Strain-energy failure criterion for rigid-plastic bodies

In the framework of the theory of strain-hardening rigid-plastic bodies, we suggest a strain-energy failure criterion. The strain states of a rigid-plastic body are depicted as points in the principal strain space and form a third-order hyperbolic surface for incompressible bodies. The strain processes are depicted by lines on this surface. The loading surface related to the strain surface lines is introduced, which permits calculating the energy dissipation as a parameter determining the failure.     

Strain rate intensity factors for a plastic material layer compressed between cylindrical surfaces

For some models of rigid-plastic bodies, the strain rate fields turn out to be singular near the maximum friction surfaces. In particular, the equivalent strain rate (the second invariant of the strain rate tensor) tends to infinity when approaching such frictions surfaces. The coefficient multiplying the leading singular term in the series expansion of the equivalent strain rate near the maximum friction surfaces is called the strain rate intensity factor. This coefficient occurs in several models predicting the development of intensive plastic deformation layers near friction surfaces and in equations describing the change in the material structure in such layers. In the present paper, the solution is constructed for the compression of a layer of a plastic material obeying the double shear model between cylindrical surfaces on each of which the maximum friction law holds. The dependence of two strain rate intensity factors on the material and process parameters is calculated and analyzed.     

Strain rate intensity factors for a plastic mass flow between two conical surfaces

The concept of strain rate intensity factor was introduced in [1], where the asymptotic expansion of the velocity field in a perfectly rigid-plastic material was obtained near the maximum friction surface, which is determined by the condition that the specific friction forces on this surface are equal to the simple shear yield strength. In particular, it was shown in this paper that near the maximum friction surface the equivalent strain rate (the second invariant of the strain rate tensor) tends to infinity inversely proportional to the square root of the distance to this surface. We note that the same result was obtained in the case of plane flow in [2]. The strain rate intensity factor is defined to be the coefficient of the leading singular number in the series expansion of the equivalent strain rate near the maximum friction surface. It was shown in [3] that there is a sufficiently complete formal analogy between the strain rate intensity factor and the stress intensity factor in mechanics of cracks [4]. In [5], it was suggested to use the concept of strain rate intensity factor to estimate the thickness of the layer near the friction surface where one should take into account viscosity effects. (Thus, this is an intensive strain layer formed as a result of a very large equivalent strain rate.) Therefore, the problem of calculating the strain rate intensity factor in specific processes is topical in the development of the general concept based on the use of the strain rate intensity factor and its applications in the theory of metal forming processes. These factors have already been calculated for several processes such as plane upsetting and drawing [3]. In the present paper, we calculate the distribution of the strain rate intensity factor in a plastic mass flow through an infinite converging channel formed by two conical surfaces on which the law of maximum friction acts (Fig. 1). A specific characteristic of this problem is the existence of two maximum friction surfaces and, accordingly, two distributions of the strain rate intensity factor. Since, according to the theory [5], the strain rate intensity factor is related to the thickness of the intensive strain layer near the friction surface, the solution of this problem may serve as a starting point for experimental confirmations of the theory. Note that the intensive strain layer thickness can be determined experimentally without any difficulties [6, 7] and the flow in an infinite channel of the shape under study can successfully model the tube drawing process [8].     

An analysis of the shear failure of rigid-linear hardening beams under impulsive loading

A theoretical rigid-plastic analysis for the dynamic shear failure of beams under impulsive loading is presented when using a travelling plastic shear hinge model which takes into account material strain hardening. The maximum dynamic shear strain and shear strain-rate can be predicted in addition to the permanent transverse deflections and other parameters. The conditions for the three modes of shear failure, i.e., excess deflection failure, excess shear strain failure and adiabatic shear failure are analyzed. The special case of an infinitesimally small plastic zone is discussed and compared with Nonaka's solution for a rigid, perfectly plastic material. The results can also be generalized to examine the dynamic response of fibre-reinforced beams.     

Plane stress state problems for notched bodies whose plastic properties depend on the form of the stress state

We consider one possible approach to the problem of describing the dependence of material plastic strain characteristics on the stress hydrostatic component arising in many porous, fractured, and other inhomogeneous materials. The plastic strain of the media under study is investigated under the plasticity assumption in the corresponding generalized form with the use of the form parameter of the stress state. The plasticity constitutive relations are stated on the basis of the plastic flow law associated with the accepted plasticity condition. For the conditions of plane stress state in the framework of the material rigid-plastic model, a system of partial differential equations is obtained and conditions for its hyperbolicity are determined. The relations for determining the stress fields and velocity fields in plastic domains are obtained, and their properties are investigated. The problem of tension of a strip with symmetric angular notches is solved, where the stress fields are determined and the continuous displacement rate field is constructed. The problem of uniform symmetric tension of a plane with a circular hole is considered. The stress fields in a strip with symmetric circular notches are examined. A comparison with solutions for plastically incompressible media whose properties are invariant with respect to the form of the stress state is performed.     

Elastoplasticity of crack loaded by two pairs of isolated shear forces in finite width plate

The failure behavior of an elastic-perfectly plastic body with a crack loaded by two pairs of concentrated shear forces is discussed. The analytical solutions of an eccentric crack in a finite plate loaded by two pairs of point shear forces are obtained. It includes the unit normal vector of the elastic-plastic boundary near the crack line, the elastic-plastic stress fields near crack line and the law of the plastic zone along the crack line with external loads. The solutions of this paper are sufficiently precise near the crack line in elastic-perfectly plastic materials. Subsequently, the present results are compared with solutions based on the minimum strain energy density theory and elastic-plastic solutions under small scale yielding condition. On the basis of the minimum strain energy density (SED) theory, the minimum values of SED in the vicinity of the crack tip are determined, the initial growth orientation of crack are determined. It is found that the normalized load under large scale yielding condition is higher than those under small scale yielding condition when the length of the plastic zone is the same.     

Plastic Extension of a Plate with Symmetric Notches

The problem of extension of a plate with symmetric edge notches of different shape is numerically solved on the basis of the theory of plane stress state of an ideally plastic body under the von Mises plasticity condition. The discontinuities of the tangential and normal components of the velocity along the rigid-plastic boundaries, which result in the plastic strain localization in the neck and the material fracture, are calculated. The obtained results are of practical interest for estimating the limit extension force for a plate with edge notches of different shape and the limit strain of a thin sheet in inhomogeneous biaxial extension.     

A unified approach for pressure and temperature effects in dynamic failure criteria

In most of the existing dynamic failure criteria, effects of temperature and pressure are ignored. In a few recent contributions, attempts to include initial temperature effects have been proposed. However, the combination of both temperature and pressure effects is still lacking in all existing criteria. The aim of the present work is to model the effects of both pressure and temperature on dynamic failure. We propose a unified approach that can be implemented within several dynamic failure criteria to simulate spallation at different initial temperatures and in a wide range of pressure. This approach is based on the scaling of the threshold failure stress similar to the scaling of the flow stress used in pressure dependent plasticity. This scaling uses the shear modulus for which we propose a thermodynamically based evolution law as function of temperature and pressure. For strain controlled failure, the threshold strain is also modified to account for triaxiality. We summarize existing dynamic failure criteria and we present our proposed approach for temperature and pressure effects as it applies to each of these criteria. Results from this approach are discussed and compared with experimental observations. As an illustration, we selected one of these failure criteria and applied it to simulate spallation in planar impact test. Predicted results for this test are also discussed in comparison with experimental observations.     

Randomness and slip avalanches in gradient plasticity

While localization of deformation at macroscopic scales has been documented and carefully characterized long ago, it is only recently that systematic experimental investigations have demonstrated that plastic flow of crystalline solids on mesoscopic scales proceeds in a strongly heterogenous and intermittent manner. In fact, deformation is characterized by intermittent bursts (‘slip avalanches’) the sizes of which obey power-law statistics. In the spatial domain, these avalanches produce characteristic deformation patterns in the form of slip lines and slip bands. Unlike to the case of macroscopic localization where gradient plasticity can capture the width and spacing of shear bands in the softening regime of the stress–strain graph, this type of mesoscopically jerky like localized plastic flow is observed in spite of a globally convex stress–strain relationship and may not be captured by standard deterministic continuum modelling. We thus propose a generalized constitutive model which includes both second-order strain gradients and randomness in the local stress–strain relationship. These features are related to the internal stresses which govern dislocation motion on microscopic scales. It is shown that the model can successfully describe experimental observations on slip avalanches as well as the associated surface morphology characteristics.     

Potential flow model of cavitation-induced interfacial fracture in a confined ductile layer

Fracture of a thin ductile layer sandwiched between stiff substrates often results from growth and coalescence of microscopic cavities ahead of an extending crack. Cavitation induced by plastic flow in a confined, ductile layer is analyzed here to evaluate the interfacial fracture toughness of such sandwich structures. For rigid-plastic materials, a new method is proposed in which the potential flow field of a fluid is used to approximate the plastic deformation. The principle of virtual work rate is applied to determine the equivalent traction-separation law. The method is demonstrated and validated for spherically symmetric cavity growth, for which an exact solution exists. We then study in detail the growth of an initially spherical cavity in a cylindrical bar of finite length subject to uniform traction at its ends. The results show that the stress-separation curves depend strongly on initial cavity size and the strain-hardening exponent, and weakly on the nominal strain. The method has clear advantages over numerical methods, such as finite-element analysis, for parametric study of cavity growth with large plastic deformation.     

细长薄壁弹体撞击钢靶屈曲的数值分析

动能深侵彻弹攻击坚硬目标时,可能会引起弹体的屈曲而导致战斗部的破坏。采用LS-DYNA3D软件,对细长薄壁弹体撞击中厚钢板的屈曲破坏进行了数值研究。数值模拟再现了各型弹体在不同撞击情形下的屈曲破坏过程,证明了实验观察得到的长、短型弹体动塑性屈曲破坏存在轴向皱褶型和轴向外翻撕裂型两种基本模式,相应的屈曲破坏存在不同的特征过载曲线。给出了相关的屈曲破坏机动场描述和比实验结果更丰富的过程分析。     

Variational method for solving problems of the plasticity of compressible media

Variational methods used in the theory of plastic flow are formulated on the assumption of the incompressibility of the deformable medium. In solving problems of the mechanics of soils and friable media and technological problems of the plastic shaping of uncompacted materials it is very important to take account of irreversible volumetric change. Extremum and variational theorems are proved in [1, 2] for rigid-plastic and viscoplastic expanding bodies. A variational equation equivalent to a complete system of differential equations is derived for a compressible plastic body.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 153–155, September–October, 1977.     

Specific features of solving the problem of compression of an orthotropic plastic material between rotating plates

An instantaneous flow with compression of a wedge-shaped layer of a rigid-plastic orthotropic material between rotating plates is considered under the assumption that the principal axes of anisotropy are rays emanating from the wedge angle and lines orthogonal to them and that the maximum friction law is valid on the plate surfaces. The solution is reduced to quadratures, and its asymptotic analysis is performed. It is found that the solution is singular near the friction surface in the general case, and conditions at which the singularity disappears are given. It is demonstrated that a rigid area can arise near the friction surface. The behavior of the resultant solution near the friction surfaces is compared with the behavior of known solutions for other models of rigid-plastic materials.     

From yield to fracture,failure initiation captured by molecular simulation

While failure of cracked bodies with strong stress concentrations is described by an energy criterion (fracture mechanics), failure of flawless bodies with uniform stresses is captured by a criterion on stress (yielding). In-between those two cases, the problem of failure initiation from flaws that moderately concentrate stresses is debated. In this paper, we propose an investigation of the process of failure initiation at the atomic scale by means of molecular simulations. We first discuss the appropriate scaling conditions to capture initiation, since system sizes that can be simulated by molecular mechanics are strongly limited. Then, we perform a series of molecular simulations of failure of a 2D model material, which exhibits strength and toughness properties that are suitable to capture initiation with systems of reasonable sizes. Transition from fracture failure to yield failure is well characterized. Interestingly, in some specific cases, failure exceeds yield failure which is in contradiction with most initiation theories. This occurs when stress are highly concentrated while little mechanical energy is stored in the material. This observation calls for a theory of initiation which requires that both stress and energy are necessary conditions of failure. Such an approach was proposed by Leguillon (2002). We show that the predictions of this theory are consistent with the molecular simulation results.     

Incorporation of yield surface distortion in finite deformation constitutive modeling of rigid-plastic hardening materials based on the Hencky logarithmic strain

In this paper a constitutive model for rigid-plastic hardening materials based on the Hencky logarithmic strain tensor and its corotational rates is introduced. The distortional hardening is incorporated in the model using a distortional yield function. The flow rule of this model relates the corotational rate of the logarithmic strain to the difference of the Cauchy stress and the back stress tensors employing deformation-induced anisotropy tensor. Based on the Armstrong–Fredrick evolution equation the kinematic hardening constitutive equation of the proposed model expresses the corotational rate of the back stress tensor in terms of the same corotational rate of the logarithmic strain. Using logarithmic, Green–Naghdi and Jaumann corotational rates in the proposed constitutive model, the Cauchy and back stress tensors as well as subsequent yield surfaces are determined for rigid-plastic kinematic, isotropic and distortional hardening materials in the simple shear deformation. The ability of the model to properly represent the sign and magnitude of the normal stress in the simple shear deformation as well as the flattening of yield surface at the loading point and its orientation towards the loading direction are investigated. It is shown that among the different cases of using corotational rates and plastic deformation parameters in the constitutive equations, the results of the model based on the logarithmic rate and accumulated logarithmic strain are in good agreement with anticipated response of the simple shear deformation.     

A theory of large-strain isotropic thermoplasticity based on metric transformation tensors

Summary A formulation of isotropic thermoplasticity for arbitrary large elastic and plastic strains is presented. The underlying concept is the introduction of a metric transformation tensor which maps a locally defined six-dimensional plastic metric onto the metric of the current configuration. This mixed-variant tensor field provides a basis for the definition of a local isotropic hyperelastic stress response in the thermoplastic solid. Following this fundamental assumption, we derive a consistent internal variable formulation of thermoplasticity in a Lagrangian as well as a Eulerian geometric setting. On the numerical side, we discuss in detail an objective integration algorithm for the mixed-variant plastic flow rule. The special feature here is a new representation of the stress return and the algorithmic elastoplastic moduli in the eigenvalue space of the Eulerian plastic metric for plane problems. Furthermore, an algorithm for the solution of the coupled problem is formulated based on an operator split of the global field equations of thermoplasticity. The paper concludes with two representative numerical simulations of thermoplastic deformation processes.     

Prediction of fracture in the vicinity of friction surfaces in metal forming processes

It is shown that the use of a fracture criterion containing a characteristic length of the flow region makes it possible to further develop the theory of fracture in the vicinity of the maximum friction surfaces in metal-forming processes, with allowance for an infinite equivalent strain rate arising near such surfaces. A model of perfectly plastic rigid solids is considered in formulating the criterion. It is noted that the approach can be extended to more complicated models of plastic solids. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 169–174, September–October, 2006.     

Homogenized rigid-plastic model for masonry walls subjected to impact

A simple rigid-plastic homogenization model for the analysis of masonry structures subjected to out-of-plane impact loads is presented. The objective is to propose a model characterized by a few material parameters, numerically inexpensive and very stable. Bricks and mortar joints are assumed rigid perfectly plastic and obeying an associated flow rule. In order to take into account the effect of brickwork texture, out-of-plane anisotropic masonry failure surfaces are obtained by means of a limit analysis approach, in which the unit cell is sub-divided into a fixed number of sub-domains and layers along the thickness. A polynomial representation of micro-stress tensor components is utilized inside each sub-domain, assuring both stress tensor admissibility on a regular grid of points and continuity of the stress vector at the interfaces between contiguous sub-domains. Limited strength (frictional failure with compressive cap and tension cut-off) of brick-mortar interfaces is also considered in the model, thus allowing the reproduction of elementary cell failures due to the possible insufficient resistance of the bond between units and joints.Triangular Kirchhoff-Love elements with linear interpolation of the displacement field and constant moment within each element are used at a structural level. In this framework, a simple quadratic programming problem is obtained to analyze entire walls subjected to impacts.In order to test the capabilities of the approach proposed, two examples of technical interest are discussed, namely a running bond masonry wall constrained at three edges and subjected to a point impact load and a masonry square plate constrained at four edges and subjected to a distributed dynamic pressure simulating an air-blast. Only for the first example, numerical and experimental data are available, whereas for the second example insufficient information is at disposal from the literature. Comparisons with standard elastic–plastic procedures conducted by means of commercial FE codes are also provided. Despite the obvious approximations and limitations connected to the utilization of a rigid-plastic model for masonry, the approach proposed seems able to provide results in agreement with alternative expensive numerical elasto-plastic approaches, but requiring only negligible processing time. Therefore, the proposed simple tool can be used (in addition to more sophisticated but expensive non-linear procedures) by practitioners to have a fast estimation of masonry behavior subjected to impact.