Different Optimization Models for Crack-Free Laser Welding

Conventional laser beam welding of aluminum alloys often leads to hot cracking. This is caused by a complex process where thermo-mechanical and metallurgical aspects are involved; cf. [3], [2]. A possibility to prevent hot crack initiation yields the multi-beam welding technique (cf. [2]), where additional laser beams are led parallelly besides the main laser beam. There by optimal positions, sizes, and powers of the additional laser beams play an important role otherwise hot cracking can even be enhanced. In [1], [4], resp., a mechanical 1D and thermal 2D model of hot cracking was derived. It provides the basis for different formulations of constrained nonlinear programming problems to identify the optimal parameters of the additional laser beams. In the present paper a comparison between these formulations and between two different optimizers for the so far best formulation are presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

FE‐formulation for plate and beam elements to simulate onset and propagation of discrete cracks in solid structures

The aim of this paper is to present some numerical aspects related to the modeling both the formation and the propagation of discrete cracks in solid structures. The presented formulation corresponds to the concept of embedded discontinuities [1], and will be applied to a plate and to a beam element. The failure of solid structures is often triggered by a highly localized pattern of inelastic deformation in the form of narrow bands. Characteristic examples are shear bands in metals and soils, or localized bands of cracking in brittle materials, like concrete or rocks. A well known difficulty associated with classical (local, rate‐independent) continuum theories with strain softening attributes is that numerical solutions are found to lack invariance with respect to the choice of spatial discretization. For quasi‐static boundary problems, this mathematical inconsistency causes the loss of ellipticity for the governing equations (material instability). To regularize this inconsistency, several strategies have been applied. In the presented formulation, additional degrees of freedom are considered. Within the concept of embedded discontinuities, the regular displacements are enriched by discontinuities. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupling effect of thickness and shear deformation on size-dependent bending of micro/nano-scale porous beams

A unified nonlocal strain gradient beam model with the thickness effect is developed to investigate the static bending behavior of micro/nano-scale porous beams. Size-dependent governing equations and corresponding analytical solutions for the bending of hinged-hinged beams are obtained by employing minimum total potential energy principle, the Navier solution method as well as the variational-consistent boundary conditions. For nonlocal strain gradient theory (NSGT) with thickness effect, virtual strain energy function of shear beams can contain additional nonlocal shear stress and high-order nonlocal shear stress related to the thickness direction in comparison with that of Euler–Bernoulli beam, so the coupling of the shear and thickness effects should be drawn huge attention. By means of detailed numerical analysis, it is found that, the stiffness-hardening effect is underestimated in NSGT without the thickness effect, and the stiffness-hardening and stiffness-softening effects of NSGT with the thickness effect can be not only length-dependent but also thickness-dependent. Interestingly, the generalized Young’s modulus depends on half-wave number, which means that the generalized Young’s modulus may be different due to applied load types. In the context of NSGT with the thickness effect, the deflection of Euler–Bernoulli beam predicted is smaller than that of shear beam, especially for thick beams. Furthermore, porosities distributed in the top or bottom of beams can possess a greater influence on the decrease of overall stiffness of beam than those distributed in the vicinity of the middle plane of beams.     

An electromechanically coupled intrinsic,mixed variational formulation for geometrically nonlinear smart composite beam

The work presented in this article is the outcome of a combined strategy of a mathematical tool for 2D cross-sectional analysis, i.e., Variational Asymptotic Method (VAM) as well as the 1D exact beam analyzer, i.e., the intrinsic mixed variational formulation for modeling and analysis of Piezoelectric-laminated composite beams. This work talks about a novel approach of mixed variational formulation to analyze a two-way electromechanically coupled piezoelectric composite beam. In a classical intrinsic mixed variational approach for a passive structure, the 1D exact beam model deals only with mechanical degrees of freedom. In the present case, an extra 1D electrical degree of freedom has been incorporated. A computational code is developed based on the present theory to solve the two-way coupled electromechanical beam problem. In the present case, we have validated the static results for sensor application. Both linear and nonlinear results have been discussed. Results obtained are very promising and are helpful in building a platform where design, optimization and nonlinear analysis of composite ‘smart’ beams in a multibody framework can be done faster while maintaining acceptable accuracy.     

关于铁木森柯梁动力优化的讨论

本文讨论受到多频率约束的铁木森柯梁和尤拉梁的最小重量设计问题.以对称的简支梁为例,本文揭示了铁木森柯梁的异常特征:如果在梁的中部适当地构造很高的一个薄条,相应于第一对称振形的频率可以高于反对称振型的频率;受到二组不同频率约束的铁木森柯梁很可能有同样的最小重量.这些异常特征说明,为了得到一个良态的问题,有必要将最大横断面积约束包括到问题提法中.     

The equivalence of the refined theory and the decomposition theorem of rectangular beams

This paper presents the decomposition theorem of rectangular beams and indicates that the general state of stress of beams can be decomposed into two parts: the interior state and the Papkovich–Fadle state (shortened form the P–F state). The refined theory of beams is derived by using Papkovich–Neuber solution (shortened form the P–N solution) and Lur’e method without ad hoc assumptions. It is shown that the displacements and stresses of the beam can be represented by the angle of rotation and the deflection of the neutral surface. Based on the refined beam theory, the exact equations for the beam without transverse surface loadings are derived and consist of two governing differential equations: the fourth-order equation and the transcendental equation. It is then proved that the refined beam theory and the decomposition beam theorem are equivalent, i.e., the fourth-order equation and the transcendental equation are equivalent to the interior state and the P–F state, respectively.     

A PIC method for solving a paraxial model of highly relativistic beams

Solving the Vlasov–Maxwell problem can lead to very expensive computations. To construct a simpler model, Laval et al. [G. Laval, S. Mas-Gallic, P.A. Raviart, Paraxial approximation of ultrarelativistic intense beams, Numer. Math. 69 (1) (1994) 33–60] proposed to exploit the paraxial property of the charged particle beams, i.e the property that the particles of the beam remain close to an optical axis. They so constructed a paraxial model and performed its mathematical analysis. In this paper, we investigate how their framework can be adapted to handle the axisymmetric geometry, and its coupling with the Vlasov equation. First, one constructs numerical schemes and error estimates results for this discretization are reported. Then, a Particle In Cell (PIC) method, in the case of highly relativistic beams is proposed. Finally, numerical results are given. In particular, numerical comparisons with the Vlasov–Poisson model illustrate the possibilities of this approach.     

A transfer matrix method capable of determining the exact solutions of a twisted Bernoulli–Euler beam with multiple edge cracks

In this paper, an improved numerical method is developed to obtain the accurate natural frequencies and mode shapes for the coupled bending vibrations of a twisted Bernoulli–Euler beam with multiple edge cracks, and this method can compute the desired number of natural frequencies by dividing the minimum number of subdivisions for a whole structural element with multiple open edge cracks. The development of a method that can simply and accurately compute the variation of the natural frequencies due to the effect of cracking is possible using the distributed mass, transcendental function, and local coordinate systems varying along the length of a twisted beam. Because the in-plane and out-of-plane bending stiffnesses are coupled in two principal planes by the effect of twisting, each crack is modeled as rotational springs in the in-plane and out-of-plane directions. With these assumptions, the effect of cracking for twisted beams is investigated using a parametric study for the various crack depths and locations.     

Validation and Limits of Finite Inflatable Beam Elements

Although nowadays inflatable tubular beams are often used in the field of civil engineering, by now there are only few publications dealing with finite deformation inflatable beam elements, see e.g. [1], [2] and [3]. All formulations of inflatable beams have several assumptions in common, as constant cross sections throughout the deformation, a constant internal gas pressure and the negligence of circumferential stresses. These assumptions have to be validated either by experiments or numerical analysis. In the current contribution beam–like structures are investigated using a finite element shell or membrane formulation and featuring a volume dependent gas loading, see e.g. [5] and [4]. In general the formulation substitutes the internal gas pressure by an energetically equivalent volume dependent loading and thus enables to check for potential gas pressure changes during the deformation process of the inflated beam as a consequence of volume changes. Further local deformations as occurring in the vicinity of supports or almost single loads can be considered. In this paper the focus will be only on the initial assumption of the beam theory that the biaxial stress state is neglected. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An exact transfer matrix expression for bending vibration analysis of a rotating tapered beam

This paper presents a transfer matrix expression that can be used to determine the eigenpairs of a rotating beam with a cross section height that linearly decreases along the length of the beam element. The proposed method considers the effect of centrifugal force, including the effects of the axial force, hub radius, and taper ratio. Differential equations are solved for the in-plane bending vibration using the Frobenius method for a power series. The effect of the rotational speed on the eigenpairs of a rotating tapered beam is first investigated, followed by an examination of the contribution rates of the bending strain and additional strain energies generated by centrifugal forces for each mode by analyzing the variation of the energies computed from the strain and kinetic energies. To compute these contribution rates, we used a shape function that was defined by the displacement at both ends of the beam elements. The effect of tapering on the eigenfrequencies of the transverse vibration of rotating beams is analyzed by using various examples, and the contribution rates are examined by using taper ratios of 0 and 0.5.     

Computational Material Forces in Micromorphic Continua

The micromorphic continuum theory is used to describe materials with significant microstructure which thus exhibit scaledependence (see e.g. [1], [2] [3]). Microcontinua are assumed to be attached to each physical point and may experience both stretch and rotation which are affine throughout the microcontinuum, nevertheless kinematically independent from the deformation on the macroscale. The additional kinematical quantities which account for the micro-deformation yield additional stresses and contributions to the balance of momentum. Additionally to the common finite-element approximation which here is a coupled problem to be solved for macro- and the micro-quantities, we apply the method of material forces, cf. [4], [5]. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A theory of heating of Voigt solids and fluids by external energy sources and flame theory

The purpose of this paper is to develop

1. a theory of laser stimulated vaporization of droplets,

2. a theory of internal heating resulting from vibration waves in linearly responding elastic material, and

3. flame theory.

There are applications to sending information through clouds on laser beams and to the control of temperature in ultrasonic welding, and improvement of the design of aircraft engines and the processes used for the destruction of toxic chemicals.

We develop a theory of thermal excursions resulting from ultrasonic welding in 3 and 7 dimensions, and interpret it as an elastic interaction with damping in a Voigt solid. It is hypothesized that with good control of temperature, one could achieve strong and uniform welds by this process and greatly reduce the cost of manufacturing aircraft, and other aluminum structures. We consider equations describing the conservation of mass, momentum, and energy coupled by an equation of state, and consider general mass, momentum, and energy transfer relationships in a compressible body subjected to external stimuli. For the Voigt solid theory, a linear elastic theory with damping forces, we show how some simple local time averaging gives us a dovetailed system consisting of the elastic wave equations whose solution provides the source term for an otherwise uncoupled heat equation. For the more general theory of droplet vaporization, we illustrate a general nonlinear energy equation which includes a radiation energy conductivity term. We get a class of exact solutions for a nonlinear flame front boundary value problem.     

Modeling of thermal phenomena in single laser beam and laser-arc hybrid welding processes using projection method

This paper concerns mathematical and numerical modeling of thermal phenomena accompanying single laser and laser-arc hybrid butt welding of steel sheets. Coupled heat transfer and fluid flow in the fusion zone were described respectively by transient heat transfer equation and Navier–Stokes equation. Laser beam and electric arc heat sources were modeled using different heat source power distributions. Latent heat associated with the material’s state changes, buoyancy forces and liquid material flow through a porous medium were taken into account in considerations. Differential governing equations were numerically solved using projection method combined with finite volume method. Elaborated solution algorithm was implemented into computer solver used for simulation of heat transfer and fluid flow during welding. The geometry of the weld and heat affected zone as well as cooling rates were estimated on the basis of numerically obtained temperature field.     

A model for resistance welding including phase transitions and Joule heating

In this paper, we introduce a new model for solid–liquid phase transitions triggered by Joule heating as they arise in the case of resistance welding of metal parts. The main novelties of the paper are the coupling of the thermistor problem with a phase‐field model and the consideration of phase‐dependent physical parameters through a mixture ansatz. The PDE system resulting from our modeling approach couples a strongly nonlinear heat equation, a non‐smooth equation for the the phase parameter (standing for the local proportion of one of the two phases) with a quasistatic electric charge conservation law. We prove the existence of weak solutions in the three‐dimensional (3D) case, whereas the regularity result and the uniqueness of solution is stated only in the two‐dimensional case. Indeed, uniqueness for the 3D system is still an open problem. Copyright © 2011 John Wiley & Sons, Ltd.     

Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories

This paper investigates the linear free vibration of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs). Two types of CNT reinforced beams, namely uniformly distributed CNT reinforced (UD-CNT) beams and functionally graded CNT reinforced (FG-CNT) beams, are considered. It is assumed that the SWCNTs are aligned along the beam axial direction and the distribution of the SWCNTs may vary through the thickness of the beam. The virtual strain and kinetic energies of the FG-CNT composite beam are obtained using the classic variational method of Hamilton’s principle and then solved by the p-Ritz method. Vibration frequency parameters for the FG-CNT beams based on the first order and third order beam theories are presented and the effects of CNT filler volume fraction, distribution, beam span to depth ratio and end support conditions on the free vibration characteristics of the beams are discussed. Comparison studies for UD-CNT and FG-CNT beams based on the first order and the third order beam theories are also performed and the differences in vibration frequencies between these two theories are highlighted.     

Reducing of birefringence of [111]-cut crystal rod using side-pumping and suitable crystal rotation

The beam quality and output power of high power solid-state lasers is influenced by birefringence. Inhomogeneous distribution of the thermal field inside the laser crystal rod occurs due to non-uniform absorption of the pump light inside the crystal and a heat sink only at boundaries. Due to the photoelastic effect, this distribution leads to inhomogeneous thermal strains and birefringence inside the rod. Plane stress and plane strain assumptions for an axially symmetric pumped crystal have been used formerly for analytical models for calculating the birefringence. This model leads in case of an [111]-cut crystal to an axially symmetric birefringence pattern. However, the shear strains in the axial-radial plane are neglected in this former models using plane stress and plane strain assumptions. This shear strains are taken into account by full 3D numerical calculations. A threefold symmetry pattern due to the anisotropic behaviour of the photoelastic tensor, which is contradictory to the ideal use of a radial or azimuthal polarized beam, is shown by results of the birefringence simulation. A laser rod pumped at three sides with threefold symmetry is analysed in order to reduce the effect of birefringence. In this case the absorption is not axially symmetric anymore. Within the crystal in regions where pumping is stronger, the pump light absorption and consequently the temperature, the strains and birefringence are higher. The degree of three-fold symmetry of birefringence will be reduced, if the region having a low birefringence due to the photoelastic effect is more strongly pumped than the rest of domain. This means the birefringence is affected by the rotation of crystal around its [111]-axis. By an optimal rotation with respect to the edges of the crystal, smallest birefringence can be obtained. For generating radial or azimuthal polarizations, the output beam of this laser device is therefore more suitable. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experimental Study on the Flexural Ductility of BFRP Bar Concrete Beams With Bamboo Fiber and Steel Wire Mesh北大核心CSCD

To study the effects of bamboo fiber and steel wire mesh on the flexural ductility of basalt fiber reinforced polymer（BFRP）bar concrete beams, 7 BFRP bar concrete beams with bamboo fiber and steel wire mesh were tested with different bamboo fiber lengths (0 mm, 30 mm and 45 mm) and different steel wire mesh layout ranges (0, 1/2 maximum bending moment point layout and full beam length layout). The flexural failure tests of the 7 beams were carried out, and the initial crack loads, the crack developments, the ultimate loads and the deformations were detected. The effects of the fiber length and the wire mesh layout range on the crack resistance and the deformation resistance of the specimens were analyzed based on the test data. With the function model, the equivalent yield points of the 7 test beams were obtained, and their ductility coefficients were calculated. The results show that, the addition of bamboo fiber and steel wire mesh increases the cracking loads of BFRP bar concrete beams by 12%~68%, decreases the crack spacings and the crack length development speed, reduces the test beam deformation under the same load, and increases the ductility coefficient by 1.58%~31.75%. © 2023 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Nonlinear bending and buckling for strain gradient elastic beams

Nonlinear bending of strain gradient elastic thin beams is studied adopting Bernoulli–Euler principle. Simple nonlinear strain gradient elastic theory with surface energy is employed. In fact linear constitutive relations for strain gradient elastic theory with nonlinear strains are adopted. The governing beam equations with its boundary conditions are derived through a variational method. New terms are considered, already introduced for linear cases, indicating the importance of the cross-section area, in addition to moment of inertia in bending of thin beams. Those terms strongly increase the stiffness of the thin beam. The non-linear theory is applied to buckling problems of thin beams, especially in the study of the postbuckling behaviour.     

Determination of the Transverse Shear Stress in an Euler-Bernoulli Beam Using Non-Admissible Virtual Displacements

The point of view that a beam can be considered as a three-dimensional continuum with a constrained position field (cf. [1]) together with the virtual work principle and the concept of perfect constraint stresses, leads to a systematic way to reduce the equilibrium equations of the continuous body to an ordinary differential equation describing the constrained displacement field of the beam. Using virtual displacements, being non-admissible with respect to the constrained beam kinematics, together with the solution of the boundary value problem, allows us to analytically determine the constraint stresses and consequently the total stresses of a beam up to a certain indeterminacy. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)