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1.
The paper proposes a modification of the mixed variational principle from which stationarity conditions are derived in the
form of a mixed system of equations resolved for the first derivatives of the displacement and stress components acting in
a plane perpendicular to one of the coordinate axes. The variational principle allows decreasing the dimension of the problem
of elasticity thus reducing the system of equations to a canonical form. The modified mixed principle helps immediately obtain
a canonical system of equations for various applied theories. This possibility is demonstrated with the example of the Timoshenko
theory of plates
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Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 55–62, May 2007. 相似文献
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分析热力学乃是用分析力学的方法来研究平衡态热力学。本文用较简单的方法证明了“熵最大”变分原理与“Gibbs自由能最小”变分原理或“Helmholtz自由能最小”变分原理是等价的;以这三个Gibbs变分原理为出发点,导出了平衡态热力学的正则方程。由平衡态热力学中的正则方程,可以证明热力学基本Poisson括号成立。本文的另一主要任务是借助于Gibbs变分原理,讨论平衡态热力学中热力学量的正则变换。可以得到热力学正则变换的四种形式。在分析(平衡态)热力学中也可提出“化准Hamiltonian为压强或容积的正则变换技术”。作为应用正则变换的实例,讨论了理想气体并得到了简明的结果。 相似文献
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Zuo-Jun Wang De-Zhong Zheng Cheng-Bo Zheng 《International Journal of Solids and Structures》2010,47(22-23):3115-3120
The fundamental equations, governing all the variables of the initial boundary value problem in fully dynamic magneto-electro-elasticity with geometrical nonlinearity, are expressed in covariant differential form. The generalized principle of virtual work is given in terms of convolutions for the present problem. Two simplified Gurtin-type generalized variational principles, directly leading to all the fundamental equations, are deduced by using He’s semi-inverse method instead of Laplace transforms. By enforcing some fundamental equations as constraint conditions, one of various constrained variational principles is given as an example. By simply dropping out selected field functions, several reduced variational principles are obtained as special forms for piezoelectricity, elastodynamics, and electromagnetics, respectively. This paper aims at providing a more complete theoretical foundation for the finite element applications for the discussed problem. 相似文献
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David C. P. Ellis François Gay-Balmaz Darryl D. Holm Vakhtang Putkaradze Tudor S. Ratiu 《Archive for Rational Mechanics and Analysis》2010,197(3):811-902
The equations of motion are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible
continuous filamentary distributions of interacting rigid charge conformations. The new feature is that these equations are
nonlocal when the screened Coulomb interactions, or Lennard–Jones potentials between pairs of charges, are included. The nonlocal
dynamics is derived in the convective representation of continuum motion by using modified Euler–Poincaré and Hamilton–Pontryagin variational formulations that
illuminate the various approaches within the framework of symmetry reduction of Hamilton’s principle for exact geometric rods.
In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods. The motion
equations in the convective representation are shown to arise by a classical Lagrangian reduction associated to the symmetry
group of the system. This approach uses the process of affine Euler–Poincaré reduction initially developed for complex fluids.
On the Hamiltonian side, the Poisson bracket of the molecular strand is obtained by reduction of the canonical symplectic
structure on phase space. A change of variables allows a direct passage from this classical point of view to the covariant
formulation in terms of Lagrange–Poincaré equations of field theory. In another revealing perspective, the convective representation
of the nonlocal equations of molecular strand motion is transformed into quaternionic form. 相似文献
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We obtain a criterion for the existence of solutions of degenerate inhomogeneous Fredholm boundary-value problems for a system
of ordinary differential equations under the assumption that the degenerate system of differential equations can be reduced
to the central canonical form. The results are illustrated by examples.
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Translated from Neliniini Kolyvannya, Vol. 10, No. 3, pp. 303–312, July–September, 2007. 相似文献
10.
何吉欢 《应用数学和力学(英文版)》2000,21(7):797-808
IntroductionIn 1 954,Hu[1,2 ]deducedHu_Washizuprinciplebyso_calledtrial_and_errormethod ,andin1 964 ,Chien[3]systematicallydiscussedtheLagrangemultipliermethod ,bywhichhesuccessfullydeducedHu_Washizuprinciple.Afterthatgeneralizedvariationalprinciplescanbearrivedat… 相似文献
11.
Generalized variational principles with several arbitrary parameters and the variable substitution and multiplier method 总被引:4,自引:0,他引:4
龙驭球 《应用数学和力学(英文版)》1987,8(7):617-629
The functional transformations of variational principles in elasticity are classified as three patterns: Ⅰ relaxation pattern, Ⅱ augmented pattern and III equivalent pattern.On the basis of pattern Ⅲ, the generalized variational principles with several arbitrary parameters are formulated and their functionals are defined. They are: the generalized principle of single variable u with several parameters, the generalized principle of two variables u, σ with several parameters, the generalized principle of two variables u, ε with several parameters, and the generalized principle of three veriables u, ε, σ with several parameters. From these principles, a series of new forms of equivalent functionals can be obtained. When the values of these parameters are properly chosen, a series of finite element models can be formulated.In this paper, the question of losing effectiveness for Lagrange multiplier method is also discussed. In order to "recover" effectiveness for multiplier method, a modified method, namely, the variable substitution and multiplier method, is proposed. 相似文献
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Hans Bufler 《Acta Mechanica Sinica》1994,10(3):227-236
A series of problems in mechanics and physics are governed by the ordinary Poisson equation which demands linearity, isotropy,
and material homogeneity. In this paper a generalization with respect to nonliearity, anisotropy, and inhomogeneity is made.
Starting from the canonical basic equations in the primal and dual formulation respectively we derive systematically the corresponding
generalized variational principles; under certain conditions they can be extended to so calle complementary extremum principles
allowing for global bounds. For simplicity a restriction to two dimensional problems is made, including twice-connected domains. 相似文献
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V. L. Berdichevsky 《Continuum Mechanics and Thermodynamics》2008,20(4):219-229
Ideal incompressible fluid is a Hamiltonian system which possesses an infinite number of integrals, the circulations of velocity
over closed fluid contours. This allows one to split all the degrees of freedom into the driving ones and the “slave” ones,
the latter to be determined by the integrals of motions. The “slave” degrees of freedom correspond to “potential part” of
motion, which is driven by vorticity. Elimination of the “slave” degrees of freedom from equations of ideal incompressible
fluid yields a closed system of equations for dynamics of vortex lines. This system is also Hamiltonian. The variational principle
for this system was found recently (Berdichevsky in Thermodynamics of chaos and order, Addison-Wesly-Longman, Reading, 1997;
Kuznetsov and Ruban in JETP Lett 67, 1076–1081, 1998). It looks striking, however. In particular, the fluid motion is set
to be compressible, while in the least action principle of fluid mechanics the incompressibility of motion is a built-in property.
This striking feature is explained in the paper, and a link between the variational principle of vortex line dynamics and
the least action principle is established. Other points made in this paper are concerned with steady motions. Two new variational
principles are proposed for steady vortex flows. Their relation to Arnold’s variational principle of steady vortex motion
is discussed.
相似文献
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Xie Dingyi 《Acta Mechanica Sinica》1985,1(2):138-146
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. 相似文献
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This paper presents a multi-scale model in phase transitions of solid materials with both macro and micro effects. This model is governed by a semi-linear nonconvex partial differential equation which can be converted into a coupled quadratic mixed variational problem by the canonical dual transformation method. The extremality conditions of this variational problem are controlled by a triality theory, which reveals the multi-scale effects in phase transitions. Therefore, a potentially useful canonical dual finite element method is proposed for the first time to solve the nonconvex variational problems in multi-scale phase transitions of solids. Applications are illustrated. Results shown that the canonical duality theory developed by the first author in nonconvex mechanics can be used to model complicated physical phenomena and to solve certain difficult nonconvex variational problems in an easy way. The canonical dual finite element method brings some new insights into computational mechanics. 相似文献
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In the present investigation the time dependent flow of an Oldroyd fluid B in a horizontal cylindrical pipe is stuided by
the variational analytical approach developed by author. The time dependent problem is mathematically reduced to a partial
differential equation of third order. Using the improved variational approach due to Kantorovich the partial differential
equation can be reduced to a system of ordinary differential equations for different approximations. The ordinary differential
equations are solved by the method of the Laplace transform which is led to an analytical form of the solutions.
Project supported by TWAS and Chinese Academy of Sciences and the National Science Foundation of China 相似文献
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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. 相似文献
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金问鲁 《应用数学和力学(英文版)》1984,5(6):1817-1823
The author gives variational principles of elastic-viscous dynamics in spectral resolving form[1], it will be extended to Laplace transformation form in this paper, mixed variational principle of shell dynamics and variational principle of dynamics of elastic-viscous-porous media are concerned, for the latter, F. E. M. formulation has been worked out.Variational principles in Laplace transformation form have concise forms, for the sake of utilizing F. E. M. conveniently it is necessary to find values of preliminary time function at some instants, when values of Laplace transformation at some points are known, but there are no efficient methods till now. In this paper, a numerical method for finding discrete values of preliminary function is presented, from numerical example we see such a method is efficient.By combining both methods stated above, variational principles in Laplace transformation form and numerical method, a quite wide district of solid dynamic problems can be solved by ths aid of digital computers. 相似文献
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Zhen-bang Kuang 《International Journal of Solids and Structures》2009,46(3-4):902-911
The first thermodynamic law contains a universal thermodynamic variational principle. The complete internal energy variational principle in the electroelastic analysis is not discussed in previous papers. In this paper this principle will be discussed. From this principle the simple complete governing equations can be deduced, and the Maxwell stress can be naturally derived from this variational principle. It is shown that the Maxwell stress has slightly different forms determined by using internal energy or electric Gibbs free energy variational principle, but substantially they are the same. In the second-order precision the Maxwell stress is uniquely determined, and its expression has the same form for all deformable and rigid dielectrics. The electroelastic analyses in the dielectric should be studied together with its environment, because the electric field exists in all materials except the ideal conductor. The complete governing equations under finite deformation in the initial configuration are also discussed. 相似文献
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In this paper,Routh’s equations for the mechanical systems of the variable masswith nonlinear nonholonomic constraints of arbitrary orders in a noninertial referencesystem have been deduced not from any variational principles,but from the dynamicalequations of Newtonian mechanics.And then again the other forms of equations fornonholonomic systems of variable mass are obtained from Routh’s equations. 相似文献