Final answer:
The compatibility equation in terms of stress ensures that strains from stress result in coherent material deformation. It stems from Hooke's law, which relates stress linearly to strain within an elastic limit. Outside this limit, the equations may become more complex, taking into account the geometry and material behavior.
Step-by-step explanation:
The compatibility equation in terms of stress refers to a condition that ensures the strains resulting from the applied stress are consistent with the continuity of material and the displacements that follow from the strains. The basic compatibility equation in linear elasticity is derived from Hooke's law, which states stress = Y X strain, where Y is the elastic modulus. This equation indicates the direct proportionality between stress and strain within the elastic limit.
Compressive stress and strain follow the same formula with absolute values considered for compressive forces. Furthermore, in the linear region, the relationship between stress and strain is simple; however, as the stress increases, the relationship may become nonlinear, requiring a more complex set of compatibility equations that could include terms from the governing equations of continuum mechanics, like the Navier-Cauchy equations.
Forces acting perpendicular to the cross-section cause deformations, resulting in tensile or compressive stress, which is defined as the force per unit area. Shear deformation, on the other hand, involves forces acting parallel to the area, and it is described with the help of the shear modulus. Compatibility in the context of adhesion and rupture rates involves interfacial stress, which takes into account both the bond strength and bond extension as factors.