 1 & \text{otherwise} \end{cases}\end{split}\]

compute_strain_increment(stress, T=None, apply_strain=True, criterion='von Mises')

Given the equivalent stress, compute the strain increment with respect to the normality rule.

Parameters:

• stress (float or StressTensor) – Equivalent stress to compute the stress from, or full stress tensor.

• T (float) – Temperature

• apply_strain (bool, optional) – If true, the JC model will be updated to account for the applied strain (hardening)

• criterion (str, optional) – Plasticity criterion to consider to compute the equivalent stress and apply the normality rule. It can be ‘von Mises’, ‘Tresca’ or ‘J2’. ‘J2’ is equivalent to ‘von Mises’.

Returns:

Increment of plastic strain. If the input stress is float, only the magnitude of the increment will be returned (float value). If the stress is of type StressTensor, the returned value will be a full StrainTensor.

Return type:

StrainTensor or float

See also

apply_strain

apply strain to the JC model and updates its hardening value

flow_stress(eps_p, eps_dot=None, T=None)

Compute the flow stress from the Johnson-Cook model

Parameters:

• eps_p (float or list or tuple or numpy.ndarray) – Equivalent plastic strain

• eps_dot (float or list or tuple or numpy.ndarray, optional) – Equivalent plastic strain rate. If float, the strain-rate is supposed to be homogeneous for every value of eps_p.

• T (float or list or tuple or np.ndarray) – Temperature. If float, the temperature is supposed to be homogeneous for every value of eps_p.

Returns:

Flow stress

Return type:

float or numpy.ndarray

reset_strain()

Reinitialize the plastic strain to 0

class Elasticipy.Plasticity.PlasticityCriterion

Bases: object

static eq_stress(stress, **kwargs)

Return the equivalent stress, with respect to the plasticity criterion.

Parameters:

• stress (StressTensor) – Stress to compute the equivalent stress from

• kwargs (dict) – keyword arguments passed to the function

Return type:

float or numpy.ndarray

normal(stress, **kwargs)

Apply the normality rule

Parameters:

• stress (StressTensor) – Stress tensor to apply the normality rule

• kwargs (dict) – Keyword arguments passed to the function

Returns:

Normalized direction of plastic flow

Return type:

StrainTensor

class Elasticipy.Plasticity.TrescaPlasticity

Bases: PlasticityCriterion

static eq_stress(stress, **kwargs)

Return the equivalent stress, with respect to the plasticity criterion.

Parameters:

• stress (StressTensor) – Stress to compute the equivalent stress from

• kwargs (dict) – keyword arguments passed to the function

Return type:

float or numpy.ndarray

static normal(stress, **kwargs)

Apply the normality rule

Parameters:

• stress (StressTensor) – Stress tensor to apply the normality rule

• kwargs (dict) – Keyword arguments passed to the function

Returns:

Normalized direction of plastic flow

Return type:

StrainTensor

class Elasticipy.Plasticity.VonMisesPlasticity

Bases: PlasticityCriterion

static eq_stress(stress, **kwargs)

Return the equivalent stress, with respect to the plasticity criterion.

Parameters:

• stress (StressTensor) – Stress to compute the equivalent stress from

• kwargs (dict) – keyword arguments passed to the function

Return type:

float or numpy.ndarray

static normal(stress, **kwargs)

Apply the normality rule

Parameters:

• stress (StressTensor) – Stress tensor to apply the normality rule

• kwargs (dict) – Keyword arguments passed to the function

Returns:

Normalized direction of plastic flow

Return type:

StrainTensor