tial Physical Configuration Current Parametric Configuration Fixed Smooth Surface x=x(5,1) Current Phsical Configuration Initial Parametric Configuration →x =E(5) Figure 1: Sketch of the initial/ current and physical/parametric configurations, where ny and Ne denote the surface normal vectors associated to the current and initial physical con- figurations respectively, igi]_I and [Gala- are local co-variant bases. The tangent space denoted by T2 is span by the co- or contra-variant basis 2 Finite Deformation theory 2.1 Configurations, Coordinates, Curvatures &z Field Operators a two dimensional fow on a fixed smooth surface is sketched in Figure 1. The surface is represented by the vector valued mapping 2= 2(r)E R, and ail is the paramet. ic coordinates. The Hlow/ deformation can be represented by the smooth diffeomorphism a (S, t)E6(2c, 2x)in the parametric space, where [=]21 and (S_1 denote Eulerian and Lagrangian coordinates respectively, and 2e and >r are termed as the initial and current para- metric configurations respectively. Correspondingly, 2: = 2(28 and 2: 2(r) are as the initial and current physical configurations respectively. The smooth surface can be tak- en as a two dimensional Riemannian manifold with the fundamental quantity of the first kind Igi=(9i,9i)i=, and the one of the second kind (big=(ari(a), nE)12i=1.Gaussian curva- ture and mean curvature are defined as Kg= det(bil/ det[gi] and H= bs respectively. Einstein summation convention is adopted throughout the paper with subscripts and superscripts repre- senting co- and contra-variant quantities respectively, and quantities with capital/ small scripts associate with initial/current configurations Generally, the surface gradient operator V= l a is defined as,sy垂∈2x), 2)°-(y918g)g-0 =Vd ( go-gi)8g+d bi(go-n)g+p i(go-gi)8n where o-can be any available algebra tensor operator, VI denotes the co-variant deriva tive/differentation of the tensor component, e. g w(1)+吗-r町 Corresponding Author: XIE XiLin, Department of Mechanics &e Engineering Science, Fudan University HanDanRoad220,Shanghai200433,China.Tel:0086-21-55664283:Email:xiexilin@fudan.edu.cnFigure 1: Sketch of the initial/current and physical/parametric configurations, where nΣ and NΣ denote the surface normal vectors associated to the current and initial physical con- figurations respectively, {gi} 2 i=1 and {GA} 2 A=1 are local co-variant bases. The tangent space denoted by T Σ is span by the co- or contra-variant basis. 2 Finite Deformation Theory 2.1 Configurations, Coordinates, Curvatures & Field Operators A two dimensional flow on a fixed smooth surface is sketched in Figure 1. The surface is represented by the vector valued mapping Σ = Σ(x) ∈ R 3 , and {x i} 2 i=1 is the paramet￾ric coordinates. The flow/deformation can be represented by the smooth diffeomorphism x = x(ξ, t) ∈ C ∞( ◦ Σξ, t Σx) in the parametric space, where {x i} 2 i=1 and {ξ A} 2 A=1 denote Eulerian and Lagrangian coordinates respectively, and ◦ Σξ and t Σx are termed as the initial and current para￾metric configurations respectively. Correspondingly, ◦ Σ := Σ( ◦ Σξ) and t Σ := Σ( t Σx) are termed as the initial and current physical configurations respectively. The smooth surface can be tak￾en as a two dimensional Riemannian manifold with the fundamental quantity of the first kind {gij , (gi , gj )} 2 i,j=1, and the one of the second kind {bij , ( ∂gj ∂xi (x), nΣ)} 2 i,j=1. Gaussian curva￾ture and mean curvature are defined as KG , det[bij ]/ det[gij ] and H , b s s respectively. Einstein summation convention is adopted throughout the paper with subscripts and superscripts repre￾senting co- and contra-variant quantities respectively, and quantities with capital/small scripts associate with initial/current configurations. Generally, the surface gradient operator Σ ∇ ≡ g l ∂ ∂xl is defined as, say Φ ∈ T 2 (T Σ), Σ ∇ ◦ −Φ ≡ ( g l ∂ ∂xl ) ◦ − ( Φ i ·jgi ⊗ g j ) , g l ◦ − ∂ ∂xl ( Φ i ·jgi ⊗ g j ) = ∇lΦ i ·j (g l ◦ −gi ) ⊗ g j + Φi ·j bli(g l ◦ −n) ⊗ g j + Φi ·j b j l (g l ◦ −gi ) ⊗ n where ◦− can be any available algebra tensor operator, ∇l denotes the co-variant deriva￾tive/differentation of the tensor component, e.g. ∇lΦ i ·j , ∂Φ i ·j ∂xl (x, t) + Γi lsΦ s ·j − Γ s ljΦ i ·s ∗Corresponding Author: XIE XiLin, Department of Mechanics & Engineering Science, Fudan University. HanDan Road 220, Shanghai 200433, China. Tel: 0086-21-55664283; Email: xiexilin@fudan.edu.cn. 2