Ch. Tsakmakis

Finite deformation viscoelasticity laws

Abstract It is proved that two different decompositions of strain may be assigned to every linear viscoelastic solid. In particular, this is true for the so-called three-parameter solids. For this case, the two decompositions of deformation are in a natural way associated with the two well known spring–dashpot models, the first one being a spring in parallel with a Maxwell element and the second model consisting of a spring in series with a Kelvin element. Furthermore, it is shown how the two decompositions of deformation may be generalized to finite deformations in the framework of a multiplicative decomposition of the deformation gradient tensor. This enables to assign to each version of the three-parameter solids a corresponding class of finite deformation counterparts. Note that the finite deformation models are derived so, that the second law of thermodynamics is satisfied for every admissible process. To this end, use is made of the so-called Mandel stress tensor. As one may expect, unlike the linear case, the finite deformation models obtained do not predict identical mechanical responses generally. This is illustrated for the loading case of uniaxial tension–compression. Also, an analysis of the model responses for simple shear is given.

Determination of constitutive properties fromspherical indentation data using neural networks. Part i:the case of pure kinematic hardening in plasticity laws

Abstract In this paper the power of neural networks in identifying material parameters fromdata obtained by spherical indentation is demonstrated for an academic problem (pure kinematichardening, given Youngs modulus) . To obtain a data basis for the training and validation of theneural network, numerous finite element simulations were carried out for various sets of materialparameters. The constitutive model describing finite deformation plasticity is formulated withnonlinear kinematic hardening of Armstrong–Frederick type. It was shown by Huber and Tsakmakis, 1998a that the depth–load response of a cyclic indentation process, consisting ofloading, unloading and reloading of the indenter displays a typical hysteresis loop for givenmaterial parameters. The inverse problem of leading the depth–load response back to the relatedparameters in the constitutive equations is solved using a neutral network.

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.

Determination of constitutive properties fromspherical indentation data using neural networks. Part ii:plasticity with nonlinear isotropic and kinematichardening

Abstract We consider materials which can be described by plasticity laws exhibiting nonlinearkinematic and nonlinear isotropic hardening effects. The aim is to show that the materialparameters governing the constitutive behavior may be determined from data obtained byspherical indentation. Note that only the measurable global quantities (load and indentationdepth) should be utilized, which are available, e.g. from depth-sensing indentation tests. For thisgoal use is made of the method of neural networks. The developed neural networks apply also tothe case of pure kinematic as well as pure isotropic hardening. Moreover it is shown how amonotonic strain–stress curve can be assigned to the spherical indentation test.

Discussion of coupled elastoplasticity and damage constitutive equations for small and finite deformations

Abstract Three small deformation plasticity models taking into account isotropic damage effects are presented and discussed. The models are formulated in the context of irreversible thermody-namics and the internal state variable theory. They exhibit nonlinear isotropic and nonlinear kinematic hardening. The aim of the paper is first to give a comparative study of the three models with reference to homogeneous and inhomogeneous deformations by using a general damage law. Secondly, and this is the main objective of the paper, we generalize the constitutive models to finite deformations by applying a thermodynamical framework based on the Mandel stress tensor. The responses of the obtained finite deformation models are then discussed for loading processes with homogeneous deformations.

A neural network tool for identifying the material parameters of a finite deformation viscoplasticity model with static recovery

In the present paper, the inverse problem of parameter identification is solved by using neural networks. In contrast to the commonly used optimization methods, neural networks represent an explicit relation between the measured strain, stress, time and the material parameters to be identified. The constitutive model under consideration describes finite deformation viscoplasticity and exhibits static recovery in both the isotropic and the kinematic hardening laws. To train the neural networks, a loading history is utilized, which consists of a homogeneous uniaxial deformation including cyclic loading and relaxation phases. It is shown that the neural networks are able to identify physically meaningful sets of material parameters so that the constitutive model may predict experimentally observed material behavior in a satisfactory manner. This is true even if complex loading histories are considered.

A comparative study of kinematic hardening rules at finite deformations

Abstract Kinematic hardening models describe a specific kind of plastic anisotropy which evolves with the deformation process. It is well known that the extension of constitutive relations from small to finite deformations is not unique. This applies also to well-established kinematic hardening rules like that of Armstrong–Frederick or Chaboche. However, the second law of thermodynamics offers some possibilities for generalizing constitutive equations so that this ambiguity may, in some extent, be moderated. The present paper is concerned with three possible extensions, from small to finite deformations, of the Armstrong–Frederick rule, which are derived as sufficient conditions for the validity of the second law. All three models rely upon the multiplicative decomposition of the deformation gradient tensor into elastic and plastic parts and make use of a yield function expressed in terms of the so-called Mandel stress tensor. In conformity with this approach, the back-stress tensor is defined to be of Mandel stress type as well. In order to compare the properties of the three models, predicted responses for processes with homogeneous and inhomogeneous deformations are discussed. To this end, the models are implemented in a finite element code (ABAQUS).

Determination of constitutive properties of thin metallic films on substrates by spherical indentation using neural networks

Abstract The indentation test has been developed into a popular method for investigating mechanical properties of thin films. However, there exist only some empirical or semi-analytical methods for determining the hardness and Young’s modulus of a film from pyramidal indentation of the film on a substrate, where the deformation of film and substrate is subjected to be elastic–plastic. The aim of the present paper is to show how constitutive properties and material parameters may be determined by using a depth-load trajectory which is related to a fictitious bulk film material. This bulk film material is supposed to possess the same mechanical properties as the real film. It is assumed that the film and the substrate exhibit elastic–plastic material properties with nonlinear isotropic and kinematic hardening. The determination of the depth-load trajectory of the bulk film is a so-called inverse problem. This problem is solved in the present paper using both the depth-load trajectory of the pure substrate and the depth-load trajectory of the film deposited on this substrate. For this, use is made of the method of neural networks. Having established the bulk film depth-load trajectory, the set of material parameters entering in the constitutive laws may be determined by using e.g. the method proposed by Huber and Tsakmakis (Huber, N., Tsakmakis, Ch., 1999. Determination of constitutive properties from spherical indentation data using neural networks. Part II: plasticity with nonlinear isotropic and kinematic hardening. J. Mech. Phys. Solids 47, 1589–1607).

A Finite Element Analysis of the Effect of Hardening Rules on the Indentation Test

Using the Finite Element Method, an analysis is given of the indentation of an elastic-plastic half-space by a rigid sphere. In particular, attention is focused on the effect of hardening rules on the material response. The materials considered are supposed to exhibit isotropic and kinematic hardening. Moreover, it is shown that the possibility of similar behavior due to effects of friction can be ruled out.

Experimental and theoretical investigation of the effect of kinematic hardening on spherical indentation

Abstract When investigating material properties, the indentation test is often used in spite of the fact that very complicated nonhomogeneous processes are involved. In order to understand this test in more detail, an analysis was given by Huber and Tsakmakis using Finite Element calculations. It was shown that there exists an analogy between uniaxial homogeneous tensile experiments and spherical indentation for cyclic loading conditions. In fact, in both cases, existence, e.g., of kinematic hardening can be identified by the existence of hysteresis loops in the strain–stress diagram and the depth–load plot, respectively. The present paper deals with an experimental verification of the existence of such hysteresis loops for the case of depth-sensing indentation tests. Further, two measures are considered in order to quantify the size of hysteresis loops. The first one is the area enclosed by the hysteresis loop while the second one is a suitable defined middle opening of the hysteresis loop. Using various Finite Element calculations, it is shown that both measures can be regarded to be correlated. These theoretical relationships are proved to be in agreement with experimental results as well. Finally, the effect of kinematic hardening on the hysteresis loops is discussed experimentally by studying the opening of the hysteresis loop as a function of the depth.