Source code for cratepy.material.models.stvenant_kirchhoff

"""St.Venant-Kirchhoff hyperelastic constitutive model.

This module includes the implementation of the St.Venant-Kirchhoff hyperelastic
constitutive model under general anisotropic elasticity. The constitutive
formulation can be found in de Vieira de Carvalho et al. (2022) [#]_.

.. [#] Vieira de Carvalho, M., de Bortoli, D., and Andrade Pires, F.M. (2022).
       Consistent modeling of the coupling between crystallographic slip and
       martensitic phase transformation for mechanically induced loadings.
       Int J Numer Methods Eng. 123(14): 3179–3236 (see
       `here <https://onlinelibrary.wiley.com/doi/full/10.1002/nme.6962>`_)

Classes
-------
StVenantKirchhoff
    St.Venant-Kirchhoff hyperelastic constitutive model.
"""
#
#                                                                       Modules
# =============================================================================
# Third-party
import numpy as np
# Local
import tensor.tensoroperations as top
import tensor.matrixoperations as mop
from material.materialoperations import first_piola_from_second_piola, \
                                        cauchy_from_second_piola, \
                                        material_from_spatial_tangent_modulus
from material.models.interface import ConstitutiveModel
from material.models.elastic import Elastic
#
#                                                          Authorship & Credits
# =============================================================================
__author__ = 'Bernardo Ferreira (bernardo_ferreira@brown.edu)'
__credits__ = ['Bernardo Ferreira', ]
__status__ = 'Stable'
# =============================================================================
#
# =============================================================================
[docs]class StVenantKirchhoff(ConstitutiveModel): """St.Venant-Kirchhoff hyperelastic constitutive model. Attributes ---------- _name : str Constitutive model name. _strain_type : {'infinitesimal', 'finite', 'finite-kinext'} Constitutive model strain formulation: infinitesimal strain formulation ('infinitesimal'), finite strain formulation ('finite') or finite strain formulation through kinematic extension (infinitesimal constitutive formulation and purely finite strain kinematic extension - 'finite-kinext'). _source : {'crate', } Material constitutive model source. _ndim : int Problem number of spatial dimensions. _comp_order_sym : list Strain/Stress components symmetric order. _comp_order_nsym : list Strain/Stress components nonsymmetric order. Methods ------- get_required_properties() Get constitutive model material properties and constitutive options. state_init(self) Get initialized material constitutive model state variables. state_update(self, inc_strain, state_variables_old, \ su_max_n_iterations=20, su_conv_tol=1e-6) Perform material constitutive model state update. """
[docs] def __init__(self, strain_formulation, problem_type, material_properties): """Constitutive model constructor. Parameters ---------- strain_formulation: {'infinitesimal', 'finite'} Problem strain formulation. problem_type : int Problem type: 2D plane strain (1), 2D plane stress (2), 2D axisymmetric (3) and 3D (4). material_properties : dict Constitutive model material properties (key, str) values (item, {int, float, bool}). """ self._name = 'stvenant_kirchhoff' self._strain_formulation = strain_formulation self._problem_type = problem_type self._material_properties = material_properties # Set strain formulation self._strain_type = 'finite' # Set source self._source = 'crate' # Get problem type parameters n_dim, comp_order_sym, comp_order_nsym = \ mop.get_problem_type_parameters(problem_type) self._n_dim = n_dim self._comp_order_sym = comp_order_sym self._comp_order_nsym = comp_order_nsym # Get elastic symmetry elastic_symmetry = material_properties['elastic_symmetry'] # Compute technical constants of elasticity if elastic_symmetry == 'isotropic': # Compute technical constants of elasticity technical_constants = Elastic.get_technical_from_elastic_modulii( elastic_symmetry, material_properties) # Assemble technical constants of elasticity self._material_properties.update(technical_constants)
# -------------------------------------------------------------------------
[docs] @staticmethod def get_required_properties(): """Get constitutive model material properties and constitutive options. *Input data file syntax*: .. code-block:: text elastic_symmetry < option > < number_of_elastic_moduli > euler_angles < value > < value > < value > Eijkl < value > Eijkl < value > ... where - ``elastic_symmetry`` - Elastic symmetry and number of elastic moduli. - ``euler_angles`` - Euler angles (degrees) sorted according with Bunge convention. Not required if ``elastic_symmetry`` is set as `isotropic`. - ``Eijkl`` - Elastic moduli. Young's modulus (``E``) and Poisson's coefficient (``v``) may be alternatively provided if ``elastic_symmetry`` is set as `isotropic`. ---- Returns ------- material_properties : list[str] Constitutive model material properties names (str). constitutive_options : dict Constitutive options (key, str) and available specifications (item, tuple[str]). """ # Get available elastic symmetries and required elastic modulii elastic_symmetries = Elastic.get_available_elastic_symmetries() # Set constitutive options and available specifications constitutive_options = \ {'elastic_symmetry': tuple(elastic_symmetries.keys())} # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # Set material properties names material_properties = () # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # Return return material_properties, constitutive_options
# -------------------------------------------------------------------------
[docs] def state_init(self): """Get initialized material constitutive model state variables. Constitutive model state variables: * ``e_strain_mf`` * Elastic deformation gradient (matricial form). * *Symbol*: :math:`\\boldsymbol{F}^{e}` * ``strain_mf`` * Deformation gradient (matricial form). * *Symbol*: :math:`\\boldsymbol{F}` * ``stress_mf`` * First Piola-Kirchhoff stress tensor (matricial form). * *Symbol*: :math:`\\boldsymbol{P}` * ``is_su_fail`` * State update failure flag. ---- Returns ------- state_variables_init : dict Initialized constitutive model material state variables. """ # Initialize constitutive model state variables state_variables_init = dict() state_variables_init['e_strain_mf'] = mop.get_tensor_mf( np.eye(self._n_dim), self._n_dim, self._comp_order_nsym) state_variables_init['strain_mf'] = mop.get_tensor_mf( np.eye(self._n_dim), self._n_dim, self._comp_order_nsym) state_variables_init['stress_mf'] = mop.get_tensor_mf( np.zeros((self._n_dim, self._n_dim)), self._n_dim, self._comp_order_nsym) state_variables_init['is_su_fail'] = False # Set additional out-of-plane strain and stress components if self._problem_type == 1: state_variables_init['e_strain_33'] = 1.0 state_variables_init['stress_33'] = 0.0 # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # Return return state_variables_init
# -------------------------------------------------------------------------
[docs] def state_update(self, inc_strain, state_variables_old, su_max_n_iterations=20, su_conv_tol=1e-6): """Perform constitutive model state update. Parameters ---------- inc_strain : numpy.ndarray (2d) Incremental strain second-order tensor. state_variables_old : dict Last converged constitutive model material state variables. su_max_n_iterations : int, default=20 State update maximum number of iterations. su_conv_tol : float, default=1e-6 State update convergence tolerance. Returns ------- state_variables : dict Material constitutive model state variables. consistent_tangent_mf : numpy.ndarray (2d) Material constitutive model consistent tangent modulus in matricial form. """ # Get last increment converged state variables e_strain_old_mf = state_variables_old['e_strain_mf'] if self._problem_type == 1: e_strain_33_old = state_variables_old['e_strain_33'] # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # Set state update failure flag is_su_fail = False # # State update # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # Build incremental deformation gradient matricial form inc_strain_mf = mop.get_tensor_mf(inc_strain, self._n_dim, self._comp_order_nsym) # # 2D > 3D conversion # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # When the problem type corresponds to a 2D analysis, perform the state # update and consistent tangent computation as in the 3D case, # considering the appropriate out-of-plain strain and stress components if self._problem_type == 4: n_dim = self._n_dim comp_order_sym = self._comp_order_sym comp_order_nsym = self._comp_order_nsym else: # Set 3D problem parameters n_dim, comp_order_sym, comp_order_nsym = \ mop.get_problem_type_parameters(4) # Build deformation gradient tensors (matricial form) by including # the appropriate out-of-plain components inc_strain_mf = mop.get_state_3Dmf_from_2Dmf(self._problem_type, inc_strain_mf, comp_33=1.0) e_strain_old_mf = mop.get_state_3Dmf_from_2Dmf(self._problem_type, e_strain_old_mf, e_strain_33_old) # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # Build last converged and incremental deformation gradient tensors e_strain_old = mop.get_tensor_from_mf(e_strain_old_mf, n_dim, comp_order_nsym) inc_strain = mop.get_tensor_from_mf(inc_strain_mf, n_dim, comp_order_nsym) # Update elastic deformation gradient tensor e_strain = np.matmul(inc_strain, e_strain_old) # Build elastic deformationg gradient tensor matricial form e_strain_mf = mop.get_tensor_mf(e_strain, n_dim, comp_order_nsym) # Compute right Cauchy-Green strain tensor right_cauchy_green = np.matmul(np.transpose(e_strain), e_strain) # Compute Green-Lagrange strain tensor green_lagrange = 0.5*(right_cauchy_green - np.eye(n_dim)) # Compute 3D elasticity tensor (matricial form) elastic_tangent_mf = Elastic.elastic_tangent_modulus( self._material_properties, elastic_symmetry=self._material_properties['elastic_symmetry']) # Compute second Piola-Kirchhoff stress tensor (matricial form) second_piola_stress_mf = np.matmul( elastic_tangent_mf, mop.get_tensor_mf(green_lagrange, n_dim, comp_order_sym)) # Build second Piola-Kirchhoff stress tensor second_piola_stress = mop.get_tensor_from_mf(second_piola_stress_mf, n_dim, comp_order_sym) # Compute first Piola-Kirchhoff stress tensor first_piola_stress = first_piola_from_second_piola(e_strain, second_piola_stress) # Build first Piola-Kirchhoff stress tensor matricial form stress_mf = mop.get_tensor_mf(first_piola_stress, n_dim, comp_order_nsym) # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # Get the out-of-plane strain and stress components if self._problem_type == 1: e_strain_33 = e_strain_mf[comp_order_nsym.index('33')] stress_33 = stress_mf[comp_order_nsym.index('33')] # # 3D > 2D Conversion # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # When the problem type corresponds to a 2D analysis, build the # 2D strain and stress tensors (matricial form) once the state update # has been performed if self._problem_type == 1: # Builds 2D strain and stress tensors (matricial form) from the # associated 3D counterparts e_strain_mf = mop.get_state_2Dmf_from_3Dmf(self._problem_type, e_strain_mf) stress_mf = mop.get_state_2Dmf_from_3Dmf(self._problem_type, stress_mf) # # Update state variables # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # Initialize state variables dictionary state_variables = self.state_init() # Store updated state variables state_variables['e_strain_mf'] = e_strain_mf state_variables['strain_mf'] = e_strain_mf state_variables['stress_mf'] = stress_mf state_variables['is_su_fail'] = is_su_fail if self._problem_type == 1: state_variables['e_strain_33'] = e_strain_33 state_variables['stress_33'] = stress_33 # # Consistent tangent modulus # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # Set required identity tensors soid, _, _, _, _, _, _ = top.get_id_operators(n_dim) # Get Cauchy stress tensor cauchy_stress = cauchy_from_second_piola(e_strain, second_piola_stress) # Get 3D elasticity tensor elastic_tangent = mop.get_tensor_from_mf(elastic_tangent_mf, n_dim, comp_order_sym) # Compute derivative of Kirchhoff stress tensor in order to the elastic # deformation gradient tensor der_kirchhoff_e_strain = \ top.dyad22_2(soid, np.matmul(e_strain, second_piola_stress)) \ + top.dyad22_3(np.matmul(e_strain, second_piola_stress), soid) \ + top.dot24_1(e_strain, top.dot24_2( e_strain, top.dot24_3(e_strain, elastic_tangent))) # Compute spatial consistent tangent modulus spatial_consistent_tangent = (1.0/np.linalg.det(e_strain)) \ * top.dot42_3(der_kirchhoff_e_strain, np.transpose(e_strain)) \ - top.dyad22_3(cauchy_stress, soid) # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # Compute material consistent tangent modulus (matricial form) material_consistent_tangent = material_from_spatial_tangent_modulus( spatial_consistent_tangent, e_strain) consistent_tangent_mf = mop.get_tensor_mf(material_consistent_tangent, n_dim, comp_order_nsym) # # 3D > 2D Conversion # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # When the problem type corresponds to a 2D analysis, build the # 2D consistent tangent modulus (matricial form) from the # 3D counterpart if self._problem_type == 1: consistent_tangent_mf = mop.get_state_2Dmf_from_3Dmf( self._problem_type, consistent_tangent_mf) # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ # Return return state_variables, consistent_tangent_mf