A constitutive model for linear hyperelastic materials with orthotropic inclusions by use of quaternions

An approximated theoretical model for linear hyperelastic materials with orthotropic inclusions is presented based on a mixture approach and orientation averaging of mechanical properties. In many modern composites, particle inclusions are used to achieve the desired properties. The properties of the resulting composite depend on the material, inclusion-density and also the shape and orientation of the particles. This paper discusses different methods to describe the orientation of the particles and formulate an orientation density distribution function. In particular, the use of quaternions is compared to Cardan angles for orientation averaging. The proposed approximating material model can be applicable to a composite made of a matrix material that contains biaxial particles or a material that is composed of particles of an orthotropic material. An example for the former could be fiber concrete with hooked-end fibers and an example for the latter could be particle boards, as wood is an orthotropic material.