restoring force whose sign is negative for L>Lo and positive for L< Lo. If the displacement is abbreviated as then Hooke's Law can be written in the usual form Energy absorbed and held in the spring can be derived using Hooke's Law to compute the restoring force as a measure of the applied force. This requires the assumption, sufficiently correct in most circumstances, that at a given moment, the magnitude of applied force, Fa is equal to the magnitude of the resultant restoring force, but its direction and thus sign is different. In other words, assume that at each point of the displacement Fa = k x, where Fa is the component of applied force along the x direction For each infinitesimal displacement dx, the applied force is simply k x and the product of these is the infinitesimal transfer of energy into the spring dU. The total elastic energy placed into the spring from zero displacement to final length L is thus the integral For a material of Young's modulus, Y (same as modulus of elasticity λ), cross sectional area, A0, initial length, l0, which is stretched by a length, : where Ue is the elastic potential energy. The elastic potential energy per unit volume is given by: where is the strain in the material. In the general case, elastic energy is given by the free energy per unit of volume f as a function of the strain tensor components εij where λ and μ are the Lamé elastic coefficients and we use Einstein summation convention. Noting the thermodynamic connection between stress tensor components and strain tensor components, where the subscript T denotes that temperature is held constant, then we find that if Hooke's law is valid, we can write the elastic energy density as Continuum systems Matter in bulk can be distorted in many different ways: stretching, shearing, bending, twisting, etc. Each kind of distortion contributes to the elastic energy of a deformed material. In orthogonal coordinates, the elastic energy per unit volume due to strain is thus a sum of contributions: where is a 4th rank tensor, called the elastic tensor or stiffness tensor which is a generalization of the elastic moduli of mechanical systems, and is the strain tensor (Einstein summation notation has been used to imply summation over repeated indices). The values of depend upon the crystal structure of the material: in the general case,