Structural Geology
A Second Derivation of the Displacement Gradient
(another look at strain)
Three examples of deformation mapping were given in the previous lecture. The
deformation can be represented in two manners. The displacement vector u was mapped in one
case leading to a displacement gradient. In the other case the new position of each particle was
mapped based on the initial position of the particle leading to a deformation gradient. In this
lecture we will show that in 2 and 3 dimensions deformation consists of rotations as well as
stretches.
We start by taking another look at strain with some simple definitions such as a change in
length of line per unit length of line.
ε = ∆l/l
This is equivalent to a stretch. A formal definition of shear strain (γ) is the change in angle (ψ)
between two initially perpendicular lines (Fig. 6-1).
γ = tan ψ
(Fig. 6-1)
A second measure of shear strain is the tensor shear strain which is half the tangent of the change
in angle between initially perpendicular lines.
Tensor shear strain = γ/2.
γ is sometimes called the engineering shear strain. Note here that shear strain is represented by
line rotations. This gives the first indication that the strain can be separated into a rotational and
irrotational component. We will deal in more detail with these concepts in the next lecture.
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