Stress and Strain Measures#

DAMASK uses a large strain formulation and, hence, the output quantities are the deformation gradient (\(\mathbf{F}\)) and the second Piola–Kirchhoff stress (\(\mathbf{P}\)). These quantities can be used to calculate derived stress and strain quantities. The following quantities are computed

  • Euler–Almansi strain: \(\boldsymbol{\varepsilon}_\mathrm{EA} := \left(\mathbf{I}-\mathbf{F}^{-\mathrm{T}}\mathbf{F}^{-1}\right)/2\)

  • Green–Lagrange strain: \(\boldsymbol{\varepsilon}_\mathrm{GL} := \left(\mathbf{F}^\mathrm{T}\mathbf{F} -\mathbf{I}\right)/2\)

  • Cauchy stress: \(\boldsymbol{\sigma} := \mathbf{P} \mathbf{F}^\mathrm{T}/\operatorname{det}(\mathbf{F})\)

  • 2nd Piola–Kirchhoff stress: \(\mathbf{S} := \mathbf{F}^{-1} \mathbf{P}\)

These quantities are then used to compute the strain energy density under the assumption of monotonous elastic loading.

[1]:

import damask
import numpy as np

Uniaxial tension, isochoric#

A single pair of stress and strain tensors is generated such that the stress state is uniaxial tension and the deformation is isochoric, i.e. without a volume change.

[2]:

F = np.eye(3)
F[0,0]+=.3
F[1,1]=F[2,2]=1/np.sqrt(F[0,0])
print('determinant of F',np.linalg.det(F))

P = np.zeros((3,3))
P[0,0] = 400.e6

determinant of F 1.0

[3]:

epsilon_EulerAlmansi = damask.mechanics.strain(F,'V',-1)
epsilon_GreenLangrange = damask.mechanics.strain(F,'U',1)

sigma = damask.mechanics.stress_Cauchy(P=P,F=F)
S = damask.mechanics.stress_second_Piola_Kirchhoff(P=P,F=F)

[4]:

# check energy density (https://doi.org/10.1016/j.ijsolstr.2004.06.008)
print(np.tensordot(sigma,epsilon_EulerAlmansi)*np.linalg.det(F))
print(np.tensordot(S,epsilon_GreenLangrange))
print(np.tensordot(P,(F-np.linalg.inv(F.T))/2))

106153846.15384617
106153846.15384616
106153846.15384617

Random#

A set of 5 stress and strain tensors is generated. Note that they are not connected by a valid constitutive law, hence some of the strain energy densities are negative.

[5]:

rng = np.random.default_rng(20191102)
F = (rng.random((5,3,3))-.5)*5e-1 + np.eye(3)
P = rng.random((5,3,3))*1e8

[6]:

epsilon_EulerAlmansi = damask.mechanics.strain(F,'V',-1)
epsilon_GreenLangrange = damask.mechanics.strain(F,'U',1)

sigma = damask.mechanics.stress_Cauchy(P=P,F=F)
S = damask.mechanics.stress_second_Piola_Kirchhoff(P=P,F=F)

[7]:

print(np.einsum('...ij,...ij',sigma,epsilon_EulerAlmansi)*np.linalg.det(F))
print(np.einsum('...ij,...ij',S,epsilon_GreenLangrange))
print(np.einsum('...ij,...ij',P,(F-np.linalg.inv(damask.tensor.transpose(F)))/2))

[   618611.99757109  18367587.046755     6535114.78263465
   -248694.05806145 -14080050.23286315]
[   618611.9975711   18367587.04675496   6535114.78263456
   -248694.05806152 -14080050.23286315]
[   618611.99757112  18367587.046755     6535114.78263462
   -248694.05806148 -14080050.23286315]

[ ]:

.