Boris Desmorat

Rodrigue Desmorat

Comptes Rendus Mécanique (CRM)

C. R. Mecanique 343 (2015) 471–475Contents lists available at ScienceDirectComptes Rendus Mecaniquewww.sciencedirect.comTensorial Polar Decomposition of 2D fourth-order tensorsDécomposition polaire tensorielle des tenseurs 2D d’ordre 4Boris Desmorat a,b,∗, Rodrigue Desmorat ca Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean-Le-Rond-d’Alembert, 75005 Paris, Franceb Université Paris-Sud, 91405 Orsay, Francec LMT-Cachan (ENS Cachan, CNRS, Université Paris-Saclay), 94235 Cachan, Francea r t i c l e i n f o a b s t r a c tArticle history:Received 26 May 2015Accepted 9 July 2015Available online 4 August 2015Keywords:Polar decompositionInvariantsHarmonic decompositionKelvin decompositionMots-clés :Décomposition polaireInvariantsDécomposition harmoniqueDécomposition de KelvinOne studies the structure of 2D symmetric fourth-order tensors, i.e. having both minor and major indicial symmetries. Verchery polar decomposition is rewritten in a tensorial form entitled Tensorial Polar Decomposition. The main result is that any 2D symmetric fourth-order tensor can be written in terms of second-order tensors only in a decomposition that makes explicitly appear invariants and symmetry classes. The link with harmonic decomposition is made thanks to Kelvin decomposition of its harmonic term.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éOn étudie la structure des tenseurs 2D symétriques d’ordre 4, c’est-à-dire : ayant aussi bien la symétrie indicielle mineure que la symétrie majeure. La décomposition polaire de Verchery est réécrite sous forme tensorielle nommée décomposition polaire tensorielle. Le résultat principal est que tout tenseur 2D symétrique d’ordre 4 peut s’écrire à l’aide de tenseurs d’ordre 2 uniquement dans une décomposition faisant apparaître explicitement les invariants et les classes de symétrie. Le lien avec la décomposition harmonique est fait en utilisant la décomposition de Kelvin de son terme harmonique.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.0. IntroductionThe structure of 3D fourth-order elasticity tensor has been intensively studied since 19th-century controversy on the number of independent elasticity constants. Major and minor symmetries reduce to 21 the number of material parameters (symmetric tensors T referred to as multi-constant tensors), when an elasticity tensor having all Cauchy indicial symmetries Ti jkl = Tikjl only has 15 material parameters (supersymmetric or rari-constant tensors).A well-known tool for the study of symmetry classes is the isomorphic harmonic decomposition 2H0 ⊕ 2H2 ⊕ H4 of symmetric tensors space [1–4], defining a scalar (real) space as H0, second-order harmonic tensors h ∈ H2 as traceless (deviatoric, ∑k hkk = 0) symmetric tensors and fourth-order harmonic tensors H ∈ H4 as traceless supersymmetric/rari-* Corresponding author.E-mail address: boris.desmorat@upmc.fr (B. Desmorat).http://dx.doi.org/10.1016/j.crme.2015.07.0021631-0721/© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.472 B. Desmorat, R. Desmorat / C. R. Mecanique 343 (2015) 471–475constant tensors (Hi jkl = Hikjl , ∑k Hkki j = ∑k Hkikj = 0). In other words, any symmetric tensor T, such as a triclinic elasticity tensor, can be represented by two Lamé isotropic constants ∈ H0, by two second-order harmonic tensors ∈ H2 and by one fourth-order harmonic tensor ∈ H4.In 2D, some simplifications arise as scalar expressions for the components Ti jkl(θ) of symmetric tensor T (having both minor and major symmetries) may be derived by making explicitly appear the dependency upon frame angle θ [5] and upon invariants [6–9]. Theses expressions do not have a complete tensorial counterpart in the literature [10].In the present note, we therefore propose a tensorial rewriting and an associated interpretation of Verchery polar decom-position for 2D fourth-order tensors with both minor and major indicial symmetries. It is shown in Section 2 that any 2D symmetric tensor T (resp. any 2D harmonic fourth-order tensor H ∈ H4(2D)) can be expressed by means of 2 scalar invari-ants and of 2 second-order deviatoric tensors ∈ H2(2D) (respectively of only one second-order deviatoric tensor h0 ∈ H2(2D)). The link with harmonic decomposition is made in Section 4. The general tensorial expression of harmonic elements ∈ H4(2D)is retrieved in Section 5 by the use of the Kelvin decomposition.Tensorial products ⊗, ⊗, ⊗ will be used. They are defined as follows: (X⊗Y)i jkl = XikY jl , (X⊗Y)i jkl = XilY jk ,X⊗Y = 12 (X⊗Y + X⊗Y).1. 2D quadratic form using the polar formalismLet us consider any 2D fourth-order tensor T with minor and major symmetries. In the polar formalism [6,7], five invariants are defined; four out of them are elastic moduli (t0, t1, r0, r1), and the last one is the angular difference ϕ0 − ϕ1(each ϕn is not an invariant by itself, see the discussion on joint invariant at the end of Section 2). A basic result of the polar formalism is the expression of the Cartesian components of T in terms of polar parameters, in a frame rotated of an angle θ :T1111(θ) = t0 + 2t1 + r0 cos 4 (ϕ0 − θ) + 4r1 cos 2 (ϕ1 − θ)T1112(θ) = r0 sin 4 (ϕ0 − θ) + 2r1 sin 2 (ϕ1 − θ)T1122(θ) = −t0 + 2t1 − r0 cos 4 (ϕ0 − θ)T1212(θ) = t0 − r0 cos 4 (ϕ0 − θ)T1222(θ) = −r0 sin 4 (ϕ0 − θ) + 2r1 sin 2 (ϕ1 − θ)T2222(θ) = t0 + 2t1 + r0 cos 4 (ϕ0 − θ) − 4r1 cos 2 (ϕ1 − θ) (1)t0 and t1 terms are frame independent (they define the isotropic part of T from a generalization of Lamé constants to anisotropy), r1 terms rotates in cos 2 (ϕ1 − θ) and sin 2 (ϕ1 − θ) as second-order tensors do (Eq. (2)), the r0 term rotates twice more in cos 4 (ϕ0 − θ) and sin 4 (ϕ0 − θ). In a given frame θ , the knowledge of the six independent coefficients of any 2D symmetric tensor T is equivalent to the knowledge of the five invariants (t0, t1, r0, r1, ϕ0 − ϕ1) and of one angle, either ϕ0 − θ or ϕ1 − θ .Still in 2D, a general expression for any symmetric second-order tensor s, making appear explicitly frame angle θ , iss = sm1 + s′ = sm1 + seq[cos 2(ϕ − θ)) sin 2(ϕ − θ)sin 2(ϕ − θ) − cos 2(ϕ − θ)]with{sm = 12 tr sseq =√12 s′ : s′ (2)with first (mean) and second (2D von Mises) invariants defined as sm and seq and where ϕ is the orientation of principal basis of s (it is not an invariant of s). The expression of the associated quadratic form is:12s : T : s = 2t0s2eq + 4t1s2m + 2r0s2eq cos 4 (ϕ0 − ϕ) + 8r1smseq cos 2 (ϕ1 − ϕ) (3)Explicit formulae giving polar invariants as a function of components Ti jkl can be found in [7].2. Proposed Tensorial Polar DecompositionIntroducing the two second-order deviatoric tensors R0, R1,R0 = R′0 =[cos 2(ϕ0 − θ) sin 2(ϕ0 − θ)sin 2(ϕ0 − θ) − cos 2(ϕ0 − θ)]R1 = R′1 =[cos 2(ϕ1 − θ) sin 2(ϕ1 − θ)sin 2(ϕ1 − θ) − cos 2(ϕ1 − θ)](4)of 2D von Mises equivalent norm R0 eq = R1 eq = 1, and of principal direction ϕ0, ϕ1, of course possibly different from principal direction ϕ of tensor s. One has first equalities concerning r1-term,(s : R′ ) tr s = tr(s · R′ · s) = 4smseq cos 2 (ϕ1 − ϕ) (5)1 1B. Desmorat, R. Desmorat / C. R. Mecanique 343 (2015) 471–475 473From (s : R′0)2 = 4s2eq cos2 2(ϕ0 − ϕ) = 2s2eq (1 + cos 4(ϕ0 − ϕ)) and 2s2eq = s′ : s′ second equality concerning r0-term is:2r0s2eq cos 4 (ϕ0 − ϕ) = r0[(s : R′0)2 − s′ : s′] (6)The quadratic form (3) can therefore be rewritten into the following intrinsic form12s : T : s = t0 s′ : s′ + t1 (tr s)2 + r0[(s : R′0)2 − s′ : s′] + 2r1 tr (s · R′1 · s) (7)From the last equation, the intrinsic form of the polar decomposition of a symmetric fourth-order tensor T is obtained in terms of polar invariants t0, t1, r0 and r1 and of the two second-order deviatoric tensors R′0 and R′1,T = 2t0J+ 2t11 ⊗ 1 + 2r0[R′0 ⊗ R′0 − J] + 2r1 (1 ⊗ R′1 + R′1 ⊗ 1) (8)It is equivalent in the present 2D case toT = 2t0J+ 2t11 ⊗ 1 + 2r0[R′0 ⊗ R′0 − J] + 2r1 (1 ⊗ R′1 + R′1 ⊗ 1) (9)thanks to the mathematical property (5) valid ∀s, which implies:1 ⊗ R′1 + R′1 ⊗ 1 = 1 ⊗ R′1 + R′1 ⊗ 1 (10)Eq. (10) is not intrinsic to tensorial products, it stands only in 2D. The tensor J = I − 12 1 ⊗ 1 (defined here in 2D) takes the deviatoric part of any second-order tensor X (i.e. J : X = X′).Equations (8)–(9) define the Tensorial Polar Decomposition of any 2D tensor T having both minor and major symmetries. As both r0- and r1-terms are found rari-constant, the rari-constancy Ti jkl = Tikjl amounts to t0 = t1.Note that joint invariant R′0 : R′1 readsR′0 : R′1 = 2 cos 2 (ϕ0 − ϕ1) (11)It is an invariant of tensor T, as is the polar angular invariant ϕ0 − ϕ1.The intrinsic form of the polar decomposition makes explicitly appear polar moduli and angles, therefore the material symmetries, including ordinary orthotropies ϕ0 −ϕ1 = k π4 , k ∈ {0, 1} [7]. For instance, if T is a 2D elasticity tensor, isotropy is r0 = r1 = 0, the square symmetry is r1 = 0, the r0-orthotropy is r0 = 0, the ordinary orthotropy, with k = 0 is R′0 = R′1 and the ordinary orthotropy with k = 1 is R′0 : R′1 = 0.3. Orthogonality of generatorsTensorial Polar Decomposition (9) can be recast as the sum of polar moduli 2gn times generators G(n) , which are fourth-order tensors (factors 2 appear for consistency with original Verchery work, polar moduli gn standing either for tn or for rn),T =3∑02gnG(n) =2∑12tnG(n)t +2∑12rnG(n)r (12)Fourth-order generator tensors G(n) are of two kinds: the G(n)t are definite positive and do not depend upon frame orienta-tion θ , while the G(n)r = G(n)r (θ) are frame dependent:G(0)t = J , G(1)t = 1 ⊗ 1 , G(0)r = R′0 ⊗ R′0 − J , G(1)r = 1 ⊗ R′1 + R′1 ⊗ 1 (13)The generators are orthogonal with respect to the scalar product :: asG(n) :: G(m) =2∑i=12∑j=12∑k=12∑l=1G(n)i jkl G(m)i jkl = 0 ∀ m = n (14)They all have a constant norm, frame independent, asG(0)t :: G(0)t = 2 G(1)t :: G(1)t = 4 G(0)r :: G(0)r = 2 G(1)r :: G(1)r = 8 (15)4. Link with harmonic decompositionIn 3D there are only two independent traces d = tr12 T = tr34 T (of components ∑3k=1 Tkki j) and v = tr13 T = tr23 T =tr14 T = tr24 T (of components ∑3k=1 Tkikj) for the symmetric tensor T. The symmetric second-order tensor d is dilatation tensor, of deviatoric part d′ , the symmetric second-order tensor v is the Voigt tensor, of deviatoric part v′ . The 3D harmonic decomposition 2H0 ⊕ 2H2 ⊕H4 of fourth-order tensor vector space reads then [2–4]:T = λ1 ⊗ 1 + 2μ1⊗1 + 1 ⊗ h1 + h1 ⊗ 1 + 1⊗h2 + h2⊗1 + 1⊗h2 + h2⊗1 +H (16)474 B. Desmorat, R. Desmorat / C. R. Mecanique 343 (2015) 471–475or in an equivalent mannerT = λ1 ⊗ 1 + 2μ I+ 1 ⊗ h1 + h1 ⊗ 1 + 2(1⊗h2 + h2⊗1) +H (17)with as constants λ = 130 (4 tr d − 2 tr v) and μ = 130 (3 tr v − tr d), as traceless symmetric second-order tensor h1 = h′1 =17 (5d′ − 4v′) ∈ H2 and h2 = h′2 = 17 (3v′ − d′) ∈H2 and as traceless rari-constant tensor H ∈ H4.In 2D (see [10]), in a consistent manner with mathematical property (10) and 2D equality v′ = d′ , if one still sets d = tr12 T, v = tr13 T of components dij = ∑2k=1 Tkki j and vij = ∑2k=1 Tkikj , the harmonic decomposition of fourth-order tensor vector space reads 2H0 ⊕H2(2D) ⊕H4(2D) orT = λ1 ⊗ 1 + 2μ I+ 1 ⊗ h + h ⊗ 1 +H (18)with as 2D constants λ = 12 (tr d − tr v) and μ = 12 (2 tr v − tr d), as 2D the harmonic tensors h = d′/2 = v′/2 ∈ H2(2D) and H ∈ H4(2D) . One easily recognizes the constant and linear terms of Tensorial Polar Decomposition (9), using I = J + 12 1 ⊗ 1, witht0 = μ, t1 = λ + μ2, 2r1 = heq , 2r1R′1 = h (19)The harmonic H4(2D)-term is given in explicit form in Section 2 thanks to polar decomposition by means of an extra traceless second-order tensor h0 = h′0 =√2r0 R′0 ∈ H2(2D) asH = 2r0[R′0 ⊗ R′0 − J] = h0 ⊗ h0 − 12h0 : h0 J tr12 H = tr13 H = 0 (20)This shows that the Tensorial Polar Decomposition of 2D symmetric fourth-order tensors is the direct sum 2H0 ⊕ 2H2(2D) .We propose in next section to use the Kelvin decomposition in order to derive the explicit r0-form of H and to prove that r0 ≥ 0, as needed.5. Retrieving the explicit r0-form of H ∈H4(2D)The harmonic fourth-order tensor H introduced in the previous section isH = T− λ1 ⊗ 1 − 2μ I− 1 ⊗ h − h ⊗ 1 (21)Let us use its harmonic properties tr12 H = tr13 H = 0 and the remark that they correspond to the orthogonality of generator G(0)r with respect to both constant generators G(0)t = J and G(1)t = 1 ⊗ 1.The Kelvin (spectral) decomposition of H [11–14,8], gives, here in 2D,H =2∑I=0I eI ⊗ eI eI : e J = δI J (22)with H : eI = I eI (no sum) defining Kelvin’s moduli I and modes eI . The traceless condition H : 1 = 1 : H = tr12 H = 0implies that 1 is an eigentensor (a Kelvin mode) of H, associated with Kelvin’s modulus 2 = 0, and that the first two eigentensors eI are deviatoric, eI = eI′ . The mathematical property of Kelvin’s projectors to give a partition of the unit tensor reads thene1′ ⊗ e1′ = I− 121 ⊗ 1 − e0′ ⊗ e0′ = J− e0′ ⊗ e0′ (23)so that Kelvin’s decomposition (22) becomesH = (0 − 1)e0′ ⊗ e0′ + 1J (24)By construction, H is orthogonal to generator G(0)t = J = I − 12 1 ⊗ 1. This gives:H :: J= (0 − 1)e0′ ⊗ e0′ :: J+ 1J :: J= 0= (0 − 1)e0′ : e0′ + 21 = (0 − 1) + 21 = 0 (25)This shows that 1 = −0 so that one just has proven that any H ∈ H4(2D) has for expressionH = 0[2e0′ ⊗ e0′ − J](26)Setting R′0 =√2 e0′ as deviatoric second-order tensor of equivalent norm R0 eq = 1, ends up toH = 2r0[R′0 ⊗ R′0 − J]r0 = 0 (27)2B. Desmorat, R. Desmorat / C. R. Mecanique 343 (2015) 471–475 475There are two possibilities for the definition of tensor R′0 and of modulus r0 as there are two Kelvin modes I = 0 and I = 1 orthogonal to Kelvin’s mode e2 = 1/√2. Only the one at positive eigenvalue, set as I = 0, 0 ≥ 0 (leaving then 1 ≤ 0for I = 1) gives a positive r0 as retained in standard polar decomposition of 2D symmetric tensors and as needed at the end of previous section. The polar modulus r0 = 0/2 ≥ 0 is therefore shown to be a half-positive eigenvalue of the harmonic fourth-order tensor H and R′0 is the associated Kelvin mode multiplied by √2.Altogether with expression (21) due to harmonic decomposition, the present derivations (and key Eq. (27)) are an al-ternate proof of Verchery polar decomposition, using tensorial mathematical tools instead of a complex variable method in case of original proof.6. ConclusionWe have proposed a tensorial intrinsic form for Verchery polar decomposition of any 2D fourth-order symmetric ten-sor T. Two proofs are given, a first one from the rewriting of quadratic form (3) associated with tensor T, a second one combining both harmonic and Kelvin decompositions.Compared to harmonic decomposition, the main results are:– the generators obtained are found orthogonal to each other (in sense of scalar product :: for fourth-order tensors) and of constant norm, independent from frame angle,– the polar invariants of tensor T explicitly appear, making easy the study of symmetry classes and sub-classes,– the structure of the harmonic fourth-order tensor H ∈ H4(2D) is given: any traceless rari-constant (harmonic) tensor His shown to be expressed thanks to a single deviatoric (harmonic) second-order tensor h0 or in an equivalent manner in the polar formalism thanks to the polar invariant r0 and to the deviatoric tensor R′0 of the unit 2D von Mises norm.As a conclusion, any 2D symmetric fourth-order tensor can be expressed thanks to two scalars and to two symmetric second-order deviatoric tensors in a decomposition that makes explicitly appear invariants and symmetry classes.References[1] R. Baerheim, Harmonic decomposition of the anisotropic elasticity tensor, Q. J. Mech. Appl. Math. 46 (3) (1993) 391–418.[2] J.P. Boehler, A.A. Kirillov Jr, E.T. Onat, On the polynomial invariants of the elasticity tensor, J. Elasticity 34 (2) (1994) 97–110.[3] S. Forte, M. Vianello, Symmetry classes for elasticity tensors, J. Elasticity 43 (2) (1996) 81–108.[4] N. Auffray, B. Kolev, M. Petitot, On anisotropic polynomial relations for the elasticity tensor, J. Elasticity 115 (1) (2014) 77–103.[5] S.W. Tsai, N.J. Pagano, Invariant properties of composite materials, in: S.W. Tsai, J.C. Halpin, N.J. Pagano (Eds.), Composite Materials Workshop, Tech-nomic Publishing Co., Lancaster, PA, USA, 1968, pp. 233–253.[6] G. Verchery, Les invariants des tenseurs d’ordre 4 du type de l’élasticité, in: Proceedings of Euromech 115, France, Villard-de-Lans, 1979, Éditions du CNRS, Paris, 1982, pp. 93–104.[7] P. Vannucci, Plane anisotropy by the polar method, Meccanica 40 (2005) 437–454.[8] G. De Saxcé, C. Vallée, Structure of the space of 2D elasticity tensors, J. Elasticity 111 (2013) 21–39.[9] B. Desmorat, P. Vannucci, An alternative to the Kelvin decomposition for plane anisotropic elasticity, Math. Methods Appl. Sci. 38 (2015) 164–175.[10] S. Forte, M. Vianello, A unified approach to invariants of plane elasticity tensors, Meccanica (2014), http://dx.doi.org/10.1007/s11012-014-9916-y.[11] J. Rychlewski, On Hooke’s law, Prikl. Mat. Meh. 48 (1984) 303–314.[12] M.M. Mehrabadi, S.C. Cowin, Eigentensors of linear anisotropic elastic materials, Q. J. Mech. Appl. Math. 43 (1990) 15–41.[13] M. François, Détermination des symétries matérielles de matériaux anisotropes, PhD thesis, Université Paris-6, 1995.[14] N. Aufray, M. Coret, M. François, B. Kolev, G. de Saxcé, J.-F. Witz, in: Lectures Notes of Summer School MatSyMat, Nantes, France, September 2014, http://matsymat.sciencesconf.org.

Tensorial Polar Decomposition of 2D fourth-order tensors

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Tensorial Polar Decomposition of 2D fourth-order tensors

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