D. X. Du’s research while affiliated with Princeton University and other places

We report the quantitative prediction of grain boundary stiffness as a function of boundary inclination using molecular dynamics simulations in a series of Sigma5 [001] tilt grain boundaries. The grain boundary stiffness exhibits a large anisotropy, which is of the same order of magnitude as that of the grain boundary mobility. Surprisingly, these two anisotropies nearly cancel, leaving the reduced mobility (product of the stiffness and boundary mobility) nearly isotropic.

The strain field ɛ(x) in an infinitely large, homogenous, and isotropic elastic medium induced by a uniform eigenstrain ɛ0 in a domain ω depends linearly upon ɛ0 : ɛij(x)=Sijklω(x)ɛkl0. It has been a long-standing conjecture that the Eshelby's tensor field Sω(x) is uniform inside ω if and only if ω is ellipsoidally shaped. Because of the minor index symmetry Sijklω=Sjiklω=Sijlkω, Sω might have a maximum of 36 or nine independent components in three or two dimensions, respectively. In this paper, using the irreducible decomposition of Sω, we show that the isotropic part S of Sω vanishes outside ω and is uniform inside ω with the same value as the Eshelby's tensor S0 for 3D spherical or 2D circular domains. We further show that the anisotropic part Aω=Sω-S of Sω is characterized by a second- and a fourth-order deviatoric tensors and therefore have at maximum 14 or four independent components as characteristics of ω's geometry. Remarkably, the above irreducible structure of Sω is independent of ω's geometry (e.g., shape, orientation, connectedness, convexity, boundary smoothness, etc.). Interesting consequences have implication for a number of recently findings that, for example, both the values of Sω at the center of a 2D Cn(n⩾3,n≠4)-symmetric or 3D icosahedral ω and the average value of Sω over such a ω are equal to S0.

The interaction direct derivative (IDD) estimate for the effective properties of composites in the matrix-inclusion type was recently proposed by the authors [1], based on the three-phase model. It has an explicit and almost the simplest structure in comparison with other existing micromechanical estimates, with clear and physically significant explanations to all the involved components. It is universally applicable for various multiphase composites in the matrix-inclusion type, regardless of the material symmetries of matrix, inclusions and effective medium, or the distributions, shapes, orientations, and concentrations of the inclusions. As a sister of our above-mentioned paper, the present one devotes to explore some further fundamental properties of the IDD estimate. It is shown that the IDD estimate is ofo(c 2)-accuracy (that is, precise up to the second order of the volume fractionc of the inclusions asc0) whenever all inclusions have ellipsoidal inclusion-matrix cells. With this, we further assess, for the first time, accuracies of other micromechanical estimates. A practical procedure to characterize the spatial distribution of inclusions in the composites is proposed. It is particularly emphasized, based on theoretical analyses and some illustrating examples, that the IDD estimate seems valid for any physically possible high concentration of inclusions; and the concept of the effective stress in the IDD estimate bridges the effective properties and local fields. The latter is required in order to characterize evolutional and irreversible (e.g., damage) mechanical processes.

For estimating the effective properties (elasticity, conductivity, piezoelectricity, etc.) of composites of the matrix-inclusion type, we develop a new micromechanical model, the effective self-consistent scheme (ESCS), based on the three-phase model. As a simplified and explicit version of the ESCS estimate, the interaction direct derivative (IDD) estimate is further proposed. The IDD estimate has an explicit and almost the simplest structure in comparison with other existing micromechanical estimates, with clear physical significance for all the involved components. It is universally applicable for various multiphase composites of the matrix-inclusion type, for any material symmetries of matrix, inclusions and effective medium, and distribution, shapes, orientations, and concentration of inclusions. Applications to effective elastic properties of composites with spherical inclusions and materials damaged due to voids of various shapes and microcracks (up to any high microcrack density) are presented, in comparison with a number of refined or accurate numerical simulation results. The IDD estimate seems to provide the best predictions in most of our examined cases. A further exploration of the proposed two estimates is given by Du and Zheng (Acta Mech. (2001), in press).

It is known from the theory of group representations that, in principle, a tensor of any finite order can be decomposed into a sum of irreducible tensors. This paper develops a simple and effective recursive method to realize such decompositions in both two- and three-dimensional spaces. Particularly, such derived decompositions have mutually orthogonal base elements. Quite a few application examples are given for generic and various physical tensors of orders up to six.

It is pointed out that to numerically estimate the effective properties and local fields of matrix-inclusion composites, a commonly adopted method is accompanied with some serious draw-backs. We call this method the nominal loading scheme (NLS), which considers the actual inclusion distribution inside a finite domain, Ω say, treats the external domain of Ω to be of the pure matrix material, and imposes the actural traction, σ ∞ say on the remote boundary. It thus gives rise to the following basic problems; (i) Can NLS be improved remarkably just by adjusting σ ∞? (ii) What is the relationship between the size of Ω and the scale of inclusions? (iii) Which choice is better in calculating the effective properties, the whole domain Ω or an appropriately selected sub-domain of Ω? Targeting these problems, the equivalent loading scheme (ELS) and equivalent matrix scheme(EMS) are proposed. It is theoretically analyzed that both ELS and EMS can be used to precisely simulating the effective properties and local fields of matrix-inclusion composites, and both ELS and EMS are self-approved. As an application, ELS combined with a so-called pseudo-dislocations method is used to evaluate the effective properties and local fields of two-dimensional two-phase composites with close-packed circular inclusions, or randomly distributed circular inclusions, or randomly distributed microcracks. The results show that substituting the remote traction σ ∞ with the effective stress field σ E suggested by IDD scheme is a simple and effective method, and the estimation of the effective properties and local fields is very close to the accurate solution.

The theory with at most 21 independent elastic parameters, called multi-constant one, corresponds to the existence of a strain energy function date to Green, or equivalently, requires that the elastic modulus tensor Cijkl possesses the major index symmetry: Cijkl = Cjikl = Cijlk = Cklij. The theory with at most 15 independent elastic parameters, called rari-constant one, requires the following six constraints: c2233 = c2323, c3311 = c3131, c1122 = c1212, c1123 = c3112, c2231 = c1223, c3312 = c2331,which are called Cauchy's relations. The multi-constant theory was supported by Green, Stokes, Kirchhoff, and others, while the rari-constant one was agreed with by Poisson, Lord Kelvin, Lame, and others. In 1887, Voigt reported a set of decisive experimental results which most likely to be right.

The recently proposed effective self-consistent method (ESCM) is a quite simple and powerful micromechanical approach of estimating the interaction effect for the overall physical properties of composite materials with multi-phase and multi-shape inclusions (involving holes and microcracks). The ESCM was developed based on the generalised self-consistent method (GSCM), while it is conveniently applicable in a similar way to the widely used Mori-Tanaka method. In this paper, as an application of the ESCM, the closed-form interacting solutions for the overall elastic moduli of an isotropic matrix with various multi-phase and multi-shape isotropic inclusions are derived. These close-form interacting solutions for the overall elastic modulus are compared with other micromechanical estimations and are shown to be perfect predictions of the up-to-date numerical and experimental observations.