SCIENCE CHIINA Physics, Mechanics Astronomy · Article· February2013Vol.56No.2:432-456 Special Topic: Fluid Mechanics doi:10.1007/s11433-012-4983-3 Some studies on mechanics of continuous mediums viewed as differential manifolds XIE XiLin, ChEn Yu shI Qian Department of Mechanics Engineering Science, Fudan University, Shanghai 200433, China Received July 2, 2012: accepted November 23, 2012: published online January 22, 2013 The continuous mediums are divided into two kinds according to their geometrical configurations, the first one is related to Eu- clidian manifolds and the other one to Riemannian manifolds/surfaces in the point of view of the modern geometry. Two kinds of inite deformation theories with respect to Euclidian and Riemannian manifolds have been developed in the present paper. Both kinds of theories include the definitions of initial and current physical and parametric configurations, deformation gradient tensors with properties, deformation descriptions, transport theories and governing equations of nature conservation laws. The essential property of the theory with respect to Euclidian manifolds is that the curvilinear coordinates corresponding to the current physical configurations include time explicitly through which the geometrically irregular and time varying physical configurations can be mapped in the diffeomorphism manner to the regular and fixed domains in the parametric space. It is quite essential to the study of the relationships between geometries and mechanics. The theory with respect to Riemannian manifolds provides the systemic ideas nd methods to study the deformations of continuous mediums whose geometrical configurations can be considered as general surfaces. The essential property of the theory with respect to Riemannian manifolds is that the thickness variation of a patch of continuous medium is represented by the surface density and its governing equation is rigorously deduced. As some applications, wakes of cylinders with deformable boundaries on the plane, incompressible wakes of a circular cylinder on fixed surfaces and axisymmetric finite deformations of an elastic membrane are numerically studied. continuous mediums, finite deformation a te bories ar Euclidian and Riemannian manifolds, intrinsic generalized Stokes for- mulas, wakes of cylinders with deformab aries, flows on surfaces, finite dynamics on deformable boundaries PACS number(s:02.10.Yy,46.05.+b,47.10.-g,47.32.c Citation: Xie X L, Chen Y, Shi Q. Some studies on mechanics of continuous mediums viewed as differential manifolds. Sci China-Phys Mech Astron, 2013. 56 432-456,doi:10.1007/11433-012-4983-3 1 Introduction sions provided through the interactions between the de- It is well known that different types of motions of different formable boundaries of birds or fishes and the surrounding gas or water. However, the intrinsic mechanisms are poorly kinds of continuous mediums has an essential role not only understood (1.21 in the natural and engineering sciences but also in the practi cal science as well. Two active aspects of modern mechanics Accompanying with the developments of the modern avi of continuous mediums will be reviewed as following ation and navigation the mechanisms of the interactions be tween the finite deformable boundaries and the surrounding 1. 1 Flows with deformable boundarie fluids have been the focus of more researches. The characters of the vortex structures with respect to cruise, start up and Birds flying in air and fishes swimming in water need propul- swerve of fishes have been systemically summarized by Tri- antafyllou et al. [3]. Furthermore, the subjective and passive *Correspondingauthor(email:xiexilin@fudan.edu.cn) controls of some kinds of fishes have been summarized by C Science China Press and Springer-Verlag Berlin Heidelberg 2013 physscichina.comwww.springerlink.com. Article . Special Topic: Fluid Mechanics SCIENCE CHINA Physics, Mechanics & Astronomy February 2013 Vol. 56 No. 2: 432–456 doi: 10.1007/s11433-012-4983-3 c Science China Press and Springer-Verlag Berlin Heidelberg 2013 phys.scichina.com www.springerlink.com Some studies on mechanics of continuous mediums viewed as differential manifolds XIE XiLin*, CHEN Yu & SHI Qian Department of Mechanics & Engineering Science, Fudan University, Shanghai 200433, China Received July 2, 2012; accepted November 23, 2012; published online January 22, 2013 The continuous mediums are divided into two kinds according to their geometrical configurations, the first one is related to Euclidian manifolds and the other one to Riemannian manifolds/surfaces in the point of view of the modern geometry. Two kinds of finite deformation theories with respect to Euclidian and Riemannian manifolds have been developed in the present paper. Both kinds of theories include the definitions of initial and current physical and parametric configurations, deformation gradient tensors with properties, deformation descriptions, transport theories and governing equations of nature conservation laws. The essential property of the theory with respect to Euclidian manifolds is that the curvilinear coordinates corresponding to the current physical configurations include time explicitly through which the geometrically irregular and time varying physical configurations can be mapped in the diffeomorphism manner to the regular and fixed domains in the parametric space. It is quite essential to the study of the relationships between geometries and mechanics. The theory with respect to Riemannian manifolds provides the systemic ideas and methods to study the deformations of continuous mediums whose geometrical configurations can be considered as general surfaces. The essential property of the theory with respect to Riemannian manifolds is that the thickness variation of a patch of continuous medium is represented by the surface density and its governing equation is rigorously deduced. As some applications, wakes of cylinders with deformable boundaries on the plane, incompressible wakes of a circular cylinder on fixed surfaces and axisymmetric finite deformations of an elastic membrane are numerically studied. continuous mediums, finite deformation theories, Euclidian and Riemannian manifolds, intrinsic generalized Stokes formulas, wakes of cylinders with deformable boundaries, flows on surfaces, finite amplitude vibrations of membranes, fluid dynamics on deformable boundaries PACS number(s): 02.10.Yy, 46.05.+b, 47.10.-g, 47.32.cCitation: Xie X L, Chen Y, Shi Q. Some studies on mechanics of continuous mediums viewed as differential manifolds. Sci China-Phys Mech Astron, 2013, 56: 432–456, doi: 10.1007/s11433-012-4983-3 1 Introduction It is well known that different types of motions of different kinds of continuous mediums has an essential role not only in the natural and engineering sciences but also in the practical science as well. Two active aspects of modern mechanics of continuous mediums will be reviewed as following. 1.1 Flows with deformable boundaries Birds flying in air and fishes swimming in water need propul- *Corresponding author (email: xiexilin@fudan.edu.cn) sions provided through the interactions between the deformable boundaries of birds or fishes and the surrounding gas or water. However, the intrinsic mechanisms are poorly understood [1,2]. Accompanying with the developments of the modern aviation and navigation, the mechanisms of the interactions between the finite deformable boundaries and the surrounding fluids have been the focus of more researches. The characters of the vortex structures with respect to cruise, start up and swerve of fishes have been systemically summarized by Triantafyllou et al. [3]. Furthermore, the subjective and passive controls of some kinds of fishes have been summarized by