Hooke's Law for Orthotropic Materials Hooke's Law: Orthotropic Materials Formula Home Mechanics of Matl. Stress Strain Hooke's Law Orthotropic Material Transverse Isotropic Isotropic Material Plane Stress Plane Strain Finding E and n Finding G and K Applications Pressure Vessels Rosette Strain Gages Failure Criteria Calculators Stress Transform Strain Transform Principal Stress Principal Strain Elastic Constants Resources Bibliography  Login   Home Membership Magazines Forum Search Member Calculators  Materials  Design  Processes  Units  Formulas  Math Orthotropic Definition Some engineering materials, including certain piezoelectric materials (e.g. Rochelle salt) and 2-ply fiber-reinforced composites, are orthotropic. By definition, an orthotropic material has at least 2 orthogonal planes of symmetry, where material properties are independent of direction within each plane. Such materials require 9 independent variables (i.e. elastic constants) in their constitutive matrices. In contrast, a material without any planes of symmetry is fully anisotropic and requires 21 elastic constants, whereas a material with an infinite number of symmetry planes (i.e. every plane is a plane of symmetry) is isotropic, and requires only 2 elastic constants. Hooke's Law in Compliance Form By convention, the 9 elastic constants in orthotropic constitutive equations are comprised of 3  Young's modulii Ex, Ey, Ez, the 3 Poisson's ratios nyz, nzx, nxy, and the 3 shear modulii Gyz, Gzx, Gxy. The compliance matrix takes the form, where . Note that, in orthotropic materials, there is no interaction between the normal stresses sx, sy, sz and the shear strains eyz, ezx, exy The factor 1/2 multiplying the shear modulii in the compliance matrix results from the difference between shear strain and engineering shear strain, where , etc.
 Hooke's Law in Stiffness Form The stiffness matrix for orthotropic materials, found from the inverse of the compliance matrix, is given by, where, The fact that the stiffness matrix is symmetric requires that the following statements hold, The factor of 2 multiplying the shear modulii in the stiffness matrix results from the difference between shear strain and engineering shear strain, where , etc.
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