Bounds on size-dependent behaviour of composites

Computational homogenisation is a powerful strategy to predict the effective behaviour of heterogeneous materials. While computational homogenisation cannot exactly compute the effective parameters, it can provide bounds on the overall material response. Thus, central to computational homogenisation is the existence of bounds. Classical first-order computational homogenisation cannot capture size effects. Recently, it has been shown that size effects can be retrieved via accounting for elastic coherent interfaces in the microstructure. The primary objective of this contribution is to present a systematic study to attain computational bounds on the size-dependent response of composites. We show rigorously that interface-enhanced computational homogenisation introduces two relative length scales into the problem and investigate the interplay between them. To enforce the equivalence of the virtual power between the scales, a generalised version of the Hill–Mandel condition is employed, and accordingly, suitable boundary conditions are derived. Macroscopic quantities are related to their microscopic counterparts via extended average theorems. Periodic boundary conditions provide an effective behaviour bounded by traction and displacement boundary conditions. Apart from the bounds due to boundary conditions for a given size, the size-dependent response of a composite is bounded, too. The lower bound coincides with that of a composite with no interface. Surprisingly, there also exists an upper bound on the size-dependent response beyond which the expected ‘smaller is stronger’ trend is no longer observed. Finally, we show an excellent agreement between our numerical results and the corresponding analytical solution for linear isotropic materials which highlights the accuracy and broad applicability of the presented scheme.     

Interaction of bursts in exponentially graded materials characterized by parametric plots

The nonlinear interaction of tone bursts in functionally graded materials with strongly variable properties is studied resorting to the five constant nonlinear theory of elasticity in the 1D setting. The problem is solved numerically for an exponentially graded material. The influence of the material properties variation on the bursts profiles evolution is traced on the boundaries of the sample. A special case of the bursts interaction by which oscillations evoked by counterpropagating bursts disappear in the homogeneous material is proposed as a reference case for the problem of nondestructive material characterization. The deviation from this special case caused by inhomogeneity in material properties is analyzed by parametric plots. Obtained results may be used by qualitative nondestructive determination of the inhomogeneity in material properties.     

Non-singular dislocation loops in gradient elasticity

Using gradient elasticity, we give in this Letter the non-singular fields produced by arbitrary dislocation loops in isotropic media. We present the ‘modified’ Mura, Peach–Koehler and Burgers formulae in the framework of gradient elasticity theory.     

Nonlinear microbeam model based on strain gradient theory

The nonlinear governing equation of microbeam based on the strain gradient theory is derived by using a combination of the strain gradient theory and the Hamilton’s principle, and the nonlinear static bending deformation, the post-bucking problem and the nonlinear free vibration are analyzed. The nonlinear term in the nonlinear governing equation is associated with the mean axial extension of the microbeam. The static bending deformation of the clamped–clamped microbeam subjected to transverse force, the critical buckling loads and the nonlinear frequencies of the simple supported microbeam with initial lateral displacement are discussed. It is shown that the size effect is significant when the ratio of characteristic thickness to internal material length scale parameters is approximately equal to one or two, but is diminishing with the increase of the ratio. The results also indicate that the nonlinearity has a great effect on the static and dynamic behavior of microbeam. To attain accurate and reliable characterization of the static and dynamic properties of microbeam, therefore, both the micro structure dependent parameters and the nonlinear term have to be incorporated in the design of micro structures in MEMS or NEMS.     

Thermal effect on vibration of non-homogenous visco-elastic rectangular plate of linearly varying thickness

An analysis for vibration of non-homogenous visco-elastic rectangular plate of linearly varying thickness subjected to thermal gradient has been discussed in the present investigation. For visco-elastic, the basic elastic and viscous elements are combined. We have taken Kelvin model for visco-elasticity that is the combination of the elastic and viscous elements in parallel. Here the elastic element means the spring and the viscous element means the dashpot. The governing differential equation of motion has been solved by Galerkin’s technique. Deflection, time period and logarithmic decrement at different points for the first two modes of vibration are calculated for various values of thermal gradients, non homogeneity constant, taper constant and aspect ratio for non-homogenous visco-elastic rectangular plate which is clamped on two parallel edges and simply supported on remaining two edges. Comparison studies have been carried out with homogeneous visco-elastic rectangular plate to establish the accuracy and versatility.     

Nonlocal elasticity and dislocation image force due to cracks

Summary   Fundamental field equations of nonlocal elasticity are presented. With these equations, the image force on a screw dislocation due to a crack is analyzed using the conformal mapping technique. Two cases are considered: one is for a finite-length crack, the other is for an infinite one. All classical singularities of the dislocation image force are eliminated when the dislocation tends to the crack tip. The maximum of the force is obtained at the crack tip. Received 10 June 1999; accepted for publication 8 February 2000     

The mechanism of asymmetric reduction catalyzed by a C2 symmetric bis-amino alcohol catalyst

By means of¹H,¹³C,¹¹B NMR, polar transfer DEPT135, DEPT90 and 2D NMR experiments, IR, etc, the structure of the diaminodihydroxyl ligand and its derivatives used in the asym-metric reduction of prochiral ketones were detected. The catalysts derived from the ligand and borane formedin situ were also studied and the structure transformations in solution were monitored. The structure of the catalyst was proved to be a new type of dual-centered catalyst — bisoxazaborolidine.     

Three-Dimensional Variational Analysis of Composite Structures Using Bernstein Polynomial Approximations. Report 1

The paper overviews the previous 3D variational analysis approaches and the numerical results obtained with the use of piecewise polynomial splines. A recently developed 3D displacement-assumed variational analysis approach is also described, where the Bernstein approximation polynomials of arbitrary degree are used as the shape functions for a 3D hexahedral discrete element of a composite body.     

Stress Analysis of an Elliptic Inclusion with Imperfect Interface in Plane Elasticity

In this paper, a semi-analytic solution of the problem associated with an elliptic inclusion embedded within an infinite matrix is developed for plane strain deformations. The bonding at the inclusion-matrix interface is assumed to be homogeneously imperfect. The interface is modeled as a spring (interphase) layer with vanishing thickness. The behavior of this interphase layer is based on the assumption that tractions are continuous but displacements are discontinuous across the interface.Complex variable techniques are used to obtain infinite series representations of the stresses which, when evaluated numerically, demonstrate how the peak stress along the inclusion-matrix interface and the average stress inside the inclusion vary with the aspect ratio of the inclusion and a representative parameter h (related to the two interface parameters describing the imperfect interface in two-dimensional elasticity) characterizing the imperfect interface. In addition, and perhaps most significantly, for different aspect ratios of the elliptic inclusion, we identify a specific value (h ^*) of the (representative) interface parameter h which corresponds to maximum peak stress along the inclusion-matrix interface. Similarly, for each aspect ratio, we identify a specific value of h (also referred to as h ^* in the paper) which corresponds to maximum peak strain energy density along the interface, as defined by Achenbach and Zhu (1990). In each case, we plot the relationship between the new parameter h ^*and the aspect ratio of the ellipse. This gives significant and valuable information regarding the failure of the interface using two established failure criteria.