Peter Haupt

Shape memory behaviour: modelling within continuum thermomechanics

Abstract A phenomenological material model to represent the multiaxial material behaviour of shape memory alloys is proposed. The material model is able to represent the main effects of shape memory alloys: the one-way shape memory effect, the two-way shape memory effect due to external loads, the pseudoelastic and pseudoplastic behaviour as well as the transition range between pseudoelasticity and pseudoplasticity. The material model is based on a free energy function and evolution equations for internal variables. By means of the free energy function, the energy storage during thermomechanical processes is described. Evolution equations for internal variables, e.g. the inelastic strain tensor or the fraction of martensite are formulated to represent the dissipative material behaviour. In order to distinguish between different deformation mechanisms, case distinctions are introduced into the evolution equations. Thermomechanical consistency is ensured in the sense that the constitutive model satisfies the Clausius–Duhem inequality. Finally, some numerical solutions of the constitutive equations for isothermal and non-isothermal strain and stress processes demonstrate that the various phenomena of the material behaviour are well represented. This applies for uniaxial processes and for non-proportional loadings as well.

On the application of dual variables in continuum mechanics

Stress and strain tensors that arise in the expression of the stress power are called “conjugate variables”. More special is the term “dual variables” which has been introduced in connection with incremental constitutive relations of hypoelasticity and plasticity, where the rates of both tensors arise. We propose a rational rule from which the most natural form of tensor-valued kinematic and dynamic variables (strain and stress tensors) including their corresponding time rates can be deduced. Dual variables and their associated dual derivatives are characterized by the property that apart from the stress power also the incremental stress power is invariant under a group of transformations that corresponds to a set of physically reasonable intermediate configurations. We outline the precursory history of these concepts and then discuss in detail how the invariance properties can be realized in the various stress and strain measures. We finally demonstrate the concept in three different applications: The rate form of the principle of virtual work, the formulation of constitutive relations in viscoelasticity and the formulation of incremental constitutive assumptions of rate-independent plasticity.

On the dynamic behaviour of polymers under finite strains : constitutive modelling and identification of parameters

Abstract In this paper we apply a model of finite viscoelasticity and propose an identification technique to represent the dynamic properties of polymers. The model is based on a multiplicative split of the deformation gradient into a thermal and a mechanical part, the latter being decomposed further into elastic and viscous parts. In order to formulate the constitutive equations we transfer the concept of discrete relaxation spectra to finite strains, specify the free energy as a function of elastic strain tensors and evaluate the dissipation principle of thermodynamics in the form of the Clausius–Duhem inequality. Then we investigate the dynamic moduli of a polyethylene melt under harmonic shear deformations and determine the material parameters. To this end we linearise the constitutive model and calculate the analytical solution of the evolution equations. In addition we formulate a second model which represents the experimental data on the basis of a fairly small number of fitting parameters. This so-called substitute model is based on the fractional calculus and corresponds to a continuous relaxation spectrum. In order to identify the material constants of the finite strain model we are looking for, we proceed as follows. We determine the parameters of the substitute model, calculate the so-called cumulative relaxation spectrum and approximate it by means of a series of step functions: the height of the steps corresponds to the stiffness parameters of the finite strain model and their locations to the inverse relaxation times.

Thermomechanical behavior of shape memory alloys

Tension and torsion as well as combined tension-torsion tests on NiTi Tubes are presented in this article. Two different specimens are used in the experiments: one is austenitic and the other is martensitic at room temperature. The experiments are performed at nearly isothermal conditions. However, non-isothermal effects occur as well because of the self-heating of the material during the phase transitions and the detwinning of the martensite. These effects can be excluded applying very small deformation rates. In contrast to this, the influence of the self- heating on the material behavior is investigated in other experiments, where temperature fields are measured by means of infrared thermography. This allows detailed observations of the temperature field on the surface of the specimen and leads to additional insight into the thermomechanical behavior of shape memory alloys. In simple tension and pure torsion experiments the various effects of the material behavior can be decoupled. In particular, relaxation and creep processes are observed as a result of self-heating, but also as a consequence of the viscosity of the material. The combined tension-torsion experiments make it possible to analyze coupling effects of the biaxial behavior. In this context, a proportional and non-proportional deformation path is carried out.

On the numerical treatment of finite deformations in elastoviscoplasticity

Abstract This paper deals with the generalization of a geometric linear viscoplastic model to finite strains and its numerical application. We owe the original formulation of the applied model to Perzyna and Chaboche ; it includes nonlinear isotropic and kinematic hardening as well as a nonlinear rate dependence. The constitutive equations are integrated numerically in the context of a finite element formulation. From theoretical considerations it is known that in the case of vanishing viscosity or slow processes rate-independent plasticity arises as an asymptotic limit. Accordingly, the numerical formulation includes this property. In fact, the stress algorithm corresponding to viscoplasticity is reduced to the asymptotic limit in a most simple way, namely by setting the viscosity parameter equal to zero. Furthermore, it is shown that the numerical integration of the constitutive model involves the solution of only one nonlinear equation for one scalar unknown. This even applies to a sum of Armstrong-Frederick terms. The algorithm incorporates the inelastic incompressibility on the level of the Gauβ points. Numerical computations of examples taken from metal forming technology show the physical significance of the model and the reliability of the numerical algorithm. These calculations have been carried out by means of the finite element program PSU .

Finite deformations of a carbon black-filled rubber. Experiment, optical measurement and material parameter identification using finite elements

The identification of material parameters occurring in constitutive equations frequently plays a more minor role in large deformations in view of the difficulties encountered when endeavouring to perform specified deformations by means of experiments. This article aims to illustrate special optical measurements obtained with the help of a CCD camera applied to tensile and compression tests, to allow a larger number of measured quantities to be incorporated into the material parameter identification process. Based on these inhomogeneous deformations, we propose an identification procedure founded on a gradient-free optimisation technique which incorporates the finite element method while also taking account of inequality constraints. The finite element method is employed to obtain a numerical solution of the underlying boundary-value problem, which has to be compared with the experimental data pertaining to carbon black-filled rubber in the sense of the least-square method. The proposed procedure is independent of the underlying finite-element programme and is being investigated from the point of view of its performance.

On the modelling of anisotropic material behaviour in viscoplasticity

Abstract A uniaxial viscoplastic deformation is motivated as a discrete sequence of stable and unstable equilibrium states and approximated by a smooth family of stable states of equilibrium depending on the history of the mechanical process. Three-dimensional crystal viscoplasticity starts from the assumption that inelastic shearings take place on slip systems, which are known from the particular geometric structure of the crystal. A constitutive model for the behaviour of a single crystal is developed, based on a free energy, which decomposes into an elastic and an inelastic part. The elastic part, the isothermal strain energy, depends on the elastic Green strain and allows for the initial anisotropy, known from the special type of the crystal lattice. Additionally, the strain energy function contains an orthogonal tensor-valued internal variable representing the orientation of the anisotropy axes. This orientation develops according to an evolution equation, which satisfies the postulate of full invariance in the sense that it is an observer-invariant relation. The inelastic part of the free energy is a quadratic function of the integrated shear rates and corresponding internal variables being equivalent to backstresses in order to consider kinematic hardening phenomena on the slip system level. The evolution equations for the shears, backstresses and crystallographic orientations are thermomechanically consistent in the sense that they are compatible with the entropy inequality. While the general theory applies to all types of lattices, specific test calculations refer to cubic symmetry (fcc) and small elastic strains. The simulations of simple tension and compression processes of a single crystal illustrates the development of the crystallographic axes according to the proposed evolution equation. In order to simulate the behaviour of a polycrystal the initial orientations of the anisotropy axes are assumed to be space-dependent but piecewise constant, where each region of a constant orientation corresponds to a grain. The results of the calculation show that the initially isotropic distribution of the orientation changes in a physically reasonable manner and that the intensity of this process-induced texture depends on the specific choice of the material constants.

Representation of cyclic hardening and softening properties using continuous variables

Abstract Our aim is the modeling of cyclic hardening, cyclic softening, cyclic mean stress relaxation, and additional nonproportional cyclic hardening. We do so by means of hardening functionals for back stress and yield stress without employing additional memory surfaces. Rather, we suppose all quantities to evolve simultaneously during elastic-plastic loading in a continuous manner. The basic idea is to formulate evolution equations for the hardening variables, which are of the “hardening/dynamic recovery” format with respect to a transformed arc length. The corresponding transformation is influenced by continuously evolving parameters, measuring strain amplitude and nonproportionality during the recent process history. Although the resulting, model has a very simple structure, it is capable of describing the basic phenomena under quite general loading conditions.

Continuous representation of hardening properties in cyclic plasticity

Abstract In this study a model of kinematic hardening is developed to represent some hardening and softening phenomena, which can be observed in the context of one-dimensional cyclic strain processes. The proposed hardening rule is based on the well-known Armstrong-Frederick equation. This equation is modified by utilizing a material-dependent transformation of the plastic arc length. The principal idea is that the transformed arc length depends on an internal variable which corresponds to a direct measure of the strain amplitude and does not depend on the mean strain. An evolution equation for a continuous change of this internal variable is motivated, developed, and investigated. As a result, the proposed hardening model is formulated as a system of two-order differential equations involving four material constants. Numerical calculations suggest that principal phenomena of cyclic hardening and softening behavior are qualitatively well represented.

Foundation of Continuum Mechanics

The following lectures give a short indroduction into modern continuum mechanics. The presentation omits many details and should motivate the reader to study the references.