The homogeneous transpression strain model formulated by Sanderson and Marchini ( Journal of Structural Geology 6, 449–458, 1984) has proved to be a useful tool in the analysis of complex three-dimensional deformation patterns. However, some of the boundary conditions introduced in the mathematical model may be unrealistic and unnecessarily restrictive. In this paper a strain matrix for unconfined transpression and transtension is derived in which material is allowed to move not only vertically, but also horizontally in and out of the deforming zone parallel to its length—‘lateral extrusion’. Three end-member plane-strain components are defined: wrench simple shear; pure shear in XY (lateral stretch); and pure shear in YZ (vertical stretch). These biaxial strains can be viewed as the apices of a ‘strain triangle’ for unconfined transpression or transtension. The edges of the triangle correspond to: triaxial pure shear; non-coaxial, biaxial lateral extrusion; and the triaxial confined transpressional or transtensional strain of Sanderson and Marchini. During unconfined transpression, the orientation and, in particular, the geometry ( k-value) of the finite-strain ellipsoid depends upon not only the amount of shortening across the zone and the amount of strike-slip parallel to the zone, but also upon the ratio of vertical to lateral stretch. This can present serious difficulties when attempting to use finite strains to infer the direction and magnitude of zone-boundary displacements. Examples of transpression zones in which there is evidence of a component of lateral extrusion are described from SW Cyprus and central Scotland. These examples illustrate that antithetic strike-slip shearing is a kinematic requirement of laterally unconfined transpression, implying that synchronous shear-sense indicators may give opposite senses of movement in shear zones. Specific geometric and mechanical boundary conditions, together with internal fault-zone rheologies, may favour the lateral extrusion of material. Kinematic partitioning can occur to form fault-bounded domains in which end-member biaxial and/or non-coaxial strains are developed. Analysis of such domains can give a clearer understanding of regional-scale triaxial deformations. These findings illustrate the importance of establishing displacement boundary conditions when qualitatively or quantitatively analysing crustal deformation zones.
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