A variational model for stress analysis in cracked laminates with arbitrary symmetric lay-up under general in-plane loading

The present research work presents a variational approach for stress analysis in a general symmetric laminate, having a uniform distribution of ply cracks in a single orientation, subject to general in-plane loading. Using the principle of minimum complementary energy, an optimal admissible stress field is derived that satisfies equilibrium, boundary and traction continuity conditions. Natural boundary conditions have been derived from the variational principle to overcome the limitations of the existing methodology on the analysis of general symmetric laminates. Thus, a systematic way to formulate boundary value problem for general symmetric laminates containing many cracked and un-cracked plies has been derived, and appropriate mathematical tools can then be employed to solve them. The obtained results are in excellent agreement with the available results in the literature. In the field of matrix cracks analysis for symmetric laminates, the present formulation is the most complete variational model developed so far.     

Stiffness reduction of cracked general symmetric laminates using a variational approach

In this paper, stiffness reduction of general symmetric laminates containing a uniform distribution of matrix cracks in a single orientation is analyzed. An admissible stress field is considered, which satisfies equilibrium and all the boundary and continuity conditions. This stress field has been used in conjunction with the principle of minimum complementary energy to get the effective stiffness matrix of a cracked general symmetric laminate. Natural boundary conditions have been derived from the variational principle to overcome the limitations of the existing variational methods on the analysis of general symmetric laminates. Therefore, the capability of analyzing cracked symmetric laminates using the variational approach has been enhanced significantly. It has been shown that the method provides a rigorous lower bound for the stiffness matrix of a cracked laminate, which is very important for practical applications. Results derived from the developed method for the properties of the cracked laminates showed an excellent agreement with experimental data and with those obtained from McCartney’s stress transfer model. The differences of the developed model with McCartney’s model are discussed in detail. It can be emphasized that the current approach is simpler than McCartney’s model, which needs an averaging procedure to obtain the governing equations. Moreover, it has been shown that the existing variational models are special cases of the current formulation.     

含界面裂纹的不可压缩双金属胶接件的解析变分－守恒积分解法

本文根据平面问题的复变函数理论推导了含界面裂纹双金属胶接件满足微分方程、开裂界面边界条件与未开裂界面连续条件的应力与位移本征函数展开式，并建立了不可压缩双金属界面裂纹的复合型守恒积分及其与应力强度因子之关系，进而利用分区广义变分原理满足其余边界条件确定包含应力强度因子在内的展开式系数，得到守恒积分并求出应力强度因子．数值计算表明，沿不同回路的在恒积分具有很好的守恒性而且由这两种方法所得应力强度因子具有很好的一致性．     

Convective stability analysis of a micropolar fluid layer by variational method

This paper studies Rayleigh-Bénard convection of micropolar fluid layer heated from below with realistic boundary conditions. A specific approach for stability analysis of a convective problem based on variational principle is applied to characterize the Rayleigh number for quite general nature of bounding surfaces. The analysis consists of replacing the set of field equations by a variational principle and the expressions for Rayleigh number are then obtained by using trial function satisfying the essential boundary conditions. Further, the values of the Rayleigh number for particular cases of large and small values of the microrotation coefficient have been obtained. The effects of wave number and micropolar parameter on the Rayleigh numbers for onset of stationary instability for each possible combination of the bounding surfaces are discussed and illustrated graphically. The present analysis establishes that the nature of bounding surfaces combination and microrotation have significant effect on the onset of convection.     

DRIVING FORCE AND DEFORMATION ANALYSIS FOR DYNAMIC CRACK PROPAGATION IN GAS PIPELINES UNDER DIFFERENT BOUNDARY CONDITIONS

In the light of a growing need for fracture control of rapid crack propagation(RCP)ingas pipelines,a program PFRAC(Pipeline FRacture Analysis Code)has been developed to analysethc various events.In this paper,by using PFRAC for the simulation of axial crack propagation in gaspipelines,a number of dynamic analysis issues rclating to boundary effects for uncracked and crackedpipes are investigated.This indicates that the boundary conditions along the length of the pipe play animportant role for fracture analysis in the pipe.In contrast.the boundary condutions at thc ends of along pipeline have little effect on the dynamic fracture events.     

Variational formulation of a rod under torsional vibration for crack identification

A variational formulation is developed for the torsional vibration of a cylindrical shaft with a circumferential crack. The work is compared with existing methods. The Hu–Washizu–Barr variational formulation was used to develop the differential equation and the boundary conditions of the cracked rod. The general variational principle and the independent assumptions about the displacement, the momentum, the strain and the stress fields of the cracked rod, and the equations of motion for a uniform rod in torsional vibration, are derived. The crack was modelled as a continuous flexibility using the displacement field in the vicinity of the crack, found with fracture mechanics methods. Rayleigh quotient was used to approximate the natural frequencies of the cracked rod. Independent evaluations of crack identification methods in rotating shafts are reported and compared with methods using the continuous crack flexibility theory.     

A step-wise variational approach to elastic-plastic analysis by boundary/interior elements

The evolutive elastic-plastic analysis problem is addressed by a time/space discretization procedure via energy methods. Using the maximum plastic work theorem in conjuction with an assumed material evolutive model, the initial/boundary value problem of rate-form plasticity is transformed into a sequence of boundary value problems of deformation-theory plasticity (weighted step problems). The main purpose of the paper is to establish a couple of boundary/domain variational principles of deformation-theory plasticity, namely a constrained stationarity principle and a saddle-point one, each of which is able to characterize the solution of the typical step problem. Through suitable space discretizations by boundary and interior elements, the aforementioned principles are then given algebraic forms and the related Kuhn-Tucker conditions are shown to provide the symmetric sign-definite solving equation system for the fully discretized step problem.     

Damage analysis by boundary element method

The response of cracked bodies subjected to loading was investigated by the boundary element method in this paper. The two-law elastic-cohesive-softening model was used for crack propagation analysis. The interface conditions for uncracked, craze, open crack, adhesive crack and slid crack parts were discussed and the corresponding incremental iteration algorithm was given. A simplified damage propagation model was presented. The technique has been applied to some specific examples which give the evidence that the method is satisfactory and efficient.     

Generalized variational principles for boundary value problem of electromagnetic field in electrodynamics

An expression of the generalized principle of virtual work for the boundary value problem of the linear and anisotropic electromagnetic field is given. Using Chien＇s method, a pair of generalized variational principles （GVPs） are established, which directly leads to all four Maxwell＇s equations, two intensity-potential equations, two constitutive equations, and eight boundary conditions. A family of constrained variational principles is derived sequentially. As additional verifications, two degenerated forms are obtained, equivalent to two known variational principles. Two modified GVPs are given to provide the hybrid finite element models for the present problem.     

A new method of analyzing surface cracks

The authors have developed a new line-spring boundary element method in thepresent paper,which combines the advantage of the line-spring model with that of theboundary element method.This method reduces the three-dimension problem of thesurface cracks into a quasi-one-dimension problem and can be used to analyze thesurface cracked plate under various loading conditions.In this paper theoreticalanalyses and numerical verifications are carried out.The calculated results arereported,which indicate that the present method is efficient and can be used to analyzethe surface crack problem on a personal computer.     

The microbuckling failure of Dyneema~? composites under compression

Two grades of Dyneema~?composite laminates with the commercial designations of HB26 and HB50 were cut into blocks with or without an edge crack and compressed in the longitudinal fiber direction. The cracked and uncracked specimens show similar compressive responses including failure pattern and failure load. The two grades of Dyneema~? composites exhibits different failure modes: a diffuse, sinusoidal buckling pattern for Dyneema~? HB50 due to its weak matrix constituent and a kink band for Dyneema~? HB26 due to its relatively stronger matrix constituent. An effective finite element model is used to simulate the collapse of Dyneema~? composites, and the sensitivity of laminate compressive responses to the overall effective shear modulus, interlaminar shear strength, thickness and imperfection angle are investigated. The change of failure mode from kink band to sinusoidal buckling pattern by decreasing the interlaminar shear strength is validated by the finite element analyses.     

应变梯度Mindlin板边值问题的讨论

实验和分子动力学计算结果表明,当材料/结构的特征尺寸降为微纳米量级时,他们将表现出明显的尺度效应,因此能否建立精确表征其力学行为的连续介质力学模型具有重要的理论和现实意义.尽管现有文献对非经典Mindlin板的力学行为进行了大量研究,但该模型的变分自洽的边值问题是近年来未攻克的科学问题之一.基于简化的应变梯度理论给出了各向同性Mindlin板应变能的表达式,通过变分原理和张量分析,得到了Mindlin板变分自洽的边值问题及其对应角点条件的位移微分表达式.本文Mindlin板模型的边值问题可退化为相应的Timoshenko梁和Kirchhoff板模型的边值问题,验证了本文结果的有效性.研究结果发现,该Mindlin板模型的控制方程是一个解耦后横向振动具有12阶的偏微分方程,因此需要每个板边提供6个边界条件.角点条件由双应力(double stress)产生,并与经典的剪力、弯矩和扭矩沿截面的法向梯度有关.本文首次澄清了应变梯度Mindlin板存在角点条件这一事实,所得的变分结果有望为其有限元法和伽辽金法等数值方法提供理论依据.     

Optimal shape of a vertical rotating column

We determine the shape of the lightest rotating column that is stable against buckling, positioned in a constant gravity field, oriented along the column axis. In deriving the optimality conditions, the Pontryagin's principle was used. Optimal cross-sectional area is obtained from the solution of a non-linear boundary value problem. For this problem a variational principle and a first integral are formulated. Also a priori estimates of the cross-sectional area at the lower end are presented. The procedure is illustrated by three concrete examples. The problem treated here may be considered as a step in the dynamic optimization procedure of a heavy rotating column.     

参变量变分原理的提出、发展与应用

参变量变分原理及其参数二次规划算法是由钟万勰院士1985年针对弹性接触边界非线性问题首次提出来的,经过将近40年的不断发展,目前参变量变分原理已经成功应用于各个领域,其中包括弹塑性分析、接触问题、润滑力学、岩土力学、变刚度杆系结构、先进材料性能分析、材料的蠕变与损伤、柔性结构力学和LQ最优控制等各个工程领域。本文首先回顾了参变量变分原理的起源,介绍了参变量变分原理的基本概念,然后以弹塑性分析问题为例,阐明建立参变量变分原理的理论模型以及实现数值参数二次规划求解原理,最后详细回顾了参变量变分原理的基本理论与相应数值算法在各个领域的发展及其工程应用,展示了参变量变分原理在求解各类非线性问题的特色与优势。     

Variational principles for Cosserat body

Two variational principles are derived for the mixed boundary value problem of Cosserat solid. These principles are a generalization of the stationary principle of potential energy and the stationary principle of complementary energy from non-linear theory of elasticity.     

The application of generalized variational principle in finite element-semianalytical method

The method developed in this paper is inspired by the viewpoint in ref. [1] that sufficient attention has not been paid to the value of the generalized variational principle in dealing with the boundary conditions in the finite element method. This method applies the generalized variational principle and chooses the series constituted by spline function multiplied by sinusoidal function and added by polynomial as the approximate deflection of plates and shells. By taking the deflection problem of thin plate, it shows that this method can solve the coupling problem in the finite element-semianalytical method. Compared with the finite element method and finite stripe method, this method has much fewer unknown variables and higher precision. Hence, it proposes an effective way to solve this kind of engineering problems by minicomputer.     

On the Optimal Shape of a Compressed Rotating Rod

By using Pontryagin's maximum principle we determine the shape of the lightest compressed rotating rod, stable against buckling. It is shown that the cross-sectional area function is determined from the solution of a nonlinear boundary value problem. A variational principle for this boundary value problem is formulated and a first integral is constructed. The optimal shape of a rod is determined by numerical integration.     

Further study of the equivalent theorem of Hellinger-Reissner and Hu-Washizu variational principles

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Free torsional vibration of cracked nanobeams incorporating surface energy effects

This paper investigates surface energy effects, including the surface shear modulus, the surface stress, and the surface density, on the free torsional vibration of nanobeams with a circumferential crack and various boundary conditions. To formulate the problem, the surface elasticity theory is used. The cracked nanobeam is modeled by dividing it into two parts connected by a torsional linear spring in which its stiffness is related to the crack severity. Governing equations and corresponding boundary conditions are derived with the aid of Hamilton's principle. Then, natural frequencies are obtained analytically, and the influence of the crack severity and position, the surface energy, the boundary conditions, the mode number, and the dimensions of nanobeam on the free torsional vibration of nanobeams is studied in detail. Results of the present study reveal that the surface energy has completely different effects on the free torsional vibration of cracked nanobeams compared with its effects on the free transverse vibration of cracked nanobeams.     

Analytical solution for the cylindrical bending vibration of piezoelectric composite plates

An analytical solution for the cylindrical bending vibrations of linear piezoelectric laminated plates is obtained by extending the Stroh formalism to the generalized plane strain vibrations of piezoelectric materials. The laminated plate consists of homogeneous elastic or piezoelectric laminae of arbitrary thickness and width. Fourier basis functions for the mechanical displacements and electric potential that identically satisfy the equations of motion and the charge equation of electrostatics are used to solve boundary value problems via the superposition principle. The coefficients in the infinite series solution are determined from the boundary conditions at the edges and continuity conditions at the interfaces between laminae, which are satisfied in the sense of Fourier series. The formulation admits different boundary conditions at the edges of the laminated piezoelectric composite plate. Results for laminated elastic plates with either distributed or segmented piezoelectric actuators are presented for different sets of boundary conditions at the edges.