pymatgen.analysis.elasticity.elastic module¶

class
ComplianceTensor
[source]¶ Bases:
pymatgen.core.tensors.Tensor
This class represents the compliance tensor, and exists primarily to keep the voigtconversion scheme consistent since the compliance tensor has a unique vscale
Create a Tensor object. Note that the constructor uses __new__ rather than __init__ according to the standard method of subclassing numpy ndarrays.
 Parameters
input_array – (arraylike with shape 3^N): arraylike representing a tensor quantity in standard (i. e. nonvoigt) notation
vscale – (N x M arraylike): a matrix corresponding to the coefficients of the voigtnotation tensor

class
ElasticTensor
[source]¶ Bases:
pymatgen.analysis.elasticity.elastic.NthOrderElasticTensor
This class extends Tensor to describe the 3x3x3x3 secondorder elastic tensor, C_{ijkl}, with various methods for estimating other properties derived from the second order elastic tensor
Create an ElasticTensor object. The constructor throws an error if the shape of the input_matrix argument is not 3x3x3x3, i. e. in true tensor notation. Issues a warning if the input_matrix argument does not satisfy standard symmetries. Note that the constructor uses __new__ rather than __init__ according to the standard method of subclassing numpy ndarrays.
 Parameters
input_array (3x3x3x3 arraylike) – the 3x3x3x3 arraylike representing the elastic tensor
tol (float) – tolerance for initial symmetry test of tensor

property
compliance_tensor
¶ returns the Voigtnotation compliance tensor, which is the matrix inverse of the Voigtnotation elastic tensor

debye_temperature_from_sound_velocities
(**kwargs)¶

directional_poisson_ratio
(n, m, tol=1e08)[source]¶ Calculates the poisson ratio for a specific direction relative to a second, orthogonal direction
 Parameters
n (3d vector) – principal direction
m (3d vector) – secondary direction orthogonal to n
tol (float) – tolerance for testing of orthogonality

classmethod
from_independent_strains
(strains, stresses, eq_stress=None, vasp=False, tol=1e10)[source]¶ Constructs the elastic tensor leastsquares fit of independent strains :param strains: list of strain objects to fit :type strains: list of Strains :param stresses: list of stress objects to use in fit
corresponding to the list of strains
 Parameters
eq_stress (Stress) – equilibrium stress to use in fitting
vasp (boolean) – flag for whether the stress tensor should be converted based on vasp units/convention for stress
tol (float) – tolerance for removing nearzero elements of the resulting tensor

classmethod
from_pseudoinverse
(strains, stresses)[source]¶ Class method to fit an elastic tensor from stress/strain data. Method uses MoorePenrose pseudoinverse to invert the s = C*e equation with elastic tensor, stress, and strain in voigt notation
 Parameters
stresses (Nx3x3 arraylike) – list or array of stresses
strains (Nx3x3 arraylike) – list or array of strains

property
g_reuss
¶ returns the G_r shear modulus

property
g_voigt
¶ returns the G_v shear modulus

property
g_vrh
¶ returns the G_vrh (VoigtReussHill) average shear modulus

get_structure_property_dict
(structure, include_base_props=True, ignore_errors=False)[source]¶ returns a dictionary of properties derived from the elastic tensor and an associated structure
 Parameters
structure (Structure) – structure object for which to calculate associated properties
include_base_props (bool) – whether to include base properties, like k_vrh, etc.
ignore_errors (bool) – if set to true, will set problem properties that depend on a physical tensor to None, defaults to False

property
homogeneous_poisson
¶ returns the homogeneous poisson ratio

property
k_reuss
¶ returns the K_r bulk modulus

property
k_voigt
¶ returns the K_v bulk modulus

property
k_vrh
¶ returns the K_vrh (VoigtReussHill) average bulk modulus

property
property_dict
¶ returns a dictionary of properties derived from the elastic tensor

property
universal_anisotropy
¶ returns the universal anisotropy value

property
y_mod
¶ Calculates Young’s modulus (in SI units) using the VoigtReussHill averages of bulk and shear moduli

class
ElasticTensorExpansion
(c_list)[source]¶ Bases:
pymatgen.core.tensors.TensorCollection
This class is a sequence of elastic tensors corresponding to an elastic tensor expansion, which can be used to calculate stress and energy density and inherits all of the listbased properties of TensorCollection (e. g. symmetrization, voigt conversion, etc.)
Initialization method for ElasticTensorExpansion
 Parameters
c_list (list or tuple) – sequence of Tensor inputs or tensors from which the elastic tensor expansion is constructed.

calculate_stress
(strain)[source]¶ Calculate’s a given elastic tensor’s contribution to the stress using Einstein summation

energy_density
(strain, convert_GPa_to_eV=True)[source]¶ Calculates the elastic energy density due to a strain

classmethod
from_diff_fit
(strains, stresses, eq_stress=None, tol=1e10, order=3)[source]¶ Generates an elastic tensor expansion via the fitting function defined below in diff_fit

get_compliance_expansion
()[source]¶ Gets a compliance tensor expansion from the elastic tensor expansion.

get_effective_ecs
(strain, order=2)[source]¶ Returns the effective elastic constants from the elastic tensor expansion.
 Parameters
strain (Strain or 3x3 arraylike) – strain condition under which to calculate the effective constants
order (int) – order of the ecs to be returned

get_ggt
(n, u)[source]¶ Gets the Generalized Gruneisen tensor for a given thirdorder elastic tensor expansion.
 Parameters
n (3x1 arraylike) – normal mode direction
u (3x1 arraylike) – polarization direction

get_gruneisen_parameter
(temperature=None, structure=None, quad=None)[source]¶ Gets the single average gruneisen parameter from the TGT.
 Parameters
temperature (float) – Temperature in kelvin, if not specified will return noncvnormalized value
structure (float) – Structure to be used in directional heat capacity determination, only necessary if temperature is specified
quad (dict) – quadrature for integration, should be dictionary with “points” and “weights” keys defaults to quadpy.sphere.Lebedev(19) as read from file

get_heat_capacity
(temperature, structure, n, u, cutoff=100.0)[source]¶ Gets the directional heat capacity for a higher order tensor expansion as a function of direction and polarization.
 Parameters
temperature (float) – Temperature in kelvin
structure (float) – Structure to be used in directional heat capacity determination
n (3x1 arraylike) – direction for Cv determination
u (3x1 arraylike) – polarization direction, note that no attempt for verification of eigenvectors is made
cutoff (float) – cutoff for scale of kt / (hbar * omega) if lower than this value, returns 0

get_stability_criteria
(s, n)[source]¶ Gets the stability criteria from the symmetric Wallace tensor from an input vector and stress value.
 Parameters
s (float) – Stress value at which to evaluate the stability criteria
n (3x1 arraylike) – direction of the applied stress

get_strain_from_stress
(stress)[source]¶ Gets the strain from a stress state according to the compliance expansion corresponding to the tensor expansion.

get_symmetric_wallace_tensor
(tau)[source]¶ Gets the symmetrized wallace tensor for determining yield strength criteria.
 Parameters
tau (3x3 arraylike) – stress at which to evaluate the wallace tensor.

get_tgt
(temperature=None, structure=None, quad=None)[source]¶ Gets the thermodynamic Gruneisen tensor (TGT) by via an integration of the GGT weighted by the directional heat capacity.
 See refs:
R. N. Thurston and K. Brugger, Phys. Rev. 113, A1604 (1964). K. Brugger Phys. Rev. 137, A1826 (1965).
 Parameters
temperature (float) – Temperature in kelvin, if not specified will return noncvnormalized value
structure (float) – Structure to be used in directional heat capacity determination, only necessary if temperature is specified
quad (dict) – quadrature for integration, should be dictionary with “points” and “weights” keys defaults to quadpy.sphere.Lebedev(19) as read from file

get_wallace_tensor
(tau)[source]¶ Gets the Wallace Tensor for determining yield strength criteria.
 Parameters
tau (3x3 arraylike) – stress at which to evaluate the wallace tensor

get_yield_stress
(n)[source]¶ Gets the yield stress for a given direction
 Parameters
n (3x1 arraylike) – direction for which to find the yield stress

omega
(structure, n, u)[source]¶ Finds directional frequency contribution to the heat capacity from direction and polarization
 Parameters
structure (Structure) – Structure to be used in directional heat capacity determination
n (3x1 arraylike) – direction for Cv determination
u (3x1 arraylike) – polarization direction, note that no attempt for verification of eigenvectors is made

property
order
¶ Order of the elastic tensor expansion, i. e. the order of the highest included set of elastic constants

thermal_expansion_coeff
(structure, temperature, mode='debye')[source]¶ Gets thermal expansion coefficient from thirdorder constants.
 Parameters
temperature (float) – Temperature in kelvin, if not specified will return noncvnormalized value
structure (Structure) – Structure to be used in directional heat capacity determination, only necessary if temperature is specified
mode (string) – mode for finding average heatcapacity, current supported modes are ‘debye’ and ‘dulongpetit’

class
NthOrderElasticTensor
[source]¶ Bases:
pymatgen.core.tensors.Tensor
An object representing an nthorder tensor expansion of the stressstrain constitutive equations
Create a Tensor object. Note that the constructor uses __new__ rather than __init__ according to the standard method of subclassing numpy ndarrays.
 Parameters
input_array – (arraylike with shape 3^N): arraylike representing a tensor quantity in standard (i. e. nonvoigt) notation
vscale – (N x M arraylike): a matrix corresponding to the coefficients of the voigtnotation tensor

GPa_to_eV_A3
= 0.006241509125883258¶

calculate_stress
(strain)[source]¶ Calculate’s a given elastic tensor’s contribution to the stress using Einstein summation
 Parameters
strain (3x3 arraylike) – matrix corresponding to strain

energy_density
(strain, convert_GPa_to_eV=True)[source]¶ Calculates the elastic energy density due to a strain

property
order
¶ Order of the elastic tensor

symbol
= 'C'¶

diff_fit
(strains, stresses, eq_stress=None, order=2, tol=1e10)[source]¶ nth order elastic constant fitting function based on centraldifference derivatives with respect to distinct strain states. The algorithm is summarized as follows:
Identify distinct strain states as sets of indices for which nonzero strain values exist, typically [(0), (1), (2), (3), (4), (5), (0, 1) etc.]
For each strain state, find and sort strains and stresses by strain value.
Find first, second .. nth derivatives of each stress with respect to scalar variable corresponding to the smallest perturbation in the strain.
Use the pseudoinverse of a matrixvector expression corresponding to the parameterized stressstrain relationship and multiply that matrix by the respective calculated first or second derivatives from the previous step.
Place the calculated nthorder elastic constants appropriately.
 Parameters
order (int) – order of the elastic tensor set to return
strains (nx3x3 arraylike) – Array of 3x3 strains to use in fitting of ECs
stresses (nx3x3 arraylike) – Array of 3x3 stresses to use in fitting ECs. These should be PK2 stresses.
eq_stress (3x3 arraylike) – stress corresponding to equilibrium strain (i. e. “0” strain state). If not specified, function will try to find the state in the list of provided stresses and strains. If not found, defaults to 0.
tol (float) – value for which strains below are ignored in identifying strain states.
 Returns
Set of tensors corresponding to nth order expansion of the stress/strain relation

find_eq_stress
(strains, stresses, tol=1e10)[source]¶ Finds stress corresponding to zero strain state in stressstrain list
 Parameters
strains (Nx3x3 arraylike) – array corresponding to strains
stresses (Nx3x3 arraylike) – array corresponding to stresses
tol (float) – tolerance to find zero strain state

generate_pseudo
(strain_states, order=3)[source]¶ Generates the pseudoinverse for a given set of strains.
 Parameters
strain_states (6xN array like) – a list of voigtnotation “strainstates”, i. e. perturbed indices of the strain as a function of the smallest strain e. g. (0, 1, 0, 0, 1, 0)
order (int) – order of pseudoinverse to calculate
 Returns
 pseudo inverses for each order tensor, these can
be multiplied by the central difference derivative of the stress with respect to the strain state
 absent_syms: symbols of the tensor absent from the PI
expression
 Return type
mis

get_diff_coeff
(hvec, n=1)[source]¶ Helper function to find difference coefficients of an derivative on an arbitrary mesh.
 Parameters
hvec (1D arraylike) – sampling stencil
n (int) – degree of derivative to find

get_strain_state_dict
(strains, stresses, eq_stress=None, tol=1e10, add_eq=True, sort=True)[source]¶ Creates a dictionary of voigtnotation stressstrain sets keyed by “strain state”, i. e. a tuple corresponding to the nonzero entries in ratios to the lowest nonzero value, e.g. [0, 0.1, 0, 0.2, 0, 0] > (0,1,0,2,0,0) This allows strains to be collected in stencils as to evaluate parameterized finite difference derivatives
 Parameters
strains (Nx3x3 arraylike) – strain matrices
stresses (Nx3x3 arraylike) – stress matrices
eq_stress (Nx3x3 arraylike) – equilibrium stress
tol (float) – tolerance for sorting strain states
add_eq (bool) – flag for whether to add eq_strain to stressstrain sets for each strain state
sort (bool) – flag for whether to sort strain states
 Returns
OrderedDict with strain state keys and dictionaries with stressstrain data corresponding to strain state

get_symbol_list
(rank, dim=6)[source]¶ Returns a symbolic representation of the voigtnotation tensor that places identical symbols for entries related by index transposition, i. e. C_1121 = C_1211 etc.
 Parameters
dim (int) – dimension of matrix/tensor, e. g. 6 for voigt notation and 3 for standard
rank (int) – rank of tensor, e. g. 3 for thirdorder ECs
 Returns
array representing distinct indices c_arr (array): array representing tensor with equivalent
indices assigned as above
 Return type
c_vec (array)