Bounds on the effective conductivity of statistically isotropic multicomponent materials and random cell polycrystals

Upper and lower bounds on the effective conductivity of statistically isotropic multicomponent materials in d dimensions (d=2 or 3) are constructed from the minimum energy principles and appropriate trial fields. The trial fields, involving harmonic potentials and free parameters to be optimized, lead to the bounds containing up to three-point correlation information about the microgeometry of a composite. The bounds are applied to give estimates for the symmetric cell materials, which are optimal over some ranges of parameters, and asymmetric multicoated spheres, which yield the exact effective conductivity in certain cases. The results also agree with many known ones. New bounds for random cell polycrystals are obtained and illustrated on a number of polycrystalline aggregates.     

Heat transfer and pressure drop in dimpled fin channels

Heat transfer characteristics are examined comparatively for four sets of dimpled fin channels with Reynolds number (Re) ranging from 1500 to 11,000 in order to determine the effects of dimple arrangement, fin length (L) to channel hydraulic diameter (d) ratio and Re on heat transfer over the dimpled fin channel. These dimpled fin channels share the identical rectangular section of a channel aspect ratio (AR) of 6 with three different L/d of 8.9, 6.2 and 3.5. The two opposite dimpled fins with four different concave and convex arrangements affect the secondary flows and vortex structures tripped by dimples that signify various heat transfer performances over each dimpled fin. Heat transfer correlations for spatially averaged Nusselt number ($\overline{\mathit{Nu}}$) over each dimpled fin are generated using Re and L/d as the controlling parameters. A set of design criteria for determining the optimal L/d that offers the maximum cooling power available from the dimpled fin for each specified dimple arrangement on two opposite fin walls is derived to assist the design activities using the dimpled fin array.     

Material characterization by dual sharp indenters

In a recent study, Le [Le, M.-Q., 2008. A computational study on the instrumented sharp indentations with dual indenters. International Journal of Solids and Structures, 45 (10), 2818–2835.] demonstrated that the yield strength Y can be replaced by the loading curvature C and hence the reduced elastic modulus-loading curvature ratio $E^1/C$ and strain hardening exponent n can be used to govern characteristic parameters of indentation load–depth curves. Extending Le’s approach and regarding dimensional analysis, it is found that $C/Y\text{and}{E}^1/Y$ can be used to investigate fundamental issues in instrumented sharp indentation. Based on extensive finite element analysis, a set of dimensionless functions are constructed for cone indenters of half included angles of 60° and 70.3°. Dimensionless relationships with respect to dual indenters are further explored. Several features of hardness are also considered. An inverse analysis procedure is suggested to estimate material properties, giving good inverse results for experimental data from the literature and representative materials. Sensitivity of inverse solution are studied and discussed. The results show that the proposed dual indenter method is quite robust and can be applied to a wide range of materials.     

Necking of an anisotropic plastic material

The two conditions for stable necking, treated as a bifurcation of stress path, are derived. The anisotropic material considered is one for which the plastic work increment = $\overline{\sigma }d\overline{e}$, where both the generalized yield stress $\overline{\sigma }$ and generalized plastic strain increment $d\overline{e}$ are invariant functions. The method is applied to the necking of a thin cylinder under internal pressure.     

Bounds for the non-local effective properties of random media—II

The paper deals with a random medium subjected to a static field with inhomogeneous mean values. Then, effective linear material parameters show dispersion, i.e. they depend on the ‘wave vector’ k of the mean field. Starting from a variational method previously developed by the authors, upper and lower bounds for k-dependent scalar effective parameters are derived in terms of two-point and three-point correlation functions of the stochastic material parameters. Taking into consideration the three-point correlation function gives a substantial improvement of the generalized Hashin-Shtrikman bounds obtained previously. In particular, composites with cell structure and arbitrary binary systems are considered. In order to illustrate the general results, numerical evaluations are carried out for effective permittivity of a binary cell material composed of nearly spherical grains of equal size.     

On some features of the effective behaviour of porous solids with J2- and J3-dependent yielding matrix behaviourSur quelques effets de l'angle de Lode sur les caractéristiques macroscopiques de matériaux en rupture ductile

Some features od the constitutive behaviour of voided materials taking into account possible effects of the Lode angle in the yielding behaviour of the matrix are discussed. The Gurson approach is used to this end. After providing a parametric representation of the effective behaviour of such materials, some closed-form results are given for pure shear stress states and also at very high stress triaxialities. In the former case corresponding to a zero macroscopic mean stress, the contour of the yield domain in the π-plane has exactly the shape of the yield surface of the matrix in the deviatoric plane, but a size reduced by a factor $1?f$, with f the porosity of the voided material. In the latter, effective yield stresses for the voided material are slightly different from the Gurson result and found to be set by the yield stress at a microscopic stress Lode angle $\frac{\pi }{3}$ for very high positive triaxiality and by the yield stress at a microscopic stress Lode angle 0 for very high negative triaxiality. This last result is extended for porous materials with yielding depending further on the hydrostatic stress, fully exhibiting the interaction between volumetric and shear interactions on the yielding behaviour of isotropic porous materials. Applications to many usual yielding criteria for the matrix are also provided.     

Scale-by-scale energy budget in a turbulent boundary layer over a rough wall

Hot-wire velocity measurements are carried out in a turbulent boundary layer over a rough wall consisting of transverse circular rods, with a ratio of 8 between the spacing (w) of two consecutive rods and the rod height (k). The pressure distribution around the roughness element is used to accurately measure the mean friction velocity ($U_{\tau }$) and the error in the origin. It is found that $U_{\tau }$ remained practically constant in the streamwise direction suggesting that the boundary layer over this surface is evolving in a self-similar manner. This is further corroborated by the similarity observed at all scales of motion, in the region $0.2⩽y/\delta ⩽0.6$, as reflected in the constancy of Reynolds number ($R_{\lambda }$) based on Taylor’s microscale and the collapse of Kolmogorov normalized velocity spectra at all wavenumbers.A scale-by-scale budget for the second-order structure function $〈{(\delta u)}^2〉$ ($\delta u=u(x+r)-u(x)$, where u is the fluctuating streamwise velocity component and r is the longitudinal separation) is carried out to investigate the energy distribution amongst different scales in the boundary layer. It is found that while the small scales are controlled by the viscosity, intermediate scales over which the transfer of energy (or $〈{(\delta u)}^3〉$) is important are affected by mechanisms induced by the large-scale inhomogeneities in the flow, such as production, advection and turbulent diffusion. For example, there are non-negligible contributions from the large-scale inhomogeneity to the budget at scales of the order of $\lambda$, the Taylor microscale, in the region of the boundary layer extending from $y/\delta =0.2$ to 0.6 ($\delta$ is the boundary layer thickness).