Finite Elements in Solid Mechanics - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 281

from the definition of the arc length via an integral. We now use it to establish the
desired result.
Since φ is locally injective, if is open, then φ() is also open. For every
x 0 ∈ , there exists a convex neighborhood U in such that the convex hull of
φ(U) is contained in φ() . The mapping φ | U is globally distance preserving, i.e.
for all pairs x,y ∈ U ,

φ(x) − φ(y) = x − y . ( 1 . 7 )

To see this, let be the line connecting the points x and y . Since φ() has the
same length,

. The equality now follows by examining
the preimage of the line connecting φ(x) and φ(y) .
Because of (1.7), the auxiliary function

φ(x) − φ(y) ≤ x − y

2 2

G(x, y) : = φ(y) − φ(x) − y − x

1
2

∂G
∂y i

vanishes on U × U . G is differentiable with respect to y , and satisfies

∂φ k
∂y i (φ k (y) − φ k (x)) − (y i − x i ) =

0 .

k

This expression is differentiable with respect to x j , and so

∂φ k
∂y i

∂φ k

−

∂x j + δ ij =

0 ,

k

φ(y) T which is just the componentwise version of ∇ ∇ φ(x) = I. Multiplying on

the left by ∇ φ(y) and using C = I , we immediately get ∇ φ(x) =∇ φ(y). Thus,

∇ φ is constant on U , and φ is a linear transformation.
Now the result follows for the entire domain by a covering argument.

The Equilibrium Equations

In mechanics we treat the influence of forces axiomatically. Euler and Cauchy
both made essential contributions. For details, see Ciarlet [1988].
We assume that the interaction of the body with the outside world is described
by two types of applied forces:
(a) applied surface forces (forces distributed over the surface),
(b) applied body forces (forces distributed over the volume).
A typical body force is the force of gravity, while the force caused by a load on a
bridge (e.g., a vehicle) is a surface force.