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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 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 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 compliance tensor is defined from the inverse stress-strain relation: = The two are related by = (+) ... are the elements of the corresponding rotation matrix. A ...
In order for the WCCA to be useful throughout the product’s life cycle, it is extremely important that the analysis be documented in a clear and concise format. This will allow for future updates and review by other than the original designer. A compliance matrix is generated that clearly identifies the results and all issues.
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 ...
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When there are degrees of freedom a matrix must be used to describe the stiffness at the point. The diagonal terms in the matrix are the direct-related stiffnesses (or simply stiffnesses) along the same degree of freedom and the off-diagonal terms are the coupling stiffnesses between two different degrees of freedom (either at the same or ...
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