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Compliance constants are the elements of an inverted Hessian matrix. The calculation of compliance constants provides an alternative description of chemical bonds in comparison with the widely used force constants explicitly ruling out the dependency on the coordinate system.
The compliance matrix is symmetric and must be positive definite for the strain energy density to be positive. This implies from Sylvester's criterion that all the principal minors of the matrix are positive, [6] i.e., := (_ _) >
The elasticity stiffness matrix has 5 independent constants, which are related to well known engineering elastic moduli in the following way. These engineering moduli are experimentally determined. The compliance matrix (inverse of the elastic stiffness matrix) is
The compliance tensor is defined from the inverse stress-strain relation: = The two are related by = (+) ... are the elements of the corresponding rotation matrix. A ...
The stiffness, , of a body is a measure of the resistance offered by an elastic body to deformation. For an elastic body with a single degree of freedom (DOF) (for example, stretching or compression of a rod), the stiffness is defined as = where,
Anelasticity is a property of materials that describes their behaviour when undergoing deformation.Its formal definition does not include the physical or atomistic mechanisms but still interprets the anelastic behaviour as a manifestation of internal relaxation processes.
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 ...
Compliance at any point in Load vs displacement curve is essentially the reciprocal of the slope of the curve that ensues if the specimen is unloaded at that point. Now the unloading curve returns to the origin for linear elastic material but not for elastic plastic material as there is a permanent deformation.