Chi-Sing Man

Towards an acoustoelastic theory for measurement of residual stress

The rudiments of an acoustoelastic theory is developed within the framework of linear elasticity with initial stress. Since no assumption is made about the origin of the initial stress, our acoustoelastic theory will be applicable to evaluation of stress in plastically deformed bodies, provided that the superimposed ultrasonic waves be hyperelastic. New universal relations are deduced. An approach to evaluation of stress which does not use calibration specimens and makes full use of universal relations in our acoustoelastic theory is advocated. Examples are given which illustrate application of our theory to evaluate residual stress in plates. Preliminary corroborations of our theory are provided by the recent experiments of King & Fortunko and Thompson et al.

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.

Constitutive Relation of Elastic Polycrystal with Quadratic Texture Dependence

Herein we consider polycrystalline aggregates of cubic crystallites with arbitrary texture symmetry. We present a theory in which we keep track of the effects of crystallographic texture on elastic response up to terms quadratic in the texture coefficients. Under this theory, the Lame constants pertaining to the isotropic part of the effective elasticity tensor of the polycrystal will generally depend on the texture. We introduce also two simple models, which we call HM-V and HM-R, by which we derive an explicit expression for the effective stiffness tensor and one for the effective compliance tensor. Each of these expressions contains a term quadratic in the texture coefficients and, in addition to three parameters given in terms of the single-crystal elastic constants, each carries an undetermined material coefficient. These two remaining coefficients can be determined by imposing the requirement that the expressions from models HM-V and HM-R be compatible to within terms linear in the texture coefficients.

A Simple Explicit Formula for the Voigt-Reuss-Hill Average of Elastic Polycrystals with Arbitrary Crystal and Texture Symmetries

The Voigt-Reuss-Hill (VRH) average provides a simple way to estimate the elastic constants of a textured polycrystal in terms of its crystallographic texture and the elastic constants of the constituting crystallites. Empirically the VRH estimates were found in most cases to have an accuracy comparable to those obtained by more sophisticated techniques such as self-consistent schemes. In this paper we determine, in the space of fourth-order tensors with major and minor symmetries, a special set of irreducible basis tensors, with which we obtain a simple explicit formula for the VRH average for elastic polycrystals with arbitrary crystal and texture symmetries. Our formula is correct to first order in the texture coefficients.

On the r-value of textured sheet metals

Abstract The r -value of a sheet metal is a measure of plastic anisotropy frequently used for prediction of performance in deep-drawing. It has also figured prominently in the literature for validation of theories where the predicted angular dependence of r is compared with the measured dependence. As plastic anisotropy in sheet metals is caused mainly by the preferred orientations of grains within the polycrystalline metal, it is natural to ask how r would depend on the orientation distribution function (ODF) w which defines the crystallographic texture of the polycrystal. In this paper a general formula relating r to w is derived for textured sheet metals whose plastic flow behavior is governed by a plastic potential f ( σ , w ), the anisotropic part of which depends linearly on the texture coefficients; here σ denotes the deviator of the Cauchy stress. Specific forms of this formula for orthorhombic sheets of cubic and of hexagonal metals are explicitly given.

Two Micromechanical Models in Acoustoelasticity: a Comparative Study

Herein we derive, under the micromechanical model we proposed earlier, Man and Paroni [14], a complete set of formulae for the twelve material constants in the acoustoelastic constitutive equation for orthorhombic aggregates of cubic crystallites. We present also a second model and compare its predictions on the material constants with those of the first model. Both these models lead to constitutive equations which are indifferent to rotation of reference placement. This allows us to appeal to a new representation theorem (Paroni and Man [15]), which greatly facilitates our derivation of the formulae for the material constants. The second model introduced in this paper is intimately related to some previous averaging theories in the literature. We explain why and in what sense our second model could be taken as a generalization of its predecessors.

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 Hartigs 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.

Explicit Formulae Showing the Effects of Texture on Acoustoelastic Coefficients

It is well known that crystallographic texture not only modifies the elastic constants of polycrystalline aggregates at (unstressed) natural states but also affects their acoustoelastic coefficients when the aggregates are stressed. While exact knowledge about the effects of texture on acoustoelastic coefficients has hitherto remained wanting, such effects are usually assumed to be negligible and are ignored in practical applications of acoustoelasticity (cf. [1] for example). Concerning this common practice, Thompson et al. [2] have urged caution: Care must be taken when [this] assumption is made since the influence of texture on acoustoelastic constants is stronger than its influence on elastic moduli or velocities.

Laser‐Ultrasonic Measurements of Residual Stresses in a 7075‐T651 Aluminum Sample Surface‐Treated with Low Plasticity Burnishing

Low‐plasticity burnishing (LPB) is used to introduce deep compressive surface residual stresses that improve the durability of parts. A non‐destructive measurement of residual stresses, their anisotropy, and distribution as a function of depth is being sought to verify initial process quality and residual stress retention over time. Laser‐ultrasonic measurements of Rayleigh wave and surface skimming longitudinal wave (SSLW) velocities were used together to evaluate the magnitudes and directions of the two principal stresses independently of LPB‐induced texture variations. The results agree with x‐ray measurements at the surface. In addition, it was found that the laser‐ultrasonic pulse generation mechanism was surface‐process dependent.

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 Strohs 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.