A direct approach to fiber and membrane reinforced bodies. Part I. Stress concentrated on curves for modelling fiber reinforced materials

An approach is outlined to the equilibrium in fiber-reinforced materials in which the fibers are modeled as curves or lines with concentrated material properties. The system of forces representing the interaction of the fibers with the bulk matter is analyzed, and equilibrium of forces is derived from global laws. The displacements of the bulk matter are assumed to have continuous extension to the fibers. This forces the set of admissible deformations superquadratically integrable. This in turn forces the energy of the bulk of superquadratic growth. The material of the bulk matrix therefore cannot be linearly elastic. The energy of fibers can have a slower growth and can be quadratic. A formal set of assumptions is given under which an equilibrium state of minimum energy exists in the given external conditions. A weak form of equilibrium equations is derived for this equilibrium state. An explicitly calculable axisymmetric example is presented with an isotropic and quadratic energy of the matrix (linear elasticity) and linearly stretchable fiber. Since the superquadratic growth assumption is not satisfied, some peculiar features of the solution arise, such as the infinite limit of the radial displacement near the fiber. Nevertheless, from the obtained solution, we can compute the normal force in the fiber and the shear stress at the interface.