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 defor￾mation 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 de￾formation, 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 consid￾ered as positive without lost of the generality and the gen￾eralized 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 ar￾rives 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 config￾urations are two dimensional Riemannian man￾ifolds 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 Cp￾diffeomorphism 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 phys￾ical configurations, respectively.