Comparative Analysis and Verification of Objective Algorithms

Simulating body deformations with initial stresses using Hooke‐like isotropic hypoelasticity models based on corotational stress rates

“…According to the procedure of step‐by‐step integration of the equations of deformable body motion for any material model implemented in some FE system, it is necessary, first, to present an algorithm for determining the components of the Cauchy stress tensor in transition from time t to time $$t+\Delta t$$ and, second, to present an expression for the tangent stiffness tensor when using the equations of quasistatic motion or implicit schemes for integrating dynamic motion equations. The algorithms implemented in our homemade FE code and used to determine the Cauchy stress tensor components (a more detailed description of these algorithms is given in [23]) are presented in Section 6.1, and expressions for the tangent stiffness tensors for all considered material models are given in Section 6.2.…”

“…Note that the tensors $$\bm{\Psi }_w$$ and $$\bm{\Psi }_{\log }$$ should be determined by numerically solving the Cauchy problem () using the tensors $$\mathbf {w}$$ and $$\bm{\omega }^{\log }$$ as the tensor ω , respectively, and the tensor $$\mathbf {R}$$ is determined directly from the polar decomposition of the tensor $$\mathbf {F}$$ (see, e.g., Appendix B in [23]).…”

“…To mathematically formulate quasi‐linear relations between stresses, strains, and their rates, one can use several theoretically different definitions of elasticity [22]: elasticity as such (Cauchy elasticity), hyperelasticity (Green elasticity), and hypoelasticity. In the numerical implementation of CRs for these material models, the first two definitions of elastic material have an obvious advantage over the last one; that is, the first two definitions specify relations between stresses and kinematic quantities explicitly, and the last definition generally requires integrating hypoelastic CRs governing the transition from one discrete time instant to another (see, e.g., [23]). However, in the presence of arbitrary initial stresses, CRs for Cauchy/Green elasticity are theoretically inapplicable, but for hypoelastic CRs, there are no such restrictions.…”

“…In 2011, Korobeynikov [45] introduced a family of continuous material spin tensors (which is a subfamily of the family of material spin tensors) with the desired additional continuity property for smooth motion and showed that the twirl tensor associated with the Gurtin–Spear corotational tensor rate does not belong to the family of continuous material spin tensors (see also [46]). In this work, we use hypoelastic models associated with classical corotational stress rates [23], namely, Zaremba–Jaumann, Green–Naghdi, and logarithmic rates of Cauchy and Kirchhoff stresses. Note that all material models considered in this paper use corotational stress rates associated with spin tensors from the family of continuous material spin tensors.…”

“…For large rotations of elemental volumes combined with large time steps and implicit integration of motion equations, direct integration of hypoelastic CRs can result in accumulation of large errors. In this study, we use the objective algorithms for integrating stresses presented in the author's book [23]. These objective algorithms are based on considering objective tensor rates in the form of Lie time derivatives [4, 47, 48], which makes it possible to integrate stresses with high accuracy when using large time steps.…”