elastodynamicsx.pde.materials package

Module contents

The materials module contains classes for a representation of a PDE of the kind:

M*a + C*v + K(u) = 0

i.e. the lhs of a PDE such as defined in the PDE class. An instance represents a single (possibly arbitrarily space-dependent) material.

The preferred way to call a material law is using the material function:

from elastodynamicsx.pde import material
mat = material( (function_space, cell_tags, marker), type_, *args, **kwargs)

elastodynamicsx.pde.materials.material(functionspace_tags_marker, type_, *args, **kwargs) → Material[source]

Builder method that instanciates the desired material type

Parameters:

functionspace_tags_marker –
Available possibilities are:
- (function_space, cell_tags, marker): meaning application
  of the material in the cells whose tag correspond to marker
- function_space: in this case cell_tags=None
  and marker=None, meaning application of the material in the entire domain
type –
Available options are:
- linear (scalar): ‘scalar’
- linear: ‘isotropic’, ‘cubic’, ‘hexagonal’, ‘trigonal’, ‘tetragonal’,
  ’orthotropic’, ‘monoclinic’, ‘triclinic’
- nonlinear, hyperelastic: ‘murnaghan’, ‘saintvenant-kirchhoff’, ‘mooney-rivlin-incomp’
*args – Passed to the required material

Keyword Arguments:

**kwargs – Passed to the required material

Returns:

An instance of the desired material

Example

# ###
# Constant / variable material parameter
# ###
rho = 2.8
rho = fem.Constant(function_space.mesh, default_scalar_type(2.8))
rho = fem.Function(scalar_function_space); rho.interpolate(lambda x: 2.8 * np.ones(len(x)))

# ###
# Subdomain(s) or entire domain?
# ###

# restricted to subdomain number 1
aluminum = material((function_space, cell_tags, 1), 'isotropic',
                     rho=2.8, lambda_=58, mu=26)

# restricted to subdomains number 1, 2, 5
aluminum = material((function_space, cell_tags, (1,2,5)), 'isotropic',
                     rho=2.8, lambda_=58, mu=26)

# entire domain
aluminum = material( function_space, 'isotropic',
                     rho=2.8, lambda_=58, mu=26)

# ###
# Available laws
# ###

# Linear elasticity
mat = material( function_space, 'isotropic'  , rho, C12, C44)
mat = material( function_space, 'cubic'      , rho, C11, C12, C44)
mat = material( function_space, 'hexagonal'  , rho, C11, C13, C33, C44, C66)
mat = material( function_space, 'trigonal'   , rho, C11, C12, C13, C14, C25,
                                                    C33, C44)
mat = material( function_space, 'tetragonal' , rho, C11, C12, C13, C16, C33, C44, C66)
mat = material( function_space, 'orthotropic', rho, C11, C12, C13, C22, C23, C33, C44,
                                                    C55, C66)
mat = material( function_space, 'monoclinic' , rho, C11, C12, C13, C15, C22, C23, C25,
                                                    C33, C35, C44, C46, C55, C66)
mat = material( function_space, 'triclinic'  , rho, C11, C12, C13, C14, C15, C16, C22,
                                                    C23, C24, C25, C26, C33, C34, C35,
                                                    C36, C44, C45, C46, C55, C56, C66)

# Hyperelasticity
mat = material( function_space, 'saintvenant-kirchhoff', rho, C12, C44)
mat = material( function_space, 'murnaghan', rho, C12, C44, l, m, n)
mat = material( function_space, 'mooney-rivlin-incomp', rho, C1, C2)

# Arbitrary
mat = material( function_space,
                'custom',
                M_fn=lambda u,v: ufl.inner(u,v) * ufl.dx,
                K_fn=lambda u,v: ... * ufl.dx)

elastodynamicsx.pde.materials.damping(type_: str, *args) → Damping[source]

Builder method that instanciates the desired damping law type

Parameters:

type –
Available options are:
- ’none’
- ’rayleigh’
*args – Passed to the required damping law

Returns:

An instance of the desired damping law

Example

from elastodynamicsx.pde import damping, material

dmp = damping('none')
dmp = damping('rayleigh', eta_m, eta_k)

mat = material(V, 'isotropic', rho, lambda_, mu, damping=dmp)

class elastodynamicsx.pde.materials.Material(functionspace_tags_marker, rho, is_linear, **kwargs)[source]

Bases: object

Base class for a representation of a material law.

Parameters:

functionspace_tags_marker –
Available possibilities are:
- (function_space, cell_tags, marker): meaning application of
  the material in the cells whose tag correspond to marker
- function_space: in this case cell_tags=None
  and marker=None, meaning application of the material in the entire domain
rho – Density
is_linear – True for linear, False for hyperelastic

Keyword Arguments:

metadata – (default=None) The metadata used by the ufl measures (dx, dS). If set to None, uses the PDECONFIG.default_metadata

labels: List[str]

property is_linear: bool

M_fn(u, v)[source]: (bilinear) mass form function

C_fn(u, v)[source]: (bilinear) damping form function

K_fn(u, v)[source]: (bilinear) Stiffness form function

K_fn_CG(u, v)[source]: Stiffness form function for a Continuous Galerkin formulation

K0_fn_CG(u, v)[source]: K0 stiffness form function for a Continuous Galerkin formulation (waveguides)

K1_fn_CG(u, v)[source]: K1 stiffness form function for a Continuous Galerkin formulation (waveguides)

K2_fn_CG(u, v)[source]: K2 stiffness form function for a Continuous Galerkin formulation (waveguides)

K_fn_DG(u, v)[source]: Stiffness form function for a Disontinuous Galerkin formulation

DG_numerical_flux(u, v)[source]: Numerical flux for a Disontinuous Galerkin formulation

property rho: Density

class elastodynamicsx.pde.materials.ElasticMaterial(functionspace_tags_marker, rho, C_21: List, **kwargs)[source]

Bases: Material

Base class for linear elastic materials, supporting full anisotropy

Parameters:

functionspace_tags_marker – See Material
rho – Density
C_21 – list of the 21 independent elastic constants

Keyword Arguments:

damping – (default=NoDamping()) An instance of a subclass of Damping

eq_IJ: ndarray = array([[ 0, 1, 2, 3, 4, 5], [ 1, 6, 7, 8, 9, 10], [ 2, 7, 11, 12, 13, 14], [ 3, 8, 12, 15, 16, 17], [ 4, 9, 13, 16, 18, 19], [ 5, 10, 14, 17, 19, 20]])

ijkm: ndarray = array([[0, 0, 0, 0], [0, 0, 1, 1], [0, 0, 2, 2], [0, 0, 1, 2], [0, 0, 0, 2], [0, 0, 0, 1], [1, 1, 1, 1], [1, 1, 2, 2], [1, 1, 1, 2], [1, 1, 0, 2], [1, 1, 0, 1], [2, 2, 2, 2], [2, 2, 1, 2], [2, 2, 0, 2], [2, 2, 0, 1], [1, 2, 1, 2], [1, 2, 0, 2], [1, 2, 0, 1], [0, 2, 0, 2], [0, 2, 0, 1], [0, 1, 0, 1]])

static Cij(C_21: List, i: int, j: int)[source]

Returns the \(C_{ij}\) coefficient from the C_21 list

i, j = 0..5

static Cijkm(C_21: List, i: int, j: int, k: int, m: int)[source]

Returns the \(C_{ijkm}\) coefficient from the C_21 list

i, j, k, m = 0..2

property epsilon: Strain function (matrix representation) \(\boldsymbol{\epsilon}(\mathbf{u})\)

property epsilonVoigt: Strain function (Voigt representation) \(\boldsymbol{\epsilon}(\mathbf{u})\)

sigma(u)[source]: Stress function (matrix representation) \(\boldsymbol{\sigma}(\mathbf{u})\)

sigmaVoigt(u)[source]: Stress function (Voigt representation) \(\boldsymbol{\sigma}(\mathbf{u})\)

sigma_n(u, n)[source]: Stress in the ‘n’ direction \(\boldsymbol{\sigma}(\mathbf{u}) \cdot \mathbf{n}\)

diamond(v1, v2)[source]

Returns the diamond product of vectors v1 and v2.

Returns:: \(\mathbf{d}\), such as \(d_{jk} = C_{ijkm} v_{1,i} v_{2,m}\)

C_fn(u, v)[source]: Damping form function

K_fn_CG(u, v)[source]: Stiffness form function for a Continuous Galerkin formulation

K0_fn_CG(u, v)[source]: K0 stiffness form function for a Continuous Galerkin formulation (waveguides)

K1_fn_CG(u, v)[source]: K1 stiffness form function for a Continuous Galerkin formulation (waveguides)

K2_fn_CG(u, v)[source]: K2 stiffness form function for a Continuous Galerkin formulation (waveguides)

K_fn_DG(u, v)[source]: Stiffness form function for a Discontinuous Galerkin formulation

select_DG_numerical_flux(variant: str = 'SIPG') → Callable[source]

property DG_numerical_flux: Callable: Numerical flux for a Disontinuous Galerkin formulation

DG_SIPG_regularization_parameter() → Constant[source]: Regularization parameter for the Symmetric Interior Penalty Galerkin methods (SIPG)

DG_numerical_flux_SIPG(u, v)[source]

DG_numerical_flux_NIPG(u, v)[source]: WARNING, instable for elasticity

DG_numerical_flux_IIPG(u, v)[source]: WARNING, instable for elasticity

property P_modulus

P-wave modulus

Returns:

\(\rho c_{max}^2\) where \(c_{max}\) is the highest wave velocity
\(\lambda + 2 \mu\) for isotropic materials
\(\mu\) for scalar materials

Cij_xyz_frame(i, j)[source]: Cij stiffness constant in the (xyz) coordinate frame

Cijkm_xyz_frame(i, j, k, m)[source]: Cijkm stiffness constant in the (xyz) coordinate frame

Cij_mat_frame(i, j)[source]: Cij stiffness constant in the material coordinate frame

Cijkm_mat_frame(i, j, k, m)[source]: Cijkm stiffness constant in the material coordinate frame

property C11: C11 stiffness constant in the material coordinate frame

property C12: C12 stiffness constant in the material coordinate frame

property C13: C13 stiffness constant in the material coordinate frame

property C14: C14 stiffness constant in the material coordinate frame

property C15: C15 stiffness constant in the material coordinate frame

property C16: C16 stiffness constant in the material coordinate frame

property C22: C22 stiffness constant in the material coordinate frame

property C23: C23 stiffness constant in the material coordinate frame

property C24: C24 stiffness constant in the material coordinate frame

property C25: C25 stiffness constant in the material coordinate frame

property C26: C26 stiffness constant in the material coordinate frame

property C33: C33 stiffness constant in the material coordinate frame

property C34: C34 stiffness constant in the material coordinate frame

property C35: C35 stiffness constant in the material coordinate frame

property C36: C36 stiffness constant in the material coordinate frame

property C44: C44 stiffness constant in the material coordinate frame

property C45: C45 stiffness constant in the material coordinate frame

property C46: C46 stiffness constant in the material coordinate frame

property C55: C55 stiffness constant in the material coordinate frame

property C56: C56 stiffness constant in the material coordinate frame

property C66: C66 stiffness constant in the material coordinate frame

class elastodynamicsx.pde.materials.HyperelasticMaterial(functionspace_tags_marker, rho, **kwargs)[source]

Bases: Material

Base class for hyperelastic materials

Parameters:

functionspace_tags_marker – See Material
rho – Density
sigma – Stress function

Keyword Arguments:

**kwargs – see Material

P(u)[source]: First Piola-Kirchhoff stress

K_fn_CG(u, v)[source]: Stiffness form function for a Continuous Galerkin formulation

K_fn_DG(u, v)[source]: (Not implemented) Stiffness form function for a Discontinuous Galerkin formulation

DG_numerical_flux(u, v)[source]: (Not implemented) Numerical flux for a Disontinuous Galerkin formulation

Submodules

elastodynamicsx.pde.materials.material module

class elastodynamicsx.pde.materials.material.Material(functionspace_tags_marker, rho, is_linear, **kwargs)[source]

Bases: object

Base class for a representation of a material law.

Parameters:

functionspace_tags_marker –
Available possibilities are:
- (function_space, cell_tags, marker): meaning application of
  the material in the cells whose tag correspond to marker
- function_space: in this case cell_tags=None
  and marker=None, meaning application of the material in the entire domain
rho – Density
is_linear – True for linear, False for hyperelastic

Keyword Arguments:

metadata – (default=None) The metadata used by the ufl measures (dx, dS). If set to None, uses the PDECONFIG.default_metadata

labels: List[str]

property is_linear: bool

M_fn(u, v)[source]: (bilinear) mass form function

C_fn(u, v)[source]: (bilinear) damping form function

K_fn(u, v)[source]: (bilinear) Stiffness form function

K_fn_CG(u, v)[source]: Stiffness form function for a Continuous Galerkin formulation

K0_fn_CG(u, v)[source]: K0 stiffness form function for a Continuous Galerkin formulation (waveguides)

K1_fn_CG(u, v)[source]: K1 stiffness form function for a Continuous Galerkin formulation (waveguides)

K2_fn_CG(u, v)[source]: K2 stiffness form function for a Continuous Galerkin formulation (waveguides)

K_fn_DG(u, v)[source]: Stiffness form function for a Disontinuous Galerkin formulation

DG_numerical_flux(u, v)[source]: Numerical flux for a Disontinuous Galerkin formulation

property rho: Density

elastodynamicsx.pde.materials.elasticmaterial module

class elastodynamicsx.pde.materials.elasticmaterial.ElasticMaterial(functionspace_tags_marker, rho, C_21: List, **kwargs)[source]

Bases: Material

Base class for linear elastic materials, supporting full anisotropy

Parameters:

functionspace_tags_marker – See Material
rho – Density
C_21 – list of the 21 independent elastic constants

Keyword Arguments:

damping – (default=NoDamping()) An instance of a subclass of Damping

eq_IJ: ndarray = array([[ 0, 1, 2, 3, 4, 5], [ 1, 6, 7, 8, 9, 10], [ 2, 7, 11, 12, 13, 14], [ 3, 8, 12, 15, 16, 17], [ 4, 9, 13, 16, 18, 19], [ 5, 10, 14, 17, 19, 20]])

ijkm: ndarray = array([[0, 0, 0, 0], [0, 0, 1, 1], [0, 0, 2, 2], [0, 0, 1, 2], [0, 0, 0, 2], [0, 0, 0, 1], [1, 1, 1, 1], [1, 1, 2, 2], [1, 1, 1, 2], [1, 1, 0, 2], [1, 1, 0, 1], [2, 2, 2, 2], [2, 2, 1, 2], [2, 2, 0, 2], [2, 2, 0, 1], [1, 2, 1, 2], [1, 2, 0, 2], [1, 2, 0, 1], [0, 2, 0, 2], [0, 2, 0, 1], [0, 1, 0, 1]])

static Cij(C_21: List, i: int, j: int)[source]

Returns the \(C_{ij}\) coefficient from the C_21 list

i, j = 0..5

static Cijkm(C_21: List, i: int, j: int, k: int, m: int)[source]

Returns the \(C_{ijkm}\) coefficient from the C_21 list

i, j, k, m = 0..2

property epsilon: Strain function (matrix representation) \(\boldsymbol{\epsilon}(\mathbf{u})\)

property epsilonVoigt: Strain function (Voigt representation) \(\boldsymbol{\epsilon}(\mathbf{u})\)

sigma(u)[source]: Stress function (matrix representation) \(\boldsymbol{\sigma}(\mathbf{u})\)

sigmaVoigt(u)[source]: Stress function (Voigt representation) \(\boldsymbol{\sigma}(\mathbf{u})\)

sigma_n(u, n)[source]: Stress in the ‘n’ direction \(\boldsymbol{\sigma}(\mathbf{u}) \cdot \mathbf{n}\)

diamond(v1, v2)[source]

Returns the diamond product of vectors v1 and v2.

Returns:: \(\mathbf{d}\), such as \(d_{jk} = C_{ijkm} v_{1,i} v_{2,m}\)

C_fn(u, v)[source]: Damping form function

K_fn_CG(u, v)[source]: Stiffness form function for a Continuous Galerkin formulation

K0_fn_CG(u, v)[source]: K0 stiffness form function for a Continuous Galerkin formulation (waveguides)

K1_fn_CG(u, v)[source]: K1 stiffness form function for a Continuous Galerkin formulation (waveguides)

K2_fn_CG(u, v)[source]: K2 stiffness form function for a Continuous Galerkin formulation (waveguides)

K_fn_DG(u, v)[source]: Stiffness form function for a Discontinuous Galerkin formulation

select_DG_numerical_flux(variant: str = 'SIPG') → Callable[source]

property DG_numerical_flux: Callable: Numerical flux for a Disontinuous Galerkin formulation

DG_SIPG_regularization_parameter() → Constant[source]: Regularization parameter for the Symmetric Interior Penalty Galerkin methods (SIPG)

DG_numerical_flux_SIPG(u, v)[source]

DG_numerical_flux_NIPG(u, v)[source]: WARNING, instable for elasticity

DG_numerical_flux_IIPG(u, v)[source]: WARNING, instable for elasticity

property P_modulus

P-wave modulus

Returns:

\(\rho c_{max}^2\) where \(c_{max}\) is the highest wave velocity
\(\lambda + 2 \mu\) for isotropic materials
\(\mu\) for scalar materials

Cij_xyz_frame(i, j)[source]: Cij stiffness constant in the (xyz) coordinate frame

Cijkm_xyz_frame(i, j, k, m)[source]: Cijkm stiffness constant in the (xyz) coordinate frame

Cij_mat_frame(i, j)[source]: Cij stiffness constant in the material coordinate frame

Cijkm_mat_frame(i, j, k, m)[source]: Cijkm stiffness constant in the material coordinate frame

property C11: C11 stiffness constant in the material coordinate frame

property C12: C12 stiffness constant in the material coordinate frame

property C13: C13 stiffness constant in the material coordinate frame

property C14: C14 stiffness constant in the material coordinate frame

property C15: C15 stiffness constant in the material coordinate frame

property C16: C16 stiffness constant in the material coordinate frame

property C22: C22 stiffness constant in the material coordinate frame

property C23: C23 stiffness constant in the material coordinate frame

property C24: C24 stiffness constant in the material coordinate frame

property C25: C25 stiffness constant in the material coordinate frame

property C26: C26 stiffness constant in the material coordinate frame

property C33: C33 stiffness constant in the material coordinate frame

property C34: C34 stiffness constant in the material coordinate frame

property C35: C35 stiffness constant in the material coordinate frame

property C36: C36 stiffness constant in the material coordinate frame

property C44: C44 stiffness constant in the material coordinate frame

property C45: C45 stiffness constant in the material coordinate frame

property C46: C46 stiffness constant in the material coordinate frame

property C55: C55 stiffness constant in the material coordinate frame

property C56: C56 stiffness constant in the material coordinate frame

labels: List[str]

property C66: C66 stiffness constant in the material coordinate frame

elastodynamicsx.pde.materials.anisotropicmaterials module

class elastodynamicsx.pde.materials.anisotropicmaterials.CubicMaterial(functionspace_tags_marker, rho, C11, C12, C44, **kwargs)[source]

Bases: ElasticMaterial

A linear elastic material with cubic symmetry -> 3 independent constants

Parameters:

functionspace_tags_marker – See Material
rho – Density
C11 – Stiffness constant
C12 – Stiffness constant
C44 – Stiffness constant

Keyword Arguments:

**kwargs – Passed to ElasticMaterial

labels: List[str] = ['cubic']

class elastodynamicsx.pde.materials.anisotropicmaterials.HexagonalMaterial(functionspace_tags_marker, rho, C11, C13, C33, C44, C66, **kwargs)[source]

Bases: ElasticMaterial

A linear elastic material with hexagonal symmetry -> 5 independent constants

Parameters:

functionspace_tags_marker – See Material
rho – Density
C11 – Stiffness constant
... – Stiffness constant
C66 – Stiffness constant

Keyword Arguments:

**kwargs – Passed to ElasticMaterial

labels: List[str] = ['hexagonal']

class elastodynamicsx.pde.materials.anisotropicmaterials.TrigonalMaterial(functionspace_tags_marker, rho, C11, C12, C13, C14, C25, C33, C44, **kwargs)[source]

Bases: ElasticMaterial

A linear elastic material with trigonal symmetry -> 7 independent constants

Parameters:

functionspace_tags_marker – See Material
rho – Density
C11 – Stiffness constant
... – Stiffness constant
C44 – Stiffness constant

Keyword Arguments:

**kwargs – Passed to ElasticMaterial

labels: List[str] = ['trigonal']

class elastodynamicsx.pde.materials.anisotropicmaterials.TetragonalMaterial(functionspace_tags_marker, rho, C11, C12, C13, C16, C33, C44, C66, **kwargs)[source]

Bases: ElasticMaterial

A linear elastic material with tetragonal symmetry -> 7 independent constants

Parameters:

functionspace_tags_marker – See Material
rho – Density
C11 – Stiffness constant
... – Stiffness constant
C66 – Stiffness constant

Keyword Arguments:

**kwargs – Passed to ElasticMaterial

labels: List[str] = ['tetragonal']

class elastodynamicsx.pde.materials.anisotropicmaterials.OrthotropicMaterial(functionspace_tags_marker, rho, C11, C12, C13, C22, C23, C33, C44, C55, C66, **kwargs)[source]

Bases: ElasticMaterial

A linear elastic material with orthotropic symmetry -> 9 independent constants

Parameters:

functionspace_tags_marker – See Material
rho – Density
C11 – Stiffness constant
... – Stiffness constant
C66 – Stiffness constant

Keyword Arguments:

**kwargs – Passed to ElasticMaterial

labels: List[str] = ['orthotropic']

class elastodynamicsx.pde.materials.anisotropicmaterials.MonoclinicMaterial(functionspace_tags_marker, rho, C11, C12, C13, C15, C22, C23, C25, C33, C35, C44, C46, C55, C66, **kwargs)[source]

Bases: ElasticMaterial

A linear elastic material with monoclinic symmetry -> 13 independent constants

Parameters:

functionspace_tags_marker – See Material
rho – Density
C11 – Stiffness constant
... – Stiffness constant
C66 – Stiffness constant

Keyword Arguments:

**kwargs – Passed to ElasticMaterial

labels: List[str] = ['monoclinic']

class elastodynamicsx.pde.materials.anisotropicmaterials.TriclinicMaterial(functionspace_tags_marker, rho, C11, C12, C13, C14, C15, C16, C22, C23, C24, C25, C26, C33, C34, C35, C36, C44, C45, C46, C55, C56, C66, **kwargs)[source]

Bases: ElasticMaterial

A linear elastic material with triclinic symmetry -> 21 independent constants

Parameters:

functionspace_tags_marker – See Material
rho – Density
C11 – Stiffness constant
... – Stiffness constant
C66 – Stiffness constant

Keyword Arguments:

**kwargs – Passed to ElasticMaterial

labels: List[str] = ['triclinic']

elastodynamicsx.pde.materials.hyperelasticmaterials module

class elastodynamicsx.pde.materials.hyperelasticmaterials.HyperelasticMaterial(functionspace_tags_marker, rho, **kwargs)[source]

Bases: Material

Base class for hyperelastic materials

Parameters:

functionspace_tags_marker – See Material
rho – Density
sigma – Stress function

Keyword Arguments:

**kwargs – see Material

P(u)[source]: First Piola-Kirchhoff stress

K_fn_CG(u, v)[source]: Stiffness form function for a Continuous Galerkin formulation

K_fn_DG(u, v)[source]: (Not implemented) Stiffness form function for a Discontinuous Galerkin formulation

DG_numerical_flux(u, v)[source]: (Not implemented) Numerical flux for a Disontinuous Galerkin formulation

class elastodynamicsx.pde.materials.hyperelasticmaterials.Murnaghan(functionspace_tags_marker, rho, lambda_, mu, l_, m_, n_, **kwargs)[source]

Bases: HyperelasticMaterial

Murnaghan’s model

Strain energy density:: \[W = \frac{\lambda}{2} \mathrm{tr}(\mathbf{E})^2 + \mu \mathrm{tr}(\mathbf{E}^2) + \frac{A}{3} \mathrm{tr}(\mathbf{E}^3) + B \mathrm{tr}(\mathbf{E}) \mathrm{tr}(\mathbf{E}^2) + \frac{C}{3} \mathrm{tr}(\mathbf{E})^3\]

with: \(l=B+C\), \(m=\frac{1}{2}A+B\), \(n=A\).

Parameters:

functionspace_tags_marker – See Material
rho – Density
lambda – Lame’s first parameter
mu – Lame’s second parameter (shear modulus)
l – Murnaghan’s third order elastic constant
m – Murnaghan’s third order elastic constant
n – Murnaghan’s third order elastic constant

Keyword Arguments:

**kwargs – Passed to HyperelasticMaterial

See:: https://en.wikipedia.org/wiki/Acoustoelastic_effect

labels: List[str] = ['murnaghan']

P(u)[source]: First Piola-Kirchhoff stress

property Z_N: (WARNING, infinitesimal strain asymptotics) P-wave mechanical impedance \(\rho c_L\)

property Z_T: (WARNING, infinitesimal strain asymptotics) S-wave mechanical impedance \(\rho c_S\)

class elastodynamicsx.pde.materials.hyperelasticmaterials.DummyIsotropicMaterial(functionspace_tags_marker, rho, lambda_, mu, **kwargs)[source]

Bases: HyperelasticMaterial

A dummy implementation of an isotropic linear elastic material based on a HyperelasticMaterial

Parameters:

functionspace_tags_marker – See Material
rho – Density
lambda – Lame’s first parameter
mu – Lame’s second parameter (shear modulus)

Keyword Arguments:

**kwargs – Passed to HyperelasticMaterial

labels: List[str] = ['dummy-isotropic']

P(u)[source]: Infinitesimal stress function; NOT the first Piola-Kirchhoff stress

property Z_N: P-wave mechanical impedance \(\rho c_L\)

property Z_T: S-wave mechanical impedance \(\rho c_S\)

class elastodynamicsx.pde.materials.hyperelasticmaterials.StVenantKirchhoff(functionspace_tags_marker, rho, lambda_, mu, **kwargs)[source]

Bases: HyperelasticMaterial

Saint Venant-Kirchhoff model

Strain energy density:: \[W = \frac{\lambda}{2} \mathrm{tr}(\mathbf{E})^2 + \mu \mathrm{tr}(\mathbf{E}^2)\]

Parameters:

functionspace_tags_marker – See Material
rho – Density
lambda – Lame’s first parameter
mu – Lame’s second parameter (shear modulus)

Keyword Arguments:

**kwargs – Passed to HyperelasticMaterial

See:: https://en.wikipedia.org/wiki/Hyperelastic_material

labels: List[str] = ['stvenant-kirchhoff', 'saintvenant-kirchhoff']

P(u)[source]: First Piola-Kirchhoff stress

property Z_N: (WARNING, infinitesimal strain asymptotics) P-wave mechanical impedance \(\rho c_L\)

property Z_T: (WARNING, infinitesimal strain asymptotics) S-wave mechanical impedance \(\rho c_S\)

class elastodynamicsx.pde.materials.hyperelasticmaterials.MooneyRivlinIncompressible(functionspace_tags_marker, rho, C1, C2, **kwargs)[source]

Bases: HyperelasticMaterial

Mooney-Rivlin model for an incompressible solid

Strain energy density:: \[W = C_1 (\overline{I}_1 - 3) + C_2 (\overline{I}_2 - 3)\]

Parameters:

functionspace_tags_marker – See Material
rho – Density
C1 – first parameter
C2 – second parameter

Keyword Arguments:

**kwargs – Passed to HyperelasticMaterial

See:: https://en.wikipedia.org/wiki/Mooney%E2%80%93Rivlin_solid

labels: List[str] = ['mooney-rivlin-incomp']

P(u)[source]: First Piola-Kirchhoff stress

property C1

property C2

class elastodynamicsx.pde.materials.hyperelasticmaterials.MooneyRivlinCompressible(functionspace_tags_marker, rho, C10, C01, D1, **kwargs)[source]

Bases: HyperelasticMaterial

Mooney-Rivlin model for a compressible solid

Strain energy density:: \[W = C_{10} (\overline{I}_1 - 3) + C_{01} (\overline{I}_2 - 3) + \frac{1}{D_1} (J - 1)^2\]

Parameters:

functionspace_tags_marker – See Material
rho – Density
C10 – First parameter
C01 – Second parameter
D1 – Third parameter

Keyword Arguments:

**kwargs – Passed to HyperelasticMaterial

See:: https://en.wikipedia.org/wiki/Mooney%E2%80%93Rivlin_solid

labels: List[str] = ['mooney-rivlin-comp']

P(u)[source]: First Piola-Kirchhoff stress

property C1

property C2

elastodynamicsx.pde.materials.hyperelasticmaterials.GreenLagrangeStrain(u)[source]

elastodynamicsx.pde.materials.dampings module

The damping module implements laws to model attenuation. A Damping instance is meant to be connected to a Material instance and thereby provides the damping form of the PDE.

The recommended way to instanciate a damping law is to use the damping function.

class elastodynamicsx.pde.materials.dampings.Damping[source]

Bases: object

Dummy base class for damping laws

labels: List[str]

C_fn(u, v)[source]: The damping form

class elastodynamicsx.pde.materials.dampings.NoDamping[source]

Bases: Damping

No damping

labels: List[str] = ['none']

C_fn(u, v)[source]

The damping form

Returns:: None

class elastodynamicsx.pde.materials.dampings.RayleighDamping(eta_m, eta_k)[source]

Bases: Damping

Rayleigh damping law, i.e. whose damping form is a linear combination of the mass and stiffness forms of the host material

\[c(u,v) = \eta_m m(u,v) + \eta_k k(u,v)\]

labels: List[str] = ['rayleigh']

property eta_m: Parameter of the mass-matrix part of the damping

property eta_k: Parameter of the stiffness-matrix part of the damping

C_fn(u, v)[source]: The damping form

property host_material: Host material from whom the mass and stiffness matrices are copied

link_material(host_material)[source]: Connects to a host material from whom the mass and stiffness matrices will be copied

elastodynamicsx.pde.materials.dampings.damping(type_: str, *args) → Damping[source]

Builder method that instanciates the desired damping law type

Parameters:

type –
Available options are:
- ’none’
- ’rayleigh’
*args – Passed to the required damping law

Returns:

An instance of the desired damping law

Example

from elastodynamicsx.pde import damping, material

dmp = damping('none')
dmp = damping('rayleigh', eta_m, eta_k)

mat = material(V, 'isotropic', rho, lambda_, mu, damping=dmp)

elastodynamicsx.pde.materials.kinematics module

The kinematics module is mainly designed for internal use and will probably not be used by an external user. It contains convenience functions to build some useful differential operators at the required ufl format, for many possible space dimensions and number of components of the function space.

elastodynamicsx.pde.materials.kinematics.epsilon_vector(u)[source]

elastodynamicsx.pde.materials.kinematics.epsilon_scalar(u)[source]

elastodynamicsx.pde.materials.kinematics.get_epsilon_function(dim, nbcomps)[source]: The strain function for a given dimension and number of components, in matrix form.

elastodynamicsx.pde.materials.kinematics.get_epsilonVoigt_function(dim, nbcomps)[source]: The strain function for a given dimension and number of components, in vector form (Voigt notation).

elastodynamicsx.pde.materials.kinematics.get_L_operators(dim, nbcomps, k_nrm=None)[source]

Parameters:

dim – space dimension
nbcomps – number of components
k_nrm –
(dim==1 only) A unitary vector (len=3) representing the phase direction.

typically: ufl.as_vector([0,ay,az]) default : ufl.as_vector([0,1,0])