A porosity-dependent inelastic criterion for engineering materials

Many criteria have been developed to describe the yielding condition, plastic potential, and failure strength of engineering materials. In this paper, the authors first review the characteristics of a few of the more common criteria used for porous materials. It is then shown that the main features of many criteria can be represented by a unique set of recently developed equations. The ensuing multiaxial criterion becomes applicable to a variety of materials and loading states. One of the advantages of the proposed criterion, named MSDP_u, is that it is explicitly porosity-dependent. The validity of this general inelastic criterion is demonstrated using experimental results obtained from various types of materials. A brief discussion follows on the advantages and limitations of the proposed equations.     

A rationally based yield criterion for work hardening materials

Summary The paper is concerned with the development of a rationally based yield criterion for work hardening materials. In an attempt to construct a single analytical function to represent yield behaviour the mathematics of continuum mechanics is combined with a statistical argument to produce a yield criterion which takes into account the inherent inhomogeneity of stress and strain distribution throughout a polycrystalline aggregate.A mathematical model is proposed which consists of spherical, anisotropic inclusions embedded in an elastic-plastic matrix. The yield criterion derived from this model is based on three main considerations, (a) arguments involving the local value of shear strain energy density, (b) Eshelby's solution for the local stress acting on a spherical inclusion in an infinite matrix and (c) the statistics of extreme values.An example is given on the fitting of the yield criterion to the results of thin-walled aluminium tubes prestrained in tensiontorsion space.

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Sommario Si considera lo sviluppo di una razionale condizione di plasticità per materiali con incrudimento. Nel tentativo di costruire una sola funzione analitica per rappresentare il comportamento plastico del materiale, la matematica della meccanica dei continui è combinata con considerazioni statiche onde definire una condizione di plasticità che tenga conto dell'implicita inomogeneità nella distribuzione di sforzi e deformazioni in un aggregato policristallino.Si propone un modello matematico che prende in considerazione inclusioni sferiche anisotrope avvolta da materiale elastoplastico. La condizione di plasticità derivata da questo modello è basata su tre considerazioni principali, (a) argomenti che involvono il valore locale della densità di energia di deformazione al taglio, (b) la soluzione di Eshelby per lo sforzo locale che agisce su ogni inclusione sferica in una matrice infinita e (c) la statistica dei valori estremi.Si dà un esempio sulla capacità del criterio di plasticità di interpretare i risultati ottenuti su tubi di alluminio sottile precaricato nello spazio tensione-torsione.

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Constitutive potential for void-containing nonlinear materials and void growth

Based on an analysis of the deformation of an isolated void in a finite nonlinear viscous material, we establish the constitutive potentials for voided nonlinearly viscous materials, from which the related curves of the macroscopic stress, the average flow stress of the matrix material and the void volume fraction f are derived. However, the theory applies equally well to small strain, rate-independent J₂ deformation theory solid. By considering the effects of the strain-hardening directly, a modifies Gurson equation are developed. Finally, we calculate the void relative growth-rates for the nonlinear materials, and in good agreement with existed numerical results.     

The size effect on void growth in ductile materials

We have extended the Rice-Tracey model (J. Mech. Phys. Solids 17 (1969) 201) of void growth to account for the void size effect based on the Taylor dislocation model, and have found that small voids tend to grow slower than large voids. For a perfectly plastic solid, the void size effect comes into play through the ratio εl/R₀, where l is the intrinsic material length on the order of microns, ε the remote effective strain, and R₀ the void size. For micron-sized voids and small remote effective strain such that εl/R₀?0.02, the void size influences the void growth rate only at high stress triaxialities. However, for sub-micron-sized voids and relatively large effective strain such that εl/R₀>0.2, the void size has a significant effect on the void growth rate at all levels of stress triaxiality. We have also obtained the asymptotic solutions of void growth rate at high stress triaxialities accounting for the void size effect. For εl/R₀>0.2, the void growth rate scales with the square of mean stress, rather than the exponential function in the Rice-Tracey model (1969). The void size effect in a power-law hardening solid has also been studied.     

A generating function for the yield criterion of isotropic and anisotropic polycrystalline materials

Summary A yield criterion for elastic pure-plastic polycrystalline materials is generated under simplified conditions by assuming that for yielding a certain fraction Q _c of the total number of slip planes in the material has to be active. This fraction Q _c is called the critical active quantity. We suppose Q _c to be independent of the state of stress. The yield criterion is mathematically expressed as an integral, which is a function of Q _c. This criterion can also be used for anisotropic materials.For isotropic materials the ratio (r) of the yield stress in torsion to that in tension is calculated as a function of Q _c. We find 0.5r0.61.The value r=0.5 (Tresca's criterion) is obtained for Q _c=0 and Q _c=1. The value r=0.577 (von Mises criterion) is obtained for Q _c=0.34 and Q _c=0.79. The difference between two criteria with the same r is the magnitude of the yield stress. We think the value Q _c=0.79 corresponds to the experiments for f.c.c. materials, since a rough estimation gives Q _c>0.75 for yielding.The independence of Q _c on the state of stress brings on that r>0.5 is more probable. This is caused by the slower increase to Q _c in torsion compared with the case of tension.From the theory follows that in the general case (Q _c0) the middle principal stress has influence on yielding.In this paper we don't determine Q _c, but adapt its value to the experimental results. However, a rough estimation of Q _c is given for isotropic materials.     

The criterion of critical void growth ratio and its applications

In the present paper, the new material parameter of ductile fracturev _gc with an obvious microscopic physical meaning is emphasized, which can be determined by macroscopic method. The theoretical and experimental background for the criterion of critical void growth ratio is expounded from both macroscopic and microscopic points of view. The relationship between the macroform parameterv _gc and microform oneR _c /R _go of critical void growth ratio is investigated. The engineering experimental procedure for determining the value ofv _gc of materials has been recommended. The values ofv _gc for some materials with different heat treatments have been estimated. It is found that the parameterv _gc can be effectively used for selecting materials and their heat treatments. A modified ASPEF criterion accounting for stress states has been proposed. Related expression of modified ASPEF andv _gc has been derived.     

A yield criterion for perforated sheets

Summary A criterion of yielding for homogenized materials with an internal structure generated by perforation is developed using the stress and the mixed stress-perforation tensor invariants. For a square pattern of perforation an experimental homogenization procedure is used to determine components of the perforation tensor introduced. A yield condition for the studied materials is proposed and compared with experiments conceived and performed for the purpose.

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Übersicht Es wird ein Fließkriterium entwickelt für Materialien mit durch Perforation erzeugter innerer Struktur. Die Herleitung gründet sich auf gemischte Invarianten von Spannungs- und Perforationstensor. Für ein rechteckiges Perforationsmuster wird eine experimentelle Homogenisierung zur Ermittlung der Komponenten des Perforationstensors durchgeführt. Der vorgeschlagene Fließort wird mit experimentellen Ergebnissen verglichen.

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Dedicated to Prof. Th. Lehmann on the occasion of his 60th birthday     

RVE-based studies on the coupled effects of void size and void shape on yield behavior and void growth at micron scales

The present paper extends the Gurson and GLD models [Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth, Part I—yield criteria and flow rules for porous ductile media. J. Mech. Phys. Solids 99, 2–15; Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing non-spherical voids—case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754; Gologanu, M., Leblond, J.B., Devaux, J., 1994. Approximate models for ductile metals containing non-spherical voids—case of axisymmetric oblate ellipsoidal cavities. J. Eng. Mater. Technol. 116, 290–297] to involve the coupled effects of void size and void shape on the macroscopic yield behavior of non-linear porous materials and on the void growth. A spheroidal representative volume element (RVE) under a remote axisymmetric homogenous strain boundary condition is carefully analyzed. A wide range of void aspect ratios covering the oblate spheroidal, spherical and prolate spheroidal void are taken into account to reflect the shape effect. The size effect is captured by the Fleck–Hutchinson phenomenological strain gradient plasticity theory [Fleck, N.A., Hutchinson, J.W., 1997. Strain gradient plasticity. In: Hutchinson, J.W., Wu, T.Y. (Eds.), Advance in Applied Mechanics, vol. 33, Academic Press, New York, pp. 295–361]. A new size-dependent damage model like the Gurson and GLD models is developed based on the traditional minimum plasticity potential principle. Consequently, the coupled effects of void size and void shape on yield behavior of porous materials and void growth are discussed in detail. The results indicate that the void shape effect on the yield behavior of porous materials and on the void growth can be modified dramatically by the void size effect and vice versa. The applied stress triaxiality plays an important role in these coupled effects. Moreover, there exists a cut-off void radius r_c, which depends only on the intrinsic length l₁ associated with the stretch strain gradient. Voids of effective radius smaller than the critical radius r_c are less susceptible to grow. These findings are helpful to our further understanding to some impenetrable micrographs of the ductile fracture surfaces.     

Limit analysis of composite materials based on an ellipsoid yield criterion

Employing repeating unit cell (RUC) to represent the microstructure of periodic composite materials, this paper develops a numerical technique to calculate the plastic limit loads and failure modes of composites by means of homogenization technique and limit analysis in conjunction with the displacement-based finite element method. With the aid of homogenization theory, the classical kinematic limit theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. Based on nonlinear mathematical programming techniques, the finite element modelling of kinematic limit analysis is then developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the limit load of a composite is then obtained. The nonlinear formulation has a very small number of constraints and requires much less computational effort than a linear formulation. An effective, direct iterative algorithm is proposed to solve the resulting nonlinear programming problem. The effectiveness and efficiency of the proposed method have been validated by several numerical examples. The proposed method can provide theoretical foundation and serve as a powerful numerical tool for the engineering design of composite materials.     

Scale-dependent plasticity potential of porous materials and void growth

In the present paper, a boundary value problem about the macroscopic response and its microscopic mechanism of a representative spherical cell with a spherical microvoid under axisymmetric triaxial tension has been theoretically investigated. To capture the size effects of local plastic deformation in the matrix, the strain gradient constitutive theory including the rotation and the stretch gradients developed by Fleck and Hutchinson [Strain gradient plasticity, in: J.W. Hutchinson, T.Y. Wu (Eds.), Advance in Applied Mechanics, vol. 33, Academic Press, New York, 1997, p. 295] is adopted. By means of the principle of minimum plasticity potential and the Lagrange multipliers method, the self-contained displacement field within the matrix has been computationally determined. Based on these, a size-dependent constitutive potential theory for porous material has been developed. The results indicate clearly that the microvoid evolution predicted by the present constitutive model displays very significant dependences on the void size especially when the radius a of microvoids is comparable with the intrinsic characteristic length l of the matrix. And when the void radius a is much lager than l, the present model can retrogress automatically to the Gurson model improved by Wang and Qin [Acta Mech. Solid. Sin. 10 (1989) 127, in Chinese].     

A stress criterion for crack growth

A stress criterion for crack growth was developed from test results with 7075-T6 aluminum-sheet specimens containing transverse machined cracks. Stress distributions near the crack tip were obtained using strain gages and by reducing the strain data to stresses with the aid of Reuss plasticity theory. These distributions indicated the biaxial nature of stress at the crack tip, the high stress gradients a short distance from the tip, and the variation in stress-concentration factor with crack length. Crack growth was found to occur when the effective stress at the crack root reached the engineering ultimate strength.     

An orthotropic yield criterion in a 3-D general stress state

An anisotropic yield criterion with a general representation was suggested. The yield criterion was derived from the use of the invariants of the stress tensor, similar in constructing an isotropic yield criterion, but which contains a “three-yield-system hypothesis” specifying the state of anisotropy. When applied to rolled sheet metals, such as high strength steels and aluminum alloys, the criterion can be treated in an analytical form to facilitate analyses of engineering problems under a general triaxial stress state. For this specified form, anisotropic properties of the predicted yield surface were characterized by seven experimental results obtained from three standard uniaxial-tension tests and one equibiaxial-tension test. When the applied material becomes isotropic it is transformed back to the form of the von-Mises’s criterion. Since the convexity of the yield criterion was proven in its general type, the characterized criterion is valid as a plastic potential in the implementation of finite element programs. It was shown, in full agreement with experimental data, that the accuracy of predicted yield surface was similar to that of predicted by the polycrystal model. Considering the equibiaxial-tension data, in general, may be not available from material supplies, a formulated relation covered variables of the equibiaxial tension and uniaxial tension was proposed. The relation can be used to calculate the equibiaxial-tension yield stress from the experimental data in uniaxial tensions. Several calculated results showed very close to the experimental results.     

A criterion for the onset of void coalescence under combined tension and shear

Depending on the relative positions of voids and on the loading conditions, shear loading components can play an important role in the void coalescence process leading to ductile fracture. Yet, most void coalescence criteria including the original criterion of Thomason, and its various extensions/improvements, take only normal loads into account and neglect the contribution from shear loads to coalescence. Shear can affect both the stress/strain at the onset of coalescence and the direction of deformation localization. In this paper, first, the predictive capabilities of different coalescence criteria without shear effect are critically assessed and the expressions involved in the original Thomason criterion are fine-tuned by comparing with 3D finite element calculations performed on a unit cell containing a spheroidal void. Then, the improved Thomason criterion is theoretically extended—by using limit load analysis—to incorporate the effect of shear. The predictions of this new coalescence criterion are in good agreement with the results produced by 3D finite element calculations, for both loadings involving or not a shear component.     

A note on the effect of surface energy and void size to void growth

Recent studies revealed that rapid void growth is the dominant failure mechanism in an elasto-plastic solid under high mean tensile stress. This paper studies the effect of the surface energy and void size to the void growth. The models of a thick spherical shell and a thick spherical column in void growth are analyzed and numerically estimated. The main conclusion from this study is that, for typical metals, the surface energy effect is negligible for voids larger than 100 nm in size, but it may become significant when the void size is on the order of 10 nm.     

A new yield criterion for soils with embedded tension cut-off

Meccanica - The object of this paper is the formulation of a new isotropic yield criterion for pressure dependent materials. The main features of the proposed formulation are: (1) variation of the...     

A method of plasticity for general aligned spheroidal void or fiber-reinforced composites

Based on the general concept of the secant moduli method, together with a new way of evaluating the average matrix effective stress originally proposed by Qiu and Weng (“A Theory of Plasticity for Porous Materials and Particle-Reinforced Composites”, ASME J. Appl. Mech. (1992), 59, 261.), a method for nonlinear effective properties of general aligned fiber or void composites is proposed. The method is capable of predicting composite (especially for porous materials) yielding under a hydrostatic load. Compared to the Tandon and Weng (“A Theory of Particle-Reinforced Plasticity,” ASME J. Appl. Mech. (1988), 55, 126.), model the proposed method always gives softer prediction in the uniaxial tension. The proposed method will predict the same nonlinear stress and strain relation as the Ponte Castaneda (“The Effective Mechanical Properties of Nonlinear Isotropic Composite,” J. Mech. Phys. Solids (1991), 39, 45.) variational model if the same estimates or bounds for the linear comparison composite are adopted.     

Prediction of localized thinning in sheet metal using a general anisotropic yield criterion

The forming limit diagram (FLD) is a useful concept for characterizing the formability of sheet metal. The ability to accurately predict the FLD for a given material has been shown to depend on the shape of the selected yield function. In addition, both experimental and numerical results have shown that the level of the FLD is strongly strain path dependent. In this work, a combination of Marciniak–Kuczynski (M–K) analysis and a general anisotropic yield criterion developed by Karafillis and Boyce (Karafillis, A.P., Boyce, M.C., 1993. A general anisotropic yield criterion using bounds and transformation weighting tensor. J. Mech. Phys. Solids 41, 1859) is used to predict localized thinning of sheet metal alloys for linear and nonlinear strain paths. A new method for determining the constants in the yield criterion is proposed. The optimal values are obtained by fitting the initial yield stresses and calculated FLD under linear strain paths with the experimental measurement. Using this approach, accurate yield functions can be defined for both Al2008-T4 and Al6111-T4. Comparisons of computed FLDs with the experimental data of Graf and Hosford (Graf, A., Hosford, W.F., 1993b. Effect of changing strain paths on forming limit diagrams of Al 2008-T4. Metall. Trans. A. 24, 2503; Graf, A., Hosford, W.F., 1994. The influence of strain path changes on forming limit diagrams of Al 6111-T4. Int. J. Mech. Sci. 36, 897) show good agreements.     

A numerical exploration of the role of void geometry on void collapse and hot spot formation in ductile materials

An Eulerian hydrocode was used to simulate the dynamic void collapse in OFHC copper, modeled with the Johnson–Cook (Johnson, G.R., Cook, W.H., 1985. Engineering Fracture Mechanics 21(1), 31) material model and the Cruneisen equation of state, to study hot spot formation and jetting. The computational techniques were first validated by a comparison to a series of two-dimensional experiments. The effect of the planar and axisymmetric geometries on the hot spot temperature and jet velocity in circular voids is explored. In addition, the effect of the apex angle in triangular voids on jetting and hot spot formation is studied. An effective apex angle for the highly nonspherical voids formed in closely packed circular particles is calculated.     

The classical pressure vessel problem for void damage materials

l.IntroductionPressurevesselbearsver}'greatpressureontheinterna1surface.Theremaybedisastrousconsequencesasaresultofthevoiddamage,andtheresearchinthepressurevesseldamageisveryactiveallthetime.IUTAMconvenedaspecia1izedconferenceonnonlineafelasticityanalysis…     

Influences of microscope size effect of matrix on plastic potential and void growth of porous materials

Based on approximate theoretical analyses on a typical spherical cell containing a spherical microvoid, the influences of matrix materials' microscopic scale on the macroscopic constitutives potential theory of porous material and microvoid growth have been investigated in detail. By assuming that the plastic deformation behavior of matrix materials follows the strain gradient (SG) plastic theory involving the stretch and rotation gradients, the ratio (λ=l/a) of the matrix materials' intrinsic characteristic lengthl to the microvoid radiusa is introduced into the plastic constitutives potential and the void growth law. The present results indicate that, when the radiusa of microvoids is comparable with the intrinsic characteristic lengthl of the matrix materials, the influence of microscopic size effect on neither the constitutive potential nor the microvoid evolution predicted can be ignored. And when the void radiusa is much lager than the intrinsic characteristic lengthl of the matrix materials, the present model can retrogress automatically to the improved Gurson model that takes into account the strain hardening effect of matrix materials. Project supported by the National Natural Science Foundation of China (No. A10102006).