A simple approach to solve boundary-value problems in gradient elasticity

We outline a procedure for obtaining solutions of certain boundary value problems of a recently proposed theory of gradient elasticity in terms of solutions of classical elasticity. The method is applied to illustrate, among other things, how the gradient theory can remove the strain singularity from some typical examples of the classical theory.

Defects in U3+: CaF2 single crystals grown under different conditions by the temperature gradient technique

Defects in as-grown U3+ : CaF2 crystals grown with or without PbF2 as an oxygen scavenger were studied using Raman spectra, thermoluminescence glow curves, and additional absorption (AA) spectra induced by heating and gamma-irradiation. The effects of heating and irradiation on as-grown U3+: CaF2 crystals are similar, accompanied by the elimination of H-type centers and production of F-type centers. U3+ is demonstrated to act as an electron donor in the CaF2 lattice, which is oxidized to the tetravalent form by thermal activation or gamma-irradiation. In the absence of PbF(2)as an oxygen scavenger, the as-grown U3+:CaF2 crystals contain many more lattice defects in terms of both quantity and type, due to the presence of O2- impurities. Some of these defects can recombine with each other in the process of heating and gamma-irradiation. (c) 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier-Stokes equations is presented. The scheme is constructed on a staggered-mesh grid system. The convection terms are discretized with a fifth-order-accurate upwind compact difference approximation, the viscous terms are discretized with a sixth-order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth-order difference approximation on a cell-centered mesh. Time advancement uses a three-stage Runge-Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented.

Application of the Conjugate Gradient Optimization Algorithm of Neural Network on Laser Machining

在应用激光技术加工复杂曲面时,通常以采样点集为插值点来建立曲面函数,然后实现曲面上任意坐标点的精确定位。人工神经网络的BP算法能实现函数插值,但计算精度偏低,往往达不到插值精确要求,造成较大的加工误差。提出人工神经网络的共轭梯度最优化插值新算法,并通过实例仿真,证明了这种曲面精确定位方法的可行性,从而为激光加工的三维精确定位提供了一种良好解决方案。这种方法已经应用在实际中。

Strain gradient theory with couple stress for crystalline solids

A new phenomenological strain gradient theory for crystalline solid is proposed. It fits within the framework of general couple stress theory and involves a single material length scale Ics. In the present theory three rotational degrees of freedom omega (i) are introduced, which denote part of the material angular displacement theta (i) and are induced accompanying the plastic deformation. omega (i) has no direct dependence upon u(i) while theta = (1 /2) curl u. The strain energy density omega is assumed to consist of two parts: one is a function of the strain tensor epsilon (ij) and the curvature tensor chi (ij), where chi (ij) = omega (i,j); the other is a function of the relative rotation tensor alpha (ij). alpha (ij) = e(ijk) (omega (k) - theta (k)) plays the role of elastic rotation reason The anti-symmetric part of Cauchy stress tau (ij) is only the function of alpha (ij) and alpha (ij) has no effect on the symmetric part of Cauchy stress sigma (ij) and the couple stress m(ij). A minimum potential principle is developed for the strain gradient deformation theory. In the limit of vanishing l(cs), it reduces to the conventional counterparts: J(2) deformation theory. Equilibrium equations, constitutive relations and boundary conditions are given in detail. For simplicity, the elastic relation between the anti-symmetric part of Cauchy stress tau (ij), and alpha (ij) is established and only one elastic constant exists between the two tensors. Combining the same hardening law as that used in previously by other groups, the present theory is used to investigate two typical examples, i.e., thin metallic wire torsion and ultra-thin metallic beam bend, the analytical results agree well with the experiment results. While considering the, stretching gradient, a new hardening law is presented and used to analyze the two typical problems. The flow theory version of the present theory is also given.

Effect of Microtopography, Slope Length and Gradient, and Vegetative Cover on Overland Flow Through Simulation

Overland flow on a hillslope is significantly influenced by its microtopography, slope length and gradient, and vegetative cover. A 1D kinematic wave model in conjunction with a revised form of the Green-Ampt infiltration equation was employed to evaluate the effect of these surface conditions. The effect of these conditions was treated through the resistance parameter in the kinematic wave model. The resistance in this paper was considered to be made up of grain resistance, form resistance, and wave resistance. It was found that irregular slopes with microtopography eroded more easily than did regular slopes. The effect of the slope gradient on flow velocity and flow shear stress could be negative or positive. With increasing slope gradient, the flow velocity and shear stress first increased to a peak value, then decreased again, suggesting that there exists a critical slope gradient for flow velocity and shear stress. The vegetative cover was found to protect soil from erosion primarily by enhancing erosion-resisting capacity rather than by decreasing the eroding capability of overland flow.

A Quantitative Analysis of the Effect of Laser Transformation Hardening on Crack Driving Force in Steels

A mechanical model of a laser transformation hardening specimen with a crack in the middle of the hardened layer is developed to quantify the effects of the residual stress and hardness gradient on crack driving force in terms of J-integral. It is assumed

The Flow Theory of Mechanism-Based Strain Gradient Plasticity

The flow theory of mechanism-based strain gradient (MSG) plasticity is established in this paper following the same multiscale, hierarchical framework for the deformation theory of MSG plasticity in order to connect with the Taylor model in dislocation mechanics. We have used the flow theory of MSG plasticity to study micro-indentation hardness experiments. The difference between deformation and flow theories is vanishingly small, and both agree well with experimental hardness data. We have also used the flow theory of MSG plasticity to investigate stress fields around a stationary mode-I crack tip as well as around a steady state, quasi-statically growing crack tip. At a distance to crack tip much larger than dislocation spacings such that continuum plasticity still applies, the stress level around a stationary crack tip in MSG plasticity is significantly higher than that in classical plasticity. The same conclusion is also established for a steady state, quasi-statically growing crack tip, though only the flow theory can be used because of unloading during crack propagation. This significant stress increase due to strain gradient effect provides a means to explain the experimentally observed cleavage fracture in ductile materials [J. Mater. Res. 9 (1994) 1734, Scripta Metall. Mater. 31 (1994) 1037; Interface Sci. 3(1996) 169].

Stress Wave Propagation in a Gradient Elastic Medium

The gradient elastic constitutive equation incorporating the second gradient of the strains is used to determine the monochromatic elastic plane wave propagation in a gradient infinite medium and thin rod. The equation of motion, together with the internal material length, has been derived. Various dispersion relations have been determined. We present explicit expressions for the relationship between various wave speeds, wavenumber and internal material length.

A new hardening law for strain gradient plasticity

A new hardening law of the strain gradient theory is proposed in this paper, which retains the essential structure of the incremental version of conventional J(2) deformation theory and obeys thermodynamic restrictions. The key feature of the new proposal is that the term of strain gradient plasticity is represented as an internal variable to increase the tangent modulus. This feature which is in contrast to several proposed theories, allows the problem of incremental equilibrium equations to be stated without higher-order stress, higher-order strain rates or extra boundary conditions. The general idea is presented and compared with the theory given by Fleck and Hutchinson (Adv. in Appl. Mech. (1997) 295). The new hardening law is demonstrated by two experimental tests i.e. thin wire torsion and ultra-thin beam bending tests. The present theoretical results agree well with the experiment results.

Deflections and Curvatures of a Film–Substrate Structure with the Presence of Gradient Stress in MEMS Applications

The close form solutions of deflections and curvatures for a film–substrate composite structure with the presence of gradient stress are derived. With the definition of more precise kinematic assumption, the effect of axial loading due to residual gradient stress is incorporated in the governing equation. The curvature of film–substrate with the presence of gradient stress is shown to be nonuniform when the axial loading is nonzero. When the axial loading is zero, the curvature expressions of some structures derived in this paper recover the previous ones which assume the uniform curvature. Because residual gradient stress results in both moment and axial loading inside the film–substrate composite structure, measuring both the deflection and curvature is proposed as a safe way to uniquely determine the residual stress state inside a film–substrate composite structure with the presence of gradient stress.

Analysis of residual stress gradient in MEMS multi-layer structure

Residual stress and its gradient through the thickness are among the most important properties of as-deposited films. Recently, a new mechanism based on a revised Thomas-Fermi-Dirac (TFD) model was proposed for the origin of intrinsic stress in solid film

Influences of strain rate and temperature variation on material intrinsic length of plasticity strain gradient theory

Cowper-Symonds and Johnson-Cook dynamic constitutive relations are used to study the influence of both strain rate effect and temperature variation on the material intrinsic length scale in strain gradient plasticity. The material intrinsic length scale decreases with increasing strain rates, and this length scale increases with temperature.

Strain Gradient Effects on Deformation Strengthening Behavior of Particle Reinforced Metal Matrix Composites

Dislocation models with considering the mismatch of elastic modulus between matrix and reinforcing particles are used to determine the effective strain gradient \ita for particle reinforced metal matrix composites (MMCp) in the present research. Based on Taylor relation and the kinetics of dislocation multiplication, glide and annihilation, a strain gradient dependent constitutive equation is developed. By using this strain gradient-dependent constitutive equation, size-dependent deformation strengthening behavior is characterized. The results demonstrate that the smaller the particle size, the more excellent in the reinforcing effect. Some comparisons with the available experimental results demonstrate that the present approach is satisfactory.