The Constitutive Library

Import of the simmit module

from simcoon import simmit as sim

np.ndarray Ireal()

Provides the fourth order identity tensor written in Voigt notation \(I_{real}\), where :

\[\begin{split}I_{real} = \left( \begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.5 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.5 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0.5 \end{array} \right)\end{split}\]

Ir = sim.Ireal()

np.ndarray Ivol()

Provides the volumic of the identity tensor \(I_{vol}\) written in the Simcoon formalism. So :

\[\begin{split}I_{vol} = \left( \begin{array}{ccc} 1/3 & 1/3 & 1/3 & 0 & 0 & 0 \\ 1/3 & 1/3 & 1/3 & 0 & 0 & 0 \\ 1/3 & 1/3 & 1/3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right)\end{split}\]

Iv = sim.Ivol()

np.ndarray Idev()

Provides the deviatoric of the identity tensor \(I_{dev}\) written in the Simcoon formalism. So :

\[\begin{split} I_{dev} = I_{real} - I_{vol} = \left( \begin{array}{ccc} 2/3 & -1/3 & -1/3 & 0 & 0 & 0 \\ -1/3 & 2/3 & -1/3 & 0 & 0 & 0 \\ -1/3 & -1/3 & 2/3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.5 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.5 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0.5 \end{array} \right)\end{split}\]

Id = sim.Idev()

np.ndarray Ireal2()

Provides the fourth order identity tensor \(\widehat{I}\) written in the form. So :

\[\begin{split}\widehat{I} = \left( \begin{array}{ccc} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 \end{array} \right)\end{split}\]

For example, this tensor allows to obtain : \(L*\widehat{M}=I\) or \(\widehat{L}*M=I\), where a matrix \(\widehat{A}\) is set by \(\widehat{A}=\widehat{I}A\widehat{I}\)

Ir2 = sim.Ireal2()

np.ndarray Idev2()

Provides the deviatoric of the identity tensor \(\widehat{I}\) written in the Simcoon formalism. So :

\[\begin{split}I_{dev2} = \left( \begin{array}{ccc} 2/3 & -1/3 & -1/3 & 0 & 0 & 0 \\ -1/3 & 2/3 & -1/3 & 0 & 0 & 0 \\ -1/3 & -1/3 & 2/3 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 \end{array} \right)\end{split}\]

Id2 = sim.Idev2()

np.ndarray Ith()

Provide the vector \(I_{th} = \left( \begin{array}{ccc} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \right)\)

It = sim.Ith()

np.ndarray Ir2()

Provide the vector \(I_{r2} = \left( \begin{array}{ccc} 1 \\ 1 \\ 1 \\ 2 \\ 2 \\ 2 \end{array} \right)\)

I2 = sim.Ir2()

np.ndarray Ir05()

Provide the vector \(I_{r05} = \left( \begin{array}{ccc} 1 \\ 1 \\ 1 \\ 0.5 \\ 0.5 \\ 0.5 \end{array} \right)\)

I05 = sim.Ir05()

np.ndarray L_iso(const double &C1, const double &C2, const std::string &conv)

Provides the elastic stiffness tensor for an isotropic material. The two first arguments are a couple of elastic properties. The third argument specifies which couple has been provided and the nature and order of coefficients. Exhaustive list of possible third argument : ‘Enu’,’nuE,’Kmu’,’muK’, ‘KG’, ‘GK’, ‘lambdamu’, ‘mulambda’, ‘lambdaG’, ‘Glambda’.

E = 210000.0
nu = 0.3;
Liso = sim.L_iso(E, nu, "Enu")

np.ndarray M_iso(const double &C1, const double &C2, const string &conv)

Provides the elastic compliance tensor for an isotropic material. The two first arguments are a couple of elastic properties. The third argument specify which couple has been provided and the nature and order of coefficients. Exhaustive list of possible third argument : ‘Enu’,’nuE,’Kmu’,’muK’, ‘KG’, ‘GK’, ‘lambdamu’, ‘mulambda’, ‘lambdaG’, ‘Glambda’.

E = 210000.0
nu = 0.3
Miso = sim.M_iso(E, nu, "Enu")

np.ndarray L_cubic(const double &C1, const double &C2, const double &C4, const string &conv)

Provides the elastic stiffness tensor for a cubic material. The last argument must be set to “Cii” if the inputs are the stiffness coefficients or to “EnuG” if the inputs are the material parameters.

E = 70000.0
nu = 0.3
G = 23000.0
Lcubic = sim.L_cubic(E, nu, G, "EnuG")

import numpy as np
C11 = np.random.uniform(10000., 100000.)
C12 = np.random.uniform(10000., 100000.)
C44 = np.random.uniform(10000., 100000.)
Lcubic = sim.L_cubic(C11, C12, C44, "Cii")

np.ndarray M_cubic(const double &C1, const double &C2, const double &C4, const string &conv)

Provides the elastic compliance tensor for a cubic material. The last argument must be set to “Cii” if the inputs are the stiffness coefficients or to “EnuG” if the inputs are the material parameters.

E = 70000.0
nu = 0.3
G = 23000.0
Lcubic = sim.L_cubic(E, nu, G, "EnuG")

C11 = np.random.uniform(10000., 100000.)
C12 = np.random.uniform(10000., 100000.)
C44 = np.random.uniform(10000., 100000.)
Mcubic = M_cubic(C11, C12, C44, "Cii")

np.ndarray L_ortho(const double &C11, const double &C12, const double &C13, const double &C22, const double &C23, const double &C33, const double &C44, const double &C55, const double &C66, const string &conv)

Provides the elastic stiffness tensor for an orthotropic material. Arguments could be all the stiffness coefficients or the material parameter. For an orthotropic material the material parameters should be : Ex,Ey,Ez,nuxy,nuyz,nxz,Gxy,Gyz,Gxz.

The last argument must be set to “Cii” if the inputs are the stiffness coefficients or to “EnuG” if the inputs are the material parameters.

C11 = np.random.uniform(10000., 100000.)
C12 = np.random.uniform(10000., 100000.)
C13 = np.random.uniform(10000., 100000.)
C22 = np.random.uniform(10000., 100000.)
C23 = np.random.uniform(10000., 100000.)
C33 = np.random.uniform(10000., 100000.)
C44 = np.random.uniform(10000., 100000.)
C55 = np.random.uniform(10000., 100000.)
C66 = np.random.uniform(10000., 100000.)
Lortho = sim.L_ortho(C11, C12, C13, C22, C23, C33, C44, C55, C66, "Cii")

np.ndarray M_ortho(const double &C11, const double &C12, const double &C13, const double &C22, const double &C23, const double &C33, const double &C44, const double &C55, const double &C66, const string &conv)

Provides the elastic compliance tensor for an orthotropic material. Arguments could be all the stiffness coefficients or the material parameter. For an orthotropic material the material parameters should be : Ex,Ey,Ez,nuxy,nuyz,nxz,Gxy,Gyz,Gxz.

The last argument must be set to “Cii” if the inputs are the stiffness coefficients or to “EnuG” if the inputs are the material parameters.

C11 = np.random.uniform(10000., 100000.)
C12 = np.random.uniform(10000., 100000.)
C13 = np.random.uniform(10000., 100000.)
C22 = np.random.uniform(10000., 100000.)
C23 = np.random.uniform(10000., 100000.)
C33 = np.random.uniform(10000., 100000.)
C44 = np.random.uniform(10000., 100000.)
C55 = np.random.uniform(10000., 100000.)
C66 = np.random.uniform(10000., 100000.)
Mortho = sim.M_ortho(C11, C12, C13, C22, C23, C33, C44, C55, C66, "Cii")

np.ndarray L_isotrans(const double &EL, const double &ET, const double &nuTL, const double &nuTT, const double &GLT, const int &axis)

Provides the elastic stiffness tensor for an isotropic transverse material. Arguments are longitudinal Young modulus EL, transverse young modulus, Poisson’s ratio for loading along the longitudinal axis nuTL, Poisson’s ratio for loading along the transverse axis nuTT, shear modulus GLT and the axis of symmetry.

EL = np.random.uniform(10000., 100000.)
ET = np.random.uniform(10000., 100000.)
nuTL = np.random.uniform(0., 0.5)
nuTT = np.random.uniform(0., 0.5)
GLT = np.random.uniform(10000., 100000.)
axis = 1
Lisotrans = sim.L_isotrans(EL, ET, nuTL, nuTT, GLT, axis)

np.ndarray M_isotrans(const double &EL, const double &ET, const double &nuTL, const double &nuTT, const double &GLT, const int &axis)

Provides the elastic compliance tensor for an isotropic transverse material. Arguments are longitudinal Young modulus EL, transverse young modulus, Poisson’s ratio for loading along the longitudinal axis nuTL, Poisson’s ratio for loading along the transverse axis nuTT, shear modulus GLT and the axis of symmetry.

EL = np.random.uniform(10000., 100000.)
ET = np.random.uniform(10000., 100000.)
nuTL = np.random.uniform(0., 0.5)
nuTT = np.random.uniform(0., 0.5)
GLT = np.random.uniform(10000., 100000.)
axis = 1
Misotrans = sim.M_isotrans(EL, ET, nuTL, nuTT, GLT, axis)

np.ndarray H_iso(const double &etaB, const double &etaS)

Provides the viscoelastic tensor H, providing Bulk viscosity etaB and shear viscosity etaS. It actually returns :

\[\begin{split}H_iso = \left( \begin{array}{ccc} \eta_B & \eta_B & \eta_B & 0 & 0 & 0 \\ \eta_B & \eta_B & \eta_B & 0 & 0 & 0 \\ \eta_B & \eta_B & \eta_B & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 \end{array} \right)\end{split}\]

etaB = np.random.uniform(0., 1.)
etaS = np.random.uniform(0., 1.)
Hiso = sim.H_iso(etaB, etaS)