Micromechanics Based 3D Framework with Second-Order Damage Tensors

Micromechanics based framework with second-order damage tensors

R. Desmorata, B. Desmoratb,c, M. Olivea, B. Kolevd

aLMT-Cachan (ENS Cachan, CNRS, UniversitéParis Saclay), F-94235 Cachan Cedex, France bSorbonne Université,UMPC Univ Paris 06, CNRS, UMR 7190, Institut d’Alembert,F-75252 Paris Cedex 05, France cUniversitéParis SUD 11, Orsay, France dAix Marseille Université,CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France

Abstract The harmonic product of tensors—leading to the concept of harmonic factorization—has been deﬁned in a previous work (Olive et al, 2017). In the practical case of 3D crack density measurements on thin or thick walled structures, this mathematical tool allows us to factorize the harmonic (irreducible) part of the fourth-order damage tensor as an harmonic square: an exact harmonic square in 2D, an harmonic square over the set of so-called mechanically accessible directions for measurements in the 3D case. The corresponding micro-mechanics framework based on second—instead of fourth—order damage tensors is derived. An illustrating example is provided showing how the proposed framework allows for the modeling of the so-called hydrostatic sensitivity up to high damage levels. Keywords: anisotropic damage, crack density, harmonic decomposition PACS: 46.50.+a, 91.60.-x, 91.60.Ba

1. Introduction a single model is partial (Ladevèze, 1983, 1995). A link with the theory of second order fabric tensors has been The damage anisotropy encountered in quasi-brittle ma- made in (Zysset and Curnier, 1995; Voyiadjis and Kattan, terials is induced by the loading direction and multi- 2006). From a theoretical point of view (Leckie and Onat, axiality. From a micro-mechanics point of view, it is 1980; Onat, 1984), second order damage frameworks are the consequence of an oriented microcracking pattern. usually seen to be restrictive compared to the fourth-order From the Continuum Damage Mechanics (CDM) point tensorial one. Nevertheless, the interpretation of a dam- of view, the anisotropic damage state is represented by age variable being simpler when a second-order tensor is a tensorial thermodynamics variable, either an eight- considered (the three principal values 푑푖 of d naturally order tensor (Chaboche, 1978, 1979), a fourth-order dam- correspond to 3 orthogonal families of microcracks), less age tensor D (Chaboche, 1978, 1979; Leckie and Onat, damage parameters are introduced and the second-order 1980; Chaboche, 1984; Lemaitre and Chaboche, 1985; An- frameworks have been widely used for either ductile or drieux et al., 1986; Ju, 1989; Kachanov, 1993; Zheng and quasi-brittle materials. Collins, 1998; Cormery and Welemane, 2010; Dormieux The recent analysis of 2D cracked media with both open and Kondo, 2016) or a symmetric second-order damage and closed microcraks has shown that the so-called irre- tensor d (Vakulenko and Kachanov, 1971; Murakami and ducible (harmonic) part H2퐷 of the damage tensor can Ohno, 1978; Cordebois and Sidoroff, 1982; Ladevèze, 1983; be decomposed by means of a second-order damage ten- Murakami, 1988). sor (Desmorat and Desmorat, 2016). More precisely, the There exist many second-order anisotropic damage standard second-order crack density tensor of Vakulenko frameworks (Murakami, 1988; Kattan and Voyiadjis, 1990; and Kachanov(1971) still represents the open cracks con- Ramtani et al., 1992; Papa and Taliercio, 1996; Halm and tribution when a novel (deviatoric) second-order dam- Dragon, 1998; Steinmann and Carol, 1998; Lemaitre et al., age tensor represents the closed—sliding—cracks (pre- arXiv:1706.00845v3 [physics.class-ph] 18 Dec 2017 2000; Carol et al., 2001; Menzel and Steinmann, 2001; viously represented by a fourth-order tensor, Andrieux Menzel et al., 2002; Brunig, 2003; Lemaitre and Desmorat, et al.(1986); Kachanov(1993)). This can be achieved 2005; Desmorat et al., 2007; Badel et al., 2007; Desmorat using Verchery’s polar decomposition of 2D fourth-order and Otin, 2008; Desmorat, 2016), as their unification into tensors (Verchery, 1979; Vannucci, 2005), which includes both (Desmorat and Desmorat, 2015):

Email addresses: [email protected] (R. Desmorat), – the harmonic decomposition of considered tensor; [email protected] (B. Desmorat), [email protected] (M. Olive), [email protected] (B. Kolev) – the harmonic factorization of its fourth-order irre-

Preprint submitted to European Journal of Mechanics A/Solids (accepted, https://doi.org/10.1016/j.euromechsol.2017.11.014)April 25, 2019 ducible (harmonic) part H2퐷, by means of a devia- T1 and T2, of respective orders 푛1 and 푛2, is the sym- toric second-order tensor h2퐷: metrization of T1 ⊗T2, defining a totally symmetric tensor of order 푛 = 푛1 + 푛2: H2퐷 = h2퐷 * h2퐷. s T1 ⊙ T2 := (T1 ⊗ T2) . The harmonic product between harmonic tensors, written as h1 *h2, is defined as the projection of the (totally) sym- Contracting two indices 푖, 푗 of a tensor T of order 푛 de- metric tensor product h1 ⊙ h2 onto the space of harmonic fines a new tensor of order 푛 − 2 denoted as tr푖푗 T. For tensors (see Sections3 and 4.4). a totally symmetric tensor T, this operation does not de- The question arises then as how to extend these results pend on a particular choice of the pair 푖, 푗. Thus, we can in 3D ? We know from (Olive et al., 2017) that any 3D refer to this contraction just as the trace of T and we will harmonic fourth-order tensor can be factorized into denote it as tr T. It is a totally symmetric tensor of order 푛 − 2. Iterating the process, we define H = h1 * h2, tr푘 T = tr(tr(··· (tr T))), i.e., represented by two (deviatoric) second-order tensors h1, h2. However, the factorization is far from being unique. which is a totally symmetric tensor of order 푛 − 2푘. To overpass these difficulties, we point out that triaxial In 3D, a totally symmetric fourth-order tensor T has mechanical testing is of high complexity, both from the no more than 15 independent components, instead of 21 experimental set-up needed (a triaxial machine) and from for a triclinic elasticity tensor (i.e. a tensor E having mi- the difficulty to measure mechanical properties in differ- nor symmetry 퐸 = 퐸 = 퐸 and major symmetry ent space directions (Calloch, 1997; Calloch and Marquis, 푖푗푘푙 푗푖푘푙 푖푗푙푘 퐸 = 퐸 ). Totally symmetric elasticity tensors were 1999). We propose, here, to restrict ourselves to a simpler, 푖푗푘푙 푘푙푖푗 called rari-constant in the nineteenth century (Navier, but still sufficiently general, situation: the case of mea- 1827; Cauchy, 1828b,a; Poisson, 1829; Love, 1905; Van- surements of a 3D crack density function on structures. nucci and Desmorat, 2016). Well-known cases are the thin walled structures, such as plates, tubes and shells for which the thinner direction is the normal 휈. But the present work also applies to 3D thick structures (as the cube of Section8), as long as an 2. Harmonic decomposition out-of-plane normal can be locally defined. Instead of considering the representation of crack den- The harmonic decomposition of tensors (Schouten, sity in any direction 푛, we shall then consider, in Section5, 1954; Spencer, 1970), introduced in geophysics by Backus its representation to a restricted set of directions (1970), has been popularized by Leckie and Onat(1980) and Onat(1984) when deriving fourth-order damage ten- ℛ(휈) := {휏 ; ‖휏 ‖ = 1 and 휏 · 휈 = 0} ∪ {휈} , sor and later by Forte and Vianello(1996) when classifying elasticity symmetries. i.e., the in-plane directions 휏 , orthogonal to 휈, and the out-of-plane direction 휈, normal to the structure itself. In the present work, we consider these directions as the me- 2.1. Harmonic tensors and corresponding polynomials chanically accessible directions for measurements. After recalling the required mathematical tools (har- An harmonic tensor is a traceless, totally symmetric ten- monic decomposition, harmonic product and Sylvester’s sor, i.e. theorem), we revisit the link between crack density func- s H = H , and tr H = 0. tion and the tensorial nature of the damage variables. This will allow us to derive a general micro-mechanics based 3D To every totally symmetric tensor H of order 푛, with com- framework with second—instead of fourth—order damage ponents 퐻푖1푖2···푖푛 , corresponds a unique homogenous poly- tensors. nomial (and conversely). More precisely,

Definitions h(푥) := H(푥푥,푥푥푥, . . . ,푥푥) = 퐻푖1푖2···푖푛 푥푖1 푥푖2 ··· 푥푖푛 We denote by Ts the totally symmetric part of a possibly non symmetric tensor T. More precisely is a homogeneous polynomial

s 1 ∑︁ h(푥) = h(푥1, 푥2, 푥3), T (푥1, . . . ,푥푥 ) := T(푥 , . . . ,푥푥 ), 푛 푛! 휎(1) 휎(푛) 휎∈S푛 of degree 푛 in the spacial coordinates 푥1, 푥2, 푥3. It is 2 where S푛 is the permutation group on the indices harmonic since ∇ h = 0, due to the traceless property {1, . . . , 푛}. The symmetric tensor product of two tensors tr H = 0. 2 1 2.2. Harmonic decomposition of a symmetric tensor where (y ⊗ z)푖푗푘푙 = 2 (푦푖푘푧푗푙 + 푦푖푙푧푗푘) so that ⊗(4) is the same as the totally symmetric tensor product ⊙: Any totally symmetric tensor T of order 푛 can be decomposed uniquely as y ⊗(4) z = y ⊙ z. ⊙푟−1 ⊙푟 T = H0 + 1 ⊙ H1 + ··· + 1 ⊙ H푟−1 + 1 ⊙ H푟 (2.1) In the harmonic decomposition (2.5), H is a fourth-order harmonic tensor, 훼, 훽 are scalars, and a′, b′ are second- where 푟 = [푛/2] is the integer part of 푛/2, H푘 is an har- order harmonic tensors (deviators) related to the dilatation monic tensor of degree 푛 − 2푘 and 1⊙푘 = 1 ⊙ · · · ⊙ 1 tensor di = tr12 E and the Voigt tensor vo = tr13 E by means the symmetrized tensorial product of 푘 copies of the formulas: the (second-order) identity tensor. For 푛 even (푛 = 2푟), 1 1 one has: 훼 = (tr di + 2 tr vo) , 훽 = (tr di − tr vo) , (2.7) 1 푛 15 6 푛 2 H푟 = H 2 = tr T, (2.2) 푛 + 1 and where H푟 = 퐻푟 is a scalar in that case. Moreover, ′ 2 (︀ ′ ′)︀ ′ (︀ ′ ′)︀ H −1,..., H0 are obtained inductively (Olive et al., 2017) a = di + 2vo , b = 2 di − vo . (2.8) 푟 7 as follows:

푟 The harmonic part of E is deﬁned as: 푘 [︁ ∑︁ ⊙푗 ]︁ H푘 = 휇(푘, 푛) tr T − 1 ⊙ H푗 (2.3) (E)0 := H = E − 1 ⊗(4) a − 1 ⊗(2,2) b (2.9) 푗=푘+1 or similarly as: (2푛 − 4푘 + 1)!(푛 − 푘)!푛! where 휇(푘, 푛) = . (2푛 − 2푘 + 1)!푘!(푛 − 2푘)!(푛 − 2푘)! 1 (E)0 := E − 1 ⊙ a − (1 ⊗ b + b ⊗ 1 − 1 ⊗ b − b ⊗ 1) , Remark 2.1. It is worth emphasizing the fact that this 3 harmonic decomposition is just a generalization to higher where a = a′ + 훼1 and b = b′ + 훽1. The scalars 훼, 훽 order symmetric tensors of the well-known decomposi- and the second-order deviators a′, b′ are given by (2.7) tion of a symmetric second-order tensor into its devia- and (2.8). toric/spheric parts:

1 3. The harmonic product and Sylvester’s theorem d = d′ + (tr d)1. 3 The harmonic product of two harmonic tensors of or- Decomposition (2.1) is an orthogonal decomposition der 푛1 and 푛2, deﬁning an harmonic tensor of order (relative to the natural Euclidean product on the space 푛 = 푛1 + 푛2, has been introduced in (Olive et al., 2017) as of symmetric tensors). The projection H0 onto the space the harmonic part of the symmetric tensor product: of highest order harmonic tensors (same order as T) will H1 * H2 := (H1 ⊙ H2)0 . be called the harmonic part of T and denoted by (T)0:

⊙푟 Note that this product is associative: (T)0 := H0 = T − 1 ⊙ H1 − · · · − 1 퐻푟. (2.4) H1 * (H2 * H3) = (H1 * H2) * H3, 2.3. Harmonic decomposition of the elasticity tensor and commutative: The harmonic decomposition of an elasticity tensor E, a fourth-order tensor having both minor and major symme- H1 * H2 = H2 * H1. tries (퐸푖푗푘푙 = 퐸푗푖푘푙 = 퐸푖푗푙푘 and 퐸푖푗푘푙 = 퐸푘푙푖푗), was first obtained by Backus(1970), as: For two vectors 푤1,푤푤2, we have ′ ′ ′ 푤1 * 푤2 =(푤1 ⊙ 푤2) E = 훼 1⊗(4) 1+훽 1⊗(2,2)1+1⊗(4) a +1⊗(2,2)b +H (2.5) 1 1 ⊗ ⊗ − · ′ 1 = (푤1 푤2 + 푤2 푤1) (푤1 푤2) 1, where (·) = (·) − 3 tr(·) 1 denotes the deviatoric part of a 2 3 second-order tensor. 푇 where 푤1 · 푤2 = 푤 푤2 is the scalar product. In formula (2.5), the Young-symmetrized tensor prod- 1 For two second-order harmonic tensors (deviators) h1, ucts ⊗(4) and ⊗(2,2), between two symmetric second-order h2, we have tensors y, z, are defined as follows: {︃ 2 ⊗ 1 (︀ ⊗ ⊗ ⊗ ⊗ )︀ h1 * h2 = h1 ⊙ h2 − 1 ⊙ (h1h2 + h2h1) y (4) z = 6 y z + z y + 2 y z + 2 z y , 7 1 (︀ )︀ y ⊗(2,2) z = 3 y ⊗ z + z ⊗ y − y ⊗ z − z ⊗ y , 2 + tr(h1h2) 1 ⊙ 1. (3.1) (2.6) 35 3 Sylvester’s theorem (Sylvester, 1909; Backus, 1970; experimental (measured) density distribution Ω(푛), F be- Baerheim, 1998) states that any harmonic tensor H of or- ing thus solution of der 푛 can be factorized as ⃦ ∫︁ ⃦2 ⃦ 4휋 ⊙푛 ⊗푛 ⃦ min ⃦ F ∙ 1 − Ω(푛)푛 d푆⃦ , H = 푤1 * 푤2 *···* 푤푛, ⃦ ⃦ F ⃦2푛 + 1 ‖푥‖=1 ⃦ i.e. as the harmonic products of 푛 (real) vectors 푤푘, the with solid angle 푆 and where so-called Sylvester-Maxwell multipoles. Note however, that this factorization is far from being unique, as discussed 푛⊗푘 := 푛 ⊗ 푛 ⊗ · · · ⊗ 푛푛. in (Olive et al., 2017). Setting h1 = 푤1 * 푤2 and h2 = 푤3 * 푤4 which are Moreover, the following equality has been used: harmonic second-order tensors (deviators), we obtain the 1 ∫︁ 1 non unique harmonic factorization of H by means of two 푛⊗2푛 d푆 = 1⊙푛. second-order tensors: 4휋 ‖푥‖=1 2푛 + 1 Note that 푛⊗푘 = 푛⊙푘 is a totally symmetric tensor. H = h1 * h2, (3.2) Comparative studies of the tensorial order, needed to as detailed in (Desmorat and Desmorat, 2016; Olive et al., represent given microcracking patterns, can be found 2017). in (Lubarda and Krajcinovic, 1993; Krajcinovic, 1996; Tikhomirov et al., 2001). Expression (4.1) is often rewritten into the finite expan- 4. Link between fourth-order crack density and sion (Kanatani, 1984; Onat, 1984; Krajcinovic, 1996): damage tensors ⊗6 Ω(푛) = F4 ∙ (푛 ⊗ 푛 ⊗ 푛 ⊗ 푛) + Ω6 ∙ (푛 ) + ··· Before formulating our main result, Theorem 5.1, we ⊗2푘 ⊗푛 ··· + Ω ∙ (푛 ) + ··· + Ω ∙ (푛 ) (4.2) summarize, in this section, the present state–of–the– 2푘 푛 art in Continuum Mechanics leading to the representa- with fourth-order part tion of damage of cracked media by a fourth-order tensor (Chaboche, 1979). We make an explicit link with the F4 ∙ (푛 ⊗ 푛 ⊗ 푛 ⊗ 푛) =Ω0 + Ω2 ∙ (푛 ⊗ 푛) harmonic decomposition and we present, by comparison (4.3) + Ω4 ∙ (푛 ⊗ 푛 ⊗ 푛 ⊗ 푛) to the 2D case, the problem of representation of damage by second-order tensors in 3D. with 푛 = 2푟 even and where Ω2푘 are totally symmetric traceless (harmonic) tensors of order 2푘. The scalar term 4.1. Crack density function and tensors Ω0 is the crack density within considered Continuum Me- chanics representative volume element The damage state of a microcracked material is classically defined by spatial arrangement, orientation and ge- 1 ∫︁ ometry of the cracks present at the microscale (Kachanov, Ω0 = Ω(푛) d푆. 4휋 ‖푥‖=1 1972; Leckie and Onat, 1980; Ladevèze, 1983; Onat, 1984; Lemaitre and Chaboche, 1985; Murakami, 1988; Crack density tensors Ω2,Ω4,...,Ω푛 are harmonic ten- Kachanov, 1993). The crack density, related to any pos- sors of even order 2, 4, . . . , 푛. They constitute independent sible 3D direction defined by a unit vector 푛, refers to crack density variables representative of the microcraking a dimensionless scalar property defined in a continuous pattern (and anisotropy), determined uniquely up to or- manner at the Representative Volume Element scale as a der 푛 from the knowledge of the 3D spatial crack density spatial crack density function Ω = Ω(푛). Owing to the distribution Ω(푛). property Ω(푛) = Ω(−푛), it is expressed by means of a totally symmetric tensor F (the so-called fabric tensor) of 4.2. Derivation of the crack density tensors from the har- even order 푛 = 2푟 (Kanatani, 1984) as: monic decomposition

Let us point out that the harmonic tensors Ω2푘 cor- Ω(푛) = F ∙ (푛 ⊗ 푛 ⊗ · · · ⊗ 푛) (4.1) respond to the tensors H푟−푘 issued from the harmonic decomposition (2.1) of the fabric tensor F: where ∙ means the contraction over the 푛 subscripts. Note ⊙푟−1 ⊙푟 that Ω(푛) corresponds to a homogeneous polynomial (see F = H0 + 1 ⊙ H1 + ··· + 1 ⊙ H푟−1 + 1 퐻푟, section 2.2) 푛 with 푟 = 푛/2, where 퐻푟 = 퐻 2 and the harmonic tensors h(푛1, 푛2, 푛3) = F(푛푛,푛푛푛, . . . ,푛푛). H푘 of degree 푛 − 2푘 are given by (2.2) and (2.3). Observe, moreover, that: The fabric tensor F, which is totally symmetric, can be ⊙푘 ⊗푛 ⊗푛−2푘 determined as the least square error approximation of an (1 ⊙ H푘) ∙ 푛 = H푘 ∙ 푛 , 4 and we get thus: Remark 4.3. An alternative framework is due to Voyiadjis and Kattan(2006). These authors extend to anisotropic Ω(푛) = 퐻푟 + H푟−1 ∙ (푛 ⊗ 푛) damage the framework of Zysset and Curnier(1995) for the ⊗푛 + H푟−2 ∙ (푛 ⊗ 푛 ⊗ 푛 ⊗ 푛) + ··· + H0 ∙ (푛 ), representation of microstructure morphology of granular materials (the considered framework neglects the fourth which is the finite expansion (4.2), where order contribution Ω4). They propose, then, a nonlinear link between the second order fabric tensor Ω2 and a fourth Ω0 = 퐻 푛 , Ω2푘 = H푟−푘. 2 order tensorial damage variable D, setting for the effective 4.3. Fourth-order damage tensor (damaged) elasticity tensor Using the decomposition (4.2) and assuming open mi- E˜ = E˜ = 휆휑휑 ⊗ 휑 + 2휇휑휑 ⊗ 휑휑, 휑 = (Ω 1 + Ω )−푘 (4.5) crocracks in an initially 3D isotropic medium, Leckie and 0 2 Onat(1980) and Onat(1984) have shown that the damage This means that they replace the identity tensor 1 in the variable defined by the coupling microcraking/elasticity is usual isotropic elasticity law by a negative power −푘 = at most a fourth-order tensor, built from F4 only, see (4.2). −0.2 of the second order crack density tensor Ω01 + Ω2 This result holds for non interacting closed—sliding with- (powers being taken in terms of the principal values, 휆, 휇 out friction—pennyshaped microcracks (Kachanov, 1993) considered as Lamé-like constants). The crack density Ω0 and, as pointed out by Cormery and Welemane(2010), must have a non zero initial value. Due to the presence for many stress based homogenization schemes, as long as of quadratic terms in crack densities, this framework does all the microcracks are in the same state, either open or not satisfy previous proportionality properties. closed. Setting: Fourth order damage tensor D is such that E˜ = (1 − D): 1 J = I − 1 ⊗ 1, E. 3 the following general definition of a fourth-order damage 4.4. 2D case tensor has then been derived for initially isotropic materi- In 2D, cracks are represented by 2D straight lines. Ex- als (Kachanov, 1993; Zheng and Collins, 1998; Cormery pression (4.2) for crack density holds, recovering a Fourier and Welemane, 2010): finite expansion (Kanatani, 1984; Burr et al., 1995):

D = 푝0Ω01 ⊗ 1 + 푝1Ω0 J + 푝2(1 ⊗ Ω2 + Ω2 ⊗ 1) ′ Ω(푛) = 휔2퐷 + 휔2퐷 ∙ (푛 ⊗ 푛) + 푝3(1 ⊗ Ω2 + Ω2 ⊗ 1) + 푝4Ω4, (4.4) + H2퐷 ∙ (푛 ⊗ 푛 ⊗ 푛 ⊗ 푛) + ··· , (4.6) where Ω0 should be interpreted as a scalar damage vari- where the unit vector 푛 is related to the possible planar able, the symmetric deviator Ω2 as a second-order damage variable, and the harmonic tensor Ω as a fourth-order direction 4 1 ∫︁ 2휋 damage variable. The expression of the scalars 푝푖 depends 휔2퐷 = Ω(푛) d휃 on the initial elasticity parameters, on the homogenization 2휋 0 ′ scheme and on the microcracks state (simultaneously open is the 2D crack density, and where 휔2퐷, H2퐷 are respec- or simultaneously closed for all cracks). tively the 2D harmonic second and the fourth-order crack density tensors. Remark 4.1. The scalars 푝푖 do not depend on Ω0,Ω2,Ω4. Verchery’s decomposition (Verchery, 1979; Vannucci, Remark 4.2. (4.4) is the harmonic decomposition (2.5) of 2005), and its rewriting into a tensorial form (Desmorat the fourth-order damage tensor D, which has the major and Desmorat, 2015), shows that any 2D harmonic fourth- and the minor indicial symmetries (퐷 = 퐷 = 퐷 ) 푖푗푘푙 푘푙푖푗 푗푖푘푙 order tensor is an harmonic square. Applied to H , this as an elasticity tensor. The deviatoric parts of the dilata- 2퐷 gives: tion and Voigt tensors are both proportional to the second- H2퐷 = h2퐷 * h2퐷, (4.7) order harmonic tensor Ω2, with the scalar factors 휅di and 휅vo depending only on the initial elastic parameters of the where h2퐷 is an harmonic second-order tensor (deviator), undamaged isotropic material: and the notation * denotes the harmonic product deﬁned

′ ′ as the orthogonal projection of the symmetrized product di (D) = (tr12 D) = 휅diΩ2, h2퐷 ⊙ h2퐷 onto 2D fourth-order harmonic tensors’ space: ′ ′ vo (D) = (tr13 D) = 휅voΩ2. h2퐷 * h2퐷 := (h2퐷 ⊙ h2퐷)0. The traces of the dilatation and the Voigt tensors are both proportional to the scalar crack density Ω0, with scalar fac- For second-order harmonic tensors, this reads: tors 푘di and 푘vo depending only on the elastic parameters 1 2 of virgin (undamaged) isotropic material: h2퐷 * h2퐷 = h2퐷 ⊙ h2퐷 − (tr h ) 1 ⊙ 1. 4 2퐷 tr di(D) = tr(tr12 D) = 푘diΩ0, This means that in 2D, any anisotropic microcracking tr vo(D) = tr(tr13 D) = 푘voΩ0. pattern can be expressed, up to order 4, exactly by means 5 of the scalar 휔2퐷 and the two independent second-order de- such as plates. This allows us to introduce the unit normal ′ 2퐷 ′ viatoric damage variables 휔 = Ω2 and h2퐷 = h2퐷. This 휈 (‖휈‖ = 1) to the walled structure and the set result is consistent with the fact that the micro-mechanics of 2D media with open and closed (sliding without fric- ℛ(휈) := {휏휏, ‖휏 ‖ = 1 and 휏 · 휈 = 0} ∪ {휈}, tion) microcracks can be represented by two second-order damage tensors only (Desmorat and Desmorat, 2015). of directions 푛 restricted to so-called mechanically acces- The question arises then whether the expansion (4.6)– sible directions for measurements. (4.7) holds in 3D, i.e. with 휔′ and h (now 3D) deviatoric As an extension of both the previous remark on fourth- second-order tensors. The answer is negative in the general order harmonic squares and the 2D result (4.6)–(4.7), we have in 3D the following theorem (the proof of which is triclinic case for which we have only Ω4 = h1 * h2 (3.2) given at the end of the present section): with usually diﬀerent second-order tensors h1, h2 (Olive et al., 2017). Furthermore, the factorization is not unique, Theorem 5.1. For a given unit vector 휈, any density forbidding to interpret h1 and h2 as damage variables. function Ω(푛) is represented, up to fourth-order, for all directions 푛 ∈ ℛ(휈) by means of a scalar 휔푚 and two har- ′ 5. 3D second-order damage tensors from walled monic (symmetric deviatoric) second-order tensors 휔 and structures h as:

′ It was noticed by Lubarda and Krajcinovic(1993) Ω(푛) = 휔푚 + 휔 ∙ (푛 ⊗ 푛) and Krajcinovic(1996) that the fourth-order crack den- + (h * h) ∙ (푛 ⊗ 푛 ⊗ 푛 ⊗ 푛) + ··· (5.2) sity tensor Ω4 = (F4)0 (the harmonic part of the totally symmetric tensor F4) induced by particular loadings is re- for all 푛 ∈ ℛ(휈). This representation is unique, up to ±h, lated, sometimes, as a square of the second-order harmonic if (휔′휈) × 휈 = h휈 = 0 . contribution Ω2. More precisely, Ω4 is proportional to the ′ harmonic square Ω2 * Ω2 in the particular situation that Remark 5.2. If we set 푒3 = 휈, the conditions (휔 휈)×휈 = 0 occurs for a family of parallel penny shaped microcracks (which is equivalent to (휔′휈) = 휆휈휈) and h휈 = 0 mean that having identical—therefore coplanar—normal 푚 (induced ⎛ ′ ⎞ 휔11 휔12 0 for example by uniaxial tension on quasi-brittle materials). ′ ′ 휔 = ⎝휔12 휔22 0 ⎠ , (5.3) An harmonic square is also present in the work of Voyi- ′ ′ adjis and Kattan(2006). Let us show that it corre- 0 0 −(휔11 + 휔22) sponds to the fourth order harmonic tensor 휑 * 휑 with and −0.2 ⎛ ⎞ 휑 = (Ω0 1 + Ω2) : ℎ11 ℎ12 0 h = ℎ −ℎ 0 . (5.4) ˜ ⎝ 12 11 ⎠ – The harmonic part (E)0 of the effective elasticity ten- 0 0 0 sor (4.5) is Applied to thinned walled structured, for which 푛 ∈ ˜ s s ℛ (E)0 = (휆 (휑 ⊗ 휑) + 2휇 (휑 ⊗ 휑) )0 , (5.1) (휈) are the accessible directions for mechanical measurements, Theorem 5.1 states that any microcracking pat- s where (.) means the totally symmetric part and (.)0 tern, possibly triclinic, can be represented, up to order 4, is the harmonic projection defined in Eq. (2.9). by means of only two symmetric second-order crack density tensors, 휔 = 휔′ + 휔 1 and h, the second one being a – By definition (휑 ⊗ 휑)s = (휑 ⊗ 휑)푠 = 휑 ⊙ 휑. 푚 deviator. In that case, we can recast (5.2) as: – The harmonic fourth order part of E˜ is thus Ω(푛) = 휔 ∙ (푛 ⊗ 푛) + (h * h) ∙ (푛 ⊗ 푛 ⊗ 푛 ⊗ 푛) + ···

(E˜)0 = (휆 + 2휇)(휑 ⊙ 휑)0 = (휆 + 2휇)휑 * 휑휑, for all 푛 ∈ ℛ(휈), where 휔 is the crack density tensor in- where * is the harmonic product (3.1). See also Ap- troduced by Vakulenko and Kachanov(1971)(Kachanov, pendix B. 1972, 1993).

The second and fourth-order crack density variables Ω2 Practical formulas – Proof of Theorem 5.1 Ω and Ω4 are independent in general case. We prefer to keep Recall that, up to order four, the crack density func- the fourth order contribution Ω4 in our further analyses tion Ω(푛) is represented by the fabric tensor F4 (4.2)– and we will use next the standard expression (4.4) (for (4.3). Consider now an orthonormal frame {푒1,푒푒2,휈휈} and instance instead of Eq. (4.5) which neglects Ω4). let (휔 ,휔휔′, h) be a triplet as in (5.3)–(5.4). Set In order to built a micromechanics based framework 푚 with second-order damage tensors, one considers as the 1[︀ general case—except for soils—that the measurements of ℎ11 + 푖ℎ12 = (F4)1111 + (F4)2222 − 6(F4)1122 2 3D crack density Ω(푛) is performed on thin or thick walled (︀ )︀]︀1/2 structures, i.e. on thin or thick tubes or on 2D structures + 4푖 (F4)1112 − (F4)1222 (5.5) 6 and from which we deduce that

1 1 1 * * 2 2 휔 = (F4)1111 + (F4)1122 + (F4)2222 (ℎ11 + 푖ℎ12) = (ℎ11 + 푖ℎ12) 푚 4 2 4 * 1 1 2 2 and therefore that h = ±h (in accordance with Remark + (F4)3333 − (ℎ11 + ℎ12), 3 15 5.3). From (5.6) and 푎0 = 푎2 = 푏2 = 0, we get 5 1 3 ′ * *′ ′ *′ ′ 휔11 = (F4)1111 + (F4)1122 − (F4)2222 ⎧ 휔 − 휔 + 휔 − 휔 + 휔 − 휔 = 0, 8 4 8 ⎪ 푚 푚 11 11 22 22 ⎪ * ′ *′ ′ *′ 1 1 2 2 ⎨ 2(휔푚 − 휔푚) + 휔11 − 휔11 + 휔22 − 휔22 = 0, − (F4)3333 + (ℎ11 + ℎ12), 3 42 휔′ − 휔*′ − 휔′ + 휔*′ = 0, ⎪ 11 11 22 22 ′ 3 1 5 ⎩⎪ * 휔 = − (F4)1111 + (F4)1122 + (F4)2222 휔12 − 휔12 = 0, 22 8 4 8 * *′ ′ 1 1 2 2 i.e. 휔 = 휔푚 and 휔 = 휔 , which achieves the proof. − (F4)3333 + (ℎ + ℎ ), 푚 3 42 11 12 휔12 =(F4)1222 + (F4)1112, 6. General micro-mechanics based framework with √ where 푖 = −1 is the pure imaginary number. It can two second-order damage variables ′ be checked by a direct computation that (휔푚,휔휔 , h) is a Using the results from Section 4.3, we deduce from (5.2) solution of (5.2) in Theorem 5.1. that the representation by means of two symmetric second- Remark 5.3. Because of the square root in (5.5), both h order tensors holds for the damage tensor itself, at least and −h are solutions. when the microcracks are all in the same state, all open We will now show the uniqueness of the solution, up to or all closed. This means that, disposing from sufficiently a sign, and under the assumption that: many in-plane measurements (along directions 푛 orthogonal to 휈) and an out-of-plane measurement (along 푛 = (휔′휈) × 휈 = h휈 = 0. 휈), the general fourth-order damage tensor of Chaboche– Leckie–Onat can be expressed by means of two symmetric * *′ * Suppose thus, that a second solution (휔푚,휔휔 , h ) to (5.2) second-order damage variables only, for example 휔 and *′ * exists, with 휔 and h as in (5.3)–(5.4). Equaling, for h (the second–one being a deviator). A general damage different directions 푛 ∈ ℛ(휈), the density function Ω(푛) framework using this feature is derived next, clarifying the ′ defined by (5.2), calculated first with (휔푚,휔휔 , h) and then link between Cordebois and Sidoroff(1982) and Ladevèze * *′ * with (휔푚,휔휔 , h ), we obtain for 푛 = 휈: (1983, 1995) phenomenological second-order damage models and micro-mechanics based framework. *′ *′ − ′ − ′ − * 35(휔11 + 휔22 휔11 휔22 + 휔푚 휔푚) We shall assume that the homogenization result (4.4) 2 2 *2 *2 + 4ℎ11 + 4ℎ12 − 4ℎ11 − 4ℎ12 = 0. (5.6) holds, where the constants 푝푖 are given (refer to the works of Kachanov(1993) and Dormieux and Kondo(2016) for Then, for 푛 = (cos 휃, sin 휃, 0), we get: comparison of different homogenization schemes). Gibbs free enthalpy density writes 푎0 + 푎2 cos 2휃 + 푏2 sin 2휃 + 푎4 cos 4휃 + 푏4 sin 4휃 = 0, (5.7) 1 1 1 where 휌휓⋆ = (tr휎)2 + 휎′ : 휎′ + 휎 : D : 휎휎, 18퐾 4퐺 2퐸 * ′ *′ ′ *′ 푎0 = 2(휔푚 − 휔푚) + 휔11 − 휔11 + 휔22 − 휔22 퐸 퐸 where 휌 is the density and 퐸, 퐺 = 2(1+휈) , 퐾 = 3(1−2휈) 3 2 *2 2 *2 are, respectively, the Young, shear and bulk moduli. The + (ℎ11 − ℎ11 + ℎ12 − ℎ12), 35 elasticity law, coupled with the anisotropic damage, writes ′ *′ ′ *′ 푎2 = 휔11 − 휔11 − 휔22 + 휔22, then as 푏 = 2(휔 − 휔* ), ⋆ 2 12 12 푒 휕휓 1 ′ 1 1 2 2 *2 *2 휖 = 휌 = 휎 + (tr휎) 1 + D : 휎휎, 푎4 = ℎ11 − ℎ12 − ℎ11 + ℎ12, 휕휎휎 2퐺 9퐾 퐸 * * 푏4 = 2(ℎ11ℎ12 − ℎ11ℎ12). or, in a more compact form, as 휖푒 = S˜ : 휎휎, Since (5.7) holds for all 휃, we have where 휖푒 is the elastic strain tensor and S˜, the effective

푎0 = 푎2 = 푏2 = 푎4 = 푏4 = 0. fourth-order compliance tensor 1 1 1 1 Since 푎4 = 푏4 = 0, we get S˜ = 1 ⊗ 1 + J + D, J = I − 1 ⊗ 1. (6.1) 9퐾 2퐺 퐸 3 {︃ *2 *2 2 2 ℎ11 − ℎ12 =ℎ11 − ℎ12, Having many in-plane and possibly one out-of-plane * * ℎ11ℎ12 =ℎ11ℎ12, measurements, allows us to use remark 4.1 and (5.2) 7 instead of (4.4), within the considered homogenization 7. A second-order anisotropic damage model in scheme. We can thus recast the fourth-order damage ten- micro-mechanics based framework sor D by substituting the scalar Ω0 by 휔푚, the second- ′ Following Cordebois and Sidoroff(1982) and Ladevèze order tensor Ω2 by the deviatoric tensor 휔 and the fourth- (1983), a symmetric second-order, unbounded damage order tensor Ω4 by the harmonic (i.e totally symmetric and traceless) tensor h * h. More precisely, we get variable Φ is introduced in the Gibbs free enthalpy (with initial value Φ = 1 for a virgin material, and with dam- ′ ′ d D = 푝0휔푚 1 ⊗ 1 + 푝1휔푚 J + 푝2 (1 ⊗ 휔 + 휔 ⊗ 1) age growth d푡Φ positive definite). The usual second-order ′ ′ damage tensor writes as: + 푝3 (1 ⊗ 휔 + 휔 ⊗ 1) + 푝4 h * h. (6.2) −2 Using (3.1), the term h * h expands as d = 1 − Φ (with initial value d = 00). 1 2 A general but phenomenological coupling of elastic- h * h = h ⊗ h + h ⊗ h 3 3 ity with second-order anisotropic damage is described 2 in (Desmorat, 2006). It reads − (︀1 ⊗ h2 + h2 ⊗ 1 + 2(1 ⊗ h2 + h2 ⊗ 1))︀ 21 2 ⋆ 푔(ΦΦΦ) 2 1 ′ ′ + (tr h2)(1 ⊗ 1 + 2 1 ⊗ 1), 휌휓 = (tr휎) + tr(ΦΦ 휎 Φ 휎 ), (7.1) 105 18퐾 4퐺 ′ 1 so that the enthalpic contribution, due to the microcracks, where 휎 = 휎− 3 (tr휎)1 is the stress deviator. The function writes 푔 was chosen as 2 ′2 휎 : D : 휎 = 푝0휔푚 (tr휎) + 푝1휔푚 tr(휎 ) 1 1 ′ ′ 2 푔(ΦΦΦ) := = −2 (7.2) + 2푝2 tr(휔 휎) tr휎 + 푝3 tr(휔 휎 ) 1 − tr d trΦΦ − 2

[︁1 ′ 2 2 ′ ′ + 푝4 (tr(h휎 )) + tr(휎 h휎 h) for metals in (Lemaitre et al., 2000), and as 3 3 8 4 ]︁ 1 − tr(h2휎′2) + tr h2 tr휎′2 . 푔(ΦΦΦ) := trΦΦ2 21 105 3 Again, as in Remark 4.2,(6.2) is nothing else but the for concrete in (Desmorat, 2016). In both models, the con- harmonic decomposition of the fourth-order damage tensor vexity with respect to 휎 and the positivity of the intrinsic D, but with the following particularities. Let dissipation are satisfied (see also Chambart et al.(2014)). The elasticity law writes di(D) := (tr12 D) 휕휓⋆ 푔(ΦΦΦ) 1 be the dilatation tensor of D and 휖푒 = 휌 = (tr휎)1 + (ΦΦ 휎′ ΦΦ)′ (7.3) 휕휎휎 9퐾 2퐺 vo(D) := (tr D) 13 with effective compliance tensor be the Voigt tensor of D. Then: 1 1 1 S˜ = 1 ⊗ 1 + J + D, (7.4) – the harmonic part H = h * h of D is factorized as an 9퐾 2퐺 퐸 harmonic square; 1 – the following proportionality relations hold between where 퐸 D has for harmonic decomposition (proof given the deviatoric parts of di(D) and vo(D): in Appendix B) di′(D) ∝ vo′(D) ∝ 휔′, (6.3) 1 푔(ΦΦΦ) − 1 2훽 − 1 D = 1 ⊗ 1 + J 퐸 9퐾 2퐺 which is equivalent for the effective compliance tensor 2 S˜ to satisfy the same conditions: − (1 ⊗ b′ + b′ ⊗ 1) 3퐺 ′ ˜ ′ ˜ ′ 1 1 di (S) ∝ vo (S) ∝ 휔 ; (6.4) + (1 ⊗ b′ + b′ ⊗ 1) + Φ * Φ (7.5) 퐺 2퐺 – the following proportionality relations hold between the traces of di(D) and vo(D): with

1 [︀ 2 2]︀ ′ 1 [︀ ′ 2 ′]︀ tr di(D) ∝ tr vo(D) ∝ 휔푚. (6.5) 훽 = tr(ΦΦ ) + 3(trΦΦΦ) , b = 3(trΦΦΦ)ΦΦ − (ΦΦ ) . 60 14 Following Cormery and Welemane(2010), who consider the scalar constants 푝푖 as material parameters, conditions The harmonic part of the fourth order damage tensor is 1 to 3 above, are the conditions for a damage model— the harmonic square, for instance built in a phenomenological manner—which should be considered as micro-mechanics based. (D)0 = (1 + 휈)ΦΦ * Φ, (7.6) 8 where 휈 is Poisson ratio of undamaged material. It satisfies second order damage variable, which vanishes for virgin the first condition on the effective compliance S˜ to be of material, is d = 1 − Φ−2. micro-mechanics based form (6.2), with Even when the damage state is anisotropic, the consti-

′ tutive equations considered in Section7 is such that the 푝4 = 1 + 휈, h = Φ . dilatation tensor of the effective compliance tensor is null. This allows for the definition of an effective bulk modulus ˜ Moreover, we have 퐾 for damaged materials (i.e. a modulus function of damage tensor and of material parameters of the undamaged ′ ′ 7(1 + 휈) ′ material) such as (tr12 D) = 0, (tr13 D) = b . 3 ˜ 푒 ′ ′ tr휎 = 3퐾 tr휖 Both deviatoric parts (tr12 D) and (tr13 D) are obviously proportional, so that the phenomenological anisotropic Calculating the trace of elasticity law (7.3), we get damage model satisfies the second condition on the effective compliance S˜ to be of micro-mechanics based form 퐾 퐾 퐾˜ = = , (8.1) (6.2), with (︀ 1 2 1 2)︀ 푔(ΦΦΦ) (1 − 휂) + 휂 10 (trΦΦΦ) + 30 trΦΦ

′ 2(1 + 휈) (︀ 2 ′ ′)︀ 퐸 푝2휔 = (ΦΦ ) − 3(trΦΦΦ) Φ , where 퐾 = is the bulk modulus for the virgin 21 3(1−2휈) (undamaged) material. Recall that 푔(1) = 1. ′ 3 ′ 푝3휔 = − 푝2휔 . 2 The measurement and therefore the identification of the material parameter 휂 are not easy tasks for quasi-brittle By identification of the isotropic part of the harmonic de- materials. This is why Discrete Elements (Cundall and composition of D, we get Strack, 1979; Herrmann and Roux, 1990; Schlangen and 1 − 2휈 Garboczi, 1997; Van Mier et al, 2002; Olivier-Leblond 푝0휔 = (푔(ΦΦΦ) − 1), 푚 3 et al, 2015) are often used as numerical experimentation (︂ )︂ for the tensile states of stresses of quasi-brittle materi- 1 2 1 2 푝1휔푚 =(1 + 휈) (trΦΦΦ) + trΦΦ − 1 . als. In such numerical tests the material is described as 10 30 a particles assembly representative of the material hetero- Recall that the material constants 푝0, 푝1 are independent geneity, the particles being here linked by elastic-brittle of ΦΦ, they are considered as material parameters, so that beams. The size 16 × 16 × 16 mm3 of Representative Vol- the proportionality requirement 푝0휔푚 ∝ 푝1휔푚 can be sat- ume Element of a micro-concrete is considered (Fig.1); isfied if—following Burr et al.(1995) and Lemaitre et al. it is representative of a micro-concrete of Young’s modu- (2000)—we define the hydrostatic sensitivity parameter 휂 lus 퐸 = 30000 MPa and Poisson’s ration 휈 = 0.2. The as the material constant number of particles is 3,072 and the number of degrees of freedom 24,576. The crack pattern obtained at the end of 3푝0(1 + 휈) 휂 = , the equi-triaxial loading (quasi-rupture) is the one given 푝 (1 − 2휈) 1 in Fig.1, with a number of beams to break before failure and set of 8,000. (︂ 1 1 )︂ It has been shown from such computations (Fig.2) that 푔(ΦΦΦ) = (1 − 휂) + 휂 (trΦΦΦ)2 + trΦΦ2 . (7.7) at low damage the dependence 퐾˜ (damage) is close to be 10 30 the same linear function Condition 3 for the model (7.1) to be considered as micro- 1 mechanics based is then fulfilled. For 휂 ≥ 0, the Gibbs free 퐾˜ ≈ 퐾(1 − 1.2 푑퐻 ), 푑퐻 = tr d, (8.2) 3 enthalpy density (7.1) is furthermore convex with respect to both the stress tensor 휎 and the damage tensor ΦΦ. of the hydrostatic damage 푑퐻 whatever the stress triaxi- Note that a full anisotropic damage model—including ality. For each mark of the Figures the components of the damage evolution—for quasi brittle materials can be natu- damage tensor d = 1−Φ−2 have been measured by means rally derived following (Desmorat, 2016). The hydrostatic of repeated numerical elastic loading-unloading sequences sensitivity just obtained in Eq. (7.7) is quantified in next performed in uniaxial tension on the 16 × 16 × 16 mm3 Section. cube (even for the triaxial loading), using then the coupling of elasticity with anisotropic damage given by elas- 8. Hydrostatic sensitivity ticity law (7.3) with one non zero principal stress 휎푖 = 휎, the two others 휎푗̸=푖 = 0. Let us consider quasi-brittle materials such as concrete Note that considering Eq. (8.2) at high damage level and the micro-mechanically based damage model just de- means that in uniaxial tension the bulk modulus 퐾˜ fully rived (Section7, Gibbs free enthalpy density (7.1), elastic- vanishes at tr d = 3/1.2 = 2.5, i.e. at maximum princi- ity law (7.3) and function 푔(ΦΦΦ), given by Eq. (7.7)). The pal damage max 푑푖 larger than 1 (퐾˜ cannot vanish then 9 Discrete 3D model as complimentary numerical testing for anisotropic damage

1 1

Triaxial loading Triaxial loading Uniaxial loading Uniaxial loading 0.8 0.8 /K /K ~ ~ K 0.6 K 0.6 s ratio s ratio u u l l u 0.4 u 0.4 Mod Mod

0.2 0.2

0 0 0 0.5 1 1.5 0 0.5 1 1.5 ¬ ||D|| √3 DH

Fig. 11 Evolution of K/K 1/h(D) versus hydrostatic damage (left, in fact versus √3D for comparison) and versus the norm D ˜ H (right) for the 16-cube sample= ∥ ∥

as principal damages 푑푖 —therefore here 푑1— are always 1 bounded by 1, see also Fig. (5) for 휂 = 1.2). This corresponds to a quite high (spurious) elastic stiﬀness 퐾˜ which is kept at rupture (see Fig.3). Enforcing then gradually 0.8 퐾˜ → 0 but allowing for damage tensor d to evolve up to second order unit tensor 1 in an adequate procedure for the numerical control of rupture is a solution which leads /K

~ 0.6 K to numerical diﬃculties in Finite Element computations (Badel et al., 2007; Ragueneau et al, 2008; Leroux, 2012). s ratio The relation (8.1) relates the eﬀective bulk modulus u l 0.4 u to damage tensor Φ and hydrostatic sensitivity parme-

Mod Fig. 13 Crack patterns for the two samples (left 8 8 8, right ter 휂 (obtained from proposed micro-mechanically based 16 16 16)Figure 1: Discrete Element sample considered× × as Representative Vol- second order damage framework by function 푔 deﬁned in 0.2 × × ume Element (16 × 16 × 16 mm3 cube, from Delaplace and Desmorat Eq. (7.7)). This relation implies that the eﬀective (dam- (2008)) and crack pattern at the end of equi-triaxial loading (high ˜ value η damage0correspondstounphysicalresponsewith level). aged) bulk modulus 퐾 vanishes exactly when the maxi- = mum eigenvalue of damage tensor d = 1 −Φ−2 is equal to 0 no damage developed in tritension. 0 0.2 0.4 0.6 0.8 Discrete 3D model as complimentary numerical testing for anisotropic damage 1, whatever the stress multiaxiality and without the need of a procedure bounding the damage eigenvalues to 1. DH 1.0 1 To illustrate this property, we describe below the three Fig. 12 Evolution of K/K 1/h(D) versus hydrostatic dam- 6Conclusion Triaxial loading particularTriaxial cases loadin ofg uniaxial, equi-biaxial and equi-triaxial ˜ = Uniaxial loading Uniaxial loading age DH for the 16-cube sample. Straight line corresponds to 0.8 0.8 tension loadings. It is shown that, at low damage in K/K 1 ηDH ˜ = − The popularity and the use of a constitutive model those three loading cases, one recovers the expression

/K ˜

~ 퐾 = 퐾(1 − 휂푑 ) due to Lemaitre et al.(2000). depend on its robustness,0.6 its simplicity, and its easiness K 0.6 퐻 to implement in aK numerical˜ code. Concerning simplic- s ratio – In uniaxial tension the damage tensor of quasi-brittle u K l η cannot be noticed (it does not affect compression). ity, the number, the physical meaning and the identifica- u 0.4 0.4 materials is classically d = diag[푑1, 0, 0] (Lubarda η Mod The sensitivity to is shown in Fig. 15, with as dif- tion easiness of material parameters is an important fea- and Krajcinovic, 1993; Krajcinovic, 1996) and Φ1 = ferent values considered η 0, η 1.25, η 3. As ture to be considered. Both the discrete and anisotropic −1/2 = = = 0.2 0.2 (1 − 푑1) ≥ 1, Φ2 = Φ3 = 1 so that the effective expected, the response in uniaxial tension is not much damage models have been developed with respect to bulk modulus (8.1) has for expression influenced by this parameter. On the other hand, tri- these considerations with quite a reduced number of 0 0.0 0.2 0.4 0.6 0.8 1.0 0 0.5 1 1.5 5퐾(1 − 3푑 ) tension response strongly depends of η.Notethatthe parameters introduced. 1 ||D||˜ 퐻 √ d = tr d 퐾 = (︀ )︀. (8.3) H 3 5 − (15 − 8휂)푑퐻 − 2휂 1 − 1 − 3푑퐻 Fig. 11 Evolution of K/K 1/h(D) versus hydrostatic damage (left, in fact versus √3D for comparison) and versus the norm D ˜ H (right) for the 16-cube sample= ∥ ∥ Figure 2: Effective bulk modulus 퐾˜ from123 Discrete Element compu- In this uniaxial loading, the maximum principal dam- tations as a1 function of hydrostatic damage 푑퐻 (from Delaplace and age is 푑1 = 3푑퐻 . Desmorat(2008)). – The equi-triaxial tension case corresponds to spherical 0.8 −1/2 Discrete 3D model as complimentary numerical testing for anisotropic damage damage tensors d = 푑퐻 1,ΦΦ = Φ퐻 1 = (1−푑퐻 ) 1, 2 1 2 with thus trΦΦ = (trΦΦΦ) = 3/(1 − 푑퐻 ), and so Eq.

/K 3

~ 0.6 K 1.0 1 (8.1) rewrites as Triaxial loading Triaxial loading

s ratio Uniaxial

u Uniaxial loading Uniaxial loading l 0.4 u 퐾(1 − 푑 ) 0.8 0.8 퐾˜ = 퐻 . (8.4) Mod Fig. 13 Crack patterns for the two samples (left 8 8 8, right − − End of uniaxial loading × × 1 (1 휂)푑퐻 at d1=1, d2=d3=0, dH=1/3 16 16 16)

0.2 /K × × ~ 0.6 K 0.6 ˜ value η 0correspondstounphysicalresponsewithIn this equi-triaxial loading, the maximum principal K = nos ratio damage developed in tritension.damages are 푑 = 푑 . Note that, then, the value 0 u 푖 퐻 K l 0 0.2 0.4 0.6 0.8 u 0.4 0.4 ˜ End of equi-triaxial loading 휂 = 1 leads to the linear law 퐾 = 퐾(1 − 푑퐻 ) over DH at d1=d2=d3=1, dH=1/1.2Mod the whole range of damage. Fig. 12 Evolution of K/K 1/h(D) versus hydrostatic dam- 6Conclusion ˜ = age D0.2H for the 16-cube sample. Straight line corresponds to 0.2 K/K 1 ηD – We can also consider equi-biaxial tension for which ˜ Equi-triaxialH The popularity and the use of a constitutive model = − −1/2 depend on its robustness, its푑 simplicity,1 = 푑2 and≥ 0, its easiness푑3 = 0, Φ1 = Φ2 = (1 − 푑1) ≥ 0, to implement0 in a numerical code. Concerning simplic- 0.0 0.2 1 0.4 0.6 0.8 1.0 0 0.5 Φ3 = 1.1 The effective1.5 bulk modulus (8.1) becomes η cannot be noticed (it does not affect compression). ity, the number, the physical meaning and the identifica- 3 1 ||D|| The sensitivity to η is shownd in Fig.= 15tr,d with as dif- tion easiness of material parametersthen is an important fea- H 3 Fig.ferent 11 valuesEvolution considered of K/K η1/h(D0,) ηversus1 hydrostatic.25, η 3. damage As (left,ture in fact to versus be considered.√3D for Both comparison) the discrete and versus and the anisotropic norm D ˜ H (right) for the 16-cube sample= = = = ∥ ∥ 5퐾(2 − 3푑 ) expected, the response in uniaxial tension is not much damage models have been developed˜ with respect to 퐻 √ influenced by this parameter. On the other˜ hand, tri- these considerations with quite퐾 a reduced= number of (︀ )︀. Figure 3: Effective bulk modulus 퐾 from initial modeling 10 − (15 − 13휂)푑퐻 − 2휂 2 − 4 − 6푑퐻 ˜ tension response strongly depends of η.Notethatthe parameters introduced. 퐾 = 퐾(1 −11.2 푑퐻 ) (dashed lines) as a function of hydrostatic dam- (8.5) age. In this equi-biaxial123 loading, the maximum principal 0.8 3 damages are 푑1 = 푑2 = 2 푑퐻 .

/K 10

~ 0.6 K s ratio u l 0.4 u

Mod Fig. 13 Crack patterns for the two samples (left 8 8 8, right 16 16 16) × × 0.2 × × value η 0correspondstounphysicalresponsewith = 0 no damage developed in tritension. 0 0.2 0.4 0.6 0.8

Fig. 12 Evolution of K/K 1/h(D) versus hydrostatic dam- 6Conclusion ˜ = age DH for the 16-cube sample. Straight line corresponds to K/K 1 ηDH ˜ = − The popularity and the use of a constitutive model depend on its robustness, its simplicity, and its easiness to implement in a numerical code. Concerning simplic- η cannot be noticed (it does not affect compression). ity, the number, the physical meaning and the identiﬁca- The sensitivity to η is shown in Fig. 15, with as dif- tion easiness of material parameters is an important fea- ferent values considered η 0, η 1.25, η 3. As ture to be considered. Both the discrete and anisotropic = = = expected, the response in uniaxial tension is not much damage models have been developed with respect to inﬂuenced by this parameter. On the other hand, tri- these considerations with quite a reduced number of tension response strongly depends of η.Notethatthe parameters introduced.

123 Figure4 shows that, for the whole range of hydrostatic damage, the loss of bulk modulus (8.1) with 휂 = 1.2 behave in a similar manner as the one of Fig.2 obtained from Discrete 3D model as complimentary numerical testing for anisotropic damage Discrete Element computations. As expected, the eﬀective (damaged) bulk modulus 퐾˜ 1.0 1

Triaxial loading vanishes exactlyTriaxial loadin wheng maximum principal damage(s) are Uniaxial loading equal to 1Uniaxial (solid loadin linesg in Fig.5) in all these loading cases. 0.8 0.8 The ﬁrst order expansions in 푑퐻 (at small damage) of the three Equations (8.3), (8.4) and (8.5) gives for all three— /K ~ 0.6 K 0.6 uniaxial, equi-biaxial and equi-triaxial—loading cases K˜ s ratio u K l ˜ 0.4 u 0.4 퐾 ≈ 퐾(1 − 휂푑퐻 ) (8.6) Mod Equi-triaxial (dashed lines in Fig.5) i.e. the exact expression intro- 0.2 0.2 Uniaxial duced by Lemaitre et al.(2000) in a fully phenomenological manner (recovering Eq. (8.2) when 휂 = 1.2 is set). 0 0.0 0.2 1 0.4 0.6 0.8 1.0 0 0.5 1 1.5 3 1 ||D|| dH = tr d 9. Conclusion 3 Fig. 11 Evolution of K/K 1/h(D) versus hydrostatic damage (left, in fact versus √3DH for comparison) and versus the norm D ˜ = ∥ ∥ (right) for the 16-cube sample Some mathematical tools such as the harmonic prod- Figure 4: Effective bulk modulus 퐾˜ from (8.1) as a function of hydro- −2 uct and the harmonic factorization into lower order ten- static damage1 푑퐻 with d = 1−Φ (for value 휂 = 1.2 representative of a micro-concrete). sors have been presented. Together with the notion of mechanically accessible directions for measurements, this 0.8 has allowed us to derive, at harmonic order 4, both a crack density expansion Ω(푛) and a micro-mechanics based dam- /K

~ 0.6 K age framework that makes use of second-order tensorial

s ratio variables only, instead of fourth-order in standard micro- u l 0.4 u mechanics based approaches. The hydrostatic sensitivity Mod Fig. 13 Crack patternsobtained for the two samples from (left 8such8 8, a right second-order damage framework is 16 16 16) × × 0.2 × × shown to have the sought property of leading to the vanish- value η 0correspondstounphysicalresponsewith = ing of the eﬀective (damaged) bulk modulus at maximum 0 no damage developed in tritension. 0 0.2 0.4 0.6 0.8 principal damage max 푑푖 exactly equal to 1.

DH 1.0 Fig. 12 Evolution of K/K 1/h(D) versus hydrostatic dam- 6Conclusion ˜ = Appendix A. Spherical/Deviatoric harmonic de- age DH for the 16-cube sample. Straight line corresponds to K/K 1 ηDH Uniaxial ˜ = − The popularity and the use of a constitutivecomposition model 0.8 depend on its robustness, its simplicity, and its easiness to implement in a numericalDifferent code. Concerning practical simplic- expressions of the harmonic decom- η cannot be noticed (it does not affect compression). ity, the number, the physical meaning and the identifica- The sensitivity0.6 to η is shown in Fig. 15, with as dif- tion easiness of materialposition parameters of is aan important fourth fea- order tensor of elasticity type ex- ferent˜ values considered η 0, η 1.25, η 3. As ture to be considered.ist Both (Backus the discrete, 1970 and anisotropic; Onat, 1984; Auffray, 2017). We use K = = = Equi-biaxial expected, the response in uniaxial tension is not much damage models havehere been the developed spherical/deviatoric with respect to harmonic decomposition in- influenced by this parameter. On the other hand, tri- these considerations with quite a reduced number of K 0.4 tension response strongly depends of η.Notethatthe parameters introduced.troduced by Auffray(2017) of a fourth order tensor T of the elasticity type

0.2 Equi-triaxial 123 T = 훼 1 ⊗ 1 + 2훽 J + 1 ⊗ c′ + c′ ⊗ 1 (︁ 2 )︁ + 2 (1 ⊗ b′ + b′ ⊗ 1) − (1 ⊗ b′ + b′ ⊗ 1) + H, 0.0 0.2 0.4 0.6 0.8 1.0 3 max di (A.1)

˜ where, di and vo being the dilatation and voigt tensors of Figure 5: Effective bulk modulus 퐾 from (8.1) (solid lines) and from ′ ′ initial modeling 퐾˜ = 퐾(1 − 휂 푑퐻 ) (dashed lines) as a function of T and di and vo their deviatoric parts, −2 hydrostatic damage 푑퐻 with d = 1 − Φ (for value 휂 = 1.2 representative of a micro-concrete). di = tr12 T, vo = tr13 T, 1 1 훼 = tr di, 훽 = (− tr di + 3 tr vo), 9 30 1 1 c′ = di′, b′ = (−2di′ + 3vo′). 3 7

11 Introducing two second symmetric second order tensors with y and x such that y = T : x, the following equalities are 1 푔(ΦΦΦ) − 1 2훽 − 1 obtained : D = 1 ⊗ 1 + J 퐸 9퐾 2퐺 2 ′ ′ ′ ′ 1 (︂ 2 )︂ x : T : x = 훼(tr x) + 2훽x : x + 2(tr x)(c : x ) + 1 ⊗ b′ + b′ ⊗ 1 − (1 ⊗ b′ + b′ ⊗ 1) 2 퐺 3 + 4b′ : x′ + x′ : H : x′, 1 + Φ * Φ. and 2퐺

′ ′ ′ ′ A straightforward calculation leads to the property y = 훼(tr x)1 + 2훽x + (c : x )1 + (tr x)c ′ ′ ′ ′ ′ ′ ′ + 2 [(b .x ) + (x .b ) ] + H : x , ′ ′ 7(1 + 휈) ′ (tr12 D) = 0, (tr13 D) = b . tr y = 3(훼 tr x + c′ : x′), 3 y′ = (tr x)c′ + 2훽x′ + 2 [(b′.x′)′ + (x′.b′)′] + H : x′.

References

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