: One encounters difficulties in the solution of integral equations, namely, the occurrence of singularities in the kernel, and the occurence of unknown-type singularities in the solution of the integral equation. Standard methods of approximation based on exactness for polynomials up to a certain degree are very poor, and frequently fail for functions having such singularities. The purpose of this contract was to develop methods for solving integral equations which work well in spite of the presence of singularities. This goal has been accomplished, in that the new approximation methods which were developed do work well in the presence of singularities. (Author)

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Numerical Solution of Singular Integral Equations.

Recently, advanced materials whose properties vary within a continuous pattern have been put to use to design and manufacture modern structures. In the current investigation, size dependencies are captured in the nonlinear free vibration characteristics of micro/nano-beams made of bi-directional functionally graded materials (2D-FGM). With the aid of the nonlocal strain gradient elasticity theory and the variational principle, the size-dependent nonlinear differential equations of motion are derived within the framework of the refined hyperbolic shear deformation beam theory. It is supposed that the material properties are distributed exponentially along longitudinal direction, and vary based on the power law function in lateral direction. Moreover, the deviation of the associated physical neutral plane from the mid-plane counterpart is taken into consideration. By employing a numerical solution methodology on the basis of the generalized differential quadrature method (GDQM) together with Galerkin technique and pseudo arc-length continuation method, the nonlocal strain gradient frequency-deflection responses of 2D-FGM micro/nano-beam are obtained corresponding to various values of longitudinal and lateral material property indexes and small scale parameters. It is revealed that the increment made by the strain gradient size dependency in the value of the nonlinear frequency is more than the reduction caused by the nonlocality, especially for the lower maximum deflection imposed to the 2D-FGM micro/nano-beam. Also, it is indicated that for lower values of the material property gradient indexes, the reduction in the nonlinear frequency caused by the lateral functionally graded pattern in the absence of the axial functionally graded pattern is more than that made by the vice versa case. However, for higher values of the material property gradient indexes, an opposite observation is seen.

Nonlinear free vibrations of bi-directional functionally graded micro/nano-beams including nonlocal stress and microstructural strain gradient size effects

Mindlin's second strain gradient continuum theory for isotropic linear elastic materials is used to model two different kinds of size-dependent surface effects observed in the mechanical behaviour of nano-objects. First, the existence of an initial higher order stress represented by Mindlin's cohesion parameter, b0, makes it possible to account for the relaxation behaviour of traction-free surfaces. Second, the higher order elastic moduli, ci, coupling the strain tensor and its second gradient are shown to significantly affect the apparent elastic properties of nano-beams and nano-films under uni-axial loading. These two effects are independent from each other and allow for separated identification of the corresponding material parameters. Analytical results are provided for the size-dependent apparent shear modulus of a nano-thin strip under shear. Finite element simulations are then performed to derive the dependence of the apparent Young modulus and Poisson ratio of nano-films with respect to their thickness, and to illustrate hole free surface relaxation in a periodic nano-porous material.

Second strain gradient elasticity of nano-objects

Summary The development of solid-state batteries has encountered a number of problems due to the complex interfacial contact conditions between lithium (Li) metal and solid electrolytes (SEs). Recent experiments have shown that applying stack pressure can ameliorate these problems. Here, we report a multi-scale three-dimensional time-dependent contact model for describing the Li-SE interface evolution under stack pressure. Our simulation considers the surface roughness of the Li and SEs, Li elastoplasticity, Li creep, and the Li metal plating/stripping process. Consistency between the very recent experiments from two different research groups indicates effective yield strength of the Li used in those experiments of 16 ± 2 MPa. We suggest that the preferred stack pressure be at least 20 MPa to maintain a relatively small interface resistance while reducing void volume.

Pressure-Driven Interface Evolution in Solid-State Lithium Metal Batteries

With the aid of advanced design techniques, functionally graded materials as promising new materials can be fabricated into various micro/nano-structures to acquire stronger mechanical performance. In this work, the size-dependent nonlinear primary resonance of periodic soft excited micro/nano-beams made of bi-directional functionally graded materials (2D-FGMs) is studied. To accomplish this end, the nonlocal strain gradient theory of elasticity is utilized within the framework of the refined hyperbolic shear deformation beam theory to construct a size-dependent beam model. On the basis of the variational approach using the principle of Hamilton, the non-classical differential equations of motion are achieved. Thereafter, a discretization scheme based numerical solving process via generalized differential quadrature method (GDQM) together with the pseudo-arclength continuation technique, the nonlocal strain gradient frequency response and amplitude response associated with the nonlinear primary resonance of 2D-FGM micro/nano-beams with different boundary conditions are obtained. It is displayed that the nonlocal size dependency makes a reduction in the oscillation amplitudes associated with both of the bifurcation points, but the strain gradient size effect causes to increase them. These patterns are more significant for the second bifurcation point and for the simply supported-simply supported boundary conditions in comparison with the clamped-clamped ones. Also, it is observed that by increasing the value of both of the axial and lateral material property gradient indexes, the peak of the oscillation amplitude and its associated excitation frequency increase.

Nonlocal strain gradient nonlinear resonance of bi-directional functionally graded composite micro/nano-beams under periodic soft excitation

Size-dependent effective elastic moduli of particulate composites with interfacial displacement and traction discontinuities

We investigate the transverse free vibration behaviour of axially moving nanobeams based on the nonlocal strain gradient theory. Considering the geometrical nonlinearity, which takes the form of von Karman strains, the coupled plane motion equations and related boundary conditions of a new size-dependent beam model of Euler-Bernoulli type are developed using the generalized Hamilton principle. Using the simply supported axially moving nanobeams as an example, the complex modal analysis method is adopted to solve the governing equation; then, the effect of the order of modal truncation on the natural frequencies is discussed. Subsequently, the roles of the nonlocal parameter, material characteristic parameter, axial speed, stiffness and axial support rigidity parameter on the free vibration are comprehensively addressed. The material characteristic parameter induces the stiffness hardening of nanobeams, while the nonlocal parameter induces stiffness softening. In addition, the roles of small-scale parameters on the flutter critical velocity and stability are explained.

Juan Liu

Papers

Size-dependent effective elastic moduli of particulate composites with interfacial displacement and traction discontinuities

Complex modal analysis of transverse free vibrations for axially moving nanobeams based on the nonlocal strain gradient theory

An efficient numerical model for predicting the torsional fretting wear considering real rough surface

Tribological behavior of polymethyl methacrylate against different counter-bodies induced by torsional fretting wear

Contact involving a functionally graded elastic thin film and considering surface effects