Generalized Hooke’s Law

While Robert Hooke’s original 1678 law (“ut tensio, sic vis” – as the extension, so the force) described a simple one-dimensional linear relationship, the generalized Hooke’s Law extends this principle to three dimensions. It forms the mathematical foundation of the theory of linear elasticity. The relationship connects the six independent components of the stress tensor to the six independent components of the infinitesimal strain tensor. This is achieved through the stiffness tensor [latex]C_{ijkl}[/latex], a fourth-order tensor containing 81 components in its most general form.

Due to the symmetry of the stress and strain tensors, the number of independent components in the stiffness tensor reduces to 36. Furthermore, assuming the existence of a strain energy density function, the stiffness tensor itself becomes symmetric ([latex]C_{ijkl} = C_{klij}[/latex]), reducing the number of independent elastic constants to 21 for the most general anisotropic material. For materials with higher degrees of symmetry, this number is further reduced. For an isotropic material, which has the same properties in all directions, only two independent elastic constants are needed, such as Young’s Modulus (E) and Poisson’s Ratio (ν). In this common case, the law simplifies significantly, allowing for direct calculation of stresses from strains and vice-versa. This law is only valid within the material’s elastic limit; beyond this point, permanent plastic deformation occurs, and other constitutive models are required.