Simple shear deformation of rate type plastic materials with combined work-hardening

The simple shear deformations of the rate type plastic materials are theoretically analyzed and numerically calculated. The materials are endowed with the combined work-hardening which is analytically represented by a scalar and a tensor internal state variable. Two types of materials are treated, that is, the Prandtl-Reuss material with a generalized Huber-von Mises yield condition and the $\text{T}$ material with a generalized Tresca yield condition. The numerically evaluated loading-unloading phenomena of the materials are very similar with those of actual metal or plastics.     

Elasto-plastic analysis for cyclic loading and tresca yield condition

Incremental theory of plasticity along with the finite element direct stiffness method is used for inelastic structural analysis. Tresca yield surface suitable for kinematic hardening formulation of the incremental theory of plasticity is presented. A uniaxial symmetric Tresca yield condition is used to formulate a small displacement theory of the incremental plasticity for material with both equal and unequal tension and compression yield strength.Constitutive laws for plane stress problem based on the linear kinematic hardening and associated flow rule are derived for sides and corners of the yield surface. In this formulation Ziegler's modification of Prager's rule has been used. Finite element formulation, numerical solution and applications are discussed.     

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.     

Generalized isotropic yield criterion for incompressible materials

Summary With the von Mises yield criterion as basis, a generalised isotropic yield function for incompressible yielding has been formulated. The proposed yield function includes as limiting forms, both the classical yield criterion of Tresca and that of von Mises. It is shown that the general yield criterion includes a twelve-sided, piece-wise linear condition of yielding. A method of relaxing the restrictions imposed by the vertices associated with a piece-wise yield criteria is described. The formulation of a composite yield condition, comprising a piece-wise, linear and continuous, yield function, together with a continuously differentiable form of yield function, is described. The use of the proposed yield criterion is reviewed in the context of the two different ypes of experiment for determining the yield condition satisfied by a given material.With 5 Figures     

A finite strain model of combined viscoplasticity and rate-independent plasticity without a yield surface

A new internal variable formulation dealing with mechanisms with different characteristic times in solid materials is proposed within a finite deformation framework. The framework relies crucially on the consistent combination of a general viscoplastic theory and a rate-independent theory (generalized plasticity) which does not involve the yield surface concept as a basic ingredient. The formulation is developed initially in a material setting and then is extended to a covariant one by applying some basic elements and results from the tensor analysis on manifolds. The covariant balance of energy is systematically employed for the derivation of the mechanical state equations. It is shown that unlike the case of finite elasticity, for the proposed formulation the covariant balance of energy does not yield the Doyle–Ericksen formula, unless a further assumption is made. As an application, by considering the material (intrinsic) metric as a primary internal variable accounting for both elastic and viscoplastic (dissipative) phenomena within the body, a constitutive model is proposed. The ability of the model in simulating several patterns of the complex response of metals under quasi-static and dynamic loadings is assessed by representative numerical examples.     

A Generalized Yield Criterion

A generalized yield criterion is proposed based on the metai plastic deformation mechanics and the fundamental formula in theory of plasticity. Using the generalized yield criterion, the reason is explained that Mises yield criterion and Tresca yield criterion do not completely match with experimental data. It has been shown that the yield criteria of ductile metals depend not only on the quadratic invariant of the deviatoric stress tensor J2, but also on the cubic invariant of the deviatoric stress tensor J3 and the ratio of the yield stress in pure shear to the yield stress in uniaxial tension κ/σs. The reason that Mises yield criterion and Tresca yield criterion are not in good agreement with the experimental data is that the effect of J3 and κ/σs is neglected.     

Numerical implementation of a J_2- and J_3-dependent plasticity model based on a spectral decomposition of the stress deviator

A new plasticity model with a yield criterion that depends on the second and third invariants of the stress deviator is proposed. The model is intended to bridge the gap between von Mises’ and Tresca’s yield criteria. An associative flow rule is employed. The proposed model contains one new non-dimensional key material parameter, that quantifies the relative difference in yield strength between uniaxial tension and pure shear. The yield surface is smooth and convex. Material strain hardening can be ascertained by a standard uniaxial tensile test, whereas the new material parameter can be determined by a test in pure shear. A fully implicit backward Euler method is developed and presented for the integration of stresses with a tangent operator consistent with the stress updating scheme. The stress updating method utilizes a spectral decomposition of the deviatoric stress tensor, which leads to a stable and robust updating scheme for a yield surface that exhibits strong and rapidly changing curvature in the synoptic plane. The proposed constitutive theory is implemented in a finite element program, and the influence of the new material parameter is demonstrated in two numerical examples.     

The influence of aluminium alloy anisotropic texture on crack-tip plasticity

Crack‐tip plasticity in textured aluminium alloys was numerically analysed using an anisotropic yield function and an isotropic hardening law as the material constitutive response. Four real textures obtained from extrusions of rectangular and square shapes as well as five ideal textures typical of rolled products were considered and predicted crack‐tip plastic zones were compared with those obtained using the isotropic von Mises yield criterion. The use of the anisotropic yield function revealed important differences in crack‐tip plasticity compared with the isotropic case. The plastic zone features obtained for different textures were compared to single crystal results published in the open literature and similar crack‐tip plastic strain localization was observed.     

The concept of physical metric in rate-independent generalized plasticity

A new internal variable formulation of rate-independent generalized plasticity is presented. The formulation relies crucially on the consideration of the physical (referential) metric as a primary internal variable and does not invoke any decomposition of the kinematical quantities into elastic and plastic parts. On the basis of a purely geometrical argument, namely the invariance of a set under the action of the flow of a vector field, the transition to classical plasticity is demonstrated. The covariant balance of energy is systematically employed for the derivation of the mechanical state equations. It is shown that unlike the case of finite elasticity, in finite plasticity the covariant balance of energy does not yield the Doyle?CEricksen formula, unless a further assumption is made. As an application, a material model is developed and is tested numerically for the solution of several problems of large-scale plastic flow.     

Thermodynamic format and heat generation of isotropic hardening plasticity

Summary Heat generation due to plastic deformation of metals and steel is studied. Whereas in many investigations it is assumed that the fraction η of the plastic work transformed into heat is constant throughout the deformation process, the fraction η is here derived from thermodynamic considerations in a large-strain setting. It is shown that for elasto-plasticity the fraction η follows as a result of the choice of free energy, potential function and yield function. Taking the stress-strain response and the dissipative properties of the material as basis for calibration, it is shown that the thermodynamic framework of a thermoplastic material is non-unique for the general situation of non-associated plasticity. In the investigation conducted here, the mechanical response and the portion of the plastic work converted into heat (or into stored energy) during plastic deformations is predicted by means of isotropic hardening von Mises plasticity. It is shown that for a situation in which the internal variable is taken as the effective plastic, close fitting to experimental data requires a non-associated format of the evolution law for the internal variable.     

油气输送管典型应力状态下的屈服行为研究

准确计算油气输送管的实际屈服强度对合理确定输送压力有重要影响。该文分析了4种应力状态下对屈服强度有明显影响的钢管轴向应力,发现管材横向小试样试验时,轴向应力为零,工厂静水压试验时,轴向应力小于零,裸露钢管静水压试验时,轴向应力为环向应力的一半,埋地服役状态时,轴向应力为0.3倍环向应力。并从理论和试验两方面分析了实物钢管屈服强度与管材小试样屈服强度的差别。理论计算表明,4种应力状态下,屈服时vonMises理论计算的钢管环向应力值大于或等于Tresca理论计算值,其中裸露钢管受内压状态下,vonMises理论计算的管道屈服环向应力为1.15倍管材屈服强度,Tresca理论计算值为1.0倍,6根钢管爆破试验及小试样拉伸试验表明,该值为1.18倍。vonMises理论比Tresca理论更适合油气输送管的屈服计算。     

Thermodynamical deduction of constitutive equations of a plastic material with combined work-hardening

Summary The constitutive equations of a plastic material are deduced by the Clausius-Duhem inequality and assumed yield restrictions. A scalar and a tensor internal state variable are introduced and we have naturally two sets of the constitutive equations, that is, those in the elastic state and those in the yield state. The latter includes the isotropic and the kinematic work-hardening case. As a simple example an isotropic linear plastic material is discussed, which follows the generalized Hooke's law with thermal expansion in the elastic state, and the generalized Huber-von Mises yield condition in the yield state.

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Thermodynamische Herleitung der Materialgleichungen eines plastischen Materials mit kombinierter Verfestigung

Zusammenfassung Die Materialgleichungen eines plastischen Werkstoffes werden aus der Clausius-Duhemschen Ungleichung und angenommenen Fließbeschränkungen hergeleitet. Eine skalare und eine tensorielle innere Zustandsvariable werden eingeführt, und wir haben natürlich zwei Mengen von Materialgleichungen, eine im elastischen Zustand und eine im Fließzustand. Letztere enthält den isotropen Fall und den Fall der kinematischen Verfestigung. Als ein einfaches Beispiel wird ein isotropes, linear plastisches Material diskutiert. Das Material gehorcht dem allgemeinen Hookeschen Gesetz mit Wärmedehnung im elastischen Zustand und dem allgemeinen Huber-von Misesschen Gesetz im Fließzustand.

     

Integration schemes for von‐Mises plasticity models based on exponential maps: numerical investigations and theoretical considerations

We consider three different exponential map algorithms for associative von‐Mises plasticity with linear isotropic and kinematic hardening. The first scheme is based on a different formulation of the time continuous plasticity model, which automatically grants the yield consistency of the method in the numerical solution. The second one is the quadratically accurate but non‐yield consistent method already proposed in Auricchio and Beirão da Veiga (Int. J. Numer. Meth. Engng 2003; 56 : 1375–1396). The third method is an improved version of the second one, in which the yield consistency condition is enforced a posteriori. We also compare the performance of the three methods with the classical radial return map algorithm. We develop extensive numerical tests which clearly show the main advantages and disadvantages of the three methods. Copyright © 2005 John Wiley & Sons, Ltd.     

基于二阶组构张量的各向异性屈服准则

将四个屈服准则:Tresca准则、Mises准则、Mohr-Coulomb准则以及Drucker-Prager准则归类为剪切屈服准则。Tresca准则和Mohr-Coulomb准则是关于最不利截面的剪切屈服准则,而Mises准则和Drucker-Prager准则是关于各方向截面的剪应力和正应力的某种综合度量的八面体剪应力和八面体正应力的剪切屈服准则。从方向函数(ODF)的概念入手,将各方向截面的剪应力和正应力综合度量直接取为所有方向截面上的剪应力和正应力的平均。对各向同性材料,提出了平均剪切屈服度准则:当平均剪应力和平均正应力的组合达到某一极限值时,材料开始屈服。研究表明,平均剪切屈服准则与Drucker-Prager准则具有相同的形式,当不考虑平均正应力对屈服的影响时,它与Mises准则具有相同的形式。针对由各向异性损伤导致的材料各向异性强度问题,定义截面上的有效正应力和有效剪应力则分别为截面上的法向力和切向力与有效承载面积之比,基于截面上的有效应力提出了各向异性材料的平均剪切屈服准则。各向异性损伤引起的截面上有效应力放大系数为方向函数,可以采用二阶组构张量来近似表示,在任意坐标系中,各向异性屈服准则为应力分量的二次齐次式,导出了其中的系数与二阶组构张量之间的显式关系式。在二阶组构张量的主轴坐标系内,各向异性屈服准则与殷有泉的拓展Hill准则形式完全相同,当不考虑正应力对屈服的影响时,它与Hill准则具有相同的形式。     

Eulerian constitutive model for finite deformation plasticity with anisotropic hardening

A constitutive model is presented for finite strain plasticity. The model incorporates both isotropic and kinematic hardening of the Ziegler type. The corotational rate used here is in line with the theory suggested by Paulun and Pecherski (1985) but not necessarily confined to the von Mises type yield criterion and the Prager hardening rule. The aspect of integration of the corotational rates is also discussed here. The use of the integration of the material rate of tensors with time as a substitute for the proper integration with time of corotational rates leads to mathematical inconsistencies of the theory of Lie derivatives. The problem of simple shear is investigated and compared with other works.     

Closed-Form Solutions for the Mode ii Crack tip Plastic Zone Shape

Analytical closed-form solutions for the Mode II crack tip plastic zone shape have been derived for a semi-infinite crack in an isotropic elastic-plastic solid under both plane stress and plane strain conditions. Two yield criteria have been applied: the Von Mises and Tresca yield criteria. The results indicate that the Tresca zone is larger in size than the Von Mises zone. The results also reveal an intricate dependence on Poisson's ratio in plane strain conditions.     

Internal-variable constitutive model for rate-independent plasticity with hardening saturation surface

Summary An elastic-plastic material model with internal variables and thermodynamic potential, not admitting hardening states out of a saturation surface, is presented. The existence of such a saturation surface in the internal variables space — a consequence of the boundedness of the energy that can be stored in the material's internal micro-structure — encompasses, in case of general kinematic/isotropic hardening, a one-parameter family of envelope surfaces in the stress space, which in turn is enveloped by a limit surface. In contrast to a multi-surface model, noad hoc rules are required to avoid the intersection between the yield and bounding/envelope surface. The flow laws of the proposed model are studied in case of associative plasticity with the aid of the maximum intrinsic dissipation theorem. It is shown that the material behaves like a standard one as long as its hardening state either is not saturated, or undergoes a desaturation from a saturated hardening state, whereas, for saturated hardening states not followed by desaturation, it conforms to a combined yielding law in which the static internal variable rates obey a nonlinear hardening rule similar to that of analogous models of the literature. Additionally, the material is shown to behave as a perfectly plastic material for a class of (critical) saturated hardening states for which the stress state is on the limit surface. For nonassociative material models, it is shown that, under a special choice of the plastic and saturation potentials and through a suitable parameter identification, the well-known Chaboche model is reproduced. A few numerical examples are presented to illustrate the associative material response under monotonic and cyclic loadings.Dedicated to Prof. Dr. Dr. h. c. Franz Ziegler on the occasion of his 60th birthday     

On a proposal for a continuum with microstructure

Summary A recently proposed model for a continuum with microstructure is further substantiated by identifying the microstructure with dislocations. In particular, the continuum is viewed as a superimposed state composed of a perfect lattice state, an immobile dislocation state, and a mobile dislocation state. It is assumed that each state evolves continuously in space-time and transitions from one state to another take place spontaneously according to the balance laws of effective mass and momentum. When the constitutive equations are subjected to the requirements of invariance, familiar statements from dislocation dynamics are deduced. When plastic strain and yield are identified in terms of the parameters characterizing the dislocation states, familiar flow rules and yield surfaces are produced. The capability of the model to predict not only Tresca and Von-Mises plastic behavior but also phenomena such as kinematic hardening, different responses in tension and compression, latent hardening, and the Bauschinger effect, is shown. Finally, the appropriateness of our equations to model creep, cyclic plasticity, and fatigue, is illustrated.With 2 Figures     

Unified solution of a borehole problem with size effect

Based on the strain gradient theory and the unified yield criterion, the borehole problem of an elasto-plastic plane strain body containing a traction-free hole subjected to uniform far-field stress is studied. The unified expressions for the plastic region radius and the stress concentration factor considering the size effect are derived on the basis of the unified yield criterion, respectively, which can be reduced to those based on the Tresca, von Mises and twin-shear criteria. The dependences of the plastic region radius and the stress concentration on the yield criteria, the material strain hardening level and the size effect are discussed. As a result, it is concluded that the influences of yield criteria and strain hardening level on the plastic region radius and the stress concentration are significant, and the size effect cannot be ignored when the hole size is in the order of tens of microns.