GEOMETRIES OF SURFACES OF NO STRETCHING, AND OF FIELDS OF SHORTENING AND EXTENSION IN THREE-DIMENSIONAL FLOW

We present a new method for representing flow geometries and apply it to geological problems. Surfaces of no stretching (SNS) are calculated from the velocity gradient tensor of any given flow, and are plotted in lower-hemisphere equal-angle projections. The diagrams show clearly the shortening and extension fields and the shapes of the SNS of the flow. The shape of the SNS depends on the flow symmetry and the (simple shear strain rate)/(pure shear strain rate) ratio. For an orthorhombic (pure shear) flow, the SNS consists of one or two planes or a circular or elliptical conical surface symmetrical about the principal planes of the strain rates. For a monoclinic flow, the SNS consists of two planes or a circular or elliptical conical surface symmetrical about the vorticity-normal section. For volume-constant plane strain flows, the SNS are mutually perpendicular planes that intersect along the principal strain rate axis with zero stretching. For triclinic strain, the SNS has no symmetry.

The diagrams can be used in combination with other results of forward modeling, or with field data. Rotation of planar and linear fabric elements during progressive deformation has been modeled previously, but the stretch or rate of stretch of these elements is not commonly addressed. Our models yield additional information about the fabric that may form. Furthermore, fields of finite shortening and extension, and the surfaces of no finite stretch, can be calculated from the position gradient tensor of any deformation, to be used in combination with finite strain studies. It can be shown that the long axis of the finite strain ellipse always rotates within the extension field of the deformation and the short axis always within the shortening field. Pre-existing fabric elements may rotate from the shortening into the extension field, but not vice versa.

The diagrams also clearly show in which orientations veins are most likely to be emplaced and in which orientations foliations and lineations are most likely to form. Pre-existing structures can be shown to exist within the shortening or extension field of the flow, and it can be predicted whether they will be shortened or extended in subsequent deformation. When field data for existing shortened and/or extended structures are available, certain constraints on the type of strain can be determined.