% Nonlinear mechanical behaviours of materials % Thomas Helfer % 2017 \newcommand{\tenseur}[1]{\underline{#1}} \newcommand{\tenseurq}[1]{\underset{=}{\mathbf{#1}}} \newcommand{\tns}[1]{{\underset{\tilde{}}{\mathbf{#1}}}} \newcommand{\transpose}[1]{{#1^{\mathop{T}}}} \newcommand{\paren}[1]{\left(#1\right)} \newcommand{\trace}[1]{\mathrm{tr}\left(#1\right)} \newcommand{\tsigma}{\underline{\sigma}} \newcommand{\Frac}[2]{{{\displaystyle \frac{\displaystyle #1}{\displaystyle #2}}}} \newcommand{\deriv}[2]{{\displaystyle \frac{\displaystyle \partial #1}{\displaystyle \partial #2}}} \newcommand{\sderiv}[2]{{\displaystyle \frac{\displaystyle \partial^{2} #1}{\displaystyle \partial #2^{2}}}} \newcommand{\dtot}{{{\mathrm{d}}}} \newcommand{\derivtot}[2]{{\displaystyle \frac{\displaystyle \dtot #1}{\displaystyle \dtot #2}}} \newcommand{\grad}[1]{{\displaystyle \overset{\longrightarrow}{\nabla} #1}} \newcommand{\Grad}[1]{\mathop{\mathrm{Grad}\,#1}} \newcommand{\diver}[1]{{\displaystyle \vec{\nabla} . #1}} \newcommand{\divergence}{\mathop{\mathrm{div}}} \newcommand{\Divergence}{\mathop{\mathrm{Div}}} \newcommand{\bts}[1]{\left.#1\right|_{t}} \newcommand{\mts}[1]{\left.#1\right|_{t+\theta\,\Delta\,t}} \newcommand{\ets}[1]{\left.#1\right|_{t+\Delta\,t}} \newcommand{\epsilonto}{\epsilon^{\mathrm{to}}} \newcommand{\tepsilonto}{\underline{\epsilon}^{\mathrm{to}}} \newcommand{\tdepsilonto}{\underline{\dot{\epsilon}}^{\mathrm{tox}}} \newcommand{\tepsilonel}{\underline{\epsilon}^{\mathrm{el}}} \newcommand{\tdepsilonel}{\underline{\dot{\epsilon}}^{\mathrm{el}}} \newcommand{\tepsilonth}{\underline{\epsilon}^{\mathrm{th}}} \newcommand{\epsilonvis}{\epsilon^{\mathrm{vis}}} \newcommand{\tepsilonvis}{\underline{\epsilon}^{\mathrm{vis}}} \newcommand{\depsilonvis}{\dot{\epsilon}^{\mathrm{vis}}} \newcommand{\tdepsilonvis}{\underline{\dot{\epsilon}}^{\mathrm{vis}}} \newcommand{\tepsilonp}{\underline{\epsilon}^{\mathrm{p}}} \newcommand{\tdepsilonp}{\underline{\dot{\epsilon}}^{\mathrm{p}}} \newcommand{\energieinterne}{e} \newcommand{\energielibre}{\Psi} \newcommand{\energielibreel}{{\Psi}^{el}} \newcommand{\energielibreine}{{\Psi}^{inél}} \newcommand{\energielibreduale}{\Psi^{\star}} \newcommand{\potentieldissip}{\Phi} \newcommand{\potentieldissipdual}{\Phi^{\star}} \newcommand{\discret}[1]{\mathbb{#1}} \newcommand{\residuEF}{\discret{\vec{R}}} \newcommand{\forceintEF}{\discret{\vec{F}}_{i}} \newcommand{\forceextEF}{\discret{\vec{F}}_{e}} \newcommand{\forceintElem}{\discret{\vec{F}}_{i}^{e}} \newcommand{\champEF}{\discret{\vec{v}}^{h}} # Preliminary mathematical results ## Tensors Second order tensors can be represented by matrices: \[ \tns{F}= \begin{pmatrix} F_{11} & F_{12} & F_{13} \\ F_{21} & F_{22} & F_{23} \\ F_{31} & F_{32} & F_{33} \\ \end{pmatrix} \] In `TFEL/MFront`, a tensor is stored as an array of values, as follows in \(3D\): \[ \tenseur{s}= \begin{pmatrix} s_{\,11}\quad s_{\,22}\quad s_{\,33}\quad s_{\,12}\quad s_{\,21}\quad s_{\,13}\quad s_{\,31}\quad s_{\,23}\quad s_{\,32} \end{pmatrix}^{T} \] The trace of a tensor is defined by: \[ \trace{\tns{F}}=\sum_{i=0}^{3}F_{ii} \] The products of two tensors is defined by: \[ \paren{\tns{F}\,.\,\tns{G}}_{ij}=\sum_{k=0}^{3}F_{ik}\,G_{kj}\neq\paren{\tns{G}\,.\,\tns{F}}_{ij} \] The contracted product of two tensors defines a scalar product: \[ \tns{F}\,\colon\,\tns{\pi}=\trace{\tns{F}\,.\,\tns{\pi}}=\sum_{i=0}^{3}\sum_{j=0}^{3}F_{ij}\,\pi_{ji}=F_{ij}\,\pi_{ji} \] ## Symmetric tensors A symmetric tensor is stored as an array of values (vector notation), as follows in \(3D\): \[ \tenseur{s}= \begin{pmatrix} s_{\,11}\quad s_{\,22}\quad s_{\,33}\quad \sqrt{2}\,s_{\,12}\quad \sqrt{2}\,s_{\,13}\quad \sqrt{2}\,s_{\,23} \end{pmatrix}^{T} \] 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: \[ \tenseur{s}=\sum_{i=0}^{3}\lambda_{i}\,\tenseur{n}_{i} \] - \(\lambda_{i}\) are the eigenvalues of the tensor. - The eigentensors \(\tenseur{n}_{i}\) are orthogonals. ### Isotropic functions of symmetric tensors This decomposition allows the definition of isotropic functions of tensors (logarithm, exponential, square root, ..): \[ \tepsilonto_{\mathrm{log}}=\sum_{i=0}^{3}f\paren{\lambda_{i}}\,\tenseur{n}_{i} \] # Kinematics ## Deformation of a body ![Initial and deformed configurations](img/Configuration.png "Initial and deformed configurations"){width=75%} - at a given time \(t\), a structure is characterised by its {\em configuration} \(\mathcal{C}_{t}\); - 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 {\bf deformation}, which associates a point \(\vec{X}\) in \(\mathcal{C}_{0}\) to its position \(\vec{x}\) in \(\mathcal{C}_{t}\): \[ \vec{x}=\phi_{t}\paren{\vec{X}}=\vec{X}+\vec{u}_{t}\paren{\vec{X}} \] - \(\vec{u}_{t}\) is the {\bf displacement field} ### Deformation gradient - Locally, the material is mechanically loaded if the deformation of the current point differs from the deformation of its neighbours - The \nom{Taylor} expansion of the deformation leads to: \[ \phi\paren{\vec{X}+\dtot\,\vec{X}}-\phi\paren{\vec{X}}=\deriv{\phi}{\vec{X}}\,.\dtot\,\vec{X}+\underbrace{\vec{X}\,.\,\Frac{\partial^{2}\,\phi}{\partial\,\vec{X}^{2}}\,.\dtot\,\vec{X}+\ldots}_{\text{Higher order terms}} \] - The deformation gradient \(\tns{F}\) is a tensor defined by: \[ \tns{F}=\deriv{\phi}{\vec{X}}=\tns{I}+\deriv{\vec{u}}{\vec{X}} \] - Volume change: \[ J=\det\paren{\deriv{\vec{x}}{\vec{X}}}=\det{\tns{F}} \] Higher order theories will not be considered in this course in this paper. ### Polar decomposition of $\tns{F}$ ![Polar decomposition (Wikipedia)](img/Polar_decomposition_of_F.png "Polar decomposition (Wikipedia)"){width=75%} - Unicity and existence of a polar decomposition of \(\tns{F}\): \( \quad\quad\tns{F}=\tns{R}\,.\tenseur{U}=\tenseur{V}\,.\tns{R} \) - \(\tns{R}\) is a rotation - \(\tenseur{V}\) and \(\tenseur{U}\) are {\bf symmetric tensors} - \(\tenseur{V}\) is the eulerian stretch tensor: - \(\bts{\tenseur{V}}\) and \(\ets{\tenseur{V}}\) are expressed in two different configuration: - \(\Delta\,\tenseur{V}=\ets{\tenseur{V}}-\bts{\tenseur{V}}\) is {\bf not} well defined - \(\tenseur{U}\) is the lagrangian stretch tensor: - expressed in the reference configuration - \(\Delta\,\tenseur{U}=\ets{\tenseur{U}}-\bts{\tenseur{U}}\) is {\bf well defined} - Volume change: \[ J=\det\,\tns{F}=\det\,\tenseur{U} \] ### Pure dilatation - Pure dilatations correspond to diagonal deformation gradients: \[ \tns{F}=\tenseur{U}= \begin{pmatrix} F_{11} & 0 & 0 \\ 0 & F_{22} & 0 \\ 0 & 0 & F_{33} \\ \end{pmatrix} \] - Example of thermal expansion. ### Rate of deformation - velocity of a point: \[ \vec{v}=\derivtot{\vec{x}}{t} \] - gradient velocity of a point: \[ \tns{L}=\deriv{\vec{\dot{x}}}{\vec{x}}=\deriv{\vec{\dot{x}}}{\vec{X}}\,.\,\deriv{\vec{X}}{\vec{x}}=\paren{\derivtot{}{t}\deriv{\vec{x}}{\vec{X}}}\,.\,\deriv{\vec{X}}{\vec{x}}=\tns{\derivtot{F}{t}}\,.\,\tns{F}^{-1} \] - rate of deformation: \[ \tenseur{D}=\Frac{1}{2}\left(\tns{L}+\transpose{\tns{L}}\right) \] - rate of : \[ \dot{J} = \trace{\tenseur{D}} \] - rotation rate: \[ \tns{\omega}=\Frac{1}{2}\left(\tns{L}-\transpose{\tns{L}}\right) \] ### Rate of deformation in pure dilatations - for pure dilatations, we have: \[ \tenseur{D}= \begin{pmatrix} \frac{\dot{F}_{11}}{F_{11}} & 0 & 0 \\ 0 & \frac{\dot{F}_{22}}{F_{22}} & 0 \\ 0 & 0 & \frac{\dot{F}_{33}}{F_{33}} \\ \end{pmatrix}= \begin{pmatrix} \frac{\dot{l}_{1}}{l_{1}\paren{t}} & 0 & 0 \\ 0 & \frac{\dot{l}_{2}}{l_{2}\paren{t}} & 0 \\ 0 & 0 & \frac{\dot{l}_{3}}{l_{3}\paren{t}} \\ \end{pmatrix} \] - \(\displaystyle\int_{0}^{t}D_{11}\,\dtot\,t=\int_{l_{0}}^{l\paren{t}}\Frac{\dtot\,l}{l}=\log\paren{\Frac{l\paren{t}}{l_{0}}}\) - \(\displaystyle\log\paren{\Frac{l\paren{t}}{l_{0}}}\) is sometimes calls the true strain, altough the meaning of this is dubious, as discussed later - The expression \(\displaystyle\int_{l_{0}}^{l\paren{t}}\Frac{\dtot\,l}{l}\) 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 ![Components of the Cauchy stress (Wikipedia)](img/CauchyStressComponents.png "Components of the Cauchy stress (Wikipedia)"){width=75%} - The stress state of a body at a given point is characterised by a second order tensor called the Cauchy stress: \(\quad\quad\quad\tsigma= \begin{pmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{pmatrix} \) - The Cauchy stress satisfies:\\ \( \quad\quad\quad\dtot\vec{T} = \tsigma\,.\,\dtot\vec{s} \) - \(\dtot\vec{s}\): oriented unit surface - \(\dtot\vec{T}\): traction acting on \(\dtot\vec{s}\) - \(\dtot\vec{s}\) is defined of the current configuration - Example of a pressure \(p\) applied to the boundary of a body, \(\vec{n}\) being the outer normal to the boundary: \[ \tsigma\,.\,\dtot\vec{n}=-p \] \paragraph{Mechanical equilibrium} - Linear momentum conservation: \[ \divergence\tsigma+\vec{f}=\rho\,\vec{a}\equiv \left\{ \begin{aligned} \deriv{\sigma_{11}}{x_{1}}+ \deriv{\sigma_{21}}{x_{2}}+ \deriv{\sigma_{31}}{x_{3}}+f_{1}=\rho\,a_{1}\\ \deriv{\sigma_{12}}{x_{1}}+ \deriv{\sigma_{22}}{x_{2}}+ \deriv{\sigma_{32}}{x_{3}}+f_{2}=\rho\,a_{2} \\ \deriv{\sigma_{13}}{x_{1}}+ \deriv{\sigma_{23}}{x_{2}}+ \deriv{\sigma_{33}}{x_{3}}+f_{3}=\rho\,a_{3} \\ \end{aligned} \right. \] with \(\vec{a}\) acceleration, \(\rho\) density, \(\vec{f}\) body forces (gravity) - Equilibrium is expressed in the unknown {\bf deformed} configuration: - geometrical {\bf non linearity} of continuum mechanics - Angular momentum conservation: without body momentum, the Cauchy stress is {\bf symmetric} \[ \tsigma=\transpose{\tsigma} \] ## Infinitesimal perturbation theory - {\bf no rotation}: \[ \tns{R}\approx\tns{I} \] - {\bf small strain (first order perturbation)}: \[ \tenseur{U}\approx\tenseur{I}+\tepsilonto \] - \(\tepsilonto\) is the {\bf linearised strain tensor}: \[ \epsilonto_{ji}=\Frac{1}{2}\paren{\deriv{u_{i}}{X_{j}}+\deriv{u_{j}}{X_{i}}} \quad\text{or}\quad \tepsilonto=\Frac{1}{2}\paren{\Grad{\vec{u}}+\transpose{\Grad{\vec{u}}}} \] - change of volume: \[ J=\det\tns{F}\approx{}=\det\tenseur{U}=1+\trace{\tepsilonto} \] - \(\Omega_{t}\) can be replaced by \(\Omega_{0}\): - no more geometrical non linearity - Equilibrium: \[ \Divergence\tsigma+\vec{f}=\rho_{0}\,\vec{a} \quad\text{and}\quad \tsigma=\transpose{\tsigma} \] # Energy, strain measures, stress measures ## Mechanical power - Power of body forces: \[ \begin{aligned} \int_{\Omega}\vec{f}\,.\,\vec{v}\,\dtot\,v &=-\int_{\partial\,\Omega}\left(\tsigma\,.\,\vec{n}\right)\,.\,\vec{v}\,\dtot\,s+\int_{\Omega}\tsigma\,\colon\,\tns{L}\,\dtot\,v\\ &=-\int_{\partial\,\Omega}\left(\tsigma\,.\,\vec{n}\right)\,.\,\vec{v}\,\dtot\,s+\int_{\Omega}\tsigma\,\colon\,\tns{D}\,\dtot\,v\quad\text{(symmetry of \(\tsigma\))}\\ \end{aligned} \] - Conservation of energy: \[ \underbrace{\int_{\Omega}\tsigma\,\colon\,\tns{D}\,\dtot\,v}_{\text{inner forces power}} = \underbrace{\int_{\Omega}\vec{f}\,.\,\vec{v}\,\dtot\,v + \int_{\partial\,\Omega}\vec{t}\,.\,\vec{v}\,\dtot\,s}_{\text{external forces power}} \] ## Strain measures - A strain measure must satisfy the following hypotheses: - tends to \(\tepsilonto\) when the infinitesimal strain theory's assumptions are satisfied; - is objective (filters finite body rotation) - is symmetric - many isotropic functions of \(\tenseur{U}\) satisfy those requirements: - Green-Lagrange strain: \[ \tepsilonto_{GL}=\Frac{1}{2}\paren{\tenseur{U}^{2}-\tns{I}}= \Frac{1}{2}\paren{\tns{F}^{T}\,.\,\tns{F}-\tns{I}} \] - Hencky strain: \[\tepsilonto_{\mathrm{log}}=\log\tenseur{U}=\displaystyle\sum_{i=0}^{3}\log\lambda_{i}\,\tenseur{n}_{i}\] where \(\lambda_{i}\) are the eigenvalues of \(\tenseur{U}\) and \(\tenseur{n}_{i}\) are its eigentensors - functions of \(\tns{V}\) will not be considered {\bf here} - \(\tepsilonto\) is {\bf not} a strain measure, \(\tns{F}\) is {\bf not} a strain measure - \(\tns{D}\) is {\bf not} the time derivative of a strain measure ## Energetic conjugates (stress measures) - Mechanical work: \[ \int_{\Omega}\tsigma\,\colon\,\tenseur{D}\,\dtot\,v= \int_{\Omega_{0}}\tsigma\,\colon\,\tenseur{D}\,J\,\dtot\,V_{0} \] - For each strain measure \(\tepsilonto_{\star}\), one may define its dual stress \(\tns{T}_{\star}\): \[ J\,\tsigma\,\colon\,\tenseur{D}=\tenseur{\tau}\,\colon\,\tenseur{D}=\tepsilonto_{\star}\,\colon\,\tns{T}_{\star} \] - The tensor \(\tenseur{\tau}=J\,\tsigma\) is called the Kirchhoff stress. - The dual of the Green-Lagrange strain is the second Piola-Kirchhoff stress \(\tenseur{S}\): \[ \tenseur{S}=\tns{F}^{-1}\,.\,\tenseur{\tau}\,.\,\tns{F}^{-T} \quad\Leftrightarrow\quad\tau = \tns{F}\,.\,\tenseur{S}\,.\,\tns{F}^{T} \] - There is no strain measure which is the dual of \(\tenseur{\tau}\), nor \(\tsigma\) ## Choice of a stress/strain couple - All strain measures are {\bf equivalent}: there is no theoretical reason to prefer one strain measure over another. - However, there are {\bf pratical} 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: \[ J \approx 1+\trace{\tepsilonto_{\mathrm{GL}}} \] \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: \[ J = \exp\paren{\trace{\tepsilonto_{\mathrm{log}}}} \] - For pure dilatations: - \(\tepsilonto_{\mathrm{log}}=\displaystyle\int_{0}^{t}\tenseur{D}\,\dtot\,t\) - \(\tenseur{T}_{\mathrm{log}}\) is equal to the Kirchhoff stress \(\tenseur{\tau}\), which is egal to the Cauchy stress \(\tsigma\) 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): \[ w\paren{0,t}=\int_{0}^{t}J\,\tsigma\,\colon\,\tenseur{D}\,\dtot\,t =\int_{0}^{t}\tenseur{T}_{\star}\,\colon\,\tepsilonto_{\star}\,\dtot\,t =\int_{\left.\tepsilonto_{\star}\right|0}^{\left.\tepsilonto_{\star}\right|t}\tenseur{T}_{\star}\,\colon\,\dtot\,\tepsilonto_{\star} \] - 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 \(\tenseur{T}_{\star}\) then satisfies : \[ \tenseur{T}_{\star} = \deriv{w}{\tepsilonto_{\star}} \] ## Linear Elasticity - Linear elasticity assumes a linear relationship between \(\tepsilonto_{\star}\) and \(\tenseur{T}_{\star}\): \[ \tenseur{T}_{\star} = \tenseurq{D}\,\colon\,\tepsilonto_{\star}\quad\Leftrightarrow\quad w=\Frac{1}{2}\tepsilonto_{\star}\,\colon\,\tenseurq{D}\,\colon\,\tepsilonto_{\star} \] - The fourth order tensor \(\tenseurq{D}\) satisfies: \[ \tenseurq{D}=\Frac{\partial^{2}w}{\partial\tepsilonto_{\star}\partial\tepsilonto_{\star}} \] - \(\tenseurq{D}\) have the following properties: - \(\tenseurq{D}\) has minor symmetries (\(\tenseur{T}_{\star}\) and \(\tepsilonto_{\star}\) are symmetric): \[ D_{ijkl}= D_{jikl}\quad\text{and}\quad D_{ijkl}= D_{ijlk} \] - \(\tenseurq{D}\) has major symmetries (Schwarz theorem): \[ D_{ijkl}=D_{klij} \] ## Isotropic elasticity - For an isotropic material, only two independant material properties remains: \[ \tenseur{T}_{\star} = \lambda\,\paren{\trace{\tepsilonto_{\star}}}\,\tenseur{I}+2\,\mu\,\tepsilonto_{\star} \] - \(\lambda\) 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: \[ \begin{aligned} \tenseur{T}_{\star}&= \lambda\,\paren{\trace{\tepsilonto_{\star}}}\,\tenseur{I}+2\,\mu\,\tepsilonto_{\star} \\ &= \paren{\lambda+\Frac{2\,\mu}{3}}\,\underbrace{\paren{\trace{\tepsilonto_{\star}}}\,\tenseur{I}}_{\text{volume change}}+2\,\mu\,\underbrace{\paren{\tepsilonto_{\star}-\Frac{\trace{\tepsilonto_{\star}}}{3}\,\tenseur{I}}}_{\text{deviator}} \\ &= K\,\paren{\trace{\tepsilonto_{\star}}}\,\tenseur{I}+2\,\mu\,\paren{\tepsilonto_{\star}-\Frac{\trace{\tepsilonto_{\star}}}{3}\,\tenseur{I}} \\ \end{aligned} \] - \(K=\lambda+\Frac{2\,\mu}{3}\) is called the bulk modulus ### Young modulus and Poisson coefficient - consider an uniaxial tensile state along \(11\): \[ \begin{aligned} \tepsilonto_{\star}& = \begin{pmatrix} \epsilonto_{\star\,11} & \epsilonto_{\star\,22} & \epsilonto_{\star\,33} & 0 & 0 & 0 \end{pmatrix}^{T}\\ \tenseur{T}_{\star}&= \begin{pmatrix} T_{\star\,11} & 0 & 0 & 0 & 0 & 0 \end{pmatrix}^{T}\\ \end{aligned} \] - The Young modulus \(E\) is defined by: \[ T_{\star\,11} = E\,\epsilonto_{\star\,11} \] - The Poisson ratio \(\nu\) is defined by: \[ \epsilonto_{\star\,22} = -\nu\,\epsilonto_{\star\,11} \] - \(\lambda=\Frac{E\,\nu}{\paren{1+\nu}\,\paren{1-2\,\nu}}\quad\mu=\Frac{E}{2\,\paren{1+\nu}}\quad K=\Frac{E}{3\,\paren{1-2\,\nu}}\) ## Orthotropy - An orthotropic material introduces a preferential material frame: \[ \tenseurq{D}= \begin{pmatrix} D_{1111} & D_{1122} & D_{1133} & 0 & 0 & 0 \\ D_{1122} & D_{2222} & D_{2233} & 0 & 0 & 0 \\ D_{1133} & D_{2233} & D_{3333} & 0 & 0 & 0 \\ 0 & 0 & 0 & D_{1212} & 0 & 0 \\ 0 & 0 & 0 & 0 & D_{1313} & 0 \\ 0 & 0 & 0 & 0 & 0 & D_{2323} \\ \end{pmatrix} \] - \(9\) independent coefficients ## Thermo-elasticty ### Thermal strain - The total strain is splitted into an elastic part and a thermal part: \[ \tepsilonto_{\star}= \tepsilonel_{\star}+ \tepsilonth_{\star} \] - The elastic part \(\tepsilonel_{\star}\) defines the stresses \(\tenseur{T}_{\star}\) through the Hooke law: \[ \tenseur{T}_{\star}=\tenseurq{D}\,\colon\,\tepsilonel_{\star} \] #### Isotropic thermal expansion - The thermal expansion is given by: \[ \Frac{\Delta\,l}{l_{T^{\alpha}}}=\Frac{l_{T}-l_{T^{\alpha}}}{l_{T^{\alpha}}}=\alpha\paren{T}\,\paren{T-T^{\alpha}} \] - If the reference temperature \(T^{\alpha}\) for the thermal expansion is different than the reference temperature \(T^{i}\) of the geometry: \[ \Frac{\Delta\,l}{l_{T^{i}}} = \Frac{1}{1+\alpha\paren{T^{i}}\,\paren{T^{i}-T^{\alpha}}}\,\left[\alpha\paren{T}\,\paren{T-T^{\alpha}}-\alpha\paren{T^{i}}\,\paren{T^{i}-T^{\alpha}}\right] \] # Isotropic damage # Visco-plasticity ## A first approach to viscoplastic behaviour - The total strain is splitted into an elastic part and a viscoplastic part: \[ \tepsilonto_{\star}= \tepsilonel_{\star}+ \tepsilonvis_{\star} \] - The plastic flow is generally isochoric\footnote{This is an approximation for all strain measures execpt the logarithmic strain}: \[ \trace{\tepsilonvis_{\star}}=0 \] - Without internal state, the mechanical dissipation associated with plasticity is: \[ \tenseur{T}_{\star}\,\colon\,\tdepsilonto_{\star}= \underbrace{\tenseur{T}_{\star}\,\colon\,\tdepsilonel_{\star}}_{\text{stored reversibly}}+ \underbrace{\tenseur{T}_{\star}\,\colon\,\tdepsilonvis_{\star}}_{\text{dissipated}} \] - The expression dissipated power can be rewritten using the deviator of the stress \(\tenseur{s}_{\star}\): \[ \tenseur{T}_{\star}\,\colon\,\tdepsilonvis_{\star}= \tenseur{s}_{\star}\,\colon\,\tdepsilonvis_{\star} \quad\text{with}\quad\tenseur{s}_{\star}=\tenseur{T}_{\star}-\Frac{1}{3}\,\trace{\tenseur{T}_{\star}}\,\tenseur{I} \] - The dissipation is maximal if the \(\tdepsilonvis_{\star}\) is colinear with \(\tenseur{s}_{\star}\). ## The Von Mises stress - The material is now assumed {\bf isotropic} - A convenient isotropic norm for deviatoric stress tensor is the Von Mises norm: \[ T^{eq}_{\star}=\sqrt{\Frac{3}{2}\tenseur{s}_{\star}\,\colon\,\tenseur{s}_{\star}} \] - The \(\Frac{3}{2}\) 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: \[ T^{eq}_{\star}=\sqrt{\Frac{1}{2}\left[\paren{T_{1}-T_{2}}^{2}+\paren{T_{1}-T_{3}}^{2}+\paren{T_{2}-T_{3}}^{2}\right]} \] ## The normal tensor - The equation: \[ T^{eq}_{\star}=\text{Cste} \] defines a sphere in the deviatoric space - The normal to this surface is: \[ \tenseur{n}_{\star}=\deriv{T^{eq}_{\star}}{\tenseur{T}_{\star}}=\Frac{3\,\tenseur{s}_{\star}}{2\,T^{eq}_{\star}} \] - \(\tenseur{n}_{\star}\colon\tenseur{n}_{\star}=\Frac{3}{2}\) - The normal is colinear to \(\tenseur{s}_{\star}\), thus an isochoric viscoplastic flow of the form: \[ \tdepsilonvis_{\star}=f\paren{\tenseur{T}_{\star}}\tenseur{n}_{\star} \] would maximise the mechanical dissipation. - In uniaxial tensile tests: \[ \tenseur{n}_{\star}= \begin{pmatrix} 1 & -\frac{1}{2} & -\frac{1}{2} & 0 & 0 & 0 \end{pmatrix}^{T} \] ## The Norton behaviour - The viscoplastic flow is: \( \tdepsilonvis_{\star}=f\paren{\tenseur{T}_{\star}}\tenseur{n}_{\star} \) - 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: \[ \tdepsilonvis_{\star}=f\paren{T^{eq}_{\star}}\tenseur{n}_{\star} \] - Restrictions: - \(f\) must be positive for the dissipation to be positive - \(f\paren{\tenseur{0}}\) must be null - The Norton behaviour correspond to a power function: \[ \tdepsilonvis_{\star}=A\,\paren{T^{eq}_{\star}}^{n}\tenseur{n}_{\star}=\dot{\varepsilon}^{0}\paren{\Frac{T^{eq}_{\star}}{T^{0}}}^{n}\tenseur{n}_{\star} \] ## Equivalent viscoplastic strain - A convenient choice for the viscoplastic strain rate norm is: \[ \dot{p}=f\paren{T^{eq}_{\star}}=\sqrt{\Frac{2}{3}\,\tdepsilonvis_{\star}\,\colon\,\tdepsilonvis_{\star}} \] - In uniaxial tensile tests: \[ \dot{p} = \left|\paren{\depsilonvis_{\star}}_{xx}\right| \] - The equivalent viscoplastic strain is defined by: \[ p=\int_{0}^{t}\dot{p}\,\dtot\,t \] - 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: \[ \tdepsilonvis_{\star}=\deriv{\potentieldissipdual}{\tenseur{T}_{\star}} \quad\text{with}\quad \potentieldissipdual\paren{T^{eq}_{\star}}=\Frac{T^{0}\dot{\varepsilon}^{0}}{n+1}\paren{\Frac{T^{eq}_{\star}}{T^{0}}}^{n+1} \] - \(\potentieldissipdual\) 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 \(\potentieldissipdual\) is {\bf convex} and {\bf minimal} 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: \[ \underbrace{\int_{\Omega}\tsigma\,\colon\,\delta\,\tns{D}^{\star}\,\dtot\,v}_{\text{virtual inner forces power}} = \underbrace{\int_{\Omega}\vec{f}\,.\,\delta\,\vec{v}^{\star}\,\dtot\,v + \int_{\partial\,\Omega_{t}}\vec{t}\,.\,\delta\,\vec{v}^{\star}\,\dtot\,s}_{\text{virtual external forces power}} \] - 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\(\Delta\discret{\vec{u}}\) such as: \[ \small \residuEF\paren{\Delta\discret{\vec{u}}}=\discret{\vec{O}}\quad\text{ avec }\quad\residuEF\paren{\Delta\discret{\vec{u}}}=\forceintEF\paren{\Delta\discret{\vec{u}}}-\forceextEF \] - element contribution to inner forces: \[ \small \begin{aligned} \forceintElem&=\int_{V^{e}}\tsigma_{t+\Delta t}\paren{\Delta\,\tepsilonto,\Delta\, t}\colon\tenseur{B}\;\dtot V \\ &= \sum_{i=1}^{N^{G}} \paren{\tsigma_{t+\Delta\,t}\paren{\Delta\tepsilonto\paren{\vec{\eta}_{i}},\Delta\, t}\colon\tenseurq{B}\paren{\vec{\eta}_{i}}}w_{i} \end{aligned} \] where \(\tenseur{B}\) gives the relationship between \(\Delta\,\tepsilonto\) and \(\Delta\discret{\vec{u}}\) \[ \forceintElem = \sum_{i=1}^{N^{G}} \paren{\tsigma_{t+\Delta\,t}\paren{\Delta\tepsilonto\paren{\vec{\eta}_{i}},\Delta\, t}\colon\tenseurq{B}\paren{\vec{\eta}_{i}}}w_{i} \] ### Resolution using the \nom{Newton-Raphson} algorithm ## Mechanical behaviours