On the convexity of the function C f(det C) on positive-definite matrices

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1 Article On the convexit of the function C fdet C) on poitive-definite matrice Mathematic and Mechanic of Solid 2014, Vol 194) The Author) 2013 Reprint and permiion: agepubcouk/journalpermiionnav DOI: / mmagepubcom Stephan Lehmich, Patrizio Neff and Johanne Lankeit Lehrtuhl für Nichtlineare Anali und Modellierung, Fakultät für Mathematik, Univerität Duiburg-Een, German Received 25 September 2012; accepted 4 October 2012 Abtract Let n 2 We prove a condition on f C 2 R +, R) for the convexit of f det on PSmn), namelthatf det i convex on PSmn) if and onl if f ) + n 1 f ) 0 and f ) 0 R + n Thi generalize the obervation that C ln det C i convex a a function of C Keword Convexit, elaticit, Neo-Hooke model, ODE, polconvexit 1 Introduction The quetion of how to chooe phicall reaonable train energ function in nonlinear elaticit ha attracted much attention and i not et completel olved The major breakthrough came with John Ball eminal contribution [1 3] introducing polconvexit, in other word, convexit of the train energ W a a function of the argument F,Cof F,detF) ee alo [8, 9]) Polconvexit reconcile the phicall reaonable growth condition WF) a det F 0 with the weak-lower-emicontinuit quaiconvexit), which in return implie ellipticit A ver imple example of a polconvex function i the uni-contant compreible Neo-Hooke model W NH F) = μ[ F T F 1, 1 2lndetF)], hear modulu μ>0 The train energ i iotropic, frame-indifferent, polconvex convex a a function of F, det F)), tre-free in the reference configuration, and W NH a det F 0 It i well known that the latter requirement exclude from the outet that F W NH F) ma be a convex function of F [6] However, rewriting W NH in term of the Cauch Green deformation tenor C = F T F, which give W NH F) = Ŵ NH C) = μ C 1, 1 ln det C), one ma readil check that C Ŵ NH C) i a convex function of C, depite it ingularit in the determinant a det C 0 We urmie that convexit of the free energ with repect to C or the tretch tenor U = C) i an additional, deirable feature of an free energ a it implie monotonicit of the tre train relation In thi hort contribution we therefore invetigate which function f C 2 R +, R) are uch that C f det C) i Correponding author: Patrizio Neff, Lehrtuhl für Nichtlineare Anali und Modellierung, Fakultät für Mathematik, Univerität Duiburg-Een, Campu Een, Thea-Lemann Str 9, Een, German patrizioneff@uni-duede

2 370 Mathematic and Mechanic of Solid 194) convex a function of C PSmn) and generalize the well-known reult that C ln det C i convex on the et of poitive-definite mmetric matrice [4, 5, 10] b proving: Theorem 11 A DIFFERENTIAL INEQUALITY CHARACTERIZATION) Let f C 2 R +, R) Then the function i convex if and onl if f det : PSmn) R, C f det C) f ) + n 1 n Proof Thi i an immediate conequence of Lemma 15, 17 and 19 f ) 0 and f ) 0 R + 1) In the following we will reformulate the condition for convexit to obtain thi reult We tart with ome preliminarie: B M n n we denote the et of all real n n matrice; Smn) tand for the et of all real mmetric n n matrice, and PSmn) for the et of all real mmetric poitive-definite n n matrice Lemma 12 CHARACTERIZATION OF CONVEXITY) Let X be a normed pace, K X open and convex, and g C 2 K, R)Then g convex D 2 gx)z, z) 0 x K, z pank) 2) Proof See [7, p27] In particular we obtain Theorem 13 CONDITION FOR CONVEXITY) For g C 2 PSm n), R) we have g convex D 2 gc)h, H) 0 C PSmn), H Smn) 3) Proof Let K := PSmn) andx := Smn) in the previou lemma PSmn) i an open convex ubet of the normed pace Smn) with operator norm): ue the characterization A PSmn) Ax, x > 0 x R n \{0}, and for convexit alo ue the Cauch Schwarz inequalit Furthermore, pank) = panpsmn)) = Smn) = X : the incluion i obviou For the other incluion, write A a a diagonal matrix the correponding tranformation preerve poitive-definitene and mmetr) and how that thi can be written a a linear combination of poitive-definite mmetric matrice B A, B =trab T ) we denote the trace inner product of the matrice A and B Theorem 14 A CONDITION FOR CONVEXITY) Let f C 2 R +, R) Then the function i convex if and onl if g := f det : PSmn) R, C f det C) C PSmn) H Smn) : [ ] f det C)detC+f det C) C 1, H 2 f det C) HC 1, C 1 H 0 4) Proof Becaue f C 2, and det C, alo g C 2 It remain to be hown that { [f D 2 gc)h, H) = det C det C) det C + f det C) ] C 1, H 2 } f det C) HC 1, C 1 H

3 Lehmich et al 371 for C PSmn) and H Smn) The claim follow b Theorem 13 Becaue det i infinitel often differentiable on M n n and D deta)h = Adj A T, H cf [6]), where Adj A denote the adjugate matrix of A, for invertible C and mmetric H we have D detc)h = det C C 1, H, and hence obtain b the chain rule DgC)H = Df det C)D detc)h = f det C) det C C 1, H, and therefore, b chain rule and the fact that D[C 1 ]H = C 1 HC 1, D 2 gc)h, H) = f det C) det C) 2 C 1, H 2 + f det C) det C C 1, H 2 + f det C) det C C 1 HC 1, H 5) = f det C) det C 2 C 1, H 2 + f det C) det C C 1, H 2 f det C) det C HC 1, C 1 H Lemma 15 The inequalit f ) 0 R + i necear for inequalit 4) Proof Aume R + atifing f ) > 0 Let C = diag,,1,1) PSmn) andh = diag1, 1, 0, 0, ) Smn) Then det C =, C 1, H = 1 1 = 0and HC 1, C 1 H = diag 1, 1,0,,0),diag 1, 1,0,,0) = 2 Together with inequalit 4) we obtain 2 f ) 0, a contradiction to f ) > 0 Lemma 16 Inequalit 4) hold if and onl if H Smn) D 1 = diagd 1,, d n ), where d 1,, d n R + and 1 := det D 1 = d 1 d n R + : f ) + f ) ) D 1, H 2 f ) D 1 H, HD 1 0 6) Proof In inequalit 4) conider an arbitrar C PSmn) Then there i an orthogonal matrix Q, uch that C = QDQ T C 1 = QD 1 Q T,whereD = diagλ 1,, λ n )andλ i are poitive B the propertie of the calar product of matrice we have C 1, H = QD 1 Q T, H = QD 1, HQ = D 1, Q T HQ For H Smn) let H := Q T HQ and note that H varie over the whole of Smn) if and onl if H doe Analogoul, Denote d i := λ 1 i 6) HC 1, C 1 H = HQD 1 Q T, QD 1 Q T H = HD 1, D 1 H = D 1 H, HD 1 and := det C = det D = n i=1 λ i, and divide inequalit 4) b > 0 to obtain condition Lemma 17 Let f C 2 R +, R) and f det be convex on PSmn)Thenf ) n 1 f ) R n + Proof According to Theorem 14 and Lemma 16, condition 6) hold for all H Smn) andd 1 = diagd 1,, d n ) Let R +, k R \{0} and H = k D 1,awellaD 1 = diag 1 n,, n 1 ) 0 k 2 f ) + f ) ) D 1, D 1 2 f ) ) D 1 ) 2,D 1 ) 2 = k 2 f ) + f ) ) trd 1 ) 2 ) 2 f ) ) trd 1 ) 4 = k 2 f ) + f ) ) n ) 2/n 2 f ) ) n 4/n = nk 2 4/n nf ) + n 1) f ) )

4 372 Mathematic and Mechanic of Solid 194) For an matrix A let diag A be the matrix obtained from A b etting all non-diagonal entrie to zero Let diag M n n be the et of all n n diagonal matrice Lemma 18 For all P diag M n n with non-negative entrie onl, and all A M n n, the following hold: P, A = P, diag A =: σ P, A), 7) PA, AP P diag A, diag AP =: σ P, A), 8) σ 2 P, A) n σ P, A) 9) Proof Let P = diagp 1,, p n ), A = a ij ) i,j and calculate P, A = n i=1 p ia ii = P, diag A Hence equation 7) hold Direct calculation of PA and PA T ield PA, AP =tr PAPA T) n n = p 2 i a2 ii + p i p k a 2 ik P diag A, diag AP, i=1 in other word, equation 8) For all P diag M n n and A M n n,wehaveσ 2 P, A) n σ P, A) To ee thi, note that P and diag A commute: σ P, A) = P diag A, P diag A = P diag A 2 hold B the Cauch Schwarz inequalit thi immediatel implie σ 2 P, A) = P, diag A 2 = P diag A, 1 2 P diag A = n σ P, A) i=1 k i Lemma 19 Inequalitie 1) are ufficient for f det to be convex Proof We will how inequalit 6) To thi end, let H Smn) and D 1 = diagd 1,, d n ), where d 1,, d n R + and := d 1 d n ) 1 = det D arbitrar Then P := D 1 and A := H atif all aumption of the previou lemma Uing the notation from Lemma 18, we can, without lo of generalit, aume σ D 1, H) = 0, becaue otherwie condition 6) become trivial b the aumption f 0 We denote σ = σ D 1, H) and σ = σ D 1, H) D 1 H, HD 1 b inequalit 8) Uing f ) 0and f ) + n 1 f ) 0, b inequalit 1) and 1 σ n 1, we obtain condition 6) from n σ 2 n f ) + f ) ) D 1, H 2 f ) D 1 H, HD 1 f ) + f ) ) σ 2 f ) σ [ = σ 2 f ) + f ) 1 σ σ 2 ) ] σ 2 f ) + n 1 f ) n ) 0 2 Solution to the differential inequalitie In thi ection we are intereted in the poible hape of the function that atif inequalit 1) To make calculation and figure more concrete, we retrict ourelve to the cae n = 3 Lemma 21 LINEAR ODE) The linear initial value problem L := + gx) = 0, ξ) = η LIVP) on J = R + and where gx) = 2 ha one and onl one olution 3x To find olution to Lf 0 under the additional contraint = f 0 which i equivalent to η 0 becaue f 0 i a olution) we conider the limiting cae : Lemma 22 LIMITING CASE FOR INEQUALITY 1)) The olution to f ) + 2 limit 3 f ) = 0 and f ) 0 R limit limit + are given b f limit : R + R, c 1/3 + d, where c 0, d R

5 Lehmich et al 373 Proof Separation of variable give the unique olution of equation LIVP) for ξ>0η: x ) 2 limit x) = η exp 3t dt = η exp 2 3 ln x ) = ηξ 2/3 x 2/3 10) ξ ξ Becaue ηξ 2/3 0, we have limit 0, hence inequalit 1) The claim follow b integration of f limit = limit with c := 3ηξ 2/3 and contant d If we conider an interval adjacent to ξ on the left hand ide, in other word, J := [ξ a, ξ], the condition for a function to be a ub- or uper-) olution to = Fx, ), ξ) = η are ) v > Fx, v)in J, vξ) η or w < Fx, w) in J, wξ) η repectivel, where ee equation LIVP)) F 2/3 x, ) := 2 3x C R + R) ield Lemma 23 Let be differentiable in R +, ξ>0η Then > > F 2/3 x, ), ξ) η = x) limit x) = ηξ 2/3 x 2/3 on Analogoul: > F 2/3 x, ), ξ) η = x) < limit x) on ξ, ) [ξ, ) 0, ξ) 0, ξ] B thee conideration, we obtain information on the qualitative hape of olution to inequalit 1) at firt dicuing the hape of = f, ee Figure 1) Note that to fulfill 0, in Lemma 23 0 ξ) η mut alo be atified For η = 0, limit 0 i the unique olution and interect in ξ, 0 ξ) 0) For 0 ξ) >η, and limit with initial value limit ξ) = ξ) interect in ξ Hence we can conider ξ) = η = limit ξ) onl Then > 2 3x = F 2/3x, ), ξ) = η implie > limit on ξ, ) [ξ, ) and < limit on 0, ξ) 0, ξ] Additionall, the graph of olution limit contain the point 1, ηξ 2/3 ) Hence there i no need to conider initial value different from 1) = η = limit 1) for η 0 Therefore all derivative of) olution to equation 1) qualitativel have the hape of the dahed line in Figure 1 0, = f 0; furthermore, f and f limit have the ame lope in 1) In 0, 1), however, f decreae more rapidl than f limit ;in1, ), le rapidl The quetion now i: are there other olution to equation 1)? Note that for example the attempt to find a olution b olving = Fx, ) := F 2/3 x, ) + ε for ome poitive ε lead to olution that atif f 0 in a bounded neighbourhood of ξ onl and not on the whole of R + However, Fx, ) := F 2/3) + a x, ) = x a) = 3a+2 3x for a 0give ax) = η exp x 1 a = 3a + 2 3x 3a+2 3t a and a 1) = η 0 dt ) = η x 2 3 +a) a olution to and hence we obtain the following famil of olution to inequalit 1): Lemma 24 FAMILY OF SOLUTIONS) For arbitrar c 0, d R, a [0, ), the famil of function that i defined on R + b d + c 3 1 a, for a [0, 1/3) f a ) := d + c ln, for a = 1/3 d c 3 1 a, for a 1/3, ) ha the propert that f a det: PSm3) R, C f a det C) i convex

6 374 Mathematic and Mechanic of Solid 194) Figure 1 The hape of = f for olution f of inequalit 1) Figure 2 Qualitative hape of olution to inequalit 1) Remark 25 Although thi condition i not necear for the convexit of f det, at leat qualitativel all olution to inequalit 1) have the hape indicated in the graph in Figure 2 a we dicued in thi ection) Acknowledgement We thank John Ball for hi interet in thi reult Conflict of interet None declared Funding Thi reearch received no pecific grant from an funding agenc in the public, commercial, or not-for-profit ector Reference [1] Ball, JM Contitutive inequalitie and exitence theorem in nonlinear elatotatic In: Knop RJ ed) Heriot-Watt Smpoium: Nonlinear Anali and Mechanic, vol 1 London: Pitman, 1977, pp [2] Ball, JM Convexit condition and exitence theorem in nonlinear elaticit Arch Rat Mech Anal 1977; 63: [3] Ball, JM Some open problem in elaticit In: Newton P et al ed) Geometr, mechanic, and dnamic New York: Springer, 2002, pp 3 59 [4] Magnu, JR, and Neudecker, H Matrix differential calculu with application in tatitic and econometric Wile Serie in Probabilit and Statitic, vol 1) New York: John Wile & Son, 1999 [5] Mirk, L An introduction to linear algebra Oxford: Clarendon Pre, 1972

7 Lehmich et al 375 [6] Neff, P Advanced chool on Pol, quai and rank one convexit in applied mechanic 5 lecture), organized b Schröder J and Neff P, Centre International de Science Mecanique CISM), Udine, Ital, September 2007 [7] Rockafellar, T Convex anali Princeton, NJ: Princeton Univerit Pre, 1970 [8] Schröder, J, and Neff, P Invariant formulation of hperelatic tranvere iotrop baed on polconvex free energ function Int J Solid Struct 2003; 402): [9] Schröder, J, Neff, P, and Ebbing V Aniotropic polconvex energie on the bai of crtallographic motivated tructural tenor J Mech Ph Solid 2008; 5612): [10] Strang, G Invere problem and derivative of determinant Arch Rat Mech Anal 1991; 114: , /BF

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