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Cumulative Damage Power Relationship

This section presents a generalized formulation of the cumulative damage model where stress can be any function of time and the life-stress relationship is based on the power relationship. Given a time-varying stress [math]\displaystyle{ x(t) }[/math] and assuming the power law relationship, the life-stress relationship is given by:

[math]\displaystyle{ L(x(t))={{\left( \frac{a}{x(t)} \right)}^{n}} }[/math]

In ALTA, the above relationship is actually presented in a format consistent with the general log-linear (GLL) relationship for the power law relationship:

[math]\displaystyle{ L(x(t))={{e}^{{{\alpha }_{0}}+{{\alpha }_{1}}\ln \left( x(t) \right)}} }[/math]

Therefore, instead of displaying [math]\displaystyle{ a }[/math] and [math]\displaystyle{ n }[/math] as the calculated parameters, the following reparameterization is used:

[math]\displaystyle{ \begin{align} {{\alpha }_{0}}=\ & \ln ({{a}^{n}}) \\ {{\alpha }_{1}}=\ & -n \end{align} }[/math]