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 explicitly434 Xie X L, et al. Sci China-Phys Mech Astron February (2013) Vol. 56 No. 2 equation. In addition, the elementary equation for infinitesi￾mal 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 in￾stituted by solids, its geometrical configuration could be re￾garded as a two dimensional surface that could be naturally taken as a Riemannian manifold in three dimensional Euclid￾ian space. Furthermore, the relationships between geome￾tries 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 medi￾ums [27] with some applications. The primary contents of the paper are organized as follows: (1) Finite deformation theory with respect to curvilinear coordinates correspond￾ing to current physical configurations including time explic￾itly; (2) Finite deformation theory with respect to continuous mediums whose geometrical configurations are two dimen￾sional Riemannian manifolds; (3) Case studies on the defor￾mations/motions of continuous mediums viewed either as Eu￾clidian manifolds or Riemannian manifolds. 2 Finite deformation theory with respect to curvilinear coordinates corresponding to cur￾rent physical configurations including time ex￾plicitly 2.1 View of the mapping As shown in Figure 1, the current physical configuration is changing its geometrical configuration as the time is vary￾ing. However, there exists so called diffeomorphism includ￾ing time explicitly through which the current physical con- figuration could be mapped bijectively onto a parametric do￾main 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 dif￾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 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 uti￾lized 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 stud￾ies, 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 re￾lated 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 configura￾tion respectively, {Xi } 3 i=1 and {xi } 3 i=1 denote the Cartesian and curvilinear coordinates corresponding to the current physi￾cal configuration respectively. Subsequently, the former two groups of coordinates are Lagrangian coordinates and the lat￾ter 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 ac￾cording 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 de￾pendent on the time explicitly.