Matrix expression of the problem ================================== .. _Ref1192116521: .. _RefNumPara__21164479: Regularized cohesive law ------------------------ Matrix writing of the linearized problem ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Using the notations of [:ref:`R5.03.54 `], and considering the unified script adopted for the regularized friction and cohesive contact laws, the linearized system as solved in Newton's :math:`k+1` iteration can be put into matrix form: .. csv-table:: "Equation of balance", ":math:`\begin{array}{c}\left\{{u}^{\text{*}}\right\}\left[{K}_{\text{méca}}\right]\left(\mathit{\delta u}\right)+\left\{{u}^{\text{*}}\right\}\left[{K}_{\mathit{coh}}^{u}\right]\left(\mathit{\delta u}\right)\\ =\left\{{u}^{\text{*}}\right\}\left({L}_{\text{méca}}^{1}\right)+\left\{{u}^{\text{*}}\right\}\left({L}_{\text{coh}}^{1}\right)\\ \end{array}`" "Normal cohesive stress post-treatment", ":math:`\left\{{\lambda }_{n}^{\text{*}}\right\}\left[C\right]\left({\text{δλ}}_{n}\right)=\left\{{\lambda }_{n}^{\text{*}}\right\}\left({L}_{\text{post}}^{1}\right)+\left\{{\lambda }_{n}^{\text{*}}\right\}\left({L}_{\text{coh}}^{2}\right)`" "Tangential cohesive stress post-treatment", ":math:`\left\{{\lambda }_{\tau }^{\text{*}}\right\}\left[{F}_{r}\right]\left({\mathit{\delta \lambda }}_{\tau }\right)=\left\{{\lambda }_{\tau }^{\text{*}}\right\}\left({L}_{\text{post}}^{2}\right)+\left\{{\lambda }_{\tau }^{\text{*}}\right\}\left({L}_{\text{coh}}^{3}\right)`" where the column vectors are denoted :math:`(x)` and the row vectors :math:`\left\{x\right\}={(x)}^{T}` .This system can be put in the following matrix form: :math:`\left[\begin{array}{c}{{\rm K}}_{\text{méca}}+{K}_{\text{coh}}^{u}\\ 0\\ 0\end{array}\begin{array}{c}0\\ C\\ 0\end{array}\begin{array}{c}0\\ 0\\ F\end{array}\right]\left(\begin{array}{c}\mathit{\delta u}\\ {\text{δλ}}_{n}\\ {\mathit{\delta \lambda }}_{\tau }\end{array}\right)=\left(\begin{array}{c}{L}_{\text{méca}}^{1}+{L}_{\text{coh}}^{1}\\ {L}_{\text{post}}^{1}+{L}_{\text{coh}}^{2}\\ {L}_{\text{post}}^{2}+{L}_{\text{coh}}^{3}\end{array}\right)` The unknown is the increment compared to the previous Newton iteration. The reference to Newton's iteration number has been deliberately omitted. :math:`{K}_{\text{méca}}` is the mechanical stiffness matrix defined in paragraph [:ref:`§3.2 <§3.2>`] of [:ref:`R7.02.12 `]. :math:`{K}_{\mathit{coh}}^{u}` is the stiffness matrix due to cohesive forces. :math:`C` is the post-processing matrix for the normal direction. :math:`F` is the post-processing matrix for the tangential direction (s). :math:`{L}_{\text{méca}}^{1}` is the second member representing internal forces and load increments. :math:`{L}_{\text{post}}^{1}` and :math:`{L}_{\text{post}}^{2}` are the second members of post-processing. :math:`{L}_{\text{coh}}^{1}`, :math:`{L}_{\text{coh}}^{2}` and :math:`{L}_{\text{coh}}^{3}` are the second members due to cohesive forces. **Note:** *We remind you that the system solved by Code_Aster is not of the type* :math:`\mathrm{[}K\mathrm{]}\mathrm{[}U\mathrm{]}\mathrm{=}\mathrm{[}F\mathrm{]}` *but of the type* :math:`\mathrm{[}K\mathrm{]}\mathrm{[}U\mathrm{]}+\mathrm{[}F\mathrm{]}\mathrm{=}0` *. There is therefore a minus sign between the second members given in this document and those coded in the Fortran files.* Expression of elementary matrices and vectors ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The expression for the :math:`C` matrix required for post-processing (the indices :math:`n` out of :math:`\lambda`, to indicate the normal component of the cohesive stress, are omitted): :math:`{\left\{{\lambda }^{\text{*}}\right\}}_{i}{\left[C\right]}_{\text{ij}}{(\text{δλ})}_{j}\mathrm{=}{\mathrm{\int }}_{\Gamma }{\psi }_{i}{\lambda }_{i}^{\text{*}}{\psi }_{j}{\text{δλ}}_{j}\mathit{d\Gamma }` The expression for the :math:`{L}_{\text{post}}^{1}` vector required for post-processing (indices :math:`n` omitted on :math:`\lambda`) is: :math:`{\left\{{\lambda }^{\text{*}}\right\}}_{i}{\left({L}_{\text{post}}^{1}\right)}_{i}={\int }_{\Gamma }{\psi }_{i}{\lambda }_{i}^{\text{*}}{\lambda }^{k-1}\mathit{d\Gamma }` The expression for the :math:`F` matrix required for post-processing is: :math:`{\left\{{\lambda }_{\tau }^{\text{*}}\right\}}_{i}{\left[F\right]}_{\text{ij}}{\left(\delta {\lambda }_{\tau }\right)}_{j}={\int }_{\Gamma }{\psi }_{i}\left\{\begin{array}{cc}{\lambda }_{\tau ,i}^{1\text{*}}& {\lambda }_{\tau ,i}^{2\text{*}}\end{array}\right\}\cdot \left[\begin{array}{cc}{\tau }_{i}^{1}{\tau }_{j}^{1}& {\tau }_{i}^{1}{\tau }_{j}^{2}\\ {\tau }_{i}^{2}{\tau }_{j}^{1}& {\tau }_{i}^{2}{\tau }_{j}^{2}\end{array}\right]\cdot {\psi }_{j}\left(\begin{array}{c}{\lambda }_{\tau ,j}^{1}\\ {\lambda }_{\tau ,j}^{2}\end{array}\right)\mathit{d\Gamma }` The expression for the :math:`{L}_{\text{post}}^{2}` vector required for post-processing is: :math:`\begin{array}{c}{\left\{{\lambda }_{\tau }^{\text{*}}\right\}}_{i}{\left({L}_{\text{post}}^{2}\right)}_{i}={\int }_{\Gamma }{\psi }_{i}\left\{{\lambda }_{\tau ,i}^{1\text{*}}{\tau }_{i}^{1}+{\lambda }_{\tau ,i}^{2\text{*}}{\tau }_{i}^{2}\right\}{\lambda }_{\tau }^{k-1}\mathit{d\Gamma }\end{array}` Now let's try to write :math:`{K}_{\mathit{coh}}^{u}`. Let two fixed base directions :math:`X` and :math:`Y` have unit vectors :math:`{e}_{X}` and :math:`{e}_{Y}`. Let's introduce the tangent matrix of the cohesive law into the fixed base :math:`{K}^{\mathit{gl}}` of coefficients :math:`{K}_{XY}^{\mathit{gl}}={e}_{X}\cdot \frac{\partial {t}_{c}}{\partial ⟦u⟧}\cdot {e}_{Y}`. With the expression for :math:`\frac{\mathrm{\partial }{t}_{c}}{\mathrm{\partial }\mathrm{〚}u\mathrm{〛}}` given in [§ :ref:`4.2 `], we have the tangent matrix of the cohesive law :math:`{K}^{\mathit{loc}}` in the local base (see doc [:ref:`R7.02.11 `]). We then get :math:`{K}^{\mathit{gl}}` by :math:`{K}^{\mathit{gl}}={Q}^{T}\cdot {K}_{\mathit{loc}}\cdot Q`, where :math:`Q` is an orthonormal transition matrix defined by: :math:`Q=\left[\begin{array}{ccc}{n}_{X}& {n}_{Y}& {n}_{Z}\\ {\tau }_{X}^{1}& {\tau }_{Y}^{1}& {\tau }_{Z}^{1}\\ {\tau }_{X}^{2}& {\tau }_{Y}^{2}& {\tau }_{Z}^{2}\end{array}\right]` Let :math:`i` and :math:`j` be two enriched nodes. Let :math:`\Gamma` be the intersection of the supports of :math:`i` and :math:`j`. The matrix :math:`[{K}_{\mathit{coh}}^{u}]` is then given by: :math:`{\{{u}^{\text{*}}\}}_{i}{[{K}_{\mathit{coh}}^{u}]}_{\mathit{ij}}{\{\delta u\}}_{j}={\int }_{\Gamma }2{\varphi }_{i}{b}_{i}^{\text{*}}\cdot {K}^{\mathit{gl}}\cdot {b}_{j}2{\varphi }_{j}d\Gamma` The second members of cohesion have the following expressions: :math:`{\{{u}_{i}\}}^{\text{*}}{\left({L}_{\mathit{coh}}^{1}\right)}_{i}=-{\int }_{\Gamma }2{\varphi }_{i}{b}_{i}^{\text{*}}\cdot {t}_{c}^{k-1}d\Gamma` where we can express :math:`{t}_{c}` in the global base as :math:`{t}_{c}^{\mathit{glo}}={Q}^{T}\cdot {t}_{c}^{\mathit{loc}}`, and: :math:`{\left\{{\lambda }_{n}^{\text{*}}\right\}}_{i}{\left({L}_{\text{coh}}^{2}\right)}_{i}=-{\int }_{\Gamma }{\psi }_{i}{\lambda }_{n,i}^{\text{*}}\left({t}_{c,n}^{k-1}\right)\mathit{d\Gamma }` :math:`\begin{array}{c}{\left\{{\lambda }_{\tau }^{\text{*}}\right\}}_{i}{\left({L}_{\text{coh}}^{3}\right)}_{i}=-{\int }_{\Gamma }{\psi }_{i}\left\{{\lambda }_{\tau ,i}^{1\text{*}}{\tau }_{i}^{1}+{\lambda }_{\tau ,i}^{2\text{*}}{\tau }_{i}^{2}\right\}\cdot {t}_{c,\tau }^{k-1}\mathit{d\Gamma }\end{array}` where :math:`k\mathrm{-}1` represents the index of the previous Newton iteration. Mixed cohesive law for quadratic elements --------------------------------------------- Matrix writing of the problem with mixed cohesive law ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The matrix system as solved in Newton's iteration :math:`k+1` can be put in the following matrix form: :math:`\left[\begin{array}{c}{{\rm K}}_{\text{méca}}+{{\rm A}}_{u}\\ {\rm A}\end{array}\begin{array}{c}{{\rm A}}^{T}\\ C\end{array}\right](\begin{array}{c}\mathit{\delta u}\\ \text{δλ}\end{array})\mathrm{=}(\begin{array}{c}{L}_{\text{méca}}^{1}+{L}_{\text{coh}}^{1}\\ {L}_{\text{coh}}^{2}\end{array})` Expression of elementary cohesion matrices: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Let two fixed base directions :math:`X` and :math:`Y` have unit vectors :math:`{e}_{X}` and :math:`{e}_{Y}`. As before, we introduce the tangent matrix of the cohesive law into the fixed base :math:`{K}^{\mathit{gl}}` of coefficients :math:`{K}_{\mathit{XY}}^{\mathit{gl}}={e}_{X}\cdot \frac{\partial \delta }{\partial p}\cdot {e}_{Y}`. Through the law of cohesive behavior, we have the tangent matrix :math:`{K}_{\mathit{loc}}` in the local base (see doc [:ref:`R7.02.11 `]). We then get :math:`{K}_{\mathit{gl}}` by :math:`{K}^{\mathit{gl}}={Q}^{T}\cdot {K}^{\mathit{loc}}\cdot Q`, where :math:`Q` is an orthonormal transition matrix defined by: :math:`Q=\left[\begin{array}{ccc}{n}_{X}& {n}_{Y}& {n}_{Z}\\ {\tau }_{X}^{1}& {\tau }_{Y}^{1}& {\tau }_{Z}^{1}\\ {\tau }_{X}^{2}& {\tau }_{Y}^{2}& {\tau }_{Z}^{2}\end{array}\right]` Having introduced these notations, we have: :math:`{\{{u}^{\text{*}}\}}_{i}{[{A}_{u}]}_{\mathit{ij}}{\{\delta u\}}_{j}={\int }_{\Gamma }2{\varphi }_{i}{b}_{i}^{\text{*}}r\left(1-r{K}^{\mathit{gl}}\right)2{\varphi }_{j}{b}_{j}d\Gamma` In addition, we choose to discretize :math:`\lambda` on a local basis. The coefficient :math:`\left(1,1\right)` of the matrix :math:`\{{\lambda }_{n}^{\text{*}}\}{[{A}_{u}]}_{\mathit{ij}}{\{\delta {u}_{X}\}}_{j}` is then :math:`{\int }_{\Gamma }{\varphi }_{i}{\lambda }_{i}^{\text{*}}n\cdot \left(1-r\frac{\partial \delta }{\partial p}\right)\cdot {e}_{X}2{\varphi }_{j}{b}_{j}d\Gamma`. For coefficients :math:`\left(2,1\right)` and :math:`\left(3,1\right)`, the formula is the same by replacing :math:`n` with :math:`{\tau }_{1}` and :math:`{\tau }_{2}`, respectively. By exploiting the notations introduced in the preceding paragraph, we deduce: :math:`{\{{\lambda }^{\text{*}}\}}_{i}{[A]}_{\mathit{ij}}{\{\delta u\}}_{j}={\int }_{\Gamma }{\varphi }_{i}{\lambda }_{i}^{\text{*}}\left(1-r{K}^{\mathit{loc}}\right)\cdot Q2{\varphi }_{j}{b}_{j}d\Gamma` As for matrix :math:`\mathrm{[}C\mathrm{]}`, it is simply written: :math:`{\{{\lambda }^{\text{*}}\}}_{i}{[C]}_{\mathit{ij}}{\{\delta \lambda \}}_{j}=-{\int }_{\Gamma }{\varphi }_{i}{\lambda }_{i}^{\text{*}}{K}^{\mathit{loc}}{\varphi }_{j}{\lambda }_{j}d\Gamma` Expression of the elementary cohesion vectors: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The coefficient according to :math:`X` of :math:`{\left({L}_{\mathit{coh}}^{1}\right)}_{i}` has the expression :math:`{\int }_{\Gamma }{\varphi }_{i}\left(-\lambda \cdot {e}_{X}-r\left(⟦u⟧-\delta \right)\cdot {e}_{X}\right)d\Gamma`. With the notations introduced in the previous part, we deduce: :math:`{\{{u}_{i}\}}^{\text{*}}{\left({L}_{\mathit{coh}}^{1}\right)}_{i}={\int }_{\Gamma }2{\varphi }_{i}{b}_{i}^{\text{*}}\cdot \left(-{Q}^{T}\cdot \lambda -r\left(⟦u⟧-{Q}^{T}\cdot \delta \right)\right)d\Gamma` where :math:`\delta` and :math:`\lambda` are given on a local basis, and :math:`⟦u⟧` on a fixed basis. The :math:`1` coefficient of :math:`{\left({L}_{\mathit{coh}}^{2}\right)}_{i}` is written as :math:`{\int }_{\Gamma }\left(-⟦u⟧\cdot n+\delta \cdot n\right)d\Gamma`. For coefficients :math:`2` and :math:`3`, the formula is the same by replacing :math:`n` with :math:`{\tau }_{1}` and :math:`{\tau }_{2}`, respectively. From this we deduce: :math:`{\{\lambda \}}_{i}^{\text{*}}{\left({L}_{\mathit{coh}}^{2}\right)}_{i}={\int }_{\Gamma }{\lambda }_{i}^{\text{*}}\cdot \left(-Q⟦u⟧+\delta \right)d\Gamma` where :math:`\delta` is given on a local basis, and :math:`⟦u⟧` on a fixed basis. Mixed cohesive law for linear elements ------------------------------------------ The components of the unknowns :math:`u` and :math:`\mu` are defined in a fixed base :math:`\left({e}_{X},{e}_{Y},{e}_{Z}\right)`, while the components of :math:`w` and :math:`\lambda` are defined in the local base :math:`\left(n,{\tau }_{1},{\tau }_{2}\right)` at the cracked surface :math:`\Gamma` at each point :math:`x\in \Gamma`, so that *:* :math:`w(x)=\sum _{i=1}^{{N}_{\lambda }}{\psi }_{I}(x)\left({w}_{I,n}n(x)+{w}_{I,\tau 1}{\tau }_{1}(x)+{w}_{I,\tau 2}{\tau }_{2}(x)\right)` A similar definition applies to :math:`\lambda`. For a degree of freedom :math:`I` of the reduced space (see § :ref:`3.3 `), it is possible to determine the :math:`{t}_{c,n}^{I},{t}_{t,\tau 1}^{I},{t}_{c,\tau 2}^{I}` components of the cohesive force from :math:`\left({w}_{I,n},{w}_{I,\tau 1},{w}_{I,\tau 2}\right)`, :math:`\left({\lambda }_{I,n},{\lambda }_{I,\tau 1},{\lambda }_{I,\tau 2}\right)` and from the cohesive law of § :ref:`2.2.3 `. These components are not intended to be associated with a particular direction :math:`I` of the degree of freedom, but intended to be linked in a weak sense to the global constraint :math:`\mu` written in a fixed base. The matrix system can take the following form: :math:`\left[\begin{array}{cccc}{K}^{\mathit{uu}}& {\left({K}^{\mu u}\right)}^{T}& 0& 0\\ {K}^{\mu u}& 0& {\left(-{K}^{w\mu }\right)}^{T}& 0\\ 0& {-K}^{w\mu }& {D}^{\mathit{ww}}& {\left({D}^{\lambda w}\right)}^{T}\\ 0& 0& {D}^{\lambda w}& {D}^{\lambda \lambda }\end{array}\right]\left(\begin{array}{c}\delta u\\ \delta \mu \\ \delta w\\ \delta \lambda \end{array}\right)=\left(\begin{array}{c}{L}_{\text{méca}}+{L}_{\mu }^{1}\\ {L}_{u}-{L}_{w}\\ -{L}_{\mu }^{2}+{L}_{\mathit{coh}}^{1}\\ -{L}_{\lambda }+{L}_{\mathit{coh}}^{2}\end{array}\right)` Where: * :math:`{K}^{uu}` is the volume stiffness matrix; * :math:`{K}^{\mu u}` and :math:`{K}^{w\mu }` are matrices discretizing the "mortar" operators, the latter also managing the base change; * the matrices :math:`D` are all block diagonals: for :math:`I` and :math:`J` two distinct DDL Lagrange numbers, they verify :math:`{D}_{\mathit{IJ}}=0`. Let us give the expressions for matrices and second members that do not depend on the cohesive law: :math:`{\{u\}}_{i}^{\text{*}}{\left[{K}^{\mu u}\right]}_{iJ}{(\delta \mu )}_{J}={b}_{i}^{\text{*}}\cdot (\delta {\mu }_{J}){\int }_{\Gamma }2{\psi }_{J}{\varphi }_{i}d\Gamma` By introducing the orthonormal base change matrix :math:`Q` defined as before, we have: :math:`{\{w\}}_{I}^{\text{*}}{\left[{K}^{w\mu }\right]}_{IJ}{(\delta \mu )}_{J}={w}_{I}^{\text{*}}\cdot \left({\int }_{\Gamma }{\psi }_{I}{\psi }_{J}Qd\Gamma \right)\cdot (\delta {\mu }_{J})` :math:`{\{u\}}_{i}^{\text{*}}{\left({L}_{\mu }^{1}\right)}_{i}=-{b}_{i}^{\text{*}}\cdot {\int }_{\Gamma }2{\varphi }_{i}\mu d\Gamma` :math:`{\{\mu \}}_{I}^{\text{*}}{\left({L}_{u}\right)}_{I}=-{\mu }_{I}^{\text{*}}\cdot {\int }_{\Gamma }{\psi }_{I}⟦u⟧d\Gamma` :math:`{\{\mu \}}_{I}^{\text{*}}{\left({L}_{w}\right)}_{I}=-{\mu }_{I}^{\text{*}}\cdot {\int }_{\Gamma }{\psi }_{I}{Q}^{T}\cdot wd\Gamma` :math:`{\{w\}}_{I}^{\text{*}}{\left({L}_{\mu }^{2}\right)}_{I}={w}_{I}^{\text{*}}\cdot {\int }_{\Gamma }{\psi }_{I}Q\cdot \mu d\Gamma` Let's now detail the discretization of interface quantities. The matrices are diagonal by blocks, and have the following expressions: :math:`{\{w\}}_{I}^{\text{*}}{\left[{D}^{ww}\right]}_{II}{(\delta w)}_{I}={w}_{I}^{\text{*}}(\delta {w}_{I})r\frac{\partial {t}_{c}}{\partial (\lambda +rw)}({\lambda }_{I}+r{w}_{I}){\int }_{\Gamma }{\psi }_{I}d\Gamma` :math:`{\{\lambda \}}_{I}^{\text{*}}{\left[{D}^{\lambda w}\right]}_{II}{(\delta w)}_{I}={\lambda }_{I}^{\text{*}}(\delta {w}_{I})\frac{\partial {t}_{c}}{\partial (\lambda +rw)}({\lambda }_{I}+r{w}_{I}){\int }_{\Gamma }{\psi }_{I}d\Gamma` :math:`{\{\lambda \}}_{I}^{\text{*}}{\left[{D}^{\lambda \lambda }\right]}_{II}{(\delta \lambda )}_{I}={\lambda }_{I}^{\text{*}}(\delta {\lambda }_{I})\frac{1}{r}\left(\frac{\partial {t}_{c}}{\partial (\lambda +rw)}({\lambda }_{I}+r{w}_{I})-1\right){\int }_{\Gamma }{\psi }_{I}d\Gamma` Elementary vectors have the following expressions: :math:`{\{w\}}_{I}^{\text{*}}{\left({L}_{\mathit{coh}}^{1}\right)}_{I}={w}_{I}^{\text{*}}{t}_{c}({\lambda }_{I}+r{w}_{I}){\int }_{\Gamma }{\psi }_{I}d\Gamma` :math:`{\{w\}}_{I}^{\text{*}}{\left({L}_{\mathit{coh}}^{2}\right)}_{I}=\frac{{w}_{I}^{\text{*}}}{r}{t}_{c}({\lambda }_{I}+r{w}_{I}){\int }_{\Gamma }{\psi }_{I}d\Gamma`