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+bg440 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 ten￾sor. 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 el￾ements between the initial and current physical configura￾tions: 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 re￾spect 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 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 .