Analytical continuum mechanics a la Hamilton-Piola least action principle for second gradient continua and capillary fluids
Analytical continuum mechanics a la Hamilton-Piola least action principle for second gradient continua and capillary fluids
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DOI：
10.1177/1081286513497616
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发表时间：
2015-04-01
影响因子：
2.6
通讯作者：
Rosi, G.
中科院分区：
文献类型：
--
作者：
Auffray, N.;dell'Isola, F.;Rosi, G.
本文从分子论证出发，证明了恒定作用原理对于毛细流体，即变形能具有所给形式的流体是成立的。我们注意到，这些流体有时也被称为Korteweg-de Vries或Cahn-Allen流体。通常，变形能取决于放置的第二梯度的连续体称为第二梯度(或Piola-Toupin、Mindlin、Green-Rivlin、Germain或二级)连续体。本文给出了二阶梯度连续体的物质描述。在物质描述和空间描述中都引入了拉格朗日作用量，得到了相应的欧拉-拉格朗日方程和边界条件。这些条件是在两种情况下用客观变形能体积密度表示的：当假定该能量依赖于C和增量C或依赖于C-1和增量C-1，其中C是柯西-格林变形张量。当细化到表征流体物质的能量时，恢复了毛管流体的演化条件。找到了适用于毛细管流体的伯努利定律的一种形式，并提出了适用于本变分公式的运动学公式。还对加布里奥·皮奥拉对分析连续介质力学的贡献进行了历史评价。
In this paper a stationary action principle is proved to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments. We remark that these fluids are sometimes also called Korteweg-de Vries or Cahn-Allen fluids. In general, continua whose deformation energy depends on the second gradient of placement are called second gradient (or Piola-Toupin, Mindlin, Green-Rivlin, Germain or second grade) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian action is introduced in both the material and spatial descriptions and the corresponding Euler-Lagrange equations and boundary conditions are found. These conditions are formulated in terms of an objective deformation energy volume density in two cases: when this energy is assumed to depend on either C and delta C or on C-1 and delta C-1, where C is the Cauchy-Green deformation tensor. When particularized to energies which characterize fluid materials, the capillary fluid evolution conditions are recovered. A version of Bernoulli's law valid for capillary fluids is found and useful kinematic formulas for the present variational formulation are proposed. Historical comments about Gabrio Piola's contribution to analytical continuum mechanics are also presented.