Template:Cd relationship

Cumulative Damage Arrhenius 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 Arrhenius life-stress relationship. Given a time-varying stress [math]\displaystyle{ x(t) }[/math] and assuming the Arrhenius relationship, the life-stress relationship is given by:

[math]\displaystyle{ L(x(t))=C{{e}^{\tfrac{b}{x(t)}}} }[/math]

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

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

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

[math]\displaystyle{ \begin{align} {{\alpha }_{0}}=\ & \ln (C) \\ {{\alpha }_{1}}=\ & b \end{align} }[/math]