Large Plastic Deformations Accompanying the Growth of an Elliptical Hole in a Thin Plate

Within the context of plane stress assumptions and approximations, an analytical solution is derived for the finite deformation of a traction-free elliptical hole in an infinite plate with tensile tractions at infinity. The plate is composed of a non-work-hardening material satisfying the Tresca yield condition under a deformation theory of plasticity. The governing partial differential equations are parabolic in nature and consequently have a single family of mathematical characteristics or slip lines associated with them. Each particle of mass follows a rectilinear path in the plane defined by its slip line which emanates orthogonally from the elliptical hole. By assuming a constant speed for each particle in the plane, a state of plane equilibrium is realized. The originally elliptical hole expands in the shape of an oval which is a parallel curve to the original ellipse. The slip lines remain orthogonal to the evolving oval hole as a necessary condition for a traction-free interior boundary. This solution also satisfies the material stability criterion that the rate of plastic work be positive throughout the entire body for all time. As this solution has some features associated with large deformation crack problems at elevated temperatures, possible applications include secondary or steady-state creep.     

Numerical plane stress elastic–perfectly plastic crack analysis under Tresca yield condition with comparison to Dugdale plastic strip model

A number of plane stress numerical analyses of the mode I elastoplastic fracture mechanics problem have been performed in the past using the Huber–Mises yield criterion. This study employs instead the Tresca yield condition using an incremental theory of plasticity for a stationary crack. A commercial finite element program is used to solve the opening mode of fracture problem (mode I) for a square plate containing a central crack under generalized plane stress loading conditions. A biaxial uniform tensile traction is applied to the edges of a thin plate composed of a linear elastic non-work hardening material under small strain assumptions. The finite element results are compared with the analytical predictions of the Dugdale plastic strip model for a crack in an infinite plate subject to a biaxial uniform load at infinity.     

Perfectly plastic caustics for the opening mode of fracture

Exact expressions for the caustics generated by the reflection of light surrounding crack tips in perfectly plastic materials under plane stress loading conditions and tensile tractions at infinity (mode I) are derived. Two individual cases are examined involving two different yield criteria. The first case uses an approximation of the Mises yield condition, where in the principal stress plane two intersecting parabolas replace the standard ellipse. The second case uses the Tresca yield condition where the mode I caustic is obtained as a limit of an elliptical hole in a perfectly plastic material. In both cases, kinematically admissible velocity fields are employed to obtain strain fields from which the theoretical caustics are predicted.     

Localization in elasto-plastic materials: Influence of the plasticity yield surface in biaxial loading conditions

The influence of the plasticity yield surface on the development of instabilities in plane plates in biaxial loading is analyzed in order to understand and simulate the localization pattern observed in an expanding hemisphere experiment. First, a criterion for the activation of slip bands is formulated in the form of a critical hardening coefficient: it is particularized to the Von Mises and Tresca surfaces. In the Von Mises case, the criterion gives a strongly negative hardening coefficient in biaxial loading conditions different from the ones of plane strain. In the Tresca case, the criterion is fulfilled for a perfectly plastic material in uniaxial and biaxial loading; besides, in equi-biaxial loading, two possible orientations for slip bands are exhibited; this can be understood, with a few approximations, by the existence of a vertex point on the Tresca yield surface which give additive degrees of freedom for the direction of the plastic strain rate. Second, the development of localization in the loading conditions met in an expanding hemisphere experiment is simulated using both plasticity yield surfaces; whereas the Von Mises simulation does not localize, the Tresca simulation exhibits a pattern composed of a network of shear bands of different orientations; this pattern is not far from the pattern observed experimentally.     

Visco-plastic flow and the tresca yield condition

This paper is concerned with a rigid visco-plastic solid which obeys the Tresca yield condition and an associated flow rule. Generalizations of both linear and nonlinear relationships between the rate of deformation and shearing stress in excess of the yield for simple shear flow are obtained from a complementary function. The visco-plastic flow of a clamped circular plate is considered as an application.     

The method of stress variation applied to minimal design for multiple loading

The method of stress variation, previously developed for the optimal design of axisymmetric sandwich plates obeying the Tresca criterion for a single loading condition[1–3], is extended to multiple loading conditions. The method consists of a systematic reduction in the design parameter until the optimum is reached. For the class of loadings treated in this paper, it is shown that repeated application of only a single stress variation is necessary to produce the optimum design for a simply supported plate. The method is such that it can easily be adapted to automatic computation, and the computer time required is simply proportional to the number of separate loads to be considered. An example of a simply supported plate for three separate loads is presented.     

Experimental evaluation of incremental theories for nonproportionate loading of thin-walled cylinders

An experimental investigation was undertaken to evaluate incremental-strain theories which have been proposed in the literature to predict the loads on thin-walled cylinders subjected to nonproportionate loading which follow prescribed strain histories. Test data were obtained for two materials, annealed SAE 1035 steel and normalized 4340 steel. Material-property tests for the SAE 1035 steel indicated that the stress-strain diagram was flat topped and the material followed the Tresca flow condition. Similar tests for the SAE 4340 steel indicated that this steel was a linear strain-hardening material that followed the von Mises flow condition. Two incremental-strain theories were developed for thin-walled cylinders made of SAE 1035 steel. Both were based on the Tresca flow condition. One theory called the Tresca-Tresca theory used the stress-strain relations for the Tresca theory. The other theory called the Tresca-Mises theory used the Prandtl-Reuss stress-strain relations. In general, the test data fell between the two theories. The incremental theory developed for thin-walled cylinders made of the SAE 4340 steel, called the Mises-Mises theory, was based on the von Mises flow condition and the Prandtl-Reuss stress-strain relations. The agreement between theory and experiment was poor.     

Contact Transform for the Biharmonic Equation Applicable to Plane Stress Elastoplastic Elliptical Hole Problems

An analytical technique is developed that reduces the unknown elastic-plastic boundary of a linear elastic-perfectly plastic material containing an elliptical hole under tensile plane stress loading conditions into an equivalent mathematical problem with known boundaries. This mathematical transformation may facilitate this problem’s solution by either analytical or numerical means. Yield is assumed to occur in this analysis under the Tresca yield criterion. An example elastic-plastic problem illustrating this method is drawn from existing literature in the form of a perturbation solution for an elliptical hole derived by a series expansion about a circular boundary.     

Non-quadratic Kelvin modes based plasticity criteria for anisotropic materials

Novel (non-quadratic) plasticity criteria based on Kelvin modes are formulated here for anisotropic materials. As an example, such a macroscopic criterion is applied with success to the case of FCC nickel-base single crystals. Indeed, relying on the cubic symmetry of the material, the Kelvin decomposition of elasticity tensor easily allows for the definition of an objective and loading independent criterion. The criterion identification is performed from different loading cases for CMSX2 single crystal superalloy. Tension-torsion yield surfaces at room temperature and yield stress dependence on crystal orientation are modeled. The Kelvin modes based criterion is compared to experimental data, to Hill and Barlat and coworkers macroscopic criteria and to Schmid law predictions. The results show that a simple three-parameter yield function built thanks to von Mises equivalent Kelvin stresses accounts for a satisfying plasticity criterion for such alloys.Non-quadratic norm ∥·∥_a plasticity framework is addressed. Intrinsic generalizations of Hershey-Hosford criterion are proposed for cubic material symmetry.     

考虑拉压强度差效应的厚壁圆筒承载能力分析

应用双剪统一强度理论，考虑材料的拉压异性和同性，推导了在内压力和轴力联合作用下的厚壁圆筒的塑性极限载荷表达式．在该表达式中，当反映中间主应力效应的系数取不同的值时，就能得到按Tresca屈服准则、线性逼近的Mises屈服准则和双剪应力屈服准则的计算结果，并且绘制了在相应准则下的极限应力线图．从而可知：在三维应力状态下，应用该理论，可以获得极限载荷分析的精确解；极限载荷线图与三种屈服准则的屈服曲线是相吻合的；计算的结果可以用于拉压异性和同性的材料，为工程应用提供了理论依据．     

New analytical criterion for porous solids with Tresca matrix under axisymmetric loadings

In this paper, a new analytic criterion for porous solids with matrix obeying Tresca yield criterion is derived. The criterion is micromechanically motivated and relies on rigorous upscaling theorems. Analysis is conducted for both tensile and compressive axisymmetric loading scenarios and spherical void geometry. Finite element cell calculations are also performed for various triaxialities. Both the new model and the numerical calculations reveal a very specific coupling between the mean stress and the third invariant of the stress deviator that results in the yield surface being centro-symmetric and void growth being dependent on the third-invariant of the stress deviator. Furthermore, it is verified that the classical Gurson’s criterion is an upper bound of the new criterion with Tresca matrix.     

Linear Elastic Solutions for Slotted Plates

Two-dimensional problems of finite-length blunted cracks cut into infinite plates subject to remote tractions are solved using complex variable theory. The slot geometry is composed of two flat surfaces connected by rounded ends. This special geometrical shape was derived by Riabouchinsky in the study of two-dimensional ideal fluid flow around parallel plates. The simpler antiplane slotted plate problem is addressed initially for this geometry. From this exact solution, the equivalent of a Westergaard stress potential is found and applied to the two other principal modes of fracture, which are plane elasticity problems. For a plate subject to uniform radial tension at infinity, an analytical solution is obtained that will reduce to the familiar mode I singular crack solution as the separation between the parallel faces of the slot becomes zero. For finite-width mode I slots, the rounded ends have tensile tractions which terminate at the adjoining flat surfaces of the slot, which remain traction-free. In this respect, the finite-width mode I slot problem resembles a Barenblatt cohesive zone model of a plane crack or a Dugdale plastic strip model of a plane crack, although the tractions will vary in magnitude along the slot ends rather than remaining uniform as in the former type of crack problems. Similarly, in the case of the finite-width mode II slot problem, the rounded ends of the slot have shear tractions, while the flat surfaces remain load-free. A distinguishing feature of the mode II slot solution over the mode I slot problem is that the maximum in-plane shear stress is constant along the rounded ends of the slot. Because of this, those particular regions of the boundary can represent incipient plastic yield based on either the Mises or Tresca yield condition under plane strain loading conditions. In this way, the problem resembles the plastic strip models of Dugdale, Cherepanov, Bilby-Cottrell-Swinden, and others. Notably, the mode III slot problem also has a constant maximum shear stress along the curved portions of the slot, while the entire slot boundary remains traction-free, unlike the mode II slot problem. Consequently, the mode III slot problem represents both a generalization of the standard mode III crack problem geometry, while simultaneously satisfying the boundary conditions of a plastic strip model.     

Evaluation of multiaxial theories for room-temperature plasticity and elevated-temperature creep and relaxation of several metals

Based on the assumption that the material satisfies the condition of isotropic hardening for either a von Mises or a Tresca material, finite-strain theories are derived for solid circular torsion members for the conditions that the inelastic deformations are either time independent or time dependent. In the latter case, both creep and relaxation theories are derived. At room temperature the theories are evaluated for each of eight metals using finite-strain data from tension, compression and torsion members. Of the six metals that are found to satisfy the condition required for the isotropic-hardening model, two are von Mises, one is Tresca, and the other three are between von Mises and Tresca. At elevated temperatures, the theories are evaluated for each of five of the latter six metals, using data from tension and torsion members. Material properties obtained from the tension specimens are used to predict creep and relaxation curves for the torsion members. Contrary to the results at room temperature, creep curves for the torsion members do not all fall within the region bounded by von Mises and Tresca theories. In the case of relaxation, either excellent agreement is obtained between the von Mises strain-hardening theory and experimental data or the theory is conservative.     

An experimental study on subsequent yield surface after finite shear prestraining

An equimodulus surface is introduced and the subsequent yield surface after large finite shear prestraining is experimentally investogated. Fully annealed, thin-walled copper tubular specimens were subjected to large torsional loading and partial unloading; strain gages were carefully mounted on the specimen after the application of pure shear loading. Specimens were then subjected to various combined tension-torsion loadings. Influences of he von Mises and Tresca equivalent offset strains on the subsequent yield surfaces are studied. On examining the experimental results reported in this article, it was found that the smaller the offset strains, the more distorted are the subsequent yield surfaces. At the torsional preloading point, a rounded corner was developed, whereas in the region opposite to the preloading point, the subsequent yield surface was flattened. When large von Mises offset strains were used, the corresponding subsequent yield surfaces passed through the von Mises loading surface. But this was not the case when Tresca offset strains were used. The subsequent yield surface determined by the back extrapolation method was almost completely outside the von Mises loading surface. On the other hand, the subsequent yield surface determined by the back extrapolation method was close to the Tresca loading surface. It is also found that the equimodulus surface is distorted and cannot simply be described by the combined kinematic and isotropic hardening rule.     

冲击荷载作用下简支圆板的塑性动力响应统一解

采用统一强度理论求解了简支圆板在中等脉冲荷载作用下的动力响应问题,得出了统一的动力塑性极限荷载、内力场和速度场,并给出了上限解和下限解。讨论了静力许可条件和运动许可条件。利用本文的解还得出了简支圆板在静力荷载作用下的极限荷载、内力场和速度场。根据选择不同的拉压比参数,本文所给出的解可以适用于各种拉压异性和拉压同性材料。Tresca解、Mohr Coulomb解和双剪统一屈服准则解是本文的特例,Mises解是本文当=1和b=0.5时的线性逼近。研究结果表明,拉压比和强度理论参数b对动力解的影响要大于对静力解的影响,所以,根据材料的不同选择合适的强度理论,对于更好的发挥材料的强度潜力,减轻结构的重量具有重要的意义。     

Stress across an elastoplastic boundary of a mode I crack: parabolic to hyperbolic plasticity transition

In previous work, the stresses of a mode I elastic–plastic fracture mechanics problem were analytically continued across a prescribed elastoplastic boundary for plane stress loading conditions involving a linear elastic/perfectly plastic material obeying the Tresca yield condition. Immediately across the elastic-plastic boundary, a nonlinear parabolic partial differential equation governs the plastic stress field. The present solution deals with stresses extending beyond the parabolic region into the hyperbolic region of the plastic zone. This analytical solution is obtained through a tranformation of the original system of nonlinear partial differential equations into a linear system with constant coefficients. The solution, so obtained, is expressible in terms of elementary transcendental functions. It also exhibits a limiting line which passes through the crack tip. This feature of the solution suggests the formation of a plastic hinge in the material.     

Plastic flow for axisymmetric-tension specimens with necking

Plastic deformation of a tension specimen bounded by a curved surface of revolution is considered. Such a configuration may occur, for example, as a result of necking. The specimen material satisfies the Tresca yield condition and the associated flow rule. Approximate solutions for the stress distribution in the neck were examined in [6]. The extension of notched bars was investigated by numerical and graphic methods in [2, 4]. Below, the problem is solved analytically for a sufficiently smooth neck; a small degree of nonuniformity of the plastic properties is taken into account.     

Stress-intensity factors for surface cracks in functionally graded materials under mode-I thermomechanical loading

This paper describes the development and application of a general domain integral method to obtain J-values along crack fronts in three-dimensional configurations of isotropic, functionally graded materials (FGMs). The present work considers mode-I, linear-elastic response of cracked specimens subjected to thermomechanical loading, although the domain integral formulation accommodates elastic–plastic behavior in FGMs. Finite element solutions and domain integral J-values for a two-dimensional edge crack show good agreement with available analytical solutions for both tension loading and temperature gradients. A displacement correlation technique provides pointwise stress-intensity values along semi-elliptical surface cracks in FGMs for comparison with values derived from the proposed domain integral. Numerical implementation and mesh refinement issues to maintain path independent J-values are explored. The paper concludes with a parametric study that provides a set of stress-intensity factors for semi-elliptical surface cracks covering a practical range of crack sizes, aspect ratios and material property gradations under tension, bending and spatially-varying temperature loads.