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 ≤i<…i≤p ) Then the identity can be proved In the case of the affine surface tensor is a symmetric t sor, its eige 3.1.3 Some properties of the affine surface tensor det(p-de)=det[o(ij)(iD]=0, In this section, we consider the general p dimensional surface where [(in) is the symmetric matrix with respect to an ar- 2 embedded in the p+ I dimensional Euclidian space RP+ bitrary orthonormal basis leili, in T2. Therefore, all of the Generally, the surface can be regraded as a Riemannian man- eigenvalues are real number and there exists an orthogonal ifold with the dimensionality p matrix Q such that Q[o(i]]e diag[dl, ...,p]. Subse- As soon as the analysis on the surface is considered, one quently, one has the representation termed usually as spec usually meets the affine tensor in the form =d gi8 trum decomposition g! whose underlying space is the tangent space 12 p=d(iDe(18e) Spang),=Spanlg',. In the presented paper, this kind (Qe()8(Qe()=:e(s)e(s) tensor is termed as the surface tensor. It is evident that the generally defined determinant of a sur- because of o(ij) a, Qis Qis, where eil is the othe following definition of the surface determinant for the affine orthonormal basis in r2 Furthermore, in the case of the affine surface tensor is pos itive define symmetric that is (:)A…:A(3}=a1A…A a·da>0,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<···<irp det ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ Φi1 · i1 ··· Φi1 · ip ··· ··· ··· Φip · i1 ··· Φip · ip ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ = 1 p! δ i1···ip j1··· jp Φj1 · i1 ··· Φjp · ip = ! 1i1<···<irp det ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ Φ· i1 i1 ··· Φ· i1 ip ... ... ... Φ· ip i1 ··· Φ· ip ip ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ = 1 p! δ i1···ip j1··· jp · Φ· j1 i1 ··· Φ· jp ip . In the case of the affine surface tensor is a symmetric tensor, its eigen-polynomial is det(Φ − λIΣ) = det[Φi j − λδi j] = 0, where [Φi j] is the symmetric matrix with respect to an arbitrary orthonormal basis {ei} p i=1 in TΣ. Therefore, all of the eigenvalues are real number and there exists an orthogonal matrix Q such that QT[Φi j]Q = diag[λ1, ··· , λp]. Subsequently, one has the representation termed usually as spectrum decomposition Φ=Φi j ei ⊗ ej = ! p s=1 λs (Qisei) ⊗ Qjsej =: ! p s=1 λseˆ(s) ⊗ eˆ(s) because of Φi j = "p s=1 λsQisQjs, where {eˆi} p i=1 is the other orthonormal basis in TΣ. Furthermore, in the case of the affine surface tensor is positive define symmetric that is a · Φ · a > 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 .