1
MT30271: EXAMPLE SHEET1 III
1.) The strains eij (r) are given throughout a body. Show that the corresponding displacement eld ui (r) is only determined to within a rigid-body displacement. Use
the following two steps: (i) Show that if two displacement e

Chapter 6
Plane strain problems
6.1
Basic equations
Denition: A deformation is said to be one of plane strain (parallel to the plane x 3 = 0) if:
u3 = 0 and
u = u (x ).
(6.1)
There are only two independent variables, (x1 , x2 ) = (x, y).
Plane strain is

Stress distribution in plate with a circular hole (From:
Eschenauer, H. & Schnell. W. Elastizitaetstheorie I,
BI Wissenschaftsverlag, 1986)
<> ; 0 =T
Note that the hoop stress at the hole is three times larger
than the stress applied at infinity. In engi

Chapter 4
Elasticity & constitutive equations
4.1
The constitutive equations
The constitutive equations determine the stress ij in the body as function of the bodys deformation.
Denition: A solid body is called elastic if
ij (xn , t) = ij (ekl (xn , t).

eij
=
ij
=
1
2
1
2
uj
ui
+
xj
xi
ui
uj
xj
xi
3
= ji
= eji
is the strain tensor and
(2.5)
is the rotation tensor.
(2.6)
where
ui
= eij + ij ,
xj
ui
xj
(2.4)
is the displacement gradient tensor:
ui
xj
1.
(2.3)
We will restrict ourselves to a linearised an

7
2 ui
ij
+ Fi = 2 ,
xj
t
(3.4)
Including inertial eects via DAlembert forces gives the equations of motion:
ij
+ Fi = 0.
xj
(3.3)
The equations of equilibrium for a body, subject to a body force (force per unit volume) F i is
3.3
The equations of equi

1
MT30271: EXAMPLE SHEET1 IV
1.) A cylinder which occupies the region x2 + x2 R2 and L x3 0 is in static
1
2
equilibrium. The stress tensor in the body is given by 11 = 12 = 22 = 0 and
13 = ax2 , 23 = ax1 , 33 = bx3 , where a and b are known (small) const

1
MT30271: EXAMPLE SHEET1 I
1.) Which one of these equations in index notation are valid? Remember the summation
convention!
a) c = ai bi
b) c = aij bi
c) ci = aij bi
d) ci = aij bj
e) ci = aji bj
f ) ij = ij T + Eijkl ekl
g) ij = kl Ti + Eijkl eij
h) kij

1
MT30271: EXAMPLE SHEET1 V
1.) Show that in a homogeneous, isotropic, linearly elastic body the principal axes of the
stress and strain tensors coincide. Derive the relation between principal strains and
principal stresses.
2.) It was shown in the lectur

1
MT30271: EXAMPLE SHEET1 II
1.) A 2D body occupying the region cfw_d : 0 x1 1, 0 x2 1 is displaced by the
following displacement eld (see problem 3 on the last sheet) u1 = (x1 + 2x2 ); u2 =
(3 + x2 ) where
1.
a) Compute the extension en1 of a line elemen