1. Localised Folding & Axial Plane Structures Alison Ord 1 and Bruce Hobbs 1,2 1 Centre for Exploration and Targeting, Earth and Environment, The University of Western Australia 2 CSIRO Earth Science and Resource Engineering, Australia.

2. The structures developed in deformed metamorphic rocks are commonly quasi-periodic although localised and this suggests a control arising from non-linear matrix response is important in many instances. As such these structures add yet another layer of richness to the complexity of the folding process observed in non-linear layered materials.

3. Localised folding with axial plane foliation, Bermagui, NSW, Australia. Photo: Mike Rubenach. Outcrop approximately 0.5m across. Localised folding, Kangaroo Island, South Australia. Outcrop about 1m across. Experimental deformation of layers of cloth (Hall, 1815). Model approximately 1m across Field sketches of folded quartz veins (Fletcher and Sherwin, 1978). Examples of non-periodic folding.

4. Folded quartz/feldspar layers (white) crossed by a metamorphic layering (S- full line) from the Jotun Nappe, Norway. Photo: Haakon Fossen Metamorphic layering (full line) developed obliquely to initial bedding (dashed line) in deformed rocks from Anglesey, UK. Outcrop approximately 1m across. Foliations developed by coupled metamorphic/deformation processes.

5. where x is the distance measured along the layer, P is a function of the mechanical properties of the layer, and F(x) is a function that represents the reaction force exerted by the embedding medium on the layer, arising from the deflection of the layer. This theory expresses the deflection, w, of a single layer embedded in a weaker medium in terms of the equation The development of folds in layered rocks is commonly analysed using Biot's theory of folding.

6. F w Biot F = kw w F Biot’s theory of folding involves a linear response of the embedding medium to the deflection. This results in strictly sinusoidal folding even at high strains with no localised deformation in the embedding medium and hence no development of axial plane structures.

7. Since there is no non-linear behaviour and the viscosity ratio is relatively small, sinusoidal fold trains develop despite a wide range in the wavelengths of initial perturbations. (a) Single layer system. In both cases the viscosity ratio between layer(s) and matrix is 100, layer viscosity 10 21 Pa s, shortening 36%, initial strain-rate 5.7x10 -13 s -1 ; constant velocity boundary conditions. Folding in visco-elastic Maxwell materials with no softening in the embedding material. (b) Three layer system. Top and bottom layers 3 units thick; central layer 2 units thick. Bulk modulus of matrix 2.3 GPa; shear modulus of matrix 1.4 GPa. Bulk modulus of layers 30 GPa; shear modulus of layers 20 GPa. (a) (b)

8. However, if the embedding medium weakens as the layer deflects or shows other more complicated deformation behaviour then the function F is no longer linear.

9. F w n = 1 n = 2 n = 3 Biot Localised F = kw w F

10. Weakening of elastic moduli with weakening of viscosity according to the velocity in the y direction Units of length are arbitrary. Folding with softening in the embedding material for a single layer system. with log (viscosity) plotted along the line marked parallel to x 1. For a simple weakening response of the embedding layer to deflection, the initiation of folds follows Biot's theory and the initial folding response is sinusoidal.

11. Units of length are arbitrary. Folding response to softening of elastic moduli and viscosity according to Folding with softening in the embedding material for a single layer system. with log (viscosity) plotted along the line marked parallel to x 1. As the folds grow and weakening develops in the embedding material the fold profile ceases to be sinusoidal and the folds localise to form packets of folding along the layer.

12. This model also involves softening according to Folding with softening in the embedding material for a three layer system. for both elastic moduli & viscosity. The variation of the logarithm of the viscosity along the line shown. Units of distance are arbitrary. This softening behaviour is reflected in the embedding medium as a series of localised deformation zones parallel to the deflection direction, w, that is, parallel to the axial plane of the folds. These zones constitute micro-lithons, or in an initially finely layered material, crenulation cleavages.

13. Examples of metamorphic layering Metamorphic layering crossing bedding (Anglesey) Quartz plus muscovite (Q) layers alternating with layers comprised almost exclusively by muscovite (Mu). (Photo: Ron Vernon. Picuris Range, New Mexico, USA.). Image is 1cm across.

14. The modified Winkler model for a matrix with metamorphic layering. Planar regions with elastic modulus and viscosity, k 1 and  1, alternate with regions with properties k 2 and  2. Examples of metamorphic layering and a model for the resulting mechanical response. The reaction force system of the matrix against the folding layer represented by a Winkler model for a homogeneous matrix (Hunt, 2006). Each reaction unit has an elastic spring with modulus, k, and a viscous dashpot with viscosity, .

15. We now explore the situation where a layering arising from metamorphic differentiation forms oblique to folding multi-layers early in the folding history.

16. 27% shortening. Localised folds with metamorphic layering in single fold systems as outlined by the distribution of viscosity. Imposition of metamorphic layering Weakening of constitutive parameters (a) 82-y (b) m=1 n=3 m=3 n=3

17. n=1, m=1, 27% shortening. Localised folding of three layers with metamorphic layering outlined by the distribution of viscosity. n=1, m=1/ , 36% shortening. The metamorphic layering is distorted on the limbs of folds so that it remains approximately normal to the folded layer. This effect is seen in natural fold systems associated with metamorphic layering.

18. The individual layers have their own intrinsic buckling modes but are connected to each other by Maxwell units that themselves weaken with respect to spring constants and viscosity as each layer deflects (a). The introduction of a metamorphic layering introduces additional spatial periodicity as shown in (b). A simplified mechanical model of the multi-layer folding problem with axial plane metamorphic layering.

19. Although the influence of initial geometrical perturbations of large wavelength undoubtedly have an influence in promoting the formation of some localised folds at large viscosity ratios an additional mechanism involves the response to weakening in the matrix between the buckling layers. Non-linearities arising from large deflections and thick beam theory using the linear non- softening matrix response of the Biot theory have not so far emerged as a mechanism for localisation.

20. The Biot theory appears as the analysis that describes the initial sinusoidal deflections in a layered system before softening of the embedding matrix appears. After the embryonic sinusoidal fold system forms, localisation of the fold system can occur and chaotic systems develop if localised packets of folds interfere. Stress Strain A B Periodic fold form Localised fold form

21. The growth of metamorphic layering parallel or approximately parallel to the axial planes of folds introduces a periodicity in matrix response along the length of the folding layer introducing the opportunity for even greater fold localisation.

22. None of the weakening behaviours proposed here produce shear instabilities in the matrix so that differentiated crenulation cleavages do not develop in the models studied. Future work needs to concentrate on matrix constitutive relations that enable such instabilities to develop. It is hoped that the numerical exploration reported here will encourage theoretical investigations of the influence of periodic or quasi-periodic weakening behaviour in the matrix on fold localisation in both single and multi-layered systems.

23. Thank you