# On the Effect of Friction on Tibiofemoral Joint Kinematics
## 3 Contact Solver: Tangential Friction and Normal Contact Forces
Finally, the resultant friction force tangentially applied to the femoral component can be obtained from the following integration that is performed over the contact area:  
\[\mathbf{F}_{\mu}=\iint\limits_{\Omega}\mathbf{F}\,dA=\sum_{k}\sum_{j}\mathbf{ F}^{k,j}dA^{k,j} \tag{21}\]  
Figure 2: A representation of the friction model A\({}_{2}\) on the right-hand side and B\({}_{2}\) on the left-hand side. where \(\Omega\) depicts the contact area, \(\mathbf{F}^{k,j}\) is friction force at each element and \(dA^{k,j}\) depicts the size of element area. The present study implements a concept proposed in [69] to evaluate contact stresses, but it is impossible to derive a closed-form mathematical formulation for the contact problem at hand. [PERSON] obtained nonlinear and promising contact models while resolving the discontinuity issue with the Kelvin-Voigt model at both the beginning and end of the contact process [69]. To this end, he simulated the contact by using a set of independent springs and dashpots, which represent the stiffness and viscous damping of the contact between articulating bodies, respectively. Readers interested are referred to the reference [69; 70; 71; 72] and references therein for further information about the model employed in this study. Such a contact model does not put any kinematic constraints on the dynamical system. It is well-known that the Kelvin-Voigt model suffers from a discontinuity at both the beginning and end of the contact process due to the form of the damping term in its formulation [73; 74]. The concept proposed in the reference [69] showed that simulating a soft contact using a set of independent Kelvin-Voigt elements spread all over the contact surfaces instead of just one spring-damper element as is used in the Kelvin-Voigt contact model can resolve the above discontinuity. The penetration depth at each node on the polyethylene surface, \(D_{k,j}\), is known from the minimum distance technique discussed later in the present paper (see Section 5.3). Its time derivative \(\dot{D}_{k,j}\) is also obtained from the normal velocity term given in Equation (17). Therefore, the normal contact pressure, i.e., \(\sigma^{k,j}\), at any node of \(Q_{T}^{k,j}\) can be determined based on the following Kelvin\(-\)Voigt contact relationship:  
\[\sigma^{k,j}=-KD_{k,j}-K\tau\dot{D}_{k,j} \tag{22}\]  
in which \(K\) is the contact stiffness and \(\tau\) the relaxation time, \(K\) is obtained from \(K=\Lambda/\Xi\) where \(\Lambda=(1-v)E/[(1+v)(1-2v)]\) and \(\Xi\) is the tibia thickness [40], \(E\) and \(v\) are [PERSON]'s modulus and [PERSON]'s ratio of the polyethylene that are, respectively, 463 MPa and 0.46 as are reported in [75; 76; 77]. Such a contact model belongs to the category of the penalty non-adhesive contact approaches and penalizes the indentation of the master body into the slave one. As independent Kelvin\(-\)Voigt elements are spread all over the contact area, the normal contact force applied to the femoral components should be computed by performing the following integration over the contact area:  
\[\mathbf{F}_{cont}=\iint\limits_{A}\sigma\mathbf{n}_{F}\;dA \tag{23}\]  
where \(\mathbf{n}_{F}\) is the normal unit vector of an element with the area \(dA\) on the femur-bearing surface. It is worth noting that the following criterion is used to assess whether a node is in contact or not. \[\left\{ \begin{array}{ll}D_{i,j}>0&\quad no\;contact\\ D_{i,j}\leq 0&\quad contact\end{array} \right. \tag{24}\]  
The resultant contact area can also be obtained according to the summation of the area associated with all elements engaged in contact area having nodes with non-zero contact pressure.