Do principal surfaces of stress and strain always exist?

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Variations of stress and strain are commonly expressed by patterns of stress or strain trajectories: three mutually orthogonal families of continuous lines, parallel to maximum, intermediate and minimum stress or strain axes. It might be assumed that there are equivalent continuous principal surfaces of stress or strain, for any state of continuously varying stress or strain. We demonstrate that this will not generally be the case for three-dimensionally varying states of stress or strain. Whether or not principal surfaces of stress or strain exist is governed by the abnormality of the vector field of principal trajectories. We consider the Z vector fields for examples of many types of three-dimensional heterogeneous deformation, and show that most of these do not lead to definable principal XY strain surfaces. An alternative geometric test is presented, termed the continuity loop, for simply demonstrating the existence (or not) of principal surfaces, using geometrical and orientational information. It is important to the understanding of geological structures to know which kinds of heterogeneous deformation give rise to principal surfaces of stress or strain. We conclude with examples of structures which might be indicative of the absence of continuous principal surfaces of stress (segmented faults, echelon veins and dykes), a discussion of the implication for strain fabrics and foliations, and a warning that foliation trace trajectories on maps or sections may not necessarily indicate the existence of real foliation surfaces in three dimensions.