Inna M. Gitman

Viscous flow between two moving parallel disks: exact solutions and stability analysis

The motion of a viscous incompressible liquid between two parallel disks, moving towards each other or in opposite directions, is considered. The description of possible conditions of motion is based on the exact solution of the Navier–Stokes equations. Both stationary and transient cases have been considered. The stability of the motion is analysed for different initial perturbations. Different types of stability were found according to whether the disks moved towards or away from each other.

Review and Critique of the Stress Gradient Elasticity Theories of Eringen and Aifantis

This chapter discusses the stress gradient theories of Eringen and Aifantis in terms of their original formulations, their differing dispersion properties in dynamics, and their finite-element implementation. A proposed combination of the two provides a dynamically consistent gradient enrichment while avoiding implementation difficulties.

Gradient Elasticity with Internal Length and Internal Inertia Based on the Homogenisation of a Representative Volume Element

In the context of microscopically-based macroscopic descriptions of material behaviour two homogenisation schemes (first-order or local, and second-order or nonlocal) are employed in this paper. They result in macroscopic expressions for the stress-strain relations and the equations of motion or equilibrium equations. In the case that second-order homogenisation is applied, it turns out that elastic behaviour at the microscale implies the appearance of second-order space and time derivatives of macroscopic strain in an otherwise elastic constitutive equation for the macroscopic stress. The coefficients of the second-order terms scale with the size of the Representative Volume Element (RVE) employed in the homogenisation scheme. Among other things, a recently popularised model of gradient elasticity is provided with a new micromechanical basis and consistently extended to include internal inertia-like effects in a dynamic context. Numerical examples demonstrate the effects of the RVE size on the macroscopic behaviour in statics and dynamics. Finally, some remarks on the size of the RVE in relation to hardening/softening behaviour and stochastic effects are given. 1 Corresponding author: H. Askes, Department of Civil and Structural Engineering, University of Sheffield, Mappin Street, Sheffield SI 3JD, email: h.askes@sheffield.ac.uk

Gradient Elastic Stress Analysis for Anisotropic Bimaterial Interface with Arbitrarily Oriented Crack

Singularities occurring at a crack tip are a well known phenomenon; gradient elasticity has been proven to have the ability to remove singularities through additional length scale parameters which at the same time provides extra microstructural information of the material. This study uses gradient elasticity to simulate singularity-free stresses at bi-material bone-implant interfaces and outlines the relationship between the fracture orientation and maximum stresses at crack tip depicted by this model.

Complex Estimation of Strength Properties of Functional Materials on the Basis of the Analysis of Grain-Phase Structure Parameters

The technique allows analysis using grain-phase structure of the functional material to evaluate its performance, particularly strength properties. The technique is based on the use of linguistic variable in the process of comprehensive evaluation. An example of estimating the strength properties of steel reinforcement, subject to special heat treatment to obtain the desired grain-phase structure.

Numerical Investigation of Thermal and Thermo-mechanical Effective Properties for Short Fibre Reinforced Composite

This study aims to investigate the thermal conductivity and the linear coefficient of thermal expansion for short fibre reinforced composites. The study combines numerical and statistical analyses in order to primarily examine the representative size and the effective properties of the volume element. Effects of various micromechanical parameters, such as fibre’s aspect ratio and fibre’s orientation, on the minimum representative size are discussed. The numerically acquired effective properties, obtained for the representative size, are presented and compared with analytical models.

A computational mechanics perspective on long-range interactions in gradient elasticity with microinertia

Abstract Penalty functions, used in computational mechanics, can provide an interpretation of the effects of the spatial gradients in continuum mechanics. In particular, the use of mass penalties and the use of microinertia terms in gradient elasticity lead to the identical systems of equations. Thus, an alternative perspective on the long-range interactions due to gradient activity is provided. This is illustrated with a numerical example.

Modelling of Size Effects with Gradient-Enriched Continuum Theories

Size-dependent mechanical behaviour in simulations with a higher-order continuum material model is studied. It is shown that size effects occur in strain concentrations when a gradient elasticity theory is used. Similarly, size effects in the peak load can be modelled with a gradient damage theory. In both cases, a comparison is made with two scaling laws available in the literature: the Multi-Fractal Scaling Law of Carpinteri and the Size Effect Law of Bažant. Finally, the energy dissipation in Elementary Volumes is shown to be size-dependent, where again a gradient damage theory has been used. This implies that Representative Volumes do not exist when the dissipated energy is considered.

A Coupled-Volume Approach to the Multi-Scale Modelling of Quasi-Brittle Materials

The hierarchical multi-scale procedure is analysed in this paper. The local multi-scale model has been studied with respect to the macro-level mesh dependency and meso-level size dependency. The material behaviour has been analysed in case of linear-elasticity, hardening and softening. Though the results show no dependency in cases of linear-elasticity and hardening, the strong dependency on both macro-level mesh and meso-level size in case of softening has been found. In order to eliminate both macro-level mesh dependency and meso-level size dependency, a new multi-scale procedure has been proposed. This procedure uniquely links the numerical parameter (macro-level mesh size) and model parameter (meso-level size). The results of this coupled-volume multi-scale model show no dependency on the macro-level mesh or meso-level size in any regimes.

The influence of random microstructure on wave propagation through heterogeneous media

In this paper the influence of mechanical and geometrical properties, both deterministic and stochastic in nature, of a heterogeneous periodic composite material on wave propagation has been analysed in terms of the occurrence of stop-bands. Numerical analyses have been used to identify those parameters that have the most significant effect on the wave filtering properties of the medium. A striking conclusion is that randomness in geometrical properties has a much larger effect than randomness in mechanical properties.