On the calculation of energy release rate for viscoelastic cracked laminates

IntroductionGenerallyspeaking ,acrackinviscoelasticmaterialswillgrowinsomeunknownspeedeveniftheappliedloadisquasistatic,thusthemaindifficultyisinducedindervingtheenergyreleaserate .Knowledgeoftheconditiongoverningthedelaminationincompositelaminateswithviscoelasticlayersisofparamountimportanceinpracticalapplications.Forexample,inplasticencapsulatedICpackages,theinterfacialdelaminationbetweentheSilicondieandviscoelasticepoxymoldingcompoundunderthethermalloadingisthemainfailuremodeofthestructure…     

On various variational principles in nonlinear theory of elasticity

In the present paper functionals for the various possible main variational principles in the nonlinear theory of e-lasticity are derived from the "full energy principle" and several of them are not found yet in the literatures available. Through the derivation of this paper we suggest a conjecture on the nonexistence of the eleventh and the sixth classes for the variational principles in Table 6.1 of H.C. Hu’s monograph [1].     

On patched variational principles in elasticity

In this paper we show how to use a combination of variational principles, each for a given subregion of the underlying region of space, so as to best utilize the chief benefits of the individual principles.     

The combined energy variational principles in linear elasticity

In this paper, a new kind of mixed energy variational principles in linear elasticity—the combined energy variational principles is presented. First, we define the conjugate body of an elastic body, which is obtained by changing the boundary conditions of the elastic body. Next, we decompose the conjugate body into two component-states, construct functionals of potential energy and complementary energy, respectively, for the component-states and define the additional hybrid energy between the component-states. Thus the functionals of combined energy can be constructed. Three typical combined energy variational principles are demonstrated and the other forms of combined energy variational principles are given. The application of the proposed principles to the calculation of thin plates with complicated boundaries is shown.     

Hamiltonian systems in elasticity and their variational principles

ＨＡＭＩＬＴＯＮＩＡＮＳＹＳＴＥＭＳＩＮＥＬＡＳＴＩＣＩＴＹＡＮＤＴＨＥＩＲＶＡＲＩＡＴＩＯＮＡＬＰＲＩＮＣＩＰＬＥＳＷａｎｇＺｈｉ－ｇｕｏ（王治国）（ＲｅｓｅａｒｃｈＩｎｓｔｉｔｕｔｅｏｆＶｉｂｒａｔｉｏｎＥｎｇｉｎｅｅｒｉｎｇ,ＮａｎｊｉｎｇＵｎ...     

含脱层层合板的能量释放率各型分量分析

本文是文献［１］的继续。文献［１］利用可动边界变分问题，导出了层合板内部脱层前缘各点处的总能量释放的表达式。在此基础上，本文对文献［１］的分析模型作了进一步的完善，导出了层板理论分析下的能量释放率各型分量的表达式，并对圆形脱层进行了计算分析。     

ENERGY RELEASE RATE COMPONENTS ANALYSIS OF LAMINATES WITH EMBEDDED DELAMINATION

This paper is the continuation of the authors' previous investigation inwhich the formula of total energy release rate at each point along the delaminationboundary iS derived by using variational method of moving boundary.In the presentpaper,the analytical model is reformed and the individual components of energyrelease rate are derived in the form of plate theory.The circular delaminationembedded in circular laminates is analysed and some conclusions are obtained.     

Semi-inverse method and generalized variational principles with multi-variables in elasticity

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Variational principles and generalized variational principles for nonlinear elasticity with finite displacement

In a previous paper (1979)^[1], the minimum potential energy principle and stationary complementary energy principle for nonlinear elasticity with finite displacement, together with various complete and incomplete generalized principles were studied. However, the statements and proofs of these principles were not so clearly stated about their constraint conditions and their Euler equations. In somecases, the Euler equations have been mistaken as constraint conditions. For example, the stress displacement relation should be considered as Euler equation in complementary energy principle but have been mistaken as constraint conditions in variation. That is to say, in the above mentioned paper, the number of constraint conditions exceeds the necessary requirement. Furthermore, in all these variational principles, the stress-strain relation never participate in the variation process as constraints, i.e., they may act as a constraint in the sense that, after the set of Euler equations is solved, the stress-strain relation may be used to derive the stresses from known strains, or to derive the strains from known stresses. This point was not clearly mentioned in the previous paper (1979)^[1]. In this paper, the high order Lagrange multiplier method (1983)^[2] is used to construct the corresponding generalized variational principle in more general form. Throughout this paper, V/.V. Novozhilov's results (1958)^[3] for nonlinear elasticity are used.     

On the Delamination of a Viscoelastic Composite Circular Plate

Delamination of a symmetric circular plate made of viscoelastic composite is studied. It is assumed that the plate has a penny-shaped crack whose edges have a minor axisymmetric imperfection. The lateral surface of the plate is clamped and is compressed by uniform radial normal forces through a rigid body. The studies are made using the exact geometrically nonlinear equations of the theory of viscoelasticity. The delamination criterion is assumed unrestrained growth of the initial imperfection. The Laplace transform and FEM are employed. In particular cases, the results are compared with those for elastic composites     

STUDY OF THE EQUIVALENT THEOREM OF GENERALIZED VARIATIONAL PRINCIPLES IN ELASTICITY

The relations of all generalized variational principles in elasticity are studied by employing the invariance theorem of field theory. The infinitesimal scale transformation in field theory was employed to investigate the equivalent theorem. Among the results found particularly interesting are those related to that all generalized variational principles in elasticity are equal to each other. Also studied result is that only two variables are independent in the functional and the stress-strain relation is the variational constraint condition for all generalized variational principles in elasticity. This work has proven again the conclusion of Prof. Chien Wei-zang.     

Generalized variational principles of symmetrical elasticity problem of large deformations

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Compound Bifurcations in the Buckling of a Delaminated Composite Strut

An analysis of the buckling and post-buckling of a delaminated composite strut is presented using a simple 4 degree of freedom nonlinear Rayleigh–Ritz formulation. Bifurcation analysis indicates that instability occurs in general at an asymmetric point of bifurcation. Depending on the depth of delamination both thin-film and overall buckling can occur in the post-buckling range, the transition being seen at a point of secondary bifurcation. For certain combinations of parameters this becomes a stellar bifurcation, associated with a double eigenvalue, where there are three possible subsequent routes for the post-buckling. The method used is fast and reliable and can be readily extended to modelling a composite with several layers.     

Two-dimensional analyses of delamination buckling of symmetrically cross-ply rectangular laminates

The conventional approach to analysis the buckling of rectangular laminates containing an embedded delamination subjected to the in-plane loading is to simplify the laminate as a beam-plate from which the predicted buckling load decreases as the length of the laminate increases. Two-dimensional analyses are employed in this paper by extending the one-dimensional model to take into consideration of the influence of the delamination width on the buckling performance of the laminates. The laminate is simply supported containing a through width delamination. A new parameter β defined as the ratio of delamination length to delamination width is introduced with an emphasis on the influence of the delamination size. It is found that (i) when the ratio β is greater than one snap-through buckling prevails, the buckling load is determined by the delamination size and depth only; (ii) as the ratio β continues to increase, the buckling load will approach to a constant value. Solutions are verified with the well established results and are found in good agreement with the latter.     

Generalized variational principles on nonlinear theory of elasticity with finite displacements

Two generalized variational principles on nonlinear theory of elasticity with finitedisplacements in which the σ_(ij),e_(i j)and u_i are all three kinds of independent functionsare suggested in this paper.It isproved that these two generalized variational principles areequivalent to each other if the stress-strain relation is satisfied as constraint.Some specialcases,i.e.generalized variational principles on nonlinear theory of elasticity with smalldeformation,on linear theory with finite deformation and on linear theory with smalldeformation together with the corresponding equivalent theorems are also obtained.All ofthem are related to the three kinds of independent variables.     

The generalized variational principles about blending energy form of non-linear elasticity

First of all, this paper gives Legendre transformation for the so-called partial corresponding variables of strain energy function Σ(E_ij) and complementary strain energy function Σc(S_ij) of the elastic materiel, and introduces the corresponding blending complementary strain energy function Σch_k! and blending strain energy function Σh_k!. Moreover, a series of generalized variational principles of the corresponding blending energy form of non-linear elasticity is given. As a special case, there exist corresponding results[1] in linear elasticity.     

含层间长裂纹的对称正交层合板的解析－分区广义变分解法

本文首先将层板横截面分为含裂尖区、韧带区以及裂纹区等三个区域，用复变函数理论与特征函数展开法，得到了各区内满足所有基本方程、裂纹表面边界条件与层间连续条件的位移与应力的特征展开式。然后利用分区广义变分原理满足裂纹表面边界以外的边界条件以及各区之间的交界条件，并由此求得奇异场控制量（广义应力强度因子）。     

复杂载荷下复合材料对称层合板的椭圆形分层屈曲

应用Ｒａｙｌｅｉｇｈ－Ｒｉｔｚ法对含有单个椭圆形分层的复合材料对称层板在压剪载荷联合作用下的脱层屈曲问题进行了分析计算。给出相应于不同的椭圆长短半轴之比ａ／ｂ值下及不同椭回主轴的偏角时，脱层屈曲的临界应变值。此计算考虑了层板的拉伸剪切藕合刚度影响及子层本身弯曲－扭转耦合刚度，和拉伸－弯曲耦合刚度的影响，给出有工程参考价值的计算结果。     

The Biot stresses in nonlinear elasticity and the associated generalized variational principles

Summary It is shown that the concept based on the polar decomposition and Reissner's concept based on a distinguished system of rotated axes lead to the same results. Especially it is pointed out how, due to the properties of the statical and kinematical operators respectively, the most general principles of virtual work and the associated generalized variational principles can be constructed systematically, allowing independent variations of the displacements, rotations, strains, stresses and reactive forces.

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Die Biot-Spannungen in der nichtlinearen Elastizitätstheorie und die zugehörigen verallgemeinerten Variationsprinzipien

Übersicht Es wird nachgewiesen, daß das auf der Polarzerlegung basierende Konzept und Reissner's Konzept, welches auf einem ausgezeichneten System gedrehter Achsen beruht, zu den gleichen Ergebnissen führen. Insbesondere wird gezeigt, wie auf Grund der Eigenschaften des statischen und kinematischen Operators die allgemeinsten Prinzipien der virtuellen Arbeiten sowie die zugehörigen verallgemeinerten Variationsprinzipien systematisch konstruiert werden können, wobei unabhängige Variationen der Verschiebungen, Drehungen, Verzerrungen, Spannungen und Reaktionskräfte möglich sind.