Erhard Krempl

Rate (time)-dependent deformation behavior: an overview of some properties of metals and solid polymers

Abstract The inelastic deformation behaviors of metals and polymers are discussed with the aim of finding a common base that would simplify academic and engineering analyses. Only monotonic loading conditions at room temperature are considered. For loading at different rates, nonlinear relations between loading rate and stress level, creep stress level and creep strain, and relaxation rate and stress were common to both type of materials. There are, of course, significant differences in elastic properties, strength levels and the strains involved. Special properties such as relaxation behaviors and creep anomalies can be qualitatively and quantitatively reproduced by the state variable model VBO (viscoplasticity theory based on overstress). Since experimental investigations typically concentrate on one particular aspect of inelastic deformation behavior such as creep or strain-rate dependence, it is often difficult to gather a comprehensive data set for a given material. In spite of this, considerable similitude in the deformation behavior of metals and polymers in various test conditions has nevertheless been established.

Extension of the viscoplasticity theory based on overstress (VBO) to capture non-standard rate dependence in solids

Abstract The viscoplasticity theory based on overstress (VBO) can model unusual loading rate effects. Included are rate-insensitivity and positive and negative rate sensitivity. An augmentation function is required that operates on the dynamic recovery term of the growth law for the equilibrium stress, which is a state variable in VBO. The augmentation function does not affect the initial, quasi-elastic region. Its full effect appears when the long-time, asymptotic solution, which corresponds to the flow stress region is reached in an experiment. The flow stress of the VBO model consists of monotonic, rate-independent hardening affected by the kinematic stress, the viscous, rate-dependent hardening through the overstress and the rate-independent hardening represented by the isotropic stress. With the augmentation function it is possible to have only rate-dependent contributions to the flow stress. Through a proper choice of the augmentation function the rate-dependent and the rate-independent contributions can be apportioned to yield positive, zero and negative rate dependence in a consistent manner. As a demonstration, the stress-strain curves of Copper, a fcc metal, are reproduced. In fcc metals the positive rate dependence increases with deformation. The rate-dependent stress-strain curves of polymethylmethacrylate (PMMA) exhibiting a distinct yield point are also reproduced. Numerical creep tests performed at the same stress level before and after the yield point lead to unlimited creep strain and primary creep, respectively. The modeling of these very different types of material behavior requires only different material constants and augmentation functions within the same general constitutive equation.

Models of viscoplasticity some comments on equilibrium (back) stress and drag stress

SummaryPhenomenological and microstructural motivations for the terms appearing in the title are found in a literature survey. Although the interpretations differ with various investigators a strong tendency is observed to consider plastic flow as rate dependent. It is stated that plastic strain takes time to develop and the existnce of an equilibrium stress is postulated at which plastic strain is fully developed. It is similar to the back stress used in materials science. The drag stress introduced from microdynamical studies performs the same function as the isotropic variable in plasticity. Most of the theories that describe the transient and steady-state behavior of metallic alloys make the inelastic strain rate a function of the over (effective) stress. It is shown that this concept has considerable advantages in the modeling of changes of viscous (time- or rate-dependent) and plastic (time- or rate-independent) contributions to hardening that are observed in cyclic loading and dynamic plasticity.

Viscoplasticity theory based on overstress. The prediction of monotonic and cyclic proportional and nonproportional loading paths of an aluminum alloy

Abstract An isotropic formulation of the viscoplasticity theory for small strain and based on overstress with a differential growth law for the equilibrium stress is introduced. The four material constants and the two material functions of the theory are determined from uniaxial tensile tests involving strain-rate changes at room temperature and performed on a 6061 T6 Aluminum Alloy. Subsequently the theory is used to predict the biaxial behavior under axialtorsion loading. All tests are under strain control and involve proportional loading and axial followed by torsional straining (and vice versa). Cyclic histories include in-phase and out-of-phase cycling. The predictions of the theory are very reasonable for this cyclically neutral alloy. For cyclic hardening or softening materials a modification of the theory is necessary and is under development.

6 – A Small-Strain Viscoplasticity Theory Based on Overstress

This chapter focuses on the viscoplasticity theory based on overstress (VBO) for small strain, isotropy, and isochoric inelastic deformation. VBO is a unified theory without a yield surface representing a solid. The total strain rate is the sum of the elastic and inelastic strain rates. The inelastic strain rate is an increasing function of the overstress, the difference between the stress and the equilibrium stress, which is a state variable of the theory. The overstress is a measure of rate dependence. The purpose of the kinematic stress, a second state variable, is to model work hardening (softening) in monotonic loading. The isotropic stress or rate independent stress, the third state variable, is constant for cyclic neutral behavior. For cyclic hardening or softening, a growth law is needed. It also changes when high homologous temperature behavior is modeled. Asymptotic solutions for constant strain rate exist and are useful in the identification of rate-dependent and rate-independent contributions to the stress.

Relaxation behavior and modeling

Abstract Load relaxation tests deliver several orders of magnitude of inelastic strain rate data while elastic strains are converted into inelastic strains [see Lemaitre and Chaboche, 1994. (Mechanics of Solid Materials, Oxford University Press, Cambridge p. 264)]. Hart used this test for providing information on the inelastic deformation behavior for modeling purposes. Characteristic relaxation curves were obtained with ductile metals and alloys at room and high temperature showing a scaling relation derived from Harts theory. Subsequent testing with servo-controlled testing machines and strain measurement on the gage length showed that an increase of prior strain rate also increased the average relaxation rate. For relaxation tests starting in the flow stress region, the relaxation curves can be independent of the stress and strain at the start of the relaxation tests. For the modeling of these newly found relaxation behaviors and other phenomena the viscoplasticity theory based on overstress (VBO) has been introduced. It is shown that VBO admits a long-term (asymptotic) solution that can be used with sufficient accuracy for the flow stress region of the stress–strain diagram. The long-term solution predicts the observed relaxation behaviors and that the relaxation curves coincide when shifted along the stress axis. This behavior is observed for the recently obtained data and is confirmed by two sets of the Hart-type data when they are plotted according to the new method.

A Theory of Viscoplasticity Based on Infinitesimal Total Strain

A viscoplasticity theory based upon a nonlinear viscoelastic solid, linear in the rates of the strain and stress tensors but nonlinear in the stress tensor and the infinitesimal strain tensor, is being investigated for isothermal, homogeneous motions. A general anisotropic form and a specific isotropic formulation are proposed. A yield condition is not part of the theory and the transition from linear (elastic) to nonlinear (inelastic) behavior is continuous. Only total strains are used and the constant volume hypothesis is not employed. In this paper Poissons ratio is assumed to be constant. The proposed equation can represent: initial linear elastic behavior; initial elastic response in torsion (tension) after arbitrary prestrain (prestress) in tension (torsion); linear elastic behavior for pure hydrostatic loading; initial elastic slope upon large instantaneous changes in strain rate; stress (strain)-rate sensitivity; creep and relaxation; defined behavior in the limit of very slow and very fast loading. Stress-strain curves obtained at different loading rates will ultimately have the same “slope” and their spacing is nonlinearly related to the loading rate.

The Influence of Cool-Down Temperature Histories on the Residual Stresses in Fibrous Metal-Matrix Composites

The vanishing fiber diameter model together with the thermoviscoplastic ity theory based on overstress including a recovery of state formulation is introduced. They are employed to analyze the effects of temperature rate and of annealing at constant temperature on the residual stresses at room temperature when unidirectional fibrous metal-matrix composites are cooled down from 1000°C during the manufacturing process. For the present analysis the fibers are assumed to be transversely isotropic thermoelastic and the matrix constitutive equation is isotropic thermoviscoplastic including recovery of state. All material functions and constants can depend on current temperature. Yield sur faces and loading/unloading conditions are not used in the theory in which the inelastic strain rate is solely a function of the overstress, the difference between stress and the equi librium stress, a state variable of the theory. Assumed but realistic material elastic and vis coplastic properties as a function of temperature which are close to W/9Cr-lMo steel com posite permit the computation of residual stresses. Due to the viscoplasticity of the matrix time-dependent effects such as creep and change of residual stresses with time are found. It is found that the residual stresses at room temperature change considerably with temper ature history. The matrix residual stress, upon reaching room temperature, is highest for the fastest cooling rate, but after 30 days rest the influence of cooling rate is hardly notice able since relaxation takes place. Annealing periods can reduce the residual stresses com pared to continuous cooling.

On the interaction of rate and history dependence in structural metals

SummaryIn slow motions, such as occur in repeated creep, relaxation and low-cycle fatigue loadings, rate and history dependence interact and the conventional metallic material idealizations do not appear to be appropriate. For later use an operational definition of aging, rate and history dependence is given. Constitutive equations capable of reproducing history dependence must not only depend on the forcing function in the present time interval but must also have a not completely fading memory of prior loadings. It is demonstrated that certain forms of integral constitutive equations and defined history dependence. The history dependence of structural metals is examined and it is shown that prior deformation can change the response of structural metals to the same forcing function permanently but only in degree and not in kind; the variable hereditary property of structural metals is limited and is caused by deformation-induced changes of the microstructure. Variant and invariant response properties under prior loading are stated.Even in a continuum approach a measure of microstructure change appears to be necessary for cyclic inelastic loading. Loading, neutral loading and unloading are defined using a properly modified differential of the second invariant of the forcing function (either stress or strain) tensor. A tensor-valued structure memory function with constant, partially or completely fading characteristics and a discontinuous growth law, operative at points of unloading, are postulated. They serve as a repository for the modelling of history dependence and can be viewed as a substitute for the yield sulface. The structure memory function and its growth law are the made part of a nonlinear integral constitutive equation. A properly constructed kernel permits the modelling of initial elastic response and deformation-induced anisotropy. It is demonstrated that the approach has enough flexibility to account for the Bauschinger Effect, history dependence in cyclic creep and cross hardening in tension-torsion experiments.ZusammenfassungBei langsamen Vorgängen, wie sie beim Kriechen, bei der Relaxation und im Zeitfestigkeitsversuch vorkommen, tritt eine Wechselwirkung zwischen den von der Vorgeschichte und von der Geschwindigkeit abhängigen Vorgängen ein, und die üblicherweise verwendeten Stoffgleichungen sind nicht mehr ganz angebracht. Eine brauchbare Definition der Begriffe Altern, Abhängigkeit von der Vorgeschichte und Geschwindigkeitsabhängigkeit wird zur späteren Verwendung eingeführt. Stoffgleichungen, die Abhängigkeit von der Vorgeschichte reproduzieren, müssen nicht nur von der Variation der Eingangsgröße im jetzigen Zeitintervall abhängen, sondern müssen auch ein nicht ganz verschwindendes Gedächtnis für frühere Belastungen aufweisen. Es wird gezeigt, daß bestimmte Integralstoffgleichungen und Theorien der verdeckten Zustandsgrößen grundsätzlich nicht in der Lage sind, Abhängigkeit von der Vorgeschichte zu reproduzieren. Die Abhängigkeit von der Vorgeschichte der Metalle wird untersucht, und es ergibt sich, daß die Reaktion auf eine bestimmte Eingangsgröße sich zwar mit der Verformung bleibend, aber nur graduell und nicht grundsätzlich verändern kann; die Veränderung der Eigenschaften der Metalle ist begrenzt und durch verformungsinduzierte Änderungen im Gefüge hervorgerufen. Die unter vorheriger Belastung varianten und invarianten Eigenschaften der Reaktionsgrößen werden angegeben.Für zyklische Belastungen muß selbst in einer Kontinuumstheorie ein Maß für Gefügeänderungen eingeführt werden. Die etwas modifizierte zweite Invariante des Tensors der Eingangsgröße (entweder Spannung oder Dehnung) wird benützt, um Belastung, neutrale Belastung und Entlastung zu definieren. Eine tensorielle Gefügegedächtnisfunktion mit teilweise oder ganz verschwindendem Gedächtnis und ein diskontinuierliches Zuwachsgesetz, das an Entlastungspunkten aktiviert werden kann, werden eingeführt. Diese können als Ersatz für die Fließgrenzfläche betrachtet werden und bilden die Grundlage für die Wiedergabe der von der Vorgeschichte abhängigen Vorgänge. Die Gefügegedächtnisfunktion und ihr Zuwachsgesetz werden dann in eine nichtlineare Integralstoffgleichung eingeführt. Ein besonders konstruierter Kern ermöglicht die Wiedergabe der verformungsinduzierten Anisotropie und des anfänglich elastischen Verhaltens. Die vorgeschlagene Methode weist genügend Spielraum auf, so daß der Bauschinger-Effekt, Abhängigkeit von der Vorgeschichte bei zyklischem Kriechen und Wechselwirkungen in Zug-Verdrehungsversuchen wiedergegeben werden können.

An orthotropic theory of viscoplasticity based on overstress for thermomechanical deformations

Abstract An infinitesimal, orthotropic theory of viscoplasticity based on overstress for thermo-mechanical loading (TVBO) is presented. The total strain rate is the sum of elastic, inelastic and thermal strain rates. An orthotropic constitutive law is postulated for each strain rate using the characteristics of orthotropic matrices and previous isotropie formulations of the viscoplasticity theory as a guide. All material functions and constants can be functions of current temperature and no influence of temperature history is modeled. Yield surfaces and loading/unloading conditions are not used in the theory in which the inelastic strain rate is solely a function of the overstress. the difference between stress and the equilibrium stress, a state variable of the theory. A comparatively simple theory is obtained which is capable of modeling important phenomena like creep, relaxation, rate sensitivity, hysteresis, tension/compression asymmetry and nearly elastic regions. It is also possible to model quasielastic behavior in one direction while the others behave viscoplastically. The theory is shown to reduce to a previously proposed formulation for inelastic incompressibility and isotropy.