23.2 Ramberg-Osgood Model

In the Ramberg-Osgood model, the relationship between shear stress and shear strain is defined by

$$\gamma {\frac{{\tau}}{{G_\mathrm{max}}}} \left(\vphantom{1+\alpha\left(\frac{\tau}{\tau_\mathrm{f}}\right)^{\!\beta-1}}\right. \alpha \left(\vphantom{\frac{\tau}{\tau_\mathrm{f}}}\right. {\frac{{\tau}}{{\tau_\mathrm{f}}}} \left.\vphantom{\frac{\tau}{\tau_\mathrm{f}}}\right)^{{\!\beta-1}}_{} \left.\vphantom{1+\alpha\left(\frac{\tau}{\tau_\mathrm{f}}\right)^{\!\beta-1}}\right)$$ (23.6)

with Gmax and $\tau_{\mathrm}^{}$f material parameters representing the maximum tangent shear modulus and the characteristic shear stress, respectively. The characteristic shear stress is defined by

$$\tau_{\mathrm}^{} \gamma_{\mathrm}^{}$$ (23.7)

where $\gamma_{\mathrm}^{}$r is the characteristic shear strain. The maximum tangent shear modulus Gmax is calculated from:

$${\frac{{E}}{{2\left(1+\nu\right)}}}$$ (23.8)

where E and $\nu$ are the initial Young's modulus and Poisson's ratio, respectively.