5. Inner forces and tangent matrix#
5.1. Pure mechanical case#
The formulation of the mechanical problem (see [R7.02.11] and [R7.01.25]) involves the work of forces along the discontinuity, which is none other than the surface energy linked to the cracking of the structure:
\({W}_{s}(\delta )=\sum _{g}{\omega }_{g}\psi ({\delta }_{g})\)
with \(\psi\) surface energy density and \({\omega }_{g}\) gauss point weight \(g\). This makes it possible to define the vector of internal forces:
\({F}_{\text{int}}^{U}=\frac{\partial {W}_{s}(\delta )}{\partial U}=\sum _{g}{\omega }_{g}\frac{\partial \psi }{\partial {\delta }_{g}}\frac{\partial {\delta }_{g}}{\partial U}\)
In the previous expression, the first term is given by the law of cohesive behavior (see [R7.02.11]). This corresponds to the stress vector \(\overrightarrow{{\sigma }_{g}}\) (or cohesive force) at the gauss point \(g\):
\(\frac{\partial \psi }{\partial {\delta }_{g}}=\overrightarrow{{\sigma }_{g}}\)
The second term comes from the definition of jump displacement in part 3:
\(\frac{\partial {\delta }_{g}}{\partial U}={M}_{g}^{U}\)
The nodal vector of internal forces is therefore expressed in the following way:
\({F}_{\text{int}}^{U}=\sum _{g}{\omega }_{g}{{M}_{g}^{U}}^{t}\overrightarrow{{\sigma }_{g}}\)
In the context of a Newton algorithm, to solve the non-linear equilibrium problem, it is useful to have the elementary tangent matrix, that is to say the derivative of the internal forces with respect to the nodal displacements. In the case of the joint element, it is expressed simply:
\({K}^{\mathrm{UU}}=\frac{\partial {F}_{\text{int}}^{U}}{\partial U}=\sum _{g}{\omega }_{g}{{M}_{g}^{U}}^{t}\frac{\partial \overrightarrow{{\sigma }_{g}}}{\partial {\delta }_{g}}{M}_{g}^{U}\)
The latter is based on the tangent operator: \(\frac{\partial \overrightarrow{{\sigma }_{g}}}{\partial {\delta }_{g}}\) specific to the law of cohesive behavior adopted (see [R7.02.11]).
5.2. Hydromechanical coupled case#
Joints HYME, in addition to the nodal forces linked to mechanics \({F}_{\text{int}}^{U}\), to which we refer [2] _ the fluid pressure at the Gauss point on the normal component \(\overrightarrow{{p}_{g}}=({p}_{g}\mathrm{,0}\mathrm{,0})\) (expressed in the local coordinate system at the crack):
\({F}_{\text{int}}^{U}=\sum _{g}{\omega }_{g}{{M}_{g}^{U}}^{t}(\overrightarrow{{\sigma }_{g}}-\overrightarrow{{p}_{g}})\)
possess nodal forces for the flow of fluid on the nodes that carry pressure ddls.
The formulation of the hydraulic problem (see [R7.01.25]) involves the work of the forces of the fluid along the flow path (inside the crack):
\({W}_{F}(\nabla p)=\sum _{g}{\omega }_{g}H(\nabla p)\)
with \(H\) surface energy density and \({\omega }_{g}\) gauss point weight \(g\). This makes it possible to define the vector of internal forces:
\({F}_{\text{int}}^{P}=\frac{\partial {W}_{F}(\nabla p)}{\partial P}=\sum _{g}{\omega }_{g}\frac{\partial H}{\partial \nabla {p}_{g}}\frac{\partial \nabla {p}_{g}}{\partial P}\)
In the previous expression, the first term is given by the law of cubic fluid behavior (see [R7.01.25]). This corresponds to hydraulic flow \(\overrightarrow{{w}_{g}}\) at the gauss point \(g\):
\(\frac{\partial H}{\partial \nabla {p}_{g}}=\overrightarrow{{w}_{g}}\)
Based on the definition of the pressure gradient in 4. The second term is given by:
\(\frac{\partial \nabla {p}_{g}}{\partial P}={M}_{g}^{P}\)
The nodal vector of internal forces is therefore expressed in the following way:
\({F}_{\text{int}}^{P}=\sum _{g}{\omega }_{g}{{M}_{g}^{P}}^{t}\overrightarrow{{w}_{g}}\)
In the context of a Newton algorithm, to solve the non-linear equilibrium problem, it is useful to have the tangent matrix, that is to say the derivative of the internal forces with respect to the degrees of freedom. The derivative of the internal forces on the degrees of freedom of movement with respect to the nodal movements gives the term identical to that of the matrix obtained in pure mechanics in 5.1:
\({K}^{\mathrm{UU}}=\frac{\partial {F}_{\text{int}}^{U}}{\partial U}=\sum _{g}{\omega }_{g}{{M}_{g}^{U}}^{t}\frac{\partial \overrightarrow{{\sigma }_{g}}}{\partial {\delta }_{g}}{M}_{g}^{U}\)
In the case of hydraulic coupling, the \({F}_{\text{int}}^{U}\) depend explicitly on the pressure (see expression above) from where:
\({K}^{\mathrm{UP}}=\frac{\partial {F}_{\text{int}}^{U}}{\partial P}=\sum _{g}{\omega }_{g}{{M}_{g}^{U}}^{t}{X}_{g}\)
with \({X}_{g}=\frac{\partial \overrightarrow{{p}_{g}}}{\partial P}=(-{N}_{g}^{i}\mathrm{,0}\mathrm{,0})\) with \(i=1\text{à}\mathrm{Nb}\), \(\mathrm{Nb}\) number of pressure knots (or even number of pressure degrees of freedom per element) and \({N}_{n}^{g}\) the value of the shape function of the node \(n\) at the Gauss point \(g\).
The derivative of the internal forces on the degrees of freedom of pressure with respect to the nodal movements gives:
\({K}^{\mathrm{PU}}=\frac{\partial {F}_{\text{int}}^{P}}{\partial U}=\sum _{g}{\omega }_{g}{{M}_{g}^{P}}^{t}\frac{\partial \overrightarrow{{w}_{g}}}{\partial {\delta }_{g}}{M}_{g}^{U}\)
Finally, the derivative of the internal forces on the degrees of freedom of pressure with respect to the nodal pressures gives:
\({K}^{\mathrm{PP}}=\frac{\partial {F}_{\text{int}}^{P}}{\partial P}=\sum _{g}{\omega }_{g}{{M}_{g}^{P}}^{t}\frac{\partial \overrightarrow{{w}_{g}}}{\partial \nabla {p}_{g}}{M}_{g}^{P}\)
The elementary tangent matrix (not symmetric) is expressed in the following way:
\(K=\left[\begin{array}{cc}{K}^{\mathrm{UU}}& {K}^{\mathrm{UP}}\\ {K}^{\mathrm{PU}}& {K}^{\mathrm{PP}}\end{array}\right]\)
The latter is based on the components of the tangent operator: \(\frac{\partial \overrightarrow{{\sigma }_{g}}}{\partial {\delta }_{g}}\), \(\frac{\partial \overrightarrow{{w}_{g}}}{\partial {\delta }_{g}}\) and \(\frac{\partial \overrightarrow{{w}_{g}}}{\partial \nabla {p}_{g}}\) that are specific to the adopted cohesive behavior law (see [R7.01.25]).