Theory of Non-linear Longitudinal Kerr Magneto-optic Effect in a Ferromagnetic Metallic Thin Film

A theory of the non-linear longitudinal Kerr magneto-optic effect, previously developed for bulk materials [1], is now applied to a magnetic thin film. The material model is based on the classical equation of motion for a free electron with a finite relaxation frequency under the action of a Lorentz force. A second harmonic current density, dependent upon three non-linear conductivity tensors, reduces to results obtained by Jha [2] in the limit of vanishing ferromagnetic state. The second harmonic reflection coefficients are approximated and are given in terms of the first harmonic reflection coefficients as derived by Smith [3].     

The application of the rate theory of deformation to the yield behaviour of wood

The yield behaviour of wood is assumed to result from the stress-assisted thermally activated motion of elementary fibrils over short-range barriers. The barrier energy is taken to be a linear decreasing function of increasing moisture content and the activation volume a linear increasing function of increasing moisture content. The flow stress equation obtained from the application of the rate theory is given by {ie264-01}The theory has been examined in relation to the strain-rate and moisture dependence of the maximum longitudinal crushing stress ofPinus radiata andKahikatea. The data for these woods follow fairly closely the general expectations of the theory and the measured activation volumes are not inconsistent with the thermally activated motion of elementary fibrils.     

A fully non-linear axisymmetrical quasi-kirchhoff-type shell element for rubber-like materials

An axisymmetrical shell element for large deformations is developed by using Ogden's non-linear elastic material law. This constitutive equation, however, demands the neglect of transverse shear deformations in order to yield a consistent theory. Therefore, the theory can be applied to thin shells only. Eventually a ‘quasi-Kirchhoff-type theory’ emerges. Within this approach the computation of the deformed director vector d is a main assumption which is essential to describe the fully non-linear bending behaviour. Furthermore, special attention is paid to the linearization procedure in order to obtain quadratic convergence behaviour within Newton's method. Finally, the finite element formulation for a conical two-node element is given. Several examples show the applicability and performance of the proposed formulation.     

Computational method for predicting the long-term performance of timber beams in variable climates

The paper describes a method for computational assessment of the long-term performance of timber beams in variable climates. The method is based on a cross-section analysis, in which the moisture content and longitudinal stress-strain distributions are calculated as function of time and ambient conditions assuming asymmetrical mechanical behaviour in the tension and compression sides of the beam. The method is implemented as a non-linear FE-program, which combines the moisture transport and structural analysis, which is based on a non-linear model of longitudinal creep in wood. The calculated stress-strain state is monitored against a failure criterion, which is based on a definition of a local damage variable and on its integration over the volume in the fashion of the Weibull theory of weakest link to give a global damage parameter. Computational results include long-term deformation (eflection) and time-to-failure (long-term capacity) predictions. Examples of time-to-failure calculation are given.     

A perturbation method used for static contact and low velocity impact

An approximate method for the solution of static and dynamic contact problems between bodies with non-linear material behaviour is described. The method is a perturbation technique based on the linear elastic quasi-static solution. Here the method is applied to the problem of a sphere in contact with a half-space which means that the Hertz solution is used. The governing equations are rewritten so that the problem for the perturbed variables is one with surface forces in the contact region and volume forces inside the bodies. The latter are due to accelerations and strain gradients calculated from the quasi-static solution and the equation of motion. The contact condition results in an integral equation for the surface forces. Results are compared with FEM calculations, which show very good agreement for the dynamic case, both with linear elastic and non-linear (plastic) material behaviour. For the static case with non-linear material behaviour the results are good approximately up to the point where the inelastic zone reaches the surface of the bodies.     

Non-linear compression stress-strain curve of pitch-based high modulus carbon fibre composites and structural responses

The longitudinal compression behaviour of unidirectional composites is studied to understand the role of the fibre compressive property in deformation and failure by systematically varying the tensile modulus of reinforcing high modulus carbon fibre. As the composites deform, their softness increases with increasing compressive strain, and the loading path is traced back when the load is removed. The intensity of softening is correlated to the fibre's tensile modulus and possible softening mechanisms are discussed in conjunction with fibre and matrix properties. Further, it is investigated how the non-linear stress-strain relation affects the stress and strain distributions and deformation when plates fabricated from these fibres are tested by the three-point bending test.     

Discrete element methods for the plastic analysis of structures subjected to cyclic loading

Methods for the analysis of complex, highly redundant structures subjected to intermittent loads causing biaxial membrane stress and stress reversal into the plastic range are presented. The Bauschinger effect in multi-axial stress is taken into account by the use of Ziegler's modification of Pragers kinematic hardening theory. The implementation of this plasticity theory in the discrete element methods involves the application of the loading in small increments. A linear relationship between increments of plastic strain and of stress, arising out of the theory, is used in conjunction with a linear matrix equation that governs the elastic behaviour of the structure. In the latter equation, plastic strains are interpreted as initial strains. A solution to the linear matrix equation, expressed in terms either of stress or of total strain, may be obtained by utilizing one of two alternative procedures. The methods are capable of treating materials which exhibit elastic–plastic behaviour involving ideal plasticity, linear or non-linear strain hardening, or limited strain hardening. Application is made to several representative structures. Comparison of some of the results with existing test data for both monotonic and reversed loading shows good correlation.     

Theory of the Non-linear Longitudinal Kerr Magneto-optic Effect in Ferromagnetic Metals

A theory of the non-linear longitudinal Kerr magneto-optic effect in ferromagnetic metals is developed. The material model is based on the classical equation of motion for a free electron with a finite relaxation frequency under the action of a Lorentz force. A second harmonic current density is found of the form: J₂ = αE₂ + β _h (H ₁)E₁ + σ (E₁? · E₁), where α, β_h (H ₁) and σ are non-liner conductivity tensors, E₁ and H₁ are the fundamental electric and magnetic fields, and E₂ is the induced second harmonic electric field. Results of this theory reduce, in the limit of a vanishing ferromagnetic state, to results obtained by Jha [1] from the Boltzmann transport equation for conduction electrons subject to a potential barrier at the metal surface. As required, the theory reduces to the linear longitudinal Kerr magneto-optic effect in the absence of second harmonic generation. The second harmonic reflection coefficients are derived. To the degree of approximations made, all four coefficients vanish at grazing incidence. Unlike the non-linear polar Kerr effect, they do not vanish at normal incidence.     

Application of method of fundamental solutions for elasto-plastic torsion of prismatic rods

In this paper torsion of prismatic bars considering elastic-plastic material behavior is studied. Based on the Saint-Venant displacement assumption and deformation theory of plasticity for stress-strain relation the boundary value non-linear problem for stress function is formulated. The purpose of our paper is application of method fundamental solution (MFS) and radial basic function (RBF) for solution of this problem. The non-linear torsion problem in plastic region is solved by means of the Picard iteration. Proposed algorithm is based on solution of the linear Poisson equation on each iteration steps.     

非线性粘弹性桩耦合运动中的混沌分析

研究轴向周期载荷作用下非线性粘弹性桩纵横向耦合运动中的混沌运动。桩体材料满足Leaderman非线性粘弹性本构关系和近似的非线性几何关系，考虑桩体发生纵横向运动的耦合，得到的方程为耦合的非线性偏微分一积分方程；利用Galerkin方法将方程简化并进行数值计算，揭示非线性粘弹性桩的混沌运动和分岔等动力学行为。     

复合材料任意壳稳定性研究（Ⅰ）──几何方程的新研究

从三维非线性理论导出，任意曲线坐标系中任意壳体的几何非线性微分方程，并修正Ｋ．Ｗａｓｈｉｚｕ的公式，以期得到更加准确的稳定方程。     

复合材料任意壳稳定性研究(Ⅰ)──几何方程的新研究

从三维非线性理论导出,任意曲线坐标系中任意壳体的几何非线性微分方程,并修正K.Washizu的公式,以期得到更加准确的稳定方程。      

Accurate integration of singular kernels in boundary integral formulations for Helmholtz equation

Boundary integral formulations for the 2D Helmholtz equation involve kernels in the form of modified Bessel functions. Accurate schemes for evaluating integrals of the kernels and their derivatives are presented. Special attention is paid to integrals involving singular and near singular kernels. Both boundary and domain integrals are considered. It is shown that, with the use of series expansion functions for the modified Bessel functions, the boundary integrals can be evaluated analytically in the neighbourhood of the singularity. For domain integrals, the behaviour of the kernels in the vicinity of the singularity is used to construct accurate numerical quadrature schemes. A transient heat conduction problem is formulated as a Helmholtz equation, solved, and compared against analytic solution to demonstrate the effectiveness of these schemes in relation to traditional methods. References are made to previous work to advocate the utility of the boundary integral method for non-linear and time-transient problems.     

A thermodynamic framework of fracture mechanics

A rigorous theory of non-linear fracture mechanics consistent with the basic laws of thermodynamics is proposed. It is shown that the local balance equation(s) can be derived from the postulated global balance equation using a generalized Reynolds transport equation appropriate to a body containing a growing crack. The necessary conditions to be fulfilled on the newly created fracture surfaces are derived. Within the framework of local theory the governing equations are applicable to crackproblems in materials with internal variables or fading memory and to geometrically and/or physically non-linear problems. As a special case, the governing equations for infinitesimal thermoviscoelastic fracture mechanics are given.     

THE INFLUENCE OF TWO-STAGE LOADING ON THE LONGITUDINAL SPLITTING OF UNIDIRECTIONAL CARBON/EPOXY LAMINATES

Abstract— A review of variable fatigue loading of carbon fibre/epoxy laminates indicates that for two-stage loading, low-to-high loading is generally more damaging than high-to-low loading. Damage accumulation under two-stage loading appears to produce linear damage in some CFRP materials and non-linear behaviour in others. The behaviour of those materials exhibiting linear damage may be predicted by the Palmgren-Miner rule. A model is presented for determining the longitudinal split growth in notched unidirectional carbon fibre/epoxy test coupons. The model is able to predict the non-linear split growth for low-to-high loading whilst the linear damage accumulation for high-to-low loading may best be determined by use of the Palmgren-Miner rule.     

Non-linear free vibrations of symmetrically laminated, slightly compressible cylindrical shell panels

A geometrically non-linear dynamic shell theory presented by the authors in an earlier work is used to study the non-linear free vibrations of symmetrically laminated cylindrical shell panels. The theory accounts for arbitrary lamination constructions, anisotropy, and slight compression across the thickness. In this paper, this theory is used to derive the equation of motion of the panel with quadratic and cubic non-linearities and symmetric lamination schemes. The symbolic manipulator Mathematica™ is used to perform the Rayleigh-Ritz procedure and derive a single-mode approximation to the vibration of the panel. The Lindstedt-Poincare perturbation technique is used to analyze the resulting non-linear differential equation of motion and study the effects of non-linearities on the dynamics of free vibrations of the panel. A numerical example of a symmetrically laminated graphite/epoxy shell panel is used to demonstrate the procedure. The numerical example shows that non-linearities are of the hardening type and are more pronounced for smaller opening angles. Moreover, it shows that the larger-amplitude motions are dominated by the lower modes.     

Compressive properties of single-filament carbon fibres

Compressive properties of mesophase pitch-based carbon fibres (NT-20, NT-40 and NT-60) were measured using the tensile recoil test and the elastica loop test. The NT-40 fibre with a 400 GPa tensile modulus showed a smaller loop compressive yield strain and a larger recoil compressive strength compared to these values obtained from the longitudinal compression test on its unidirectional composites. Further, the recoil compressive strength of this fibre was higher than that of PAN-based carbon fibre with a corresponding modulus. Under the ideal conditions in the tensile recoil test, the strain energy was conserved before and after recoil, and the initial tensile stress and the recoil compressive stress do not coincide when fibre stress-strain behaviour is non-linear, and the non-linearity in compression and in tension is different. The difference between the composite compressive strength and the recoil compressive strength of NT-40 was quantitatively explained by taking account of the fibre compressive stress-strain non-linear relation. The difference between the loop compressive yield strain and the composite compressive strain to failure was also explained by this non-linearity.     

Evaluating longitudinal vibration characteristics of bonded strands embedded in prestressed concrete beams by a system identification approach

This study is to investigate the wire rope governing equation in order to establish the longitudinal vibration characteristics of bonded PSC strands. For this purpose, the longitudinal vibration tests have been conducted for six bonded PSC specimens with the different stress levels. Subsequently, the wave velocities of the strands have been estimated from the test results. Then, a sensitivity-based system identification algorithm has been applied to evaluate the constitutive constants of the strands using the measured wave velocities. The present study demonstrates that the longitudinal dynamic behaviour of the strands is governed by coupled extensional-torsional oscillations. The study also reveals that longitudinal elastic wave velocity of not only the bonded strands but also unbounded strands is nonlinearly increased as the applied tensile stress level increases.     

Influence of dislocations and plasticity on the tensile behaviour of flax and hemp fibres

This paper presents finite element analyses of the tensile behaviour of elementary fibres of flax and hemp, focusing on the non-linear tensile behaviour of the fibres and the relationship between the stiffness and the diameter of the fibre. The non-linear tensile behaviour is modelled by introducing dislocations in the helical structure of the cellulose fibrils and assuming that the hemicellulose has an elastoplastic constitutive relation. The relationship between the elastic stiffness and the diameter of the fibres is analysed similarly, using an elastic constitutive behaviour of the hemicellulose. The results agree with experimental results found in the literature.