Rigid polygons in shear

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Rigid polygons in shear D A N I E L W. S C H M I D Physics of Geological Processes, University of Oslo, Pb 1048 Blindern, 0316 Oslo, Norway (e-mail: [email protected]) Abstract: Clasts, inclusions and intrusions in shear are potential recorders of strain, stress, rheology and metamorphism. In order to extract the recorded information, it is essential to have analytical and numerical theories that describe the deformation mechanics of such bodies. To overcome the simplifications of the commonly employed ellipsoid-based shape approximation, a combination of Muskhelishvili-type analytical solutions and finite-element method calculations is used to study the behaviour of (quasi) rigid polygons in shear. The results confirm that the polygon rotation and the pressure perturbation outside rigid polygonal clasts are well approximated by ellipse-based theories. However, this observation does not hold for the inside of these polygons, which show strongly varying values of pressure perturbation and maximum shear stress. For example, pressure perturbations inside the polygons are usually the opposite of the neighbouring matrix values across the polygonmatrix interface. This complex behaviour is summarized in the ellipse decomposition rule that allows for a qualitative understanding of the pressure perturbation in and around a wide range of polygons in shear. Other quantities studied include maximum values of overpressure relative to the shortening stress, and the area that undergoes overpressure with respect to the clast size. The results demonstrate that overpressure can be twice as large as the rock strength.

Particles, ranging in size from clasts to plutons, that have been subjected to shear potentially record important information about the geological past and may be used to decipher the kinematic history, metamorphosis and the mechanical behaviour of a certain outcrop or region. To achieve this, a sound understanding of the mechanics of particles in pure and simple shear is required. The available analytical theories (Bilby et al. 1975; Eshelby 1957, 1959; Ghosh & Ramberg 1976; Jeffery 1922; Schmid & Podladchikov 2003) approximate the geometry of particles with ellipsoidal shapes. Arbaret et al. (2001) have shown by means of analogue modelling that the ellipsoidal-shape simplification is justified in regard to kinematics. Yet, a theoretical foundation for this observation is lacking (cf. Treagus & Lan 2003) and, even more importantly, it remains unclear to what extent the ellipsoidal-shape simplification holds for dynamic key parameters, such as pressure and maximum shear stress, that drive metamorphic reactions and determine the deformation mechanism. The consequences of imperfect or nonellipsoidal geometry can be significant. For example, rhomboidal particle shapes may enhance the development of shape preferred

orientations (SPOs) (Ceriani et al. 2003), which renders the existing analytical theories unsuitable and, consequently, the far-field flow conditions cannot be reconstructed based on the theory derived by Ghosh & Ramberg (1976). Other examples include the understanding of mineral growth in rocks (Fletcher & Merino 2001), kinetics of phase transitions (Perrillat et al. 2003) and strength of minerals (Mosenfelder et al. 2000), all of which employ ellipsoidal-shape simplifications for cases where local perturbations caused by imperfect geometries could shift the system to a different equilibrium. In particular, non-ellipsoidal shapes drive local pressure perturbations around particles that are likely to deviate even more strongly from lithostatic values than previously established (Kenkmann & Dresen 1998; Tenczer et al. 2001; Schmid & Podladchikov 2003). Hence, the barometric interpretation of a mineral assemblage requires an analytical theory for the dynamics in and around non-ellipsoidal particles. While the ellipsoidal-shape simplification may hold for the dynamic parameters in the vicinity of a clast, it is almost certainly an oversimplification for the inside of the clast. Perfectly ellipsoidal clasts have the exceptional property that all stress and strain-rate components inside the

From: BRUHy,D. & Bt;RLINJ,L. (eds) 2005. High-Strain Zones: Structure and Physical Properties. Geological Society, London, Special Publications, 245, 421--431. 0305-8719/05/$15.00 ~ The Geological Society of London 2005.


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clast are constant under homogenous far-field boundary conditions such as pure or simple shear. This is unlikely to be the case for general polygonal shapes. A combination of two different approaches is employed in this paper to illustrate the complexities that arise with natural, non-ellipsoidal shapes: analytical solutions and finite-element models. Both have distinct advantages. Analytical solutions are extremely valuable as they allow for large parameter studies and generally help to understand a problem, but they can only be derived for a small range of problems. On the other hand, finite-element models can deal with a large range of problems, but must be rerun for every parameter change and therefore allow for the understanding of individual cases but are less useful for capturing the general character of the problem. In the context of this paper, an analytical solution based on Muskhelishvili's (1953) method is presented that is valid outside a special class of rigid triangles and squares. The inside of these triangles and squares is calculated with finite-element models, which are also applied to deal with clasts of more complex shape. Both analytics and numerics are two dimensional and assume plane strain. The material behaviour is linear viscous (Newtonian) and incompressible. The term 'rigid' employed in this paper designates the situation where the viscosity contrasts between clast and matrix is

so high that a further increase in viscosity contrast does not change the stress fields. The restriction to rigid (or quasi-rigid) clasts does not cause much loss of generality. As shown by Schmid & Podladchikov (2003), the infinite viscosity contrast limit can be taken as representative for clast-matrix viscosity contrasts as low as 10:1.

Methods Analytical solution The most flexible method for obtaining analytical solutions to complex geometries in twodimensional elastic or incompressible viscous problems was developed by Muskhelishvili (1953). Co-ordinate transformations are a basic part of this method and allow for solution finding in geometrically simpler image domains. For example, a square in the physical domain can be transformed into a circle in the image domain, facilitating the derivation of the analytical solution for the square (Fig. 1). In this paper, the hypotrochoid (cf. mathworld. wolfram.comkHypotrochoid.html) transformation is used: z= 8

-plane

+m~'"

.

z-plane

3

tY

X

Fig. 1. Hypotrochoid mapping example. Setting R = I, m = - 1/6, n = 3 maps the inside of the unit circle in the (-plane to the outside of an approximated square in the z-plane. Note the correspondence between the bold solid lines and the dashed lines in the two planes. Since z ~ co for ( ~ 0, the inside of the unit circle in the (-plane corresponds to the entire plane outside of the hypotrochoid in the physical plane z.

(1)


RIGID POLYGONS IN SHEAR Here z is the complex co-ordinate in the physical plane, ~" the complex co-ordinate in the image plane, if m is non-zero n + 1 is the number of vertexes (n is a positive, non-zero integer) and R is used to scale the size of the hypotrochoid. An example of hypotrochoid mapping is given in Figure 1. The hypotrochoid transform does not only map the inside of the unit circle in ~" to the outside of the hypotrochoid in z, but also maps the outside of the unit circle in ~"to the outside of the hypotrochoid in z. Therefore, possible solutions are restricted to the outside of the hypotrochoid in the physical domain and must implement the behaviour of the interior of the hypotrochoid through the interface boundary condition. In this paper, the hypotrochoids are modelled as rigid objects and, hence, the possible velocities at the interface are restricted to rigidbody rotation with a rotation rate that is determined through the condition that the resultant moment acting on the rigid hypotrochoid must be zero (Muskhelishvili 1953, p. 349). Savin (1961, pp. 281-293) gives the solution for rigid triangles and squares in uniaxial tension for an elastic matrix. As the instantaneous, incompressible elastic and viscous problem are mathematically identical and it is admissible to perform linear solution superposition, it is straightforward to translate Savin's (1961) solutions to rigid triangles and squares embedded in a viscous matrix and subjected to pure shear conditions (general shear is discussed in the subsection on 'Kinematics'). The resulting expressions are:

r __ ~e2iO~ 21~kR

(2)

llttri __ ~ -le-2ia + ~3 ..ff m e2ia 2txkR 2msr3 - 1 me- 2iot e2ia q~squ =_____~.(. ~__~2i )

(3)

2/.tkR

(4)

t/tsqu __ ~--1e-2ia 2/xkR (3m2+ 1)r (me - 2 m - e2i"'~ 4 (3m~-l) \ ~-fl--1- ]

(5)

where/x is the viscosity of the matrix, k the pure shear strain rate, i = v'~-]-, ce is the inclination of the clast with respect to the far-field flow, and the subscripts tri and squ denote solutions for the rigid hypotrochoid triangle and square, respectively. The Muskhelishvili solution of biharmonic

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problems, such as the one solved here, consists of two different complex potentials, & and 0. From these, all stress, strain rates and velocities can be deduced with a set of rules that can be found in various works, including Muskhelishvili (1953), Savin (1961), Jaeger & Cook (1979) and Schmid & Podladchikov (2003). For example, the expression for the pressure perturbation p (for the given boundary conditions, pressure can only be determined up to a lithostatic constant) is:

p = - 2 ~ R --~

(6)

where ~ means the real part and the minus sign is due to the convention that compressive pressure is positive. Substituting 4~t~i and OS~q u into equation (6) we obtain:

( ~2e2iC~"~

Ptri 2~ = -2~

1Z

2-m-m~'3J

(7)

Psqu __ 29t( ~2(2 me-2ia -- 2e2i~)'~ 2/~k

\ 0 _ - - ~ m ~ "~ 7 i)J"

(8)

These relatively simple expressions determine the entire pressure perturbation field outside isolated rigid triangles and squares, respectively. By substituting i~rl = 1 into equations (7) and (8) the pressure at the clast interface is obtained, setting m = 0 the hypotrochoid transform yields a circle and the corresponding pressure perturbations are found. The second key parameter of interest for the understanding of clasts in shear zones is the effective or maximum shear stress, r, which is calculated as (e.g. Ranalli 1995):

(