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with origin O is ¬xed in space and if in the deformed state the symmetry with
respect to the y0 -axis is maintained. For the given problem an exact analytical
solution can be calculated. It is easy to verify that the solution has the following
form:
1 y0
ux ( x0 , y0 ) = ±+β x0
E b
y2 2
x0
1 0
uy ( x0 , y0 ) = ’ ν±y0 + νβ +β
E 2b 2b

1 y0
µxx ( x0 , y0 ) = ±+β
E b
ν y0
µyy ( x0 , y0 ) = ’ ±+β
E b
µxy ( x0 , y0 ) = 0
y0
σxx ( x0 , y0 ) = ± + β
b
σyy ( x0 , y0 ) = 0
σxy ( x0 , y0 ) = 0.
It has to be considered as an exception, when for a speci¬ed plane stress problem
an analytical solution exists. In general, only approximate solutions can be deter-
mined. A technique to do this is the Finite Element Method, which is the subject
of the Chapters 14 to 18. Chapter 18 is especially devoted to the solution of linear
elasticity problems as described in the present chapter.


13.2.6 Boundary conditions

In the previous sections only simply formulated boundary conditions have been
considered. In the case of dynamic boundary conditions, the components of the
stress vector p (see Fig. 8.1) are prescribed at a point along the boundary of
the volume of the continuum (in case of plane stress along the boundary of the
con¬guration surface). In case of kinematic boundary conditions the displace-
ment vector u is prescribed. Sometimes the boundary conditions are less explicitly
de¬ned, however, for example when the considered continuum interacts with its
environment. In the following an example of such a situation will be outlined.
219 13.2 Solution strategies for deforming solids


s



punch
y0
x0



Figure 13.4
Rigid indenter impressing a deforming continuum.



Consider a plane stress continuum of which the midplane (coinciding with the
x0 y0 -plane) has a rectangular shape in the reference con¬guration. The current
state arises because the top edge is indented by means of a rigid punch. The
punch displacement is speci¬ed by s (Fig. 13.4 shows the deformation process,
the displacements are magni¬ed). At the location of the contact between inden-
ter and continuum, the interaction is described with a friction model according to
Coulomb (which can be considered to be a constitutive description for the contact
interaction). For a material point at the top contour of the continuum the following
three distinguishable situations may arise:
(i) There is no contact between the point of the continuum and the indenter. In this case
the boundary conditions are

σyy = 0, σxy = 0, (13.22)

with as an extra constraint that in the current state the vector x0 + u does not cross
the edge of the indenter.
(ii) There is contact between the point of the continuum and the indenter, and that with
˜stick™ boundary conditions (no relative tangential displacement between continuum
and indenter):

ux = 0, uy = ’s, (13.23)

with as extra constraints σyy ¤ 0 and |σxy | ¤ ’μσyy with μ the friction coef¬cient.
(iii) There is contact between the continuum and the indenter, and that with ˜slip™
boundary conditions:
ux
uy = ’s, σxy = μ σyy , (13.24)
|ux |
with as additional constraint σyy ¤ 0.
The principal problem in accounting for the interaction between the continuum
and the indenter is that it is not a priori known to which of the three categories
described above the points of the top layer of the continuum belong. In general an
Solution strategies for solid and fluid mechanics problems
220

estimation is made for this that is updated in case the constraints are violated. In
this way an iterative solution can eventually be determined.




13.3 Solution strategies for viscous fluids

Consider a ¬xed volume V in three-dimensional space, through which (or within
which) a certain amount of material ¬‚ows, while this material can be considered
as an incompressible viscous ¬‚uid (see Section 12.5). Because, for such a ¬‚uid a
(possibly de¬ned) reference state is not of interest at all, an Eulerian description
is used for relevant ¬elds within the volume V.
The velocity ¬eld has to ful¬l the incompressibility constraint at each point in
time. The current velocity ¬eld fully determines the deviatoric part of the stress
state (via the constitutive modelling). The hydrostatic part of the stress ¬eld cannot
be determined on the basis of the velocity ¬eld. The stress ¬eld (the combination
of the hydrostatic and deviatoric part) has to satisfy the momentum balance equa-
tion (see Section 11.3). The problem de¬nition is completed by means of initial
conditions and boundary conditions. The initial velocity ¬eld has to be described
in consistency with the incompressibility constraint and along the boundary of V
for every point in time velocities and/or stresses have to be in agreement with real-
ity. It should be emphasized explicitly, that considering a ¬xed volume in space
implies a serious limitation for the prospects to apply the theory.
The goal of the present section is to outline a routine to formally determine the
velocity ¬eld and the (hydrostatic) stress ¬eld, both as a function of time, such
that all the above mentioned equations are satis¬ed. However, for (almost all)
realistic problems it is not possible to derive an exact analytical solution, not even
via assumptions that simplify the mathematical description drastically. A global
description will be given of strategies to derive approximate solutions.
In Section 13.3.1 the general (complete) formulation of the problem will be
given, including the relevant equations. Thereupon, in the following sections the
complexity of the formulation will be gradually reduced. For this, ¬rstly in Section
13.3.2 the material will be modelled as a Newtonian ¬‚uid (see Section 12.6). This
leads to the so-called Navier“Stokes equation (an equation with the pressure ¬eld
and the velocity ¬eld as unknowns) that has to be solved in combination with the
continuity equation (mass balance). In section 13.3.3 the limitation for a stationary
¬‚ow is dealt with (the time dependency, including the need for initial conditions
is no longer relevant). After a section on boundary conditions, a few elementary
analytical solutions of the equations for a stationary viscous ¬‚ow are presented in
Section 13.3.5.
221 13.3 Solution strategies for viscous fluids

13.3.1 General equations for viscous flow

Consider the ¬‚ow of an incompressible ¬‚uid through (or within) a spatially
¬xed, time-independent volume V. For general viscous ¬‚ow problems it can be
stated that, in an Eulerian description, the following physical ¬elds have to be
determined:
• the velocity ¬eld: v( x, t) for all x in V and all t and
• the stress ¬eld: σ ( x, t) for all x in V and all t.
Starting from the incompressibility condition the velocity ¬eld has to satisfy the
continuity equation (mass balance) for all x in V for all t:
1 T
tr( D) = 0 with D= ∇v + ∇v , (13.25)
2
while next to that the velocity ¬eld and the stress ¬eld should be related for all x in
V and for all times t according to the local constitutive equation (see Section 12.5):

σ = ’p I + σ d ( D) . (13.26)

Also the local balance of momentum (see Section 11.5 for the Eulerian descrip-
tion) has to be satis¬ed; so for all x in V and for all t:
δv
T
∇ · σ + ρq = ρ ∇v ·v+ , (13.27)
δt
with the (mass) density ρ constant.
The equations above form a set of coupled partial differential equations. Conse-
quently, for a unique solution of the velocity ¬eld v( x, t) and the stress ¬eld σ ( x, t)
boundary conditions and initial conditions are indispensable.
With respect to boundary conditions it can be stated that for each t at every
point on the outer surface of V three (scalar) relations have to be speci¬ed: either
completely formulated in stresses (dynamic boundary conditions), or completely
expressed in velocities (kinematic boundary conditions) or in a mixed format.
A detailed description of the interpretation of the initial conditions is not
considered.




13.3.2 The equations for a Newtonian fluid

For a Newtonian ¬‚uid, see Section 12.5, the stress tensor can be written as

σ = ’pI + 2·D, (13.28)
Solution strategies for solid and fluid mechanics problems
222

where the viscosity · is constant. Substitution of this constitutive equation into
the local momentum balance leads to the equation:
δv
T T
’ ∇p + ·∇ · ∇v + ∇v + ρq = ρ ∇v ·v+ . (13.29)
δt
The left-hand side of this equation can be simpli¬ed by using the following
identities (to be derived by elaboration in components):
T
∇ · ∇v =∇ ∇ ·v (13.30)

∇ · ∇v = ∇ · ∇ v. (13.31)
This leads to:
δv
T
·v+
’ ∇p + · ∇ ∇ · v + ∇ · ∇ v + ρq = ρ ∇v . (13.32)
δt
Using the expression for incompressibility of the ¬‚uid:
∇ · v = 0, (13.33)
results in the so-called Navier“Stokes equation:
δv
T
∇v
’ ∇p + · ∇ · ∇ v + ρq = ρ ·v+ . (13.34)
δt
The last two relations, the incompressibility condition (continuity equation)
and the Navier“Stokes equation, together form a set that allows the deter-
mination of the velocity ¬eld v( x, t) and the pressure ¬eld p( x, t). For the
solution boundary conditions and initial conditions have to be supplied to the
equations.



13.3.3 Stationary flow of an incompressible Newtonian fluid

For a stationary ¬‚ow the relevant ¬eld variables are only a function of the position
vector x within the volume V and no longer a function of time. To determine the
velocity ¬eld v( x) and the pressure ¬eld p( x) the set of equations that has to be
solved is reduced to:
∇ ·v=0 (13.35)

T
’ ∇p + · ∇ · ∇ v + ρq = ρ ∇v · v. (13.36)
In addition, it is necessary to specify a full set of boundary conditions. Initial con-
ditions do not apply for stationary problems. Note that the equation is non-linear
223 13.3 Solution strategies for viscous fluids

as a consequence of the term on the right-hand side of the last equation; this has
a seriously complicating effect on the solution process. Exact analytical solutions
can only be found for very simple problems.



13.3.4 Boundary conditions

In the present section the attention is focussed on the formulations of simple
boundary conditions with respect to an arbitrary point on the outer surface of
the considered (¬xed) volume V, with local outward unit normal n. A number of
different possibilities will separately be reviewed.
• Locally prescribed velocity v along the boundary, i.e. the component v · n in normal
direction as well as the component v’( v · n) n in tangential direction. A well-known
example of this, is the set of boundary conditions for ˜no slip™ contact of a ¬‚uid with a
¬xed wall: v = 0. In fact, the impermeability of the wall (there is no ¬‚ux through the
outer surface, also see Fig. 7.9) is expressed by v · n = 0, while suppressing of slip is
expressed by v’( v · n) n = 0.
• A locally prescribed stress vector σ · n along the boundary, i.e. the component n · σ · n
in normal direction as well as the components of σ · n’( n · σ · n) n in tangential direc-
tion. Boundary conditions of this type can be transformed into boundary conditions
expressed in v and p by means of the constitutive equations. A known example of
this is the set of boundary conditions at a free surface: the normal component of the
stress vector is related to the atmospheric pressure (equal with opposite sign) and the
tangential components are equal to zero.
• For a frictionless ¬‚ow along a ¬xed wall it should be required that v · n = 0 combined
with σ · n’( n · σ · n) n = 0. Again the last condition can, by using the constitutive
equation, be expressed in v and p.



13.3.5 Elementary analytical solutions

Figure 13.5 visualizes a stationary ¬‚ow of a ¬‚uid between two ˜in¬nitely
extended™ stationary parallel ¬‚at plates (mutual distance h). The ¬‚ow in the pos-
itive x-direction is activated by means of an externally applied pressure gradient.
We consider that part between the plates (the speci¬c domain with x- and z-
coordinates) where the ¬‚ow is fully developed. This means that no in¬‚uence is

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