sfepy.terms.terms_hyperelastic_tl module¶

class sfepy.terms.terms_hyperelastic_tl.BulkActiveTLTerm(*args, **kwargs)[source]¶

Hyperelastic bulk active term. Stress $S_{ij} = A J C_{ij}^{-1}$ , where $A$ is the activation in $[0, F_{\rm max}]$ .

Definition:

$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$

Call signature:

dw_tl_bulk_active

(material, virtual, state)

Arguments:

material : $A$
virtual : $\ul{v}$
state : $\ul{u}$

family_data_names = ['det_f', 'sym_inv_c']¶

name = 'dw_tl_bulk_active'¶

static stress_function(out, mat, det_f, vec_inv_cs)¶

static tan_mod_function(out, mat, det_f, vec_inv_cs)¶

class sfepy.terms.terms_hyperelastic_tl.BulkPenaltyTLTerm(*args, **kwargs)[source]¶

Hyperelastic bulk penalty term. Stress $S_{ij} = K(J-1)\; J C_{ij}^{-1}$ .

Definition:

$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$

Call signature:

dw_tl_bulk_penalty

(material, virtual, state)

Arguments:

material : $K$
virtual : $\ul{v}$
state : $\ul{u}$

family_data_names = ['det_f', 'sym_inv_c']¶

name = 'dw_tl_bulk_penalty'¶

static stress_function(out, mat, det_f, vec_inv_cs)¶

static tan_mod_function(out, mat, det_f, vec_inv_cs)¶

class sfepy.terms.terms_hyperelastic_tl.BulkPressureTLTerm(*args, **kwargs)[source]¶

Hyperelastic bulk pressure term. Stress $S_{ij} = -p J C_{ij}^{-1}$ .

Definition:

$\int_{\Omega} S_{ij}(p) \delta E_{ij}(\ul{u};\ul{v})$

Call signature:

dw_tl_bulk_pressure

(virtual, state, state_p)

Arguments:

virtual : $\ul{v}$
state : $\ul{u}$
state_p : $p$

arg_geometry_types = {('state_p', None): {'facet_extra': 'facet'}}¶

arg_shapes = {'state': 'D', 'state_p': 1, 'virtual': ('D', 'state')}¶

arg_types = ('virtual', 'state', 'state_p')¶

compute_data(family_data, mode, **kwargs)[source]¶

family_data_names = ['det_f', 'sym_inv_c']¶

get_eval_shape(virtual, state, state_p, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

get_fargs(virtual, state, state_p, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

name = 'dw_tl_bulk_pressure'¶

static stress_function(out, pressure_qp, det_f, vec_inv_cs)¶

static tan_mod_u_function(out, pressure_qp, det_f, vec_inv_cs)¶

static weak_dp_function(out, mtx_f, vec_inv_cs, det_f, cmap_s, cmap_v, transpose, mode)¶

static weak_function(out, stress, tan_mod, mtx_f, det_f, cmap, is_diff, mode_ul)¶

class sfepy.terms.terms_hyperelastic_tl.DiffusionTLTerm(*args, **kwargs)[source]¶

Diffusion term in the total Lagrangian formulation with linearized deformation-dependent permeability $\ull{K}(\ul{u}) = J \ull{F}^{-1} \ull{k} f(J) \ull{F}^{-T}$ , where $\ul{u}$ relates to the previous time step $(n-1)$ and $f(J) = \max\left(0, \left(1 + \frac{(J - 1)}{N_f}\right)\right)^2$ expresses the dependence on volume compression/expansion.

Definition:

$\int_{\Omega} \ull{K}(\ul{u}^{(n-1)}) : \pdiff{q}{\ul{X}} \pdiff{p}{\ul{X}}$

Call signature:

dw_tl_diffusion

(material_1, material_2, virtual, state, parameter)

Arguments:

material_1 : $\ull{k}$
material_2 : $N_f$
virtual : $q$
state : $p$
parameter : $\ul{u}^{(n-1)}$

arg_shapes = {'material_1': 'D, D', 'material_2': '1, 1', 'parameter': 'D', 'state': 1, 'virtual': (1, 'state')}¶

arg_types = ('material_1', 'material_2', 'virtual', 'state', 'parameter')¶

family_data_names = ['mtx_f', 'det_f']¶

static function(out, pressure_grad, mtx_d, ref_porosity, mtx_f, det_f, cmap, mode)¶

get_eval_shape(perm, ref_porosity, virtual, state, parameter, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

get_fargs(perm, ref_porosity, virtual, state, parameter, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

name = 'dw_tl_diffusion'¶

class sfepy.terms.terms_hyperelastic_tl.GenYeohTLTerm(*args, **kwargs)[source]¶

Hyperelastic generalized Yeoh term [1]. Effective stress $S_{ij} = 2 p K (I_1 - 3)^{p-1} J^{-\frac{2}{3}}(\delta{ij} - \frac{1}{3}C_{kk}C{ij}^{-1})$ .

Definition:

$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$

Call signature:

dw_tl_he_genyeoh

(material, virtual, state)

Arguments:

material : $p, K$
virtual : $\ul{v}$
state : $\ul{u}$

[1] Travis W. Hohenberger, Richard J. Windslow, Nicola M. Pugno, James J. C. Busfield. Aconstitutive Model For Both Lowand High Strain Nonlinearities In Highly Filled Elastomers And Implementation With User-Defined Material Subroutines In Abaqus. Rubber Chemistry And Technology, Vol. 92, No. 4, Pp. 653-686 (2019)

arg_shapes = {'material': '1, 2', 'state': 'D', 'virtual': ('D', 'state')}¶

family_data_names = ['det_f', 'tr_c', 'sym_inv_c']¶

geometries = ['3_4', '3_8']¶

name = 'dw_tl_he_genyeoh'¶

stress_function(out, mat, *fargs, **kwargs)[source]¶

tan_mod_function(out, mat, *fargs, **kwargs)[source]¶

class sfepy.terms.terms_hyperelastic_tl.HyperElasticSurfaceTLBase(*args, **kwargs)[source]¶

Base class for all hyperelastic surface terms in TL formulation family.

get_family_data = HyperElasticSurfaceTLFamilyData¶

class sfepy.terms.terms_hyperelastic_tl.HyperElasticSurfaceTLFamilyData(**kwargs)[source]¶

Family data for TL formulation applicable for surface terms.

cache_name = 'tl_surface_common'¶

data_names = ('mtx_f', 'det_f', 'inv_f')¶

static family_function(mtx_f, det_f, mtx_fi, state, cmap, fis, conn)¶

class sfepy.terms.terms_hyperelastic_tl.HyperElasticTLBase(*args, **kwargs)[source]¶

Base class for all hyperelastic terms in TL formulation family.

The subclasses should have the following static method attributes: - stress_function() (the stress) - tan_mod_function() (the tangent modulus)

The common (family) data are cached in the evaluate cache of state variable.

get_family_data = HyperElasticTLFamilyData¶

hyperelastic_mode = 0¶

static weak_function(out, stress, tan_mod, mtx_f, det_f, cmap, is_diff, mode_ul)¶

class sfepy.terms.terms_hyperelastic_tl.HyperElasticTLFamilyData(**kwargs)[source]¶

Family data for TL formulation.

cache_name = 'tl_common'¶

data_names = ('mtx_f', 'det_f', 'sym_c', 'tr_c', 'in2_c', 'sym_inv_c', 'green_strain')¶

static family_function(mtx_f, det_f, vec_cs, tr_c, in_2c, vec_inv_cs, vec_es, state, cmap, conn)¶

class sfepy.terms.terms_hyperelastic_tl.MooneyRivlinTLTerm(*args, **kwargs)[source]¶

Hyperelastic Mooney-Rivlin term. Effective stress $S_{ij} = \kappa J^{-\frac{4}{3}} (C_{kk} \delta_{ij} - C_{ij} - \frac{2}{3 } I_2 C_{ij}^{-1})$ .

Definition:

$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$

Call signature:

dw_tl_he_mooney_rivlin

(material, virtual, state)

Arguments:

material : $\kappa$
virtual : $\ul{v}$
state : $\ul{u}$

family_data_names = ['det_f', 'tr_c', 'sym_inv_c', 'sym_c', 'in2_c']¶

name = 'dw_tl_he_mooney_rivlin'¶

static stress_function(out, mat, det_f, tr_c, vec_inv_cs, vec_cs, in_2c)¶

static tan_mod_function(out, mat, det_f, tr_c, vec_inv_cs, vec_cs, in_2c)¶

class sfepy.terms.terms_hyperelastic_tl.NeoHookeanTLTerm(*args, **kwargs)[source]¶

Hyperelastic neo-Hookean term. Effective stress $S_{ij} = \mu J^{-\frac{2}{3}}(\delta_{ij} - \frac{1}{3}C_{kk}C_{ij}^{-1})$ .

Definition:

$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$

Call signature:

dw_tl_he_neohook

(material, virtual, state)

Arguments:

material : $\mu$
virtual : $\ul{v}$
state : $\ul{u}$

family_data_names = ['det_f', 'tr_c', 'sym_inv_c']¶

name = 'dw_tl_he_neohook'¶

static stress_function(out, mat, det_f, tr_c, vec_inv_cs)¶

static tan_mod_function(out, mat, det_f, tr_c, vec_inv_cs)¶

class sfepy.terms.terms_hyperelastic_tl.OgdenTLTerm(*args, **kwargs)[source]¶

Single term of the hyperelastic Ogden model [1] with the strain energy density

$W = \frac{\mu}{\alpha} \, \left( \lambda_1^{\alpha} + \lambda_2^{\alpha} + \lambda_3^{\alpha} - 3 \right) \; ,$

where $\lambda_k, k=1, 2, 3$ are the principal stretches, whose squares are the principal values of the right Cauchy-Green deformation tensor $\mathbf{C}$ .

Effective stress (2nd Piola-Kirchhoff) is [2]

$S_{ij} = 2 \, \frac{\partial W}{\partial C_{ij}} = \sum_{k=1}^3 S^{(k)} \, N^{(k)}_i \, N^{(k)}_j \; ,$

where the principal stresses are

$S^{(k)} = J^{-2/3} \, \left( \mu \, \bar\lambda^{\alpha - 2} -\sum_{j=1}^3 \frac{\mu}{3} \frac{\lambda_j^{\alpha}}{\lambda_k^2} \right) \; , \quad k = 1, 2, 3 \; .$

and $\mathbf{N}^{(k)}$ , $k=1, 2, 3$ are the eigenvectors of $\mathbf{C}$ .

Definition:

$\int_{\Omega} S_{ij}(\ul{u}) \delta E_{ij}(\ul{u};\ul{v})$

Call signature:

dw_tl_he_ogden

(material, virtual, state)

Arguments:

material : $p, K$
virtual : $\ul{v}$
state : $\ul{u}$

[1] Ogden, R. W. Large deformation isotropic elasticity - on the correlation of theory and experiment for incompressible rubberlike solids. Proceedings of the Royal Society A, Vol. 326, No. 1567, Pp. 565-584 (1972), DOI 10.1098/rspa.1972.0026.

[2] Steinmann, P., Hossain, M., Possart, G. Hyperelastic models for rubber-like materials: Consistent tangent operators and suitability for Treloar’s data. Archive of Applied Mechanics, Vol. 82, No. 9, Pp. 1183-1217 (2012), DOI 10.1007/s00419-012-0610-z.

arg_shapes = {'material': '1, 2', 'state': 'D', 'virtual': ('D', 'state')}¶

family_data_names = ['det_f', 'sym_c', 'tr_c', 'sym_inv_c']¶

geometries = ['3_4', '3_8']¶

name = 'dw_tl_he_ogden'¶

stress_function(out, mat, *fargs, **kwargs)[source]¶

tan_mod_function(out, mat, *fargs, **kwargs)[source]¶

class sfepy.terms.terms_hyperelastic_tl.SurfaceFluxTLTerm(*args, **kwargs)[source]¶

Surface flux term in the total Lagrangian formulation, consistent with DiffusionTLTerm.

Definition:

$\int_{\Gamma} \ul{\nu} \cdot \ull{K}(\ul{u}^{(n-1)}) \pdiff{p}{\ul{X}}$

Call signature:

ev_tl_surface_flux

(material_1, material_2, parameter_1, parameter_2)

Arguments:

material_1 : $\ull{k}$
material_2 : $N_f$
parameter_1 : $p$
parameter_2 : $\ul{u}^{(n-1)}$

arg_shapes = {'material_1': 'D, D', 'material_2': '1, 1', 'parameter_1': 1, 'parameter_2': 'D'}¶

arg_types = ('material_1', 'material_2', 'parameter_1', 'parameter_2')¶

family_data_names = ['det_f', 'inv_f']¶

static function(out, pressure_grad, mtx_d, ref_porosity, mtx_fi, det_f, cmap, mode)¶

get_eval_shape(perm, ref_porosity, pressure, displacement, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

get_fargs(perm, ref_porosity, pressure, displacement, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

integration = 'facet_extra'¶

name = 'ev_tl_surface_flux'¶

class sfepy.terms.terms_hyperelastic_tl.SurfaceTractionTLTerm(*args, **kwargs)[source]¶

Surface traction term in the total Lagrangian formulation, expressed using $\ul{\nu}$ , the outward unit normal vector w.r.t. the undeformed surface, $\ull{F}(\ul{u})$ , the deformation gradient, $J = \det(\ull{F})$ , and $\ull{\sigma}$ a given traction, often equal to a given pressure, i.e. $\ull{\sigma} = \pi \ull{I}$ .

Definition:

$\int_{\Gamma} \ul{\nu} \cdot \ull{F}^{-1} \cdot \ull{\sigma} \cdot \ul{v} J$

Call signature:

dw_tl_surface_traction

(opt_material, virtual, state)

Arguments:

material : $\ull{\sigma}$
virtual : $\ul{v}$
state : $\ul{u}$

arg_shapes = [{'opt_material': 'D, D', 'state': 'D', 'virtual': ('D', 'state')}, {'opt_material': None}]¶

arg_types = ('opt_material', 'virtual', 'state')¶

family_data_names = ['det_f', 'inv_f']¶

static function(out, traction, det_f, mtx_fi, bf, cmap, fis, mode)¶

get_fargs(mat, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

integration = 'facet_extra'¶

name = 'dw_tl_surface_traction'¶

class sfepy.terms.terms_hyperelastic_tl.VolumeSurfaceTLTerm(*args, **kwargs)[source]¶

Volume of a $D$ -dimensional domain, using a surface integral in the total Lagrangian formulation, expressed using $\ul{\nu}$ , the outward unit normal vector w.r.t. the undeformed surface, $\ull{F}(\ul{u})$ , the deformation gradient, and $J = \det(\ull{F})$ . Uses the approximation of $\ul{u}$ for the deformed surface coordinates $\ul{x}$ .

Definition:

$1 / D \int_{\Gamma} \ul{\nu} \cdot \ull{F}^{-1} \cdot \ul{x} J$

Call signature:

ev_tl_volume_surface

(parameter)

Arguments:

parameter : $\ul{u}$

arg_shapes = {'parameter': 'D'}¶

arg_types = ('parameter',)¶

family_data_names = ['det_f', 'inv_f']¶

static function(out, coors, det_f, mtx_fi, bf, cmap, conn)¶

get_eval_shape(parameter, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

get_fargs(parameter, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

integration = 'facet_extra'¶

name = 'ev_tl_volume_surface'¶

class sfepy.terms.terms_hyperelastic_tl.VolumeTLTerm(*args, **kwargs)[source]¶

Volume term (weak form) in the total Lagrangian formulation.

Definition:

$\begin{array}{l} \int_{\Omega} q J(\ul{u}) \\ \mbox{volume mode: vector for } K \from \Ical_h: \int_{T_K} J(\ul{u}) \\ \mbox{rel\_volume mode: vector for } K \from \Ical_h: \int_{T_K} J(\ul{u}) / \int_{T_K} 1 \end{array}$

Call signature:

dw_tl_volume

(virtual, state)

Arguments:

virtual : $q$
state : $\ul{u}$

arg_geometry_types = {('virtual', None): {'facet_extra': 'facet'}}¶

arg_shapes = {'state': 'D', 'virtual': (1, None)}¶

arg_types = ('virtual', 'state')¶

family_data_names = ['mtx_f', 'det_f', 'sym_inv_c']¶

static function(out, mtx_f, vec_inv_cs, det_f, cmap_s, cmap_v, transpose, mode)¶

get_eval_shape(virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

get_fargs(virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

name = 'dw_tl_volume'¶