Finite deformation analysis has been one important subject of nonlinear stress analysis. Formulation of such phenomenon has been followed through the Eulerian and Lagrangean approaches. Lagrangian approach is mainly formulated based upon the multiplicative decomposition of the deformation gradient into its elastic and plastic parts. In the Eulerian approach however, an additive decomposition is mainly used. Both approaches have been debated based on the assumptions used to decompose the deformation state. Hypoelasticity is an Eulerian approach introduced first by Truesdell to set up an objective rate form of elasticity for the finite deformation analysis. However, use of the proper objective rates in the formulation and model integrability has been debated for several decades. The integrability conditions of hypoelastic model showed that in general the model does not result in a Cauchy/Green elastic model. This motivated the use of Hyperelasticity in the formulation of large deformation elastoplasticity. A multiplicative decomposition was proposed by Lee to be used for the Lagrangian formulation of Hyperelasticity with the assumption of an intermediate stress free configuration. The Lagrangian formulation provided an exactly integrable model for the elastic part of the deformation.
The problem of non-integrability and elastic dissipation in hypoelasticity was considered by several authors. Mainly two different interpretations were followed to solve such issues. Following the Hill’s statement several efforts were made in search for a specific rate of stress which makes the model integrable. Lehmann et al. and Reinhardt and Dubey searched for a specific frame of reference in which the observed rate of strain is the stretching tensor. Therefore, the D rate of stress (D frame) was discovered and introduced. Another interpretation was to search for an exactly integrable model resulting into the Hookean response of the material. Following this approach the logarithmic rate was introduced by Xiao et al. as the unique rate of stress which results in an exactly integrable rate model. The current research focuses on utilizing other physically objective spins to be used in finite deformation analysis. Simplicity of some corotational spins over others motivates the introduction of a new formulation of elastoplasticity. Different applications such as viscoplasticity and damage analysis of ductile materials can be formulated based on such formulation.
References:
-
Eshraghi A., Papoulia K., Jahed H., 2012, Eulerian Framework for Inelasticity Based on the Jaumann Rate and a Hyperelastic Constitutive Relation. Part I. Rate-Form Hyperelasticity, Journal of Applied Mechanics, in press.
-
Eshraghi A., Jahed H., Papoulia K., 2012, Eulerian Framework for Inelasticity Based on the Jaumann Rate and a Hyperelastic Constitutive Relation. Part II. Finite Strain Elastoplasticity, Journal of Applied Mechanics, in press.
-
Eshraghi M.A., Jahed H., Lambert S., “A Lagrangian Model for Hardening Behaviour of Materials at Finite Deformation based on the Right Plastic Stretch Tensor”, Journal of Materials and Design, Vol. 31, 2010, pp. 2342-2354
-
Eshraghi A, " Finite Strain Elastoplasticity: Consistent Eulerian and Lagrangian Approaches ", PhD thesis, Univeristy of Waterloo, May 2009.
-
Eshraghi A, Jahed H and Lambert S , " A Finite Deformation Constitutive Model for the Prediction of Cyclic Behaviour of Shape Memory Alloys ," 12th International Conference on Fracture , Ottawa, Canada, June 2009.
-
Amin Eshraghi, Hamid Jahed and Steve Lambert , " Rate Type Constitutive Models and Their Applications in Finite Elasto-Plasticity ," 3rd Canadian Conference on Nonlinear Solid Mechanics, CanCNSM 2008 , June 2008, Toronto, Canada, pp. 303-311.
-
Amin Eshraghi, Hamid Jahed and Steve Lambert , " Generalized Hypo-Elasticity for Arbitrary Stress Rates ," The international conference of plasticity 2008 , January 08, Kona, Hawaii.