Linear Elastic Behavior (Constitutive Relations) (§ 2.3.0)

What are our variables?
ε	Axial Strain
γ	Shear Strain
σ	Normal Stress
τ	Shear Stress
E	Young's Modulus
ν	Poisson's Ratio
G	Modulus of Rigidity

The linear elastic behavior, used to describe the material properties of an element, assumes that strain is proportional to the stress on the element. This assumption explains how deformations appear when stress is applied and how they disappear when stress is removed. Hooke's Law, which mathematically expresses this linear relationship, allows us to express the strains in terms of the stresses:

εxx = 1E (σxx - νσyy)

(22)

εyy = 1E (σyy - νσxx)

(23)

γxy = τxyG

(24)

G = E 2 (1 + ν)

(25)

Explanation: Due to the Poisson effect, an expanding normal strain in the x-direction will cause a proportional compressive normal strain in the y-direction, and the constant of proportionality is Poisson's ratio, ν. The equations above are merely different expressions of Hooke's law for plane stress, with the strains being added together by means of the principle of superposition. It should be noted that the expressions are valid only if the material is linear-elastic.