Structural softening of the lithosphere

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1 Structural softening of the lithosphere doi: /j x Stefan M. Schmalholz, Yuri Y. Podladchikov and Bjørn Jamtveit PGP, University of Oslo, PO Box 1048, Blindern, 0316 Oslo, Norway ABSTRACT Structural softening is a decrease in the amount of stress needed to deform the lithosphere at a particular rate because of its structural reorganization while all true rheological properties remain constant. Structural softening is fundamentally different than material softening, where the decrease in stress is generated by a change in rheological properties with progressive deformation, such as grain size reduction resulting from large shearing strain. We study structural softening generated by folding of the crust-mantle boundary, which is a structural instability that inevitably develops during compression of the mechanically layered lithosphere. For ductile rheologies, the stress decrease represents a decrease of the effective lithospheric viscosity, which is proportional to the ratio of stress to lithospheric shortening strain rate. We present analytical and numerical results quantifying the decrease in stress and effective viscosity that occur during shortening at a constant rate. The decrease in effective viscosity can be up to 10-fold. Terra Nova, 17, 66 72, 2005 Introduction Rheological properties of the lithosphere vary significantly with depth because of variations in mineral rock composition and temperature dependence of rheological properties (Brace and Kohlstedt, 1980; Carter and Tsenn, 1987; Kohlstedt et al., 1995; Fig. 1). Therefore, both oceanic and continental lithospheres are considered mechanically layered. These layers are never perfectly flat and geometrical perturbations along lithospheric layers are expected because of characteristics such as lateral variations of the depth of the crust-mantle boundary. If a compression acts parallel to the mechanical layering of the lithosphere, the deformation is sensitive to lateral perturbations and will inevitably generate structural instabilities, such as folding (also often termed buckling; Biot, 1961; Zuber, 1987; Martinod and Davy, 1992; Burov et al., 1993; Zuber and Parmentier, 1996; Burg and Podladchikov, 1999; Gerbault, 2000; Schmalholz et al., 2002). However, the deformation of the lithosphere is frequently modelled using thin-sheet models, which assume that strain rates are constant with depth and, hence, neglect structural instabilities such as folding (England and McKenzie, 1982, 1983; Vilotte and Daigniere, 1982; Houseman and England, 1986; Correspondence: Stefan M. Schmalholz, PGP, University of Oslo, PO Box 1048, Blindern, 0316 Oslo, Norway. Tel.: ; fax: ; s.m.schmalholz@fys.uio.no Flesch et al., 2001). The constant strain rate assumption is also often applied to construct yield strength envelopes used to interpret the mechanical strength of the lithosphere (Brace and Kohlstedt, 1980; Kohlstedt et al., 1995; Fig. 1). Yield strength envelopes plot differential stress vs. depth, and it is generally assumed that differential stress can be directly related to mechanical strength. In this study, we restrict the simplifying constant strain rate assumptions to the far field only while allowing for near field strain rate variations to account for emerging lithospheric structures. To keep the analysis simple, only ductile rheologies are considered to quantify the evolution of the differential stress during lithospheric shortening. In this case, the appropriate rheological property is the effective lithospheric viscosity, which is defined by the ratio of differential stress to twice the lithospheric shortening strain rate. A decrease in this effective viscosity with progressive shortening strain is referred to here as softening. A decrease in the effective viscosity during progressive shortening while all rheological properties (e.g. the Newtonian viscosity) remain constant is termed here structural softening. Structural softening is fundamentally different to material softening, where a decrease in stress results from a change in rheological properties during deformation (e.g. Tackley, 1998; Braun et al., 1999; Bercovici, 2003; Huismans and Beaumont, 2003). The feasibility of structural softening is investigated in the case of structural instabilities, such as folding, emerging around the crust-mantle boundary and lithospheric conditions for which the upper crust is mechanically decoupled from the lithospheric mantle. Single-layer folding is analysed as a representative process to quantify the evolution of the effective viscosity. Numerical simulations for localized perturbations considering gravitational stresses, depth-dependent effective viscosities and power-law rheologies show that the analytical solutions for single layer folding capture the essential mechanical behaviour of lithospheric structural instabilities. The main goals of this paper are to present the basic mechanics of structural softening, to emphasize the importance of structural instabilities for the evolution of differential stress during lithospheric shortening and to show that differential stress is not necessarily the suitable parameter that quantifies the mechanical strength of ductile rocks. Structural softening: analytical results The fundamental structural softening mechanism is sketched in Fig. 2. Single-layer folding under pure shear of a competent Newtonian layer embedded in less competent Newtonian material is considered (Biot, 1961; Fletcher, 1974; Schmalholz et al., 2002). The finite amplitude solution (FAS) derived by Schmalholz and Podladchikov (2000) describes the evolution of the fold amplitude during shortening. The main feature of the 66 Ó 2005 Blackwell Publishing Ltd

2 Terra Nova, Vol 17, No. 1, S. M. Schmalholz et al. Structural softening of the lithosphere Fig. 1 Sketch of a typical yield strength envelope for the continental lithosphere. The differential stress is calculated from experimentally derived flow laws assuming a constant strain rate. The vertical profile of the differential stress corresponds to the vertical profile of the rheological properties corresponding to mechanical strength. Areas of high differential stress are considered to be mechanically strong whereas areas of low differential stress are considered weak. Dashed line shows qualitatively the effect of high-stress flow laws generating smaller variations in rheological properties with depth. FAS is that strain rates within the folding layer can vary during shortening, although the overall background shortening strain rate e B is constant. The strain rate e L within the layer (constant along and averaged across the layer) is given by e L ¼ )(@L/@t)/L, where L is the fold arc length approximated by the relation L/k ¼ 1+b(A/ k) 2, with b ¼ p 2 /[1 + 3(A/k) 2 ], k being the fold wavelength and A being the fold amplitude (see Schmalholz and Podladchikov, 2000). Transforming derivatives with respect to time into derivatives with respect to strain (see Schmalholz and Podladchikov, 2000) and approximating the resulting expression for e L provides a ratio between e L and the background shortening strain rate e B given by e L ¼ e B 33þp 2 A 4þ k 6 þ p 2 A 2þ1! k 4þ 2þ1 33þ ð p 2 Þ A k ð 2ap2 þ 6 þ p 2 Þ A k 1 2ap 2 A 2 ð1þ k where a is the maximum amplification rate (corresponding to the dominant Fig. 2 Schematic illustration of structural softening. A Newtonian layer, embedded in material with smaller Newtonian viscosity, is shortened parallel to its layering. If interfaces between layer and embedding material are perfectly straight, the layer shortens and thickens with the same background shortening strain rate as the embedding material. The deformation is homogeneous despite the viscosity contrast between layer and embedding material and the stress (r) profile across the layer represents the profile of the Newtonian viscosity. If small perturbations exist along the layer (e.g. a sinusoidal perturbation), the shortened layer is mechanically unstable, it folds and the strain rate within the folded layer is smaller than the background shortening strain rate, because the fold arc length (L) shortens slower than the fold wavelength (k). The deformation is heterogeneous and the stress profile across the layer does not represent the profile of the Newtonian viscosity, because the strain rate is not constant. Stress profiles for homogeneous and heterogeneous deformations are different although in both cases the rheological properties are identical. wavelength and derived by linear stability analysis) normalized by e B (Biot, 1961; Fletcher, 1974). The ratio e L /e B is identical to the stress ratio P/P 0, where P is the deviatoric layerparallel stress within the folding layer (P ¼ 2 l L e L, with l L being the Newtonian viscosity of the layer) and P 0 is the stress for pure shear deformation of the layer (P 0 ¼ 2 l L e B ). We can alternatively express the ratio e L /e B as e L ¼ 2l Le L ¼ 1 P ¼ l eff ¼ P : e B 2l L e B l L 2e B l L P 0 ð2þ The effective viscosity l eff is defined as the ratio of stress within the layer to twice background shortening strain rate (P/2e B ). In the analytical solution, the stress and strain rate within the embedding material remain constant and, therefore, a change in l eff during shortening represents a change in the effective viscosity of the total system including layer and embedding material. In eqn (1), the ratio A/k increases during shortening (e ¼ 100(k 0 ) k)/k 0, with k 0 being the initial wavelength, Fig. 2) representing fold amplification (Fig. 3). If the ratio A/k is zero (no perturbation) then e B ¼ e L and the deformation is homogeneous. In addition, if a ¼ 0 then geometrical perturbations do not grow (no instability) and again e B ¼ e L. The equation shows that both a competence contrast (to provide a > 0) and a geometrical perturbation (to provide A/k > 0) are necessary to generate a stress decrease (Fig. 3B). For a comparison with elastic rheologies, we consider the classical solution for post-buckling of a compressed elastic bar, known as the Elastica (e.g. Euler, 1744; Bazant and Cedolin, 1991). An approximate relationship between amplitude and shortening for the Elastica is derived in Bazant and Ó 2005 Blackwell Publishing Ltd 67

3 Structural softening of the lithosphere S. M. Schmalholz et al. Terra Nova, Vol 17, No. 1, Fig. 3 Amplification and structural softening during folding of a single layer. The legend for both graphs A and B is displayed in figure B. The numbers displayed in the legend represent the viscosity contrast (vc) between layer and embedding material. The shortcut FAS represents the analytical finite amplitude solution (derived in Schmalholz and Podladchikov, 2000) and the shortcut FEM represents a numerical finite element solution. (A) The evolution of the fold limb dip (h ¼ arctan(2pa/k)180/p, see Fig. 2) is plotted vs. shortening (e). The Elastica is the solution for post-buckling of an elastic column and represents an upper bound for viscous folding solutions. (B) The averaged layer-parallel stress P (normalized by the value for pure shear deformation P 0 ) is plotted vs. e times vc 2/3 using eqn (1). Multiplying by the viscosity contrast to the power 2/3 collapses the results onto a narrow zone in the space P/P 0 -evc 2/3. The ratio P/P 0 also represents the ratio of the effective layer viscosity to the true Newtonian layer viscosity (see eqn 2). Note that the decrease in stress and effective viscosity can be more than one order of magnitude. Cedolin (1991, their equation ) and is also plotted in Fig. 3A. Interestingly, a comparison between the FAS and the Elastica shows that the Elastica provides an upper bound for viscous folding and corresponds to folding with infinitely high viscosity contrast. Effects of gravity on folding are quantified using analytical results of Schmalholz et al. (2002). Gravity effects on lithospheric shortening are best evaluated using dimensionless numbers such as the Argand number. For folding processes, the Argand number represents the ratio of gravitational stress acting against folding to compressive stress driving folding. The Argand number is given by Ar ¼ DqgH/(2 l L e B ), where Dq, g and H are the density difference between the material below and above the folding layer, the gravitational acceleration and the layer thickness, respectively. The parameters in the Argand number depend on the assumed deformation behaviour. Originally, the Argand number was derived for thin-sheet models using different parameters (referred to here as Ar TS ; England and McKenzie, 1982). Thin-sheet models neglect structural instabilities and consider homogeneous thickening only, and Ar TS represents the ratio of gravitational stress acting against thickening to compressive stress driving thickening. The characteristic length scale in Ar is the layer thickness, whereas the characteristic length scale in Ar TS is the thickness of the entire lithosphere. Therefore, Ar and Ar TS can be considerably different although the same geological setting is considered. Values of a larger than 10 indicate that vertical, lithospheric velocities as a result of structural instabilities are one order of magnitude larger than vertical, lithospheric velocities for homogeneous thickening (Fig. 4). Figure 4A shows that for values of Ar smaller than 0.25 and viscosity contrasts larger than around 40 values of a are always larger than 10. Assuming a density difference between lithospheric mantle and lower crust of 450 kg m )3 and folding of the top 10 km of the mantle lithosphere, the gravitational stresses are around 45 MPa. To reach values of Ar smaller than 0.25 the compressive stress must be larger than 180 MPa, which is a stress value expected in compressional settings (e.g. Gerbault, 2000; Fig. 4B). Structural softening: numerical results Structural instabilities are studied in the framework of continuum mechanics (Mase, 1970; Sedov, 1994). The equilibrium equations and the continuity equation for incompressible flow are solved numerically using the finite element method (e.g. Thomasset, 1981; Cuvelier et al., 1986; Girault and Raviart, 1986). Pure shear shortening in the horizontal x-direction is applied. Lateral and bottom model boundaries are kept straight and shear stresses at the bottom and lateral sides are zero. The top boundary is a free surface. The rheology is power-law using an effective power-law viscosity given by g eff ¼ BE (1/n)1) (England and McKenzie, 1982). B is a material Fig. 4 Analytical solutions showing the effects of gravity on folding. (A) The maximal amplification rate (a, normalized by the background shortening strain rate, e B )is plotted vs. the effective viscosity contrast (vc) for different values of the Argand number (Ar, figures printed next to the lines). A value for a of 10 means that vertical lithospheric velocities because of folding are 10 times larger than vertical lithospheric velocities because of homogeneous thickening. (B) Contours of Ar are plotted in the space gravitational stress (DqgH) compressive stress (2 le B ). 68 Ó 2005 Blackwell Publishing Ltd

4 Terra Nova, Vol 17, No. 1, S. M. Schmalholz et al. Structural softening of the lithosphere Fig. 5 Numerical solutions for single-layer folding (power law, n ¼ 3) showing effects of gravity. (A) The limb dip (h) is plotted vs. shortening (e). Fold amplification with gravity (Ar ¼ 0.3) is slower than without gravity. (B) The averaged differential stress within the layer Dr av (normalized by the value for pure shear deformation Dr av0 ) is plotted vs. e. To generate the same amount of stress decrease, folding with gravity has to accommodate more shortening than folding without gravity. (C) The ratio Dr av /Dr av0 is plotted vs. h. The stress decrease for folding with and without gravity is more similar if h and not e is used as a measure for deformation. constant, n is the power-law exponent and E is the second invariant of the strain rate tensor (E equals one for pure shear). For n ¼ 1, B corresponds to the Newtonian viscosity using s xx ¼ 2g eff e xx, with s xx being the deviatoric stress tensor component and e xx being the deviatoric strain rate component, both in the horizontal, x-direction. Single-layer folding with and without gravity is simulated for n ¼ 3. The layer exhibits initially a sinusoidal perturbation corresponding to the dominant wavelength (Fletcher, 1974; Schmalholz et al., 2002) and the initial ratio of amplitude to layer thickness is The initial contrast between the effective power-law viscosities is 100. For folding with gravity, Ar equals 0.3, which corresponds to a situation where gravitational stresses and compressive stresses have an equal impact on folding (i.e. S ¼ 1, where S ¼ 6 1/2 9/4l m n/(g eff Ar 3/2 ) with l m being the viscosity of the matrix; see Schmalholz et al., 2002). Despite the strong effect of gravity, the compressive stresses decrease significantly during shortening (Fig. 5). A layer of light material rests on a layer of denser material (Fig. 6A). The initial model height is normalized to one and the initial width is 10 times the initial height. The initial height of the bottom layer is 0.7. Both layers exhibit n ¼ 3 with different values of B decreasing exponentially with depth (Fig. 6B). B decreases with depth because of its temperature dependence (Fletcher and Hallet, 1983) using B ¼ B 0 exp()z/f), where B 0 is the effective viscosity at the top of the layer, z is the vertical distance from the top of the layer and f is the e-fold length (Fletcher and Hallet, 1983). f for the top layer is 1/30th of the total model thickness and f for the bottom layer Fig. 6 Numerical results for initial localized perturbations. (A) Model set-up. (B) Profile of the rheological property B (see text) vs. z (vertical coordinate). (C) Values of the vertically averaged differential stress (Dr av, see text) vs. shortening (e). Dr av is normalized by Dr av0, the value corresponding to homogeneous pure shear deformation. Structural softening becomes significant after a shortening of about 30 for the given perturbation. The strongest stress decrease occurs for the vertical section across the initial perturbation, where the fold develops (see Fig. 7). The evolution of the decrease in differential stress for structural instabilities arising from localized perturbations is similar to the stress decrease for single-layer folding (Figs 3 and 5). Note that the decrease in differential stress occurs along the entire model domain, although the perturbation is localized, and that the state of stress in areas far from the initial perturbation is affected by the structural instability. Ó 2005 Blackwell Publishing Ltd 69

5 Structural softening of the lithosphere S. M. Schmalholz et al. Terra Nova, Vol 17, No. 1, Fig. 7 Numerical results for the model shown in Fig. 6. Contours of the basis 10 logarithm of the second invariant of the strain rate tensor, log 10 (E), are displayed for different amounts of shortening (e). log 10 (E) ¼ 0 corresponds to pure shear. Results show that E varies more than 1.5 orders of magnitude. The black line represents the boundary between top and bottom layer. With increasing shortening, strain rate variations become significant and occur within a domain significantly larger than the domain of the initial perturbation. is 1/24th of the total thickness (if the total thickness represents a 120 km thick lithosphere then f for the bottom layer is 5 km). The viscosity profile does not exhibit contrasts in effective power-law viscosities larger than 100, which is achieved by limiting the value of the minimum viscosity, which is normalized to one. Ar ¼ 0.1 using the effective power-law viscosity at the top of the bottom layer to calculate the compressive stress, f of the bottom layer as characteristic length scale and the densities of the top and bottom layer to calculate the density difference (Schmalholz et al., 2002). In the middle of the model domain a small geometrical perturbation is introduced (Fig. 6A). Differential stresses are calculated using Dr ¼ 2(0.25(s xx )s zz ) 2 þ s 2 xz )1/2 (e.g. Turcotte and Schubert, 1982). The averaged differential stress (Dr av ) is calculated at a specific horizontal x-position by integrating Dr vertically along the z-direction and dividing by the thickness of the integrated domain. During shortening, values of Dr av decrease at every horizontal x-position representing structural softening of the entire model lithosphere (Fig. 6C). Similar to eqn (2), the ratio of Dr av to Dr av0 (the constant value for pure shear deformation) corresponds to the ratio of the effective lithospheric viscosity to the power-law viscosity of the layer. The second invariant of the strain rate tensor, E, measures the deviation from homogeneous, pure shear deformation for which E ¼ 1, and is displayed in Fig. 7. E ¼ (e 2 xx + e2 xz )1/2, using e xx ¼ )e zz for incompressible materials. Discussion Folding is a type example for structural instabilities. This study shows that differential stresses during folding of the crust-mantle boundary can be significantly (10-fold) smaller than stress estimates based on constant strain rate assumptions, such as applied for constructing yield strength envelopes or thin-sheet models. In a recent paper, Jackson (2002) explored compositional (wet vs. dry) dependences of rock properties to evaluate the Ômechanical strengthõ of the Indian mantle lithosphere. Our results show that a competent lithospheric layer around the crust-mantle boundary in a compressive tectonic setting may exhibit a significant decrease of its stress level without any loss of the Ômechanical strengthõ, which is because of structural rather than material softening. Neglecting horizontal deformation, Jackson (2002) concluded that stresses are small and, therefore, the mantle lithosphere is Ômechanically weakõ. The analysis of Jackson has been criticized by Lamb (2002) for not considering any compressive stresses, which are expected to be present during a continental collision. The magnitude of these compressive stresses can be accurately estimated by rheology independent force balance calculations to be around 100 MPa at average in order to maintain the topographic elevation of the Himalaya (Jeffreys, 1976). This high level of average compressive stress is sufficient to drive large scale, localized lithospheric folding of the Indian lithosphere (similar to scenarios presented by Burg and Podladchikov, 1999). Moreover, it is essential for discussions about the strength of rocks to identify the property which actually quantifies the strength. For ductile rheologies, the parameter quantifying the relative strength or weakness is viscosity, which is proportional to the ratio of stress to strain rate. Therefore, both stress and strain rate values are needed to estimate the effective viscosity (i.e. the mechanical strength) of the lithosphere, which can vary significantly during lithospheric deformation as demonstrated in this study (see also Watts and Burov, 2004). Folding is often neglected in lithospheric shortening models, because lithospheric viscosities vary exponentially with depth and not discontinuously as considered for single-layer folding. However, several studies showed that shortening layers with exponentially decreasing viscosities generates folding (Biot, 1961; Zuber and Parmentier, 1996; Schmalholz et al., 2002). In addition, flow laws for high stresses indicate a smaller decrease of rheological properties with depth compared with an exponential decrease (e.g. Regenauer-Lieb and Yuen, 2003; Fig. 1) and generate, together with power-law flow laws, a distribution of rheological properties closer to the single-layer folding setup. Finally, developing 70 Ó 2005 Blackwell Publishing Ltd

6 Terra Nova, Vol 17, No. 1, S. M. Schmalholz et al. Structural softening of the lithosphere structural instabilities is preferable to homogeneous thickening, because lithospheric deformation involving structural instabilities requires less compressive stress than lithospheric deformation involving thickening only. Only ductile rheologies have been considered in this study. Leroy et al. (2002) showed that for an elastic layer over an inviscid and buoyant fluid the lateral compressive force decreases during shortening, indicating structural softening also occurs with pure elastic rheologies. Conclusions Structural instabilities are likely to emerge around the crust-mantle boundary during lithospheric shortening and decrease the effective lithospheric viscosity while all rheological properties remain constant (Fig. 6). This structural softening is also felt far from the lithospheric domain where the structural instability actually develops (Figs 6 and 7). Effects of gravity, depth-dependent rheological properties and limited strength of rocks do not prohibit structural instabilities and structural softening. Neglecting structural instabilities a priori in models for lithospheric shortening is not justified. Dependences of effective rheological properties on shortening strain, similar to those presented in Fig. 3B, must be introduced to account for the softening caused by structural instabilities. Acknowledgements We thank M. Gerbault and E. Burov for their helpful and constructive reviews, Boris Kaus for discussions and Nate Onderdonk for improving the English of the manuscript. This study was supported by a centre of excellence grant to PGP from the Norwegian Research Council. References Bazant, Z.P. and Cedolin, L., Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories. The Oxford Engineering Science Series. Oxford University Press, New York. 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7 Structural softening of the lithosphere S. M. Schmalholz et al. Terra Nova, Vol 17, No. 1, Vilotte, J.P. and Daigniere, M., Numerical modeling in intraplate deformation: simple mechanical models of continental collision. J. Geophys. Res., 87, Watts, A.B. and Burov, E.B., Lithospheric strength and its relationship to the elastic and seismogenic layer thickness. Earth Planet. Sci. Lett., 213, Zuber, M.T., Compression of the oceanic lithosphere: an analysis of intraplate deformation in the Central Indian Basin. J. Geophys. Res., 92, Zuber, M.T. and Parmentier, E.M., Finite amplitude folding of a continuously viscosity-stratified lithosphere. J. Geophys. Res., 101, Received 26 August 2004; revised version accepted 27 October Ó 2005 Blackwell Publishing Ltd