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
Fish and Lauder [4]. The vortex structures introduced by the manifold could be considered as a subspace in three dimen finite deformations of fins have been studied through com- sional Euclid space with a certain bulk. Thus, a group of putational fluid dynamics(CFD) by other researchers [5,6]. three numbers/ coordinates independently must be needed to In addition, the swim rules with respect to a single fish or describe its configuration. Certainly, the Cartesian coordi- a group of fishes have been studies based on CFD and con- nates is the naturally choose trol theory by Wu and Wang (1, 7]. Conversely, Du and Sun 8] discovered that the flying performances of insects could ensional motions be improved drastically by the finite deformations of wings as compared to the solid vibrations of wings. Lu and Yin Two dimensional motions are generally referred to as the mo- [9] pointed out that the separation could be effectively sup- tions on the plane or on a curved surface embedded in the pressed as the boundary of a two dimensional tunnel does three dimensional Euclid space. Physically, a two dimen travailing wave deformation. Wu et al. [10] put forward the sional motions should be considered as a model for the mo- concept of fluid roller bearing that could be utilized in the tion of continuous medium that is limited to a quite thin layer drag reduction of wings and so on. In addition, it is discov- as described by Irion [ 19] ered that the street of Karman vortices of a circular cylinder The typical two dimensional motion could be considered could be suppressed completely as some kind of travailing as the flowing soap film for fluid mechanics. Zhang et al. [20) wave deformations is trigged on its boundary [ll] reported the motions of flexible filaments with their wakes in On numerical studies, the popular methods of CFD utilized a flowing soap film in which the filaments should be consid to simulate and investigate the flows with deformable bound- ered as parts of the boundaries of the soap film. In addition aries could be concluded into three kinds. The first is termed the soap film flows around a circular cylinder generally po of the whole flow fields in which certain kind of additional flow visualizations using light interference techniques. It can body forces should be added to the general Navier-Stokes be noted that the discoveries of elegant vortical structures of equation(NSE). Correspondingly, the Cartesian coordinates the flows of soap films are accompanied with the considerable is commonly adopted to discrete NSE and boundaries. The surface density or film thickness variations. The amplitude of second is the transformation of the reference systems as usu- the density variation could be about 20% of its characteristic ally used in the system constituted by rigid bodies. Unfortu- or mean value but the mechanism of the density variation is nately, this kind of methods is only effective in few cases. The unreported. third is the transformation of coordinates that is widely used Two dimensional motions usually related to the motions in CFD that generally transform the differential equations of thin enough membranes but their amplitudes could take with respect to Cartesian coordinates to the curvilinear coor- finite values in regard to solid mechanics. Gutierrez [21]re- dinates in order to detail with the curved boundaries. Usually, ported the results from a series of numerical simulations that the constructed curvilinear coordinates are time-independent. put emphasis on the determination of the lower natural fre Luo and Bewley [13] studied the contra-variant form of nse quencies for the transverse vibration of annular membranes with respect to the time-dependent curvilinear coordinates in including the special case of a solid circular membrane when the point of view of the coordinate transformation. Compar- the mass per unit area varies linearly, quadratically, and cubi atively, the experimental studies are seldom reported due to cally with the radial coordinate. Buchanan[22]studied den the difficulties in the controls of the deformable boundaries sity which was assumed to vary linearly along the diameter and the measurements associated of the membrane and could vary circumferentially. Tsiatas On theoretical studies, the primary achievements usually and Katsikadelis [23] presented a solution for the problem originated from the point of view of vorticity and vortex dy- of initially non-flat membranes based on a new formulation namics could be concluded as three aspects, mainly devel- of the governing differential equations in terms of displace oped by Prof Wu J.Z. with his collaborators. Firstly is on the ments. They solved the problem by direct integration of the representation of the velocity gradient, subsequently one differential equations. In addition, the nonlinear membrane attain more advanced representation of the deformation rate model has been used to simulate the nonlinear vibrations of tensor on the deformable boundary [14]. Secondly is the in- single layer graphene sheets with large amplitudes [24]. The tegral representations of the resultant force and moment of related research indicates the promising applications of the momentum imposing on the deformable body surrounded by theory of membranes in nano-technology. fluids [15,16]. Thirdly is the coupling representations of the The vibration analysis of composite circular and annular boundary fluxes of the vorticity and dilation [17] membranes has been carried out with some results obtained Although a flow with deformable boundary can be pos- through exact solutions, energy methods and finite element essed of abundant attractive dynamics, its geometrical con- analysis. Although its variations are usually considered, the figuration is generally Euclidian manifold as viewed from surface density of a membrane is usually assumed as a cer- modern differential geometry [18]. The reputed Euclidian tain function of the position referred seldom to its governing
Xie X L, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 433 Fish and Lauder [4]. The vortex structures introduced by the finite deformations of fins have been studied through computational fluid dynamics (CFD) by other researchers [5,6]. In addition, the swim rules with respect to a single fish or a group of fishes have been studies based on CFD and control theory by Wu and Wang [1,7]. Conversely, Du and Sun [8] discovered that the flying performances of insects could be improved drastically by the finite deformations of wings as compared to the solid vibrations of wings. Lu and Yin [9] pointed out that the separation could be effectively suppressed as the boundary of a two dimensional tunnel does travailing wave deformation. Wu et al. [10] put forward the concept of fluid roller bearing that could be utilized in the drag reduction of wings and so on. In addition, it is discovered that the street of Karman vortices of a circular cylinder could be suppressed completely as some kind of travailing wave deformations is trigged on its boundary [11]. On numerical studies, the popular methods of CFD utilized to simulate and investigate the flows with deformable boundaries could be concluded into three kinds. The first is termed as immersed boundary method [12]. Its essential idea is to regard the solid bodies embedded in the flow field as some parts of the whole flow fields in which certain kind of additional body forces should be added to the general Navier-Stokes equation (NSE). Correspondingly, the Cartesian coordinates is commonly adopted to discrete NSE and boundaries. The second is the transformation of the reference systems as usually used in the system constituted by rigid bodies. Unfortunately, this kind of methods is only effective in few cases. The third is the transformation of coordinates that is widely used in CFD that generally transform the differential equations with respect to Cartesian coordinates to the curvilinear coordinates in order to detail with the curved boundaries. Usually, the constructed curvilinear coordinates are time-independent. Luo and Bewley [13] studied the contra-variant form of NSE with respect to the time-dependent curvilinear coordinates in the point of view of the coordinate transformation. Comparatively, the experimental studies are seldom reported due to the difficulties in the controls of the deformable boundaries and the measurements associated. On theoretical studies, the primary achievements usually originated from the point of view of vorticity and vortex dynamics could be concluded as three aspects, mainly developed by Prof.Wu J.Z. with his collaborators. Firstly is on the representation of the velocity gradient, subsequently one can attain more advanced representation of the deformation rate tensor on the deformable boundary [14]. Secondly is the integral representations of the resultant force and moment of momentum imposing on the deformable body surrounded by fluids [15,16]. Thirdly is the coupling representations of the boundary fluxes of the vorticity and dilation [17]. Although a flow with deformable boundary can be possessed of abundant attractive dynamics, its geometrical con- figuration is generally Euclidian manifold as viewed from modern differential geometry [18]. The reputed Euclidian manifold could be considered as a subspace in three dimensional Euclid space with a certain bulk. Thus, a group of three numbers/coordinates independently must be needed to describe its configuration. Certainly, the Cartesian coordinates is the naturally choose. 1.2 Two dimensional motions Two dimensional motions are generally referred to as the motions on the plane or on a curved surface embedded in the three dimensional Euclid space. Physically, a two dimensional motions should be considered as a model for the motion of continuous medium that is limited to a quite thin layer as described by Irion [19]. The typical two dimensional motion could be considered as the flowing soap film for fluid mechanics. Zhang et al. [20] reported the motions of flexible filaments with their wakes in a flowing soap film in which the filaments should be considered as parts of the boundaries of the soap film. In addition, the soap film flows around a circular cylinder generally positioned perpendicular to the planar soap have been studied. Currently, the flows of soap films are generally studied by the flow visualizations using light interference techniques. It can be noted that the discoveries of elegant vortical structures of the flows of soap films are accompanied with the considerable surface density or film thickness variations. The amplitude of the density variation could be about 20% of its characteristic or mean value but the mechanism of the density variation is unreported. Two dimensional motions usually related to the motions of thin enough membranes but their amplitudes could take finite values in regard to solid mechanics. Gutierrez [21] reported the results from a series of numerical simulations that put emphasis on the determination of the lower natural frequencies for the transverse vibration of annular membranes including the special case of a solid circular membrane when the mass per unit area varies linearly, quadratically, and cubically with the radial coordinate. Buchanan [22] studied density which was assumed to vary linearly along the diameter of the membrane and could vary circumferentially. Tsiatas and Katsikadelis [23] presented a solution for the problem of initially non-flat membranes based on a new formulation of the governing differential equations in terms of displacements. They solved the problem by direct integration of the differential equations. In addition, the nonlinear membrane model has been used to simulate the nonlinear vibrations of single layer graphene sheets with large amplitudes [24]. The related research indicates the promising applications of the theory of membranes in nano-technology. The vibration analysis of composite circular and annular membranes has been carried out with some results obtained through exact solutions, energy methods and finite element analysis. Although its variations are usually considered, the surface density of a membrane is usually assumed as a certain function of the position referred seldom to its governing
Xie XL, et aL. Sci China- Phys Mech Astron February (2013) VoL. 56 No. 2 equation. In addition, the elementary equation for infinitesi- field with respect to the canonical basis. In the present stud- mal amplitude vibration is usually adopted with some modi- ies, any tensor field is represented with respect to the local fications such as letting the density be a function of position covariant or contra-variant bases that is of benefit to set up and so on the relationships between mechanics and geometries Either a thin enough layer of fluids or a membrane in- stituted by solids, its geometrical configuration could be re- 2.2 Physical and parametric configurations garded as a two dimensional surface that could be naturally taken as a Riemannian manifold in three dimensional euclid- The initial and current physica cal configurations with their re- ian space. Furthermore, the relationships between geome lated parametric configurations is shown in Figure 2, where tries and mechanics may be become more active particularly (X-I and Is -i denote the Cartesian and curvilinear in the case that the configuration of the continuous medium coordinates corresponding to the initial physical configura is Riemannian manifold. Yin et al. pointed out that the tion respectively, (Xi),, and (ria, denote the Cartesian and some kinds of gradients of curvatures could be considered curvilinear coordinates corresponding to the current phy as some novel kinds of forces on bio-membrances [25] and cal configuration respectively. Subsequently, the former two micro/nano curved surfaces [26] groups of coordinates are Lagrangian coordinates and the lat ter ones are Eulerian coordinates. In the whole paper, the in the point of view of the mechanics of continuous medi- aE 2 13 mary ral Einstein summatio Two kinds of the motions of continuous mediums as reviewed he velocity of a fluid partial is defined generally as the above have been theoretically studied in the present paper of the change of its position with respect to the time that ums [27] with some applications. The primary contents of he paper are organized as follows: (1) Finite deformation vax(x0)1(,)+a(x0)=g;+a1(x,D heory with respect to curvilinear coordinates correspond- ing to current physical configurations including time explic- where i: =dx/at( t) and the additional term ax/at(x, t)is itly: (2) Finite deformation theory with respect to continuous due to the curvilinear coordinates x=X(x, t) including time mediums whose geometrical configurations are two dimen- explicitly. Consequently, the general material sional Riemannian manifolds; (3) Case studies on the defor- any tensor field (x, t) takes the following form mations/motions of continuous mediums viewed either as Eu- clidian manifolds or Riemannian manifolds d-(,t)=-(x,t)+i-(x,t) 2 Finite deformation theory with respect to (x,0)+('8). 8 curvilinear coordinates corresponding to cur- rent physical configurations including time ex pli city (,1)+VB)(aa山) 2.1 View of the mapping where o a(x)g denotes the full gradient operator with As shown in Figure 1. the cu physical configuration is espect to Eulerian coordinates changing its geometrical configuration as the time is vary ing. However, there exists so called diffeomorphism includ- 2.3 Deformation gradient tensor ing time explicitly through which the current physical con- As shown in Figure 2, the relationship between the vectors figuration could be mapped bijectively onto a parametric do- connecting the same point a and b in the current physical main that is geometrical regular and is keeping unchanged in configuration and initial physical configuration, denoted its configuration as the time is varying. Subsequently, one rabl, and rable respectively, can be represented as follows ac can set up in the parametric domain the related partial dif- cording to differential calculus ferential equations(PDEs) that are the representations of the natural laws such as the mass or momentum conservation. It can be realized through the tensor filed analysis with respe rab=X(x(5+△,1,1)-X(x,1),1)=(5,n)g:(x1)·△ to the general curvilinear coordinates It should be pointed out that the diffeomorphism could be a(1(8()(“ce)=:Frap ust regarded as the curvilinear coordinates that is widely uti- lized in the computational fluid and solid mechanics always where F termed as the deformation gradient tensor. Its form with the aim to transfer the PDEs defined originally in the is same to the general case but the base vectors with respect general Cartesian coordinates to the curvilinear ones. In other to the current physical configuration such as gi(r, t) are de words, the variables are usually the components of a tensor pendent on the time explicitly
434 Xie X L, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 equation. In addition, the elementary equation for infinitesimal amplitude vibration is usually adopted with some modi- fications such as letting the density be a function of position and so on. Either a thin enough layer of fluids or a membrane instituted by solids, its geometrical configuration could be regarded as a two dimensional surface that could be naturally taken as a Riemannian manifold in three dimensional Euclidian space. Furthermore, the relationships between geometries and mechanics may be become more active particularly in the case that the configuration of the continuous medium is Riemannian manifold. Yin et al. pointed out that the some kinds of gradients of curvatures could be considered as some novel kinds of forces on bio-membrances [25] and micro/nano curved surfaces [26]. 1.3 Summary Two kinds of the motions of continuous mediums as reviewed above have been theoretically studied in the present paper in the point of view of the mechanics of continuous mediums [27] with some applications. The primary contents of the paper are organized as follows: (1) Finite deformation theory with respect to curvilinear coordinates corresponding to current physical configurations including time explicitly; (2) Finite deformation theory with respect to continuous mediums whose geometrical configurations are two dimensional Riemannian manifolds; (3) Case studies on the deformations/motions of continuous mediums viewed either as Euclidian manifolds or Riemannian manifolds. 2 Finite deformation theory with respect to curvilinear coordinates corresponding to current physical configurations including time explicitly 2.1 View of the mapping As shown in Figure 1, the current physical configuration is changing its geometrical configuration as the time is varying. However, there exists so called diffeomorphism including time explicitly through which the current physical con- figuration could be mapped bijectively onto a parametric domain that is geometrical regular and is keeping unchanged in its configuration as the time is varying. Subsequently, one can set up in the parametric domain the related partial differential equations (PDEs) that are the representations of the natural laws such as the mass or momentum conservation. It can be realized through the tensor filed analysis with respect to the general curvilinear coordinates. It should be pointed out that the diffeomorphism could be just regarded as the curvilinear coordinates that is widely utilized in the computational fluid and solid mechanics always with the aim to transfer the PDEs defined originally in the general Cartesian coordinates to the curvilinear ones. In other words, the variables are usually the components of a tensor field with respect to the canonical basis. In the present studies, any tensor field is represented with respect to the local covariant or contra-variant bases that is of benefit to set up the relationships between mechanics and geometries. 2.2 Physical and parametric configurations The initial and current physical configurations with their related parametric configurations is shown in Figure 2, where { ◦ XA} 3 A=1 and {ξA} 3 A=1 denote the Cartesian and curvilinear coordinates corresponding to the initial physical configuration respectively, {Xi } 3 i=1 and {xi } 3 i=1 denote the Cartesian and curvilinear coordinates corresponding to the current physical configuration respectively. Subsequently, the former two groups of coordinates are Lagrangian coordinates and the latter ones are Eulerian coordinates. In the whole paper, the general Einstein summation convention is adopted. The velocity of a fluid partial is defined generally as the rate of the change of its position with respect to the time that is V ∂X ∂xi (x, t) ∂xi ∂t (ξ, t) + ∂X ∂t (x, t) =: x˙ i gi + ∂X ∂t (x, t), where x˙i := ∂xi /∂t(ξ, t) and the additional term ∂X/∂t(x, t) is due to the curvilinear coordinates X = X(x, t) including time explicitly. Consequently, the general material derivative of any tensor field Φ(x, t) takes the following form Φ˙ ∂Φ ∂t (ξ, t) = ∂Φ ∂t (x, t) + x˙ i ∂Φ ∂xi (x, t) =∂Φ ∂t (x, t) + x˙ i gi · gl ⊗ ∂Φ ∂xl (x, t) =∂Φ ∂t (x, t) + V − ∂X ∂t (x, t) · ( ⊗ Φ) where := ∂ ∂xs (x)gs denotes the full gradient operator with respect to Eulerian coordinates. 2.3 Deformation gradient tensor As shown in Figure 2, the relationship between the vectors connecting the same point a and b in the current physical configuration and initial physical configuration, denoted by rab| t V and rab| o V respectively, can be represented as follows according to differential calculus: rab| t V =X(x(ξ + ξ, t), t) − X(x(ξ, t), t) = ∂xi ∂ξA (ξ, t)gi(x, t) · ξA = ∂xi ∂ξA (ξ, t)gi(x, t) ⊗ GA(ξ) · (ξBGB(ξ)) =: F · rab| o V , where F termed as the deformation gradient tensor. Its form is same to the general case but the base vectors with respect to the current physical configuration such as gi(x, t) are dependent on the time explicitly.
L, et al. Sci China-Phys Mech Astron February (2013) VoL. 56 (x,n) X(x1):D~xR3(x母={0}→(x=x( x,f):(sR(8.en)/@sing Figure 1( Color online) Sketch of the diffeomorphism including time explicitly 92(x(D.0 9,(x(ea, t). t) x- curve 93(x(D.0 G3(5a (=X(2) x3-curve Current parametric Figure 2( Color online) Sketch of the physical and parametric configurations for continuous mediums considered as Euclidian manifolds where the curvilinear coordinates corresponding to the current physical configurations include time explicitly. The general properties of the deformation gradient tensor could be concluded as follows )(G,g1)g18g1=F818g Proposition 1( Properties of deformation gradient tensor). Subsequently, the determinant can be calculated readily detF=一de=F, =du1=d(0x小G,g F=(VeD v,「ax det --(5, t) F 6Fl, where= v.a=口.V (2)In the present case, the representation of the velocity gra- dient is Proof (1) According to the definition of the deformation gradient tensor with the transformation between the base vectors with L=8=(+线g respect to the initial and current physical configurations, one where x(x,1)=(x,D8(xD)=-(,1)g(x,D) F(5,1);(x,1)8G(E To consider. 2((x,D8|,g)g aEA(SD8:8G
Xie X L, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 435 Figure 1 (Color online) Sketch of the diffeomorphism including time explicitly. Figure 2 (Color online) Sketch of the physical and parametric configurations for continuous mediums considered as Euclidian manifolds where the curvilinear coordinates corresponding to the current physical configurations include time explicitly. The general properties of the deformation gradient tensor could be concluded as follows. Proposition 1 (Properties of deformation gradient tensor). detF = √g √ G · det ∂xi ∂ξA (ξ, t) := |F|, F˙ = (V ⊗ ) · F, ˙ |F| = θ|F|, where θ V · = · V. Proof: (1) According to the definition of the deformation gradient tensor with the transformation between the base vectors with respect to the initial and current physical configurations, one has F ∂xi ∂ξA (ξ, t)gi(x, t) ⊗ GA(ξ) = ∂xi ∂ξA (ξ, t)gi(x, t) ⊗ (GA, gj )R3 gj = ∂xi ∂ξA (ξ, t)(GA, gj)R3 gi ⊗ gj = Fi · j gi ⊗ gj . Subsequently, the determinant can be calculated readily |F| = det[Fi · j ] = det ∂xi ∂ξA (ξ, t) · (GA, gj )R3 = √g √ G det ∂xi ∂ξA (ξ, t) . (2) In the present case, the representation of the velocity gradient is L := V ⊗ ∂V ∂xl (x, t) ⊗ gl = ∂ ∂xl x˙ + ∂X ∂t (x, t) ⊗ gl , where x˙(x, t) x˙ i (x, t)gi(x, t) = ∂xi ∂t (ξ, t) gi(x, t). To consider, F˙ = ∂x˙ i ∂ξA (ξ, t) gi ⊗ GA
436 Xie XL, et aL. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 aer(s, p oj(x 0).N ms 4, o oGA 2.4 Deformation descriptions ax (E,D) gradient tensor as studied in the previous subsection, could deduce the deformation descriptions through the gen- ax ar cx, laG eral analysis ways [27]. The whole descriptions of deforma- ns are divided into fou in the present noted by Gl to G4 respectively (r, t)@GA Gl Transformations of the vectored curve, surface and vol ax ume elements between the initial and current physical config (x,D)⑧ DEx(.08,8G=L (=F. ()One has the following identities aX a×|(,)=(F”) d a=),=a(xd( a可(y=闪a动可 =[B1,g2,g3 办(x,D+x1g8+ X():[a,b3A→X(X(x((y),D),D) x(O:[a,b]3A→x(±r() are the vector valued maps of the material curves embedded =s+元g{1g23 in the initial and current physical configurations respectively 下+小 The maps of the material surfaces X(, u), X(, u) and th ones of material material volumes X(, 4, y), X(, u, y) with sely, the divergence of the velocity can be represented respect to the initial and current physical configurations are follows similarly defined. G2 Transformations of thethe norms of curve, surface el (x,t)·g (x,1)·g ements between the initial and current physical configura tions: vii n)·g, Vr=-(x, n)+r (p)={F And the following identity is keeping valid a×w D× m0x=kn=“c(63M加 1a√ (x,D) Then the identity is proved. da d)=l. dr As a summary, in the present case that the curvilinear co- ordinates corresponding to the current physical configuration ncluding time explicitly, the representations of the velocity o=p(ar e] A, w)=B.or and the material derivative of any tensor field are differ from he ones in the general case, but the fundamental properties of the deformation gradient tensor presented in the intrinsic forms are the same as the general ones 碗可)=0aa dn du dy
436 Xie X L, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 + ∂xi ∂ξA (ξ, t) ∂gi ∂xj (x, t) · x˙j + ∂gi ∂t (x, t) ⊗ GA = ∂xs ∂ξA (ξ, t) ∂x˙ i ∂xs (x, t)gi + x˙j ∂gj ∂xs (x, t) + ∂ ∂xs ∂X ∂t (x, t) ⊗ GA = ∂xs ∂ξA (ξ, t) ∂ ∂xs x˙ + ∂X ∂t (x, t) ⊗ GA = ∂xs ∂ξA (ξ, t) ∂V ∂xs (x, t) ⊗ GA = ∂V ∂xt (x, t) ⊗ gt · ∂xs ∂ξA (ξ, t)gs ⊗ GA = L · F. (3) One has the following identities d dt det ∂xi ∂ξA (ξ, t) = ∂x˙s ∂xs (x, t) det ∂xi ∂ξA (ξ, t). d dt √g = d dt [g1, g2, g3] = ∂g1 ∂xs (x, t)x˙ s + ∂g1 ∂t (x, t), g2, g3 + ··· + g1, g2, dg3 dt =Γs stx˙ t √g + ∂ ∂t g1, g2, g3 = √g Γs stx˙ t + 1 √g ∂ √g ∂t (x, t) . Conversely, the divergence of the velocity can be represented as follows: V · ∂V ∂xl (x, t) · gl = ∂ ∂xl x˙ + ∂X ∂t (x, t) · gl , =∇lx˙l + ∂gl ∂t (x, t) · gl , where ∇lx˙l = ∂x˙ l ∂xl (x, t) + Γl lsx˙ s . And the following identity is keeping valid ∂gl ∂t (x, t) · gl = glkgk · ∂gl ∂t (x, t) = 1 2 glk ∂glk ∂t (x, t) = 1 √g ∂ √g ∂t (x, t). Then the identity is proved. As a summary, in the present case that the curvilinear coordinates corresponding to the current physical configuration including time explicitly, the representations of the velocity and the material derivative of any tensor field are differ from the ones in the general case, but the fundamental properties of the deformation gradient tensor presented in the intrinsic forms are the same as the general ones. 2.4 Deformation descriptions Based on the fundamental properties of the deformation gradient tensor as studied in the previous subsection, one could deduce the deformation descriptions through the general analysis ways [27]. The whole descriptions of deformations are divided into four groups in the present paper, denoted by G1 to G4 respectively. G1 Transformations of the vectored curve, surface and volume elements between the initial and current physical configurations: d t X dλ (λ) = F · d o X dλ (λ), ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ ∂ t X ∂λ × ∂ t X ∂μ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ) = (|F|F−∗) · ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ ∂ o X ∂λ × ∂ o X ∂μ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ), ⎡ ⎢⎢⎢⎢⎢⎢⎢⎣ ∂ t X ∂λ , ∂ t X ∂μ , ∂ t X ∂γ ⎤ ⎥⎥⎥⎥⎥⎥⎥⎦ (λ, μ, γ) = |F| ⎡ ⎢⎢⎢⎢⎢⎢⎢⎣ ∂ o X ∂λ , ∂ o X ∂μ , ∂ o X ∂γ ⎤ ⎥⎥⎥⎥⎥⎥⎥⎦ (λ, μ, γ), where t X(λ):[a, b] λ → t X(λ) X(x(ξ(γ), t), t), o X(λ):[a, b] λ → o X(λ) o X(ξ(λ)) are the vector valued maps of the material curves embedded in the initial and current physical configurations respectively. The maps of the material surfaces t X(λ, μ), o X(λ, μ) and the ones of material material volumes t X(λ, μ, γ), o X(λ, μ, γ) with respect to the initial and current physical configurations are similarly defined. G2 Transformations of the the norms of curve, surface elements between the initial and current physical configurations: d t X dλ (λ) R3 = (F∗ · F) 1 2 · d o X dλ (λ) R3 , ∂ t X ∂λ × ∂ t X ∂μ R3 (λ, μ) = |F| (F∗ · F) −1 2 · ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ ∂ o X ∂λ × ∂ o X ∂μ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ) R3 . G3 Material derivatives of the vectored curve, surface and volume elements in the current physical configuration: ˙ d t X dλ (λ) = L · d t X dλ (λ), ⎛ ˙ ⎜⎜⎜⎜⎜⎜⎜⎝ ∂ t X ∂λ × ∂ t X ∂μ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ) = B · ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ ∂ t X ∂λ × ∂ t X ∂μ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ), ⎡ ˙ ⎢⎢⎢⎢⎢⎢⎢⎣ ∂ t X ∂λ , ∂ t X ∂μ , ∂ t X ∂γ ⎤ ⎥⎥⎥⎥⎥⎥⎥⎦ (λ, μ, γ) = θ ⎡ ⎢⎢⎢⎢⎢⎢⎢⎣ ∂ t X ∂λ , ∂ t X ∂μ , ∂ t X ∂γ ⎤ ⎥⎥⎥⎥⎥⎥⎥⎦ (λ, μ, γ).
Xie XL, et aL. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 437 where B= 61-0. v is generally termed as the surface defor- where the value of the volume transformation term is consid mation gradient tensor. ered as positive without lost of the generality and the gen G4 Material derivatives of the norms of the curve, surface eralized Gauss-Ostrogradskii formula is adopted in the last elements in the current physical configuration identity Accompanying the deformation description denoted as G4 with the curve and surface integrals of the first kind, one aa=(rD Dla a), rives readily at the following transport theories of the first kind The curve transport of the first kind dX aX /( Cod =d c ()da where D=(L+L)/2 is the rate of the change of the de formation, T and n are denoted for the unit tangent vector of curve element and the normal vector of the surface element ddn+,Φ(r:D.r)dl respectively The surface transport of the first kind 2.5 Transport theories d d with the curve and surface integrals of the second kinds one arrives readily at the following transport theories that d do+ aedo-.a(nDn)do are termed as the transport theories of the second kind in the present paper. The denotation o- represents any meaningful field operation. ka1)+口:(sΦ)dr The curve transport of the second kind X (x,1)(8Φ)d-.Φ(m:D.n)dσ d add 3 Finite deformation theory with respect to do-rdl+,Φ。-(L.r)dl. continuous mediums whose geometrical config urations are two dimensional riemannian man- The surface transport of the second kind ifolds d 3.1 Kinematics and kinetics of the finite deformation 3.1.1 Physical and pe (,g)d an au As shown in Figure 3, the general moving surface could represented by the following vectored valued map d。-(B·n)da The volume transport 2(s, 0): DE 3xs= ∑(x,1)4X|(x,t)∈R where De C R is termed as the parametric domain. One aX aX can define the motion of continuous medium that is limited d an du ay on the surface in the parametric domain as the following CP diffeomorphism ddr+.eΦdr (E,D)∈CP(V,Vx) x,D)+口·(WΦ)dn where Va is termed as the initial parametric configuration, (x,D)·(口Φ)d corresponding domains of actions denoted as VE: =2(Ve, to) (x,t)dr+d,Φ(V·n)d and VE:=2(Ve, t)are termed as the initial and current phys-
Xie X L, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 437 where B θI− ·V is generally termed as the surface deformation gradient tensor. G4 Material derivatives of the norms of the curve, surface elements in the current physical configuration: ˙ d t X dλ R3 (λ) = (τ · D · τ) d t X dλ R3 (λ), ˙ ∂ t X ∂λ × ∂ t X ∂μ R3 (λ, μ) = (θ − n · D · n) ∂ t X ∂λ × ∂ t X ∂μ R3 (λ, μ). where D (L + L∗ )/2 is the rate of the change of the deformation, τ and n are denoted for the unit tangent vector of curve element and the normal vector of the surface element respectively. 2.5 Transport theories Accompanying the deformation description denoted as G3 with the curve and surface integrals of the second kinds, one arrives readily at the following transport theories that are termed as the transport theories of the second kind in the present paper. The denotation ◦− represents any meaningful field operation. The curve transport of the second kind d dt t C Φ ◦ −τdl = d dt b a Φ ◦ −d t X dλ (λ)dλ = t C Φ˙ ◦ −τdl + t C Φ ◦ −(L · τ)dl. The surface transport of the second kind d dt t Σ Φ ◦ −ndσ = d dt Dλμ Φ ◦ − ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ ∂ t X ∂λ × ∂ t X ∂μ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ)dσ = t Σ Φ˙ ◦ −ndσ + t Σ Φ ◦ −(B · n)dσ. The volume transport d dt t V Φdσ = d dt Dλμγ Φ ⎡ ⎢⎢⎢⎢⎢⎢⎢⎣ ∂ t X ∂λ , ∂ t X ∂μ , ∂ t X ∂γ ⎤ ⎥⎥⎥⎥⎥⎥⎥⎦ (λ, μ, γ)dτ = t V Φ˙ dτ + t V θ Φdτ = t V ∂Φ ∂t (x, t) + · (V ⊗ Φ) dτ − t V ∂X ∂t (x, t) · ( ⊗ Φ)dτ = t V ∂Φ ∂t (x, t) dτ + ∂ t V Φ(V · n)dτ − t V ∂X ∂t (x, t) · ( ⊗ Φ)dτ. where the value of the volume transformation term is considered as positive without lost of the generality and the generalized Gauss-Ostrogradskii formula is adopted in the last identity. Accompanying the deformation description denoted as G4 with the curve and surface integrals of the first kind, one arrives readily at the following transport theories of the first kind. The curve transport of the first kind d dt t C Φ dl= d dt b a Φ d t X dλ R3 (λ)dλ = t C Φ˙ dl + t C Φ (τ · D · τ)dl. The surface transport of the first kind d dt t Σ Φ dσ= d dt Dλμ Φ ∂ t X ∂λ × ∂ t X ∂μ R3 (λ, μ)dσ = t Σ Φ˙ dσ + t Σ Φ θ dσ − t Σ Φ (n · D · n)dσ = t Σ ∂Φ ∂t (x, t) + · (V ⊗ Φ) dσ − t Σ ∂X ∂t (x, t) · ( ⊗ Φ) dσ − t Σ Φ (n · D · n)dσ. 3 Finite deformation theory with respect to continuous mediums whose geometrical configurations are two dimensional Riemannian manifolds 3.1 Kinematics and kinetics of the finite deformation 3.1.1 Physical and parametric configurations As shown in Figure 3, the general moving surface could be represented by the following vectored valued map Σ(xΣ, t) : DΣ xΣ = ⎡ ⎢⎢⎢⎢⎢⎣ x1 Σ x2 Σ ⎤ ⎥⎥⎥⎥⎥⎦ → Σ(xΣ, t) ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ X1 Σ X2 Σ X3 Σ ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (xΣ, t) ∈ R3 , where DΣ ⊂ R2 is termed as the parametric domain. One can define the motion of continuous medium that is limited on the surface in the parametric domain as the following Cpdiffeomorphism xΣ = xΣ(ξΣ, t) ∈ Cp ( ◦ VξΣ , t VxΣ ). where ◦ VξΣ is termed as the initial parametric configuration, t VxΣ is the current parametric configuration. Subsequently, the corresponding domains of actions denoted as ◦ VΣ := Σ( ◦ VξΣ , t0) and t VΣ := Σ( t VξΣ , t) are termed as the initial and current physical configurations, respectively.
438 Xie XL, et aL. Sri China-Phys Mech Astron February (2013) Vol. 56 No. 2 Initial physical (x, x2=x(52,D configuration Current parametric Figure 3(Color online) Sketch of the physical and parametric configurations for a continuous mediums whose geometrical configurations can be considered as general surfaces, in other words riemannian The velocity of a mass particle on the surface is defined in the same meaning as the change of the rate of its position (5s,1)g(x,1)G′(x)△s2GB(x) with respect to the time 一(x(,D),D) (x(,D),t) ∑ =(x(,D),1)+g,(x(E,D,1 %个(+4)-远小 where i: ax/at(S, t). Subsequently, the representation of the material derivative of any tensor filed defined on the ae(5x. 18 (xE, D)8G(E)ET(R continuous medium takes the following form is the deformation gradient tensor in the present case (x,D),)+(x(5,1,1 The fundamental properties of the deformation gradient tensor could be concluded as follows ot e,t)+ Proposition 2(Properties of deformation gradient tensor). (x,1)+ F=(s品),F whereas os a =a(x,)+ det F=edet F. where= where d: =(xelg'denotes the full gradient operator on Proof the surface with respect to Eulerian coordinates (1)In the present case, one just has the rate of the change of any tensor field along a certain curve on the surface, there 3.1.2 Deformation gradient tensor fore the so called full gradient with respect to the Eulerian As in the general case, the deformation gradient tensor can coordinates, say d, is defined as: also be defined as the transformation between the differential the initial and current physical configurations, that is spect to segments connecting the same pairing points with (x2,8g 2(x+A,1)-(,1)(x,D)·(E2,1)·A The proof of this property is a verbatim repeat of the one in the sect. 2.4 (5,1)(x,1A (2) It is evident that the determinant of the deformation gradi ent tensor in the present case is naturally equal to naught due
438 Xie X L, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 Figure 3 (Color online) Sketch of the physical and parametric configurations for a continuous mediums whose geometrical configurations can be considered as general surfaces, in other words Riemannian manifolds. The velocity of a mass particle on the surface is defined in the same meaning as the change of the rate of its position with respect to the time Σ V Σ˙ ∂Σ ∂t (xΣ(ξΣ, t), t) + x˙ i Σ ∂Σ ∂xi Σ (xΣ(ξΣ, t), t) = ∂Σ ∂t (xΣ(ξΣ, t), t) + x˙ s Σ Σ gs (xΣ(ξΣ, t), t) , where x˙s Σ := ∂xs Σ/∂t(ξΣ, t). Subsequently, the representation of the material derivative of any tensor filed defined on the continuous medium takes the following form Φ˙ ∂Φ ∂t (ξΣ, t) = ∂Φ ∂t (xΣ(ξΣ, t), t) + x˙ s Σ ∂Φ ∂xs Σ (xΣ(ξΣ, t), t) = ∂Φ ∂t (xΣ, t) + x˙ s Σ Σ gs · ⎛ ⎜⎜⎜⎜⎝ Σ gl ⊗ ∂Φ ∂xl Σ (xΣ, t) ⎞ ⎟⎟⎟⎟⎠ = ∂Φ ∂t (xΣ, t) + x˙ s Σ Σ gs · Σ ⊗ Φ = ∂Φ ∂t (xΣ, t) + Σ V − ∂Σ ∂t (xΣ, t) · Σ ⊗ Φ , where := ∂ ∂xs Σ (xΣ) Σ gs denotes the full gradient operator on the surface with respect to Eulerian coordinates. 3.1.2 Deformation gradient tensor As in the general case, the deformation gradient tensor can also be defined as the transformation between the differential segments connecting the same pairing points with respect to the initial and current physical configurations, that is Σ(ξΣ + ΔξΣ, t) − Σ(ξΣ, t) ∂Σ ∂xi Σ (xΣ, t) · ∂xi Σ ∂ξA Σ (ξΣ, t) · ΔξA Σ = ∂xi Σ ∂ξA Σ (ξΣ, t) Σ gi(xΣ, t) · ΔξA Σ = ⎡ ⎢⎢⎢⎢⎣ ∂xi Σ ∂ξA Σ (ξΣ, t) Σ gi(xΣ, t) ⊗ Σ GA(xΣ) ⎤ ⎥⎥⎥⎥⎦ · ΔξB Σ Σ GB(xΣ) ⎡ ⎢⎢⎢⎢⎣ ∂xi Σ ∂ξA Σ (ξΣ, t) Σ gi(xΣ, t) ⊗ Σ GA(xΣ) ⎤ ⎥⎥⎥⎥⎦ · ◦ Σ(ξΣ + ΔξΣ) − ◦ Σ(ξΣ) , where Σ F ∂xi Σ ∂ξA Σ (ξΣ, t) Σ gi(xΣ, t) ⊗ Σ GA(xΣ) ∈ T2 (R3 ) is the deformation gradient tensor in the present case. The fundamental properties of the deformation gradient tensor could be concluded as follows. Proposition 2 (Properties of deformation gradient tensor). d dt Σ F = ( Σ V ⊗ Σ ) · Σ F, where Σ Σ gs ∂ ∂xs Σ , d dt det Σ F = Σ θ det Σ F, where Σ θ Σ V · Σ = Σ · Σ V. Proof: (1) In the present case, one just has the rate of the change of any tensor field along a certain curve on the surface, therefore the so called full gradient with respect to the Eulerian coordinates, say Σ Φ, is defined as: Σ Φ ⊗ Σ ∂ Σ Φ ∂xs Σ (xΣ, t) ⊗ Σ gs . The proof of this property is a verbatim repeat of the one in the sect. 2.4. (2) It is evident that the determinant of the deformation gradient tensor in the present case is naturally equal to naught due
Xie XL, et aL. Sci China-Phys Mech Astron February (2013) Vol. 56 No to the basis of the surface. such as , Is a basis of the lin where(ai le, can be any basis of TZ. It could be deduced that ear subspace in three dimensional Euclidian space. Therefore det a=det[a ]=det[ l, where [.]. ]E RPXP one define its determinant as follows One can say that an affine surface tensor is nonsingular in the case of its surface determinate does not vanish. For any 题1e nonsingular surface tensor therefore, there uniquely exists its inverse one d- in the following mean As similar to the previous related proof refer to the sect. Φ=Φ.Φ=I÷6{gg 2.4, the following identity is keeping valid d where l is termed as the unity affine surface tensor. Furthermore, the right and left eigenvalue problems can be defined as: (x,n2+1(x,1,2,n where br, bl E TM are termed as the right and left eigenvec +31,a(x,D)运+A,(x,D),n tors respectively. The corresponding eigen-polynomial can be represented 2苏x(x,l+(x,D detd-)=(-1P+1(-yP-1 lp-1(-1)+lp=0 √+ where the rth-primary invariant can be determined through accompanying with the similar relation g (xE,1)·g ≤i0,Va≠0∈T∑
Xie X L, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 439 to the basis of the surface, such as Σ gl 2 l=1 , is a basis of the linear subspace in three dimensional Euclidian space.Therefore, one define its determinant as follows det Σ F √gΣ √ GΣ · det ⎡ ⎢⎢⎢⎢⎣ ∂xi Σ ∂ξA Σ ⎤ ⎥⎥⎥⎥⎦(ξΣ, t). As similar to the previous related proof refer to the sect. 2.4, the following identity is keeping valid: d dt √gΣ = d dt [ Σ g1, Σ g2, n] = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎣ ∂ Σ g1 ∂xs Σ (x, t)x˙ s Σ + ∂ Σ g1 ∂t (xΣ, t), Σ g2, n ⎤ ⎥⎥⎥⎥⎥⎥⎥⎦ + ⎡ ⎢⎢⎢⎢⎢⎢⎢⎣ Σ g1, ∂ Σ g2 ∂xs Σ (x, t)x˙ s Σ + ∂ Σ g2 ∂t (xΣ, t), n ⎤ ⎥⎥⎥⎥⎥⎥⎥⎦ + Σ g1, Σ g2, ∂n ∂xs Σ (x, t)x˙ s Σ + ∂n ∂t (xΣ, t) = Σ Γs stx˙ t Σ √gΣ + ∂ ∂t Σ g1, Σ g2, n = √gΣ Σ Γs stx˙ t Σ + 1 √gΣ ∂ √gΣ ∂t (xΣ, t) , accompanying with the similar relation ∂ Σ gl ∂t (xΣ, t) · Σ gl = Σ glkΣ gk · ∂ Σ gl ∂t (xΣ, t) = 1 2 Σ glk ∂ Σ glk ∂t (xΣ, t) = 1 √gΣ ∂ √gΣ ∂t (xΣ, t). Then the identity can be proved. 3.1.3 Some properties of the affine surface tensor In this section, we consider the general p dimensional surface Σ embedded in the p + 1 dimensional Euclidian space Rp+1. Generally, the surface can be regraded as a Riemannian manifold with the dimensionality p. As soon as the analysis on the surface is considered, one usually meets the affine tensor in the form Φ = Φi · j Σ gi ⊗ Σ gj whose underlying space is the tangent space TΣ Span{ Σ gi} p i=1 = Span{ Σ gi } p i=1. In the presented paper, this kind tensor is termed as the surface tensor. It is evident that the generally defined determinant of a surface tensor is naturally naught. Therefore, one introduce the following definition of the surface determinant for the affine surface tensor. (Φ · a1) ∧···∧ (Φ · ap) (a1 · Φ) ∧···∧ (ap · Φ) =: det Φ · a1 ∧···∧ ap, where {ai} p i=1 can be any basis of TΣ. It could be deduced that det Φ = det[Φi · j ] = det[Φ· j i ], where [Φi · j ], [Φ· j i ] ∈ Rp×p. One can say that an affine surface tensor is nonsingular in the case of its surface determinate does not vanish. For any nonsingular surface tensor Φ, therefore, there uniquely exists its inverse one Φ−1 in the following mean Φ−1 · Φ = Φ · Φ−1 = Σ I δj i Σ gi ⊗ Σ gj, where Σ I is termed as the unity affine surface tensor. Furthermore, the right and left eigenvalue problems can be defined as: Φ · bR = λ bR, bL · Φ = λ bL, where bR, bL ∈ TM are termed as the right and left eigenvectors respectively. The corresponding eigen-polynomial can be represented as: det(Φ − λ Σ I) =(−λ) p + I1(−λ) p−1 + ··· + Ir(−λ) p−r + ··· + Ip−1(−λ) + Ip = 0, where the rth-primary invariant can be determined through Ir = ! 1i1 0, ∀ a 0 ∈ TΣ.
440 Xie XL, et aL. Sri China-Phys Mech Astron February (2013) Vol. 56 No. 2 It is equivalently that all of its eigenvalues are positive. Then G3 Material derivatives of the vectored curve, surface and one can define the power operation denoted as volume elements in the current physical configuration e(s)@创(s),Ya∈R, (A)= (),Lv∞口 d based on its spectrum decomposition. It is evident has the property.Φ=Φ.Φ=Φ+. Conse ,p)=B.方×=(p) or any nonsingular affine su tensor o is an au definite then it is valid that d=Φ()→Φ={Φ(ΦΦ)]·(Φ"Φ)t, where L and B in the present paper are termed similarly as the velocity gradient tensor and surface deformation gradient where the first term on the right hand side is orthogonal and tensor respectively the second one is positive definite symmetric. That is just the The first relation could be readily deduced. For the second polar decomposition for any nonsingular affine surface ten- one, one can conside As soon as the deformation gradient tensor is considered, it does not a surface tensor, but F.F and FF"are positive an du ,k)=detF ax动am definite symmetric surface tensors. 3.1.4 Deformation descriptions aam(,p)·n(Ap Similarly, the whole descriptions of deformations in the present case can be divided into four groups still denoted by =0. p+ax oa,p-n(p) Gl to G4 respectively GI Transformations of the vectored curve, surface/volume el- ements between the initial and current physical configura- a×(,) a×(,p tIonS. where the following lemma is adopted. lamma Proof: To consider the relation n(, u): =n(xe(se(, u), t),t) where one has Σ():[a,b3是台Σ(A)兰Σ(x(x(y),1),D) 1)+x一(x,D=一(x,1)-·bg ∑():[a,b]3A→Σ()Σ(5() where are the vector valued maps of the material curves embedded in the initial and current physical configurations respectively 1,, The maps of the material surfaces 2(, u), 2(, u) with re spect to the initial and current physical configurations are og milarly defined G2 Transformations of the the norms of curve, surface el ements between the initial and current physical configura- aa(,)=ce、正 (,p) (x,1)8g+bg
440 Xie X L, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 It is equivalently that all of its eigenvalues are positive. Then one can define the power operation denoted as Φα := ! p s=1 λα s eˆs ⊗ eˆs, ∀ α ∈ R, based on its spectrum decomposition. It is evident that one has the property Φα · Φβ = Φβ · Φα = Φα+β. Consequently, for any nonsingular affine surface tensor Φ, Φ∗Φ is positive definite then it is valid that Φ∗ Φ = (Φ∗ Φ) 1 2 (Φ∗ Φ) 1 2 ⇒ Φ = [Φ−∗(Φ∗ Φ) 1 2 ] · (Φ∗ Φ) 1 2 , where the first term on the right hand side is orthogonal and the second one is positive definite symmetric. That is just the polar decomposition for any nonsingular affine surface tensor. As soon as the deformation gradient tensor is considered, it does not a surface tensor, but Σ F∗ · Σ F and Σ F · Σ F∗ are positive definite symmetric surface tensors. 3.1.4 Deformation descriptions Similarly, the whole descriptions of deformations in the present case can be divided into four groups still denoted by G1 to G4 respectively. G1 Transformations of the vectored curve, surface/volume elements between the initial and current physical configurations: d t Σ dλ (λ) = Σ F · d ◦ Σ dλ (λ) ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ ∂ t Σ ∂λ × ∂ t Σ ∂μ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ) = det Σ F · ∂ o Σ ∂λ × ∂ o Σ ∂μ R3 (λ, μ) · t n(λ, μ), where t Σ(λ):[a, b] λ → t Σ(λ) Σ(xΣ(ξΣ(γ), t), t), ◦ Σ(λ):[a, b] λ → ◦ Σ(λ) ◦ Σ(ξΣ(λ)) are the vector valued maps of the material curves embedded in the initial and current physical configurations respectively. The maps of the material surfaces t Σ(λ, μ), ◦ Σ(λ, μ) with respect to the initial and current physical configurations are similarly defined. G2 Transformations of the the norms of curve, surface elements between the initial and current physical configurations: d t Σ dλ (λ) R3 = ( Σ F∗ · Σ F) 1 2 · d o Σ dλ (λ) R3 , ∂ t Σ ∂λ × ∂ t Σ ∂μ R3 (λ, μ) = det Σ F ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ ∂ ◦ Σ ∂λ × ∂ ◦ Σ ∂μ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ) R3 . G3 Material derivatives of the vectored curve, surface and volume elements in the current physical configuration: ˙ d t Σ dλ (λ) = Σ L · d t Σ dλ (λ), Σ L Σ V ⊗ Σ , ⎛ ˙ ⎜⎜⎜⎜⎜⎜⎜⎝ ∂ t Σ ∂λ × ∂ t Σ ∂μ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ) = Σ B · ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ ∂ t Σ ∂λ × ∂ t Σ ∂μ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ), Σ B Σ θ Σ I − Σ ⊗ Σ V. where Σ L and Σ B in the present paper are termed similarly as the velocity gradient tensor and surface deformation gradient tensor respectively. The first relation could be readily deduced. For the second one, one can consider ˙ ∂ t Σ ∂λ × ∂ t Σ ∂μ (λ, μ) = ˙ det Σ F · ∂ ◦ Σ ∂λ × ∂ ◦ Σ ∂μ (λ, μ) R3 · t n(λ, μ) + det Σ F · ∂ ◦ Σ ∂λ × ∂ ◦ Σ ∂μ (λ, μ) R3 · ˙ t n(λ, μ) = Σ θ · ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ ∂ t Σ ∂λ × ∂ t Σ ∂μ (λ, μ) ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ + ∂ t Σ ∂λ × ∂ t Σ ∂μ (λ, μ) R3 · ˙ t n(λ, μ) = Σ θ Σ I − Σ ⊗ Σ V · ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ ∂ t Σ ∂λ × ∂ t Σ ∂μ (λ, μ) ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ =: Σ B · ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ ∂ t Σ ∂λ × ∂ t Σ ∂μ (λ, μ) ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠, where the following lemma is adopted. lamma: ˙ t n(λ, μ) = − Σ ⊗ Σ V · t n(λ, μ). Proof: To consider the relation t n(λ, μ) := n(xΣ(ξΣ(λ, μ), t), t), one has ˙ t n(λ, μ) = ∂n ∂t (xΣ, t) + x˙ i Σ ∂n ∂xi Σ (xΣ, t) = ∂n ∂t (xΣ, t) − x˙ i Σ · bis Σ gs , where ∂n ∂t (xΣ, t) = ∂n ∂t (xΣ, t), Σ gi R3 Σ gi = − ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ t n, ∂ Σ gi ∂t (xΣ, t) ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ R3 Σ gi = − ⎛ ⎜⎜⎜⎜⎝ t n, ∂ ∂xi Σ ( ∂Σ ∂t )(xΣ, t) ⎞ ⎟⎟⎟⎟⎠ R3 Σ gi = − ⎛ ⎜⎜⎜⎜⎝ t n, ∂ ∂xi Σ ( Σ V − x˙ s Σ Σ gs)(xΣ, t) ⎞ ⎟⎟⎟⎟⎠ R3 Σ gi = − ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎝ t n, ∂ Σ V ∂xi Σ (xΣ, t) ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎠ R3 Σ gi + ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ t n, x˙ s Σ ∂ Σ gs ∂xi Σ (xΣ, t) ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ R3 Σ gi = − t n · ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∂ Σ V ∂xi Σ (xΣ, t) ⊗ Σ gi ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎠ + x˙ s Σbis Σ gi .
Xie XL, et aL. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 This ends the proof 3.1.6 Definition of vorticity G4 Material derivatives of the norms of the curve, surface elements in the current physical configuration Firstly, one can consider s V,Vg'ef-bsv86g+av+ bis In8g az ( )=ax It results in the following representations of the strain rate tensor where D is the rate of the change of the deformation t and n are denoted for the unit tangent vector of curve element and 口+v= he normal vector of the surface element respectively 3.1.5 Transport theories 2a+Disv(nag+g8n)-bsVg8g Accompanying the deformation description denoted as G3 with the curve and surface integrals of the second kind, one and the vorticity tensor once can readily arrive at the following transport theories of the second kind The curve transport of the second kind ∑∑ ()dn as+bis v n g-gen Φ。-rdl+ Because of the following equivalent relations The surface transport of the second kind VV=:日s3台=eVv=3ov d do-ndσr there exists the familiar relation that i =.o-nd+.Φ。-(B:nldσ, 1g5sgl]·b=a×b,b Accompanying the deformation description denoted as G4 where A a'n is defined as the vorticity on the surface with the curve and surface integrals of the first kind, one ar- ential of the velocity between two neighboring matenalpcr Based on the above discussion, one can consider the differ rives at the following transport theories of the first kind The curve transport of the first kind ticle points d V(Ex+A,1)-V(￡x,1) ()d (vb)·(FA)=D+g|·(△xg)=: Φd+ (T The surface transport of the first kind V3(K.△∑|+v3+v.K,△∑n aa ( u)do where ka biggis termed as the curvature tensor of the Φd+,6Φdo- As compared to the general Helmholtz velocity decom- position for the flow on the plane, the last additional two
Xie X L, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 441 This ends the proof. G4 Material derivatives of the norms of the curve, surface elements in the current physical configuration: ˙ d t Σ dλ R3 (λ) = (τ · Σ D · τ) d t Σ dλ R3 (λ), Σ D ( Σ L + Σ L∗ )/2, ˙ ∂ t Σ ∂λ × ∂ t Σ ∂μ R3 (λ, μ) = Σ θ ∂ t Σ ∂λ × ∂ t Σ ∂μ R3 (λ, μ). where Σ D is the rate of the change of the deformation, τ and n are denoted for the unit tangent vector of curve element and the normal vector of the surface element respectively. 3.1.5 Transport theories Accompanying the deformation description denoted as G3 with the curve and surface integrals of the second kind, one once can readily arrive at the following transport theories of the second kind. The curve transport of the second kind d dt t C Σ Φ ◦ −τdl = d dt b a Σ Φ ◦ −d t Σ dλ (λ)dλ = t C Σ˙ Φ ◦ −τdl + t C Σ Φ ◦ −( Σ L · τ)dl, The surface transport of the second kind d dt t Σ Σ Φ ◦ −ndσ= d dt Dλμ Σ Φ ◦ − ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ ∂ t Σ ∂λ × ∂ t Σ ∂μ ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ (λ, μ)dσ = t Σ Σ˙ Φ ◦ −ndσ + t Σ Σ Φ ◦ −( Σ B · n)dσ, Accompanying the deformation description denoted as G4 with the curve and surface integrals of the first kind, one arrives at the following transport theories of the first kind. The curve transport of the first kind d dt t C Σ Φ dl= d dt b a Σ Φ d t Σ dλ R3 (λ)dλ = t C Σ˙ Φ dl + t C Σ Φ (τ · Σ D · τ)dl. The surface transport of the first kind d dt t Σ Σ Φ dσ= d dt Dλμ Σ Φ ∂ t Σ ∂λ × ∂ t Σ ∂μ R3 (λ, μ)dσ = t Σ Σ˙ Φ dσ + t Σ Σ θ Σ Φ dσ. 3.1.6 Definition of vorticity Firstly, one can consider Σ V ⊗ Σ = ∂ Σ V ∂xl Σ ⊗ Σ gl = ∂ ∂xl Σ ⎛ ⎜⎜⎜⎜⎝ Σ Vi Σ gi + Σ V3 n ⎞ ⎟⎟⎟⎟⎠ ⊗ Σ gl = Σ ∇l Σ Vs Σ gs ⊗ Σ gl − bsl Σ V3 Σ gs ⊗ Σ gl + ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∂ Σ V3 ∂xl Σ + bls Σ Vs ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ n ⊗ Σ gl . It results in the following representations of the strain rate tensor Σ D=1 2 Σ V ⊗ Σ + Σ ⊗ Σ V = 1 2 Σ ∇l Σ Vs + Σ ∇s Σ Vl Σ gs ⊗ Σ gl + 1 2 ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∂ Σ V3 ∂xl Σ + bls Σ Vs ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (n ⊗ Σ gl + Σ gl ⊗ n) − bsl Σ V3 Σ gs ⊗ Σ gl , and the vorticity tensor Σ Ω 1 2 Σ V ⊗ Σ − Σ ⊗ Σ V =1 2 Σ ∇l Σ Vs − Σ ∇s Σ Vl Σ gs ⊗ Σ gl + 1 2 ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ ∂ Σ V3 ∂xl Σ + bls Σ Vs ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎛ ⎜⎜⎜⎜⎝ n ⊗ Σ gl − Σ gl ⊗ n ⎞ ⎟⎟⎟⎟⎠. Because of the following equivalent relations Σ ∇l Σ Vs − Σ ∇s Σ Vl =: ls3ω3 ⇔ ω3 = 3ls Σ ∇l Σ Vs = 3ls ∂ Σ Vs ∂xl Σ , there exists the familiar relation that is [( Σ ∇l Σ Vs − Σ ∇s Σ Vl) Σ gs ⊗ Σ gl ] · Σ b = Σ ω × b, ∀ Σ b ∈ TΣ, where Σ ω ω3n is defined as the vorticity on the surface. Based on the above discussion, one can consider the differential of the velocity between two neighboring material particle points Σ V(ξΣ + ΔξΣ, t) − Σ V(ξΣ, t) ( Σ V ⊗ Σ ) · ( Σ F · Δ ◦ Σ) Σ D + Σ Ω · (Δxt Σ Σ gt) =: Σ D + Σ Ω · (Δ t Σ) = 1 2 Σ ∇l Σ Vs + Σ ∇s Σ Vl Δxl Σ Σ gs + 1 2 Σ ω × Δ t Σ − Σ V3 K · Δ t Σ + ⎛ ⎜⎜⎜⎜⎝ Σ ∇ Σ V3 + Σ V · K, Δ t Σ ⎞ ⎟⎟⎟⎟⎠ R3 n, where K bi j Σ gi ⊗ Σ gj is termed as the curvature tensor of the surface. As compared to the general Helmholtz velocity decomposition for the flow on the plane, the last additional two
