58
CHAPTER 5. TWODIMENSIONAL ELASTICITY
where the last condition is concluded from the fact 33 = 0. This is inserted to Hookes
law (4.19) and taken to express 33 . The tensors look
11 12 0
= 12 22 0
(5.16)
0
0 33
11 12 0
= 12 22 0 .
(5.17)
0
00
The zer
A.1. CHAPTER 1
83
(b)
2
f = f,ii = (f,i ),i =
f (r) xi
r
,i
(f (r) xi ),i r f (r) xi r,i
r2
f (r),i xi r + f (r) xi,i r f (r) xi r,i
=
r2
r
r
f (r) xi xi r + f (r) xi,i r f (r) xi xi
=
r2
xi
f (r) r xi r + f (r) xi,i r f (r) xi xi
r
=
r2
1
1
= f (r) + 3f
78
CHAPTER 6. ENERGY PRINCIPLES
1D:
Strain energy:
1
1
E
E
E
uk,l Cklij ui,j =
xx xx = (u,x )2 = (z (x)2 = z 2 (w (x)2
2
2
2
2
2
with dV = A dx = b dz dx:
h
2
1
uk,l Cklij ui,j dV
2
V
=
z = h
2
Eb
2
=
(w (x)2 dx =
z 2 dz
h
2
EIy
2
=
x=0
h
2
E2
z (w (x)2
6.3. APPROXIMATIVE SOLUTIONS
73
with the element-length le . So, the approximation in e is
e
w(e) (x) = w1 N1
x xe
+w1e N2
le
x xe
e
+w2 N3
le
x xe
+w2e N4
le
x xe
le
(6.72)
The test functions Ni are
N1 ( ) = 1 3 2 + 2 3
(6.73)
N2 ( ) = le ( 2 2 + 3 )
(6.
68
CHAPTER 6. ENERGY PRINCIPLES
also the surface A of the elastic body will not change due to the virtual displacement,
and, also, that the prescribed boundary traction ti as well as the body forces f i will not
change, the variation, i.e., the sign , can
6 Energy principles
Energy principles are another representation of the equilibrium and boundary conditions
of a continuum. They are mostly used for developing numerical methods as, e.g., the
FEM.
6.1
Work theorem
Starting from the strain energy density o
3.8. EXERCISE
43
2. A continuum body undergoes the displacement
3x2 4x3
u = 2x1 x3 .
4x2 x1
Determine the displaced position of the vector joining particles A(1, 0, 3) and
B (3, 6, 6).
3. A displacement eld is given by u1 = 3x1 x2 , u2 = 2x3 x1 and u3 = x
38
CHAPTER 3. DEFORMATION
are equal to onehalf of the familiar engineering shear strains ij . However, only with
the denitions above the strain tensor
11 12 13
= 12 22 23
(3.33)
13 23 33
has tensor properties. By the denition of the strains the symmetry
3 Deformation
3.1
Position vector and displacement vector
Consider the undeformed and the deformed conguration of an elastic body at time t = 0
and t = t, respectively (g. 3.1).
deformed
undeformed
x3 X 3
P (x)
p(X)
u
x2 X 2
x
X
t=0
t=t
x1 X 1
Figure 3.1:
28
CHAPTER 2. TRACTION, STRESS AND EQUILIBRIUM
In indical notation (I = ij : itendity-matrix (3x3):
kk
sij = ij ij
3
(2.60)
where kk /3 are the components of the hydrostatic stress tensor and sij the components
of the deviatoric stress tensor.
The princip
2.2. EQUILIBRIUM
23
or
ui dV
(fi + ji,j )dV =
(2.25)
V
V
using the divergence theorem (1.56). The above equation must be valid for every element
in V , i.e., the dynamic equilibrium is fullled. Consequently, because V is arbitrary the
integrand vanishes.
18
CHAPTER 2. TRACTION, STRESS AND EQUILIBRIUM
At any element An in or on the body (n indicates the orientation of this area) a resultant
force Fn and/or moment Mn produces stress.
Fn
dFn
=
= tn
An 0 An
dAn
lim
Mn
dM n
=
= Cn
An 0 An
dAn
lim
stress vector
1.8. SUMMARY OF CHAPTER 1
13
Addition of two matrices:
A + B = B + A = (Aik ) + (Bik ) = (Aik + Bik )
e.g.:
A11 A12
A21 A22
B11 B12
B21 B22
+
A11 + B11 A12 + B12
A21 + B21 A22 + B22
=
Rules for addition of matrices:
(A + B) + C = A + (B + C) = A + B + C
M
1 Introduction and mathematical
preliminaries
1.1
Vectors and matrices
A vector is a directed line segment. In a cartesian coordinate system it looks like
depicted in gure 1.1,
x3
z
az
a3
P
P
a
a
y
ez
ay
x2
e3
a2
ey
e2
ex
e1
ax
a1
x
x1
Figure 1.1: Vector
108
A.6
APPENDIX A. SOLUTIONS
Chapter 6
1-D-Beam:
xx = E xx = E u,x
u = z tan z (x)
z
xx = Ez (x)
u
xx = Ezw (x)
tan
z 2 dA
zxx dA = Ew (x)
My =
A
A
:=Iy (moment of inertia)
My = EIy w (x)
Principle of minimum total potential energy:
l
(w)beam
EIy
=
2