5. Matrix expression of the problem#

5.1. Regularized cohesive law#

5.1.1. Matrix writing of the linearized problem#

Using the notations of [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 \(k+1\) iteration can be put into matrix form:

Equation of balance	\(\begin{array}{c}\left\{{u}^{\text{}}\right\}\left[{K}_{\text{méca}}\right]\left(\mathit{\delta u}\right)+\left\{{u}^{\text{}}\right\}\left[{K}_{\mathit{coh}}^{u}\right]\left(\mathit{\delta u}\right)\\ =\left\{{u}^{\text{}}\right\}\left({L}_{\text{méca}}^{1}\right)+\left\{{u}^{\text{}}\right\}\left({L}_{\text{coh}}^{1}\right)\\ \end{array}\)
Normal cohesive stress post-treatment	\(\left\{{\lambda }_{n}^{\text{}}\right\}\left[C\right]\left({\text{δλ}}_{n}\right)=\left\{{\lambda }_{n}^{\text{}}\right\}\left({L}_{\text{post}}^{1}\right)+\left\{{\lambda }_{n}^{\text{*}}\right\}\left({L}_{\text{coh}}^{2}\right)\)
Tangential cohesive stress post-treatment	\(\left\{{\lambda }_{\tau }^{\text{}}\right\}\left[{F}_{r}\right]\left({\mathit{\delta \lambda }}_{\tau }\right)=\left\{{\lambda }_{\tau }^{\text{}}\right\}\left({L}_{\text{post}}^{2}\right)+\left\{{\lambda }_{\tau }^{\text{*}}\right\}\left({L}_{\text{coh}}^{3}\right)\)

where the column vectors are denoted \((x)\) and the row vectors \(\left\{x\right\}={(x)}^{T}\) .This system can be put in the following matrix form:

\(\left[\begin{array}{c}{{\rm K}}_{\text{méca}}+{K}_{\text{coh}}^{u}\\ 0\\ 0\end{array}\begin{array}{c}0\\ C\\ 0\end{array}\begin{array}{c}0\\ 0\\ F\end{array}\right]\left(\begin{array}{c}\mathit{\delta u}\\ {\text{δλ}}_{n}\\ {\mathit{\delta \lambda }}_{\tau }\end{array}\right)=\left(\begin{array}{c}{L}_{\text{méca}}^{1}+{L}_{\text{coh}}^{1}\\ {L}_{\text{post}}^{1}+{L}_{\text{coh}}^{2}\\ {L}_{\text{post}}^{2}+{L}_{\text{coh}}^{3}\end{array}\right)\)

The unknown is the increment compared to the previous Newton iteration. The reference to Newton’s iteration number has been deliberately omitted.

\({K}_{\text{méca}}\) is the mechanical stiffness matrix defined in paragraph [§3.2] of [R7.02.12].

\({K}_{\mathit{coh}}^{u}\) is the stiffness matrix due to cohesive forces.

\(C\) is the post-processing matrix for the normal direction.

\(F\) is the post-processing matrix for the tangential direction (s).

\({L}_{\text{méca}}^{1}\) is the second member representing internal forces and load increments.

\({L}_{\text{post}}^{1}\) and \({L}_{\text{post}}^{2}\) are the second members of post-processing.

\({L}_{\text{coh}}^{1}\), \({L}_{\text{coh}}^{2}\) and \({L}_{\text{coh}}^{3}\) are the second members due to cohesive forces.

Note:

We remind you that the system solved by Code_Aster is not of the type \(\mathrm{[}K\mathrm{]}\mathrm{[}U\mathrm{]}\mathrm{=}\mathrm{[}F\mathrm{]}\) but of the type \(\mathrm{[}K\mathrm{]}\mathrm{[}U\mathrm{]}+\mathrm{[}F\mathrm{]}\mathrm{=}0\) . There is therefore a minus sign between the second members given in this document and those coded in the Fortran files.

5.1.2. Expression of elementary matrices and vectors#

The expression for the \(C\) matrix required for post-processing (the indices \(n\) out of \(\lambda\), to indicate the normal component of the cohesive stress, are omitted):

\({\left\{{\lambda }^{\text{*}}\right\}}_{i}{\left[C\right]}_{\text{ij}}{(\text{δλ})}_{j}\mathrm{=}{\mathrm{\int }}_{\Gamma }{\psi }_{i}{\lambda }_{i}^{\text{*}}{\psi }_{j}{\text{δλ}}_{j}\mathit{d\Gamma }\)

The expression for the \({L}_{\text{post}}^{1}\) vector required for post-processing (indices \(n\) omitted on \(\lambda\)) is:

\({\left\{{\lambda }^{\text{*}}\right\}}_{i}{\left({L}_{\text{post}}^{1}\right)}_{i}={\int }_{\Gamma }{\psi }_{i}{\lambda }_{i}^{\text{*}}{\lambda }^{k-1}\mathit{d\Gamma }\)

The expression for the \(F\) matrix required for post-processing is:

\({\left\{{\lambda }_{\tau }^{\text{*}}\right\}}_{i}{\left[F\right]}_{\text{ij}}{\left(\delta {\lambda }_{\tau }\right)}_{j}={\int }_{\Gamma }{\psi }_{i}\left\{\begin{array}{cc}{\lambda }_{\tau ,i}^{1\text{*}}& {\lambda }_{\tau ,i}^{2\text{*}}\end{array}\right\}\cdot \left[\begin{array}{cc}{\tau }_{i}^{1}{\tau }_{j}^{1}& {\tau }_{i}^{1}{\tau }_{j}^{2}\\ {\tau }_{i}^{2}{\tau }_{j}^{1}& {\tau }_{i}^{2}{\tau }_{j}^{2}\end{array}\right]\cdot {\psi }_{j}\left(\begin{array}{c}{\lambda }_{\tau ,j}^{1}\\ {\lambda }_{\tau ,j}^{2}\end{array}\right)\mathit{d\Gamma }\)

The expression for the \({L}_{\text{post}}^{2}\) vector required for post-processing is:

\(\begin{array}{c}{\left\{{\lambda }_{\tau }^{\text{*}}\right\}}_{i}{\left({L}_{\text{post}}^{2}\right)}_{i}={\int }_{\Gamma }{\psi }_{i}\left\{{\lambda }_{\tau ,i}^{1\text{*}}{\tau }_{i}^{1}+{\lambda }_{\tau ,i}^{2\text{*}}{\tau }_{i}^{2}\right\}{\lambda }_{\tau }^{k-1}\mathit{d\Gamma }\end{array}\)

Now let’s try to write \({K}_{\mathit{coh}}^{u}\). Let two fixed base directions \(X\) and \(Y\) have unit vectors \({e}_{X}\) and \({e}_{Y}\). Let’s introduce the tangent matrix of the cohesive law into the fixed base \({K}^{\mathit{gl}}\) of coefficients \({K}_{XY}^{\mathit{gl}}={e}_{X}\cdot \frac{\partial {t}_{c}}{\partial ⟦u⟧}\cdot {e}_{Y}\). With the expression for \(\frac{\mathrm{\partial }{t}_{c}}{\mathrm{\partial }\mathrm{〚}u\mathrm{〛}}\) given in [§ 4.2], we have the tangent matrix of the cohesive law \({K}^{\mathit{loc}}\) in the local base (see doc [R7.02.11]). We then get \({K}^{\mathit{gl}}\) by \({K}^{\mathit{gl}}={Q}^{T}\cdot {K}_{\mathit{loc}}\cdot Q\), where \(Q\) is an orthonormal transition matrix defined by:

\(Q=\left[\begin{array}{ccc}{n}_{X}& {n}_{Y}& {n}_{Z}\\ {\tau }_{X}^{1}& {\tau }_{Y}^{1}& {\tau }_{Z}^{1}\\ {\tau }_{X}^{2}& {\tau }_{Y}^{2}& {\tau }_{Z}^{2}\end{array}\right]\)

Let \(i\) and \(j\) be two enriched nodes. Let \(\Gamma\) be the intersection of the supports of \(i\) and \(j\). The matrix \([{K}_{\mathit{coh}}^{u}]\) is then given by:

\({\{{u}^{\text{*}}\}}_{i}{[{K}_{\mathit{coh}}^{u}]}_{\mathit{ij}}{\{\delta u\}}_{j}={\int }_{\Gamma }2{\varphi }_{i}{b}_{i}^{\text{*}}\cdot {K}^{\mathit{gl}}\cdot {b}_{j}2{\varphi }_{j}d\Gamma\)

The second members of cohesion have the following expressions:

\({\{{u}_{i}\}}^{\text{*}}{\left({L}_{\mathit{coh}}^{1}\right)}_{i}=-{\int }_{\Gamma }2{\varphi }_{i}{b}_{i}^{\text{*}}\cdot {t}_{c}^{k-1}d\Gamma\)

where we can express \({t}_{c}\) in the global base as \({t}_{c}^{\mathit{glo}}={Q}^{T}\cdot {t}_{c}^{\mathit{loc}}\), and:

\({\left\{{\lambda }_{n}^{\text{*}}\right\}}_{i}{\left({L}_{\text{coh}}^{2}\right)}_{i}=-{\int }_{\Gamma }{\psi }_{i}{\lambda }_{n,i}^{\text{*}}\left({t}_{c,n}^{k-1}\right)\mathit{d\Gamma }\)

\(\begin{array}{c}{\left\{{\lambda }_{\tau }^{\text{*}}\right\}}_{i}{\left({L}_{\text{coh}}^{3}\right)}_{i}=-{\int }_{\Gamma }{\psi }_{i}\left\{{\lambda }_{\tau ,i}^{1\text{*}}{\tau }_{i}^{1}+{\lambda }_{\tau ,i}^{2\text{*}}{\tau }_{i}^{2}\right\}\cdot {t}_{c,\tau }^{k-1}\mathit{d\Gamma }\end{array}\)

where \(k\mathrm{-}1\) represents the index of the previous Newton iteration.

5.2. Mixed cohesive law for quadratic elements#

5.2.1. Matrix writing of the problem with mixed cohesive law#

The matrix system as solved in Newton’s iteration \(k+1\) can be put in the following matrix form:

\(\left[\begin{array}{c}{{\rm K}}_{\text{méca}}+{{\rm A}}_{u}\\ {\rm A}\end{array}\begin{array}{c}{{\rm A}}^{T}\\ C\end{array}\right](\begin{array}{c}\mathit{\delta u}\\ \text{δλ}\end{array})\mathrm{=}(\begin{array}{c}{L}_{\text{méca}}^{1}+{L}_{\text{coh}}^{1}\\ {L}_{\text{coh}}^{2}\end{array})\)

5.2.1.1. Expression of elementary cohesion matrices:#

Let two fixed base directions \(X\) and \(Y\) have unit vectors \({e}_{X}\) and \({e}_{Y}\). As before, we introduce the tangent matrix of the cohesive law into the fixed base \({K}^{\mathit{gl}}\) of coefficients \({K}_{\mathit{XY}}^{\mathit{gl}}={e}_{X}\cdot \frac{\partial \delta }{\partial p}\cdot {e}_{Y}\). Through the law of cohesive behavior, we have the tangent matrix \({K}_{\mathit{loc}}\) in the local base (see doc [R7.02.11]). We then get \({K}_{\mathit{gl}}\) by \({K}^{\mathit{gl}}={Q}^{T}\cdot {K}^{\mathit{loc}}\cdot Q\), where \(Q\) is an orthonormal transition matrix defined by:

\(Q=\left[\begin{array}{ccc}{n}_{X}& {n}_{Y}& {n}_{Z}\\ {\tau }_{X}^{1}& {\tau }_{Y}^{1}& {\tau }_{Z}^{1}\\ {\tau }_{X}^{2}& {\tau }_{Y}^{2}& {\tau }_{Z}^{2}\end{array}\right]\)

Having introduced these notations, we have:

\({\{{u}^{\text{*}}\}}_{i}{[{A}_{u}]}_{\mathit{ij}}{\{\delta u\}}_{j}={\int }_{\Gamma }2{\varphi }_{i}{b}_{i}^{\text{*}}r\left(1-r{K}^{\mathit{gl}}\right)2{\varphi }_{j}{b}_{j}d\Gamma\)

In addition, we choose to discretize \(\lambda\) on a local basis. The coefficient \(\left(1,1\right)\) of the matrix \(\{{\lambda }_{n}^{\text{*}}\}{[{A}_{u}]}_{\mathit{ij}}{\{\delta {u}_{X}\}}_{j}\) is then \({\int }_{\Gamma }{\varphi }_{i}{\lambda }_{i}^{\text{*}}n\cdot \left(1-r\frac{\partial \delta }{\partial p}\right)\cdot {e}_{X}2{\varphi }_{j}{b}_{j}d\Gamma\). For coefficients \(\left(2,1\right)\) and \(\left(3,1\right)\), the formula is the same by replacing \(n\) with \({\tau }_{1}\) and \({\tau }_{2}\), respectively. By exploiting the notations introduced in the preceding paragraph, we deduce:

\({\{{\lambda }^{\text{*}}\}}_{i}{[A]}_{\mathit{ij}}{\{\delta u\}}_{j}={\int }_{\Gamma }{\varphi }_{i}{\lambda }_{i}^{\text{*}}\left(1-r{K}^{\mathit{loc}}\right)\cdot Q2{\varphi }_{j}{b}_{j}d\Gamma\)

As for matrix \(\mathrm{[}C\mathrm{]}\), it is simply written:

\({\{{\lambda }^{\text{*}}\}}_{i}{[C]}_{\mathit{ij}}{\{\delta \lambda \}}_{j}=-{\int }_{\Gamma }{\varphi }_{i}{\lambda }_{i}^{\text{*}}{K}^{\mathit{loc}}{\varphi }_{j}{\lambda }_{j}d\Gamma\)

5.2.1.2. Expression of the elementary cohesion vectors:#

The coefficient according to \(X\) of \({\left({L}_{\mathit{coh}}^{1}\right)}_{i}\) has the expression \({\int }_{\Gamma }{\varphi }_{i}\left(-\lambda \cdot {e}_{X}-r\left(⟦u⟧-\delta \right)\cdot {e}_{X}\right)d\Gamma\). With the notations introduced in the previous part, we deduce:

\({\{{u}_{i}\}}^{\text{*}}{\left({L}_{\mathit{coh}}^{1}\right)}_{i}={\int }_{\Gamma }2{\varphi }_{i}{b}_{i}^{\text{*}}\cdot \left(-{Q}^{T}\cdot \lambda -r\left(⟦u⟧-{Q}^{T}\cdot \delta \right)\right)d\Gamma\)

where \(\delta\) and \(\lambda\) are given on a local basis, and \(⟦u⟧\) on a fixed basis.

The \(1\) coefficient of \({\left({L}_{\mathit{coh}}^{2}\right)}_{i}\) is written as \({\int }_{\Gamma }\left(-⟦u⟧\cdot n+\delta \cdot n\right)d\Gamma\). For coefficients \(2\) and \(3\), the formula is the same by replacing \(n\) with \({\tau }_{1}\) and \({\tau }_{2}\), respectively. From this we deduce:

\({\{\lambda \}}_{i}^{\text{*}}{\left({L}_{\mathit{coh}}^{2}\right)}_{i}={\int }_{\Gamma }{\lambda }_{i}^{\text{*}}\cdot \left(-Q⟦u⟧+\delta \right)d\Gamma\)

where \(\delta\) is given on a local basis, and \(⟦u⟧\) on a fixed basis.

5.3. Mixed cohesive law for linear elements#

The components of the unknowns \(u\) and \(\mu\) are defined in a fixed base \(\left({e}_{X},{e}_{Y},{e}_{Z}\right)\), while the components of \(w\) and \(\lambda\) are defined in the local base \(\left(n,{\tau }_{1},{\tau }_{2}\right)\) at the cracked surface \(\Gamma\) at each point \(x\in \Gamma\), so that :

\(w(x)=\sum _{i=1}^{{N}_{\lambda }}{\psi }_{I}(x)\left({w}_{I,n}n(x)+{w}_{I,\tau 1}{\tau }_{1}(x)+{w}_{I,\tau 2}{\tau }_{2}(x)\right)\)

A similar definition applies to \(\lambda\). For a degree of freedom \(I\) of the reduced space (see § 3.3), it is possible to determine the \({t}_{c,n}^{I},{t}_{t,\tau 1}^{I},{t}_{c,\tau 2}^{I}\) components of the cohesive force from \(\left({w}_{I,n},{w}_{I,\tau 1},{w}_{I,\tau 2}\right)\), \(\left({\lambda }_{I,n},{\lambda }_{I,\tau 1},{\lambda }_{I,\tau 2}\right)\) and from the cohesive law of § 2.2.3. These components are not intended to be associated with a particular direction \(I\) of the degree of freedom, but intended to be linked in a weak sense to the global constraint \(\mu\) written in a fixed base.

The matrix system can take the following form:

\(\left[\begin{array}{cccc}{K}^{\mathit{uu}}& {\left({K}^{\mu u}\right)}^{T}& 0& 0\\ {K}^{\mu u}& 0& {\left(-{K}^{w\mu }\right)}^{T}& 0\\ 0& {-K}^{w\mu }& {D}^{\mathit{ww}}& {\left({D}^{\lambda w}\right)}^{T}\\ 0& 0& {D}^{\lambda w}& {D}^{\lambda \lambda }\end{array}\right]\left(\begin{array}{c}\delta u\\ \delta \mu \\ \delta w\\ \delta \lambda \end{array}\right)=\left(\begin{array}{c}{L}_{\text{méca}}+{L}_{\mu }^{1}\\ {L}_{u}-{L}_{w}\\ -{L}_{\mu }^{2}+{L}_{\mathit{coh}}^{1}\\ -{L}_{\lambda }+{L}_{\mathit{coh}}^{2}\end{array}\right)\)

Where:

\({K}^{uu}\) is the volume stiffness matrix;
\({K}^{\mu u}\) and \({K}^{w\mu }\) are matrices discretizing the « mortar » operators, the latter also managing the base change;
the matrices \(D\) are all block diagonals: for \(I\) and \(J\) two distinct DDL Lagrange numbers, they verify \({D}_{\mathit{IJ}}=0\).

Let us give the expressions for matrices and second members that do not depend on the cohesive law:

\({\{u\}}_{i}^{\text{*}}{\left[{K}^{\mu u}\right]}_{iJ}{(\delta \mu )}_{J}={b}_{i}^{\text{*}}\cdot (\delta {\mu }_{J}){\int }_{\Gamma }2{\psi }_{J}{\varphi }_{i}d\Gamma\)

By introducing the orthonormal base change matrix \(Q\) defined as before, we have:

\({\{w\}}_{I}^{\text{*}}{\left[{K}^{w\mu }\right]}_{IJ}{(\delta \mu )}_{J}={w}_{I}^{\text{*}}\cdot \left({\int }_{\Gamma }{\psi }_{I}{\psi }_{J}Qd\Gamma \right)\cdot (\delta {\mu }_{J})\)

\({\{u\}}_{i}^{\text{*}}{\left({L}_{\mu }^{1}\right)}_{i}=-{b}_{i}^{\text{*}}\cdot {\int }_{\Gamma }2{\varphi }_{i}\mu d\Gamma\)

\({\{\mu \}}_{I}^{\text{*}}{\left({L}_{u}\right)}_{I}=-{\mu }_{I}^{\text{*}}\cdot {\int }_{\Gamma }{\psi }_{I}⟦u⟧d\Gamma\)

\({\{\mu \}}_{I}^{\text{*}}{\left({L}_{w}\right)}_{I}=-{\mu }_{I}^{\text{*}}\cdot {\int }_{\Gamma }{\psi }_{I}{Q}^{T}\cdot wd\Gamma\)

\({\{w\}}_{I}^{\text{*}}{\left({L}_{\mu }^{2}\right)}_{I}={w}_{I}^{\text{*}}\cdot {\int }_{\Gamma }{\psi }_{I}Q\cdot \mu d\Gamma\)

Let’s now detail the discretization of interface quantities. The matrices are diagonal by blocks, and have the following expressions:

\({\{w\}}_{I}^{\text{*}}{\left[{D}^{ww}\right]}_{II}{(\delta w)}_{I}={w}_{I}^{\text{*}}(\delta {w}_{I})r\frac{\partial {t}_{c}}{\partial (\lambda +rw)}({\lambda }_{I}+r{w}_{I}){\int }_{\Gamma }{\psi }_{I}d\Gamma\) \({\{\lambda \}}_{I}^{\text{*}}{\left[{D}^{\lambda w}\right]}_{II}{(\delta w)}_{I}={\lambda }_{I}^{\text{*}}(\delta {w}_{I})\frac{\partial {t}_{c}}{\partial (\lambda +rw)}({\lambda }_{I}+r{w}_{I}){\int }_{\Gamma }{\psi }_{I}d\Gamma\)

\({\{\lambda \}}_{I}^{\text{*}}{\left[{D}^{\lambda \lambda }\right]}_{II}{(\delta \lambda )}_{I}={\lambda }_{I}^{\text{*}}(\delta {\lambda }_{I})\frac{1}{r}\left(\frac{\partial {t}_{c}}{\partial (\lambda +rw)}({\lambda }_{I}+r{w}_{I})-1\right){\int }_{\Gamma }{\psi }_{I}d\Gamma\)

Elementary vectors have the following expressions:

\({\{w\}}_{I}^{\text{*}}{\left({L}_{\mathit{coh}}^{1}\right)}_{I}={w}_{I}^{\text{*}}{t}_{c}({\lambda }_{I}+r{w}_{I}){\int }_{\Gamma }{\psi }_{I}d\Gamma\)

\({\{w\}}_{I}^{\text{*}}{\left({L}_{\mathit{coh}}^{2}\right)}_{I}=\frac{{w}_{I}^{\text{*}}}{r}{t}_{c}({\lambda }_{I}+r{w}_{I}){\int }_{\Gamma }{\psi }_{I}d\Gamma\)