Description:

_Journal of Structural Geology 32 (2010) 131–134_ _Contents lists available at ScienceDirect_ _Journal of Structural Geology_ _journal homepage: www.elsevier.com locate jsg_ _Reply_ _Folding with thermal mechanical feedback: Another reply_ _Keywords: Folding Coupled processes Shear heating Strain-rate softening_ _Abstract_ _In response to Schmid et al. (2010): (i) The linear Biot theory assumes fold wavelengths grow independently of each other; this is the ‘‘Biot process’’. (ii) The Biot theory predicts that only one wavelength grows to finite amplitudes; a spread of wavelengths at finite amplitudes indicates non-Biot processes operate. (iii) Boundary conditions control the wavelength that grows. (iv) Non-linear behaviour can result in non-Biot behaviour such as localised folding with no dominant wavelength. (v) Strain-rate softening is one form of non-linearity and leads to folding and boudinage at all scales; thermal–mechanical feedback leads to strain-rate softening producing folding and boudinage at the kilometre scale. (vi) The larger the viscosity ratio the larger the feedback effect. (vii) The Biot process may be important in some deformed rocks but others perhaps dominate._ _Crown Copyright ? 2009 Published by Elsevier Ltd. All rights reserved._ _1. What is and is not part of the Biot theory?_ _Equations describing the finite deformation of a layer or series of layers are discussed by Hunt et al. (1997) and Muhlhaus et al. (1994, 1998). In the general case these equations are partial differential equations, fourth order in spatial variables but also a function of time and may or may not be solvable using Fourier expansion methods. If the constitutive equations are linear and the deflections are small then many of the complicated terms (in particular the time derivatives) vanish and one arrives at the classical Biot expression. These governing equations can still represent elastic, viscous or plastic behaviour, as Biot (1965) points out, so the theory is powerful. We have the highest respect for Biot’s work and have never implied that this is at fault or should be neglected. The essence of Biot’s theory is that the deformation of layered materials is intrinsically unstable and folds begin to grow depending on the mechanical contrast between layers. The analysis assumes (true for linear systems and small deflections) that the wavelengths that begin to grow can be represented by a Fourier series and that the growth of each wavelength is independent of all others. The emphasis is on discovering which wavelength grows fastest and this becomes the dominant wavelength. The analysis is strictly applicable only to the moment the instabilities begin to grow and only for certain linear systems can one extrapolate this result to finite amplitudes. In general, the growth of initial perturbations to finite size cannot be predicted from the results of a linear perturbation analysis. It is possible, for instance, that other instabilities can develop sequentially after the first instability (Hunt et al., 2006). Extensions of the small deflection theory to large deflections (Muhlhaus et al., 1994, 1998) are still part of the Biot approach so long as growth-independence of the unstable modes is preserved. If non-linear behaviour is included then the assumption of growth-independence may not be true and approaches other than the linear Fourier analysis need to be implemented._ _There are now two questions to be asked of the strict small amplitude Biot theory: (i) What happens at large deflections? (ii) What happens if the system is non-linear as arises from geometrical softening, from strain softening or from strain-rate softening?_ _1.1 Finite amplitude folding_ _As far as we are aware, the only strictly analytical solutions to large amplitude viscous folding are those by Muhlhaus et al. (1994, 1998) which show that the Biot theory at small deflections can be extended to large deflections but only one wavelength grows to large amplitudes and this wavelength is governed by the boundary conditions. This result is supported by all computer models we know of (including those reported in Schmid et al., 2010). Thus, the assertion by Schmid et al. (2010) that a spectrum of wavelengths is to be expected at large amplitudes from the Biot theory is at odds with the analytical solutions and assumes that the dispersion of wavelengths at the moment of instability is reflected in the subsequent finite amplitude folds. The analytical solutions show that the dominant wavelength grows preferentially, is the only one preserved after relatively small strains and depends on the boundary conditions as we discussed in Hobbs et al. (2008). We take the data that show a spread of wavelength to thickness ratios in natural folds to indicate that processes other than the strict Biot process are operating in nature._ _1.2 Influence of non-linearities_ _Hunt et al. (1997) emphasise that the Biot theory is a special case of a general approach to folding in which non-linearities are relevant. The general outcome of such considerations is that the wavelengths that grow are not independent of each other, with interference between modes, and the result can be localisation of the folding behaviour; the concept of a dominant wavelength need no longer exist and complicated wave packets can form (Hunt and Wadee, 1991). This behaviour is beyond the reach of linear theories such as those considered by Biot. A vast literature on non-linear bifurcation theory has appeared since Biot’s important 1965 book allowing one to analyse some forms of non-linear behaviour. One way of establishing the nature of such non-linear behaviour is through modern continuum thermodynamics which we started to explore in Hobbs et al. (2008, in press) and Regenauer-Lieb et al. (2009)._ _An important example of non-Biot folding is the development of kink, chevron and concentric folds in a multilayer stack of thin elastic layers (Wadee et al., 2004; Edmunds et al., 2005) where the non-linearity stems from geometrical softening associated with large rotations. The critical load required to initiate instabilities is modelled using the Maxwell stability criterion; folding initiates after the stored elastic energy matches the energy required for slip on the layers. Such behaviour is non-linear and relies on bifurcation theory to define the initial and subsequent initiation and growth of instabilities. The concept of a dominant wavelength is meaningless here. The difference between this and the Biot approach is that Biot (1965, p204; and others) regarded the Maxwell stability criterion as defining the initiation of kink folds in anisotropic materials whereas bifurcation theory regards instability as developing after the energy minimum is passed. The Biot theory gives no information on when folds nucleate and how the system evolves._ _Hobbs et al. (2008) present another example of non-linear, non-Biot behaviour during buckling. The constitutive behaviour of a temperature dependent Newtonian or power-law viscous material, with thermal–mechanical feedback, is modified such that the effective viscosity becomes a decreasing function of strain-rate. For a Newtonian material, the effective viscosity decreases with the square of the strain-rate (Fleitout and Froidevaux, 1980). Similar strain-rate softening relationships arise from other feedback processes including mineral reaction-mechanical coupling (Hobbs et al., in press) and diffusion-mechanical feedback (Regenauer-Lieb et al., 2009). Strain-rate perturbations are amplified during deformation leading to greater localised strain-rates and self-enhancing feedback._ _The general theory for instabilities in rate dependent materials with strain and strain-rate softening is presented in many papers; examples are Anand et al. (1987) and Needleman (1988). One conclusion is that instability is sensitive to geometrical or physical heterogeneities. With reference to folding, heterogeneities in strain-rate introduced by incipient buckling of a layer are amplified by strain-rate softening. These heterogeneities may be those predicted by the Biot theory but from the moment instabilities grow, the process is not one of Biot-type wavelength selection but involves localisation of deformation. In Hobbs et al. (2008) we selected constitutive relations for quartz and feldspar that give realistic but small (<20) viscosity ratios between layers under mid-to lower-crustal conditions. This selection is based on careful and critical analysis by Hirth et al. (2001). Of course, other constitutive relations exist but as Schmid et al. (2010) point out they can give very large viscosity ratios, so large in fact that one has to question the validity of extrapolating the experimental data to geological conditions. A viscosity ratio of 103 at 1000 K in a homogeneously shortening layered material implies magnitudes of differential stress in the crust of Gigapascals if the strength of weak layers is as low as 1 MPa. Such high values seem unlikely. An additional issue is that the higher the viscosity ratio, the higher the viscous dissipation in the competent layer(s) leading to thermal–mechanical feedback effects even more dramatic than are considered by Hobbs et al. (2008). In the presence of strain-rate softening feedback, the larger the initial viscosity ratio the more intense the fold localisation._ _Fig. 1 shows shortening of a single layer with a relatively high (200) viscosity ratio to the embedding matrix. With no strain-rate softening (Fig. 1a and b) the Biot result of a sinusoidal wave form develops; with strain-rate softening (Fig. 1c and d) localised folding develops. The effects discussed in Hobbs et al. (2008) are conservative compared to what is expected at larger viscosity ratios and tend to support conclusions that these feedback effects_ Ключевые слова: hirth, evolution, material, strain-rate softening, deections, mesh, considered, equation, press, law, preserved, velocity, ?nite, homogeneous, diffusion, scale, time, wadee, natural fold, non-linear, independent, geol, thermal–mechanical feedback, behaviour, mech phys, layer, journal structural, solids, earth, thermalmechanical feedback, high, non-linear behaviour, elsevier, fold, coupled, soc, approach, size, boudinage, softening, general, geometrical softening, spread, de?ections, process, instability, struct geol, grows, model, veveakis, form, analytical solution, depends, muhlhaus, strain-rate, folding, deformation, struct, non-biot, viscous material, heterogeneity, biot process, wavelength, predicted, mech, hobbs press, coupling, spectrum, localisation, larger, thickness ratio, constant, needleman, dominant wavelength, soc lond, localised, homogeneous shortening, single layer, biot approach, feedback, journal, moment, rate, shortening, vii, small, edmunds, linear, constitutive, boundary, arc length, wavelength grows, ratio, nite amplitude, expected, hobbs, uid pressure, nite size, growth-independence, point, vast literature, lond, phys, natural deformation, result, chemical, strain, mesh sensitivity, method, large amplitude, fourier series, thermal–mechanical, viscous, structural, hunt, dominant, structural geology, viscosity ratio, length, discussion, rock, issue, discussed, initial, mode, small deections, geology, viscosity, reply, natural, froidevaux, mechanical, inter, schmid, order fold, shawki, concept, thermalmechanical coupling, growth, biot, condition, inuence, biot theory, question, boundary condition, regenauer-lieb, represented, proc, linear theory, nature, lead, unstable mode, amplitude, length scale, grow, theory, buckling, thermal, governed, phys solids, viscous dissipation, strain rate, analysis, large, shear