Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 280

The word deformation suggests that subdomains with positive volume are mapped
into subdomains with positive volume. Deformations are injective mappings lo-
cally.
The mapping φ induces

φ(x + z) − φ(x) =∇ φ(x) · z + o(z).

In terms of the Euclidean distance,

2 2 2 )

φ(x + z) − φ(x) =∇ φ · z + o( z

( 1 . 3 )

= z ∇ φ T

2 ).

∇ φz + o( z

Thus, the matrix

=∇ φ T C : ∇ φ ( 1 . 4 )

describes the transformation of the length element. It is called the (right) Cauchy-
Green strain tensor. The deviation

1
2 (C − I)

E : =

from the identity is called the strain , and is one of the most important concepts
in the theory. Frequently, we will work with matrix representations of C and E .
These matrices are obviously symmetric. Inserting (1.1) into (1.4) gives

∂u i

2
k

1
2

∂u j
∂x i

1 ∂u k

∂x i

∂u k

E ij =

∂x j +

+

∂x j .

( 1 . 5 )

In the linear theory we neglect the quadratic terms, leading to the following sym-
metric gradient as an approximation:

∂u i
∂x j +

.

∂u j
∂x i

1
2

ε ij : = ( 1 . 6 )

1.1 Remark. Let be connected. If the strain tensor associated with the defor-
mation φ ∈ C 1 () satisfies the relation

C(x) = I for all x ∈ ,

then φ describes a rigid body motion, i.e., φ(x) = Qx + b , where Q is an orthogonal
matrix.

Sketch of a proof . Let be a smooth curve in . In view of (1.3) and C(x) = I , the
rectifiable curves and φ() always have the same length. This follows directly