Next, we turn our attention on the sub-structural characterization of the random,two-scale,gran- ular composite of Fig. 2.1. Towards this end, we make use of the work of Smyshlyaev and Willis [136] on martensitic polycrystals along with the development of the previous subsection.
Recall that in the context of the discussion of the previous subsection for single-scale granular systems it was assumed that the grains of Fig. 2.1(a) are homogeneous. Here, we remove this restriction and each grain of Fig. 2.1(a) is taken to be a polycrystal, as shown in Fig. 2.1(b). Hence, in the present context, the grains of Fig. 2.1(a) are referred to as composite-grains and, similarly, the single-crystals of Fig. 2.1(b) are referred to as homogeneous-grains. In general, different composite-grains are assumed to be different polycrystals. The term composite-grain- family (or composite-phase) is used for the set of all composite-grains that are made out of the same polycrystal (i.e., all grains shown in Fig. 2.1(a) with a specific color). In analogy, the term
homogeneous-grain-family (orhomogeneous-phase) refers to the set of all single-crystals of a given orientation (Fig. 2.1(b)). The sub-structures of Figs. 2.1(a) and2.1(b) are respectively referred to as the meso-structure andmicro-structure of the two-scale composite. Similarly, the termsmeso- scale ℓ1 and micro-scale ℓ2 are used respectively for the characteristic length-scale of the size of a typical composite-grain (Fig. 2.1(a)) and a typical homogeneous-grain (Fig. 2.1(b)), while the termmacro-scale 𝐿1is used to characterize the size of the RVE Ω of the two-scale medium. The RVE Ω of Fig. 2.1is taken to be made out of an arbitrary number𝑁 of composite-grain-families occupying subregions Ω(𝑟), with𝑟= 1, ..., 𝑁, of Ω, and each composite-grain-family𝑟is assumed to consist of a large number 𝑁(𝑟) of single-crystal orientations occupying subregions Ω(𝑟,𝑝), with
𝑝= 1, ..., 𝑁(𝑟), of Ω.
Towards the definition of the sub-structure of the two-scale granular composite of Fig. 2.1, we consider the following characteristic functions associated with the homogeneous-phases (𝑟, 𝑝), with
𝑟= 1, ..., 𝑁 and𝑝= 1, ..., 𝑁(𝑟), ˜ 𝜒(𝑟,𝑝)(x) = ⎧ ⎨ ⎩ 1, if x∈Ω(𝑟,𝑝) 0, otherwise , (2.9)
which are required to be compatible with the corresponding composite-phase functions𝜒(𝑟), given by (2.1), i.e., 𝑁(𝑟) ∑ 𝑝=1 ˜ 𝜒(𝑟,𝑝)(x) =𝜒(𝑟)(x). (2.10) The functions ˜𝜒(𝑟,𝑝), as defined by (2.9) along with the constraints (2.10), account both for the heterogeneity at the micro-scaleℓ2, i.e., at the level of the homogeneous-grains (Fig. 2.1(b)), and the heterogeneity at the meso-scale ℓ1, i.e., at the level of the composite-grains (Fig. 2.1(a)), and, therefore, fully characterize the sub-structure underlying thespecific two-scale system of Fig.
2.1. As implicitly suggested from the discussion of the previous subsection, our intention in this work is to account for the functions 𝜒(𝑟) and ˜𝜒(𝑟,𝑝) only through the associated one- and two- point probabilities. The one-point probability function ˜𝑝(𝑟,𝑝)(x) associated with ˜𝜒(𝑟,𝑝), i.e., the
expectation that ˜𝜒(𝑟,𝑝)(x, 𝛼) = 1, is defined by ˜ 𝑝(𝑟,𝑝)(x) = ∫ 𝑆 ˜ 𝜒(𝑟,𝑝)(x, 𝛼)𝑝(𝛼)𝑑𝛼, (2.11) and the two-point probability ˜𝑝(𝑟,𝑝;𝑠,𝑞)(x,x′), i.e., the expectation that ˜𝜒(𝑟,𝑝)(x, 𝛼) ˜𝜒(𝑠,𝑞)(x′, 𝛼) = 1,
is similarly defined by ˜ 𝑝(𝑟,𝑝;𝑠,𝑞)(x,x′) = ∫ 𝑆 ˜ 𝜒(𝑟,𝑝)(x, 𝛼) ˜𝜒(𝑠,𝑞)(x′, 𝛼)𝑝(𝛼)𝑑𝛼. (2.12) The compatibility requirement (2.10) for the functions ˜𝜒(𝑟,𝑝)implies analogous restrictions on ˜𝑝(𝑟,𝑝) and ˜𝑝(𝑟,𝑝;𝑠,𝑞). For example, from (2.10) and (2.11) it follows that
𝑁(𝑟)
∑
𝑝=1 ˜
𝑝(𝑟,𝑝)(x) =𝑝(𝑟)(x), (2.13)
while from (2.10) and (2.12) it can be seen that 𝑁(𝑟) ∑ 𝑝=1 𝑁(𝑠) ∑ 𝑞=1 ˜ 𝑝(𝑟,𝑝;𝑠,𝑞)(x,x′) =𝑝(𝑟𝑠)(x,x′), (2.14)
where we recall that𝑝(𝑟)(x) and𝑝(𝑟𝑠)(x,x′) denote respectively the one- and two-point correlation
functions associated with𝜒(𝑟)(x), which have been already considered in the previous subsection. The two-point correlation functions ˜𝑝(𝑟,𝑝;𝑠,𝑞) defined in this way are very general to be of any practical use. Furthermore, given the constraints imposed to these functions because of (2.10), we do not see an obvious way in which simplifying assumptions, such as the ones used in the context of𝑝(𝑟𝑠), could help in restricting the class of possible functions ˜𝑝(𝑟,𝑝;𝑠,𝑞). It is remarked that higher point statistics could be defined similarly.
In order to make further progress, we restrict our attention to composite systems that are con- sistent with the separation of the length-scales hypothesis (2.4), as well as with the corresponding hypothesis in the composite-grains
ℓ2<< 𝐿2, (2.15)
where we recall that the length-scale𝐿2 characterizes the size of the mesoscopic RVE in a typical composite-grain (see Fig. 2.1(b)), and we further assume that
𝐿2<< ℓ1. (2.16)
In addition, we restrict our consideration to two-scale composites with micro-structures (Fig.
2.1(b)) that arestatistically independent from the associated meso-structures (Fig. 2.1(a)). More specifically, we assume that the characteristic functions ˜𝜒(𝑟,𝑝) of a two-scale composite may be expressed in the form
˜
where, for a given 𝑟= 1, ..., 𝑁, we assume that the functions 𝜒(𝑟,𝑝)(x), defined over Ω(𝑟), exhibit
statistical uniformity,ergodicity andno long-range order in the entire space, such that 𝑁(𝑟)
∑
𝑝=1
𝜒(𝑟,𝑝)(x) = 1. (2.18)
In other words, for any given composite-phase 𝑟 = 1, ..., 𝑁, the set of functions 𝜒(𝑟,𝑝), with 𝑝= 1, ..., 𝑁(𝑟), defines a single-scale composite material (i.e., the material of Fig. 2.1(b)) independently of the characteristic functions 𝜒(𝑟) (defining the “single-scale” composite of Fig. 2.1(a)). Hence, relation (2.17) interprets the sub-structure ˜𝜒(𝑟,𝑝) of the two-scale composite of Fig. 2.1 as the “intersection” of the sub-structure 𝜒(𝑟) of the “single-scale” composite of Fig. 2.1(a) with the sub-structures𝜒(𝑟,𝑝)of the𝑁 different single-scale composites of Fig. 2.1(b).
Now, recalling that the characteristic functions 𝜒(𝑟,𝑝) have been assumed to be statistically uniform and ergodic, it makes sense to consider the microscopic probability functions 𝑝(𝑟,𝑝)(x) and𝑝(𝑟,𝑝𝑞)(x,x′) defined as the expectations that𝜒(𝑟,𝑝)(x, 𝛼) = 1 and 𝜒(𝑟,𝑝)(x, 𝛼)𝜒(𝑟,𝑞)(x′, 𝛼) = 1,
respectively. Note that, due to the ergodicity of 𝜒(𝑟,𝑝), the average of a quantity at a point x over the sample may be replaced by the corresponding average in a particular specimen over any neighborhood of x that is sufficiently large compared to the size of the heterogeneity (i.e., the size of a single-crystal). For the case of 𝜒(𝑟,𝑝), it is convenient to chose this neighborhood to be the entire volume Ω(𝑟) occupied by the composite-grain-phase 𝑟 in the RVE Ω of the two-scale composite. Thus, the one-point probability functions𝑝(𝑟,𝑝)reduce to
𝑝(𝑟,𝑝)(x) = 1 ∣Ω(𝑟)∣ ∫ Ω(𝑟) 𝜒(𝑟,𝑝)(x′)𝑑x′ = ∣Ω (𝑟,𝑝) ∣ ∣Ω(𝑟)∣ ≡𝑐 (𝑟,𝑝), (2.19)
where the single-crystal concentrations 𝑐(𝑟,𝑝) are such that∑𝑁(𝑟)
𝑝=1 𝑐(𝑟,𝑝) = 1 for each𝑟= 1, ..., 𝑁. The two-point probability functions𝑝(𝑟,𝑝𝑞) are given by
𝑝(𝑟,𝑝𝑞)(x−x′) = 1
∣Ω(𝑟)∣
∫
Ω(𝑟)
𝜒(𝑟,𝑝)(x+x′′)𝜒(𝑟,𝑞)(x+x′′)𝑑x′′. (2.20) For practical purposes, it is also important to assume that the two-point correlation functions
𝑝(𝑟,𝑝𝑞) are characterized by ellipsoidal symmetry, so that𝑝(𝑟,𝑝𝑞)(x−x′) =𝑝(𝑟,𝑝𝑞)(∣Z(𝑟,𝑝𝑞)
𝑑 (x−x′)∣), where Z(𝑑𝑟,𝑝𝑞) are the associated shape tensors defining the following microscopic distributional ellipsoids (centered atx𝐶)
Ω(𝑑𝑟,𝑝𝑞)={x∣(x−x𝐶)⋅Z(𝑑𝑟,𝑝𝑞)(x−x𝐶)≤1}. (2.21) Furthermore, in the applications of this work we will only consider composites for whichZ(𝑑𝑟,𝑝𝑞)≡ Z(𝑑𝑟) for all pairs (𝑝𝑞), with 𝑝, 𝑞= 1, ..., 𝑁(𝑟), but for different values of 𝑟the tensors Z(𝑟)
𝑑 will be taken to be different, in general.
Based on the preceding considerations, we are now in a position to simplify further the probabil- ity functions ˜𝑝(𝑟,𝑝)and ˜𝑝(𝑟,𝑝;𝑠,𝑞)associated with the two-scale characteristic functions ˜𝜒(𝑟,𝑝)defined by (2.9). The micro-meso-structural statistical independence assumed for ˜𝜒(𝑟,𝑝)in the context of (2.17) consists in the following two conditions: (𝑖) for a fixed value of 𝑟, the functions 𝜒(𝑟) and
𝜒(𝑟,𝑝) are mutually independent and (𝑖𝑖) for different values of 𝑟 and 𝑠, the functions 𝜒(𝑟,𝑝) and
𝜒(𝑠,𝑞) are also mutually independent. Conditions (𝑖) and (𝑖𝑖) imply that the probability functions ˜
𝑝(𝑟,𝑝)and ˜𝑝(𝑟,𝑝;𝑠,𝑞)for these special types of sub-structures are respectively given by ˜ 𝑝(𝑟,𝑝)=𝑝(𝑟)𝑝(𝑟,𝑝)≡𝑐(𝑟)𝑐(𝑟,𝑝), (2.22) and ˜ 𝑝(𝑟,𝑝;𝑠,𝑞)(z) =𝑝(𝑟𝑠)(z)[𝛿𝑟𝑠𝑝(𝑟,𝑝𝑞)(z) + (1−𝛿𝑟𝑠)𝑐(𝑟,𝑝)𝑐(𝑠,𝑞) ] . (2.23)
The quantities 𝑝(𝑟) and 𝑝(𝑟𝑠) in the above expressions correspond respectively to the one- and two-point correlation functions associated with 𝜒(𝑟). Furthermore, the concentrations𝑝(𝑟) =𝑐(𝑟) and𝑝(𝑟,𝑝)=𝑐(𝑟,𝑝)are given by (2.5) and (2.19), respectively. In expression (2.23),𝛿
𝑟𝑠= 1 if𝑟=𝑠 and 0 otherwise, and no summation is implied for the repeated indexes𝑟and 𝑠.
In summary, the sub-structure of a random, two-scale granular composite—for the purposes of this work—is completely determined by means of the probability functions ˜𝑝(𝑟,𝑝)and ˜𝑝(𝑟,𝑝;𝑠,𝑞)given by (2.22) and (2.23), respectively. Expressions (2.22) and (2.23) relate thetwo-scale functions ˜𝑝(𝑟,𝑝)
and ˜𝑝(𝑟,𝑝;𝑠,𝑞) with the corresponding single-scale microscopic, 𝑝(𝑟,𝑝) and 𝑝(𝑟,𝑝𝑞), and mesoscopic,
𝑝(𝑟)and𝑝(𝑟𝑠), probability functions. The one-point probability functions𝑝(𝑟,𝑝), given by (2.19), are identified with the volume fractions𝑐(𝑟,𝑝) of the single-crystal orientations (𝑟, 𝑝) in the polycrystal
𝑟, while the two-point probability functions𝑝(𝑟,𝑝𝑞)are given in terms of the associated shape tensor
Z(𝑑𝑟), characterizing the distribution of the pairs of single-crystals (𝑝𝑞) in the polycrystal 𝑟. The corresponding mesoscopic probabilities𝑝(𝑟),𝑝(𝑟𝑠)are obtained from the discussion of the previous subsection. In particular, assuming that the functions 𝜒(𝑟) are statistically uniform, ergodic and with no long-range order, the one-point probabilities 𝑝(𝑟) are given by (2.5) and correspond to the volume fractions 𝑐(𝑟) of the composite-grains. Assuming, in addition, that the two-point probabilities𝑝(𝑟𝑠) are characterized by ellipsoidal symmetry with the same features for all pairs of composite-grains, i.e., choosing the shape tensorsZ(𝑑𝑟𝑠)≡Z𝑑 for all pairs (𝑟𝑠), with𝑟, 𝑠= 1, ..., 𝑁, in the definition (2.8) of the distributional ellipsoids Ω(𝑑𝑟𝑠), the two-point correlations𝑝(𝑟𝑠)reduce to a function of ∣Z𝑑(x−x′)∣ only.
1
L
Ω
1ℓ
( )rΩ
( )
a( )
b ( )r p,Ω
2L
2ℓ
Figure 2.2: A two-scale particulate composite with random distributions of ellipsoidal symmetry (dotted ellipses). (𝑎) Macroscopic RVE Ω: composite-inclusion families in a composite-matrix (distinguished by color). (𝑏) Mesoscopic RVE of the composite-inclusion family 𝑟: homogeneous-inclusion families in a homogeneous-matrix.