Extension of a V-notched bar and failure of plastic bodies

The effect of plastic strain localization near the domains of sharp variation in shape and transverse cross-section of bodies is well known. But such processes have not yet been studied analytically well enough. On the basis of the model of an ideally rigid-plastic body, we propose an approach for determining the strain fields near the concentrators on the basis the motion of the displacement velocity field (near surfaces or discontinuity lines in the form of rigid-plastic boundaries and centers of the fan of slip lines under plane strain). We consider the problem on plastic flow with failure for a V-notched bar. We show that the plastic flow is not unique (in the framework of the solution completeness).We propose to use the strain criterion for choosing the preferable plastic flow. On the basis of the solutions thus obtained, we state an approach to studying failure processes for more complicated models of bodies.     

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.     

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.     

On the plane stress state under plastic strain

We consider the plane stress state used in the analysis of strains of thin bodies. Methods for solving problems under the assumption that three out of six components of the stress tensor are zero can be found in [1–3].In the present paper, the strain of a perfect rigid-plastic medium in the plane stress state is studied in curvilinear orthogonal coordinates. We show that in any coordinate system there exists an exact solution for which the stresses depend only on the metric function.     

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.     

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.     

Influence of dilatancy on the type of the system of equation in the plane problem of plasticity

In the classical model of an ideal rigid-plastic material without hardening, the governing system of equations is a system of hyperbolic type, and, if hardening is taken into account, the type of the system of equations changes from hyperbolic to elliptical. In this case, the correlation between the experimentally observed strain localization lines and the characteristics of the quasilinear system is violated. It is shown that if dilatancy is taken into account, the system of equations remains hyperbolic.     

Compression of an axisymmetric layer on a rigid mandrel in creep

An approximate solution describing the compression of an axisymmetric layer ofmaterial on a rigid mandrel under the equations of the creep theory is constructed. The constitutive equation is introduced so that the equivalent stress tends to a finite value as the equivalent strain rate tends to infinity. Such a constitutive equation leads to a qualitatively different asymptotic behavior of the solution near the mandrel surface, on which the maximum friction law is satisfied, compared with the well-known solution for the creep model based on the power-law relationship between the equivalent stress and the equivalent strain rate. It is shown that the solution existence depends on the value of one of the parameters contained in the constitutive equations. If the solution exists, then the equivalent strain rate tends to infinity as the maximum friction surface is approached, and the qualitative asymptotic behavior of the solution depends on the value of the same parameter. There is a range of variation of this parameter for which the qualitative behavior of the equivalent strain rate near the maximum friction surface coincides with the behavior of the same variable in ideally rigid-plastic solutions.     

Free-surface roughness of thin-walled tubes in reduction

The problem of the stress-strain state of thin-walled tubes in axisymmetric steady-state deformation is solved using the membrane theory of shells and the model of an ideal rigid-plastic material satisfying the Mises yield condition and the associated flow law. The obtained solution is used together with the empirical relation between the strain state at an arbitrary point of the free surface and the surface roughness parameters at the same point to determine the influence of some tube reduction parameters on the surface roughness parameters in the product. The employed empirical relation is derived assuming that the free surface roughness parameters depend on two independent kinematic variables.     

Stability of the Couette flow of ideal rigid-plastic bodies

The generalized Rayleigh stability problem is studied for the plane flows of ideal rigid-plastic bodies. The stationary scattering theory is used for the Couette flow to describe the structure of continuous and point spectra and to construct an expansion in eigenfunctions and in generalized eigenfunctions. Some integral estimates are proposed for the domain containing the spectrum of the problem to prove the stability of this flow.     

Estimate of strength of anisotropic bars of arbitrary cross-section in the general case of their combined stress

We consider bars of arbitrary shape made of a homogeneous anisotropic material. In the general case, all six internal force factors (three forces and three moments) are simultaneously nonzero in the transverse cross-sections of the bar. We consider the case of small displacements and strains of the bar. Using the rigid-plastic model of a strained rigid body, the associated strain law, and the traditional hypotheses of static and kinematic character for the bars, we derive parametric equations for the limit surface (the strength surface) in the space of internal forces and moments acting in the the transverse cross-section. We present several versions of the obtained equations in specific cases (for orthotropy, transversal isotropy, and isotropy) and some numerical examples.     

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。     

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].     

A combined necking and shear localization analysis for aluminum sheets under biaxial stretching conditions

A combined necking and shear localization analysis is adopted to model the failures of two aluminum sheets, AA5754 and AA6111, under biaxial stretching conditions. The approach is based on the assumption that the reduction of thickness or the necking mode is modeled by a plane stress formulation and the final failure mode of shear localization is modeled by a generalized plane strain formulation. The sheet material is modeled by an elastic-viscoplastic constitutive relation that accounts for the potential surface curvature, material plastic anisotropy, material rate sensitivity, and the softening due to the nucleation, growth, and coalescence of microvoids. Specifically, the necking/shear failure of the aluminum sheets is modeled under uniaxial tension, plane strain tension and equal biaxial tension. The results based on the mechanics model presented in this paper are in agreement with those based on the forming limit diagrams (FLDs) and tensile tests. When the necking mode is suppressed, the failure strains are also determined under plane strain conditions. These failure strains can be used as guidances for estimation of the surface failure strains on the stretching sides of the aluminum sheets under plane strain bending conditions. The estimated surface failure strains are higher than the failure strains of the forming limit diagrams under plane strain stretching conditions. The results are consistent with experimental observations where the surface failure strains of the aluminum sheets increase significantly on the stretching sides of the sheets under bending conditions. The results also indicate that when a considerable amount of necking is observed for a sheet metal under stretching conditions, the surface failure strains on the stretching sides of the sheet metal under bending conditions can be significantly higher.     

齿轮系统传动轴受横向冲击的响应分析

将齿轮传动系统的齿轮轴承简化为具有集中质量的固支梁,将齿轮受到啮合齿轮的意外撞击看成是质量块对梁的冲击。给出弯扭组合的Mises屈服条件,指出传动轴受冲击时不能忽略扭矩作用。分析了弯曲和扭转作用下的结构响应,进行了应变率修正,给出特殊情况下弯扭响应的简化分析。算例表明,弯扭冲击下传动轴的横向位移和扭转角都较大,不可忽略应变率效应;传动轴直径是影响横向位移的重要因素。     

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.     

Singular Solutions in an Axisymmetric Flow of a Medium Obeying the Double Shear Model

An asymptotic analysis of equations of an axisymmetric flow of a rigid-plastic material obeying the double shear model in the vicinity of surfaces with the maximum friction is performed. It is shown that the solution is singular if the friction surface coincides with the envelope of the family of characteristics. A possible character of the behavior of singular solutions in the vicinity of surfaces with the maximum friction is determined. In particular, the equivalent strain rate in the vicinity of the friction surface tends to infinity in an inverse proportion to the square root from the distance to this surface. Such a behavior of the equivalent strain rate is also observed in the classical theory of plasticity of materials whose yield condition is independent of the mean stress.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 180–186, September–October, 2005.     

Numerical and experimental study of elastoplastic tension-torsion processes in axisymmetric bodies under large deformations

We consider a numerical solution technique for generalized axisymmetric problems with torsion for elastoplastic bodies of revolution of arbitrary shape under large strains, as well as simple or complex loading, and the conditions of inhomogeneous stress-strain state. The processes of elastoplastic deformation, strain localization, and fracture of solid axisymmetric steel samples of variable thickness are studied experimentally and numerically for the cases of proportional and nonproportional kinematic torsional and/or tensile loading until failure. The mutual influence of torsion and tension on the deformation and failure under large strains is estimated.     

Large strain plastic deformation by crystallographic slio in metals

Large strain plastic deformation of metallic materials is investigated. The rate of change of the resolved shear stress applied to a slip system in a grain is expressed as a linear function of the slip rates. For this purpose, using an elastic-plastic analysis of the inclusion problem, the Jaumann rate of change of the stress state applied to the grain, together with its total spin, are obtained as a function of its deformation rate in the rigid-plastic case. It is shown that the existing stress has a significant influence on the accommodation tensors involved in the solution.     

Damage accumulation and crack growth in bilinear materials with softening: Application of strain energy density theory

A pseudo-elastic damage-accumulation model is developed by application of the strain energy density theory. The three-point bending specimen is analyzed to illustrate the crack growth characteristics according to a linear elastic softening constitutive law that is typical of concrete materials. Damage accumulation is accounted for by the decrease of elastic modulus and fracture toughness. Both of these effects are assessed by means of the strain energy density functions in the elements around a slowly moving crack. The rate of change of the strain energy density factor S with crack growth as expressed by the relation dS/da = constant is shown to describe the failure behavior of concrete. Results are obtained for different loading steps that yield different slopes of lines in an S versus a (crack length) plot. The lines rotate about the common intersect in an anti-clockwise direction as the load steps are increased. The intersect shifts upward according to increase in the specimen size. In this way, the combined interaction of material properties, load steps and specimen geometry and size are easily analyzed in terms of the failure mode or behavior that can change from the very brittle to the ductile involving stable crack growth. An upper limit on specimen or structural size is established beyond which stable crack growth ceases to occur and failure corresponds to unstable crack propagation or catastrophic fracture. The parameters that control the failure mode are the threshold values of the strain energy density function (dW/dV)_c and the strain energy density factor S_c.