Experimental study of vector and scalar properties of 9Kh2 steel in deformation along flat curvilinear trajectories. Part 2. Astroid-shaped trajectories

The paper gives the results of an experimental investigation of complex deformation of 9Kh2 steel along stroid-shaped trajectories. We study the behavior of vector and scalar properties of the material along this trajectory. We generalize the results of an experimental investigation of complex unloading of the material and determine the locations of limiting surfaces. Tver' State Technical University, Tver', Russia. Translated from Problemy Prochnosti, No. 6, pp. 50–55, November–December, 1999.     

Deformational theory of plasticity of transversely isotropic media

We propose a version of the theory of plasticity of transversely isotropic media for the case of simple loading. Our version is based on the concept of yield surface. We use a quadratic condition of yield that takes into account the partial effects of equivalent stresses computed according to von Mises and Hill and according to Tresca on the plastic deformation of the material. In the general case, this condition can be interpreted as a singular surface in the space of stresses. On the basis of the assumptions concerning the linearity of trajectories of plastic deformation and their normality to the initial yield surface under simple loading as well as concerning the existence of a relationship between the introduced equivalent stresses and equivalent plastic strains independent of the type of the stressed state, we deduce reversible master equations of plasticity. The adequacy of the proposed model is confirmed by the good agreement between the results of numerical analysis and experimental data. Translated from Problemy Prochnosti, No. 1, pp. 5 – 14, January – February, 1998.     

On the theory of plasticity with smooth surfaces of loading

The well-known theories of plastic yield with smooth surfaces of loading are developed and generalized on the basis of a model of a nonlinear anisotropically hardened medium based on the concept of slip. Unlike the commonly used procedure of application ofa priori known laws of hardening (variation of the surface of loading in the process of plastic deformation), we suggest a method for the experimental evaluation of a universal function of the material appearing in the constitutive equations for arbitrary complex loading processes including elastic unloading and plastic deformation in the direction opposite to the initial one. The constitutive equations are relatively simple and, hence, can easily be used for the statement and solution of boundary-value problems in the theory of plastic yield. Dnepropetrovsk State University, Dnepropetrovsk. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 33, No. 6, pp. 63–70, November–December, 1997.     

On deformation of metals along flat curvilinear trajectories of variable curvature. Part 2. Process functions

The paper gives the results of experimental investigation of complex loading and unloading of 45 steel along “Archimedean spiral” type trajectories in the plane Э₁-Э₂ in the Il'yushin's deviator strain subspace Э_i (i=1, 2, 3), which was carried out using an SN-éVM computerized calculation and testing system. We obtain experimentally the regularities of variation of the determining functions of plasticity, which were constructed within the Il'yushin hypothesis of coplanarity. Tver' State Technical University, Tver' Russia. Translated from Problemy Prochnosti, No. 4, pp. 12–18, July–August, 1999.     

Experimental study of vector and scalar properties of 9Kh2 steel in deformation along flat curvilinear trajectories. Part 1. Logarithmic-spiral trajectories

We report results of an experimental investigation of complex loading and unloading of 9Kh2 steel along logarithmic-spiral trajectories that was carried out using an SN-éVM computerized calculation and testing system. We consider the regularities of variation of vector and scalar properties of the material along these trajectories. Tver' State Technical University, Tver', Russia. Translated from Problemy Prochnosti, No. 5, pp. 36–41, September–October, 1999.     

On the Elastic and Plastic Strain Decomposition in Granular Materials

An impressive number of constitutive relations have been developed in the past few decades. With respect to the class of elastoplastic phenomenological models, elastic and plastic strain decomposition is generally stated as a basic assumption, so as to treat the elastic (i.e., recoverable) and plastic (i.e., unrecoverable) parts of the strains separately. For incrementally nonlinear relations, this decomposition is not possible. In the first part of this paper a detailed discussion of elastic and plastic decomposition is presented. Then the paper expands the debate on this crucial point by addressing the question of defining elastic (or plastic) deformation specifically for granular materials, considering two complementary approaches. An incrementally nonlinear model is used first and then a multi-scale approach is considered to examine the compatibility of this partition from a micromechanical point of view, with the usual definition of both elastic and plastic incremental strains. Finally, micro-structural considerations show that only a fraction of the elastic strain energy can be recovered, whatever the unloading path, after an incremental loading path inducing both elastic and plastic mechanisms.     

Reserves of strength in plastic deformation. Report 1

The possibility of increasing the limiting deformation and strength properties of a material in one of the main directions of the tensor of stresses relative to similar characteristics in simple loading is considered. The required change in properties is accomplished by directed strain hardening in complex loading accompanied by active loadings in single directions and partial unloading with hardening in the other directions (in the terminology of S. A. Khristianovich and E. I. Shemyakin). The results of experiments on thin-walled tubular specimens under conditions of biaxial tension, which made it possible to determine the types of trajectories of preliminary complex loading leading to increased deformation and strength properties, and also the conditions of loading with which an isotropic loading surface is obtained, are presented.Translated from Problemy Prochnosti, No. 11, pp. 3–8, November, 1991.     

Isoclinic versus arbitrary rotated intermediate configuration for gradient plasticity applications

Every multiplicative decomposition of the deformation gradient tensor into elastic and plastic parts is related to a local unloading process, which determines a so-called plastic intermediate configuration. Two kinds of such configurations are commonly used, at least in metal plasticity. The first one is connected with some fixed directions and is known as isoclinic configuration. The second may be chosen to be arbitrarily rotated and is called here arbitrary rotated intermediate configuration. We call formulation of plasticity from the point of view of these configurations respectively the isoclinic intermediate configuration approach (IIC-approach) and the arbitrary rotated intermediate configuration approach (ARIC-approach). It is known that in classical (local) plasticity, the IIC- and the ARIC-approaches may be viewed as alternative but equivalent. Under the assumptions made in the present paper, we show that this is generally no longer true, whenever gradient effects are involved.     

On the concept of the yield surface

Summary In this paper we introduce in the theory of plasticity the concept of a family of loading surfaces that are distinct from and enclose the yield surface.The yield surface is the boundary of a region in stress space in which both unloading and reloading produce elastic strains only. In the region of stress space occupied by the family of loading surfaces, reloading produces plastic strains, while unloading produces only elastic strains. Some implications of this concept for the theory of plasticity are considered. In particular, it is shown that the existing proof of convexity of a yield surface is not applicable to the loading surfaces.

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Zusammenfassung Der Begriff der Belastungsflächen, welche verschieden von der Fließfläche sind und diese einschließen, wird in die Plastizitätstheorie eingeführt.Die Fließfläche begrenzt den Bereich im Spannungsraum, in dem Entlastung und Wiederbelastung nur elastische Verzerrungen hervorrufen. Die Belastungsflächen dagegen bedecken den Bereich des Spannungsraumes, in dem Wiederbelastung plastische, Entlastung dagegen nur elastische Verzerrungen erzeugt. Einige Folgerungen für die Plastizitätstheorie werden gezogen. Im besonderen wird gezeigt, daß der bekannte Beweis für die Konvexität einer Fließfläche nicht auf die Belastungsflächen anwendbar ist.

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With 8 Figures

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This work was sponsored by the National Science Foundation of the United States Government.     

On the kinematics of finite-deformation plasticity

Summary A theory of finite deformation plasticity is developed which involves a multiplicative decomposition of the deformation gradient through the assumption that there exists a stress-free configuration which can be used to separate the elastic and plastic components of the response. By using the polar decomposition on the usual indeterminate elastic and plastic deformation tensors, two uniquely defined stress-free configurations can be identified. The structure of this theory is compared with that of a spatial theory involving the polar decomposition of the total deformation gradient. It is shown that for the special case of linear response between the stress and the elastic strain, the two theories are indistinguishable in terms of their stress responses.With 1 Figure     

Prediction of the fracture of metals in processes of cold plastic deformation. Part 2. Effect of anisotropic hardening and experimental verification of the model of plastic deformation and fracture

On the basis of the experimental investigation of the Bauschinger effect, in steels with developed prestrains, we generalize the model of plastic deformation to the case of anisotropic hardening. Within the framework of the model, we show that for high strains, the Bauschinger effect is caused by dislocations. We present the results of the experimental verification of the developed semiphenomenological model of the joint process of plastic deformation and fracture of metals under the conditions of cold deformation. It is shown that this model gives adequate predictions (in good agreement with the experimental data) of the probability of fracture of the metal caused by exhaustion of the plasticity margin in the processes of plastic deformation realized under the conditions of both simple and complex loading. Ufa State Aviation Technical University, Ufa, Russia. Translated from Problemy, Prochnosti, No. 2, pp. 74–84, March–April, 1999.     

Analysis of boundary-value problems describing the non-isothermal processes of elastoplastic deformation taking into account the loading history

We discuss the theory and approximate methods for solving boundary-value problems of thermoplasticity in a quasi-static formulation when the process of non-isothermal elastoplastic deformation of a body is a sequence of equilibrium states. In this case, the stress-strain state depends on the loading history, and the process of inelastic deformation is to be observed over the whole time interval being studied. The boundary-value problem is stated as a non-linear operator equation in the Hilbertian space. The conditions that provide the existence, uniqueness and continuous dependence of the generalized solution on the applied loads and initial strains are defined. A convergence of the methods of elastic solutions and variable elastic parameters is studied to solve the boundary-value problems describing the non-isothermal processes of active loading taking into account the initial strains dependent on the deformation history and heating. __________ Translated from Problemy Prochnosti, No. 1, pp. 69–99, January–February, 2006.     

Constitutive model and finite element formulation for large strain elasto-plastic analysis of shells

This contribution presents a refined constitutive and finite element formulation for arbitrary shell structures undergoing large elasto-plastic deformations. An elasto-plastic material model is developed by using the multiplicative decomposition of the deformation gradient and by considering isotropic as well as kinematic hardening phenomena in general form. A plastic anisotropy induced by kinematic hardening is taken into account by modifying the flow direction. The elastic part of deformations is considered by the neo-Hookean type of a material model able to deal with large strains. For an accurate prediction of complex through-thickness stress distributions a multi-layer shell kinematics is used built on the basis of a six-parametric shell theory capable to deal with large strains as well as finite rotations. To avoid membrane locking in bending dominated cases as well as volume locking caused by material incompressibility in the full plastic range the displacement based finite element formulation is improved by means of the enhanced assumed strain concept. The capability of the algorithms proposed is demonstrated by various numerical examples involving large elasto-plastic strains, finite rotations and complex through-thickness stress distributions.     

Birefringent effects in the cyclic loading for an elastic-viscoplastic polymer

The general plastic deformation should be considered by the elastic-viscoplastic theory, which contains the viscosity in addition to the elastic am plastic properties. In this paper, the birefringen effects are experimentally investigated by using celluloid as elastic-viscoplastic polymer under cyclic loadings. The optical relations in the loading and the unloading processes are semi-empirically deduced for the elastic-viscoplastic solid. The deduced optical relations give good agreement with the actual observations under the cyclic loadings for various cyclic conditions.     

Effects of elastic unloading on multiple necking in tensile bars

The deformation of round tensile bars subject to dynamic loading is studied numerically using two different plasticity theories: J₂-flow theory and J₂-deformation theory. Time development of neck localization is visualized and a comparison of the necking patterns obtained using the two different constitutive models is presented. The possibility of using J₂-deformation theory as an alternative to J₂-flow theory to model the dynamic behaviour of tensile bars subject to high rate deformation is investigated and the effects of elastic unloading are discussed. Influence of specimen geometry and initial thickness imperfections is also considered. The effects of elastic unloading on multiple necking are of particular interest, since elastic unloading may serve as a restricting factor regarding the growth of multiple neck localizations. In the present analysis, multiple necking is found to appear using both J₂-flow theory and J₂-deformation theory. The use of stubby bars leads to similar necking patterns when comparing the results obtained with J₂-flow theory and J₂-deformation theory. For slender bars subject to certain loading conditions a larger number of necks is obtained when J₂-deformation theory is applied rather than J₂-flow theory. In addition, results of simulations near the ideal-plastic limit are discussed as well as results where the yield stress is varied.     

Irreversible deformation of metal matrix composites: A study via the mechanism-based cohesive zone model

An elastic–plastic interface model at finite deformations is utilized to predict the irreversible deformation of metal matrix composites (MMCs) under the transverse loading and unloading conditions. The associated benefit of the cohesive model is to provide a physical insight on the main irreversible deformation mechanisms, i.e., the geometrically nonlinear, localized plastic deformation and damage induced debonding, at the interface of MMCs. The extensive parametric study is conducted using this cohesive model to investigate the effects of the cohesive parameters on the stress–strain response of MMCs under transverse loading. Further, the ductile mechanism of the matrix is considered to characterize the competition between the plastic flow of the matrix and the inelastic interface induced irreversible deformation. Moreover, the predictions using the cohesive model are compared with those available experimental data in the literature to demonstrate the inelastic behaviors, including the interfacial plasticity and damage induced debonding, as well as the plastic flow of the matrix. The numerical results of the stress–strain responses for both loading and unloading conditions show good agreements with those obtained by the experiment. The deformation and failure modes of MMCs predicted by the model are also consistent with the observations of the experiment.     

Discrete element methods for the plastic analysis of structures subjected to cyclic loading

Methods for the analysis of complex, highly redundant structures subjected to intermittent loads causing biaxial membrane stress and stress reversal into the plastic range are presented. The Bauschinger effect in multi-axial stress is taken into account by the use of Ziegler's modification of Pragers kinematic hardening theory. The implementation of this plasticity theory in the discrete element methods involves the application of the loading in small increments. A linear relationship between increments of plastic strain and of stress, arising out of the theory, is used in conjunction with a linear matrix equation that governs the elastic behaviour of the structure. In the latter equation, plastic strains are interpreted as initial strains. A solution to the linear matrix equation, expressed in terms either of stress or of total strain, may be obtained by utilizing one of two alternative procedures. The methods are capable of treating materials which exhibit elastic–plastic behaviour involving ideal plasticity, linear or non-linear strain hardening, or limited strain hardening. Application is made to several representative structures. Comparison of some of the results with existing test data for both monotonic and reversed loading shows good correlation.     

Evaluation of the Elastic-Plastic Mixity Parameters on the Base of Different Crack Propagation Criteria. Part 2. Solution and Results

We propose a new scheme of mixed-mode problem solution based on the deformation theory of plasticity with a power-law hardening stress-strain response and on application of elastic and plastic mixity parameters. Depending on the mixed-mode loading conditions and the initial crack line direction, this approach allows one to analyze a wide range of possible crack propagation paths controlled by shear and tensile mechanisms. The equilibrium equation with Airy function is used for a two-dimensional problem in the polar coordinate system. The Ramberg-Osgood model is applied to a material with power-law hardening behavior. Using the finite difference method we obtained a numerical solution of the mixed-mode loading problem with boundary conditions corresponding to two cases of crack propagation. Within the framework of the proposed approach we estimated the dependencies between mixity parameters and various loading parameters and crack inclination angle for a range of strain hardening exponent values, which dependencies closely fit the experimental data. __________ Translated from Problemy Prochnosti, No. 4, pp. 46 – 63, July – August, 2005.     

基于SMP破坏准则的土体弹塑性动力本构模型

利用土体的塑性流动理论,提出了用于描述饱和砂土动力反应性质的弹塑性本构模型。土体总的变形由三部分组成:即弹性应变、与体积屈服机制相关的塑性应变和与剪切屈服机制相关的塑性应变。土体在初始加载与卸载和重新加载阶段性质的差别通过采用不同的模型参数加以反映。该模型能够较为准确地描述饱和砂土在单调加载和循环加载条件下的反应性质。