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Article 149677178







Article 149677541







#dynamics #element #explicit #finite #for #introduction #method #nonlinear #the #to #transient
8 INTRODUCTION Special tensors and their functionalities are adopted: δ ij =  0ifi = j 1ifi = j : Kronecker delta, ∈ ijk = ⎧ ⎨ ⎩ 0 if any two indices are equal 1ifi, j, k = 1, 2, 3 or 2, 3, 1 or 3, 1, 2 −1ifi, j, k = 3, 2, 1 or 2, 1, 3 or 1, 3, 2 : 3D permutation tensor. Part of their operational functionalities is listed below for later reference: δ ij u j = u i , δ jj = 3,δ ij δ jk = δ ik ,δ ij δ ij = δ jj = 3, ∈ ijk ∈ imn = δ jm δ kn −δ jn δ km , ∈ ijk ∈ ijn = 2δ kn , c k =∈ ijk a i b j : c = a × b: vector product of two vectors in 3D space, ∈ ijk a i b j c k = a × b • c: mixed product of three vectors in 3D space. This threefold summation represents a mixed product of three vectors, which is equivalent to the volume framed by the vectors a, b, and c. The 2D Kronecker delta and permutation tensor are defined with α and β ranging from1to2: δ αβ =  0ifα = β 1ifα = β : 2D Kronecker delta, ∈ αβ = ⎧ ⎨ ⎩ 0ifα = β 1ifα, β = 1, 2 −1ifα, β = 2, 1 : 2D permutation tensor. The related properties are, for example, δ ββ = 2, ∈ αβ a α b β = a 1 b 2 −a 2 b 1 =     a 1 a 2 b 1 b 2     : 2D determinant, ∈ αβ ψ ,β :(ψ ,2 , −ψ ,1 ) : differential operator of curl on a scalar function. 1.2.2 Constitutive Relations of Elasticity Elasticity is the foundation of structural mechanics. Here we summarize the con- stitutive relations for later reference, but we would not provide detailed review for

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