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.     

An upper bound approach on yield surfaces of porous materials

Summary Triaxial deformation of a porous material is analyzed. A material model is proposed where the material consists of spherical cells each of which has a spherical pore in the center of the matrix. The velocity field in the matrix is assumed and the upper bound approach is attempted. The yield surfaces of the porous materials with various volume fractions of pores v _f are obtained as ellipsoids whose axes become smaller with increasing v _f.They coincide with the yield surfaces which have been proposed by the authors earlier.

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Ein Obere-Schranken-Ansatz für den Fließort von porösem Material

Übersicht Analysiert wird die dreiachsige Deformation eines porösen Werkstoffs, der durch kugelförmige Zellen mit sphärischen Löchern im Zentrum der Matrix modelliert wird. Für die Anwendung des Obere-Schranken-Satzes wird ein Geschwindigkeitsfeld in der Matrix angesetzt. Damit erhält man als Fließort von porösen Werkstoffen mit verschiedenen Porenvolumen-Anteilen v _fEllipsoide, deren Hauptachsen mit zunehmendem v _fkleiner werden. Sie stimmen mit den Fließorten, welche die Autoren schon früher halbexperimentell und halb-analytisch ermittelt haben, überein.

     

An analytical homogenization method for heterogeneous porous materials

The objective of this work is to develop an analytical homogenization method to estimate the effective mechanical properties of fluid-filled porous media with periodic microstructure. The method is based on the equivalent inclusion concept of homogenization applied earlier for solid–solid mixture. It is assumed that porous media are described by the poroelastic constitutive law developed by Biot where porosity is a material parameter. By solving the governing equations of poroelasticity in Fourier transformed domain, the relation between periodic strain and eigenstrain in porous media is established. This relation is subsequently used in an average consistency condition involving both solid and fluid phase stresses and strains. The geometry of the porous microstructure is captured in the g-integral. This homogenization method can also be applied to estimate the equivalent properties of solid–fluid mixture where a pure solid and fluid can be modeled by assuming very low and high porosity, respectively. Several examples are considered to establish this new method by comparing with other existing analytical and numerical methods of homogenization. As an application, poroelastic properties of cortical bone fibril are estimated and compared with previously computed values.     

Auxetic mechanics of crystalline materials

In the present paper, we analyze uniaxial deformation of crystals of different systems with negative Poisson’s ratios, known as auxetics. The behavior of auxetic crystals is studied on the basis of extensive knowledge on the experimental values of elastic constants of different crystals, gathered in the well-known Landolt-Börnstein tables. The competition between the anisotropy of crystal structures and the orientation of deformable samples results in the dependence of the elastic characteristics of deformation, such as Young’s modulus and Poisson’s ratio, on the orientation angles. In the special case of a single angle, a large number of auxetics were found among crystals of cubic, hexagonal, rhombohedral, tetragonal, and orthorhombic systems and the character of variations in their response due to changes in orientation was determined.     

A multiscale model for the ductile fracture of crystalline materials

In this paper, a multiscale model that combines both macroscopic and microscopic analyses is presented for describing the ductile fracture process of crystalline materials. In the macroscopic fracture analysis, the recently developed strain gradient plasticity theory is used to describe the fracture toughness, the shielding effects of plastic deformation on the crack growth, and the crack tip field through the use of an elastic core model. The crack tip field resulting from the macroscopic analysis using the strain gradient plasticity theory displayes the 1/2 singularity of stress within the strain gradient dominated region. In the microscopic fracture analysis, the discrete dislocation theory is used to describe the shielding effects of discrete dislocations on the crack growth. The result of the macroscopic analysis near the crack tip, i.e. a new K-field, is taken as the boundary condition for the microscopic fracture analysis. The equilibrium locations of the discrete dislocations around the crack and the shielding effects of the discrete dislocations on the crack growth at the microscale are calculated. The macroscopic fracture analysis and the microscopic fracture analysis are connected based on the elastic core model. Through a comparison of the shielding effects from plastic deformation and the discrete dislocations, the elastic core size is determined.     

A study on yield and flow of orthotropic materials

On the assumption that the yield criterion of orthotropic materials is isomorphic with Huber-Mises criterion of isotropic materials, we put forward a dimensionless stress yield criterion, and obtained the associated plastic flow law. Using experimental stress-strain curves in various simple stress states, generalized effective stress-strain formulae may be derived correspondingly in various forms.     

Suitability of yield functions for the approximation of subsequent yield surfaces

Many consitutive models in the plasticity of metals are based on the existence of a yield function, which is not only used to mark the elastic limit but also as a potential function for the plastic strain rate. The construction of this function therefore deserves the utmost interest. Measurements of the elastic limit show the essential features of how the geometry of the yield function contour lines should change with further straining. Against these typical geometric forms of the yield surface existing proposals for invariant formulations of the yield function taking into account isotropic, kinematic and formative hardening are tested. Even if no evolution equations for the constitutive variables contained in the yield functions are specified, best approximations of measured yield surfaces can be computed by optimisation of a quality function. It can be shown that most of the representations are not even able to describe the experimental results qualitatively. The numeric results show further that the yield function is essentially of grade three in the deviatoric stresses. The evolution of internal variables can be deduced from the approximations of the measurements.     

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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A simple model for diffusion-induced dislocations during the lithiation of crystalline materials

Assuming that the lithiation reaction occurs randomly in individual small particles in the vicinity of the reaction front, a simple model of diffusion-induced dislocations was developed. The diffusion-induced dislocations are controlled by the misfit strain created by the diffusion of solute atoms or the phase transformation in the vicinity of the reaction front. The dislocation density is proportional to the total surface area of the “lithiated particle” and inversely proportional to the particle volume. The diffusion-induced dislocations relieve the diffusion-induced stresses.     

The study of relations between loading history and yield surfaces in powder materials using discrete finite element simulations

Only a few studies in the literature have applied the finite-element method to analyse assemblies of meshed particles. These studies illustrated the relevance of this method for granular materials. Here, the compaction of ductile metal powders was studied through a numerical assembly of elastic–plastic and rate-independent spherical particles. The aim of this paper was to understand the evolution of yield surfaces with complex loading paths up to high relative density at the macroscopic scale and at the granular scale. Simulation results revealed that yield surfaces evolved with both isotropic and kinematic hardening mechanisms, depending on the compaction stage. An analysis of the sample microstructure was proposed, and a detailed study of contacts between particles revealed some of the mechanisms that led to the observed evolution of yield surfaces.     

混凝土靶抗贯穿的一种新工程分析方法

在土盘模型的基础上,通过计入冲击波衰减的影响及改进浮动锁应变的计算,发展了一种对贯穿问题的新的工程分析方法。最后通过一些对混凝土靶贯穿的算例说明了方法的有效性和实用性。     

双周期圆截面纤维复合材料平面问题的解析法

结合双准周期Riemann边值问题理论与Eshelby等效夹杂原理，为双周期圆截面纤维复合材 料平面问题发展了一个实用有效的解析方法，获得了问题的全场级数解并与有限元结果进行 了比较. 该方法为非均匀材料的力学性质分析和复合材料等新材料的微结构设计提供了 一个有效的计算工具，也可用来评估有限元等数值与近似方法的精度.     

An experimental and analytical investigation of the rail shear-test method as applied to composite materials

This report presents the results from an experimental and analytical investigation of the stress distributions occurring in a rail shear test. The effects of nonuniform stresses induced by differential thermal expansion, rail flexibility and specimen aspect ratio on measured shear modulus and ultimate strength of composite laminates are shown. A two-dimensional linearly elastic finite-element model was used to analytically determine how various geometric parameters influenced the magnitude and distribution of inplane normal and shear stresses in a tensile-rail-shear specimen. Rail shear tests were conducted at room temperature and 589 K (600°F) on selected graphite-polyimide composite laminates using two titanium rail configurations. The analysis and test methods are discussed, and the results of the effects of the various parameters on shear modulus and ultimate strength are presented.     

Void growth and interaction in crystalline materials

An inelastic rate-dependent crystalline constitutive formulation and specialized computational schemes have been developed and used to obtain a detailed understanding of the interrelated physical mechanisms that can result in ductile material failure in rate-dependent porous crystalline materials subjected to finite inelastic deformations. The effects of void growth and interaction and specimen necking on material failure have been investigated for a single material cell, with a discrete cluster of four voids, where geometrical parameters have been varied to result in seven unique periodic and random void arrangements. The interrelated effects of void distribution and geometry, strain hardening, geometrical softening, localized plastic strains and slip-rates, and hydrostatic stresses on failure paths and ligament damage in face centered cubic (f.c.c.) crystalline materials have been studied. Results from this study are consistent with experimental observations that ductile failure can occur either due to void growth parallel to the stress axis, which results in void coalescence normal to the stress axis, or void interaction along bands, which are characterized by intense shear-strain localization and that intersect the free surface at regions of extensive specimen necking.     

A yield function for anisotropic materials Application to aluminum alloys

A phenomenological yield function is proposed to represent the plastic anisotropy of aluminum sheets. It is an extension of the functions given by Barlat et al. [Int. J. Plasticity 7 (1991) 693] and Karafillis and Boyce [J. Mech. Phys. Solids 41 (1993) 1859]. The anisotropy is represented by 12 parameters in the form of two fourth order symmetric tensors. Four other parameters influence the shape of the yield surface uniformly. The role of each parameter is described in detail. The convexity of the yield surface is proved. The implementation of the proposed yield function is done in the 3D general case in an object-oriented finite element code. It is used to represent the anisotropy of a 2024 aluminum thin sheet and the adjustment is excellent. Other anisotropic materials from the literature are also well described by the proposed yield function.