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Several challenges exist for studying deformation of polyphase aggregates. Of primary importance is understanding stress and strain or strain rate partitioning between phases. This can affect both the interpretation of stress levels measured in the individual phases as well as attempts to estimate the bulk mechanical properties of the aggregate. Based on observations of naturally deformed crustal rocks and analogs deformed in laboratory experiments, Handy (1990, 1994) outlined a method to place bounds on the mechanical properties of a two‐phase aggregate with non‐Newtonian rheology and a strength contrast. Others have also outlined methods to study two‐phase deformation either analytically (e.g., Takeda (1998) for Newtonian rheology) or numerically (e.g., Jessell et al., 2009; Takeda & Griera, 2006; Treagus, 2002; Tullis et al., 1991). However Handy’s phenomenological description remains a useful approach to addressing the problem of polyphase deformation. Handy’s description will be briefly summarized below.

In Handy’s formulation the strength of a two‐phase mixture depends not only on the rheological properties of the individual phases but also on the microstructure and phase proportions. In Handy’s formalism bounds can be place on the mechanical behavior of a two phase aggregate as the bulk properties lie between two end members. These end members are based on the degree of interconnectivity of the weaker or the stronger phase. If the weak phase is interconnected, the microstructure is termed an interconnected weak layer (IWL) (Figure 2.6a), and if the strong phase is connected, it is termed a load‐bearing framework (LBF) (Figure 2.6b).

In the case of an IWL microstructure, the aggregate approaches an iso‐stress state. That is to say that both phases experience similar stress levels but strain at very different rates. For a pure iso‐stress condition, the soft phase modulates the stress levels and strain is partitioned into the weak phase (Figure 2.6c). Thus, the bulk properties of the aggregate approach those of the softer phase. In cases where yield strength contrast is large, stress levels in the hard phase may be too low to result in yielding as stress levels in the strong phase are limited by yielding of the soft phase. Although IWL approaches that of an iso‐stress state, these are not identical and there are several important differences. For IWL deformation, if the aggregate is composed of small to moderate amounts of the weak phase and/or the strong phase has a significantly higher stress exponent than the weak phase, then the weak phase will deform at higher stresses and strain rates than the iso‐stress condition and the stronger phase will deform at lower stress and strain rate than the iso‐stress condition. However, if there are large proportions of the softer phase and/or the stress exponent of the harder phase is similar or smaller than the soft phase, then stress levels in the harder phase will exceed the soft phase.

Figure 2.6 Schematic of (a) interconnected weak layer and (b) load‐bearing framework microstructures. Light gray indicates the weaker phase and dark gray/black indicates the stronger phase. During deformation of the interconnected weak layer microstructure (c), strain is partitioned into the weaker phase and strong inclusions remain relatively undeformed. Stresses in the two phases will be similar but large strain heterogenaity is induced in the weak phase as the weaker material must flow around the rigid inclusions. During deformation of the load‐bearing framework microstructure (d) strain is distributed ~homogeneously as the strong matrix imposes its strain and strain rate on the weaker inclusions. Stresses will be lower in the weaker phase as it flows at lower stress than the stronger matrix.

In contrast, when a LBF microstructure exists, the soft phase is not interconnected and the rheology of the aggregate is controlled by the harder phase. Stress is supported primarily by the stronger phase and the stronger phase imposes its strain rate on the softer phase (Figure 2.6d). In this case, stress levels in the two phases diverge. Although local heterogeneous strain rate occurs at phase boundaries, the aggregate as a whole behaves in a manner very close uniform strain rate in the two phases.

Determining conditions where IWL versus LBF are stable depends on phase proportions, strength contrast and total strain. Strength contrast will be determined by the contrast in parameters in the flow laws for the phases (e.g., activation energy and volume and stress exponent). Larger strength contrasts between the phases and a lower volume fraction of the softer phase promotes stability of a LBF. Generally speaking IWL dominates when the volume fraction of the weak phase exceeds ~30–40% (i.e., percolation threshold). However, large strains also tend to favor IWL microstructures as the weaker inclusions coalesce to form interconnected microstructures resulting in a transition from LBF to IWL behavior (Handy, 1990, 1994).