In this chapter, we have developed variational procedures for the determination of the instantaneous effective response oftwo-scale composites with fairly generalviscoplasticconstituents andrandom
sub-structures under arbitrary loading conditions. These procedures are based on two equivalent variational formulations of the associated homogenization problem for the two-scale composite, namely, the “direct” (2.44) and the “sequential” (2.42), together with (2.45), variational principles, and they constitute the generalized versions of the corresponding “linear comparison composite” (LCC) methods of Ponte Casta˜neda [114], [117], [118] for single-scale systems, referred to as the “secant”, “tangent second-order” (TSO) and “generalized-secant second-order” (GSO) methods, respectively. The main idea behind the LCC methods is to construct an LCC with the same sub-structure as the nonlinear two-scale medium and express the effective stress-potential ˜𝑢 of the composite material of interest in terms of the effective stress-potential ˜𝑢𝑇 of the LCC and an appropriately defined error function. In this connection, it is relevant to remark that the terms “secant”, “tangent” and “generalized-secant” are used here to indicate the type of linearization scheme utilized in each case. The secant method is capable of delivering bounds for ˜𝑢that are exact to first-order in the heterogeneity contrast for small contrast systems, while both the TSO and GSO methods deliver estimates for ˜𝑢that are exact to second-order in the heterogeneity contrast. Results based on the implementation of these schemes on the direct (2.44) and sequential (2.42), with (2.45), problems are termed “direct linearization scheme” (DLS) and “sequential linearization scheme” (SLS) estimates, respectively.
For completeness, in this chapter we have also reviewed estimates from the literature for the effective behavior of single-scale linear thermoelastic composites and we have discussed in detail the appropriate generalization of these estimates to the corresponding two-scale systems, which are being utilized in the context of the various estimates for the nonlinear composites of interest. Results were presented both for particulate and granular random systems. These estimates are fairly simple to compute and, at the same time, they incorporate fine sub-structural information (i.e., they depend both on one- and two-point statistics) which allow consideration of a wide range of composite materials of practical interest in a realistic manner. This later feature will prove particularly helpful in the applications considered in later parts of this thesis, where fairly complex sub-structures, such as semi-crystalline polymers, will be modeled by accounting for crucially important, fine morphological features of the underlying sub-structure at both the meso- and micro-scale level.
More specifically, by applying the secant method to the direct (2.44) and the sequential (2.42), with (2.45), problems for the class of two-scale composites with isotropic constituents, we developed the estimates (2.107) and (2.119), respectively, which have been shown to be identical, as they should. An important property of the secant estimates, which was used in proving the equivalence of (2.107) and (2.119), is that they are stationary with respect to all material parameters defining the local properties of the associated LCCs. In this connection, it is relevant to remark here that the corresponding TSO and GSO estimates are not fully stationary. Furthermore, it was shown that the secant estimate constitutes a rigorous lower bound for the two-scale nonlinear medium when a corresponding lower bound is used for the LCC.
By implementing the TSO method to the direct (2.44) and the sequential (2.42), with (2.45), problems, in this chapter we have generated the DLS-TSO estimate (2.128) and SLS-TSO estimate (2.67), with (2.142), respectively. These estimates are completely general in the sense that they are applicable to two-scale viscoplastic composites with anisotropic homogeneous-phases and random sub-structures of any type. They are completely determined in terms of the stress phase-averages in the associated LCCs and, therefore, they are fairly simple to compute. It should be remarked, however, that since the corresponding TSO estimate for single-scale systems, which is a special case of the more general estimates developed in this work, is known to be inconsistent with the rigorous secant bounds in certain special cases, the predictions of the DLS-TSO and SLS-TSO estimates, at least for these special cases, arenot expected to be accurate.
By applying the GSO method to the direct (2.44) and the sequential (2.42), with (2.45), prob- lems we generated the DLS-GSO estimate (2.148) and SLS-GSO estimate (2.74), with (2.163), respectively. These estimates are also applicable to random sub-structures of any type and, po- tentially, to general anisotropic homogeneous-phases. Given, however, the fact that these GSO procedures provide generalizations of the corresponding GSO procedure for single-scale systems, which thus far has been developed in full detail for composites with isotropic and crystalline phases, the DLS-GSO estimate can be practically computed for systems with isotropic and crystalline homogeneous-phases, which is a fairly general class of two-scale composites, while the SLS-GSO estimate can be calculated for systems which are further restricted to be made out of composite- phases with isotropic effective behavior. Thus, by further specializing the DLS-GSO estimate
(2.148) to composites with isotropic and crystalline homogeneous-phases we obtained the DLS-
GSO estimates (2.154) and (2.160), respectively. The SLS-GSO estimate (2.74), with (2.163), was computed explicitly for the case of combined isotropic homogeneous-phases and composite-phases with isotropic effective behavior, and it is given by (2.81), with (2.167). In any case, the GSO estimates depend explicitly both on the stress phase-averages and the stress phase-field-fluctuation in the associated LCCs, and they are superior to the corresponding TSO estimates in that they are always consistent with the bounds.
In conclusion, the most important results of this chapter are the DLS-GSO estimate (2.154) for two-scale composites with isotropic homogeneous-phases and the DLS-GSO estimate (2.160) for two-scale composites with crystalline phases, to which we will mainly focus our attention in subsequent chapters. In chapter 3, however, we provide a thorough investigation of the predictions of all estimates derived here for special types of 2-dimensional model problems.
Chapter 3
Applications to 2-D model problems
In this chapter, we consider specific applications of the “secant”, “tangent second-order” (TSO) and “generalized-secant second-order” (GSO) methods for two-scale viscoplastic composites de- veloped in chapter 2. The main objective of these applications is to compare the predictions of the corresponding “direct linearization scheme” (DLS) and “sequential linearization scheme” (SLS) second-order estimates for the instantaneous effective response of particulate and granular systems, as well as to highlight the differences in the effective behavior of a two-scale composite and a corresponding single-scale composite. To this end, we focus our attention on the effective in-plane response of severalrigidly-reinforced systems with transversely isotropic sub-structures. Specifically, we consider the two-scale particulate composites of Figs. 3.1(b) and 3.4(b), which for convenience are labeled by P1 and P2, respectively, the single-scale particulate composite of Figs. 3.1(a), labeled by P0, thetwo-scale granular system of Fig. 3.8(b), labeled by G2, and the
single-scale granular system of Fig. 3.8(a), labeled by G0.
The structure of this chapter is as follows. In section3.1, we recall the basic features of the estimates developed in the previous chapter and introduce some definitions that facilitate the discussion in the subsequent sections. Sections 3.2, 3.3 and 3.4 deal with the effective in-plane response of the two-scale composites P1, P2 and G2, while the single-scale composites P0 and G0 are treated as special cases of the corresponding two-scale composites. In each of the later sections, we first specialize the relevant secant, TSO and GSO estimates to the corresponding two-scale composite and then, in the context of specific results, we investigate their predictions and highlight some of their important features. The effective properties of the “linear comparison composites” (LCCs) involved in the calculation of these estimates are computed by means of the results provided in appendix I for general two-phase single-scale linear thermoelastic composites. Finally, in section 3.5we summarize the main findings and conclusions of this chapter.
3.1
Preliminaries
With a slight abuse in notation, in the applications of this chapter we make use of the notation
𝝈 to denote both the macroscopic and the mesoscopic applied stress1 and 𝝈(𝑟,𝑝) to denote the LCC homogeneous-phase-average stress tensors for both the DLS- and SLS-based2 second-order
estimates of the previous chapter, and let their interpretation be inferred by the context. Further- more, we make use of the notation𝐴(⋅), where the superscript (⋅) indicates the phase of the LCC to which the quantity𝐴is referred to.
In this chapter, we focus our attention on the two-scale composite systems P1, P2 and G2 shown schematically in Figs. 3.1(b), 3.4(b) and3.8(b), respectively. A common feature of these systems is that each of them is characterized by transversely isotropic micro- and meso-structure. In other words, making contact with the discussions of subsections 2.1.2and 2.2.2, the relevant meso-structural shape tensor Z and micro-structural shape tensors Z(𝑟) for the composites of interest will be taken to be such thatZ(𝑟)=Z, where
Z=I−(1−𝜖)e3⊗e3, as 𝜖→0, (3.1) withe3denoting the preferred direction of transverse isotropy andIbeing the second-order identity tensor. In addition, these composites are constituted by a rigid-material, with stress-potential equal to infinity, reinforcing a deformable isotropic homogeneous-material, which is characterized by the following viscoplastic power-law stress-potential
𝑢(𝝈) =𝜓(𝜎𝑒) = 𝜖0𝜎0 𝑛+ 1 (𝜎 𝑒 𝜎0 )𝑛+1 , (3.2) where𝜎𝑒= √
3𝝈𝑑⋅𝝈𝑑/2, the subscript𝑑is used to indicate the deviatoric part of a tensor,𝑛is the nonlinearity exponent (the inverse of the rate-sensitivity𝑚= 1/𝑛) and𝜖0,𝜎0are reference strain- rate and stress measures, respectively. Furthermore, we will restrict our attention to plane-stress loading conditions, i.e., loadings normal to the direction of transverse isotropye3. For these loading conditions, it turns out that the corresponding effective stress-potential ˜𝑢of these composites is also power-law with the same exponent𝑛and reference stain𝜖0as those of the deformable material in (3.2), i.e., ˜ 𝑢(𝝈) = ˜𝜓(¯𝜎𝑒) = 𝜖0𝜎˜0 𝑛+ 1 ( 𝜎𝑒 ˜ 𝜎0 )𝑛+1 , (3.3)
where 𝝈denotes the applied stress, 𝜎𝑒=
√
3𝝈𝑑⋅𝝈𝑑/2 and ˜𝜎0 is the in-plane effective yield-stress of the composite. The effective quantity ˜𝜎0 for each of the composite systems P1, P2 and G2 is determined by means of the five different estimates of the previous chapter, namely, the secant
1The mesoscopic applied stress in the context of the SLS methods of chapter 2 is denoted by𝝈.
estimate (2.107), the DLS-TSO estimate (2.128), the SLS-TSO estimate (2.67), with (2.142), the DLS-GSO estimate (2.154) and the SLS-GSO estimate (2.81), with (2.167). It is recalled that these estimates are given in terms of the local and effective properties of LCCs with the same sub- structure as the actual nonlinear composites and defined in terms of the associated phase reference stress𝝈(⋅)and complianceM(⋅)tensors.
In the context of the TSO methods, the reference stresses𝝈(⋅)are such that𝝈(⋅)=𝝈(⋅), where we recall that 𝝈(⋅)denotes the volume-average of the stress field over phase (⋅) of the LCC, while the reference compliances M(⋅) are set equal to the tangent modulus tensors associated with the corresponding phase (⋅) of the LCC, evaluated at𝝈(⋅). Therefore, the TSO estimates are completely determined in terms of the average stress tensors𝝈(⋅)in the phases of the associated LCCs, which for the applications of interest in this chapter can be shown [107] to be proportional to the applied stress𝝈, i.e.,
𝝈(⋅)=𝜔(⋅)𝝈, (3.4)
where𝜔(⋅)denotes the associated proportionality constants. Taking into account the above result, it can be easily shown that the tangent modulus tensorsM(⋅)may be expressed in the form
M(⋅)= 1 2𝜆(E⋅) E+ 1 2𝜆(F⋅) F, E= 3 2 𝝈𝑑 ¯ 𝜎𝑒 ⊗ 𝝈𝑑 ¯ 𝜎𝑒 , F=K−E, (3.5)
where K is the forth-order identity tensor in the deviatoric space and the scalar moduli 𝜆(E⋅)and 𝜆(F⋅) associated withM(⋅)are functions of𝜔(⋅) and ¯𝜎𝑒=
√
3𝝈𝑑⋅𝝈𝑑/2.
Following Idiart et. al [64], all reference stress tensors of the LCCs involved in the calculations of the GSO estimates are set equal to the applied stress, i.e., 𝝈(⋅)=𝝈. These prescriptions have the merit that, when any one of the two-scale composites considered here reduces to the associated single-scale composite, the corresponding DLS-GSO and SLS-GSO estimates for the two-scale system reduce accordingly to a unique GSO estimate for the single-scale system. In addition, with these prescriptions the associated reference compliances M(⋅)are given by a relation of the form (3.5) in terms of the associated scalar variables𝜆(E⋅) and𝜆
(⋅)
F, corresponding to generalized-secant moduli that arenot known in advance. Furthermore, the phase averages𝝈(⋅) in these LCCs can also be shown [107] to be proportional to the applied stress𝝈, i.e., they are given by (3.4) in terms of the corresponding proportionality constants 𝜔(⋅) (which are different from the corresponding variables in the context of the TSO estimates). In addition to the determination of the phase- average constants𝜔(⋅), the GSO estimates require the computation of the corresponding fluctuation variables ˆ𝜎∥(⋅)and ˆ𝜎⊥(⋅)in the phases of the associated LCC. Recall that, in the context of the DLS- GSO estimate (2.154) the variables ˆ𝜎∥(⋅) and ˆ𝜎⊥(⋅)are defined by (2.151) and determined by means of the corresponding fluctuation equations (2.153), while in the context of the SLS-GSO estimate (2.81), with (2.167), these variables are defined by (2.151) and computed by (2.80), (2.166), as
appropriate. For later reference, we introduce at this point the following normalized fluctuation variables𝑥(∥⋅)and 𝑥⊥(⋅), and anisotropy ratios𝑘(⋅),
𝑥(∥⋅)=𝜎ˆ (⋅) ∥ ¯ 𝜎𝑒 , 𝑥 (⋅) ⊥ = ˆ 𝜎⊥(⋅) ¯ 𝜎𝑒 , 𝑘 (⋅)=𝜆 (⋅) F 𝜆(E⋅) , (3.6)
where, again, the superscript (⋅) indicates the phase of the LCC to which a quantity is referred to. With these definitions, taking the ratio of the generalized-secant conditions (2.152)1and (2.152)2 we obtain the following equation
ℰ(⋅)≡(1−𝑘(⋅))𝑥(∥⋅)+𝑘(⋅)−((𝑥(∥⋅))2+ (𝑥(⊥⋅))2) 1−𝑛
2
= 0. (3.7)
Finally, following Idiart et. al [64], the root in the fluctuation equations (2.153), (2.80) and (2.166) is chosen according to the sign of (𝜔(⋅)−1). Note that this prescription has the advantage of recovering the exact result 𝑥(∥⋅) = 𝜔(⋅) and 𝑥(⊥⋅) = 0 when it so happens that the stress field is constant in the given phase (⋅) of the LCC.