A RECTANGULAR ELEMENT OF THIN PLATES BASED UPON THE GENERALIZED VARIATIONAL PRINCIPLES

Based on generalized variational principles,an element called MR-12 was constructed for the static and dynamic analysis of thin plates with orthogonal anisotropy.Numerical results showed that this incompatible clement converges very rapidly and has good accuracy.It was demonstrated that generalized variational principles are useful and effective in founding incompatible element.Moreover,element MR-12 is easy for implementation since it does not differ very much from the common rectangular element R-12 of thin plate.     

VARIATIONAL PRINCIPLES AND GENERALIZED VARIATIONAL PRINCIPLES FOR NONLINEAR ELASTICITY WITH FINITE DISPLACEMENT

In a previous paper (1979), 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     

SUBREGION GENERALIZED VARIATIONAL PRINCIPLES FOR ELASTIC THICK PLATES

In this paper the subregion generalized variational principle forelastic thick plates is proposed.Its main points may be stated asfollows:1.Each subregion may be assigned arbitrarily as a potential re-gion or complementary region.The subregion variational principlesof potential energy,complementary energy and mixed enerqy repre-sent three special forms of this principle.2.The number of independent variational variables in each sub-region may be assigned arbitrarily.Any one of the subregions maybe assigned as a one-variable-region,two-variable-region or three-variable-region.3.The conjunction conditions of displacements and stresses oneach interline of neighbouring subregions may be relaxed.On thebasis of this principle the finite element analysis of non-confor-ming elements for thick plates can be formulated.Different finite element models for thick plates can be obtain-ed by different applications of this principle.In particular.thesubregion mixed variational principle for thick plates may be ap-plied     

VARIATIONAL PRINCIPLES AND GENERALIZED VARIATIONAL PRINCIPLES IN HYDRODYNAMICS OF VISCOUS FLUIDS

In this paper, the variational principles of hgdrodynamic problems for the incompressibleand compressible viscous fluids are established. These principles are principles of maximum power losses. Their generalized variational principles are also discussed on the basis of Lagrangian multiplier methods.     

GENERALIZED VARIATIONAL PRINCIPLESFOR VISCOELASTIC THIN AND THICK PLATES WITH DAMAGE

From the constitutive model with generalized force fields for a viscoelastic body with damage, the differential equations of motion for thin and thick plates with damage are derived under arbitrary boundary conditions. The convolution-type functionals for the bending of viscoelastic thin and thick plates with damage are presented, and the corresponding generalized variational principles are given. From these generalized principles, all the basic equations of the displacement and damage variables and initial and boundary conditions can be deduced. As an example, we compare the difference between the dynamical properties of plates with and without damage and consider the effect of damage on the dynamical properties of plates.     

COUPLED VARIATIONAL PRINCIPLES AND GENERALIZED COUPLED VARIATIONAL PRINCIPLES IN PHOTOELASTICITY

ＦｕＢａｏｌｉａｎ（付宝连）（ＲｅｃｅｉｖｅｄＮｏｖ．２２，１９９３；ＣｏｍｍｕｎｉｃａｔｅｄｂｙＣｈｉｅｎＷｅｉｚａｎｇ）ＣＯＵＰＬＥＤＶＡＲＩＡＴＩＯＮＡＬＰＲＩＮＣＩＰＬＥＳＡＮＤＧＥＮＥＲＡＬＩＺＥＤＣＯＵＰＬＥＤＶＡＲＩＡＴＩＯＮＡＬＰＲＩ...     

THE SOLID VARIATIONAL PRINCIPLES OF THE DISCRETE FORM—THE VARIATIONAL PRINCIPLES OF THE DISCRETE ANALYSIS BY THE FINITE ELEMENT METHOD

This paper suggests a new solid variational prin-ciple of discrete form.Basing on the true case of thediscrete analysis by the finite element method and con-sidering the variable boundaries of the elements andthe unknown functions of piecewise approximation,theunknown functions have various discontinuities at theinterfaces between successive elements.Thus,we have used mathematical technique of vari-able boundary with discontinuity of the unknown func-tions,based on the conditions that the first varia-tion vanishes immediately,to establish the solid vari-ation principles of discrete form.It generalizes theclassical and non-classical variational principles.suc-cessive equations that have to be satisfied by the un-known functions are the convergency necessary conditionsfor the finite elements method(including conformingand non-conforming).They expand that convergency ne-cessary-conditions of the compatibility conditions inthe internal interfaces.     

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.     

PRINCIPLES OF MINIMUM TRANSFORMED ENERGY AND MINIMUM PRINCIPLES FOR DYNAMICS OF PLATES

In the present paper,we first by Laplace transform present a derivation of principle oftransformed virtual work, three principles of minimum transformed energy with influenceof rotatory enertia for dynamics of anisotropic linear elastic plates with three generalizeddisplacements.Moreover,the forms with the original in place-time domain correspondingthese variational principles are presented.Then by the introduction of the set of abmissible weight functions the three minimumprinciples for the original place-time domain are derivedIn each of the preceding groups of the variational principles there are two which are thedynamic counterparts to the static principles of minimum potetial energy and minimumcomplementary energy;the other principles are formulated in terms of the internal forcealone,but have no counterpart in elastostatics of plates.     

GENERALIZED BIHARMONIC OPERATOR AND ITS APPLICATION TO THE BENDING OF ELASTIC THIN PLATES

ＧＥＮＥＲＡＬＩＺＥＤＢＩＨＡＲＭＯＮＩＣＯＰＥＲＡＴＯＲＡＮＤＩＴＳＡＰＰＬＩＣＡＴＩＯＮＴＯＴＨＥＢＥＮＤＩＮＧＯＦＥＬＡＳＴＩＣＴＨＩＮＰＬＡＴＥＳＹｕＺｈｏｎｅｔｔ－ｚｈｉ（俞中直）（Ｄｅｐｔ．ｏｆＥｎｇ．Ｍｅｃｈａｎｉｃｓ,ＤａｌｉａｎＵｎ...     

Variational principles of perforated thin plates and finite element method for buckling and post-buckling analysis

On the basis of the general theory of perforated thin plates under large deflections^{[1, 2]}, variational principles with deflectionw and stress functionF as variables are stated in detail. Based on these principles, finite element method is established for analysing the buckling and post-buckling of perforated thin plates. It is found that the property of element is very complicated, owing to the multiple connexity of the region. Project supported by National Natural Science Foundation of China.     

The applications of generalized variational principles in nonlinear structural analysis

This paper discusses the generalized variational principles founded by thetechnique of Lagrangian multipliers in structural mechanics and analyzes the nonlinearstatically indeterminate structures.It is assumed that the stress-strain relationship ofthe materials of structures has the form ofσ=Bε~(1/m)orτ=Cγ~(1/m),namely,thephysical equations of structures have the shape of exponential functions.Severalexamples are given to illustrate the statically indeterminate structures such as thetrusses,beams,frames and torsional bars.     

Region-wise variational principles and generalized variational principles on large strain for consolidation theory

Introduction Thedeformationofsaturatedsoftclayisoneofimportantquestionsingeotechnical engineering.Thenotablecharacteristicisthatthedegreeofitsstrainisgenerallylargerthan10percent.Oneofthecauses[1],Whichleadtoaquantitativedifferencebetweenthe numericalsimu…     

A variational principle of perturbed motion on viscoelastic thin plates with applications

In this paper, in the light of the Boltzmann superposition principle in linear viscoelasticity, a mathematical model of perturbed motion on viscoelastic thin plates is established. The corresponding variational principle is obtained in a convolution bilinear form. For application the problems of free vibration, forced vibration and stability of a viscoelastic simply-supported rectangular thin plate are considered. The results show that numerical solutions agree well with analytical solutions. The project was supported by the National Natural Science Foundation of China (No. 19772027) and the Shanghai Municipal Development Foundation of Science and Technology (No. 98JC14032).     

电磁弹性动力学初边值问题12类变量广义变分原理

基于Yao建立的电磁弹性固体广义变分原理,运用关于非传统Hamilton型广义变分原理的方法,建立了电磁弹性动力学初边值问题的12类变量广义变分原理,可反映该问题的全部特征,其独立变分变量为该问题的全部变量,即位移、速度、动量、应变、应力、电位移、磁感应强度、电场强度、磁场强度、电标量势、磁标量势和磁矢量势.本文建立的...     

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

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ 1 954,Ｈｕ[1,2 ]ｄｅｄｕｃｅｄＨｕ＿Ｗａｓｈｉｚｕｐｒｉｎｃｉｐｌｅｂｙｓｏ＿ｃａｌｌｅｄｔｒｉａｌ＿ａｎｄ＿ｅｒｒｏｒｍｅｔｈｏｄ ,ａｎｄｉｎ1 964 ,Ｃｈｉｅｎ[3]ｓｙｓｔｅｍａｔｉｃａｌｌｙｄｉｓｃｕｓｓｅｄｔｈｅＬａｇｒａｎｇｅｍｕｌｔｉｐｌｉｅｒｍｅｔｈｏｄ ,ｂｙｗｈｉｃｈｈｅｓｕｃｃｅｓｓｆｕｌｌｙｄｅｄｕｃｅｄＨｕ＿Ｗａｓｈｉｚｕｐｒｉｎｃｉｐｌｅ.Ａｆｔｅｒｔｈａｔｇｅｎｅｒａｌｉｚｅｄｖａｒｉａｔｉｏｎａｌｐｒｉｎｃｉｐｌｅｓｃａｎｂｅａｒｒｉｖｅｄａｔ…     

The variational principles and application of nonlinear numerical manifold method

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｆｉｎｉｔｅｃｏｖｅｒｔｅｃｈｎｉｑｕｅｓ,ｓｉｍｉｌａｒｔｏｔｈｅｃｏｎｃｅｐｔｉｏｎｏｆｆｉｎｉｔｅｃｏｖｅｒｕｓｅｄｉｎｍａｎｉｆｏｌｄａｎａｌｙｓｉｓｏｆｍｏｄｅｒｎｍａｔｈｅｍａｔｉｃｓ,ａｒｅｉｎｔｒｏｄｕｃｅｄｉｎｔｏＮｕｍｅｒｉｃａｌＭａｎｉｆｏｌｄｍｅｔｈｏｄ (ＮＭＭ)ａｎｄｔｈｅｆｉｎｉｔｅｃｏｖｅｒｓｃｏｎｓｉｓｔｏｆｍａｔｈｅｍａｔｉｃａｌｃｏｖｅｒｓａｎｄｐｈｙｓｉｃａｌｃｏｖｅｒｓｗｈｉｃｈｃａｎｂｅｓｅｐａｒａｔｅｄ .ＴｈｅＮＭＭｉｓｃｏ…     

Formulation of a semi-analytical approach based on Gurtin variational principle for dynamic response of general thin plates

4 semi-analytical approach for the dynamic response of general thin plates which employes finite element discretization in space domain and a series of representation in time domain is developed on the basis of Gurtin variational principles. The formulation of time series is also investigated so that the dynamic response of plates with arbitrary shape and boundary constraints can be achieved with adequate accuracy.Project supported by the National Natural Science Foundation of China