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Figure 4. Iso-frame model of chalk (Fabricius, 2003). The iso-frame model is an effective medium model based on modified upper Hashin-Shtrikman (MUHS) bounds of Nur et al. (1998). Sketch to the right shows how a part of the solid (including calcite (white) and silicates (gray)) are suspended in pore-fluid (black), and how the suspension is embedded in the supporting frame of calcite (white) and silicates (grey). Samples with the same fraction of solid in frame but with varying porosity fall along one iso-frame curve in an elastic modulus-porosity plot (left). The MUHS bounds are calculated from:


 MHS+ = KHS+ + 4/3GHS± , where K1 and K2 are bulk moduli of individual phases G1 and G2 are shear moduli of individual phases. f1 and f2 are volume fractions of individual phases normalized to a critical porosity, φc. The upper bound is defined when 1 is the stiffest component and 2 the softest, For the IF model, the suspension (termed 2) is imbedded in the mineral frame (termed 1): f1 = (IF)(φc-φ)/φc f2 = (φ + (1−IF)(φc-φ))/φc, where IF is a fraction between 0 and 1.