Elasticipy.Plasticity module

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, optional) – 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)

eq_stress(stress)

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, 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

Elasticipy.Plasticity.normality_rule(stress, criterion='von Mises')

Apply the normality rule for plastic flow, given a yield criterion.

The stress can be a single tensor, or an array of tensors.

Parameters:

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

• criterion (str, optional) – Name of the criterion to use. Can be either ‘von Mises’ or ‘Tresca’

Returns:

If a single stress tensor is passed, the returned array will be of shape

Return type:

StrainTensor

Notes

The singular points for the Tresca criterion are treated as the von Mises criterion, which is equivalent to the average of the two adjacent normals of the domain.