pymatgen.analysis.elasticity.strain — pymatgen 2020.7.3 documentation

# 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"

    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)

        def is_independent(self, tol=1e-8):
            """
            checks to determine whether the deformation is independent
            """
            return len(self.get_perturbed_indices(tol)) == 1

        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)

        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

        @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)

    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]

    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)

        @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)))

        @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")

        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)

    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)