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The first constitutive equation (constitutive law) was developed by Robert Hooke and is known as Hooke's law.It deals with the case of linear elastic materials.Following this discovery, this type of equation, often called a "stress-strain relation" in this example, but also called a "constitutive assumption" or an "equation of state" was commonly used.
In physics, Hooke's law is an empirical law which states that the force (F) needed to extend or compress a spring by some distance (x) scales linearly with respect to that distance—that is, F s = kx, where k is a constant factor characteristic of the spring (i.e., its stiffness), and x is small compared to the total possible deformation of the spring.
These integral expressions are a generalization of Hooke's law in the case of anelasticity, and they show that material acts almost as they have a memory of their history of stress and strain. These two of equations imply that there is a relation between the J(t) and M(t).
The defining equation can be written as ... This relationship can be interpreted as a generalization of Hooke's law to a 3D ... is defined from the stress-strain ...
Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are: [1]. Equation of motion: , + = where the (), subscript is a shorthand for () / and indicates /, = is the Cauchy stress tensor, is the body force density, is the mass density, and is the displacement.
In homogeneous and isotropic materials, these define Hooke's law in 3D, = + (), where σ is the stress tensor, ε the strain tensor, I the identity matrix and tr the trace function. Hooke's law may be written in terms of tensor components using index notation as σ i j = 2 μ ε i j + λ δ i j ε k k , {\displaystyle \sigma _{ij}=2\mu ...
The stress is proportional to the strain, that is, obeys the general Hooke's law, and the slope is Young's modulus. In this region, the material undergoes only elastic deformation. The end of the stage is the initiation point of plastic deformation. The stress component of this point is defined as yield strength (or upper yield point, UYP for short
In linear elasticity, the relation between stress and strain depend on the type of material under consideration. This relation is known as Hooke's law. For anisotropic materials Hooke's law can be written as [3] =