Adaptive integration of constitutive rate equations

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Abstract

In finite element calculations the constitutive model plays a key role. The evaluation of the stress response of the constitutive relation for a given strain increment, which is a time integration in the case of models of the rate type, is a typical sub task in such calculations. Adaptive behaviour of the time integration is essential to assure numerical stability and to control the accuracy of the solution. An adaptive second order semi-implicit method is developed in this paper. Its numerical behaviour is compared with an adaptive second order explicit scheme. The two proposed methods control the local error and guarantee numerical stability of the time integration. We include several numerical geotechnical element tests using hypoplasticity with intergranular strain. The element tests simulate the behaviour of a finite element method based on the displacement formulation.

Introduction

The computation of the stress response of a constitutive relation for a given strain increment is a typical sub task in finite element calculations. Constitutive models of the rate type require a time integration procedure for this task. In order to save computational time, the given strain increments used in finite element programs are usually rather large. Therefore, the time integration has to employ some sub stepping in order to account for the nonlinearity of the constitutive model.

Hypoplasticity [20] is a framework for constitutive models of the rate type specialised for soil behaviour. Early versions of finite element implementations of hypoplasticity were based on the explicit Euler method with constant step size, e.g. [16], [34]. Later implementations [5], [13], [30] used the implicit Euler method. Semi-implicit Rosenbrock methods were employed in [1]. In all these methods, the integration error is not controlled. In our opinion, however, a reliable error control is essential for the integration of nonlinear constitutive relations as it is the basis for an adaptive step size selection. Higher order explicit methods with step size control were applied in [8], [42].

Adaptive explicit schemes were also employed for elasto-plastic constitutive models [3], [39], [40], [47]. Recently an adaptive implicit method [32] and a mixed implicit/explicit scheme [2] were proposed in this field.

A key issue in all adaptive methods is the estimation of the local error. This error is usually derived from the difference of two numerical solutions of the integrated state variables, one obtained with higher accuracy than the other. Due to the different physical dimension of various internal variables of the constitutive models it is necessary to use a scaled error measurement. Various scaling methods have been proposed [3], [8], [32], [39], [40], [47].

Displacement driven implicit finite element methods require the so called consistent (algorithmic) tangent operator for fast convergence [38]. In order to build this consistent tangent operator, the knowledge of the Jacobian of the model (derivative of the stress increment with respect to the strain increment) is essential. When the Jacobian is obtained from variational equations [8], derivatives of state variables are integrated together with state variables. The required accuracy of these derivatives is relatively low compared with the accuracy of the state variables. Using weighting factors to account for different accuracy requirements in the scaled error measurement is indispensable in such a case.

The scaling should also introduce an absolute tolerance to avoid unnecessary small time steps if state variables approach zero. This can be the case for some additional (internal) state variables of the constitutive models (intergranular strain in hypoplasticity, e.g.) or in regions where the stress level is low, which is always the case near the soil surface in geotechnical applications. To our knowledge, absolute error tolerances have not yet been applied for geotechnical material models.

In this paper, we develop an adaptive second order semi-implicit method with a simple and appropriately scaled error estimate and compare its numerical behaviour with that of the explicit method of [8]. We show that the derivative of the stress increment with respect to the strain increment (Jacobian) can be calculated with the method of [8] in our semi-implicit method. Therefore, fast convergence can be expected in displacement driven implicit finite element calculations. Hypoplasticity with intergranular strain [31] is chosen for this comparison. Under certain circumstances, this constitutive model can exhibit a numerically stiff behaviour for which implicit methods are known to be superior. We use a semi-implicit extrapolation scheme. In contrast to fully implicit methods, semi-implicit methods are much faster in computation, although they require slightly more sub steps.

The present article is organized as follows. In Section 2 we shortly introduce the notation of constitutive models of the rate type. In Section 3 we present two time integration methods and describe an error control on which the adaptive sub step size strategy is based. Section 4 illustrates our two approaches with various numerical examples. We have chosen four typical geotechnical element tests: the confined compression test, the drained and the undrained triaxial test, as well as the constant volume simple shear test. A multi element test is presented in Section 5. Our main conclusions are finally given in Section 6. To make our paper self-contained, we outline the used constitutive model in Appendix A and give some implementation details in Appendix B.

Section snippets

Constitutive models of the rate type

A core part of any mechanical model of a continuum, mathematically formulated as an initial boundary value problem, is the description of the mechanical behaviour of the material. The latter can be provided by a constitutive model of the rate type. Such models are relations of the objective stress rate T of the effective Cauchy stress [43] with the Eulerian stretching D, the effective Cauchy stress T and some additional state or internal variables QT=h(T,D,Q).The additional state variables

Adaptive integration

Collecting the components of the stress tensor T, the additional state variables Q, and all derivatives with respect to the stretching in a vector y, we obtain a nonlinear system of differential equationsy(t)=f(y(t))y(0)=y0with m components. In this section, we explain in detail how we solve this initial value problem numerically.

In many scientific applications, the explicit Euler methodyn+1=yn+τnf(yn)is still the method of choice for integrating (3.1a), (3.1b). Notwithstanding its

Element tests

The performance of the proposed time integration schemes is shown for typical element tests from geotechnical engineering. The computations were performed with MATLAB. We have chosen three tests with rectilinear extensions: the confined compression test, the drained and the undrained triaxial test. The fourth test is the constant volume simple shear test in which rotations of the stress tensor occur.

Finite element example

Furthermore, the performance of the proposed time integration schemes is shown with a shear band calculation carried out with ABAQUS 6.6-1. We use the default convergence criteria and load incrementation of ABAQUS, and we set the error tolerances in the constitutive time integration like stated in Section 4.2.

The application was taken from Hügel [16]. A soil specimen of 0.04m width and 0.14m height is laterally compressed with a constant stress of size 400kN/m2. The specimen is compressed

Conclusions and outlook

An adaptive step size selection is an essential feature of time integration methods. It assures numerical stability and it is the basis for controlling the accuracy of the solution. Our proposed step size control works for explicit and implicit methods. It guarantees numerical stability of the time integration of the investigated constitutive model of the rate type in the framework of a displacement version of the finite element method. Our approach for calculating the consistent tangent

Hypoplastic model

For the sake of completeness, we outline the used hypoplastic model and the parameters used for our calculations. Tensors of second order are denoted with bold letters (e.g. D,T,δ,N) and tensors of fourth order with calligraphic letters (e.g. L,M). Different kinds of tensorial multiplication are used: TD=TijDkl,T:D=TijDij,L:D=LijklDkl,T·D=TijDjk. The Euclidian norm of a tensor is D=DijDij. Unit tensors of second and fourth order are denoted by I and I, respectively.

System of differential equations

The constitutive subroutine has to integrate the following evolution equations simultaneously in each strain controlled load step for a given strain increment Δε and a given time increment Δt, with the initial states T(0) and Q(0). Large strain effects are handled by the main program (e.g. the finite element code), thus, the objective rates of the state variables are equal to the time rates in the subroutine. Thus the evolution equations areddtT=h(T,D,Q),ddtQ=k(T,D,Q).

The constitutive equation

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