Page 232 - Intro to Tensor Calculus

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Figure 2.3-14. Displacement field due to state of strain

The quantity

∂u ∂v
α + β = + =2e 12 =2e 21 (2.3.50)
∂y ∂x
1
is the change from a ninety degree angle due to the deformation and hence we can write (α+β)= e 12 = e 21
2
as representing a change from a 45 angle due to the deformation. The quantities e 21 ,e 12 are called the
◦
shear strains and the quantity
(2.3.51)
γ 12 =2e 12
is called the shear angle.
In the equation (2.3.45), the quantities ω 21 = −ω 12 are the elements of the rigid body rotation matrix
and are interpreted as angles associated with a rotation. The situation is analogous to the transformations
and figures for the deformation of the unit square which was considered earlier.

Strain in Three Dimensions

The development of strain in three dimensions is approached from two different viewpoints. The first
approach considers the derivation using Cartesian tensors and the second approach considers the derivation
of strain using generalized tensors.

Cartesian Tensor Derivation of Strain.

Consider a material which is subjected to external forces such that all the points in the material undergo
a deformation. Let (y 1 ,y 2 ,y 3 ) denote a set of orthogonal Cartesian coordinates, fixed in space, which is
used to describe the deformations within the material. Further, let u i = u i (y 1 ,y 2 ,y 3),i =1, 2, 3denote a
displacement field which describes the displacement of each point within the material. With reference to the
figure 2.3-14 let P and Q denote two neighboring points within the material while it is in an unstrained state.
These points move to the points P and Q when the material is in a state of strain. We let y i ,i =1, 2, 3
0
0
represent the position vector to the general point P in the material, which is in an unstrained state, and
denote by y i + u i ,i =1, 2, 3 the position vector of the point P when the material is in a state of strain.
0