Optimal lower bounds on the stress inside random two-phase composites Bacim Alali and Robert Lipton Department of Mathematics, Louisiana State University, Baton Rouge {alali,lipton}@math.lsu.edu Abstract We consider a composite material made from two perfectly bonded isotropic linear elastic components. A prescribed uniform stress is applied to the composite. We develop lower bounds on the max- imum stress generated inside the composite for all mixtures made with two elastic materials in fixed volume fractions. We outline various cases in which these lower bounds are optimized by simple configurations of the two materials. 1. Introduction Many composite structures are hierarchical in nature and are made up of substructures distributed across several length scales. Exam- ples include fiber reinforced laminates as well as naturally occurring structures like bone. From the perspective of failure initiation it is crucial to quantify the load transfer between length scales. It is common knowledge that the load transfer can result in local stresses that are significantly greater than the applied macroscopic stress. Figure 1: Trabecular Bone Sand Grains Quantities useful for the study of load transfer include the higher order moments of the local stress. The higher moments are sen- sitive to local stress concentrations generated by the interaction between the microstructure and the macroscopic load. In this work we find optimal lower bounds on all higher moments of the local stress inside random composites made from two isotropic elastic materials in prescribed proportions. These bounds provide the min- imum amount of stress amplification that can be expected from this class of composites. In this work we consider • the moments of the local stress σ (x) inside each phase <χ 1 |σ (x)| r > 1/r and <χ 2 |σ (x)| r > 1/r for 2 ≤ r ≤∞ • and the maximum stress inside the composite ‖σ ‖ L ∞ (Q) = lim r →∞ < |σ (x)| r > 1/r 2. Elastic boundary value problem for composite materials - The elastic stress and strain fields σ (x) and ǫ(x) satisfy ǫ ij (x)= ∂ j u i (x)+ ∂ i u j (x) 2 and σ (x)= C (x)ǫ(x). - Here u is the displacement field and C is the local elasticity tensor C (x)= χ 1 (x)C 1 + χ 2 (x)C 2 Here χ i is the indicator function of the i-th material. - The two materials are isotropic C i =2µ i Π D + dκ i Π H , for i =1, 2. - Shear modulus µ i and bulk modulus κ i in the i-th material. - Here d = 2 (3) for planar (three dimensional) elastic problems. - The projection onto the hydrostatic part of the stress Π H is given by Π H ijkl = 1 d δ ij δ kl hence, Π H σ (x)= tr σ (x) d I. - The projection onto the deviatoric part of the stress is given by Π D = I − Π H , where I is the fourth order identity. - The equation of elastic equilibrium inside each phase is given by div σ =0 - Continuity of the displacement u and the traction σ n , n being the unit normal, across the interface u | 1 = u | 2 and σ | 1 n = σ | 2 n. - It is assumed that the composite is periodic with period cell Q (unit square or cube). - The composite is subjected to an imposed average macroscopic stress σ = 〈σ 〉. - The effective elastic tensor C e relates the applied stress to the average strain 〈ǫ〉 and is given by 〈σ 〉 = C e 〈ǫ〉 . 3. Optimal lower bounds on the local stress 3.1 Deviatoric applied stress For a deviatoric macroscopic stress Π D σ = σ , optimal lower bounds on the local stress inside the composite are given by the following results. Optimal lower bounds on the moments of the local stress In what follows, θ i = 〈χ i 〉 denotes the volume fraction of the i-th material. For fixed values of θ 1 and θ 2 the stress field inside the i-th material for which µ i = max{µ 1 ,µ 2 } satisfies 〈χ i |σ (x)| r 〉 1/r ≥ θ 1/r i | σ | , (1) for 2 ≤ r ≤∞. Optimal lower bound on the maximum stress For fixed values of θ 1 and θ 2 the stress field inside the composite satisfies ‖|σ (x)|‖ L ∞ (Q) ≥| σ | . (2) Optimality: For d = 2, the lower bounds (1) and (2) are attained by the stress fields inside a rank-one laminate with layering direction n = 1 √ 2 (e 1 + e 2 ), where e 1 and e 2 are the eigenvectors of the macroscopic stress σ . e 1 e 2 Figure 2: Rank one layered material 3.2 Same bulk modulus in the two phases For a composite in which κ 1 = κ 2 = κ, optimal lower bounds on the Von Mises equivalent stress and the local stress are given by the following results. Optimal lower bounds on the moments of the Von Mises equivalent stress For fixed values of θ 1 and θ 2 the stress field inside the i-th material for which µ i = max{µ 1 ,µ 2 } satisfies  χ i |Π D σ (x)| r  1/r ≥ θ 1/r i    Π D σ    , (3) for 2 ≤ r ≤∞. Optimal lower bound on the maximum stress For fixed values of θ 1 and θ 2 the stress field inside the composite ‖|σ (x)|‖ L ∞ (Q) ≥| σ | . (4) Optimality: For d =2, the lower bounds (3) and (4) are attained by the stress fields inside a rank-one laminate with layering direc- tion n = 1 √ 2 (e 1 + e 2 ), where e 1 and e 2 are the eigenvectors of the macroscopic stress σ . 3.3 Same shear modulus in the two phases For a composite in which µ 1 = µ 2 = µ, optimal lower bounds on the local hydrostatic stress are given by the following results. Optimal lower bounds on the moments of the local hy- drostatic stress For fixed values of θ 1 and θ 2 the stress field inside the i-th material satisfies  χ i |Π H σ (x)| r  1/r ≥ θ 1/r i κ 1 κ 2 +2 d−1 d µκ i κ 1 κ 2 +2 d−1 d µ(θ 1 κ 1 + θ 2 κ 2 ) |Π H σ |, (5) for 2 ≤ r ≤∞. Optimal lower bound on the maximum hydrostatic stress For fixed values of θ 1 and θ 2 the stress field inside the composite satisfies ‖|Π H σ (x)|‖ L ∞ (Q) ≥ κ 1 κ 2 +2 d−1 d µ(κ + ) κ 1 κ 2 +2 d−1 d µ(θ 1 κ 1 + θ 2 κ 2 ) |Π H σ |, (6) where κ + = max{κ 1 ,κ 2 }. Optimality: For d = 2 (3), the stress field inside any rank one laminate attains the bounds (5) and (6). 3.4 Hydrostatic applied stress For a hydrostatic macroscopic stress Π H σ = σ and assuming that the materials properties are well-ordered (i.e. µ 1 >µ 2 ,κ 1 >κ 2 or µ 2 >µ 1 ,κ 2 >κ 1 ), optimal lower bounds on the local stress inside the composite are given by the following results. Optimal lower bounds on the moments of the local stress inside material one For fixed values of θ 1 and θ 2 the stress field inside material one satisfies 〈χ 1 |σ (x)| r 〉 1/r ≥ θ 1/r 1 κ 1 κ 2 +2 d−1 d (µ − )(κ + ) κ 1 κ 2 +2 d−1 d (µ − )(θ 1 κ 1 + θ 2 κ 2 ) | σ | , (7) for 2 ≤ r ≤∞. Here µ − = min{µ 1 ,µ 2 }. Optimality: For d = 2 (3), the stress field inside material one for the coated cylinder (sphere) assemblage with core of material one and coating of material two attains the lower bound (7). Optimal lower bounds on the moments of the local stress inside material two For fixed values of θ 1 and θ 2 the stress field inside material two satisfies 〈χ 2 |σ (x)| r 〉 1/r ≥ θ 1/r 2 κ 1 κ 2 +2 d−1 d (µ − )(κ − ) κ 1 κ 2 +2 d−1 d (µ − )(θ 1 κ 1 + θ 2 κ 2 ) | σ | , (8) for 2 ≤ r ≤∞. Optimality: For d = 2 (3), the stress field inside material two for the coated cylinder (sphere) assemblage with core of material two and coating of material one attains the lower bound (8). Optimal lower bound on the maximum stress For fixed values of θ 1 and θ 2 the stress field inside the composite satisfies ‖|σ (x)|‖ L ∞ (Q) ≥ κ 1 κ 2 +2 d−1 d (µ − )(κ + ) κ 1 κ 2 +2 d−1 d (µ − )(θ 1 κ 1 + θ 2 κ 2 ) | σ | . (9) Optimality: For d = 2 (3), the stress field inside the coated cylinder (sphere) assemblage with core of material one and coating of material two attains the lower bound (9). Figure 3: Hashin-Shtrikman coated cylinders assemblage Refrences 1. Alali B. and Lipton R.,“Optimal lower bounds on stress and strain amplifications inside random two-phase elastic composites.” In preperation. 2. Lipton R., 2005 “Optimal lower bounds on the hydrostatic stress amplification inside random two-phase elastic composites.” Journal of the Mechanics and Physics of Solids, 53, pp. 2471– 2481. 3. Lipton R., 2004“Optimal lower bounds on the electric-field con- centration in composite media.” Journal of Applied Physics, 96, pp. 2821–2827.