An Indirect Method to Determine Nonlinear-Elastic Shear Stress-Strain Constitutive Relationships for Nonlinear Torsional Deformation of Al-1%Si Shaft

In this paper, an effective method to indirectly determine nonlinear elastic shear stress-strain constitutive relationships for nonlinear elasticity materials is proposed to study the nonlinear torsional deformation of shafts made of nonlinear elasticity materials. BoxLucas1 model is utilized to fit the nonlinear elastic normal stress-strain curve of Al-1%Si, a typical case of isotropic nonlinear elasticity materials, and its corresponding nonlinear elastic shear stress-strain constitutive relationships are derived indirectly on the basis of BoxLucas1 model, general equations of plane-stress transformation, and plane-strain transformation. When the shaft made of Al-1%Si is subjected to a couple of static torques, a nonlinear differential governing equation for static torsional deformation is obtained with the help of the equilibrium equation of torque. DQM is employed to convert the governing equation and boundary conditions to a series of algebraic equations of discretization which are solved numerically by LMM. Numerical results are in good agreement with the analytical solution, and MATLAB's FSOLVE. A detailed parametric study is conducted to analyze the influence of taking into consideration nonlinear-elasticity property to substitute for approximately-simplified linear-elasticity property, and a change in one of external torque T and geometry dimension (length L and radius R) coupling with nonlinear-elasticity property on rotation deformation of Al-1%Si shaft, respectively. In addition, for different kinds of nonlinear elasticity materials with the same tangent at zero-strain as Al-1%Si, we introduce a nonlinear elasticity factor delta into the BoxLucas1 model for describing nonlinear elastic shear stress-strain constitutive relationships and study its influence on rotation deformation of the shafts made of the different types of nonlinear elasticity materials. Results show that (1) For torsional deformation of shafts made of one kind of nonlinear elasticity materials, compared with the utilization of approximately-simplified linear elastic shear constitutive relationships, taking advantage of nonlinear elastic shear stress-strain constitutive relationships instead of linear elastic model causes a negative torsional-stiffness difference, which refers to nonlinear elastic torsional stiffness minus linear elastic torsional stiffness, (2) and T and L have a positive-correlation relationship with absolute value of the torsional-stiffness difference, while R has an opposite and negative-correlation relationship with it; (3) and show that for the different kinds of nonlinear elasticity materials, delta also has a positive-correlation relationship with absolute value of torsional-stiffness difference. The study could provide a reference for indirectly determining nonlinear elastic shear stress-strain constitutive relationships for nonlinear elasticity materials, and for structure design of torsional shaft made of nonlinear elasticity materials.