The Fisher-Burmeister complementary function \(\Phi{\left(x,y\right)}\) is defined as follows:
\[ \Phi{\left(x,y\right)}=x+y-\sqrt{x^{2}+y^{2}} \]
\(\Phi{\left(x,y\right)}\) has the following property:
\[ \Phi{\left(x,y\right)}=0 \quad\Leftrightarrow\quad \left\{ \begin{aligned} x&\geq 0\\ y&\geq 0\\ x\,y&=0 \end{aligned} \right. \qquad{(1)}\]
Equation (1) clearly reflects the classical Karush-Kuhn-Tucker condition of plasticity:
\[ \left\{ \begin{aligned} \dot{\lambda}&\geq 0\\ f&\leq 0\\ \dot{\lambda}\,f&=0 \end{aligned} \right. \qquad{(2)}\]
where:
In other words, finding a plastic multiplier \(\dot{\lambda}\) satisfying Conditions (2) is equivalent to finding the root of the following equation:
\[ \Phi{\left(\dot{\lambda},-f\right)}=0 \qquad{(3)}\]
If several plastic surfaces are involved, each plastic multiplier must satisfy an equation similar to (3). In an implicit formulation, this removes the need of using a status method to determine which mechanisms are active or inactive.
\(\Phi{\left(x,y\right)}\) is differentiable everywhere, except at \(0,0\). For this reason, following [1], we introduce a regularized version denoted \(\Phi_{\varepsilon}{\left(x,y\right)}\):
\[ \Phi_{\varepsilon}{\left(x,y\right)}=x+y-\sqrt{x^{2}+y^{2}+\varepsilon^{2}} \]
where \(\varepsilon\) is a small regularization parameter. \(\Phi_{\varepsilon}{\left(x,y\right)}\) has the following property:
\[ \Phi_{\varepsilon}{\left(x,y\right)}=0 \quad\Leftrightarrow\quad \left\{ \begin{aligned} x&\geq 0\\ y&\geq 0\\ 2\,x\,y&=\varepsilon^{2} \end{aligned} \right. \qquad{(4)}\]
The authors are grateful to Y. Chemisky, main developer of the
Simcoon library, for introducing them to the use of the
Fisher-Burmeister complementary function in multi-surface
plasticity.