Material with Volume Interactions at a Distance. Fibre Reinforced Material

Abstract

Let us consider a solid containing a large number of long fibres. Those fibres apply actions at a distance. On a macroscopic scale, the engineering scale we choose, the solid fibre mixture appears as an homogeneous continuum medium. We decide to describe the deformations due to the fibres by the variations in the distances between the points of the continuum medium. In a subdomain D of the part Ω of R ³ occupied by the material, the actions at a distance on D are the actions at a distance of the points of D on D itself and the actions at a distance of the points of R ³ — D on D (the exterior nonlocal actions). The interior forces resulting from those actions are defined by their power (Sect. 10.1). Those nonlocal actions between two points are forces pointing toward the two points. The definition of the exterior forces (Sect. 10.2) and the virtual power principle give the equations of motion (Sect. 10.4). A mixture of sand and long textile fibres is an example of such a material. The fibres are active only when they are under traction (Sect. 10.9.1). The fibres break when they are subjected to too large a traction (Sect. 10.9.2). By using the volume fraction of unbroken fibres as a state quantity, a macroscopic predictive theory is established for the mixture. With the constitutive laws chosen in this theory, the rupture of the fibres is irreversible: the fibres cannot mend by themselves. A material of this type is used in civil engineering, Texsol [159, 244], a mixture of granular material, for instance, sand and textile fibres (Fig. 10.1).