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
T
F, where F = ∂
x
/∂
X 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 π 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 π — may be determined.
The features of the strain ellipses are determined analytically and also geometrically through use of a ruler and compass.
