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The use of the classical Coulomb law of friction in the formulation of contact problems in elasticity leads to both physical and mathematical difficulties; the former arises from the fact that this law provides a poor model of frictional stresses at points on metallic surfaces in contact, and the latter is due to the fact that the existence of solutions of the governing equations can be proved only for very special situations. In the present paper, nonclassical friction laws are proposed in an attempt to overcome both of these difficulties. We consider a class of contact problems involving the equilibrium of linearly elastic bodies in contact on surfaces on which nonlocal and nonlinear friction laws are assumed to hold. The physics of friction between metallic bodies in contact is discussed and arguments in support of the theory are presented. Variational principles for boundary-value problems in elasticity in which such nonlinear nonlocal laws hold are then developed. A brief discussion of the questions of existence and uniqueness of solutions to the nonlocal and nonlinear problems is given.

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This article describes research involving finite element simulations of women's pelvic floor, undertaken in the engineering schools of Lisbon and Oporto, in collaboration with the medical school of Oporto. These studies are motivated by the pelvic floor dysfunctions that lead namely to urinary incontinence and pelvic organ prolapse. This research ultimately aims at: (i) contributing to clarify the primary mechanism behind such disorders; (ii) providing tools to simulate the pelvic floor function and the effects of its dysfunctions; (iii) contributing to planning and performing surgeries in a more controlled and reliable way. The finite element meshes of the levator ani are based on a publicly available geometric data set, and use triangular thin shell or special brick elements. Muscle and soft tissues are assumed as (quasi-)incompressible hyperelastic materials. Skeletal muscles are transversely isotropic with a single fiber direction, embedded in an isotropic matrix. The fibers considered in this work may be purely passive, or active with input of neuronal excitation and consideration of the muscle activation process. The first assumption may be adequate to simulate passive deformations of the pelvic muscles and tissues (namely, under the extreme loading conditions of childbirth). The latter may be adequate to model faster contractions that occur in time intervals of the same order as those of muscle activation and deactivation (as in preventing urinary incontinence in coughing or sneezing). Numerical simulations are presented for the active deformation of the levator ani muscle under constant pressure and neural excitation, and for the deformation induced by a vaginal childbirth.

The pelvic floor gives support to the organs in the abdominal cavity. Using the dataset made public in [11] we have reconstructed the geometry of one of the most important parts of the pelvic floor, the levator ani, using NURB surfaces. Once the surface is triangulated, this mesh is used in a finite element analysis with shell elements.Based on the 3D behavior of the muscle we have constructed a shell that takes into account the direction of the muscle fibers and the incompressibility of the tissue. The constitutive model for the isotropic strain energy and the passive strain energy stored in the fibers are adapted from Humphrey's model for cardiac muscles. To this the active behavior of the skeletal muscle is added.We present preliminary results of a simulation of the levator ani muscle under pressure and with active contraction. This research aims at helping predict the damages to the pelvic floor that can occur during childbirth.