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 dy436 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 co￾ordinates 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 gen￾eral analysis ways [27]. The whole descriptions of deforma￾tions are divided into four groups in the present paper, de￾noted by G1 to G4 respectively. G1 Transformations of the vectored curve, surface and vol￾ume elements between the initial and current physical config￾urations: 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 el￾ements between the initial and current physical configura￾tions:         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).