An approach is presented that enables the calculation of elastic strain energy in linear and nonlinear elastic solids during arbitrary thermomechanical load cycles. The approach uses the simple fact that the variation of both strain and complementary energies always forms a rectangular shape in stress–strain space, hence integration is no longer required to calculate the energy. Furthermore, the approach considers the mean stress effect so that predictions of fatigue damage are more realistically representative of real-life experimental observations. By doing so, a parameter has been proposed to adjust the mean stress effect. This parameter α is based on the well-known Smith–Watson–Topper energy criterion, but allows consideration of other arbitrary mean stress effects, e.g. the Bergmann type criterion.The approach has then been incorporated into a numerical method which can be applied to uniaxial and multiaxial, proportional and non-proportional loadings to predict fatigue damage. The end result of the method is the cyclic evolution of accumulated damage. Numerical examples show how the method presented in this paper could be applied to a nonlinear elastic material

Magnesium is one of the lightest metals for structural components. It has been used for producing various lightweight cast components, but the application of magnesium sheet plates is less widespread. There are two reasons for this: (i) its poor formability at ambient temperatures; and (ii) insufficient data on its durability, especially for dynamic loading. In this article, an innovative approach to predicting the fatigue life of the AZ31 magnesium alloy is presented. It is based on an energy approach that links the strain–energy density with the fatigue life. The core of the presented methodology is a comprehensive new model for tensile and compressive loading paths, which makes it possible to calculate the strain–energy density of closed hysteresis loops. The model is universal for arbitrary strain amplitudes. The material parameters are determined from several low-cycle fatigue tests. The presented approach was validated with examples of variable strain histories.