Source code for pymatgen.analysis.elasticity.strain

# coding: utf-8
# Copyright (c) Pymatgen Development Team.
# Distributed under the terms of the MIT License.


"""
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.
"""

import numpy as np
import scipy
import itertools

import collections

from pymatgen.core.lattice import Lattice
from pymatgen.core.tensors import SquareTensor, symmetry_reduce

__author__ = "Joseph Montoya"
__copyright__ = "Copyright 2012, The Materials Project"
__credits__ = "Maarten de Jong, Mark Asta, Anubhav Jain"
__version__ = "1.0"
__maintainer__ = "Joseph Montoya"
__email__ = "montoyjh@lbl.gov"
__status__ = "Production"
__date__ = "July 24, 2018"


[docs]class Deformation(SquareTensor): """ Subclass of SquareTensor that describes the deformation gradient tensor """ symbol = "d" 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().__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() old_latt = def_struct.lattice.matrix new_latt = np.transpose(np.dot(self, np.transpose(old_latt))) def_struct.lattice = Lattice(new_latt) 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 """ 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. Args: structure (Structure): structure to undergo deformation norm_strains (list of floats): strain values to apply to each normal mode. shear_strains (list of floats): strain values to apply to each shear mode. symmetry (bool): whether or not to use symmetry reduction. """ norm_strains = norm_strains or [-0.01, -0.005, 0.005, 0.01] shear_strains = shear_strains or [-0.06, -0.03, 0.03, 0.06] self.undeformed_structure = structure self.deformations = [] self.def_structs = [] # Generate deformations for ind in [(0, 0), (1, 1), (2, 2)]: for amount in norm_strains: strain = Strain.from_index_amount(ind, amount) self.deformations.append(strain.get_deformation_matrix()) for ind in [(0, 1), (0, 2), (1, 2)]: for amount in shear_strains: strain = Strain.from_index_amount(ind, amount) self.deformations.append(strain.get_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 __len__(self): return len(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. """ symbol = "e" def __new__(cls, strain_matrix): """ 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().__new__(cls, strain_matrix, vscale=vscale) 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._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)))
[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. Args: idx (tuple or integer): index to be perturbed, can be voigt or full-tensor notation amount (float): amount to perturb selected index """ 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")
[docs] def get_deformation_matrix(self, shape="upper"): """ returns the deformation matrix """ return convert_strain_to_deformation(self, shape=shape)
@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: Args: strain (3x3 array-like): strain matrix shape: (string): method for determining deformation, supports "upper" produces an upper triangular defo "lower" produces a lower triangular defo "symmetric" produces a symmetric defo """ 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)