sfepy.terms.terms_elastic module¶

class sfepy.terms.terms_elastic.CauchyStrainSTerm(name, arg_str, integral, region, **kwargs)[source]¶

Evaluate Cauchy strain tensor on a surface region.

See CauchyStrainTerm.

Supports ‘eval’, ‘el_avg’ and ‘qp’ evaluation modes.

Definition:

$\int_{\Gamma} \ull{e}(\ul{w})$

$\mbox{vector for } K \from \Ical_h: \int_{T_K} \ull{e}(\ul{w}) / \int_{T_K} 1$

$\ull{e}(\ul{w})|_{qp}$

Call signature:

ev_cauchy_strain_s (parameter)

Arguments:	parameter : $\ul{w}$

arg_types = (‘parameter’,)¶

integration = ‘surface_extra’¶

name = ‘ev_cauchy_strain_s’¶

class sfepy.terms.terms_elastic.CauchyStrainTerm(name, arg_str, integral, region, **kwargs)[source]¶

Evaluate Cauchy strain tensor.

It is given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as $[11, 22, 33, 12, 13, 23]$ , in 2D it has 3 components with the indices ordered as $[11, 22, 12]$ . The last three (non-diagonal) components are doubled so that it is energetically conjugate to the Cauchy stress tensor with the same storage.

Supports ‘eval’, ‘el_avg’ and ‘qp’ evaluation modes.

Definition:

$\int_{\Omega} \ull{e}(\ul{w})$

$\mbox{vector for } K \from \Ical_h: \int_{T_K} \ull{e}(\ul{w}) / \int_{T_K} 1$

$\ull{e}(\ul{w})|_{qp}$

Call signature:

ev_cauchy_strain (parameter)

Arguments:	parameter : $\ul{w}$

arg_shapes = {‘parameter’: ‘D’}¶

arg_types = (‘parameter’,)¶

static function(out, strain, vg, fmode)[source]¶

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]¶

name = ‘ev_cauchy_strain’¶

class sfepy.terms.terms_elastic.CauchyStressETHTerm(name, arg_str, integral, region, **kwargs)[source]¶

Evaluate fading memory Cauchy stress tensor.

It is given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as $[11, 22, 33, 12, 13, 23]$ , in 2D it has 3 components with the indices ordered as $[11, 22, 12]$ .

Assumes an exponential approximation of the convolution kernel resulting in much higher efficiency.

Supports ‘eval’, ‘el_avg’ and ‘qp’ evaluation modes.

Definition:

$\int_{\Omega} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}$

$\mbox{vector for } K \from \Ical_h: \int_{T_K} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau} / \int_{T_K} 1$

$\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}|_{qp}$

Call signature:

ev_cauchy_stress_eth (ts, material_0, material_1, parameter)

Arguments:	ts : `TimeStepper` instance material_0 : $\Hcal_{ijkl}(0)$ material_1 : $\exp(-\lambda \Delta t)$ (decay at $t_1$ ) parameter : $\ul{w}$

arg_shapes = {‘material_0’: ‘S, S’, ‘material_1’: ‘1, 1’, ‘parameter’: ‘D’}¶

arg_types = (‘ts’, ‘material_0’, ‘material_1’, ‘parameter’)¶

get_eval_shape(ts, mat0, mat1, parameter, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

get_fargs(ts, mat0, mat1, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

name = ‘ev_cauchy_stress_eth’¶

class sfepy.terms.terms_elastic.CauchyStressTHTerm(name, arg_str, integral, region, **kwargs)[source]¶

Evaluate fading memory Cauchy stress tensor.

It is given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as $[11, 22, 33, 12, 13, 23]$ , in 2D it has 3 components with the indices ordered as $[11, 22, 12]$ .

Supports ‘eval’, ‘el_avg’ and ‘qp’ evaluation modes.

Definition:

$\int_{\Omega} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}$

$\mbox{vector for } K \from \Ical_h: \int_{T_K} \int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau} / \int_{T_K} 1$

$\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{w}(\tau)) \difd{\tau}|_{qp}$

Call signature:

ev_cauchy_stress_th (ts, material, parameter)

Arguments:	ts : `TimeStepper` instance material : $\Hcal_{ijkl}(\tau)$ parameter : $\ul{w}$

arg_shapes = {‘material’: ‘.: N, S, S’, ‘parameter’: ‘D’}¶

arg_types = (‘ts’, ‘material’, ‘parameter’)¶

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

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

name = ‘ev_cauchy_stress_th’¶

class sfepy.terms.terms_elastic.CauchyStressTerm(name, arg_str, integral, region, **kwargs)[source]¶

Evaluate Cauchy stress tensor.

It is given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as $[11, 22, 33, 12, 13, 23]$ , in 2D it has 3 components with the indices ordered as $[11, 22, 12]$ .

Supports ‘eval’, ‘el_avg’ and ‘qp’ evaluation modes.

Definition:

$\int_{\Omega} D_{ijkl} e_{kl}(\ul{w})$

$\mbox{vector for } K \from \Ical_h: \int_{T_K} D_{ijkl} e_{kl}(\ul{w}) / \int_{T_K} 1$

$D_{ijkl} e_{kl}(\ul{w})|_{qp}$

Call signature:

ev_cauchy_stress (material, parameter)

Arguments:	material : $D_{ijkl}$ parameter : $\ul{w}$

arg_shapes = {‘material’: ‘S, S’, ‘parameter’: ‘D’}¶

arg_types = (‘material’, ‘parameter’)¶

static function(out, coef, strain, mat, vg, fmode)[source]¶

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

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

name = ‘ev_cauchy_stress’¶

class sfepy.terms.terms_elastic.ElasticWaveCauchyTerm(name, arg_str, integral, region, **kwargs)[source]¶

Elastic dispersion term involving the wave strain $g_{ij}$ , $g_{ij}(\ul{u}) = \frac{1}{2}(u_i \kappa_j + \kappa_i u_j)$ , with the wave vector $\ul{\kappa}$ and the elastic strain $e_{ij}$ . $D_{ijkl}$ is given in the usual matrix form exploiting symmetry: in 3D it is $6\times6$ with the indices ordered as $[11, 22, 33, 12, 13, 23]$ , in 2D it is $3\times3$ with the indices ordered as $[11, 22, 12]$ .

Definition:

$\int_{\Omega} D_{ijkl}\ g_{ij}(\ul{v}) e_{kl}(\ul{u}) \;, \int_{\Omega} D_{ijkl}\ g_{ij}(\ul{u}) e_{kl}(\ul{v})$

Call signature:

dw_elastic_wave_cauchy	`(material_1, material_2, virtual, state)`
	`(material_1, material_2, state, virtual)`

Arguments 1:	material_1 : $D_{ijkl}$ material_2 : $\ul{\kappa}$ virtual : $\ul{v}$ state : $\ul{u}$
Arguments 2:	material_1 : $D_{ijkl}$ material_2 : $\ul{\kappa}$ state : $\ul{u}$ virtual : $\ul{v}$

arg_shapes = {‘material_1’: ‘S, S’, ‘material_2’: ‘.: D’, ‘state’: ‘D’, ‘virtual’: (‘D’, ‘state’)}¶

arg_types = ((‘material_1’, ‘material_2’, ‘virtual’, ‘state’), (‘material_1’, ‘material_2’, ‘state’, ‘virtual’))¶

static function(out, out_qp, geo, fmode)[source]¶

geometries = [‘2_3’, ‘2_4’, ‘3_4’, ‘3_8’]¶

get_fargs(mat, kappa, gvar, evar, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

modes = (‘ge’, ‘eg’)¶

name = ‘dw_elastic_wave_cauchy’¶

class sfepy.terms.terms_elastic.ElasticWaveTerm(name, arg_str, integral, region, **kwargs)[source]¶

Elastic dispersion term involving the wave strain $g_{ij}$ , $g_{ij}(\ul{u}) = \frac{1}{2}(u_i \kappa_j + \kappa_i u_j)$ , with the wave vector $\ul{\kappa}$ . $D_{ijkl}$ is given in the usual matrix form exploiting symmetry: in 3D it is $6\times6$ with the indices ordered as $[11, 22, 33, 12, 13, 23]$ , in 2D it is $3\times3$ with the indices ordered as $[11, 22, 12]$ .

Definition:

$\int_{\Omega} D_{ijkl}\ g_{ij}(\ul{v}) g_{kl}(\ul{u})$

Call signature:

dw_elastic_wave (material_1, material_2, virtual, state)

Arguments:	material_1 : $D_{ijkl}$ material_2 : $\ul{\kappa}$ virtual : $\ul{v}$ state : $\ul{u}$

arg_shapes = {‘material_1’: ‘S, S’, ‘material_2’: ‘.: D’, ‘state’: ‘D’, ‘virtual’: (‘D’, ‘state’)}¶

arg_types = (‘material_1’, ‘material_2’, ‘virtual’, ‘state’)¶

static function(out, out_qp, geo, fmode)[source]¶

geometries = [‘2_3’, ‘2_4’, ‘3_4’, ‘3_8’]¶

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

name = ‘dw_elastic_wave’¶

class sfepy.terms.terms_elastic.LinearElasticETHTerm(name, arg_str, integral, region, **kwargs)[source]¶

This term has the same definition as dw_lin_elastic_th, but assumes an exponential approximation of the convolution kernel resulting in much higher efficiency. Can use derivatives.

Definition:

$\int_{\Omega} \left [\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{u}(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v})$

Call signature:

dw_lin_elastic_eth (ts, material_0, material_1, virtual, state)

Arguments:	ts : `TimeStepper` instance material_0 : $\Hcal_{ijkl}(0)$ material_1 : $\exp(-\lambda \Delta t)$ (decay at $t_1$ ) virtual : $\ul{v}$ state : $\ul{u}$

arg_shapes = {‘material_0’: ‘S, S’, ‘material_1’: ‘1, 1’, ‘state’: ‘D’, ‘virtual’: (‘D’, ‘state’)}¶

arg_types = (‘ts’, ‘material_0’, ‘material_1’, ‘virtual’, ‘state’)¶

static function()¶

get_fargs(ts, mat0, mat1, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

name = ‘dw_lin_elastic_eth’¶

class sfepy.terms.terms_elastic.LinearElasticIsotropicTerm(name, arg_str, integral, region, **kwargs)[source]¶

Isotropic linear elasticity term.

Definition:

$\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u}) \mbox{ with } D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) + \lambda \ \delta_{ij} \delta_{kl}$

Call signature:

dw_lin_elastic_iso	`(material_1, material_2, virtual, state)`
	`(material_1, material_2, parameter_1, parameter_2)`

Arguments:	material_1 : $\lambda$ material_2 : $\mu$ virtual : $\ul{v}$ state : $\ul{u}$
Arguments 2:	material : $D_{ijkl}$ parameter_1 : $\ul{w}$ parameter_2 : $\ul{u}$

arg_shapes = {‘virtual’: (‘D’, ‘state’), ‘state’: ‘D’, ‘parameter_2’: ‘D’, ‘material_1’: ‘1, 1’, ‘material_2’: ‘1, 1’, ‘parameter_1’: ‘D’}¶

arg_types = ((‘material_1’, ‘material_2’, ‘virtual’, ‘state’), (‘material_1’, ‘material_2’, ‘parameter_1’, ‘parameter_2’))¶

geometries = [‘2_3’, ‘2_4’, ‘3_4’, ‘3_8’]¶

get_eval_shape(mat1, mat2, virtual, state, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

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

name = ‘dw_lin_elastic_iso’¶

class sfepy.terms.terms_elastic.LinearElasticTHTerm(name, arg_str, integral, region, **kwargs)[source]¶

Fading memory linear elastic (viscous) term. Can use derivatives.

Definition:

$\int_{\Omega} \left [\int_0^t \Hcal_{ijkl}(t-\tau)\,e_{kl}(\ul{u}(\tau)) \difd{\tau} \right]\,e_{ij}(\ul{v})$

Call signature:

dw_lin_elastic_th (ts, material, virtual, state)

Arguments:	ts : `TimeStepper` instance material : $\Hcal_{ijkl}(\tau)$ virtual : $\ul{v}$ state : $\ul{u}$

arg_shapes = {‘state’: ‘D’, ‘material’: ‘.: N, S, S’, ‘virtual’: (‘D’, ‘state’)}¶

arg_types = (‘ts’, ‘material’, ‘virtual’, ‘state’)¶

static function()¶

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

name = ‘dw_lin_elastic_th’¶

class sfepy.terms.terms_elastic.LinearElasticTerm(name, arg_str, integral, region, **kwargs)[source]¶

General linear elasticity term, with $D_{ijkl}$ given in the usual matrix form exploiting symmetry: in 3D it is $6\times6$ with the indices ordered as $[11, 22, 33, 12, 13, 23]$ , in 2D it is $3\times3$ with the indices ordered as $[11, 22, 12]$ . Can be evaluated. Can use derivatives.

Definition:

$\int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})$

Call signature:

dw_lin_elastic	`(material, virtual, state)`
	`(material, parameter_1, parameter_2)`

Arguments 1:	material : $D_{ijkl}$ virtual : $\ul{v}$ state : $\ul{u}$
Arguments 2:	material : $D_{ijkl}$ parameter_1 : $\ul{w}$ parameter_2 : $\ul{u}$

arg_shapes = {‘parameter_2’: ‘D’, ‘state’: ‘D’, ‘material’: ‘S, S’, ‘parameter_1’: ‘D’, ‘virtual’: (‘D’, ‘state’)}¶

arg_types = ((‘material’, ‘virtual’, ‘state’), (‘material’, ‘parameter_1’, ‘parameter_2’))¶

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

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

modes = (‘weak’, ‘eval’)¶

name = ‘dw_lin_elastic’¶

set_arg_types()[source]¶

class sfepy.terms.terms_elastic.LinearPrestressTerm(name, arg_str, integral, region, **kwargs)[source]¶

Linear prestress term, with the prestress $\sigma_{ij}$ given in the usual vector form exploiting symmetry: in 3D it has 6 components with the indices ordered as $[11, 22, 33, 12, 13, 23]$ , in 2D it has 3 components with the indices ordered as $[11, 22, 12]$ . Can be evaluated.

Definition:

$\int_{\Omega} \sigma_{ij} e_{ij}(\ul{v})$

Call signature:

dw_lin_prestress	`(material, virtual)`
	`(material, parameter)`

Arguments 1:	material : $\sigma_{ij}$ virtual : $\ul{v}$
Arguments 2:	material : $\sigma_{ij}$ parameter : $\ul{u}$

arg_shapes = {‘material’: ‘S, 1’, ‘parameter’: ‘D’, ‘virtual’: (‘D’, None)}¶

arg_types = ((‘material’, ‘virtual’), (‘material’, ‘parameter’))¶

d_lin_prestress(out, strain, mat, vg, fmode)[source]¶

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

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

modes = (‘weak’, ‘eval’)¶

name = ‘dw_lin_prestress’¶

set_arg_types()[source]¶

class sfepy.terms.terms_elastic.LinearStrainFiberTerm(name, arg_str, integral, region, **kwargs)[source]¶

Linear (pre)strain fiber term with the unit direction vector $\ul{d}$ .

Definition:

$\int_{\Omega} D_{ijkl} e_{ij}(\ul{v}) \left(d_k d_l\right)$

Call signature:

dw_lin_strain_fib (material_1, material_2, virtual)

Arguments:	material_1 : $D_{ijkl}$ material_2 : $\ul{d}$ virtual : $\ul{v}$

arg_shapes = {‘material_1’: ‘S, S’, ‘material_2’: ‘D, 1’, ‘virtual’: (‘D’, None)}¶

arg_types = (‘material_1’, ‘material_2’, ‘virtual’)¶

static function()¶

get_fargs(mat1, mat2, virtual, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

name = ‘dw_lin_strain_fib’¶

class sfepy.terms.terms_elastic.NonsymElasticTerm(name, arg_str, integral, region, **kwargs)[source]¶

Elasticity term with non-symmetric gradient. The indices of matrix $D_{ijkl}$ are ordered as $[11, 12, 13, 21, 22, 23, 31, 32, 33]$ in 3D and as $[11, 12, 21, 22]$ in 2D.

Definition:

$\int_{\Omega} \ull{D} \nabla\ul{u} : \nabla\ul{v}$

Call signature:

dw_nonsym_elastic	`(material, virtual, state)`
	`(material, parameter_1, parameter_2)`

Arguments 1:	material : $\ull{D}$ virtual : $\ul{v}$ state : $\ul{u}$
Arguments 2:	material : $\ull{D}$ parameter_1 : $\ul{w}$ parameter_2 : $\ul{u}$

arg_shapes = {‘parameter_2’: ‘D’, ‘state’: ‘D’, ‘material’: ‘D2, D2’, ‘parameter_1’: ‘D’, ‘virtual’: (‘D’, ‘state’)}¶

arg_types = ((‘material’, ‘virtual’, ‘state’), (‘material’, ‘parameter_1’, ‘parameter_2’))¶

geometries = [‘2_3’, ‘2_4’, ‘3_4’, ‘3_8’]¶

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

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

modes = (‘weak’, ‘eval’)¶

name = ‘dw_nonsym_elastic’¶

set_arg_types()[source]¶

class sfepy.terms.terms_elastic.SDLinearElasticTerm(name, arg_str, integral, region, **kwargs)[source]¶

Sensitivity analysis of the linear elastic term.

Definition:

$\int_{\Omega} \hat{D}_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u})$

$\hat{D}_{ijkl} = D_{ijkl}(\nabla \cdot \ul{\Vcal}) - D_{ijkq}{\partial \Vcal_l \over \partial x_q} - D_{iqkl}{\partial \Vcal_j \over \partial x_q}$

Call signature:

d_sd_lin_elastic (material, parameter_w, parameter_u, parameter_mesh_velocity)

Arguments:	material : $D_{ijkl}$ parameter_w : $\ul{w}$ parameter_u : $\ul{u}$ parameter_mesh_velocity : $\ul{\Vcal}$

arg_shapes = {‘parameter_mesh_velocity’: ‘D’, ‘material’: ‘S, S’, ‘parameter_w’: ‘D’, ‘parameter_u’: ‘D’}¶

arg_types = (‘material’, ‘parameter_w’, ‘parameter_u’, ‘parameter_mesh_velocity’)¶

function()¶

geometries = [‘2_3’, ‘2_4’, ‘3_4’, ‘3_8’]¶

get_eval_shape(mat, par_w, par_u, par_mv, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

get_fargs(mat, par_w, par_u, par_mv, mode=None, term_mode=None, diff_var=None, **kwargs)[source]¶

name = ‘d_sd_lin_elastic’¶

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sfepy.terms.terms_elastic module¶