Abstract
This paper describes a general method for deriving the plastic corrections and the consistent tangent modulus for a wide range of arbitrary non-linear hardening models within the framework of standard small strains elastoplasticity. The features of the proposed formulation are: (i) the local solution is obtained through an iterative procedure. The plastic corrections are given in closed forms exhibiting one scalar function denoted by G alg and three fourth-order tensors D alg , G alg , L alg , which are shown to be the algorithmic discrete counterparts of usual theoretical continuum quantities, (ii) the consistent tangent modulus has a symmetrical expression involving the same quantities. Finite element computations are performed using a particular non-linear kinematic hardening model and allow to exhibit the ratcheting phenomenon usually observed on mechanical components subjected to cyclic loadings.
Highlights
The numerical response of an elastoplastic structure is usually determined by an incremental method where the solution is computed at every time step by using the finite element method combined with a Newton-type iterative scheme
Within each iteration it is necessary to perform the two following important tasks characteristic of any elastoplastic algorithm: first, the local integration of the non-linear constitutive equations in order to compute the stress for a given strain increment and to build up the internal force vector, second, the computation of the consistent tangent modulus in order to form the structural tangent matrix
Later, dealing with the plane stress situation, Simo and Govindjee [29] showed that the linear kinematic hardening leads to an algebraic equation of degree four which can be solved analytically
Summary
The numerical response of an elastoplastic structure is usually determined by an incremental method where the solution is computed at every time step by using the finite element method combined with a Newton-type iterative scheme. Within each iteration it is necessary to perform the two following important tasks characteristic of any elastoplastic algorithm: first, the local integration of the non-linear constitutive equations in order to compute the stress for a given strain increment and to build up the internal force vector, second, the computation of the consistent tangent modulus in order to form the structural tangent matrix. The complete stress state is obtained by solving a governing scalar equation called the effective stress function equation and the consistent tangent matrix is derived thereof This method is applicable for more complex inelastic situations, like Drucker–Prager soil model, thermoelastoplasticity and creep solutions, as well as large strain elastoplasticity [9]. The finite element computations carried out on the example of a particular non-linear kinematic hardening model allow to assess the validity of the proposed algorithm and to exhibit different types of ratcheting usually observed on mechanical components subjected to cyclic loadings
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