Constrained thickness optimization of rectangular orthotropic fiber-reinforced plate for fundamental frequency maximization

This paper addresses the thickness optimization of rectangular orthotropic fiber-reinforced composite plates by using mathematical programming theory of nonlinear constrained optimization. We optimize the thickness distribution of various composite plates for reaching a maximum in the plate natural frequency, subject to an equality constraint in a plate volume and a number of inequality constraints on the lower and upper bounds of allowable range of thicknesses. For the modeling part of our optimization algorithm, we formulate the kinematical hypothesis of the classical plate theory (CPT), reduced plane stress theory of linearized elasticity, and Hamilton’s principle. For the solution part, we implement the Rayleigh-Ritz semi-analytical solution technique to spatially discretize the weak form of the plate partial differential equation (PDE), and transforming it to a generalized algebraic eigen-problem. For the design optimization part, we use a conjugate Rosen’s gradient projection method. The design variables of the optimization design space are thicknesses of an arbitrary but rectangular partition of the plate into constant-thickness subdivisions. After simulating our algorithm, we noticed many interesting and impressive results, and tried to justify them. The most impressive finding is that, as the fiber orientation of the composite plate rotates, the plate thickness distribution rotates also in the same way, but with different manifestations for different boundary conditions. For parametric studies, we analyze the sensitivity of the algorithm against parameters, like, the degree of fineness in the number of plate partitions into constant-thickness sub-plates, the degree of plate anisotropy, the fiber orientation, the rectangular plate aspect ratio, the upper to lower bounds of allowable thickness ratio, and boundary conditions.     

Effect of base plate thickness on wave size and wave morphology in explosively welded couples

The influence of plate thickness on the wave morphology generated in explosive welding is analysed by the use of mild steel base plates machined into steps of different thicknesses and a constant-thickness mild steel flyer plate, to ensure similar impacting conditions. It is found that wave shape, at the same distance from the collision point, remains unchanged for the different thicknesses. However, wavelength and wave amplitude are observed to decrease for thinner base plates, while the incubation distance for stable wave generation is observed to increase with decreasing base plate thickness. Previously suggested mechanism for explosive wave generation are analysed in terms of the present experiments. It is apparent that most of the theories proposed to date for wave formation are in one way or another incomplete.     

Stress-Strain State and Energy Fluxes in a Plate Containing a Stationary Crack under Impulsive Loading

We study the stress-strain state and energy fluxes in a continuous infinite plate and in a plate containing a stationary crack of finite length under impulsive loading. For this purpose, we develop a model of formation of the zone of elevated stresses and the zone of unloading as well as a numerical procedure for the evaluation of their parameters. We deduce analytic expressions for the quantitative analysis of the stress-strain state and energy fluxes in the plate near the tip of the stationary crack under arbitrary impulsive loading. The theoretical results are compared with the experimental data on the initiation of main cracks in plane specimens made of solid polymers. We formulate a quantitative criterion for the difference between the dynamic and quasistatic loading of a crack and show that the dynamic and static fracture processes can be described within the framework of a single approach.     

On modelling cracks in orthotropic plates under biaxial loading: synthesis and summary

The model of a crack with a process zone is considered and generalized to orthotropic materials. It is assumed that a material in the process zone satisfies a strength condition of arbitrary form. Based on the crack model, the fracture of an orthotropic cracked plate under biaxial loading is studied. The crack is directed along one of the anisotropy axes with external loads being applied in parallel and perpendicularly to it. The influence of the biaxiality of external loading on the critical state of the cracked plate is analysed within the framework of the critical crack opening displacement and critical J ‐integral criteria. Numerical solution is obtained using the Mises‐Hill and Gol’denblat‐Kopnov strength criteria. Theoretical results are compared with experimental data obtained by testing specimens made of structural metals.     

The realization of long focal depth with a linear varied-area zone plate

We report a linear varied-area zone plate, in which arbitrary long focal depth can be achieved by properly adjusting the corresponding parameters. Meanwhile, the lateral focal spot and side lobes can be kept very small. Numeral simulations are carried out to verify the performance of our zone plate through Fresnel–Kirchhoff diffraction theory, and the results are in good accord with the experimental verifications. The influences of our zone plate’s parameters to the intensity distribution in focal region are discussed in detail. Comparisons are made with the behaviour of a linear varied-line-space grating, and we find that the behaviour of our novel zone plate along optical axis is just like a reverse transformation of the focusing behaviour of a linear varied-line-space grating.     

有限宽偏心裂纹板在裂纹面受两对集中拉力作用时裂纹线的弹塑性解析解

在实际中偏心裂纹板的受力问题比中心裂纹板受力问题更为普遍。利用裂纹线场分析法简化了弹塑性断裂力学问题的复杂性和数学上的困难,求得了偏心裂纹板在裂纹面上受两对集中拉力作用时裂纹线附近弹塑性边界上的单位法向量、裂纹线附近的弹塑性应力场以及裂纹线上的塑性区长度随荷载的变化规律。在理想弹塑性情况下,该文中的理论解在裂纹线场附近是足够精确的。     

Arbitrary Lagrangian–Eulerian finite element analysis of strain localization in transient problems

Non-local models guaranty that finite element computations on strain softening materials remain sound up to failure from a theoretical and computational viewpoint. The non-locality prevents strain localization with zero global dissipation of energy, and consequently finite element calculations converge upon mesh refinements to non-zero width localization zones. One of the major drawbacks of these models is that the element size needed in order to capture the localization zone must be smaller than the internal length. Hence, the total number of degrees of freedom becomes rapidly prohibitive for most engineering applications and there is an obvious need for mesh adaptivity. This paper deals with the application of the arbitrary Lagrangian–Eulerian (ALE) formulation, well known in hydrodynamics and fluid–structure interaction problems, to transient strain localization in a non-local damageable material. It is shown that the ALE formulation which is employed in large boundary motion problems can also be well suited for non-linear transient analysis of softening materials where localization bands appear. The remeshing strategy is based on the equidistribution of an indicator that quantifies the interelement jump of a selected state variable. Two well known one-dimensional examples illustrate the capabilities of this technique: the first one deals with localization due to a propagating wave in a bar, and the second one studies the wave propagation in a hollow sphere.     

Theory of stress coining technique

This paper describes the process of stress coining that improves fatigue performance at the hole. In stress coining, residual compressive stresses are induced at the edge of the hole, the elastic-plastic boundary in the plate material is located slightly away from the hole, the minimum and maximum pressures which establish the residual stresses are determined, the residual stresses and their distributions in the partially elastic zone and partially plastic zone are analyzed, and the minimum and maximum mandrel sizes are determined as well as the lower and upper limits of the mandrel-hole interference in the mandrel design. Numerical solutions are given for varieties of different mandrel materials, plate materials, and hole sizes.     

强动载荷下焊接钢板力学性能及本构模型研究

为研究强动载荷下船用焊接钢板的力学性能。开展了典型船用焊接钢板母材、焊缝和热影响区的准静态拉伸试验、高温拉伸试验及SHPB动态压缩试验,分析了焊接钢板材料在不同应力状态下的力学行为,基于力学性能试验结果拟合了焊接钢板母材、焊缝和热影响区材料的本构模型。结果表明:准静态条件下,与母材相比,焊缝和热影响区材料的屈服强度与抗拉强度偏大,延伸率偏小;高应变率下,热影响区材料抵抗塑性变形的能力明显强于其他两种材料,且随着应变率的增加抵抗塑性变形的能力呈增强趋势;焊接板母材、焊缝与热影响区材料均表现出应变率效应和温度效应;热影响区是焊接板抗冲击性能相对薄弱的区域。建立的Johnson-Cook模型可以描述强动载荷下焊接钢板的力学性能。     

Generalized Zone Plates Focusing Light into Arbitrary Line Segments

Abstract

A new computer-generated generalized zone plate is proposed, which is able to concentrate the incident beam into a line segment of various length and arbitrary inclination to the optical axis, as well as unrestricted distribution of the longitudinal intensity. The approach is based on the energy conservation principle and equations of the paraxial ray tracing, both considered in the cartesian coordinate system. Hyberbolic, elliptic, linear, and conical zone plates are shown to be limiting cases of the proposed element. Other specific cases, e.g. a new element focusing into a segment of the optical axis, are also derived.     

On the propagation of waves in layered anisotropic media in generalized thermoelasticity

In view of the increased usage of anisotropic materials in the development of advanced engineering materials such as fibers and composite and other multilayered, propagation of thermoelastic waves in arbitrary anisotropic layered plate is investigated in the context of the generalized theory of thermoelasticity. Beginning with a formal analysis of waves in a heat-conducting N-layered plate of an arbitrary anisotropic media, the dispersion relations of thermoelastic waves are obtained by invoking continuity at the interface and boundary conditions on the surfaces of layered plate. The calculation is then carried forward for more specialized case of a monoclinic layered plate. The obtained solutions which can be used for material systems of higher symmetry (orthotropic, transversely isotropic, cubic, and isotropic) are contained implicitly in our analysis. The case of normal incidence is also considered separately. Some special cases have also been deduced and discussed. We also demonstrate that the particle motions for SH modes decouple from rest of the motion, and are not influenced by thermal variations if the propagation occurs along an in-plane axis of symmetry. The results of the strain energy distribution in generalized thermoelasticity are useful in determining the arrangements of the layer in thermal environment.     

Plastic zone near the crack tip in a material with strain anisotropy caused by loading along spatial rectilinear trajectories

The equation of the boundary of the plastic zone near the tip of a mode I crack is deduced for the case of a plate made of a material with strain anisotropy. It is assumed that the anisotropy is caused by hardening in the process of plastic deformation performed prior to the appearance of the crack under loading along arbitrary rectilinear trajectories in the space of the stress tensor. An analysis of this equation demonstrates that the main factors affecting the shape and size of the plastic zone are the level of plastic strains accumulated in the process of preloading, their sign, and the orientation of the crack relative to the axes of anisotropy.     

On Determination of the Stressed State of a Annular Orthotropic Plate

We propose a method of solving a plane problem for thin orthotropic plates using a parameter expansion technique. A solution to the respective isotropic problem serves as a null approximation. For particular materials, the results obtained by this method are shown to agree well with the known solution for an infinite plate with a hole. We study the stressed state of an annular orthotropic plate, derive the stress distribution functions, and compare the results obtained with those available for a similar isotropic plate.     

A nonlinear theory of wake development

Summary A simple new series, using an expansion of the velocity profile in parabolic cylinder functions, has been developed to describe the nonlinear evolution of a steady, laminar, incompressible wake from a given arbitrary initial profile. The first term in this series is itself found to provide a very satisfactory prediction of the decay of the maximum velocity defect in the wake behind a flat plate or aft of the recirculation zone behind a symmetric blunt body. A detailed analysis, including higher order terms, has been made of the flat plate wake with a Blasius profile at the trailing edge. The same method yields, as a special case, complete results for the development of linearized wakes with arbitrary initial profile under the influence of arbitrary pressure gradients. Finally, for purposes of comparison, a simple approximate solution is obtained using momentum integral methods, and found to predict satisfactorily the decay of the maximum velocity defect.     

Long Range Detection of Defects in Composite Plates Using Lamb Waves Generated and Detected by Ultrasonic Phased Array Probes

This article presents a technique for the generation and detection of Lamb waves guided along large plate-like structures made from various types of materials (metal, polymer, fibre-reinforced composite, etc.). A multi-element matrix ultrasonic probe is driven using the well-known phased array principle, for launching and detecting pure Lamb modes in/from specific directions along the plate, which are arbitrary for isotropic materials and limited to specific directions for anisotropic materials, e.g. principal directions or directions for which both phase and group velocities are collinear. The probe is gel-coupled to the tested specimen and allows quick inspection of large area from its fixed position, even of zones with limited access. The technique, which takes into account the frequency dispersive effects, is different than SHM-like (Structural Health Monitoring) inspection, since all transmitting or receiving elements are grouped together in a localized area defined by the active surface of the probe, and not permanently attached to the tested structure. The use of a multi-element probe, for long range Lamb waves-based inspection, is also distinctive from that usually performed, which consists of very local inspection of a material by steering the ultrasonic beam below and nearby the probe. A prototype is presented, as well as measurements of its performances in terms of modal selectivity and directivity. Finally the detection and localisation of a through-thickness hole in a large aluminium plate, of a delamination-like defect in a carbon epoxy composite plate and of an impact damage on a stiffened composite curved plate are shown.     

Stress analysis for bonded materials with a graded interfacial layer under a rigid punch

It is well known that functionally graded materials can be used to eliminate the stress discontinuity that is often encountered in multilayer composites. In this article, the stress analysis for the coating-functionally graded interfacial layer-substrate structure under a rigid spherical punch is investigated. A linear multi-layer model is used to model the graded interfacial layer with arbitrary varying materials properties along the thickness direction. The spherical indentation problem is formulated in terms of a singular integrate equation with the method of transfer matrix and Hankel integral transform technique. The stress components in the coating-graded interfacial layer-substrate structure are calculated by solving the equation numerically. The results show that stiffens ratio and the gradient index of the graded interfacial layer has a significant effect on the distribution of stress components.     

Numerical limit analysis of perforated plates

Full limit analysis is preseted of a thin plate under plane stress, perforated with circular holes arranged in a regular penetration pattern. The material is assumed to be elastic–perfectly plastic and to obey the Huber–Mises yield condition together with its associated flow law. The finite element tangential stiffness method is employed. Triangular constant-strain elements are used, the load being generated by means of constant edge displacements at the periphery of a suitably selected subregion. Load-displacement diagrams are obtained for various edge displacement programs and then an interaction curve is constructed for the perforated plate considered. Each computational step is associated with a certain plastic zone which develops until the limit state configuration is reached. The plate is provided with equilateral triangular configuration of holes with arbitrary cutout coefficients. The interaction curves are plotted against the Huber–Mises yield condition for the plate without openings.     

Temperature, stress and microstructure in 10Ni5CrMoV steel plate during air–arc cutting process

Although the air–arc cutting process has been widely used in the material processing engineering, little information about temperature, stress and microstructure in the plate air–arc cut is known. A three-dimensional finite element model including the material removal and the thermal effect of the arc is developed to study the temperature and stress fields of 10Ni5CrMoV steel plate during air–arc cutting process in this paper. The microstructures and micro-mechanical properties of the parts near the groove especially in heat affected zone (HAZ) are studied by experimental methods, and they also can be used as a method to verify the numerical results. Effects of stresses induced by air–arc cutting process on the initial residual stress fields of base materials are also researched. The results show that the cooling velocity in HAZ is higher than the one of the welding process for the same base material, and the zone with high temperature is very narrow, which means that the temperature gradients near the groove are very steep during the air–arc cutting process; this special temperature field depresses multiphase transformations and coarse microstructures. The evolution of the stress during the air–arc cutting is described, and the numerical results indicate that the characteristics of the evolution of stresses and the residual stresses distribution in the plate in air–arc cutting process seem to be similar to the ones of the butt welding for flat plates. The influences of air–arc cutting process on initial stress fields present two aspects: thermal effect and material removal effect, and the former plays a primary role. Both numerical temperature and stress fields are compared with the experimental ones. It is very important for researchers to clarify the temperatures, stresses and microstructures in the plate during air–arc cutting process, and understand fully the mechanism of influences of air–arc cutting on the plate; it is also very valuable for engineering application of the air–arc cutting process.     

Fracture of coated plates and shells under thermal shock

The main interest in this study is in the subcritical crack propagation and fracture of coated materials, specifically of cylindrical shells under repeated thermal shock. First it is shown that the circumferential crack problem in a cylindrical shell may be approximated by a plate on an elastic foundation under plane strain conditions. The thermal shock problem for a layered plate supported by an elastic foundation containing a crack in each layer of arbitrary sizes and locations is then considered. An additional factor studied is the influence of the cooling rate of the plate surface on the stress intensity factors at the crack tips. The problem is formulated in terms of a pair of singular integral equations which are solved for a number of typical crack geometries such as an edge crack, a crack terminating at the interface, an undercoat crack, and a crack crossing the interface. The main results of this paper are the stress intensity factors.     

An exact solution for an anisotropic plate with an elliptic hole under arbitrary remote uniform moments

The problem of an anisotropic plate containing an elliptic hole subjected to remote bending or twisting moments is considered. In contrast with the previous works on the problem, the requirement that the deflection be a single-valued function is satisfied by introducing a correction constant. An exact solution for general anisotropic materials under arbitrary uniform loading conditions is derived. Explicit expressions for the deflection and moments on the edge of an elliptic hole in an orthotropic plate subjected to bending or twisting moments are obtained. The moment intensity factors as the elliptic hole degenerates into a crack are given.