Quasilinear parabolic problems with nonlinear boundary conditions

where f represents external forces acting on the body, like gravity or electromagnetic
forces. In components (5.2) reads
ρ0(x)üi(t, x) =
3�
j=1
∂xj Sij(t, x) + ρ0(x)fi(t, x), i = 1, 2, 3.
We have to supplement (5.2) by boundary conditions. Possible boundary conditions are
either ’prescribed displacement’ or ’prescribed normal stress’. Let ∂Ω = Γd ∪ Γs with
◦
Γd= Γd, ◦
Γs= Γs and ◦
Γd ∩ ◦
Γs= ∅. The boundary conditions then read as follows.
u(t, x) = ud(t, x), t ∈ R, x ∈ ◦
Γd, (5.3)
S(t, x)n(x) = gs(t, x), t ∈ R, x ∈ ◦
Γs, (5.4)
where n(x) denotes the outer normal of Ω at x ∈ Ω.
A material is called incompressible, if there are no changes of volume in the body Ω
during a deformation, i.e. if
det (I + ∇u(t, x)) = 1, t ∈ R, x ∈ Ω, (5.5)
is fulfilled; otherwise the material is called compressible. For the linear theory, the
nonlinear constraint (5.5) can be simplified to the linear condition
div u(t, x) = 0, t ∈ R, x ∈ Ω. (5.6)
In the sequel, we shall consider compressible materials.
We still have to describe how the stress S(t, x) depends on the strain E. This is done
by a constitutive law or a stress-strain relation. Such an equation completes the system
inasmuch as it relates the stress S(t, x) to the unknown u and its derivatives. If the
material is purely elastic, then the stress S(t, x) will depend (linearly) only on the strain
E(t, x). However, the stress may also depend on the history of the strain and its time
derivative; in this case the material is called viscoelastic. The general constitutive law
for compressible materials is given by
� ∞
S(t, x) =
0
dA(τ, x) ˙
E(t − τ, x) dτ, t ∈ R, x ∈ Ω, (5.7)
where A : R+ ×Ω → B(Sym{3}) is locally of bounded variation w.r.t. t ≥ 0. The symbol
Sym{n} denotes the space of n-dimensional real symmetric matrices. As a consequence
of this, the symmetry relations
Aijkl(t, x) = Ajikl(t, x) = Aijlk(t, x), t ∈ R+, x ∈ Ω, (5.8)
have to be satisfied for all i, j, k, l ∈ {1, 2, 3}. The function A is called the relaxation
function of the material. Its component functions Aijkl, the so-called stress relaxation
moduli, have to be determined in experiments.
In the following we want to consider the case where the material is isotropic, which
by definition means that the constitutive law is invariant under the group of rotations.
It can be shown that the general isotropic stress relaxation tensor takes the form
Aijkl(t, x) = 1
3 (3b(t, x) − 2a(t, x))δijδkl + a(t, x)(δikδjl + δilδjk), (5.9)
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