Preliminary mathematical results

Tensors

Second order tensors can be represented by matrices: \[ {\underset{\tilde{}}{\mathbf{F}}}= \begin{pmatrix} F_{11} & F_{12} & F_{13} \\ F_{21} & F_{22} & F_{23} \\ F_{31} & F_{32} & F_{33} \\ \end{pmatrix} \]

In TFEL/MFront, a tensor is stored as an array of values, as follows in \(3D\): \[ \underline{s}= \begin{pmatrix} s_{\,11}\quad s_{\,22}\quad s_{\,33}\quad s_{\,12}\quad s_{\,21}\quad s_{\,13}\quad s_{\,31}\quad s_{\,23}\quad s_{\,32} \end{pmatrix}^{T} \]

The trace of a tensor is defined by:

\[ \mathrm{tr}\left({\underset{\tilde{}}{\mathbf{F}}}\right)=\sum_{i=0}^{3}F_{ii} \]

The product of two tensors is defined by:

\[ \left({\underset{\tilde{}}{\mathbf{F}}}\,.\,{\underset{\tilde{}}{\mathbf{G}}}\right)_{ij}=\sum_{k=0}^{3}F_{ik}\,G_{kj}\neq\left({\underset{\tilde{}}{\mathbf{G}}}\,.\,{\underset{\tilde{}}{\mathbf{F}}}\right)_{ij} \]

The contracted product of two tensors defines a scalar product:

\[ {\underset{\tilde{}}{\mathbf{F}}}\,\colon\,{\underset{\tilde{}}{\mathbf{\pi}}}=\mathrm{tr}\left({\underset{\tilde{}}{\mathbf{F}}}\,.\,{\underset{\tilde{}}{\mathbf{\pi}}}\right)=\sum_{i=0}^{3}\sum_{j=0}^{3}F_{ij}\,\pi_{ji}=F_{ij}\,\pi_{ji} \]

Symmetric tensors

A symmetric tensor is stored as an array of values (vector notation), as follows in \(3D\):

\[ \underline{s}= \begin{pmatrix} s_{\,11}\quad s_{\,22}\quad s_{\,33}\quad \sqrt{2}\,s_{\,12}\quad \sqrt{2}\,s_{\,13}\quad \sqrt{2}\,s_{\,23} \end{pmatrix}^{T} \]

The contracted product of two symmetric tensors is the scalar product of their vector forms (hence the \(\sqrt{2}\)).

Diagonalisation of a symmetric tensor

Symmetric tensors are diagonalisable: \[ \underline{s}=\sum_{i=0}^{3}\lambda_{i}\,\underline{n}_{i} \]

• \(\lambda_{i}\) are the eigenvalues of the tensor.
• The eigentensors \(\underline{n}_{i}\) are orthogonal.

Isotropic functions of symmetric tensors

This decomposition allows the definition of isotropic functions of tensors (logarithm, exponential, square root, ..):

\[ \underline{\epsilon}^{\mathrm{to}}_{\mathrm{log}}=\sum_{i=0}^{3}f\left(\lambda_{i}\right)\,\underline{n}_{i} \]

Kinematics

Deformation of a body

• at a given time \(t\), a structure is characterised by its {} \(\mathcal{C}_{t}\);
• the initial configuration \(\mathcal{C}_{0}\) is often taken as the reference configuration
• the motion is described by a family of mappings \(\phi_{t}\), the {}, which associates a point \(\vec{X}\) in \(\mathcal{C}_{0}\) to its position \(\vec{x}\) in \(\mathcal{C}_{t}\): \[ \vec{x}=\phi_{t}\left(\vec{X}\right)=\vec{X}+\vec{u}_{t}\left(\vec{X}\right) \]
• \(\vec{u}_{t}\) is the {}

Deformation gradient

• Locally, the material is mechanically loaded if the deformation of the current point differs from the deformation of its neighbours
• The expansion of the deformation leads to: \[ \phi\left(\vec{X}+{{\mathrm{d}}}\,\vec{X}\right)-\phi\left(\vec{X}\right)={\displaystyle \frac{\displaystyle \partial \phi}{\displaystyle \partial \vec{X}}}\,.{{\mathrm{d}}}\,\vec{X}+\underbrace{\vec{X}\,.\,{{\displaystyle \frac{\displaystyle \partial^{2}\,\phi}{\displaystyle \partial\,\vec{X}^{2}}}}\,.{{\mathrm{d}}}\,\vec{X}+\ldots}_{\text{Higher order terms}} \]
• The deformation gradient \({\underset{\tilde{}}{\mathbf{F}}}\) is a tensor defined by: \[ {\underset{\tilde{}}{\mathbf{F}}}={\displaystyle \frac{\displaystyle \partial \phi}{\displaystyle \partial \vec{X}}}={\underset{\tilde{}}{\mathbf{I}}}+{\displaystyle \frac{\displaystyle \partial \vec{u}}{\displaystyle \partial \vec{X}}} \]
• Volume change: \[ J=\det\left({\displaystyle \frac{\displaystyle \partial \vec{x}}{\displaystyle \partial \vec{X}}}\right)=\det{{\underset{\tilde{}}{\mathbf{F}}}} \]

Higher order theories will not be considered in this course or in this paper.

Polar decomposition of \({\underset{\tilde{}}{\mathbf{F}}}\)

• Uniqueness and existence of a polar decomposition of \({\underset{\tilde{}}{\mathbf{F}}}\): \(\quad\quad{\underset{\tilde{}}{\mathbf{F}}}={\underset{\tilde{}}{\mathbf{R}}}\,.\underline{U}=\underline{V}\,.{\underset{\tilde{}}{\mathbf{R}}}\)
• \({\underset{\tilde{}}{\mathbf{R}}}\) is a rotation
• \(\underline{V}\) and \(\underline{U}\) are {}
• \(\underline{V}\) is the eulerian stretch tensor:
  • \(\left.\underline{V}\right|_{t}\) and \(\left.\underline{V}\right|_{t+\Delta\,t}\) are expressed in two different configurations:
  • \(\Delta\,\underline{V}=\left.\underline{V}\right|_{t+\Delta\,t}-\left.\underline{V}\right|_{t}\) is {} well defined
  • \(\underline{U}\) is the lagrangian stretch tensor:
    • expressed in the reference configuration
  • \(\Delta\,\underline{U}=\left.\underline{U}\right|_{t+\Delta\,t}-\left.\underline{U}\right|_{t}\) is {}
  • Volume change: \[ J=\det\,{\underset{\tilde{}}{\mathbf{F}}}=\det\,\underline{U} \]

Pure dilatation

• Pure dilatations correspond to diagonal deformation gradients: \[ {\underset{\tilde{}}{\mathbf{F}}}=\underline{U}= \begin{pmatrix} F_{11} & 0 & 0 \\ 0 & F_{22} & 0 \\ 0 & 0 & F_{33} \\ \end{pmatrix} \]
• Example of thermal expansion.

Rate of deformation

• velocity of a point: \[ \vec{v}={\displaystyle \frac{\displaystyle {{\mathrm{d}}}\vec{x}}{\displaystyle {{\mathrm{d}}}t}} \]
• gradient velocity of a point: \[ {\underset{\tilde{}}{\mathbf{L}}}={\displaystyle \frac{\displaystyle \partial \vec{\dot{x}}}{\displaystyle \partial \vec{x}}}={\displaystyle \frac{\displaystyle \partial \vec{\dot{x}}}{\displaystyle \partial \vec{X}}}\,.\,{\displaystyle \frac{\displaystyle \partial \vec{X}}{\displaystyle \partial \vec{x}}}=\left({\displaystyle \frac{\displaystyle {{\mathrm{d}}}}{\displaystyle {{\mathrm{d}}}t}}{\displaystyle \frac{\displaystyle \partial \vec{x}}{\displaystyle \partial \vec{X}}}\right)\,.\,{\displaystyle \frac{\displaystyle \partial \vec{X}}{\displaystyle \partial \vec{x}}}={\underset{\tilde{}}{\mathbf{{\displaystyle \frac{\displaystyle {{\mathrm{d}}}F}{\displaystyle {{\mathrm{d}}}t}}}}}\,.\,{\underset{\tilde{}}{\mathbf{F}}}^{-1} \]
• rate of deformation: \[ \underline{D}={{\displaystyle \frac{\displaystyle 1}{\displaystyle 2}}}\left({\underset{\tilde{}}{\mathbf{L}}}+{{\underset{\tilde{}}{\mathbf{L}}}^{\mathop{T}}}\right) \]
• rate of : \[ \dot{J} = \mathrm{tr}\left(\underline{D}\right) \]
• rotation rate: \[ {\underset{\tilde{}}{\mathbf{\omega}}}={{\displaystyle \frac{\displaystyle 1}{\displaystyle 2}}}\left({\underset{\tilde{}}{\mathbf{L}}}-{{\underset{\tilde{}}{\mathbf{L}}}^{\mathop{T}}}\right) \]

Rate of deformation in pure dilatations

• for pure dilatations, we have: \[ \underline{D}= \begin{pmatrix} \frac{\dot{F}_{11}}{F_{11}} & 0 & 0 \\ 0 & \frac{\dot{F}_{22}}{F_{22}} & 0 \\ 0 & 0 & \frac{\dot{F}_{33}}{F_{33}} \\ \end{pmatrix}= \begin{pmatrix} \frac{\dot{l}_{1}}{l_{1}\left(t\right)} & 0 & 0 \\ 0 & \frac{\dot{l}_{2}}{l_{2}\left(t\right)} & 0 \\ 0 & 0 & \frac{\dot{l}_{3}}{l_{3}\left(t\right)} \\ \end{pmatrix} \]
• \(\displaystyle\int_{0}^{t}D_{11}\,{{\mathrm{d}}}\,t=\int_{l_{0}}^{l\left(t\right)}{{\displaystyle \frac{\displaystyle {{\mathrm{d}}}\,l}{\displaystyle l}}}=\log\left({{\displaystyle \frac{\displaystyle l\left(t\right)}{\displaystyle l_{0}}}}\right)\)
• \(\displaystyle\log\left({{\displaystyle \frac{\displaystyle l\left(t\right)}{\displaystyle l_{0}}}}\right)\) is sometimes called the true strain, although the meaning of this is dubious, as discussed later
• The expression \(\displaystyle\int_{l_{0}}^{l\left(t\right)}{{\displaystyle \frac{\displaystyle {{\mathrm{d}}}\,l}{\displaystyle l}}}\) is sometimes used to justify an incremental framework of the mechanics of deformable body at finite strain (updated lagrangian formulations, hypoelasticity): it’s a common pitfall of the \(80's\) which is still present in major finite element solvers.

Mechanical equilibrium

Cauchy stress

• The stress state of a body at a given point is characterised by a second order tensor called the Cauchy stress: \(\quad\quad\quad\underline{\sigma}= \begin{pmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{pmatrix}\)
• The Cauchy stress satisfies:\ \(\quad\quad\quad{{\mathrm{d}}}\vec{T} = \underline{\sigma}\,.\,{{\mathrm{d}}}\vec{s}\)
  • \({{\mathrm{d}}}\vec{s}\): oriented unit surface
  • \({{\mathrm{d}}}\vec{T}\): traction acting on \({{\mathrm{d}}}\vec{s}\)
  • \({{\mathrm{d}}}\vec{s}\) is defined in the current configuration
• Example of a pressure \(p\) applied to the boundary of a body, \(\vec{n}\) being the outer normal to the boundary: \[ \underline{\sigma}\,.\,{{\mathrm{d}}}\vec{n}=-p \]

• Linear momentum conservation: \[ \mathop{\mathrm{div}}\underline{\sigma}+\vec{f}=\rho\,\vec{a}\equiv \left\{ \begin{aligned} {\displaystyle \frac{\displaystyle \partial \sigma_{11}}{\displaystyle \partial x_{1}}}+ {\displaystyle \frac{\displaystyle \partial \sigma_{21}}{\displaystyle \partial x_{2}}}+ {\displaystyle \frac{\displaystyle \partial \sigma_{31}}{\displaystyle \partial x_{3}}}+f_{1}=\rho\,a_{1}\\ {\displaystyle \frac{\displaystyle \partial \sigma_{12}}{\displaystyle \partial x_{1}}}+ {\displaystyle \frac{\displaystyle \partial \sigma_{22}}{\displaystyle \partial x_{2}}}+ {\displaystyle \frac{\displaystyle \partial \sigma_{32}}{\displaystyle \partial x_{3}}}+f_{2}=\rho\,a_{2} \\ {\displaystyle \frac{\displaystyle \partial \sigma_{13}}{\displaystyle \partial x_{1}}}+ {\displaystyle \frac{\displaystyle \partial \sigma_{23}}{\displaystyle \partial x_{2}}}+ {\displaystyle \frac{\displaystyle \partial \sigma_{33}}{\displaystyle \partial x_{3}}}+f_{3}=\rho\,a_{3} \\ \end{aligned} \right. \] with \(\vec{a}\) acceleration, \(\rho\) density, \(\vec{f}\) body forces (gravity)
• Equilibrium is expressed in the unknown {} configuration:
  • geometrical {} of continuum mechanics
  • Angular momentum conservation: without body momentum, the Cauchy stress is {} \[ \underline{\sigma}={\underline{\sigma}^{\mathop{T}}} \]

Infinitesimal perturbation theory

• {}: \[ {\underset{\tilde{}}{\mathbf{R}}}\approx{\underset{\tilde{}}{\mathbf{I}}} \]
• {}: \[ \underline{U}\approx\underline{I}+\underline{\epsilon}^{\mathrm{to}} \]
• \(\underline{\epsilon}^{\mathrm{to}}\) is the {}: \[ \epsilon^{\mathrm{to}}_{ji}={{\displaystyle \frac{\displaystyle 1}{\displaystyle 2}}}\left({\displaystyle \frac{\displaystyle \partial u_{i}}{\displaystyle \partial X_{j}}}+{\displaystyle \frac{\displaystyle \partial u_{j}}{\displaystyle \partial X_{i}}}\right) \quad\text{or}\quad \underline{\epsilon}^{\mathrm{to}}={{\displaystyle \frac{\displaystyle 1}{\displaystyle 2}}}\left(\mathop{\mathrm{Grad}\,\vec{u}}+{\mathop{\mathrm{Grad}\,\vec{u}}^{\mathop{T}}}\right) \]
• change of volume: \[ J=\det{\underset{\tilde{}}{\mathbf{F}}}\approx{}=\det\underline{U}=1+\mathrm{tr}\left(\underline{\epsilon}^{\mathrm{to}}\right) \]
• \(\Omega_{t}\) can be replaced by \(\Omega_{0}\):
  • no more geometrical non linearity
  • Equilibrium: \[ \mathop{\mathrm{Div}}\underline{\sigma}+\vec{f}=\rho_{0}\,\vec{a} \quad\text{and}\quad \underline{\sigma}={\underline{\sigma}^{\mathop{T}}} \]

Energy, strain measures, stress measures

Mechanical power

• Power of body forces: \[ \begin{aligned} \int_{\Omega}\vec{f}\,.\,\vec{v}\,{{\mathrm{d}}}\,v &=-\int_{\partial\,\Omega}\left(\underline{\sigma}\,.\,\vec{n}\right)\,.\,\vec{v}\,{{\mathrm{d}}}\,s+\int_{\Omega}\underline{\sigma}\,\colon\,{\underset{\tilde{}}{\mathbf{L}}}\,{{\mathrm{d}}}\,v\\ &=-\int_{\partial\,\Omega}\left(\underline{\sigma}\,.\,\vec{n}\right)\,.\,\vec{v}\,{{\mathrm{d}}}\,s+\int_{\Omega}\underline{\sigma}\,\colon\,{\underset{\tilde{}}{\mathbf{D}}}\,{{\mathrm{d}}}\,v\quad\text{(symmetry of \(\underline{\sigma}\))}\\ \end{aligned} \]
• Conservation of energy: \[ \underbrace{\int_{\Omega}\underline{\sigma}\,\colon\,{\underset{\tilde{}}{\mathbf{D}}}\,{{\mathrm{d}}}\,v}_{\text{inner forces power}} = \underbrace{\int_{\Omega}\vec{f}\,.\,\vec{v}\,{{\mathrm{d}}}\,v + \int_{\partial\,\Omega}\vec{t}\,.\,\vec{v}\,{{\mathrm{d}}}\,s}_{\text{external forces power}} \]

Strain measures

• A strain measure must satisfy the following hypotheses:
  • tends to \(\underline{\epsilon}^{\mathrm{to}}\) when the infinitesimal strain theory’s assumptions are satisfied;
  • is objective (filters finite body rotation)
  • is symmetric
    • many isotropic functions of \(\underline{U}\) satisfy those requirements:
    • Green-Lagrange strain: \[ \underline{\epsilon}^{\mathrm{to}}_{GL}={{\displaystyle \frac{\displaystyle 1}{\displaystyle 2}}}\left(\underline{U}^{2}-{\underset{\tilde{}}{\mathbf{I}}}\right)= {{\displaystyle \frac{\displaystyle 1}{\displaystyle 2}}}\left({\underset{\tilde{}}{\mathbf{F}}}^{T}\,.\,{\underset{\tilde{}}{\mathbf{F}}}-{\underset{\tilde{}}{\mathbf{I}}}\right) \]
  • Hencky strain: \[\underline{\epsilon}^{\mathrm{to}}_{\mathrm{log}}=\log\underline{U}=\displaystyle\sum_{i=0}^{3}\log\lambda_{i}\,\underline{n}_{i}\] where \(\lambda_{i}\) are the eigenvalues of \(\underline{U}\) and \(\underline{n}_{i}\) are its eigentensors
  • functions of \({\underset{\tilde{}}{\mathbf{V}}}\) will not be considered {}
• \(\underline{\epsilon}^{\mathrm{to}}\) is {} a strain measure, \({\underset{\tilde{}}{\mathbf{F}}}\) is {} a strain measure
• \({\underset{\tilde{}}{\mathbf{D}}}\) is {} the time derivative of a strain measure

Energetic conjugates (stress measures)

• Mechanical work: \[ \int_{\Omega}\underline{\sigma}\,\colon\,\underline{D}\,{{\mathrm{d}}}\,v= \int_{\Omega_{0}}\underline{\sigma}\,\colon\,\underline{D}\,J\,{{\mathrm{d}}}\,V_{0} \]
• For each strain measure \(\underline{\epsilon}^{\mathrm{to}}_{\star}\), one may define its dual stress \({\underset{\tilde{}}{\mathbf{T}}}_{\star}\): \[ J\,\underline{\sigma}\,\colon\,\underline{D}=\underline{\tau}\,\colon\,\underline{D}=\underline{\epsilon}^{\mathrm{to}}_{\star}\,\colon\,{\underset{\tilde{}}{\mathbf{T}}}_{\star} \]
• The tensor \(\underline{\tau}=J\,\underline{\sigma}\) is called the Kirchhoff stress.
• The dual of the Green-Lagrange strain is the second Piola-Kirchhoff stress \(\underline{S}\): \[ \underline{S}={\underset{\tilde{}}{\mathbf{F}}}^{-1}\,.\,\underline{\tau}\,.\,{\underset{\tilde{}}{\mathbf{F}}}^{-T} \quad\Leftrightarrow\quad\tau = {\underset{\tilde{}}{\mathbf{F}}}\,.\,\underline{S}\,.\,{\underset{\tilde{}}{\mathbf{F}}}^{T} \]
• There is no strain measure which is the dual of \(\underline{\tau}\), nor \(\underline{\sigma}\)

Choice of a stress/strain couple

• All strain measures are {}: there is no theoretical reason to prefer one strain measure over another.
• However, there are {} reasons to do so.

The Green-Lagrange strain

• The Green-Lagrange strain and its dual are:
  • easy to compute (no computational penalty);
  • there is no straight-forward relation between the Green-Lagrange strain and the change of volume - for small strain however: \[ J \approx 1+\mathrm{tr}\left(\underline{\epsilon}^{\mathrm{to}}_{\mathrm{GL}}\right) \] \begin{center}
• The Green-Lagrange strain framework is indeed well suited for extending behaviours identified in the infinitesimal perturbation theory to finite rotation*

The logarithmic strain framework

• There is a straight-forward relation between the Hencky strain strain and the change of volume: \[ J = \exp\left(\mathrm{tr}\left(\underline{\epsilon}^{\mathrm{to}}_{\mathrm{log}}\right)\right) \]
• For pure dilatations:
  • \(\underline{\epsilon}^{\mathrm{to}}_{\mathrm{log}}=\displaystyle\int_{0}^{t}\underline{D}\,{{\mathrm{d}}}\,t\)
  • \(\underline{T}_{\mathrm{log}}\) is equal to the Kirchhoff stress \(\underline{\tau}\), which is equal to the Cauchy stress \(\underline{\sigma}\) for isochoric deformation
  • However, the Hencky strain and its dual are costly to compute (see Miehe et al., 2005);

** The logarithmic strain framework is indeed well suited for plasticity and/or viscoplasticity **

Elasticity

• The mechanical work during the deformation process is (time integral of the mechanical power): \[ w\left(0,t\right)=\int_{0}^{t}J\,\underline{\sigma}\,\colon\,\underline{D}\,{{\mathrm{d}}}\,t =\int_{0}^{t}\underline{T}_{\star}\,\colon\,\underline{\epsilon}^{\mathrm{to}}_{\star}\,{{\mathrm{d}}}\,t =\int_{\left.\underline{\epsilon}^{\mathrm{to}}_{\star}\right|0}^{\left.\underline{\epsilon}^{\mathrm{to}}_{\star}\right|t}\underline{T}_{\star}\,\colon\,{{\mathrm{d}}}\,\underline{\epsilon}^{\mathrm{to}}_{\star} \]
• Elasticity assumes that the previous integral is path-independent, i.e. the mechanical work only depends on the current state of deformation and not the history of the deformation process
• The dual stress \(\underline{T}_{\star}\) then satisfies : \[ \underline{T}_{\star} = {\displaystyle \frac{\displaystyle \partial w}{\displaystyle \partial \underline{\epsilon}^{\mathrm{to}}_{\star}}} \]

Linear Elasticity

• Linear elasticity assumes a linear relationship between \(\underline{\epsilon}^{\mathrm{to}}_{\star}\) and \(\underline{T}_{\star}\): \[ \underline{T}_{\star} = \underset{=}{\mathbf{D}}\,\colon\,\underline{\epsilon}^{\mathrm{to}}_{\star}\quad\Leftrightarrow\quad w={{\displaystyle \frac{\displaystyle 1}{\displaystyle 2}}}\underline{\epsilon}^{\mathrm{to}}_{\star}\,\colon\,\underset{=}{\mathbf{D}}\,\colon\,\underline{\epsilon}^{\mathrm{to}}_{\star} \]
• The fourth order tensor \(\underset{=}{\mathbf{D}}\) satisfies: \[ \underset{=}{\mathbf{D}}={{\displaystyle \frac{\displaystyle \partial^{2}w}{\displaystyle \partial\underline{\epsilon}^{\mathrm{to}}_{\star}\partial\underline{\epsilon}^{\mathrm{to}}_{\star}}}} \]
• \(\underset{=}{\mathbf{D}}\) have the following properties:
  • \(\underset{=}{\mathbf{D}}\) has minor symmetries (\(\underline{T}_{\star}\) and \(\underline{\epsilon}^{\mathrm{to}}_{\star}\) are symmetric): \[ D_{ijkl}= D_{jikl}\quad\text{and}\quad D_{ijkl}= D_{ijlk} \]
  • \(\underset{=}{\mathbf{D}}\) has major symmetries (Schwarz theorem): \[ D_{ijkl}=D_{klij} \]

Isotropic elasticity

• For an isotropic material, only two independant material properties remains: \[ \underline{T}_{\star} = \lambda\,\left(\mathrm{tr}\left(\underline{\epsilon}^{\mathrm{to}}_{\star}\right)\right)\,\underline{I}+2\,\mu\,\underline{\epsilon}^{\mathrm{to}}_{\star} \]
• \(\lambda\) and \(\mu\) are called the Lamé coefficients
• \(\mu\) is also called the shear modulus
• In the infinitesimal perturbation hypothesis, this is the Hooke law.
• In the Green-Lagrange strain framework, this is the Saint-Venant Kirchhoff law.
• In the logarithmic strain framework, this is the Hencky-Biot law.

Bulk modulus

• Separate change of volume from deviatoric part: \[ \begin{aligned} \underline{T}_{\star}&= \lambda\,\left(\mathrm{tr}\left(\underline{\epsilon}^{\mathrm{to}}_{\star}\right)\right)\,\underline{I}+2\,\mu\,\underline{\epsilon}^{\mathrm{to}}_{\star} \\ &= \left(\lambda+{{\displaystyle \frac{\displaystyle 2\,\mu}{\displaystyle 3}}}\right)\,\underbrace{\left(\mathrm{tr}\left(\underline{\epsilon}^{\mathrm{to}}_{\star}\right)\right)\,\underline{I}}_{\text{volume change}}+2\,\mu\,\underbrace{\left(\underline{\epsilon}^{\mathrm{to}}_{\star}-{{\displaystyle \frac{\displaystyle \mathrm{tr}\left(\underline{\epsilon}^{\mathrm{to}}_{\star}\right)}{\displaystyle 3}}}\,\underline{I}\right)}_{\text{deviator}} \\ &= K\,\left(\mathrm{tr}\left(\underline{\epsilon}^{\mathrm{to}}_{\star}\right)\right)\,\underline{I}+2\,\mu\,\left(\underline{\epsilon}^{\mathrm{to}}_{\star}-{{\displaystyle \frac{\displaystyle \mathrm{tr}\left(\underline{\epsilon}^{\mathrm{to}}_{\star}\right)}{\displaystyle 3}}}\,\underline{I}\right) \\ \end{aligned} \]
• \(K=\lambda+{{\displaystyle \frac{\displaystyle 2\,\mu}{\displaystyle 3}}}\) is called the bulk modulus

Young modulus and Poisson coefficient

• consider an uniaxial tensile state along \(11\): \[ \begin{aligned} \underline{\epsilon}^{\mathrm{to}}_{\star}& = \begin{pmatrix} \epsilon^{\mathrm{to}}_{\star\,11} & \epsilon^{\mathrm{to}}_{\star\,22} & \epsilon^{\mathrm{to}}_{\star\,33} & 0 & 0 & 0 \end{pmatrix}^{T}\\ \underline{T}_{\star}&= \begin{pmatrix} T_{\star\,11} & 0 & 0 & 0 & 0 & 0 \end{pmatrix}^{T}\\ \end{aligned} \]
• The Young modulus \(E\) is defined by: \[ T_{\star\,11} = E\,\epsilon^{\mathrm{to}}_{\star\,11} \]
• The Poisson ratio \(\nu\) is defined by: \[ \epsilon^{\mathrm{to}}_{\star\,22} = -\nu\,\epsilon^{\mathrm{to}}_{\star\,11} \]
• \(\lambda={{\displaystyle \frac{\displaystyle E\,\nu}{\displaystyle \left(1+\nu\right)\,\left(1-2\,\nu\right)}}}\quad\mu={{\displaystyle \frac{\displaystyle E}{\displaystyle 2\,\left(1+\nu\right)}}}\quad K={{\displaystyle \frac{\displaystyle E}{\displaystyle 3\,\left(1-2\,\nu\right)}}}\)

Orthotropy

• An orthotropic material introduces a preferential material frame: \[ \underset{=}{\mathbf{D}}= \begin{pmatrix} D_{1111} & D_{1122} & D_{1133} & 0 & 0 & 0 \\ D_{1122} & D_{2222} & D_{2233} & 0 & 0 & 0 \\ D_{1133} & D_{2233} & D_{3333} & 0 & 0 & 0 \\ 0 & 0 & 0 & D_{1212} & 0 & 0 \\ 0 & 0 & 0 & 0 & D_{1313} & 0 \\ 0 & 0 & 0 & 0 & 0 & D_{2323} \\ \end{pmatrix} \]
• \(9\) independent coefficients

Thermo-elasticty

Thermal strain

• The total strain is splitted into an elastic part and a thermal part: \[ \underline{\epsilon}^{\mathrm{to}}_{\star}= \underline{\epsilon}^{\mathrm{el}}_{\star}+ \underline{\epsilon}^{\mathrm{th}}_{\star} \]
• The elastic part \(\underline{\epsilon}^{\mathrm{el}}_{\star}\) defines the stresses \(\underline{T}_{\star}\) through the Hooke law: \[ \underline{T}_{\star}=\underset{=}{\mathbf{D}}\,\colon\,\underline{\epsilon}^{\mathrm{el}}_{\star} \]

Isotropic thermal expansion

• The thermal expansion is given by: \[ {{\displaystyle \frac{\displaystyle \Delta\,l}{\displaystyle l_{T^{\alpha}}}}}={{\displaystyle \frac{\displaystyle l_{T}-l_{T^{\alpha}}}{\displaystyle l_{T^{\alpha}}}}}=\alpha\left(T\right)\,\left(T-T^{\alpha}\right) \]
• If the reference temperature \(T^{\alpha}\) for the thermal expansion is different than the reference temperature \(T^{i}\) of the geometry: \[ {{\displaystyle \frac{\displaystyle \Delta\,l}{\displaystyle l_{T^{i}}}}} = {{\displaystyle \frac{\displaystyle 1}{\displaystyle 1+\alpha\left(T^{i}\right)\,\left(T^{i}-T^{\alpha}\right)}}}\,\left[\alpha\left(T\right)\,\left(T-T^{\alpha}\right)-\alpha\left(T^{i}\right)\,\left(T^{i}-T^{\alpha}\right)\right] \]

Isotropic damage

Visco-plasticity

A first approach to viscoplastic behaviour

• The total strain is splitted into an elastic part and a viscoplastic part: \[ \underline{\epsilon}^{\mathrm{to}}_{\star}= \underline{\epsilon}^{\mathrm{el}}_{\star}+ \underline{\epsilon}^{\mathrm{vis}}_{\star} \]
• The plastic flow is generally isochoric: \[ \mathrm{tr}\left(\underline{\epsilon}^{\mathrm{vis}}_{\star}\right)=0 \]
• Without internal state, the mechanical dissipation associated with plasticity is: \[ \underline{T}_{\star}\,\colon\,\underline{\dot{\epsilon}}^{\mathrm{tox}}_{\star}= \underbrace{\underline{T}_{\star}\,\colon\,\underline{\dot{\epsilon}}^{\mathrm{el}}_{\star}}_{\text{stored reversibly}}+ \underbrace{\underline{T}_{\star}\,\colon\,\underline{\dot{\epsilon}}^{\mathrm{vis}}_{\star}}_{\text{dissipated}} \]
• The expression dissipated power can be rewritten using the deviator of the stress \(\underline{s}_{\star}\): \[ \underline{T}_{\star}\,\colon\,\underline{\dot{\epsilon}}^{\mathrm{vis}}_{\star}= \underline{s}_{\star}\,\colon\,\underline{\dot{\epsilon}}^{\mathrm{vis}}_{\star} \quad\text{with}\quad\underline{s}_{\star}=\underline{T}_{\star}-{{\displaystyle \frac{\displaystyle 1}{\displaystyle 3}}}\,\mathrm{tr}\left(\underline{T}_{\star}\right)\,\underline{I} \]
• The dissipation is maximal if the \(\underline{\dot{\epsilon}}^{\mathrm{vis}}_{\star}\) is colinear with \(\underline{s}_{\star}\).

The Von Mises stress

• The material is now assumed {}
• A convenient isotropic norm for deviatoric stress tensor is the Von Mises norm: \[ T^{eq}_{\star}=\sqrt{{{\displaystyle \frac{\displaystyle 3}{\displaystyle 2}}}\underline{s}_{\star}\,\colon\,\underline{s}_{\star}} \]
• The \({{\displaystyle \frac{\displaystyle 3}{\displaystyle 2}}}\) factor is here so that in uniaxial tensile tests: \[ T^{eq}_{\star}=\left|T_{xx}\right| \]
• The Von Mises norm is one the three invariants of the stress (the other ones are the pressure and the determinant)
• In term of eigen values: \[ T^{eq}_{\star}=\sqrt{{{\displaystyle \frac{\displaystyle 1}{\displaystyle 2}}}\left[\left(T_{1}-T_{2}\right)^{2}+\left(T_{1}-T_{3}\right)^{2}+\left(T_{2}-T_{3}\right)^{2}\right]} \]

The normal tensor

• The equation: \[ T^{eq}_{\star}=\text{Cste} \] defines a sphere in the deviatoric space
• The normal to this surface is: \[ \underline{n}_{\star}={\displaystyle \frac{\displaystyle \partial T^{eq}_{\star}}{\displaystyle \partial \underline{T}_{\star}}}={{\displaystyle \frac{\displaystyle 3\,\underline{s}_{\star}}{\displaystyle 2\,T^{eq}_{\star}}}} \]
• \(\underline{n}_{\star}\colon\underline{n}_{\star}={{\displaystyle \frac{\displaystyle 3}{\displaystyle 2}}}\)
• The normal is colinear to \(\underline{s}_{\star}\), thus an isochoric viscoplastic flow of the form: \[ \underline{\dot{\epsilon}}^{\mathrm{vis}}_{\star}=f\left(\underline{T}_{\star}\right)\underline{n}_{\star} \] would maximise the mechanical dissipation.
• In uniaxial tensile tests: \[ \underline{n}_{\star}= \begin{pmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} & 0 & 0 & 0 \end{pmatrix}^{T} \]

The Norton behaviour

• The viscoplastic flow is: \(\underline{\dot{\epsilon}}^{\mathrm{vis}}_{\star}=f\left(\underline{T}_{\star}\right)\underline{n}_{\star}\)
• The material being isotropic, \(f\) must be a function of the invariants of the stresses: the pressure, the Von Mises stress, the determinant.
• Experimentally, viscoplastic behaviour is found to be pressure insensitive.
• The effect of the third invariant is neglected in general.
• Thus, a simple viscoplastic model for an isotropic incompressible material is: \[ \underline{\dot{\epsilon}}^{\mathrm{vis}}_{\star}=f\left(T^{eq}_{\star}\right)\underline{n}_{\star} \]
• Restrictions:
  • \(f\) must be positive for the dissipation to be positive
  • \(f\left(\underline{0}\right)\) must be null
  • The Norton behaviour correspond to a power function: \[ \underline{\dot{\epsilon}}^{\mathrm{vis}}_{\star}=A\,\left(T^{eq}_{\star}\right)^{n}\underline{n}_{\star}=\dot{\varepsilon}^{0}\left({{\displaystyle \frac{\displaystyle T^{eq}_{\star}}{\displaystyle T^{0}}}}\right)^{n}\underline{n}_{\star} \]

Equivalent viscoplastic strain

• A convenient choice for the viscoplastic strain rate norm is: \[ \dot{p}=f\left(T^{eq}_{\star}\right)=\sqrt{{{\displaystyle \frac{\displaystyle 2}{\displaystyle 3}}}\,\underline{\dot{\epsilon}}^{\mathrm{vis}}_{\star}\,\colon\,\underline{\dot{\epsilon}}^{\mathrm{vis}}_{\star}} \]
• In uniaxial tensile tests: \[ \dot{p} = \left|\left(\dot{\epsilon}^{\mathrm{vis}}_{\star}\right)_{xx}\right| \]
• The equivalent viscoplastic strain is defined by: \[ p=\int_{0}^{t}\dot{p}\,{{\mathrm{d}}}\,t \]
• This quantity is a convenient measure of the viscoplastic history of the material and is widely used as a damage criterium.

Dissipation potential

• The Norton behaviour can be expressed as: \[ \underline{\dot{\epsilon}}^{\mathrm{vis}}_{\star}={\displaystyle \frac{\displaystyle \partial \Phi^{\star}}{\displaystyle \partial \underline{T}_{\star}}} \quad\text{with}\quad \Phi^{\star}\left(T^{eq}_{\star}\right)={{\displaystyle \frac{\displaystyle T^{0}\dot{\varepsilon}^{0}}{\displaystyle n+1}}}\left({{\displaystyle \frac{\displaystyle T^{eq}_{\star}}{\displaystyle T^{0}}}}\right)^{n+1} \]
• \(\Phi^{\star}\) is called the dissipation potential
• Other expressions of the dissipation potential defines used the define other viscoplastic behaviours.
• The resulting viscoplastic behaviours will lead to a positive dissipation if \(\Phi^{\star}\) is {} and {} at zero.
• The introduction of dissipation potentials is the departure of a theorical developments which can ease the formulation of mechanical behaviours:
  • thermodynamical consistent behaviours
  • numerically efficient behaviours

Thermodynamics

First principle

The Finite Element Method

Principle of virtual power

• Let \(\partial_{u}\Omega\) the boundary part where displacements are prescribed
• \(\partial_{t}\Omega=\partial\Omega\setminus\partial_{u}\Omega\) is the boundary part where tractions are prescribed
• Let \(\vec{v}^{\star}\) be a vector field compatible with prescribed displacement and \(\delta\,\vec{v}^{\star}=\vec{v}^{\star}-\vec{v}\), then: \[ \underbrace{\int_{\Omega}\underline{\sigma}\,\colon\,\delta\,{\underset{\tilde{}}{\mathbf{D}}}^{\star}\,{{\mathrm{d}}}\,v}_{\text{virtual inner forces power}} = \underbrace{\int_{\Omega}\vec{f}\,.\,\delta\,\vec{v}^{\star}\,{{\mathrm{d}}}\,v + \int_{\partial\,\Omega_{t}}\vec{t}\,.\,\delta\,\vec{v}^{\star}\,{{\mathrm{d}}}\,s}_{\text{virtual external forces power}} \]
• this is the principle of virtual power which is the basis of the Finite Element Method (FEM)

Finite element method

• The principle of virtual power is used to find the best approximation of the solution on a finite space.
• Finite elements are a widely used way of defining such a finite space by discretizing the real geometry by subdomains called finite elements:
  • Given values at specified points of the finite element (the nodes), the function value is approximated by interpolation functions.

Resolution

• Mechanical equilibrium: find\(\Delta\mathbb{\vec{u}}\) such as: \[ \small \mathbb{\vec{R}}\left(\Delta\mathbb{\vec{u}}\right)=\mathbb{\vec{O}}\quad\text{ avec }\quad\mathbb{\vec{R}}\left(\Delta\mathbb{\vec{u}}\right)=\mathbb{\vec{F}}_{i}\left(\Delta\mathbb{\vec{u}}\right)-\mathbb{\vec{F}}_{e} \]

• element contribution to inner forces: \[ \small \begin{aligned} \mathbb{\vec{F}}_{i}^{e}&=\int_{V^{e}}\underline{\sigma}_{t+\Delta t}\left(\Delta\,\underline{\epsilon}^{\mathrm{to}},\Delta\, t\right)\colon\underline{B}\;{{\mathrm{d}}}V \\ &= \sum_{i=1}^{N^{G}} \left(\underline{\sigma}_{t+\Delta\,t}\left(\Delta\underline{\epsilon}^{\mathrm{to}}\left(\vec{\eta}_{i}\right),\Delta\, t\right)\colon\underset{=}{\mathbf{B}}\left(\vec{\eta}_{i}\right)\right)w_{i} \end{aligned} \] where \(\underline{B}\) gives the relationship between \(\Delta\,\underline{\epsilon}^{\mathrm{to}}\) and \(\Delta\mathbb{\vec{u}}\)

  \[ \mathbb{\vec{F}}_{i}^{e}= \sum_{i=1}^{N^{G}} \left(\underline{\sigma}_{t+\Delta\,t}\left(\Delta\underline{\epsilon}^{\mathrm{to}}\left(\vec{\eta}_{i}\right),\Delta\, t\right)\colon\underset{=}{\mathbf{B}}\left(\vec{\eta}_{i}\right)\right)w_{i} \]

Resolution using the algorithm

Mechanical behaviours