Aggregates Of Cubic Crystallites Research Articles

Strict convexity of yield surfaces of some weakly-textured materials

Let Sym0 be the space of traceless symmetric second-order tensors. We say that a polycrystalline elastic–plastic material is weakly-textured if its yield function f:Sym0→R is the sum of a texture-independent isotropic part fiso and an anisotropic part which is linear in the relevant texture coefficients. Let c>0 and S≔f−1(c)⊂Sym0 be the yield surface of the weakly-textured material in question. We present a sufficient condition (*), namely that ∇2f(S) be positive definite for each S∈S, for a smooth yield surface S to be strictly convex in Sym0. We apply this sufficient condition to weakly-textured materials with yield functions that satisfy the following conditions: (i) the yield functions f and fiso are smooth; (ii) ∇2fiso(S) is positive definite for each S in Siso≔fiso−1(c)⊂Sym0. We prove that the yield surface S⊂Sym0 of such weakly-textured material is strictly convex if the texture coefficients in f are sufficiently small. As illustration for practical applications, by appealing to condition (*) we study the strict convexity of the yield surface pertaining to a weakly-textured orthorhombic aggregate of cubic crystallites which has a quadratic yield function of the type proposed by Hill in 1948. Moreover, we show that all 35 samples of cold-rolled and annealed low-carbon steel sheets studied by Stickels and Mould have their quadratic yield functions and corresponding yield surfaces strictly convex.

Nonlinearly Elastic Constitutive Relation of Anisotropic Aggregate of Cube Crystallites

Under Voigt model, Barsch and Johnson gave the formula of nonliearly elastic constitutive relations for isotropic aggregates of cubic crystallites and orthorhambic aggregates of cubic crystallies, respectively. In this paper, a nonlinear elastic constitutive relation based on Voigt model, which is more general than Barsch's and Johnson's results, is derived for the set of anisotropic cubic grains. The anisotropy of metals is described by the texture coefficient.

Monitoring near-surface depth profile of residual stress in weakly anisotropic media by Rayleigh-wave dispersion

Herein we study the inverse problem on inferring depth profile of near-surface residual stress in a weakly anisotropic medium by boundary measurement of Rayleigh-wave dispersion if all other relevant material parameters of the elastic medium are known. Our solution of this inverse problem is based on a recently developed algorithm by which each term of a high-frequency asymptotic formula for dispersion relations can be computed for Rayleigh waves that propagate in various directions along the free surface of a vertically-inhomogeneous, prestressed, and weakly anisotropic half-space. As a prime example of possible applications we focus on a thick-plate sample of AA 7075-T651 aluminum alloy, which has one face treated by low plasticity burnishing (LPB) that induced a depth-dependent prestress at and immediately beneath the treated surface. We model the sample as a prestressed, weakly-textured orthorhombic aggregate of cubic crystallites and assume that by nondestructive and/or destructive measurements we have ascertained everything about the sample, including the LPB-induced prestress, before it is put into service. Under the supposition that the prestress be partially relaxed but other material parameters remain unchanged after the sample undergoes a period of service, we examine the possibility of inferring the depth profile of the partially relaxed stress by boundary measurement of Rayleigh-wave dispersion.

The Hosford yield function for weakly textured sheets of cubic crystal orthotropic metals in any stress state

Most rolled sheet metals are orthotropic aggregates of cubic crystallites. The texture coefficients, characterized by the preferred orientation of the crystallites, are important to set up the yield function. Although the Hosford yield function is more suitable than the Hill yield function for describing both the yielding and plastic deformation of orthotropic material, it suffers from the restriction that the three principal stresses must be coaxial with the orthotropy of materials. This paper proposes the new Hosford yield function for weakly textured sheets of cubic crystal orthotropic metals in any stress state by expanding the introduced orientation-dependent functions to its sixth-order Taylor series expansion. Also, the new yield function, which covers three material parameters and seven texture coefficients, is more general than the existing Hosford yield function. Finally, both the plastic anisotropy of the q-value and the yield stress obtained from the new yield function agree well with experimental results. This yield function can lay a theoretical foundation for analyzing the mechanical properties of metal materials.

Dispersion of Rayleigh waves in weakly anisotropic media with vertically-inhomogeneous initial stress

Herein we present a procedure by which a high-frequency asymptotic formula can be derived for dispersion relations of Rayleigh waves that propagate in various directions along the free surface of a vertically-inhomogeneous, prestressed, and generally anisotropic half-space. The procedure is based on three assumptions, namely: (i) the incremental elasticity tensor of the material half-space can be written as the sum of a homogeneous isotropic part CIso and a depth-dependent perturbative part A; (ii) at the free surface both the initial stress and A are small as compared with CIso; (iii) the mass density, the initial stress, and A are smooth functions of depth from the free surface. We derive formulas and Lyapunov-type equations that can iteratively deliver each term of an asymptotic expansion of the surface impedance matrix, which leads to the aforementioned high-frequency asymptotic formula for Rayleigh-wave dispersion. As illustration we consider a thick-plate sample of AA 7075-T651 aluminum alloy, which has one face treated by low plasticity burnishing that induced a (depth-dependent) prestress at and immediately beneath the treated surface. We model the sample as a prestressed, weakly-textured orthorhombic aggregate of cubic crystallites and work out explicitly, up to the third order, the dispersion relations that pertain to Rayleigh waves propagating in several directions along the treated face of the sample.

Determination of all elastic and plastic parameters for sheets of cubic metals only by uniaxial tension tests

Most of sheet metals belong to orthorhombic aggregates of cubic crystallites. Since both the elasticity tensor and the yield function of sheet metals depend on the crystalline orientation distribution, some correlations between elasticity and plasticity must exist. Employing the correlations, herein we derive the computational expressions of determining all elastic and plastic parameters in the elasticity tensor and the general Hosford yield function only by the data of uniaxial tension tests. By the elastic FEM numerical simulations and the plastic Taylor-Bishop-Hill model's numerical simulations, we verify the correlations and the computational expressions, and compare the general Hosford yield function with the Hill quadratic yield function and the Hill 1990 yield function. The main goal of this paper is to give a method of determining all the elastic and plastic parameters in the elasticity tensor and the general Hosford yield function only by uniaxial tension tests.

Explicit Bounds of Effective Stiffness Tensors for Textured Aggregates of Cubic Crystallites

For isotropic aggregates of cubic crystallites, Hashin and Shtrikman derived lower and upper bounds for the effective stiffness tensor, which are tighter than the lower and upper bound provided by the Reuss and Voigt model, respectively. In this paper we consider anisotropic aggregates of cubic crystallites with arbitrary texture. We model the elastic polycrystal in question as an assemblage of space-filling spherical grains. Moreover, we assume that every point within one grain has the same crystallographic orientation, whereas the orientations of different grains are uncorrelated. Under this model, we appeal to the variational principles of Hashin and Shtrikman and derive explicit lower and upper bounds for the effective stiffness tensor, which are quadratic in texture coefficients and carry parameters given in terms of the single-crystal elastic constants. For weakly-textured aggregates of cubic crystallites, several examples suggest that our bounds for the effective elastic tensor provide estimates much tighter than those delivered by the Reuss lower bound and the Voigt upper bound.

Constitutive relation of weakly anisotropic polycrystal with microstructure and initial stress

Man (Nondestr Test Eval 15:191–214, 1999) derived the constitutive relation of a weakly-textured orthorhombic aggregate of cubic crystallites with effects of microstructure and initial stress. In this paper, a computational expression on the integration \({{\int_{\rm SO(3)} Q^{\otimes h}D^{l}_{m0}{\rm dg}}}\) is given. Then, by means of the computational expression, the general constitutive relation of a weakly-textured anisotropic polycrystal with the consideration of microstructure and initial stress is derived. As special cases of our general constitutive relation, two constitutive relations are given for an isotropic polycrystal and a weakly-textured anisotropic aggregate of cubic crystallites. The acoustoelastic tensor of the reference cubic crystal is derived to determine the material constants of the polycrystal. Two examples are given for understanding the physical meaning of the texture coefficients and the constitutive relations.

A finite-element study on constitutive relation HM-V for elastic polycrystals

Through a semi-phenomenological approach, we have recently derived a simple constitutive relation (HM-V) for aggregates of cubic crystallites with arbitrary texture symmetry. This constitutive relation is quadratic in the texture coefficients of the polycrystal, and it carries four elastic constants. The derivation delivers four explicit formulae which express the four elastic constants of the polycrystal in terms of the three elastic constants of the single cubic crystal and one undetermined parameter. In this paper we use finite-element calculations to check the adequacy of constitutive relation HM-V and to determine the undetermined parameter for copper and iron polycrystals, respectively.

The n-point orientation correlation function and its application

The properties of a polycrystal can be treated from a statistical point of view. Herein the concept of the n-point orientation correlation function (the n-OCF) is presented in detail. The relations among the 2-point microtexture coefficients, statistical symmetry, and crystallographic symmetry are given for an orthorhombic aggregate of cubic crystallites. As an application of the n-OCF, we obtain a formula for the effective elastic stiffness tensor pertaining to an orthorhombic aggregate of cubic crystallites.

Perturbation approach to elastic constitutive relations of polycrystals

Herein we obtain a formula for the effective elastic stiffness tensor C eff of an orthorhombic aggregate of cubic crystallites by the perturbation method. The effective elastic stiffness tensor of the polycrystal gives the relationship between volume average stress and volume average strain. Under Voigt's model, Reuss’ model and Man's theory, the elastic constitutive relation accounts for the effect of the orientation distribution function (ODF) up to terms linear in the texture coefficients. However, the formula derived in this paper delineates the effect of crystallographic texture on elastic response and shows quadratic texture dependence. The formula is very simple. We also consider the influence of grain shape to elastic constitutive relations of polycrystals. Some examples are given to compare computational results of the formula with those given by Voigt's model, Reuss's model, the finite element method, and the self-consistent method. In Section 3, we also present an expression of the perturbation displacement field, in which Green's function for an orthorhombic aggregate of cubic crystallites is included.

Green's Function for Orthorhombic Aggregates of Weakly Anisotropic Cubic Crystals

Herein a closed but approximate formula of the Green's function is obtained for orthorhombic aggregates of cubic crystallites. This formula, which includes three material constants and three texture coefficients, accounts for the effects of the orientation distribution function (ODF) up to terms linear in the texture coefficients. Thus it is expected that our formula would be applicable to aggregates with weak texture or to materials such as aluminum whose single crystal has weak anisotropy. The approximate formula remains valid and assumes a simpler form when the polycrystal reduces to a weakly anisotropic cubic crystal. Two examples are presented to compare predictions from our formula with those from Nishioka and Lothe's formula and from Synge's contour integral through numerical integration.

Elastic constants of a polycrystal with an orthorhombic texture

Herein we describe a self-consistent method on eigenstrain, by which we obtain a formula for the effective elasticity stiffness tensor pertaining to an orthorhombic aggregate of cubic crystallites. This formula, given in a closed form, delineates the effect of crystallographic texture on elastic response. A simpler form of the formula, which is obtained by truncating a Taylor expansion, is expected to be applicable to aggregates of crystallites with weak texture or to materials such as aluminum whose single crystal has weak anisotropy.

Surface Impedance Tensors of Textured Polycrystals

A formula for the surface impedance tensors of orthorhombic aggregates of cubic crystallites is given explicitly in terms of the material constants and the texture coefficients. The surface impedance tensor is a Hermitian second-order tensor which, for a homogeneous elastic half-space, maps the displacements given at the surface to the tractions needed to sustain them. This tensor plays a fundamental role in Stroh's formalism for anisotropic elasticity. In this paper we account for the effects of crystallographic texture only up to terms linear in the texture coefficients and give an explicit formula for the terms in the surface impedance tensor up to those linear in the texture coefficients.

Angular Dependence of Rayleigh-Wave Velocity in Prestressed Polycrystalline Media with Monoclinic Texture

This article is concerned with Rayleigh waves propagating along the free surface of a macroscopically homogeneous, prestressed half-space. In the meso-scale, the half-space in question is taken to be a textured polycrystalline aggregate of cubic crystallites, which has the normal to its free surface being a 2-fold axis of monoclinic sample symmetry. Under the theoretical framework of linear elasticity with initial stress, an angular dependence formula, which shows explicitly how the phase velocity of Rayleigh waves depends on the propagation direction, the prestress, and the crystallographic texture, is derived from a constitutive equation motivated by Hartig's law. This velocity formula includes terms which describe the effects of texture on acoustoelastic coefficients, and it is correct to within terms linear in the initial stress and in the anisotropic part of the incremental elasticity tensor. Since its derivation makes no presumption on the origin of the initial stress, this velocity formula is meant to be applicable when the prestress is induced by plastic deformations such as those incurred during the surface enhancement treatment of low plasticity burnishing. The angular dependence formula assumes a simpler form when the texture of the prestressed half-space is orthorhombic.

Homogenization of Polycrystalline¶Aggregates

The effective elastic properties of a polycrystalline material depend on the single crystal elastic constants of the crystallites comprising the polycrystal and on the manner in which the crystallites are arranged. In this paper we apply the techniques of homogenization to put the problem of determining effective elastic constants in a precise mathematical framework that permits us to derive an expression for the effective elasticity tensor. We also study how the homogenized elasticity tensor changes as the probability characterizing the ensemble changes. Under the assumption that the field of orientations of the crystallographic axes of the crystallites is an independent random field, we show that our theory is compatible with the formulation used in texture analysis. In particular, we are able to prove that the physical assumption made by [10] in his study of weakly-textured polycrystals holds true. In addition, we establish some elementary bounds on the material constants that characterize the effective elasticity tensor of weakly-textured orthorhombic aggregates of cubic crystallites.

On the separation of stress-induced and texture-induced birefringence in acoustoelasticity

In this paper we develop a simple micromechanical model of a prestressed polycrystalline aggregate, in which the texture-induced and stress-induced anisotropies of the aggregate are precisely defined; here the word ‘texture’ always refers to the texture of the aggregate at the given prestressed configuration, not to that of a perhaps fictitious natural state of the aggregate. We use this model to derive, for a prestressed orthotropic aggregate of cubic crystallites, a birefringence formula which shows explicitly the effects of the orthotropic texture on the acoustoelastic coefficients. From this formula we observe that, generally speaking, we cannot separate the total birefringence into two distinct parts, one reflecting purely the influence of stress on the birefringence, and the other encompassing all the effects of texture. The same formula, on the other hand, provides for each material specific quantitative criteria under which the ‘separation of stress-induced and texture-induced birefringence’ would become meaningful in an approximate sense.

Diffusivity of ultrasound in polycrystals

T he diffusivity of ultrasound in an untextured aggregate of cubic crystallites is studied theoretically with a view towards nondestructive characterization of microstructures. Multiple scattering formalisms for the mean Green's dyadic and for the covariance of the Green's dyadic (and therefore for the energy density) based upon the method of smoothing are presented. The first-order smoothing approximation used is accurate to leading order in the anisotropy of the constituent crystallites. A further, Born, approximation is invoked which limits the validity of the calculation to frequencies below the geometrical optics regime. Known result for the mean field attenuations are recovered. The covariance is found to obey an equation of radiative transfer for which a diffusion limit is taken. The resulting diffusivity is found to vary inversely with the fourth power of frequency in the Rayleigh, long wavelength, regime and inversely with the logarithm of frequency on the short wavelength, stochastic, asymptote. The results are found to fit the experimental data.

Coincidence of bounds for some elastic moduli of polycrystals with [100] fibre texture

Abstract Kroner and Wawra (1978) have demonstrated that the Voigt and Reuss bounds for the elastic constants C′13, C′33, and C′44 of a polycrystalline aggregate of cubic crystallites with [100] fibre texture coincide. In this paper it is shown in addition that the quantity C′11+C′12 has the same value in the Voigt and Reuss schemes, and that the difference in C′66=(C′11-C′12)/2 varies with the square of the anisotropy factor C=(C 11-C 12-2C 44) of the crystallites from which the aggregate is composed for small anisotropy C. Expressions for the elastic stiffness and compliance constants in the Voigt and Reuss schemes are given which correct those recently given by Ledbetter (1982).

Elastic moduli of perfectly disordered composite materials

Using the concept of correlation functions the overall elastic moduli of perfectly disordered composite materials are derived from the local elastic moduli in a rigorous manner. Explicit results which compare well with experiment are obtained for the shear modulus of aggregates of cubic crystallites. The results are close to those derived by the self-consistent method.