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