Source code for schrodinger.application.matsci.elasticity.strain
"""
This module provides classes and methods used to describe deformations and
strains, including applying those deformations to structure objects and
generating deformed structure sets for further calculations.
Copyright Schrodinger, LLC. All rights reserved.
"""
# Based on pymatgen/analysis/elasticity/strain.py (LEGAL-413)
# last updated from upstream: 8767b89 on Aug 7, 2017
import collections.abc
import itertools
import numpy as np
import scipy
from schrodinger.application.matsci.elasticity.tensors import SquareTensor
from schrodinger.application.matsci.elasticity.tensors import symmetry_reduce
from schrodinger.application.matsci.nano import xtal
[docs]class Deformation(SquareTensor):
"""
Subclass of SquareTensor that describes the deformation gradient tensor
"""
def __new__(cls, deformation_gradient):
"""
Create a Deformation object. Note that the constructor uses __new__
rather than __init__ according to the standard method of subclassing
numpy ndarrays.
Args:
deformation_gradient (3x3 array-like): the 3x3 array-like
representing the deformation gradient
"""
obj = super(Deformation, cls).__new__(cls, deformation_gradient)
return obj.view(cls)
[docs] def is_independent(self, tol=1e-8):
"""
checks to determine whether the deformation is independent
"""
return len(self.get_perturbed_indices(tol)) == 1
[docs] def get_perturbed_indices(self, tol=1e-8):
"""
Gets indices of perturbed elements of the deformation gradient,
i. e. those that differ from the identity
"""
indices = list(zip(*np.where(abs(self - np.eye(3)) > tol)))
return indices
@property
def green_lagrange_strain(self):
"""
calculates the euler-lagrange strain from
the deformation gradient
"""
return Strain.from_deformation(self)
[docs] def apply_to_structure(self, structure):
"""
Apply the deformation gradient to a structure.
Args:
structure (Structure object): the structure object to
be modified by the deformation
"""
def_struct = structure.copy()
vecs = np.array(xtal.get_vectors_from_chorus(def_struct))
new_vecs = np.array([np.dot(self, vec) for vec in vecs])
xtal.set_pbc_properties(def_struct, new_vecs.flat)
frac = xtal.trans_cart_to_frac_from_vecs(def_struct.getXYZ(), *vecs)
new_cart = xtal.trans_frac_to_cart_from_vecs(frac, *new_vecs)
def_struct.setXYZ(new_cart)
return def_struct
[docs] @classmethod
def from_index_amount(cls, matrixpos, amt):
"""
Factory method for constructing a Deformation object
from a matrix position and amount
Args:
matrixpos (tuple): tuple corresponding the matrix position to
have a perturbation added
amt (float): amount to add to the identity matrix at position
matrixpos
"""
f = np.identity(3)
f[matrixpos] += amt
return cls(f)
[docs]class DeformedStructureSet(collections.abc.Sequence):
"""
class that generates a set of independently deformed structures that
can be used to calculate linear stress-strain response
"""
NORM_INDICES = [(0, 0), (1, 1), (2, 2)]
NORM_STRAINS = [-0.01, -0.005, 0.005, 0.01]
SHEAR_INDICES = [(0, 1), (0, 2), (1, 2)]
SHEAR_STRAINS = [-0.06, -0.03, 0.03, 0.06]
[docs] def __init__(self,
structure,
norm_strains=None,
shear_strains=None,
symmetry=False):
"""
constructs the deformed geometries of a structure. Generates
m + n deformed structures according to the supplied parameters.
:param structure: structure to undergo deformation
:type structure: Structure
:param: norm_strains: strain values to apply to each normal mode.
:type norm_strains: list[float]
:param shear_strains: strain values to apply to each shear mode.
:type shear_strains: list[float]
:param symmetry: whether or not to use symmetry reduction.
:type symmetry: bool
"""
norm_strains = norm_strains or self.NORM_STRAINS
shear_strains = shear_strains or self.SHEAR_STRAINS
self.undeformed_structure = structure
self.strains = []
self.deformations = []
self.def_structs = []
self.sym_dict = {}
# Generate deformations
for ind in self.NORM_INDICES:
for amount in norm_strains:
strain = Strain.from_index_amount(ind, amount)
self.strains.append(strain)
self.deformations.append(strain.deformation_matrix)
for ind in self.SHEAR_INDICES:
for amount in shear_strains:
strain = Strain.from_index_amount(ind, amount)
self.strains.append(strain)
self.deformations.append(strain.deformation_matrix)
# Perform symmetry reduction if specified
if symmetry:
self.sym_dict = symmetry_reduce(self.deformations, structure)
self.deformations = list(self.sym_dict.keys())
self.deformed_structures = [
defo.apply_to_structure(structure) for defo in self.deformations
]
def __iter__(self):
return iter(self.deformed_structures)
def __getitem__(self, ind):
return self.deformed_structures[ind]
[docs]class Strain(SquareTensor):
"""
Subclass of SquareTensor that describes the Green-Lagrange strain tensor.
"""
def __new__(cls, strain_matrix, dfm=None, dfm_shape="upper"):
"""
Create a Strain object. Note that the constructor uses __new__
rather than __init__ according to the standard method of
subclassing numpy ndarrays. Note also that the default constructor
does not include the deformation gradient
Args:
strain_matrix (3x3 array-like): the 3x3 array-like
representing the Green-Lagrange strain
"""
vscale = np.ones((6,))
vscale[3:] *= 2
obj = super(Strain, cls).__new__(cls, strain_matrix, vscale=vscale)
if dfm is None:
obj._dfm = convert_strain_to_deformation(obj, dfm_shape)
else:
dfm = Deformation(dfm)
gls_test = 0.5 * (np.dot(dfm.trans, dfm) - np.eye(3))
if (gls_test - obj > 1e-10).any():
raise ValueError("Strain and deformation gradients "
"do not match!")
obj._dfm = Deformation(dfm)
if not obj.is_symmetric():
raise ValueError("Strain objects must be initialized "
"with a symmetric array or a voigt-notation "
"vector with six entries.")
return obj.view(cls)
def __array_finalize__(self, obj):
if obj is None:
return
self.rank = getattr(obj, "rank", None)
self._dfm = getattr(obj, "_dfm", None)
self._vscale = getattr(obj, "_vscale", None)
[docs] @classmethod
def from_deformation(cls, deformation):
"""
Factory method that returns a Strain object from a deformation
gradient
Args:
deformation (3x3 array-like):
"""
dfm = Deformation(deformation)
return cls(0.5 * (np.dot(dfm.trans, dfm) - np.eye(3)), dfm)
[docs] @classmethod
def from_index_amount(cls, idx, amount):
"""
Like Deformation.from_index_amount, except generates
a strain from the zero 3x3 tensor or voigt vector with
the amount specified in the index location. Ensures
symmetric strain.
:param idx: index to be perturbed, can be voigt or
full-tensor notation
:type idx: tuple or integer
:param amount: amount to perturb selected index
:type amount: float
"""
if np.array(idx).ndim == 0:
v = np.zeros(6)
v[idx] = amount
return cls.from_voigt(v)
elif np.array(idx).ndim == 1:
v = np.zeros((3, 3))
for i in itertools.permutations(idx):
v[i] = amount
return cls(v)
else:
raise ValueError("Index must either be 2-tuple or integer "
"corresponding to full-tensor or voigt index")
@property
def deformation_matrix(self):
"""
returns the deformation matrix
"""
return self._dfm
@property
def von_mises_strain(self):
"""
Equivalent strain to Von Mises Stress
"""
eps = self - 1 / 3 * np.trace(self) * np.identity(3)
return np.sqrt(np.sum(eps * eps) * 2 / 3)
[docs]def convert_strain_to_deformation(strain, shape="upper"):
"""
This function converts a strain to a deformation gradient that will
produce that strain. Supports three methods:
:param strain: (3x3 array-like) strain matrix
:param shape: method for determining deformation, supports:
- "upper" produces an upper triangular defo
- "lower" produces a lower triangular defo
- "symmetric" produces a symmetric defo
:type shape: str
"""
strain = SquareTensor(strain)
ftdotf = 2 * strain + np.eye(3)
if shape == "upper":
result = scipy.linalg.cholesky(ftdotf)
elif shape == "symmetric":
result = scipy.linalg.sqrtm(ftdotf)
else:
raise ValueError("shape must be \"upper\" or \"symmetric\"")
return Deformation(result)