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Using an unsheared pair to construct the strain ellipse

par Boulanger, Philippe ;Hayes, Michael
Référence Mathematics and mechanics of solids, 16, 7, page (739-752)
Publication Publié, 2011-09

Article révisé par les pairs

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Résumé :

In a finite deformation x = x(X), a particle initially at X is displaced to x. Fundamental to the description of strain are the two Cauchy-Green strain tensors B = FF T and C = F TF, where F = θx/θX is the deformation gradient. Both are real, symmetric and positive definite, so that an ellipsoid at X may be associated with C and an ellipsoid at x with B -1. The purpose of this note is to consider the strain of infinitesimal material line elements in a typical plane π at X and dually in a plane φ at x. It is shown that if an unsheared pair of material line elements in π at X is known together with the stretches along the arms of the unsheared pair, then all the features of the strain ellipse at X - that is the elliptical section of the C-ellipsoid by the plane π - may be determined. Dually, all the features of the strain ellipse at x - that is the section of the B -1-ellipsoid by the plane φ - may be determined. The features of the strain ellipses are determined analytically and also geometrically through use of a ruler and compass. ©The Author(s) 2011.