====== ORTHO3D ====== ===== Description ===== Orthotropic elastic constitutive law for solid elements at constant temperature. ==== The model ==== This law is used for mechanical analysis of orthotropic elasticity undergoing large strains. === Orthotropic elasticity === There are 9 independent parameters : $E_1$, $E_2$, $E_3$, $\nu_{12}$, $\nu_{13}$, $\nu_{23}$, $G_{12}$, $G_{13}$, $G_{23}$. The elastic compliance tensor is : \[D^e_{ijkl} = \begin{bmatrix} \frac{1}{E_1} & \frac{-\nu_{21}}{E_2} & \frac{-\nu_{31}}{E_3} & & & \\ \frac{-\nu_{12}}{E_1} & \frac{1}{E_2} & \frac{-\nu_{32}}{E_3} & & & \\ \frac{-\nu_{13}}{E_1} & \frac{-\nu_{23}}{E_2} & \frac{1}{E_3} & & & \\ & & & \frac{1}{2G_{12}} & & \\ & & & & \frac{1}{2G_{13}} & \\ & & & & & \frac{1}{2G_{23}}\\ \end{bmatrix}\] The compliance elastic tensor is symmetric (Love, 1944), thus the equalities blow must be satisfied : \[\frac{\nu_{21}}{E_2} = \frac{\nu_{12}}{E_1} \quad , \quad \frac{\nu_{31}}{E_3} = \frac{\nu_{13}}{E_1} \quad , \quad \frac{\nu_{23}}{E_2} = \frac{\nu_{32}}{E_3} \] Moreover, the strain energy function must be positive (i.e. the quadratic form is said positive definite) : \[\mathbf{U} = \frac{1}{2} \varepsilon_{ij}\varepsilon_{kl}C_{ijkl} > 0\] Thus, the condition below must be satisfied : \[1 - \nu_{12}\nu_{21} > 0 \quad ; \quad 1 - \nu_{13}\nu_{31} > 0 \quad ; \quad 1 - \nu_{23}\nu_{32} > 0\]\[1-\nu_{12}\nu_{23}\nu_{31}-\nu_{21}\nu_{13}\nu_{32} - \nu_{12}\nu_{21} - \nu_{13}\nu_{31} - \nu_{23}\nu_{32} > 0 \]\[E_1 > 0 \quad ; \quad E_2 > 0 \quad ; \quad E_3 > 0\]\[G_1 > 0 \quad ; \quad G_2 > 0 \quad ; \quad G_3 > 0\] By inverting the matrix, the elastic tensor is then : \[C^e_{ijkl} = \begin{bmatrix} \frac{1-\nu_{23}\nu_{32}}{E_2E_3det} & \frac{\nu_{21}+\nu_{31}\nu_{23}}{E_2E_3det} & \frac{\nu_{21}\nu_{32}+\nu_{31}}{E_2E_3det} & & & \\ \frac{\nu_{12}+\nu_{13}\nu_{32}}{E_1E_3det} & \frac{1-\nu_{13}\nu_{31}}{E_1E_3det} & \frac{\nu_{32}+\nu_{31}\nu_{12}}{E_1E_3det} & & & \\ \frac{\nu_{13}+\nu_{23}\nu_{12}}{E_1E_2det} & \frac{\nu_{23}+\nu_{21}\nu_{13}}{E_1E_2det} & \frac{1-\nu_{21}\nu_{12}}{E_1E_2det} & & & \\ & & & 2G_{12} & & \\ & & & & 2G_{13} & \\ & & & & & 2G_{23} \\ \end{bmatrix}\] with $det=\dfrac{1-\nu_{31}\nu_{13}-\nu_{21}\nu_{12}-\nu_{32}\nu_{23}-2\nu_{31}\nu_{12}\nu_{23}}{E_1E_2E_3}$ \\ === Cross-anisotropic elasticity === There are 5 independent parameters : ${E_{\parallel}}$, ${E_{\perp}}$, ${\nu_{\parallel\parallel}}$, ${\nu_{\parallel\perp}}$, ${G_{\parallel,\perp}}$.\\ From orthotropic elasticity let us consider ($e_1$,$e_2$) as the isotropic plane (bedding plane for sedimentary rock) and $e_3$ the normal to this plane. The subscripts ${\parallel}$ and $\perp$ indicates, respectively, the direction parallel to bedding and perpendicular to bedding. \[{E_1=E_2=E_{\parallel}}\quad , \quad {E_3=E_{\perp}}\] The elastic compliance tensor becomes : \[D^e_{ijkl} = \begin{bmatrix} \frac{1}{E_{\parallel}} & \frac{-\nu_{\parallel\perp}}{E{\parallel}} & \frac{-\nu_{\perp\parallel}}{E_{\perp}} & & & \\ \frac{-\nu_{\parallel\parallel}}{E_{\parallel}} & \frac{1}{E_{\parallel}} & \frac{-\nu_{\perp\parallel}}{E_{\perp}} & & & \\ \frac{-\nu_{\parallel\perp}}{E_{\parallel}} & \frac{-\nu_{\parallel\perp}}{E_{\parallel}} & \frac{1}{E_{\perp}} & & & \\ & & & \frac{1}{2G_{\parallel\parallel}} & & \\ & & & & \frac{1}{2G_{\parallel\perp}} & \\ & & & & & \frac{1}{2G_{\parallel\perp}}\\ \end{bmatrix}\] The compliance elastic tensor is symmetric (Love, 1944), thus the equalities blow must be satisfied : \[\frac{\nu_{\perp\parallel}}{E_{\perp}} = \frac{\nu_{\parallel\perp}}{E_{\parallel}}\] In the isotropic plane, the shear modulus is obtained as follow : \[G_{\parallel\parallel} = \frac{E_{\parallel}}{2(1+\nu_{\parallel\parallel})}\] Because of the symmetry of the stress and strain tensors : \[G_{\parallel\perp}=G_{\perp\parallel}\] === Rotation === The Hooke’s law is defined in the orthotropic axes for orthotropic elasticity. As a result, a change of the reference system is needed to obtain the stress in the global axes. In the purpose of estimating the stresses in the global axes, a relation taking into account this change in the reference system is proposed. This relation is (Cescotto, 1995) : \[\sigma_{ij} = R_{ik}R_{jl}\sigma'_{kl}\] where $R_{ij}$ is a component of the matrix of rotation, $\sigma_{ij}$ the stress in the orthotropic axes and $\sigma'_{ij}$ the stresses in the current configuration. \\ Generally, the matrix of rotation is characterized by the Euler’s angles. The positive direction of rotation is counter-clockwise.\\ {{ :laws:ortho3d.png?400 |}} The (X,Y,Z) space represents the current configuration (or global configuration) while the ($\underline{e_1}$,$\underline{e_2}$,$\underline{e_3}$) space represents the orthotropic configuration. To define the rotation, let consider that the cartesian system are equal. The rotation is decomposed in 3 steps and the definition of the angles will be: * The angle $\alpha$ defines a rotation around the axis Z, thus the axis $\underline{e_2}$ and $\underline{e_1}$ rotates from an angle of amplitude $\alpha$ in the (X,Y) plane. * The angle $\phi$ defines a rotation around the axis $\underline{e_2}$. * The angle $\theta$ defines a rotation around the axis $\underline{e_1}$.\\ The matrix which defines the rotation may be written : \[R = \begin{bmatrix} \cos\alpha\cos\phi & \sin\alpha\cos\phi & -\sin\phi \\ -\sin\alpha\cos\theta+\sin\theta\sin\phi\cos\alpha & \cos\alpha\cos\theta+\sin\theta\sin\phi\sin\alpha & \sin\theta\cos\phi \\ \sin\theta\sin\alpha +\cos\alpha\sin\phi\cos\theta & \sin\phi\sin\alpha\cos\theta-\sin\theta\cos\alpha & \cos\phi\cos\theta \end{bmatrix}\] ==== Files ==== Prepro : LORTHO.F \\ Lagamine: ORTHO3D.F ===== Availability ===== |Plane stress state| NO | |Plane strain state| NO | |Axisymmetric state| NO | |3D state| YES| |Generalized plane state| NO | ===== Input file ===== ==== Parameters defining the type of constitutive law ==== ^ Line 1 (2I5, 60A1)^^ |IL|Law number| |ITYPE| 605| |COMMENT| Any comment (up to 60 characters) that will be reproduced on the output listing| ==== Integer parameters ==== ^ Line 1 (2I5) ^^ |NINTV| $\neq$ 0 : Number of sub-steps used to integrate numerically the constitutive equation in a time step | |:::| = 0 : Number of sub-steps is based on the norm of the deformation increment and on DIV | |ISOL| = 0 : Use of total stresses in the constitutive law | |:::| $\neq$ 0 : Use of effective stresses in the constitutive law. See [[appendices:a8|Appendix 8]] | ==== Real parameters ==== ^ Line 1 (3G10.0) ^^ |ALPHA| Angle of rotation of the anisotropic axis around Z axis (see figure below) | |THETA| Angle of rotation of the anisotropic axis around $e_1$ axis (see figure below) | |PHI| Angle of rotation of the anisotropic axis around $e_2$ axis (see figure below) | ^ Line 2 (6G10.0) ^^ |E1| Elastic Young modulus E($e_1$) | |E2| Elastic Young modulus E($e_2$) | |E3| Elastic Young modulus E($e_3$) | |G12| Elastic shear modulus G($e_1e_2$) | |G13| Elastic shear modulus G($e_1e_3$) | |G23| Elastic shear modulus G($e_2e_3$) | ^ Line 3 (5G10.0) ^^ |ANU12| Poisson ration NU($e_1e_2$) | |ANU13| Poisson ration NU($e_1e_3$) | |ANU23| Poisson ration NU($e_2e_3$) | |RHO| Specific mass | |DIV| Size of sub-steps for computation of NINTV (only if NINTV = 0, Default value = 5.D-3 ) | ===== Stresses ===== ==== Number of stresses ==== 6 for 3D analysis \\ ==== Meaning ==== The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates. \\ For the 3-D state: |SIG(1)|$\sigma_{xx}$| |SIG(2)|$\sigma_{yy}$| |SIG(3)|$\sigma_{zz}$| |SIG(4)|$\sigma_{xy}$| |SIG(5)|$\sigma_{xz}$| |SIG(6)|$\sigma_{yz}$| ===== State variables ===== ==== Number of state variables ==== 0 (see [[laws:ela|ELA3D]]) \\ ==== List of state variables ==== |Q(1)| = 0 |