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The three-dimensional form of Hooke's law can be derived using Poisson's ratio and the one-dimensional form of Hooke's law as follows. Consider the strain and stress relation as a superposition of two effects: stretching in direction of the load (1) and shrinking (caused by the load) in perpendicular directions (2 and 3 ...
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 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 ...
This relationship is commonly known as the generalised Hooke's law. The linear elastic theory involves second order elastic constants (e.g. λ {\displaystyle \lambda } and μ {\displaystyle \mu } ) and yields constant longitudinal and shear sound velocities in an elastic material, not affected by an applied stress.
Mechanical strains are caused by mechanical stress, see stress-strain curve. The relationship between stress and strain is generally linear and reversible up until the yield point and the deformation is elastic. Elasticity in materials occurs when applied stress does not surpass the energy required to break molecular bonds, allowing the ...
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
The elasticity tensor is a fourth-rank tensor describing the stress-strain relation in a ... This relationship can be interpreted as a generalization of Hooke's law ...
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] =