Page 352 - Caldera Volcanism Analysis, Modelling and Response

Magma-Chamber Geometry, Fluid Transport, Local Stresses and Rock Behaviour 327

Figure 12 Schematic illustration of a nested collapse caldera similar to the pair Askja-Lake
O º skuvatn in central Iceland (Figure11).

Farmer, 1983; Hudson and Harrison, 1997; Bell, 2000; Scho ¨n, 2004). At low
temperature and pressure and reasonably high strain rates, most solid rocks behave as
approximately linear elastic up to about 1% strain. For a completely anisotropic
(triclinic) linear elastic material, there are 21 independent elastic constants to be
considered (Love, 1927; Nye, 1957; Hudson and Harrison, 1997).
It is not feasible to model rocks as completely anisotropic, but quite a few
numerical solutions have been obtained from models where the rock body is
regarded as transversely isotropic (layered), in which case three independent
constants must be specified (Hudson and Harrison, 1997). The most common
approach for layered rocks, however, is to regard each individual layer as
homogeneous and isotropic, for which two independent constants are needed,
and then introduce the anisotropy through layers, contacts and discontinuities with
different mechanical properties.
The two elastic constants or moduli most commonly used in rock physics are
Young’s modulus and Poisson’s ratio. Young’s modulus is a measure of the rock
stiffness, and is often referred to as stiffness. When determining which Young’s moduli
to use for rock layers in a numerical model, there are certain aspects as to its
measurements and values that must be taken into account. First, for a given rock body,
particularly at shallow depths in a composite volcano or rift zone, the dynamic
modulus is normally much higher than the static modulus (Goodman, 1989; Scho ¨n,
2004). For a highly fractured and porous rock at shallow depths, the dynamic modulus
may be as much as 10–15 times higher than the static modulus of the same rock.
Second, small-sample laboratory measurements, dynamic and static, yield
stiffness values that are commonly 1.5–5 times greater than those of the field
modulus of the same rock (Heuze, 1980). For igneous and metamorphic rocks, the
laboratory modulus is commonly three times the field modulus.
Third, with increasing depth or mean stress Young’s modulus generally increases
(Heuze, 1980). Fourth, with increasing temperature, porosity and water content, or
a combination of these, Young’s modulus decreases.