Limiting stress of metals and the limiting state under creep conditions

We show that, as long as one deals with a plane stress state, the Coulomb-Mohr fracture criterion traditionally used in soil and rock mechanics is in a rather good agreement with the results of fracture experiments for metal materials under the conditions of creep caused by a long-term action of a constant load. We perform statistical analysis to compare the proposed theoretical limiting-stress-fracture-time dependencies with the experimental results given in [1–3] and the results of computations by other authors [1, 4, 5]. Just as in [6], the limiting state arising under the conditions of creep caused by long-term loading is considered in the case of plane strain. We use the solutions of the rigid-plastic problems on the pressure exerted by a flat die on a half-plane and the extension of a strip with angular notches to find the fracture time for the corresponding problems in the setting of a perfectly rigid-creeping body.     

Sufficient criterion of fracture in the case with a complex stress state and non-proportional deformation of the material in the pre-fracture zone

A general case of proportional loading with a complex stress state of the material in the pre-fracture zone, which is typical for polycrystalline solids with plastic deformation, is considered. A sufficient criterion of fracture is proposed for the case of a complex stress state with non-proportional deformation of the material in the pre-fracture zone. Critical parameters of fracture (pre-fracture zone length and load) for cracks propagating in quasi-brittle materials are obtained with the use of a modified Leonov-Panasyuk-Dugdale model. The pre-fracture zone width is determined by solving the problem of the plasticity theory in the vicinity of the crack tip. The proposed modification of the Leonov-Panasyuk-Dugdale model makes it possible to estimate the critical opening of the crack and the critical displacement of the crack flanks. Inequalities that describe different mechanisms of material fracture under proportional loading (predominantly shear fracture mechanism and fracture mechanism through cleavage) are derived.     

Formulierung des mechanischen Werkstoffverhaltens bei zeitabhängiger Spannung-Verformung-Beziehung

Zusammenfassung In der vorliegenden Arbeit werden Rechenverfahren untersucht, mit deren Hilfe das mechanische Verhalten nichtlinearer viskoelastischer Werkstoffe bei unterschiedlicher Beanspruchungsart beschrieben werden kann.Da der einachsige isotherme Kriechversuch am einfachsten Auskunft über das zeitabhängige Werkstoffverhalten gibt, sind mehrere empirische Kriechgesetze angegeben. Eine zweckmäßige Funktion für Kurzzeitbeanspruchung ist der multiplikative Potenzansatz nachNutting. Zur Verringerung des experimentellen Aufwands werden unter Verwendung verschiedener Annahmen die Ergebnisse unterschiedlicher Belastungsgeschichten denen des Kriechversuchs mit konstanter Spannung zugeordnet.Unter der Voraussetzung der Existenz einer mechanischen Zustandsgleichung wird nachgewiesen, daß die meisten empirischen Kriechgesetze zu einer allgemeinen Zustandsgleichung führen. Die Übereinstimmung zwischen der Rechnung nach dieser Gleichung und den Ergebnissen aus Zugversuchen mit konstanter Dehngeschwindigkeit läßt die Existenz einer mechanischen Zustandsgleichung auch für die untersuchte Gruppe von Werkstoffen vermuten, allerdings nur für Sonderfälle der Beanspruchung.Der Vergleich einiger Kriechhypothesen für variable Belastungsgeschichte zeigt, daß eine nichtlineare Erinnerungsfunktion das zeitabhängige Verhalten nichtlinearer viskoelastischer Polymere bei Kurzzeitbeanspruchung richtig wiedergibt. Damit und im Vergleich mit Meßergebnissen ist eine geschlossene Darstellung des Werkstoffverhaltens bei unterschiedlicher Beanspruchungsart erreicht worden.Eine Methode zur Vorhersage nichtlinearer Spannungsrelaxation aus den Ergebnissen einachsiger Kriechversuche mit jeweils konstanter Spannung wird angegeben. Bei Verwendung der Zeitverfestigungshypothese und einer Näherungsfunktion kann das Relaxationsverhalten auch bei mehrstufiger Dehnungsgeschichte in guter Übereinstimmung mit experimentellen Ergebnissen beschrieben werden.Über die Theorie des plastischen Potentials können empirische Kriechgesetze für den mehrachsigen Spannungszustand verallgemeinert werden. Die sich daraus ergebenden Beziehungen werden zur Berechnung des Kriechverhaltens bei kombinierter Zug-Torsions-Beanspruchung mit spontaner Spannungsänderung verwendet.Keine der untersuchten Kriechtheorien beschreibt das Werkstoffverhalten vollständig. Schon im Zustand mechanischer Stabilität werden unterschiedliche Rechenverfahren benötigt. Jedoch hat dieses Konzept den Vorteil, daß allein die Ergebnisse einer Beanspruchungsart in Verbindung mit einfachen Näherungsmethoden genügen, um das Werkstoffverhalten in anderen Beanspruchungssituationen zu berechnen.

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Summary In this paper calculation procedures are investigated in order to formulate the mechanical behaviour of nonlinear viscoelastic materials at different loading history.Isothermal creep under constant uniaxial stress is the easiest way of gathering relevant data of time-dependent materials. Therefore various empirical expressions for creep are presented. A useful function for short loading times is theNutting equation. As it is desirable that other loading forms are related to this test some alternative theories are employed.Assuming the existence of a mechanical equation of state it is proved that a general equation can be derived from most of the empirical creep laws. The agreement between the prediction of this equation and the tension test data at different strain-rate lends support to the existence of a mechanical equation of state also for this group of materials, but only for a special case of loading form.A comparison of some creep theories for variable stress indicates that a nonlinear hereditary function represents the memory dependent behaviour in the short time range. A close agreement between the experimental response to different loading histories and the response predicted by the hereditary representation is found.A method is described for predicting nonlinear stress relaxation from nonlinear creep data under constant uniaxial stress. This method utilizes the time hardening theory and as a first approximation an inversion of the function describing creep at constant stress. Using this concept and the hereditary representation the behaviour during a multistep strain history is computed and found to be in good agreement with relaxation data.By employing a yield criterion based on the plastic potential uniaxial creep laws may be generalized to represent creep under combined stress. The relations deduced are applied to predicted creep behaviour in combined tension-torsion loading with abrupt changes in stress.For variable loading history none of the theories which have been described is totally satisfactory. In the absence of thermal softening and structural changes the results indicate that different calculation procedures are necessary to compute the mechanical behaviour of nonlinear viscoelastic materials But this concept has the advantage that the results from one test form and simple curve fitting technique alone are sufficient to formulate the behaviour of a material under another situation where the stress or strain are different.

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Vortrag, gehalten auf der Jahrestagung der Deutschen Rheologischen Gesellschaft in Berlin vom 8.–10. Mai 1978.

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Mit 13 Abbildungen und 2 Tabellen     

Theory of creep of initially strain-hardening isotropic materials

Conclusions We analyzed the status of the problem of developing governing equations of isotropic creep within the framework of a mechanical equation of state. The feasibility of using equations of the hereditary type to describe the third stage of creep was evaluated.We also constructed a creep theory based on a refinement of the principle of the similarity of isochronic curves. This can be regarded as an attempt to generalize the concepts of mechanical equation of state and nonlinear heredity. The theory makes it possible to consider the initial strain-hardening of the medium, evaluate the third stage of creep, and take into account the history and cyclicity of loading.It was shown that nonsteady creep develops in media which exhibit linear strain-hardening, while the development of all three stages of creep is possible in media characterized by exponential strain-hardening. It was discovered that there is a sudden increase in the rate of nonsteady creep under constant stress. The creep of certain structural materials under steady, stepped, and cyclic loading was calculated and satisfactory agreement was obtained with experimental results.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 25, No. 2, pp. 90–100, February, 1989.     

Modelling effects of degree of crystallinity on mechanical behavior of semicrystalline polymers

Viscoplasticity theory based on overstress (VBO) which is one of the unified state variable theories is extended to account for crystallinity ratio () on mechanical behavior of semicrystalline polymers. The modifications on VBO are done considering the semicrystalline polymeric materials somewhat as a composite material since it consists of amorphous and crystalline phases. Amorphous and crystalline phase resistances are arranged in two different analog models: amorphous stiffness and flow are in parallel and series with crystalline phase. Apart from many existing work in the literature, not only uniaxial loading are modeled but also creep and relaxation behaviors are simulated for a hypothetical material. It is shown that when amorphous and crystalline phase resistances acting in parallel are considered in the model, creep, relaxation and uniaxial loading and unloading behaviors can be simulated well using the modified VBO. In addition, uniaxial compression loading and unloading behavior of highly crosslinked ultra-high molecular weight polyethylene (UHMWPE) and creep behavior of polytetrafluoroethylene (PTFE) with different crystallinity ratios are simulated using the proposed VBO model where amorphous and crystalline phases are parallel. Simulation results are compared to the experimental data by Kurtz et al. (2002) and Sun et al. (2005) [Kurtz, S.M., Villarragaa, M.L., Herra, M.P., Bergström, J.S., Rimnacc, C.M., Edidin, A.A., 2002. Thermomechanical behavior of virgin and highly crosslinked ultra-high molecular weight polyethylene used in total joint replacements. Biomaterials 23, 3681–3697; Sun, H., Cooke, R. S., Bates, W. D., Wynne, K.J., 2005. Supercritical CO₂ processing and annealing of polytetrafluoroethylene (PTFE) and modified PTFE for enhancement of crystallinity and creep resistance. Polymer 46, 8872–8882] respectively and good match with experimental data is obtained.     

Law of damage accumulation and fracture criteria in highly filled polymer materials

We present the results of a large series of experiments aimed at the study of laws of damage accumulation and fracture in highly filled polymer materials under loading conditions of various types: monotone, repeated, low- and high-cycle, with varying type of stress state, dynamic (in general, more than 50 programs implemented on specimens from one lot of material). The data obtained in these test allow one to make conclusions about the constitutive role of the attained maximum of strain intensity when estimating the accumulated damage in the process of uniaxial tension by various programs (in particular, an additional cyclic deformation below the preliminary attained strain maximum does not affect the limit values of strain and stress in the subsequent active extension), about the strong influence of the stress state on the deformation and fracture, about the specific features of the nonlinear behavior of the material under the shock loading conditions and its influence on the repeated deformation. All tests are described (with an accuracy acceptable in practical calculations, both with respect to stresses and strains in the process of loading and at the moment of fracture) in the framework of the same model of nonlinear viscoelasticity with the same set of constants. The constants of the proposed model are calculated according to a relatively simple algorithm by using the results of standard uniaxial tension tests with constant values of the strain rate and hydrostatic pressure (each test for 2–3 levels of these parameters chosen from the ranges proposed in applications, each loading lasts until the fracture occurs, and one of the tests contains an intermediate interval of total loading and repeated loading) and one axial shock compression test if there are dynamic problems in the applications. The model is based on the use of the criterion fracture parameter which, in the class of proportional loading processes, is the sum of partial increments of the strain intensity on active segments of the process (where the strain intensity is at its historical maximum) with the form of the stress state and the intensity of strain rates taken into account.     

Results of studying creep and long-term strength of metals at the Institute of Mechanics at the Lomonosov Moscow State University (To Yu. N. Rabotnov’s Anniversary)

Basic results of experimental and theoretical research of creep processes and long-term strength of metals obtained by researchers of the Institute of Mechanics at the Lomonosov Moscow State University are presented. These results further develop and refine the kinetic theory of creep and long-duration strength proposed by Yu. N. Rabotnov. Some problems arising in formulating various types of kinetic equations and describing experimental data for materials that can be considered as statically homogeneous materials (in studying the process of deformation and rupture of such materials, there is no need to study the evolution of individual cracks) are considered. The main specific features of metal creep models at constant and variable stresses, in uniaxial and complex stress states, and with allowance for one or two damage parameters are described. Criterial and kinetic approaches used to determine long-term strength under conditions of a complex stress state are considered. Methods of modeling the metal behavior in an aggressive medium are described. A possibility of using these models for solving engineering problems is demonstrated.     

金属蠕变与蠕变破坏理论综述

本文对金属在均匀单向与复杂应力状态下的蠕变与蠕变破坏行为及结构物蠕变下持久性发展了系统的试验与理论研究。 在试验研究中本文考虑了各种金属在恒应力与变应力下的行为,应力集中,加载方式对蠕变破坏的影响;同时还提出了材料的结构损伤的新的量测方法。     

Applied creep theory for bodies with anisotropy due to plastic prestrain

Plastic strains in structures at the stages of manufacturing, testing, and approaching the operation regime cause anisotropic variations in the mechanical properties of materials, including creep strength. We consider the following special but practically important class of loading processes for originally isotropic materials: a simple active plastic strain is followed by a long-term steady-state loading within the elastic limits. To describe the second stage, we present the creep strain deviator in the form of an additive orthogonal decomposition in the directions of the repeated loading and the vector anisotropy. The coefficients in the decomposition are material functions of time, of the intensities of the preliminary and repeated loadings, and of the angle between the directions of these loadings. We obtain conditions on the material functions under which, at any given time instant, there is a one-to-one continuous correspondence between the stress and strain tensors for the model proposed and the boundary-value problem in the generalized statement has a unique solution; we also prove the convergence of the iteration method of elastic solutions used to find this unique solution. The model is identified according to the creep diagrams (under steady-state stresses of different values) determined for the material in the original state and after the plastic prestrain at an angle (zero, extended, and intermediate) to the direction of the repeated loading. We show that our results are in good agreement with the results available in the literature concerning experiments in this class of processes for stainless steel at high temperature. We propose an engineering version of the theory in which only the experimental data for uniaxial tension are used. We discuss the versions of the model for the cases in which the plastic preloading is cyclic (one-dimensional or circular) and the repeated loading is unsteady.     

近场动力学研究进展与岩石破裂过程模拟

近场动力学（Peridynamics，PD）作为一种新兴的非局部性理论，在非连续处不需要任何处理，能够很好表述模型从连续到非连续的过程。首先，在PD基本理论简介的基础上，系统回顾了PD的国内外研究现状。其次，采用键型PD理论对非均匀性的圆孔岩板单轴拉伸破裂过程进行了二维数值模拟，采用态型PD理论对单轴、常规三轴以及真三轴等不同压缩条件下的岩石破裂过程进行了三维数值模拟，并以加拿大Mine-by隧洞为例对现场岩体破裂过程进行了模拟，结果表明PD在岩石破裂过程模拟上具有较强适用性。最后，指出当前PD在岩石破裂过程模拟中存在的主要问题和未来值得开展的若干研究课题。     

Experimental study of D16AT alloy deformation under periodic loading

An experimental procedure and results of an experimental study of the kinetics of the strained state of a material under periodic uniaxial loading are reported. Diagrams of the principal components of the total strain tensor for an asymmetric periodic “soft” regime of tension-compression of the samples are presented. Specific features of deformation revealed under periodic asymmetric loading of the samples can be used to formulate a criterion of fatigue fracture of materials and to identify the mechanism of strain hardening of pre-fracture zones in machine elements.     

砂岩短时单轴直接拉伸蠕变特性试验研究

采用自行设计加工的挂重型岩石材料直接拉伸装置,对重庆市某地红砂岩进行了短时分级加载单轴直接拉伸蠕变试验,并对该砂岩的蠕变特性进行了分析。结果表明,砂岩在该试验条件下具有明显的蠕变特性,其蠕变表现为衰减蠕变和稳态蠕变两个阶段,砂岩蠕变量与稳态蠕变速率随着荷载的增加而增加。应用三参量广义Kelvin模型与Burgers蠕变模型描述该砂岩的蠕变规律,结果表明,两个模型均可以准确描述该砂岩的蠕变特性,其中,Burgers蠕变模型效果更佳。本文所得结论可为从事相关研究的工作人员提供一定的借鉴。     

考虑晶体滑移面分解正应力的细观损伤模型

材料内部的解理、滑移面剥离等细观损伤是引起宏观失效的根源, 从细观尺度研究损伤的发生和发展有助于深入认识材料的变形和失效过程. 本文基于晶体塑性理论, 从滑移系的受力和变形出发研究材料的细观损伤, 建立了考虑滑移面分解正应力的细观损伤模型, 为晶体材料解理断裂的分析提供了新方法. 首先, 在晶体弹塑性变形构型的基础上引入损伤变形梯度张量的概念, 从变形运动学着手建立了考虑损伤能量耗散的本构方程, 并推导了塑性流动方程与损伤演化方程; 然后, 建立了相应的数值计算方法, 给出了应力与状态变量的更新算法, 推导了Jacobian矩阵的表达式; 接着, 以$[100]$取向的单晶铜材料为例, 通过有限元计算与试验结果的对比, 并采用粒子群优化算法标定了11个材料细观参数; 最后, 将所提细观损伤模型应用于RVE单轴拉伸过程的模拟, 得到了考虑损伤影响的应力应变曲线, 并分析了材料的塑性流动与损伤演化过程. 结果表明, 本文所提模型能够计算材料在受载过程中的损伤累积效应, 合理反映晶体材料的细观损伤机理.      

Deformations of an elastic,internally constrained material. Part 1: Homogeneous deformations

The nonlinear elastic response of a class of materials for which the deformation is subject to an internal material constraint described in experiments by James F. Bell on the finite deformation of a variety of metals is investigated. The purely kinematical consequences of the Bell constraint are discussed, and restrictions on the full range of compatible deformations are presented in geometrical terms. Then various forms of the constitutive equation relating the stress and stretch tensors for an isotropic elastic Bell material are presented. Inequalities on the mechanical response functions are introduced. The importance of these in applications is demonstrated in several examples throughout the paper.This paper focuses on homogeneous deformations. In a simple illustration of the theory, a generalized form of Bell's empirical rule for uniaxial loading is derived, and some peculiarities in the response under all-around compressive loading are discussed. General formulae for universal relations possible in an isotropic elastic, Bell constrained material are presented. A simple method for the determination of the left stretch tensor for essentially plane problems is illustrated in the solution of the problem of pure shear of a materially uniform rectangular block. A general formula which includes the empirical rule found in pure shear experiments by Bell is derived as a special case. The whole apparatus is then applied in the solution of the general problem of a homogeneous simple shear superimposed on a uniform triaxial stretch; and the great variety of results possible in an isotropic, elastic Bell material is illustrated. The problem of the finite torsion and extension of a thin-walled cylindrical tube is investigated. The results are shown to be consistent with Bell's data for which the rigid body rotation is found to be quite small compared with the gross deformation of the tube. Several universal formulas relating various kinds of stress components to the deformation independently of the material response functions are derived, including a universal rule relating the axial force to the torque.Constitutive equations for hyperelastic Bell materials are derived. The empirical work function studied by Bell is introduced; and a new constitutive equation is derived, which we name Bell's law. On the basis of this law, we then derive exactly Bell's parabolic laws for uniaxial loading and for pure shear. Also, form Bell's law, a simple constitutive equation relating Bell's deviatoric stress tensor to his finite deviatoric strain tensor is obtained. We thereby derive Bell's invariant parabolic law relating the deviatoric stress intensity to the corresponding strain intensity; and, finally, Bell's fundamental law for the work function expressed in these terms is recovered. This rule is the foundation for all of Bell's own theoretical study of the isotropic materials cataloged in his finite strain experiments on metals, all consistent with the internal material constraint studied here.     

Theory of creep deformation with kinematic hardening for materials with different properties in tension and compression

A constitutive model for creep deformation that describes the loading-history-dependent behavior of initially isotropic materials with different properties in tension and compression under stress vector rotations limited by 50–60° is presented within a thermodynamic framework. In the proposed constitutive model a kinematic hardening rule is adopted. This model also introduces an effective equivalent stress in the creep potential that is based on the first and second invariants of the effective stress tensor, and on the joint invariant of the effective stress tensor and eigenvector associated with the maximum principal Cauchy stress. The formulation of the kinematic hardening rule is presented and discussed. All the material parameters in the model have been obtained from a series of proposed basic experiments with constant stresses. These model parameters are then used to predict the creep deformation of the aluminum alloy under multiaxial loading with constant stresses, and under non-proportional uniaxial and non-proportional multiaxial loadings for both isothermal and nonisothermal processes.     

金属材料在复杂应力状态下的塑性流动特性及本构模型

工程应用中,金属材料和结构往往处于复杂应力状态。材料的塑性行为会受到应力状态的影响,要精确描述材料在复杂应力状态下的塑性流动行为,必须在本构模型中考虑应力状态效应的影响。然而,由于在动态加载下材料的应变率效应和应力状态效应相互耦合、难以分离,给应力状态效应的研究和模型的建立造成很大困难。通过对Ti-6Al-4V钛合金材料开展不同加载条件下的力学性能测试,提出了一个包含应力三轴度和罗德角参数影响的新型本构模型,并通过VUMAT用户子程序嵌入ABAQUS/Explicit软件。分别采用新提出的塑性模型和Johnson-Cook模型对压剪复合试样的动态实验进行了数值模拟。结果表明,新模型不仅在对材料本构曲线的拟合方面具有较强的优势,而且由该模型所得到的透射脉冲和载荷-位移曲线均更加准确。因此,该模型能够更精确地描述和预测金属材料在复杂应力状态下的塑性流变行为。     

Multiaxial and non-proportional loading responses,anisotropy and modeling of Ti–6Al–4V titanium alloy over wide ranges of strain rates and temperatures

Results from a series of multiaxial loading experiments on the Ti–6Al–4V titanium alloy are presented. Different loading conditions are applied in order to get the comprehensive response of the alloy. The strain rates are varied from the quasi-static to dynamic regimes and the corresponding material responses are obtained. The specimen is deformed to large strains in order to study the material behavior under finite deformation at various strain rates. Torsional Kolsky bar is used to achieve shear strain rates up to 1000 s⁻¹. Experiments are performed under non-proportional loading conditions as well as dynamic torsion followed by dynamic compression at various temperatures. The non-proportional loading experiments comprise of an initial uniaxial loading to a certain level of strain followed by biaxial loading, using a channel-type die at various rates of loadings. All the non-proportional experiments are carried out at room temperature. Experiments are also performed to investigate the anisotropic behavior of the alloy. An orthotropic yield criterion [proposed by Cazacu, O., Plunkett, B., Barlat, F., 2005. Orthotropic yield criterion for hexagonal closed packed metals. International Journal of Plasticity 22, 1171–1194.] for anisotropic hexagonal closed packed materials with strength differential is used to generate the yield surface. Based on the definition of the effective stress of this yield criterion, the observed material response for the different loading conditions under large deformation is modeled using the Khan–Huang–Liang (KHL) equation assuming isotropic hardening. The model constants used in the present study, were pre-determined from the extensive uniaxial experiments presented in the earlier paper [Khan, A.S., Suh, Y.S., Kazmi R., 2004. Quasi-static and dynamic loading responses and constitutive modeling of titanium alloys. International Journal of Plasticity 20, 2233–2248]. The model predictions are found to be extremely close to the observed material response.