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(SD8:8GXie 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 gra￾dient 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