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对偶互补
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  dual complementarity
     According to the basic idea of classical yin yang complementarity and modern dual complementarity, in a simple and unified way proposed by the author, Hamilton type quasi variational principles in coupled thermoelastodynamics can be established systematically.
     根据古典阴阳互补和现代对偶互补的基本思想 ,通过作者早已提出的一条简单而统一的途径 ,系统地建立了耦合热弹性动力学的各类Hamilton型拟变分原理 .
短句来源
  dual-complementarity
     According to the basic idea of classical yin-yang complementarity and modern dual-complementarity, in a simple and unified new way proposed by Luo, the unconventional Gurtin-type variational prinicples for finite deformation elastodynamics can be established systematically.
     根据古典阴阳互补和现代对偶互补的基本思想 ,通过作者早已提出的一条简单而统一的途径 ,系统地建立了有限变形弹性动力学的各类非传统Gurtin型变分原理 .
短句来源
     According to the basic idea of classical yin-yang complementarity and modern dual-complementarity, in a simple and unified new way proposed by Luo, the unconventional simplified Gurtin-type variational prinicples for finite deformation elastodynamics can be established systematically.
     根据古典阴阳互补和现代对偶互补的基本思想 ,通过作者提出的一条简单而统一的新途径 ,建立了有限变形弹性动力学的另一种单卷积形式的变分原理—各类非传统简化Gurtin型变分原理 .
短句来源
     According to the basic idea of classical yin-yang complementarity and modern dual-complementarity,in a simple and unified way proposed by Luo(1987), some unconventional Hamilton-type variational principles for dynamics of Reissner sandwich plate can be established systematically.
     根据古典阴阳互补和现代对偶互补的基本思想,通过早已提出的一条简单而统一的新途径,系统地建立了Reissner夹层板动力学的各类非传统Hamilton型变分原理.
短句来源
     According to the basic idea of classical yin-yang complementarity and modern dual-complementarity,the principle of virtual work,eight-field and two-field simplified Gurtin-type variational principles for elastodynamics of piezoelectric thin plate are established systematically.
     根据现代对偶互补的基本思想,系统地建立了压电弹性薄板动力学的虚功原理和8类变量与2类变量简化Gurtin型变分原理.
短句来源
     According to the basic idea of classical yin-yang complementarity and modern dual-complementarity,the principle of virtual work,unconventional Hamilton-type variational principles for elastodynamics of piezoelectric plane probˉlem are established systematically.
     根据古典阴阳互补和现代对偶互补的基本思想,系统地建立了压电弹性平面问题动力学的虚功原理和各类变量非传统Hamilton型变分原理。
短句来源
  “对偶互补”译为未确定词的双语例句
     According to the basic idea of classical yin_yang complementarity and modern dual_complementarity, the unconventional Hamilton_type variational principle in phase space for dynamics of elastic beam with linear damping is established, which can fully characterize the initial_boundary_value problem of this dynamics.
     根据古典阴阳互补和现代对偶互补的基本思想 ,首次建立了线性阻尼情形下弹性梁动力学的相空间(挠度、动量 )非传统Hamilton型变分原理。
短句来源
     According to the basic idea of classical yin_yang complementarity and modern dual_complementarity, in a new, simple and unified way proposed by Luo, the unconventional Hamilton_type variational principles for geometrically nonlinear elastodynamics can be established systematically.
     根据古典阴阳互补和现代对偶互补的基本思想,通过作者早已提出的一条简单而统一的新途径,系统地建立了几何非线性弹性动力学的各类非传统Hamilton型变分原理。
短句来源
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  dual-complementarity
According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author[1], various energy principles in theory of elastic materials with voids can be established systematically.
      
According to the basic idea of dual-complementarity, in a simple and unified way proposed by the author[1], some basic principles in dynamic theory of elastic materials with voids can be established systematically.
      


According to the basic idea of dual-complementarity, in a simple and unified new way proposed by the author ̄[1] , some basic principles in dynamic theory of thermoelastic materials with voids can be established systematically. In this paper, an important integral relation in terms of convolutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem in dynamic...

According to the basic idea of dual-complementarity, in a simple and unified new way proposed by the author ̄[1] , some basic principles in dynamic theory of thermoelastic materials with voids can be established systematically. In this paper, an important integral relation in terms of convolutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem in dynamic theory of thermoelastic materials with voids, but also to derive systematically the complementary functionals for the eleven-field, nine-field, six-field and three-field simplified Gurtin-type variational principles. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly.

根据对偶互补的基本思想,通过作者在文[1]中所提出的一条简单而统一的新途径,系统地建立了有孔隙的耦合热弹性体动力学的一些基本原理.文中首先给出一个重要的以卷积表示的积分关系式,可以认为,在力学上它是一个广义虚功原理的表式.然后从该式出发,不仅可以得到有孔隙的耦合热弹性体动力学的虚功原理和互等定理,而且能系统地导出成互补关系的11类变量、9类交量、6类变量及3类变量简化Gurtin型变分原理.同时,通过这条新途径,还能清楚地阐明这些原理之间的内在联系.

According to the basic idea of classical yin yang complementarity and modern dual complementarity, in a simple and unified way proposed by the author, Hamilton type quasi variational principles in coupled thermoelastodynamics can be established systematically. The Hamilton type quasi variational principles in terms of variation of unitary functional can fully characterize the initial boundary value problem of coupled thermoelastodynamics. In this paper, an important relation is given, which can be considered...

According to the basic idea of classical yin yang complementarity and modern dual complementarity, in a simple and unified way proposed by the author, Hamilton type quasi variational principles in coupled thermoelastodynamics can be established systematically. The Hamilton type quasi variational principles in terms of variation of unitary functional can fully characterize the initial boundary value problem of coupled thermoelastodynamics. In this paper, an important relation is given, which can be considered as the generalized principle of quasi virtual work in mechanics. Based on this relation, the eight field, six field, four field and two field Hamilton type quasi variational principles by the generalized Legendre transformations given in this paper can be derived systematically. Furthermore, with this approach, the intrinsic relationship among these principles can be explained clearly.

根据古典阴阳互补和现代对偶互补的基本思想 ,通过作者早已提出的一条简单而统一的途径 ,系统地建立了耦合热弹性动力学的各类Hamilton型拟变分原理 .这种以单一泛函的变分式表示的Hamilton型拟变分原理 ,能精确反映耦合热弹性动力学初值 边值问题的全部特征 .文中首先给出一个在力学上可以认为是广义拟虚功原理的表式 .然后从该式出发 ,通过所给出的一系列广义Legendre变换 ,系统地推导出耦合热弹性动力学的 8类变量、6类变量、4类变量和 2类变量Hamilton型拟变分原理 .同时 ,通过这条途径还能阐明这些原理的内在联系

The dynamic theory of piezoelectric materials with voids is intended for applications to natural and artifical materials with distributed voids, some bioengineering materials and intelligent materials. Therefore, it plays an important role in the development and application of modern new materials. But the energy principles in dynamic theory of piezoelectric materials with voids, which the principle of virtual work, the reciprocal theorem and various variational principles are not yet established systematically....

The dynamic theory of piezoelectric materials with voids is intended for applications to natural and artifical materials with distributed voids, some bioengineering materials and intelligent materials. Therefore, it plays an important role in the development and application of modern new materials. But the energy principles in dynamic theory of piezoelectric materials with voids, which the principle of virtual work, the reciprocal theorem and various variational principles are not yet established systematically. According to the basic idea of classical yin-yang complementarity and modern dual- complementarity, in a simple and unified way proposed by Luo[6,7], the energy principles in dynamic theory of piezoelectric materials with voids can be established systematically. In this paper, an important integral relation in terms of convolutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem in dynamic theory of piezoelectric materials with voids, but also to derive systematically the complementary functionals for eleven-field, nine-field, six-field and three-field simplified Gurtin-type variational principles by the generalized Legendre transformations given in this paper. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly. In this paper first applying the principle of virtual work, one may obtain the reciprocal theorem in dynamic theory of piezoelectric materials with voids avoiding both the use of the Laplace transform and the incorporation of the initial conditions into the equations of motion. Obviously this method given in this paper is even simpler and directer than the conventional method. The energy principles (the principle of virtual work, the reciprocal theorem and various simplified Gurtin-type variational principles) established in this paper are an important part of dynamic theory of piezoelectric materials with voids. Obviously, the simplified Gurtin-type variational principle can fully characterize the initial-boundary-value problem of this dynamics. Consequently, the energy principles proposed in this paper will be of great value both in theoretical studies and in the establishment of various approximate methods and approximate engineering theories.

根据古典阴阳互补和现代对偶互补的基本思想,通过作者早已提出的一条简单而统一的新途径:系统地建立了微孔压电弹性动力学的能量原理。给出一个重要的以卷积表示的积分关系式,可以认为,在力学上它是广义虚功原理的表式。从该式出发,不仅能得到微孔压电弹性动力学的虚功原理和互等定理,而且通过作者所给出的一系列广义Legendre变换,能系统地导出成互补关系的11类变量、9类变量、6类变量和3类变量简化Gurtin型变分原理的泛函。同时,通过这条新途径,还能清楚地阐明这些原理之间的内在联系。

 
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