Wavelet-Galerkin method for one-dimensional elastoplasticity and damage problems: Constitutive modeling and computational aspects

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Abstract

This work presents an analysis of the wavelet-Galerkin method for one-dimensional elastoplastic-damage problems. Time-stepping algorithm for non-linear dynamics is presented. Numerical treatment of the constitutive models is developed by the use of return-mapping algorithm. For spacial discretization we can use wavelet-Galerkin method instead of standard finite element method. This approach allows to locate singularities. The discrete formulation developed can be applied to the simulation of one-dimensional problems for elastic–plastic-damage models.

Section snippets

Formulation of the problem

The governing equations obtained from the study of dynamic response of an elastoplastic-damage problem are represented by the mathematical model that we describe below.

Let Ω a body with boundary ∂Ω in a one-dimensional space R. The space variable is denoted by x ∈ Ω, the time variable by t ∈ I = ]0, T[ and u(x,t)∈R represent the displacement of a point x in the body. The linearized strain is represented by ϵ = u,x(x, t) and the Cauchy stress σ is related to ϵ through a constitutive relation (specified in

Variational formulation

To define the variational formulation of the dynamic problem (Eqs. (1), (2), (3), (4), (5)), we need to characterize two classes of functions.

For each t, let St denote the space of trial solutions,St={u(.,t):u(x,t)=g(x,t),x∈∂Ωg,u(.,t)∈H1(Ω)},where H1(Ω) is a Hilbert space and let V, the usual space of weighting functions, or test functions, satisfying zero-displacement Dirichlet boundary conditions, namelyV={w:w(x)=0,x∈∂Ωg,w∈H1(Ω)}.The weak formulation for the problem (Eqs. (1), (2), (3), (4),

Galerkin formulation

Galerkin methods are projections methods. In the Galerkin method we seek the solution which minimizes the error in an integral sense over the entire domain. In this section we describe the formulation of the (Bubnov-) Galerkin method for the governing equations. For our problem, it is defined via finite-dimensional subspaces approximations Sth and Vh to St and V, respectively.

We assume all members of Vh vanish, or vanish approximately, on boundary ∂Ωg and that each member of Sth admits the

Wavelet analysis

Wavelet transform and wavelet bases were originally used as a powerful tool for signal and image processing. It was developed as an extension of short time Fourier transform in order to decompose the frequency content of a signal in both time and frequency domain. The fundamental purpose of transformation techniques like Fourier transform is to give the frequency information of the time dependent signal. However, they do not provide any information about the frequencies in the time domain,

Wavelet-Galerkin method

When solving partial differential equations by the Galerkin method, the approximating spaces requires good approximation properties and they must also allow easy and fast computations. Besides, if the goal is the development of a multilevel method to detect and follow local singularities then hierarchial bases are necessary.

With the advent of wavelet analysis, a new approach has been successfully applied: the wavelet-Galerkin method.

Good approximation properties can be achieved using wavelet

One-dimensional elastoplastic-damage model

The material models have a linear dependence for elasticity problems and non-linear for elastoplasticity problems. For the linear range, the elastic constitutive equation relates linearly stress with strain asσ=Eu,x,where the material parameter E is the Young’s Modulus which is assumed to be constant over domain Ω¯. For materials with non-linear characteristics (plasticity), the stress is a non-linear function (possibly history-dependent) of ϵ. The total strain ϵ splits into two parts ϵ = ϵe + ϵp,

Numerical issues

In this section, we present the numerical treatment for elastoplastic-damage problem. We describe the discretization in space and time, and we also detail the elastic-predictor, plastic-damage-corrector method.

Final considerations

We present a description of the wavelet-Galerkin method for the resolution of elastoplastic-damage problem arising from damage mechanics. In the spatial discretization multiscale wavelet bases can be used as trial and test functions. From this point of view we can realize experiments with several wavelet functions. Besides, this approach allows to locate discontinuities and to capture strain localization phenomena. The procedure proposed in this work can be extended to multidimensional

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