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Tensors, Gradients,Continuum Mechanics QUARTERLY OF APPLIED MATHEMATICS VOLUME , NUMBER 0 XXXX XXXX, PAGES 000–000 S 0033-569X(XX)0000-0 HIGHER DERIVATIVES AND THE INVERSE DERIVATIVE OF A TENSOR-VALUED FUNCTION OF A TENSOR By ANDREW N. NORRIS Mechanical and Aerospace Engineering, Rutgers University, Piscataway NJ 08854-8058, USA Abstract. The nth derivative of a tensor valued function of a tensor is defined by a finite number of coefficients each with closed form expression. 1. Introduction. We consider tensor functions on symmetric second order tensors, Sym Sym, defined by a scalar function f(x) of a single variable according to → d f(A)= f(αi)Ai, (1.1) i=1 X where d is the eigen-index of A. The tensor A Sym is arbitrary with spectral decom- ∈ position d d Ai i = j, A = αiAi, AiAj = Ai = I. (1.2) i=1 (0, i = j, i=1 X 6 X The specific case of tensors acting on 3-dimensional vectors, d 3, is discussed in this ≤ paper, although the results can be readily generalized. Derivatives of f(A) are defined by the expansion 1 1 f(A + X)=f(A)+ f(A)X + (2)f(A): XX + (3)f(A): XXX ∇ 2∇ 3!∇ 1 + ... + (n)f(A): XX ... X + ... (1.3) arXiv:0707.0115v3 [math.SP] 29 Oct 2007 n!∇ n terms The nth derivative (n)f(A) is a tensor of order| {z 2(n }+ 1) which contracts n-times with ∇ the second order tensor X to produce a second order tensor. The first derivative, or gradient, is ′ d f (αi), i = j, (1) (1) f(A)= f A ⊠A , f = (1.4) ∇ ij i j ij i,j=1 f(αi)−f(αj ) X , i = j, αi−αj 6 E-mail address: [email protected] c XXXX Brown University 1 2 ANDREW N. NORRIS where ⊠ denotes the outer tensor product, defined in Section 2. The identity (1.4) is well known and has appeared in various formats. The first explicit presentation I am aware of is due to Ogden [10] who defines a fourth order tensor L1 = ∂f(A)/∂A (with slight change in notation). Ogden gives the components 1 in terms of the eigenvectors a Lijkl i of A, i.e. L1 = 1 a a a a . These coefficients are related to those of (1.4) Lijkl i ⊗ j ⊗ k ⊗ l by the fact that A = a a (no sum) when d = 3 and hence 1 = f (1), 1 = f (1). i i ⊗ i Liiii ii Lijij ij The fundamental result (1.4) was also derived by Carlson and Hoger [1], although the present notation is based on [14]. Ogden [10] (Section 3.4) also presented the second derivative. In the present notation it is d 1 (2)f(A)= f (2)A ⊠A ⊠A , (1.5) 2∇ ijk i j k i,j,kX=1 1 f (2) = f ′′(α ), iii 2 i f(α ) f(α ) (α α )f ′(α ) f (2) = j − i − j − i i , iij (α α )2 j − i [f(α ) f(α )](α + α 2α ) [f(α )+ f(α ) 2f(α )](α α ) f (2) = j − i i j − k − i j − k j − i . ijk 2(α α )(α α )(α α ) i − j j − k k − i Ogden’s expressions (1.4) and (1.5) are special cases of the more general formulae for derivatives of isotropic tensor functions derived by Chadwick and Ogden [2]. It is interesting to note that the tensor gradient function (first derivative) involves finite differences of the function f as well as its derivative evaluated at the eigenvalues of A. Similarly, the second derivative contains second order finite differences. The general result derived here shows that the coefficients for the nth derivative are related to an interpolating polynomial. Our main result is the following: Theorem 1.1. The nth derivative of the tensor function f(A) is given by d 1 (n) (n) f(A)= f Ai ⊠Ai ⊠...⊠Ai . (1.6) n!∇ i1,i2,...,in+1 1 2 n+1 i ,i ,...,i =1 1 2 Xn+1 (n) The coefficients fi1,i2,...,in+1 are unaltered under all permutations of the n+1 indices. The 2 (n+2)(n+3)/2 distinct coefficients can be classified into1 (n+4) +4 expressions f νi,νj ,νk ⌊ 12 ⌋ ijk where νi,νj ,νk, are the number of occurrences of distinct indices i, j, k respectively, with i = j = k = i and ν + ν + ν = n + 1. The coefficient is 6 6 6 i j k νl−1 νi,νj ,νk 1 d f(x) fijk = ν −1 ν . (1.7) (νl 1)! d x l (x αm) m l=i,j,k x=αl − m=i,j,k − νXl>0 m=6 l Q Alternatively, the coefficient can be found from the unique interpolating polynomial P (x) of degree n that fits the data at the three points x = α , α , α defined by the n +1 { i j k} 1 ⌊x⌋ is the floor function. HIGHER DERIVATIVES AND THE INVERSE DERIVATIVE OF A TENSOR-VALUED FUNCTION OF A TENSOR3 (I) (J) (K) values f (αi), f (αj ), and f (αk) for 0 I νi 1, 0 J νj 1, 0 K νk 1 (l) th ≤ ≤n − ≤n−1≤ − ≤ ≤ − where f (x) is the l derivative. Let P (x)= pnx + pn−1x + ... + p0, then νi,νj ,νk fijk = pn. (1.8) The first few expressions for the coefficients are 1 f 0,0,n+1 = f (n)(α ), (1.9a) ijk n! k n−1 0,1,n 1 1 l (l) fijk = n−1 f(αj ) (αj αk) f (αk) , (1.9b) (αj αk) − l! − − Xl=0 n−1 0,2,n−1 1 (n l) l (l) ′ fijk = n−1 − (αj αk) f (αk) + (αj αk)f (αj ) nf(αj ) , (αj αk) l! − − − − l=0 X (1.9c) 1 f(α ) f(α ) f 1,1,n−1 = i j ijk (α α ) (α α )n−1 − (α α )n−1 i − j i − k j − k n−2 1 (l) 1 1 f (αk) n−1−l n−1−l , − l! (αi αk) − (αj αk) l=0 − − X (1.9d) 1 f(α ) f(α ) (α α ) f 1,2,n−2 = i j 1 + (n 2) j − i ijk (α α )2 (α α )n−2 − (α α )n−2 − (α α ) i − j i − k j − k j − k n−3 (αj αi) ′ 1 (l) 1 + − n−2 f (αj ) f (αk) n−2−l (αj αk) − l! (αi αk) − Xl=0 − 1 (α α ) 1 + (n l 2) j − i . (1.9e) − (α α )n−2−l − − (α α ) j − k j − k These are sufficient to determine all derivatives up to and including the fourth. Thus, 0,0,2 0,1,1 the gradient tensor (n = 1) requires the two expressions fijk and fijk evident from 0,0,3 0,1,2 1,1,1 (1.4); the second derivative (n = 2) requires fijk , fijk and fijk , which may be read 0,0,4 0,1,3 off from (1.5); the third derivative (n = 3) involves four distinct formulas for fijk , fijk , 0,2,2 1,1,2 0,0,5 fijk and fijk ; and the fourth derivative (n = 4) requires the five expressions fijk , 0,1,4 0,2,3 1,1,3 1,2,2 fijk , fijk , fijk and fijk . Note that (1.9b) and (1.9c) reduce to (1.9a) in the limit as α α . Similarly, (1.9d) and (1.9e) reduce to (1.9c) in the limit. i → k The main objective of this paper is to prove Theorem 1.1. Some new results concerning the properties of the fourth order gradient tensor f are also presented and the inverse ∇ fourth order tensor −1f is introduced. Both the gradient and its inverse are discussed ∇ with application to strain measure functions [6, 12]. The proof of Theorem 1.1 begins with a new derivation of the well known expression for the gradient f. The essential ∇ structure of the second and higher order derivatives is shown to depend on a general algebraic identity. This identity also reveals the appearance of the characteristic finite difference terms. The proof is completed by making connections with contour integrals and with interpolation polynomials. The results are presented in terms of Kronecker products of tensors which makes the expressions more transparent. 4 ANDREW N. NORRIS The paper is laid out as follows. Notation is introduced in Section 2 followed by the derivation of Theorem 1.1 in Section 3. The inverse gradient tensor is introduced and its properties discussed in Section 4. 2. Notation and preliminaries. We consider second order tensors acting on vectors in a three dimensional inner product space, x Ax with transpose At such that y Ax = → · x Aty. Spaces of symmetric and skew-symmetric tensors are distinguished, Lin = Sym · Skw where A Sym (Skw) iff At = A (At = A). Products AB Lin are defined ⊕ ∈ − ∈ by y ABx = (Aty) Bx. · · Psym is the space of positive definite second order tensors. Functions of a symmetric tensor can be phrased in terms of its spectral decomposition (1.1) where A Psym and i ∈ the distinct eigenvalues αi, i = 1 ...,d 3 are real numbers. The single function f(A) ≤ 3 is a special case of isotropic tensor functions of the form T(A) = fi(α1, α2, α3)Ai, i=1 involving three functions of three variables. Chadwick and Ogden [2P] derived first and second derivatives for more general situation (see [3] for a recent overview).
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