Criterion for predicting the morphology of crystalline cubic precipitates in a cubic matrix

The present work is concerned with the configuration of precipitates having cubic crystal symmetry in a cubic matrix. The shape and orientation of the precipitates were determined by minimizing the elastic strain energy, while neglecting surface energy effects. Lee, Barnett, and Aaronson7 have shown that only spherical or plate-shaped precipitates are associated with minimum strain energy. Equating the exact expression for the energy of an infinite coherent plate-shaped precipitate with an approximation suggested for the energy of a spherical precipitate, a simple criterion is derived. The criterion enables the prediction of the shape and orientation of the precipitate associated with minimum strain energy, and allows identification of the basic elastic parameters which determine this configuration. When compared to exact numerical results, good agreement was obtained. The criterion predicts that the minimum strain energy is associated with a plate-shaped precipitate, parallel to its 100 planewhen HC ₄₄ >C* ₄₄/A* and the anisotropy factor of the precipitateA * > 1, and parallel to its {111} plane whenHC ₄₄ > F*(111)C*₄₄/A* andA* < 1. In all other cases, a spherical precipitate is associated with minimum strain energy.H is a parameter which depends on the anisotropy of the matrix.F * is an orientation factor which depends on the anisotropy of the precipitate. Sabbatical leave with the Department of Welding Engineering, The Ohio State University, 190 West 19th Avenue, Columbus, OH 43210.     

Criterion for predicting the morphology of crystalline cubic precipitates in a cubic matrix

The present work is concerned with the configuration of precipitates having cubic crystal symmetry in a cubic matrix. The shape and orientation of the precipitates were determined by minimizing the elastic strain energy, while neglecting surface energy effects. Lee, Barnett, and Aaronson7 have shown that only spherical or plate-shaped precipitates are associated with minimum strain energy. Equating the exact expression for the energy of an infinite coherent plate-shaped precipitate with an approximation suggested for the energy of a spherical precipitate, a simple criterion is derived. The criterion enables the prediction of the shape and orientation of the precipitate associated with minimum strain energy, and allows identification of the basic elastic parameters which determine this configuration. When compared to exact numerical results, good agreement was obtained. The criterion predicts that the minimum strain energy is associated with a plate-shaped precipitate, parallel to its 100 planewhen HC ₄₄ C* ₄₄/A* and the anisotropy factor of the precipitateA * > 1, and parallel to its {111} plane whenHC ₄₄ F*(111)C*₄₄/A* andA* 1. In all other cases, a spherical precipitate is associated with minimum strain energy.H is a parameter which depends on the anisotropy of the matrix.F * is an orientation factor which depends on the anisotropy of the precipitate. Sabbatical leave with the Department of Welding Engineering, The Ohio State University, 190 West 19th Avenue, Columbus, OH 43210.     

Precipitate-induced plastic anisotropy: Explicit solutions of the plastic anisotropy due to plate-shaped precipitates

In some aluminum alloys, the observed plastic anisotropy cannot be explained solely by the measured Taylor factor variation. Qualitatively, it has been suggested that this difference results from a secondary effect due to plate-shaped precipitates. Models addressing the effect of plastically-deforming and elastically-deforming precipitates have been previously proposed. In the present article, explicit solutions of the anisotropic strengthening increment are presented for the case of plate-shaped precipitates. These solutions allow a quantitative consideration of the effect of precipitates on different habit planes and of the effect due to stress aging. Generally, in fcc materials, precipitates on {100} habit planes are predicted to minimize the anisotropy due to texture; precipitates on {111} habit planes are predicted to accentuate the anisotropy due to texture; and precipitates on other habit planes are predicted to produce a minor effect resulting from an averaging over a greater number of crystallographcally equivalent habit planes. Stress aging to alter the relative orientation distribution of a single precipitate type is predicted to produce only slight changes in the plastic anisotropy. Larger effects on the yield variation will be observed when stress aging alters the relative volume fractions of two precipitate types on different habit planes.     

Elastic and plastic anisotropy in sheets of cubic metals

Methods are described for the calculation of the elastic and plastic properties of textured polycrystalline metals. The calculations involve the crystallite orientation distribution function derived from X-ray texture data together with data which describe the single crystal behavior. Subsequently, it is shown how a knowledge of the elastic properties can be used to make an analytical prediction of the plastic properties. In making this prediction use is made of the fact that for cubic metals both the elastic and plastic properties are influenced predominantly through the zeroth and fourth order coefficients of the crystallite distribution function. The methods are illustrated by application to the analysis of the data of Stickels and Mould. The statistical correlation they observed is shown to have a justifiable analytical basis.     

Effect of precipitates on plastic anisotropy for polycrystalline aluminum alloys

The effects of crystallographic texture and precipitate distribution on macroscopic anisotropy in aluminum alloys were investigated. In order to simultaneously consider the effects of crystallographic texture and precipitate distribution on macroscopic anisotropy, predictions of plastic properties were carried out using an anisotropic yield function based on the material texture and a combined isotropic-kinematic hardening rule. The input to the model was a single stress-strain curve, the crystallographic texture, and the precipitate volume fraction, shape, and habit planes. It was shown that the kinematic hardening rule, which expresses a translation of the yield surface in stress space, was a function of all the parameters describing the precipitate distribution. The model was applied to the case of an extruded and recrystallized binary Al-3 wt pct Cu alloy deformed in uniaxial compression in different directions. Excellent agreement was observed between the experimental and predicted yield stress anisotropy and the specimen cross section shape anisotropy. Gaussian distributions of grain orientations around ideal texture components typical of aluminum alloys were generated using computer simulations. These textures were combined with the isotropic-kinematic hardening rule determined for the Al-3 wt pct Cu binary alloy to theoretically assess the influence of precipitates on the r-value (the width-to-thickness plastic strain ratio in uniaxial tension) and yield stress anisotropy for aluminum sheets. It was shown that, for these textures, the precipitate distribution had the effect of reducing plastic anisotropy, in agreement with the trends generally observed in practice.     

第二相粒子与织构对高强Cu-Ni-Si系合金薄板各向异性的影响

利用利用X射线衍射、电子背散射衍射和透射电子显微镜等手段研究了Cu-Ni-Si系合金在不同固溶温度下第二相与织构对其平面各向异性的影响.结果表明:随固溶温度的升高,合金强度和伸长率均出现先升高后降低的趋势,且存在明显的各向异性;800℃固溶时,Cu'和S'为主织构,部分形变晶粒诱发少量Brass、Goss和{011}〈511〉取向的形成,使得合金各向异性减弱;高温固溶时(≥ 850℃),晶粒发生完全再结晶,Cu'和S'织构强度显著增加,Brass等织构减弱甚至消失,各向异性增强;850℃固溶时效后形成δ-Ni₂Si析出相,并与基体满足[001]_Cu//[相似文献   

Elastic interaction energy of two spherical precipitates in an anisotropic matrix

Following the difference method of Eshelby, the elastic interaction energy between two spherical precipitates embedded in an infinite matrix of cubic anisotropy is studied as a function of their distance of separation and alignment direction. When the precipitates are positioned along the [100] direction of the matrix phase, the elastic interaction is found to be attractive and often to exhibit a maximum value at an intercenter distance of two to three radii. For the [110] and [111] alignments, the results depend on the sign of the anisotropic factor,H=2C₄₄+C₁₂−C₁₁, of the matrix phase. When it is positive as in Cu and Ni, the interaction is found to be repulsive. In the reverse case, the situation is substantially different; for the [111] alignment with a Mo matrix, the interaction is found to be of an attractive nature.     

Symmetry-breaking transitions in equilibrium shapes of coherent precipitates: Effect of elastic anisotropy and inhomogeneity

We examine the symmetry-breaking transitions in equilibrium shapes of coherent precipitates in two-dimensional (2-D) systems under a plane-strain condition with the principal misfit strain components ε* _xx and ε* _yy . For systems with cubic elastic moduli, we first show all the shape transitions associated with different values of t=ε* _yy /ε* _xx . We also characterize each of these transitions, by studying its dependence on elastic anisotropy and inhomogeneity. For systems with dilatational misfit (t=1) and those with pure shear misfit (t=−1), the transition is from an equiaxed shape to an elongated shape, resulting in a break in rotational symmetry. For systems with nondilatational misfit (−1<t<1; t ≠ 0), the transition involves a break in mirror symmetries normal to the x- and y-axes. The transition is continuous in all cases, except when 0<t<1. For systems which allow an invariant line (−1≤t<0), the critical size increases with an increase in the particle stiffness. However, for systems which do not allow an invariant line (0<t≤1), the critical size first decreases, reaches a minimum, and then starts increasing with increasing particle stiffness; moreover, the transition is also forbidden when the particle stiffness is greater than a critical value.     

Analysis of grain boundary movement analogous to dislocation movement – both with a matrix having precipitates

In the present investigation it has been shown exemplarily for an austenite microstructure containing carbonitrides that the specific restraining force of a grain boundary due to the distribution of precipitates can be described by a modified Orowan's law, so that the grain boundary movement is considered to be analogous to the dislocation movement. This is achieved by using a modified restraining force F_r (corresponding to the cubic geometry of the particles), a detailed thermodynamic analysis of the precipitates (in this case, carbonitrides) and selected relation for the interparticle spacing L_a. The specific restraining force and the interparticle spacing depend on temperature within the austenite range. Each temperature is attached to a couple (F_r/L_a). The plot F_r against shows straight lines for each steel and from their average slope, a shear module G' of grain boundaries, which is about 64300 N·cm^?2, is obtained. Hence, it is possible to describe the specific restraining force of a grain boundary with the help of Orowan's law or as it is conventionally done by using Zener's modell. At a critical temperature T*, which is different for each steel, the specific restraining force is equal to the driving force of the grain boundary. The corresponding values of L_a are also different for each steel and the steel which is the most resistant to grain coarsening is allowed to have the greatest value of L_a which still retains the capacity of impeding grain coarsening.