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## Über dieses Buch

Recognized authors contributed to this collection of original papers from all fields of research in continuum mechanics. Special emphasis is given to time dependent and independent permanent deformations, damage and fracture. Part of the contributions is dedicated to current efforts in describing material behavior with regard to, e.g., anisotropy, thermal effects, softening, ductile and brittle fracture, porosity and granular structure. Another part deals with numerical aspects arising from the implementation of material laws in the calculations of forming processes, soil mechanics and structural mechanics. Applications of theory and numerical methods belong to the following areas: Comparison with experimental results from material testing, metal forming under thermal and dynamic conditions, failure by damage, fracture and localized deformation modes. The variety of treated topics provides a survery of the actual research in these fields; therefore, the book is addressed to those interested in special problems of continuum mechanics as well as to those interested in a general knowledge.

## Inhaltsverzeichnis

### Some Remarks on the Phenomenological Description of Anisotropic Behaviour of Elastic-Plastic Solids

Considering deformation processes of polycrystalline solid bodies from a phenomenological (macroscopic) point of view we may observe different phenomena giving rise to an anisotropic behaviour of the material. The main reasons for the existence of an anisotropy of polycrystalline materials are: a)anisotropic distribution of lattice defects (like dislocations etc.) inside the single crystal grainsb)occurrence of a crystallographic texture which is characterized by a non-random distribution of the orientation of the crystal axes of the grains, andc)appearance of a morphological texture due to an oriented distribution of the shape of the grains.

Th. Lehmann

### On the Choice of Integrity Base of Strain Invariants for Constitutive Equations of Isotropic Materials

Using the well known representation theorem for isotropic tensor functions, the stress-strain law for a thermoelastic material can be written as: 1$$\sigma ={{\phi }_{0}}1+{{\phi }_{1}}B+{{\phi }_{2}}{{B}^{2}},$$ where σ and B are appropriate tensors of stress and strain respectively. Although not yet necessary at the moment, we will define for later use σ as the Cauchy stress and B as the left Cauchy-Green tensor. The coefficients ø0, ø1, ø2 depend on three orthogonal invariants IB, IIB, IIIB of B forming an integrity base and on the temperature T: 2$${{\phi }_{i}}+{{\phi }_{i}}\left( {{I}_{B}},I{{I}_{B}},II{{I}_{B}},T \right)i=0,1,2$$ In many cases the principal invariants of B are taken to be IB, IIB, IIIB and for the present we will do the same. However, any other integrity base of strain invariants is admissible, as long as no further restrictions are imposed on (1).

H. Bednarczyk, C. Sansour

### Modification of the Normality Rule

In the theory of elasticity an elastic potential (strain-energy function W) is assumed, from which the constitutive equations can be derived by using the relation σij = ∂W/εij, where σij and εij are the components of appropriately defined stress and strain tensors, respectively. In the isotropic special case, when the elastic constitutive equation can be represented as an isotropic tensor function 1$${{\sigma }_{ij}}={{\phi }_{0}}{{\delta }_{ij}}+{{\phi }_{1}}{{\varepsilon }_{ij}}+{{\phi }_{2}}\varepsilon _{ij}^{\left( 2 \right)},$$ the elastic potential is a scalar-valued function only of the strain tensor and can be represented in the form W =S1, S2, S3), where S1, S2, S3 are the basic invariants of the strain tensor (finite or infinitesimal strain tensor). In (1) it has been shown in detail that the scalar coefficients in (1) can be expressed through the elastic potential: 2a,b,c$${{\phi }_{0}}\equiv \partial W/\partial {{S}_{1}},{{\phi }_{1}}\equiv 2\partial W/\partial {{S}_{2}},{{\phi }_{2}}\equiv 3\partial W/\partial {{S}_{3}}.$$ Eliminating the elastic potential, one can find the following sufficient and necessary conditions 3$$2\partial {{\phi }_{0}}/\partial {{\phi }_{2}}=\partial {{\phi }_{1}}/\partial {{S}_{1}},3\partial {{\phi }_{1}}/\partial {{S}_{3}}=2\partial {{\phi }_{2}}/\partial {{S}_{2}},3\partial {{\phi }_{_{0}}}/\partial {{S}_{3}}=\partial {{\phi }_{2}}/\partial {{S}_{1}},$$ i.e., the elastic potential is “compatible” with the tensor function theory (1), if the conditions (3) have been fulfiled (1).

J. Betten

### Plastic Behaviour of Saturated Porous Media

Porous media with empty or fluid-saturated pores play an important role in many branches of engineering, e.g., in material science, the petrol industry, chemical engineering and soil mechanics. Although porous media are of considerable practical significance, the description of their mechanical behaviour was unsatisfactory for a considerable time, and has only recently reached well-founded conclusions. The main topic of research today lies in the development of constitutive relations for single constituents and the interaction effects of saturated porous solids. In this range, constitutive equations for fluids and interaction effects are derived which, in several cases, are sufficient for application in engineering. This, however, is not true for solid skeletons. Even if the skeletons behave elastically, many problems arise in describing this property. This becomes even more difficult if plastic effects occur; their description sometimes leads to invincible difficulties. Thus, rigorous assumptions must be introduced in order to develop a mathematical model for which consistent constitutive equations can be derived. In this paper, the assumptions refer to the restriction of small deformations, to the neglect of elastic deformations, and to the treatment of a liquid-saturated porous solid with an incompressible viscous liquid and an incompressible solid material in the plastic range.

R. de Boer

### Compressible Plastic Deformation of Porous Metals

This paper describes some theoretical and practical aspects in formulating of basic equations of compressible metal-plasticity mainly in view of sintered materials. The yield condition for anisotropic and isotropic material behaviour is discussed, and a modified isotropic yield condition which is in good agreement for stress states with high triaxiality is introduced. Using the associated flow rule and the modified yield condition the constitutive law and evolutional equations are derived. The FEM implementation based on a weak formulation in a MIMD multiprocessing system is mentioned. Finally some numerical results of the deformation of a sintered component are presented.

H. Günther

### Application of the Mathematical Theory of Plasticity for Developing New Methods of Material Testing

The further development of metallic materials demands the determination of characteristics for strength and ductility. Often the volume of metallurgically produced new material compositions is rather small and expensive, so that the use of standardized tension or compression tests is excluded. It is, therefore, desirable to develop new methods of materials testing allowing for relatively small specimens. Stimulated by a research project of the Max-Planck-Institut für Eisenforschung on the development of new intermetallic compounds for high temperature service, three new test methods have been developed, which will be described here in detail: the small-cup drawing test, the mantle compression test, and the ball compression test. In order to analyze the stresses and strains involved in these three deformation processes some of the current methods of the mathematical theory of plasticity have been used successfully.

O. Pawelski, R. Hop, K. E. Hagedorn

### Stochastic Models for the Plasticity of Metals

Plastic deformations of metals in the low- and high temperature range are caused by the movement of lattice defects, which can be of line-shape (dislocations) or of point shape. The activation of these defects by mechanical or thermal energy and their movements in the lattice are basic mechanisms causing macroscopic plastic deformations and damage. In addition to this, rearrangement processes in the lattice result in recovery mechanisms, which are, especially in the high temperature range, of strong influence on the macroscopic behaviour of crystalline materials.

E. Steck

### A Concept for Modelling and Computation of Finite Inelastic Deformations

Recent developments in the formulation of inelastic material behaviour at finite strains based on a multiplicative decomposition of the deformation gradient, as proposed by Lee & Liu [10], Lee [9], are summarized. The approach proposed by Simo [21], [22] for the case of finite strain elastoplasticity is extended to finite strain elastoviscoplasticity, see MÜller-Hoeppe&Stein [14], [15], employing at the outset hyperelastic relations and leading at the outcome to an algorithm suitable for large scale computation. From a theoretical standpoint novel steps that differentiate the proposed approach from previous formulations are the following: The role of isotropy. Applying the representation theorem of isotropic tensor functions, e.g. Truesdell & Noll [28], and the spatial versions of the Doyle-Ericksen formula, Doyle&Ericksen [1], Marsden & Hughes [13], the derivation of the underlying theory is carried out completely in current configuration, bypassing pull-back and push-forward transformations. The initial configuration only is necessary to define the kinematic relations.The role of maximum inelastic dissipation Once a constraint is postulated, the associative flow rule compatible with the multiplicative decomposition is derived as optimality condition arising from maximum inelastic dissipation. The resulting flow rule is sixdimensional exactly as in the infinitesimal theory, and not nine-dimensional, Lubliner [11], [12].

E. Stein, N. Müller-Hoeppe

### A Theoretical and Experimental Modelling of the Mechanics of Ductile Fracture in Metals

The mechanics of ductile fracture is complex and not yet fully examined. Depending on the temperature, the rate of deformation and the structure of the metal various mechanisms may be responsible for the softening and fracture. Coalescence of voids, interaction between voids and the external surface of the body, interaction between variously oriented cracks may contribute to the progressing process of ductile fracture. These processes cannot be analysed in terms of the standard fracture mechanics based on the assumption of the elastic model of the material. The model of an elastic-plastic material has been used in numerous works for numerical calculation of the propagation of plastic zones at the front of the crack. On the other hand the possibilities of application of the theory of plastic flow based on the assumption of a rigid plastic model of the material to the analysis of ductile fracture mechanics have been relatively unexplored.

W. Szczepinski

### A Numerical Cosserat-Approach Predicting the Statical Stability of a Plane Frictionless Block-Structure, and its Boundary Conditions

The idea of Cosserat-continua was published at the end of the last century [1,2] and re-formulated by Günther [3] and Grioli [4] in 1958 and 1960, resp. These papers were followed by lots of articles concerning theoretical aspects and generalisations (see [5,6]), whereas applications were usually not mentioned. Lippmann [7] and his school showed that predictions of rotations in a plane system of needles are possible [8, 9].

D. Besdo

### Evolution of Anisotropy in Sheet-Steels Submitted to Off-Axes Large Deformations

A proper understanding and a suitable description of the mechanical behavior of anisotropic solids, in relation with the strain-induced modifications of the oriented internal structures, are essential in modern technology for the manufacturing of materials with adequate mechanical properties, the optimization of forming processes and the analysis of engineering structures subjected to extreme loads. The observed experimental facts reveal the complexity of anisotropic phenomena and show the necessity of a rational and unified formulation of anisotropic constitutive laws. Such a formulation must be able to take into account the invariance conditions resulting from the symmetries of the material internal structures, to describe the strong non-linearities of the actual mechanical behavior of anisotropic solids, and to specify the independent variables to be observed in experiments, in order to be able to identify particular constitutive laws.

J.-P. Boehler, S. Koss

### On the Plane Strain Deformations of Critical State Models for Sands

The deformation and flow of sands and similar granular materials are characterised by (a) the fact that shearing is frequently accompanied by volume changes - either positive or negative and (b) the fact that unlike metals and other crystalline materials the deformation behaviour is strongly dependent on the ambient hydrostatic pressure. Thus, in constructing plasticity models to describe their behaviour, it is necessary to allow the yield function and flow potential to depend on the current density and mean pressure in addition to the deviatoric components of stress. The critical state models described by Schofield and Wroth [1] are examples of these types of plasticity theories. However, their theory assumes a normal or associated flow rule which works well for normally consolidated clays but is inadequate to explain the observed behaviour of sands and granular materials. We are hence concerned with plasticity models with distinct yield functions f and flow potential g, both of which are functions of the stress tensor σij (compressive normal stresses taken as positive) and the specific volume v, defined as the ratio of the total volume to the volume of the solid grains in a material element. Such materials possess a critical state in which the material behaves incompressibly and the associated deformation is isochoric. The state of stress in such a critical state is obtained by eliminating the specific volume v between the yield condition and the condition of zero volume change arising from the flow rule, l.e. between1$$f(\sigma ij,v)=0and\partial g(\sigma ij,v)/\partial {{\sigma }_{kk}}=0$$ Any successful phenomenological theory in continuum mechanics must be validated by parallel considerations of conceptual micromechanical models. In the flow of granular materials this amounts to looking for planes or surfaces on which some critical condition necessary for local slip to occur is achieved. In the primitive Coulomb model of a cohesionless material, slip is assumed to occur when the ratio of the shear traction ti too, the normal traction on a plane, reaches a critical value. The effect of dilatancy - either positive or negative- can be incorporated in a crude manner by considering the sliding of serrated rather than smooth blocks over each. Models of this sort have been discussed by Taylor [2], de Josselin de Jong [3] and Atkinson and Bransby [4]. The latter authors show that this model leads to a stress-dilatancy condition of the form: 2$$(\tau /\sigma )=(-d{{\varepsilon }_{v}}/dy)+\mu$$ where dεv and dγ are the normal and shear strain increments across the slip plane, the former being positive when compressive. The constant p. is the value of the stress ratio in the critical state. A very similar expression has been derived by Matsuoka [5] from observations of the behaviour of aluminium rods in model tests. The difference being the presence of a factor x which can have values between 1.1 and 1.5, in front of the strain-increment ratio term in (2). This relation is the basis of the comprehensive ‘spatially mobilised plane’ theory developed by Matsuoka and collaborators - see reference [6] for a recent review of this theory. A number of more detailed studies of the deformation of assemblies of granular particles using statistical arguments which predict dilatancy relations similar to (2) have been made by Nemat-Nasser and co-workers, e.g. [7]. Since (2) involves strain-increments as well as stress components it will be related to the flow rule of the corresponding phenomenological plasticity theory.

I. F. Collins

### On the Quasi-Analytical Solutions of Elastic-Plastic Problems with Nonlinear Hardening

Elastic-plastic equilibrium problems accessible to analytical treatment are mainly one-dimensional ones exhibiting spherical or cylindrical symmetry. For perfectly plastic behaviour as well as material with linear isotropic hardening, Tresca’s yield criterion and the flow rule associated to it lead to linear differential equations for stresses and displacement. (In the case of spherical symmetry the Tresca and the von Mises criterion coincide.) The image points of the plastic region may lie on different edges or sides of the Tresca prism in stress space. Correspondingly, the plastic region is composed of several parts with different mathematical forms of the yield condition (e.g. [1]). Continuity of radial stress and displacement at the borders separating these parts and the boundary conditions constitute a system of equations that is linear in the integration constants and the load parameter but nonlinear in the border radii. It can be solved without problems, in general. Solutions of this type are termed analytical. Nevertheless, they are not restricted to perfectly plastic behaviour or linear hardening but exist also in cases of special nonlinear hardening laws [2].

V. Gamer

### Some Mechanisms of a Granular Mass in a Silo — Model Tests and a Numerical Cosserat Approach

After opening a silo bottom outlet some characteristic phenomena can be observed: The lateral pressure on the silo wall increases immediately after the beginning of emptying and can be a multiple of the one during filling.Shear zones come into being along the silo wall and, for a dense fill, also inside the flowing granular material. The dilatancy constraint in the shear zones (caused by the stiffness of neighbouring material) results in a scale effect in model silos, i.e. an additional increase of dimensionless wall stresses.Stress fluctuations are registered during the whole emptying process in big and in model silos as well.

G. Gudehus, J. Tejchman

### Experimental Investigation on Plastic Deformation of Stainless Steel at Low Temperature

Low temperature structures in aerospace technology or super conduct technology have become more popular in recent years. The structures need high reliability in their service life. The strength design should be realized on well understanding the characteristic property of low temperature deformation. But there are not so many experimental investigations on plastic deformation at low temperature. The review is reported by one of the authors [1] and the shortage of experimental results under combined stress state is pointed out.

K. Ikegami, Y. Niitsu

### Upper Bound Approach for Prediction of Occurence of Ductile Fracture in Metal Forming

With a view to prediction of occurrence of fracture, ductile fracture criteria have been proposed [1,2]. Since ductile fracture is the final stage of void growth, i.e. coalescence of voids, there are quite a few reports on calculation of void volume fraction based on the evolution equations in various metal forming processes [3–5], although the occurrence of central bursting cannot be predicted.

S. Shima, M. Oka

### Plasticity-Related Critical States and Failure Criteria

In applied solid mechanics the calculation of stresses and strains is a fundamental step in assessing the behavior of mechanical components when subjected to loadings either by external and internal forces or thermal effects. However those quantities have no interest in themselves unless comparison is made with physical properties of the materials.

F. A. Leckie, J. Lemaitre

### Physical Argument for Nonlocality of Microcracking Damage in a Continuum

The concept of nonlocal continuum, which was introduced into continuum mechanics by Eringen [1,2], Kröner [3] and others, has recently proven to be an effective approach to endow a continuum model for strain-softening damage with a localization limiter which prevents spurious localization instabilities [4–6]. Physical micromechanical justification of this concept, however, has been almost nonexistent.

Z. P. Bažant

### Plastic Localization in Damaged Materials

Bands of localized plastic deformation may initiate during metal forming processes. These are bands, inside which plastic strains are much higher than in the rest part of the deformed body.

N. Bontcheva, A. Baltov

### Some Remarks to the Application of Damage Mechanics to Low Cycle Fatigue

In Low Cycle Fatigue two variables are usually applied to describe the state of a material : The damage variable D, 0≤D≤DF, defining the damage of the material, often without any detailed physical explanation. DF is the damage variable at fracture, DF≤1.0.The number of cycles N. NF is the number of cycles to fracture for a given stress range ∆σ or strain range ∆ε. Manson and Coffin gave for a strain controlled test the relation 1$${{\text{N}}_{\text{F}}}\text{ }\Delta {{\varepsilon }^{\beta }}=\text{const}\text{., 1}\text{.5}\le \beta \le \text{2}\text{.0}\text{.}$$ see e.g. the review by Manson and Halford, [1].

F. D. Fischer, H. Stüwe

### A New Integral Equation Approach for the Curved Crack Problem in a Circular Plate

Integral equation approaches are an important tool for the solution of crack problems in plane elasticity [1–5]. The description generally used in the mentioned papers is the singular integral equation approach. Recently, alternatives were suggested. In [6,7], taking the tractions and the dislocation function respectively along the crack to be polynomials, the multiple crack problem and the curved crack problem in an infinite plate were solved. The regularization problem of the singular integral equation for the crack problem was considered in [8,9].

D. Gross, Y. Z. Chen

### Crack Analysis in Fibrous Composites with Partially Plastified Matrix Materials

A brief review is given concerning the crack analysis of fibrous composites with partially plastified matrix materials. Various microstructural models are introduced and the fracture mechanisms of fibrous composites are discussed from different standpoints.

K. P. Herrmann, Y. Q. Wang

### On Isolated Point Singularities in Classical Elasticity

This paper is concerned with isolated point singularities in an otherwise smooth solution to the mixed boundary value problem of classical linear elasticity on bounded regions. The singularity, which may be located either on the surface of the region or at an interior póint, is due to the application of point loads, material defects, non-smoothness of the surface or some other effect. The precise cause of the singularity is irrelevant to the purpose of the present treatment, which aims at determining a lower bound for the order of magnitude of all possible isolated point singularities that can occur in the solution. The results obtained are widely applicable but this same generality implies that the results are unlikely to be the best possible for any individual singularity e.g. that produced by point loads. Nevertheless, the present investigation may be regarded as complementing previous studies of singularities arising in elasticity or elliptic systems of differential equations, where the emphasis has tended towards establishing removability of singularities rather than estimating possible order of magnitudes. (Compare, for example, Serrin [3,11], Aviles [1]; Oleinik et al [7,8,9] and Grisvard [4,5] are amongst those who have examined order of magnitudes. See also Castellani Rizzonelli [2].)

R. J. Knops

### Localization within the Framework of Micropolar Elasto-Plasticity

Recently increasing attention is paid to continuum models which describe failure and post-peak behaviour in terms of localized deformations in failure bands of finite width. The classical Boltzmann description of local continua signals loss of ellipticity as soon as localization occurs causing strong mesh size dependence. To remedy this pathological behaviour an internal length scale was introduced by Bazant, Belytschko and Chang [1] via non-local concepts, or alternatively, by higher order gradient approaches involving an internal length scale, as proposed by Coleman and Hodgdon [2], Triantafyllidis and Aifantis [3]. In contrast, the micropolar theories, as advocated recently by Mühlhaus [4], Mühlhaus and Vardoulakis [5] and de Borst [6] possess an internal length scale to start with, which tends to regularize the impending loss of ellipticity near localization. For wave propagation problems Belytschko and Lasry [7] demonstrated that non-local and higher order gradient approaches exhibit a stable response to short wave length inputs and an unstable response to long wave lengths.

P. Steinmann, K. Willam

### Bifurcation Phenomena of Compressible Materials in the Plane Tension or Compression Test

The variety of possible bifurcation phenomena from a state of homogeneous in-plane tension for an incompressible rectangular block constrained to plane deformations has been documented in Hill and Hutchinson’s [1] basic investigation which was extended by Young [2] to the corresponding compression problem.

K. Thermann

### Impact Load on Elastic Rod Embedded in a Rigid Plastic Medium

The problem to be discussed in the present paper has a counterpart in evereday life: the familiar tap on the barometer glass window. Dry friction in the bearings prevents the barometer to record accurately every small change in the atmospheric pressure. Hence small internal friction forces will be set up in the barometer due to pressure fluctuations. A slight tap on the glass relieves these forces and lets the needle move to give a more correct reading. The way to check this is, of course, to make a second tap, which is usually found to be unnecessary.

J. Hult

### Micromechanical Foundations of Dynamic Plasticity with Applications to Damaging Structures

Macroscopic inelastic deformations generally are a result of dissipative processes taking place in the materials microstructure. Therefore, it appears to be desirable to include — at least in an averaged sense — the microscopic behavior in the formulation of the macroscopic constitutive relations. In this paper we develop a method which allows a consistent description of the microscopic as well as the macroscopic behavior of polycrystalline solids with micro-cavities. In this method dissipative mechanisms are considered as defects in a background material of time invariant constitutive properties. These defects are mechanically described by fields of distortions (or eigenstrains in the notation of Mura [1]). On the microlevel (for determination of disturbance fields due to cavities) this concept is equivalent to Eshelby’s equivalent inclusion method [2] applied to an appropriate averaging procedure. On the macroscale this theory leads to the concept of internal excitation which is described in detail by Fotiu [3].

P. Fotiu, H. Irschik, F. Ziegler

### The Elastic Continuum with a Cylindrical Hole Subjected to a Moving Time Depending Load

The basis for all investigations concerning spatial elastodymamics are the fundamental equations of the elastic continuum as formulated by Navier, Cauchy and Poisson [1]. For the solution of these equations the introduction of the Lamé potentials proved to be extremely advantageous. Sternberg [2] has discussed in some detail the further transformation of the Lamé equations into wave equations.

H. Grundmann, A. Konrad, G. Zirwas

### Elastic-Plastic Wave Propagation of Combined Generalized Forces in a Timoshenko Beam

The propagation of elastic and plastic waves of combined generalized forces in a Timoshenko beam under symmetrical bending and tension or compression is investigated for isotropic work-hardening materials. The governing system of differential equations has the same structure as that of a certain class of problems dealing with the elastic-plastic wave propagation of stresses in a two or three dimensional medium. In contrast to the latter problem, however, the compliance matrix associated with the wave propagation in a Timosiienko beam is unsymmetrical. The eigenvalues and the eigenvectors are determined. The eigenvalues represent the velocities of the waves, whereas the eigenvectors yield the corresponding jumps of the generalized forces.

M. Müller, W. Hauger

### On the Limit Analysis of High Speed Forming Processes in Cold or Hot Conditions

This work presents a general.procedure to asses the dynamic stress that prevails in the deforming zone during high-speed forming processes. A particular emphasis is given to materials which behave in a viscoplastic manner (akin to high-temperature forming) to which a stress potential exists (Rice[1]). Thus a limit analysis approach seems possible, though not yet tried.

J. Tirosh, D. Iddan

### Thermomechanical Simulation of Some Types of Steady Continuous Casting Processes

Continuous casting technique is one of the most promissing processes to produce slabs, ingots or plates of many kinds of metals. Difficulties still remain, however, to optimize the operating condition since the phase transformation, or solidification, is essential in the process, and the material close to the solidifying region behaves in a rate dependent inelastic manner due to high temperature.

T. Inoue, D. Ju

### Some Aspects of Nonlinear Thermomechanical Structure Analysis

One of the main topics in continuum mechanics is modelling and analysis of solid mechanics problems with respect to physical and geometrical nonlinearities in connection with the internal processes in the material. Such internal processes in engineering materials are based on structural transformations, irreversible deformations, damage etc. with mechanical, thermical, elctromagnetic or other effects. The present computational possibilities allow the solving of real problems in structural analysis. References [1] to [8] describe the state of the art in continuum mechanics as well as some computational aspects.

J. Altenbach, H. Altenbach, C. Muench

### On the Thermodynamics of Rate-Independent Elastoplastic Materials

Rate-independent material behavior is characterized by hysteresis effects, which do not depend on the rate of cyclic processes. In this case, infinitely slow thermodynamic processes cannot be interpreted to be reversible in the sense of classical thermodynamics. In spite of this well known fact, which has already been noticed by Carathlodory [1], it turns out, that it is possible to model a quasistatic hysteresis behavior as a. sequence of equilibrium states. However, as a basic difference to thermoviscoelasticity in the sense of Coleman and Gurtin [2], the stability properties of the equilibrium states change in dependence of the applied straining process.

P. Haupt

### Some Remarks on Thermo-Mechanical Hysteresis

Hysteresis is an ubiquitous phenomenon; it occurs in simple relais, plasticity, shape memory, ferromagnetism, ferroelectricity, and is often related to phase transitions. Wherever it occurs, it indicates non-convex potential energies, non-monotone load-displacement curves and hence possible instability.

I. Müller

### Analysis of Recovery Stress and Cyclic Deformation in Shape Memory Alloys

Thanks to intensive metallurgical investigations the microscopic mechanism of the shape memory effect has clearly been disclosed[1–3]. Thermomechanical behavior of shape memory alloys is due to one or some of the following microscopic elementary processes: The thermoelastic martensitic transformation, the reorientation of the martensite plate, and the transformation from one martensite phase to the other which has a different crystallographic structure than the first.

K. Tanaka

### On a Hybrid Method to Analyse Viscoelastic Problems

To analyze problems of plane stress states when the material shows time-depending response, i.e. linear viscoelastic response, experimental procedures can be applied similar to photoelasticity. However, not only the mechanical, but also the optical rheological effects must be considered. Beside the isochromatic fringe pattern, the isoclinics are to be recorded over time. Obviously, recording the isoclinics is difficult and the reason of unavoidable incertainty and inaccuracy. On the other hand, the two different fringe patterns are to be evaluated separately by means of automatic digital image processing and the data to combine in order to determine the components of the stress-tensor according to algorithms, which are not appropriate to computer calculations. To avoid the difficulties, in order to obtain more reliable results and to save time for recording and evaluating the optical phenomena as well as computer capacity, a procedure will be described, which enables the principal stress components to be calculated, although the principal directions of the stress tensor, i.e. the isoclinics, are still unknown.

K. H. Laermann

### Energetical Aspects of Polymer Failure

Failure of polymers does not mean only fracture or rupture. Phenomena, such as transition from linear to nonlinear viscoelasticity, crazing and yielding are also failure phenomena. While crazing and fracture can be directly observed and yielding becomes visible from the stress-strain representation of the loading history, the transition from linear to nonlinear viscoelasticity can be detected only by using more sophisticated procedures. On the other hand, observation of the first crazes is relatively simple in long term experiments (such as creep and stress relaxation), but very difficult in short term tests, whereas yielding and fracture can be studied better in short term test (for example in constant strain rate experiments). All these occurences make the problem of failure of plastics much more complicated than that of time independent materials.

O. S. Brüller

### Creep Behavior of SUS 304 after Cyclic Plasticity

In the last decade, research on the constitutive formulation for cyclic plasticity or cyclic viscoplasticity has been developed so extensively that many models have been proposed in the literatures. Recently, ratchetting has attracted the interest of researchers[1–3]. Chaboche and Nouailhas[1] pointed out that the description of ratchetting in term of constitutive equations is mainly related to the kinematic hardening, and that the constitutive formulations which are presently available either underestimate or overestimate the ratchetting effect. Similarly, Buggles and Krempl[2] showed that under the selected test conditions at room temperature ratchetting of SUS304 stainless steel is viscous in nature, and suggested that predictions of ratchetting could be improved with the help of a viscoplasticity theory.

H. Ishikawa

### The Time History Analysis of Viscoelastic Structures by Mathematical Programming

The analysis of viscous or viscoelastic structures has to be based on suitable mathematical and mechanical backgrounds. Regarding mathematics, there are two main directions of the solution. One is describing the viscous system by several superposition integrals and the other one is using the Laplace’s transformation. The advantages and disadvantages of these methods are described elsewhere [3,4,61.

S. Kaliszky, A. Vásárhelyi, J. Lógó
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