Elasticipy.plasticity

class elasticipy.plasticity.DruckerPrager(alpha)

Bases: PlasticityCriterion

Drucker-Prager pressure-dependent plasticity criterion, with associated normality rule

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:

\[\alpha I_1 + \sqrt{J_2}\]

where \(I_1\) is the first invariant of the stress tensor, and \(J_2\) is the second invariant of the deviatoric stress tensor.

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

name = 'Drucker-Prager'

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 plasticity 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

Examples

As an example, we consider the Johnson-Cook plasticity model:

>>> from elasticipy.plasticity import JohnsonCook
>>> JC = JohnsonCook(A=792, B=510, n=0.26)
>>> print(JC)
Johnson-Cook plasticity model
 type: Isotropic
 criterion: von Mises
 current strain: 0.0

>>> stress = JC.apply_strain(0.1)
>>> print(stress)
1072.2658456673885
>>> print(JC)
Johnson-Cook plasticity model
 type: Isotropic
 criterion: von Mises
 current strain: 0.1

Obvisously, the applied strain is cumulative:

>>> stress = JC.apply_strain(0.1)
>>> print(stress)
1127.612381818713
>>> print(JC)
Johnson-Cook plasticity model
 type: Isotropic
 criterion: von Mises
 current strain: 0.2

compute_strain_increment(stress, criterion='von Mises', apply_strain=True, **kwargs)

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.

• apply_strain (bool, optional) – If true, the plasticity 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’.

• kwargs – Keyword arguments passed to the model

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

Examples

As an example, we consider the Johnson-Cook plasticity model:

>>> from elasticipy.plasticity import JohnsonCook
>>> JC = JohnsonCook(A=792, B=510, n=0.26)

The yield stress is equal to A here. So consider a tensile stress whose magnitude below A:

>>> from elasticipy.tensors.stress_strain import StressTensor
>>> sigma = StressTensor.tensile([1,0,0], 700)
>>> strain_inc = JC.compute_strain_increment(sigma)
>>> print(strain_inc)
Strain tensor
[[ 0.  0.  0.]
 [ 0. -0.  0.]
 [ 0.  0. -0.]]

whereas if the stress is larger than A:

>>> sigma = StressTensor.tensile([1,0,0], 800)
>>> strain_inc = JC.compute_strain_increment(sigma)
>>> print(strain_inc)
Strain tensor
[[ 1.14733854e-07  0.00000000e+00  0.00000000e+00]
 [ 0.00000000e+00 -5.73669268e-08  0.00000000e+00]
 [ 0.00000000e+00  0.00000000e+00 -5.73669268e-08]]

Check out that the JC model has been updated:

>>> print(JC)
Johnson-Cook plasticity model
 type: Isotropic
 criterion: von Mises
 current strain: 1.1473385353505149e-07

Therefore, the yield stress has increased because of hardening. For instance, if we apply the same stress has before, we get:

>>> JC.compute_strain_increment(sigma)
Strain tensor
[[ 0.  0.  0.]
 [ 0. -0.  0.]
 [ 0.  0. -0.]]

flow_stress(strain, **kwargs)

Compute the stress from the cumulative plastic strain

Parameters:

• strain (float) – Equivalent Plastic strain

• kwargs – Additional arguments passed to the function

Return type:

float or np.ndarray

Examples

As an example, we consider a Jonhson-Cook model:

>>> from elasticipy.plasticity import JohnsonCook
>>> JC = JohnsonCook(A=792, B=510, n=0.26)
>>> print(JC)
Johnson-Cook plasticity model
 type: Isotropic
 criterion: von Mises
 current strain: 0.0

>>> JC.flow_stress(0.0) # Check that the yield stress = A
792.0

In order to get the full tensile curve in 0 to 10% strain range:

>>> import numpy as np
>>> JC.flow_stress(np.linspace(0,0.1,5)) # Check that the yield stress = B
array([ 792.        ,  987.44950657, 1026.04662195, 1052.067513  ,
       1072.26584567])

name = 'Generic'

reset_strain()

Update the internal variable so that the plastic strain is reset to zero.

Return type:

None

Examples

As an example, we consider the Johnson-Cook plasticity model:

>>> from elasticipy.plasticity import JohnsonCook
>>> JC = JohnsonCook(A=792, B=510, n=0.26)

First apply a strain increment:

>>> stress = JC.apply_strain(0.1)
>>> print(JC)
Johnson-Cook plasticity model
 type: Isotropic
 criterion: von Mises
 current strain: 0.1

If one wants to reset the JC, without recreating it:

>>> stress = JC.reset_strain()
>>> print(JC)
Johnson-Cook plasticity model
 type: Isotropic
 criterion: von Mises
 current strain: 0.0

type = 'Isotropic'

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

Special case of isotropic hardening with an underlying Johnson Cook hardening evolution rule

Constructor for a Johnson-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

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

apply_strain(strain, **kwargs)

Apply strain to the current plasticity 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

Examples

As an example, we consider the Johnson-Cook plasticity model:

>>> from elasticipy.plasticity import JohnsonCook
>>> JC = JohnsonCook(A=792, B=510, n=0.26)
>>> print(JC)
Johnson-Cook plasticity model
 type: Isotropic
 criterion: von Mises
 current strain: 0.0

>>> stress = JC.apply_strain(0.1)
>>> print(stress)
1072.2658456673885
>>> print(JC)
Johnson-Cook plasticity model
 type: Isotropic
 criterion: von Mises
 current strain: 0.1

Obvisously, the applied strain is cumulative:

>>> stress = JC.apply_strain(0.1)
>>> print(stress)
1127.612381818713
>>> print(JC)
Johnson-Cook plasticity model
 type: Isotropic
 criterion: von Mises
 current strain: 0.2

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.

• apply_strain (bool, optional) – If true, the plasticity 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’.

• kwargs – Keyword arguments passed to the model

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

Examples

As an example, we consider the Johnson-Cook plasticity model:

>>> from elasticipy.plasticity import JohnsonCook
>>> JC = JohnsonCook(A=792, B=510, n=0.26)

The yield stress is equal to A here. So consider a tensile stress whose magnitude below A:

>>> from elasticipy.tensors.stress_strain import StressTensor
>>> sigma = StressTensor.tensile([1,0,0], 700)
>>> strain_inc = JC.compute_strain_increment(sigma)
>>> print(strain_inc)
Strain tensor
[[ 0.  0.  0.]
 [ 0. -0.  0.]
 [ 0.  0. -0.]]

whereas if the stress is larger than A:

>>> sigma = StressTensor.tensile([1,0,0], 800)
>>> strain_inc = JC.compute_strain_increment(sigma)
>>> print(strain_inc)
Strain tensor
[[ 1.14733854e-07  0.00000000e+00  0.00000000e+00]
 [ 0.00000000e+00 -5.73669268e-08  0.00000000e+00]
 [ 0.00000000e+00  0.00000000e+00 -5.73669268e-08]]

Check out that the JC model has been updated:

>>> print(JC)
Johnson-Cook plasticity model
 type: Isotropic
 criterion: von Mises
 current strain: 1.1473385353505149e-07

Therefore, the yield stress has increased because of hardening. For instance, if we apply the same stress has before, we get:

>>> JC.compute_strain_increment(sigma)
Strain tensor
[[ 0.  0.  0.]
 [ 0. -0.  0.]
 [ 0.  0. -0.]]

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

name = 'Johnson-Cook'

reset_strain()

Update the internal variable so that the plastic strain is reset to zero.

Return type:

None

Examples

As an example, we consider the Johnson-Cook plasticity model:

>>> from elasticipy.plasticity import JohnsonCook
>>> JC = JohnsonCook(A=792, B=510, n=0.26)

First apply a strain increment:

>>> stress = JC.apply_strain(0.1)
>>> print(JC)
Johnson-Cook plasticity model
 type: Isotropic
 criterion: von Mises
 current strain: 0.1

If one wants to reset the JC, without recreating it:

>>> stress = JC.reset_strain()
>>> print(JC)
Johnson-Cook plasticity model
 type: Isotropic
 criterion: von Mises
 current strain: 0.0

type = 'Isotropic'

class elasticipy.plasticity.PlasticityCriterion

Bases: object

Template class for plasticity criteria

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

name = 'generic'

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

Tresca plasticity criterion, with associated normality rule

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

name = 'Tresca'

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

von Mises plasticity criterion, with associated normality rule

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

name = 'von Mises'

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