Ultrasonic scattering in polycrystals with orientation clusters of orthorhombic crystallites

Scattering-induced ultrasonic attenuation and backscattering in a polycrystalline medium with orientation clusters composed of orthotropic crystallites are studied theoretically, aiming to improve understanding of ultrasonic wave interaction with such clustered microstructures for application to the modeling of titanium alloys. Both orthorhombic crystallites and their arrangements in orientation clusters (also termed microtexture regions, MTRs) are of general ellipsoidal shape. The preferred orientation of orthotropic crystallites in the clusters is represented by a generalized Gaussian orientation distribution function with three independent texture parameters. The effective elastic properties of the clusters, which have in general orthorhombic symmetry, are determined by a volume average of elastic constants weighted by orientation distribution functions and then used to obtain the cluster-scattering-induced attenuation and backscattering in the polycrystalline medium. In the model the wave propagation direction is arbitrary relative to the ellipsoidal axes of the clusters. The contribution of crystallite-scattering-induced attenuation is estimated by the untextured attenuation coefficient factored by a texture transition function. The total attenuation and backscattering are determined by combining scattering by the clusters and crystallites. Drastic effects of clustering and the transition to unclustered polycrystals are demonstrated. Reasonable agreement is observed between the model’s prediction and measurements on two Ti-alloy samples with different crystallite clustering.     

Correlation of elastic wave attenuation and scattering with volumetric grain size distribution for polycrystals of statistically equiaxed grains

This study establishes an explicit relation between spatial two-point correlation function (TPCF) and volumetric (or three-dimensional) grain size distribution for aggregates of statistically equiaxed grains by extending a prior study (Sha, JASA, 2018). This relation is further validated by applying it to available TPCF and volumetric grain size distribution in the literature. Based on this relation, analytical attenuation coefficients for longitudinal and transverse waves, accounting for volumetric grain size distribution, are derived under Born approximation for macroscopically isotropic polycrystals of equiaxed triclinic grains. These attenuation models are applicable for whole frequency range except geometric region. Moreover, scattering coefficients for a polycrystal of equiaxed triclinic grains with a volumetric grain size distribution are obtained. Finally, the analytical attenuation model for the longitudinal wave is verified by comparison with existing 3D finite element simulation results in the literature. This theoretic study has practical applications to the inverse determination of volumetric grain size distribution from ultrasonic measurements.     

On the modeling of evolving anisotropy and large strains in pearlitic steel

A phenomenological model for deformation induced evolution of anisotropy at large strains in pearlitic steel is proposed. The modeled anisotropy is based on a homogenization of an ideal pearlitic microstructure. An areal affine type of reorientation is assumed for the individual grains. Furthermore, a yield criterion of the Hill type is proposed and motivated from the grain reorientation. In each pearlitic grain the cementite lamellas have a privileged direction. The symmetry group of each individual grain is therefore considered transversally isotropic. In a virgin material, the privileged directions of the different grains are randomly oriented, which allows for the interpretation that the material on the macroscopic length scale is initially isotropic. However, the cementite lamellas in the grains tend to align after large stretching or shearing deformation. The modeled evolution of anisotropy on the macroscopic length scale shows a saturation characteristics under large deformations.     

CONSTITUTIVE MODELING OF ROLLED SHAPE MEMORY ALLOY SHEETS TAKING INTO ACCOUNT PRE-TEXTURE AND ANISOTROPIC HARDENING

The drawing or rolling process endows polycrystal shape memory alloy with a crys- tallographic texture, which can result in macroscopic anisotropy. The main purpose of this work is to develop a constitutive model to predict the thermomechanical behavior of shape memory alloy sheets, which accounts for the crystallographic texture. The total macroscopic strain is decom- posed into elastic strain and macro-transformation strain under isothermal condition. Considering the transformation strain in local grains and the orientation distribution function of crystallo- graphic texture, the macro-transformation strain and the effective elastic modulus of textured polycrystal shape memory alloy are developed by using tensor expressions. The kinetic equation is established to calculate the volume fraction of the martensite transformation under given stress. Furthermore, the Hill＇s quadratic model is developed for anisotropic transformation hardening of textured SMA sheets. All the calculation results are in good agreement with experimental data, which show that the present model can accurately describe the macro-anisotropic behaviors of textured shape memory alloy sheets.     

A selfconsistent formulation for the prediction of the anisotropic behavior of viscoplastic polycrystals with voids

In this work we consider the presence of ellipsoidal voids inside polycrystals subjected to large strain deformation. For this purpose, the originally incompressible viscoplastic selfconsistent (VPSC) formulation of Lebensohn and Tomé (Acta Metall. Mater. 41 (1993) 2611) has been extended to deal with compressible polycrystals. In doing this, both the deviatoric and the spherical components of strain-rate and stress are accounted for. Such an extended model allows us to account for the void and for porosity evolution, while preserving the anisotropy and crystallographic capabilities of the VPSC model. The formulation can be adjusted to match the Gurson model, in the limit of rate-independent isotropic media and spherical voids. We present several applications of this extended VPSC model, which address the coupling between texture, plastic anisotropy, void shape, triaxiality, and porosity evolution.     

Anisotropic yield and plastic flow of polycrystalline solids

Deformation induced anisotropy in polycrystalline solids results mainly from crystallographic slip due to dislocation motion at the grain level and texture development due to grain rotation at the aggregate level. To describe these characteristics, the so-called scale invariance approach is adopted which allows information and constitutive relations pertaining to single slip to be cast in a form of macroscopic constitutive equations. An orientation distribution function (ODF) and a texture tensor are introduced into the earlier version (based on the hypotheses of single slip at the grain level and isotropic distribution of the crystallites at the aggregate level) of the scale invariance framework to describe texture effects in plastically deformed polycrystals. The texture tensor is calculated either directly through the solution of ODF, or indirectly through an appropriate set of evolution equations for the orientation tensors and the use of a closure approximation. Theoretical predictions for anisotropic yield and plastic flow behavior compare well with available experimental data.     

On the accuracy of self-consistent elasticity formulations for directionally solidified polycrystal aggregates

In this work, the elastic properties of directionally solidified (DS) polycrystal aggregates are investigated through a combination of analytical and numerical approaches. The effects of crystallographic misorientations and grain aspect ratios of aggregates with ellipsoidal shaped grains are first examined following a self-consistent approach. Finite element techniques are then used to examine the effects of grain size on the elastic properties of the aggregate and to assess the accuracy of the self-consistent predictions. To that purpose, a finite element procedure is presented to generate numerically realistic 3D DS microstructures from electron back-scatter diffraction (EBSD) lattice orientation measurements on an arbitrary cross-section of a DS material. The elastic stiffnesses predicted numerically and analytically are then compared with experimental data on a Ni-base DS alloy tested uniaxially along arbitrary orientations. The general trend predicted analytically was found to be consistent with the numerical and experimental results. Furthermore, an increase in the misorientation between the [0 0 1] axis of each DS grain with respect to the grain growth direction was found to decrease the elastic anisotropy of the DS material.     

Influence of Anisotropy Generated by Rolling on the Stress Measurement by Ultrasound in 7050 T7451 Aluminum

Stress applied to a material can be evaluated using ultrasonic waves. This practice is based on acoustoelastic theory, which relates the stress to the velocity of a wave traveling through the body. How the stress affects the wave velocity is determined by the material’s acoustoelastic constant. This constant can be experimentally measured or calculated from the material’s elastic constants. However, ultrasonic techniques have yet to be adopted as an inspection tool in the field. A factor contributing to this fact is the non-uniformity of materials, mostly associated with grain alignment or texture. As researchers consider this factor, they should take into account the anisotropy generated by rolling. The common practice, however, when relating strain and wave velocity is to ignore anisotropy and to simply utilize isotropic models. No studies have been performed to evaluate the effect of anisotropy on the stress measurement by ultrasound, especially for methods using critically refracted longitudinal waves. The aim of this study is to evaluate how the anisotropy generated by rolling affects the acoustoelastic effect for 7050 T7451 aluminum alloy. We compare the value of the acoustoelastic constant obtained experimentally for rolled samples to the constant calculated with measured elastic constants when the material is assumed to be isotropic. The results show that the methods yield different results, suggesting that the simplified isotropic model should be applied with caution. Since no true known value for elastic constants exists, the results can be used to approach the uncertainty when employing the isotropic model to evaluate stresses in aluminum alloys.     

Propagation of wave at the boundary surface of transversely isotropic thermoelastic material with voids and isotropic elastic half-space

The purpose of this research is to study the effect of voids on the surface wave propagation in a layer of a transversely isotropic thermoelastic material with voids lying over an isotropic elastic half-space. The frequency equation is derived after developing a mathematical model for welded and smooth contact boundary conditions. The dispersion curves giving the phase velocity and attenuation coefficient via wave number are plotted graphically to depict the effects of voids and anisotropy for welded contact boundary conditions. The specific loss and amplitudes of the volume fraction field, the normal stress, and the temperature change for welded contact are obtained and shown graphically for a particular model to depict the voids and anisotropy effects. Some special cases are also deduced from the present investigation.     

Numerical evaluation of overestimation of the interface thickness around ellipsoidal particle

When interfacial layers are viewed as a separate phase, the interface thickness plays an essential role in assessing physico-mechanical properties of particulate materials. However, the interface thickness from sectional analysis is often overestimated, due to the irregularity of surface textures of grains in opaque materials that gives rise to the normal of a cross-sectional plane non-perpendicular to the surface of grains. Hence, the determination of the overestimation degree is very critical to precisely obtain the interface thickness. This article develops a numerical model for the overestimation degree of the interface thickness around an ellipsoidal grain with an arbitrary aspect ratio, by applying an accurate sectional analysis algorithm, and quantitative stereology and geometrical probability theories. Furthermore, on the basis of the developed numerical model, the influence of ellipsoidal particle shape on the overestimation degree is quantitatively characterized.     

Acoustics of two-component porous materials with anisotropic tortuosity

The paper is devoted to the analysis of monochromatic waves in two-component poroelastic materials described by a Biot-like model whose stress–strain relations are isotropic but the permeability is anisotropic. This anisotropy is induced by the anisotropy of the tortuosity which is given by a second order symmetric tensor. This is a new feature of the model while in earlier papers only isotropic permeabilities were considered. We show that this new model describes four modes of propagation. For our special choice of orientation of the direction of propagation these are two pseudo longitudinal modes P1 and P2, one pseudo transversal mode S2 and one transversal mode S1. The latter becomes also pseudo transversal in the general case of anisotropy. We analyze the speeds of propagation and the attenuation of these waves as well as the polarization properties in dependence on the orientation of the principal directions of the tortuosity. We indicate the practical importance of different shear (transversal) modes of propagation in a possible new nondestructive test of geophysical materials.     

各向异性粘弹性多孔截止中平面波的传播

This study discusses wave propagation in perhaps the most general model of a poroelastic medium. The medium is considered as a viscoelastic, anisotropic and porous solid frame such that its pores of anisotropic permeability are filled with a viscous fluid. The anisotropy considered is of general type, and the attenuating waves in the medium are treated as the inhomogeneous waves. The complex slowness vector is resolved to define the phase velocity, homogeneous attenuation, inhomogeneous attenuation, and angle of attenuation for each of the four attenuating waves in the medium. A non-dimensional parameter measures the deviation of an inhomogeneous wave from its homogeneous version. An numerical model of a North-Sea sandstone is used to analyze the effects of the propagation direction, inhomogeneity parameter, frequency regime, anisotropy symmetry, anelasticity of the frame, and viscosity of the pore-fluid on the propagation characteristics of waves in such a medium.     

Modeling transients in the mechanical response of copper due to strain path changes

When copper is deformed to large strains its texture and microstructure change drastically, leading to plastic anisotropy and extended transients when it is reloaded along a different strain path. For predicting these transients, we develop a constitutive model for polycrystalline metals that incorporates texture and grain microstructure. The directional anisotropy in the single crystals is considered to be induced by variable latent hardening associated with cross-slip, cut-through of planar dislocation walls, and dislocation-based reversal mechanisms. These effects are introduced in a crystallographic hardening model which is, in turn, implemented into a polycrystal model. This approach successfully explains the flow response of OFHC Cu pre-loaded in tension (compression) and reloaded in tension (compression), and the response of OFHC Cu severely strained in shear by equal channel angular extrusion and subsequently compressed in each of the three orthogonal directions. This new theoretical framework applies to arbitrary strain path changes, and is fully anisotropic.     

Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers

A micromechanically based constitutive model for the elasto-viscoplastic deformation and texture evolution of semi-crystalline polymers is developed. The model idealizes the microstructure to consist of an aggregate of two-phase layered composite inclusions. A new framework for the composite inclusion model is formulated to facilitate the use of finite deformation elasto-viscoplastic constitutive models for each constituent phase. The crystalline lamellae are modeled as anisotropic elastic with plastic flow occurring via crystallographic slip. The amorphous phase is modeled as isotropic elastic with plastic flow being a rate-dependent process with strain hardening resulting from molecular orientation. The volume-averaged deformation and stress within the inclusions are related to the macroscopic fields by a hybrid interaction model. The uniaxial compression of initially isotropic high density polyethylene (HDPE) is taken as a case study. The ability of the model to capture the elasto-plastic stress-strain behavior of HDPE during monotonic and cyclic loading, the evolution of anisotropy, and the effect of crystallinity on initial modulus, yield stress, post-yield behavior and unloading-reloading cycles are presented.     

Analysis of plane waves in anisotropic thermoelastic diffusive medium

This paper concentrates on the study of the propagation of harmonic plane waves in a homogeneous anisotropic thermoelastic diffusive medium in the context of different theories of thermoelastic diffusion. It is found that five types of waves propagate in an anisotropic thermoelastic diffusive medium, namely a quasi-elastodiffusive (QED-mode), two quasi-transverse (QSH-mode and QSV-mode), a quasi-mass diffusive (QMD-mode) and a quasi-thermo diffusive (QTD-mode) wave. The governing equations for homogeneous transversely isotropic diffusive medium in different theories of thermoelastic diffusion are taken as a special case. It is noticed that when plane waves propagate in one of the planes of transversely isotropic thermoelastic diffusive solid, purely quasitransverse wave mode(QSH) decouples from rest of the motion and is not affected by the thermal and diffusion vibrations. On the other hand, when plane waves propagate along the axis of solid, two quasi-transverse wave modes (QSH and QSV) decouple from the rest of the motion and are not affected by the thermal and diffusion vibrations. From the obtained results, the different characteristics of waves like phase velocity, attenuation coefficient, specific loss and penetration depth are computed numerically and presented graphically for a single crystal of magnesium. The effects of diffusion and relaxation times on phase velocity, attenuation coefficient, specific loss and penetration depth has been studied. Some particular cases are also discussed.     

Effect of pore shape on effective porothermoelastic properties of isotropic rocks

The present work is devoted to the determination of the macroscopic poroelastic and porothermoelastic properties of geomaterials or rock-like composites constituted by an isotropic matrix with embedded ellipsoidal inhomogeneities and/or pores randomly oriented. By considering the solution of a single ellipsoidal inhomogeneity in the homogenization problem it is possible to observe the significant influence of the shape of inhomogeneities on the effective porothermoelastic properties. In the particular case of microscopic and macroscopic isotropic behaviors, a closed form solution based on analytical integrate of the Eshelby solution for the single ellipsoidal inhomogeneity can be obtained for the randomly oriented distribution. This result completes the well known solutions in the limiting cases of spherical and penny shape inhomogeneities. Based on recent works on porous rock-like composites such as shales or argillites, an application of the developed solution to a two-level microporomechanics model is presented. The microporosity in homogenized at the first level, and multiple solid mineral phase inclusions are added at the second level. The overall porothermoelastic coefficients are estimated in the particular context of heterogeneous solid matrix. Numerical results are presented for data representative of isotropic rock-like composites.     

Finite element modeling of tube hydroforming of polycrystalline aluminum alloy extrusions

Finite element modeling of tube hydroforming requires information about the anisotropy of the extruded aluminum tube. Unlike sheet metals, the complex geometry of extruded tubes makes it difficult, except in extrusion direction, to directly measure material properties. Therefore, polycrystalline models provide a good alternative for calculating the anisotropy of the tube in all directions and under various loading conditions. Using a rate-independent single crystal yield surface and rigid plasticity, a Taylor-type polycrystalline model was developed and implemented into ABAQUS/Explicit finite element (FE) code using VUMAT. The constitutive model was then used to calculate the crystallographic texture evolution during the hydroforming of an extruded aluminum tube. Initial crystallographic texture measured using orientation imaging microscopy (OIM) and uniaxial tensile test data obtained along the extrusion direction were input to this FEA model. In order to efficiently and practically simulate the tube hydroforming process using the polycrystalline model, sensitivity to the number of grain orientation, total simulation time, and number of finite elements were studied. Predicted results agreed very well with experimentally measured strain obtained from tube hydroforming process.     

A simplified “polycrystalline” model for viscoplastic and damage finite element analyses

In the framework of classical polycrystalline models, drastic reductions of the numbers of slip systems and of “grains” are proposed. With a number of “grains” representing the texture of the material smaller than 10, good results are obtained either for initially isotropic fcc steel or anisotropic hcp zirconium alloy, with some predictive capacity despite the partial loss of physical relevance. Finite element analyses CPU times are not significantly increased as compared to advanced macroscopic models. Novel extensions of the polycrystalline model are developed for intergranular creep or void growth damage. This methodology increases the field of application of the polycrystalline approach in plastic anisotropy, cyclic plasticity, plastic instability and fracture, and in corresponding industrial problems.     

Simulation of an optically induced asymmetric deformation of a liquid–liquid interface

Deformations of liquid interfaces by the optical radiation pressure of a focused laser wave were generally expected to display similar behavior, whatever the direction of propagation of the incident beam. Recent experiments showed that the invariance of interface deformations with respect to the direction of propagation of the incident wave is broken at high laser intensities. In the case of a beam propagating from the liquid of smaller refractive index to that of larger one, the interface remains stable, forming a nipple-like shape, while for the opposite direction of propagation, an instability occurs, leading to a long needle-like deformation emitting micro-droplets. While an analytical model successfully predicts the equilibrium shape of weakly deformed interface, very few work has been accomplished in the regime of large interface deformations. In this work, we use the Boundary Integral Element Method (BIEM) to compute the evolution of the shape of a fluid–fluid interface under the effect of a continuous laser wave, and we compare our numerical simulations to experimental data in the regime of large deformations for both upward and downward beam propagation. We confirm the invariance breakdown observed experimentally and find good agreement between predicted and experimental interface hump heights below the instability threshold.