Placements and deformations
In distinguishing the kinds of physical quantities, it is of great importance to know
how they are related to the directions of those coordinate axes which we usually em-
ploy in dening the positions of things. The introduction of coordinate axes into ge-
ometry by Des Cartes was one of the greatest steps in mathematical progress, for it
reduced the methods of geometry to calculations performed on numerical quantities.
[...]
But for many purpose of physical reasoning, as distinguished from calculation, it is
desirable to avoid explicitly introducing the Cartesian coordinates, and to x the mind
at once on a point of space instead of its three coordinates, and on the magnitude
and direction of a force instead of its three components. This mode of contemplating
geometrical and physical quantities is more primitive and more natural than the other
[...].
[Maxwell J. C., A Treatise on Electricity and Magnetism, Oxford University Press, 1892,
vol. I, p. 9]
Contents
1 Placement and deformation 2
2 Rigid deformations and afne deformations 2
3 Composition of deformations 3
4 Afne deformations in coordinate form 4
5 Parameterizations 4
6 Deformation gradient 5
7 Displacement gradient 7
8 Local deformation 8
8.1 Stretch and shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 PLACEMENTS AND DEFORMATIONS
1 Placement and deformation
Denoting by B a set {A, B, . . . } made up of body points, a placement is a map
p : B E (1)
which assigns a position to each point, in such a way that different body points (or simply points)
take different positions. The shape of the body B is the set
R := imp. (2)
We call conguration of B the collection of the couples
A, p(A)
A B , (3)
where p is a placement. For any two placements p and p, corresponding to shapes
R and R there
exists a bijective map, called deformation,
:
R R, (4)
dened as := p ( p)
1
, which moves every point A B from position p(A) to position
p(A) = ( p(A)). (5)
We can dene also the displacement eld,
u : p(A) (p(A) p(A)) A B. (6)
2 Rigid deformations and afne deformations
If the deformation is an isometry which leaves the orientation unchanged we call it a rigid
deformation. In a rigid deformation, however we choose a body point A the position of any other
body point B in the placement p is given by the following expression
( p
B
) = ( p
A
) + R( p
B
p
A
), (7)
where R is a rotation of V.
If is an afne map which leaves the orientation unchanged then we call it an afne deformation
or a homogenous deformation. However we choose a body point A the position of any other body
point B in the placement p is given by the following expression
( p
B
) = ( p
A
) + F( p
B
p
A
), (8)
where F is an endomorphism of V, such that det F > 0. Note that the line
c(h) = p
A
+ h a (9)
is transformed, through (8), into the curve
c(h) := ( c(h)) = ( p
A
) + F( c(h) p
A
) = ( p
A
) + h F a. (10)
This curve is nothing but the line
c(h) = ( p
A
) + h a (11)
whose tangent vector is
a = F a. (12)
Hence an afne deformation transforms lines into lines.
DISAT, University of LAquila, April 23, 2011 (1432) A. Tatone Mechanics of Solids and Materials.
PLACEMENTS AND DEFORMATIONS 3
3 Composition of deformations
Let us consider the following composition of two afne deformations
:=
[2]
[1]
. (13)
For any two body points A and B, the rst deformation is such that
[1]
( p
B
) =
[1]
( p
A
) + F
[1]
( p
B
p
A
). (14)
Applying the second deformation we get
[2]
[1]
( p
B
)
=
[2]
[1]
( p
A
)
+ F
[2]
(
[1]
( p
B
)
[1]
( p
A
)) (15)
and, by substituting (14),
[2]
[1]
( p
B
)
=
[2]
[1]
( p
A
)
+ F
[2]
F
[1]
( p
B
p
A
)
. (16)
Hence for the composition (13) is described by
( p
B
) = ( p
A
) + F( p
B
p
A
) (17)
where
F = F
[2]
F
[1]
. (18)
Note that in general neither composition (13) nor composition (18) are commutative.
By using the polar decomposition
F = RU, (19)
every afne deformation can be expressed, after choosing any body point A, as
( p
B
) = ( p
A
) + RU( p
B
p
A
) (20)
and then decomposed into a translation
[0]
( p
B
) =
[0]
( p
A
) + ( p
B
p
A
), (21)
such that
[0]
( p
A
) = ( p
A
), followed by a stretch, while holding ( p
A
) xed,
[1]
(
[0]
( p
B
)) =
[1]
(
[0]
( p
A
)) + U(
[0]
( p
B
)
[0]
( p
A
)) = ( p
A
) + U( p
B
p
A
), (22)
followed in turn by a rotation, with center ( p
A
),
[2]
(
[1]
(
[0]
( p
B
))) = ( p
A
) + R(
[1]
(
[0]
( p
B
)) ( p
A
)). (23)
The orthogonal lines through ( p
A
) generated by the eigenvectors of U, called the principal di-
rections of the stretch, are invariant under the deformation
[1]
. The distance between any two
positions along the principal directions changes by a factor equal to the corresponding eigen-
value of U, called the principal stretches. The line through ( p
A
) generated by the eigenvector of
R, called rotation axis, is invariant under the deformation
[2]
.
DISAT, University of LAquila, April 23, 2011 (1432) A. Tatone Mechanics of Solids and Materials.
4 PLACEMENTS AND DEFORMATIONS
4 Afne deformations in coordinate form
Given a Cartesian coordinate system in a Euclidean space E of dimension two, the positions of
any two body points A, B B in the placement p can be described by the expressions
p
A
= o + x
1A
e
1
+ x
2A
e
2
,
p
B
= o + x
1B
e
1
+ x
2B
e
2
.
(24)
The positions of the same two body points in the placement p can be described by the expressions
( p
A
) = o + x
1A
e
1
+ x
2A
e
2
,
( p
B
) = o + x
1B
e
1
+ x
2B
e
2
.
(25)
For a rigid deformation, by substituting (24) and (25) into (7), we get
o + x
1B
e
1
+ x
2B
e
2
= o + x
1A
e
1
+ x
2A
e
2
+ R
( x
1B
x
1A
)e
1
+ ( x
2B
x
2A
)e
2
, (26)
which implies
x
1B
e
1
+ x
2B
e
2
= x
1A
e
1
+ x
2A
e
2
+ ( x
1B
x
1A
)Re
1
+ ( x
2B
x
2A
)Re
2
. (27)
We can parameterize the rotation tensor through an angle in the following way
Re
1
= cos e
1
+ sin e
2
, (28)
Re
2
= sin e
1
+ cos e
2
. (29)
Then, by equating components in the vector equation (27) we obtain the coordinate description
of the rigid deformation
x
1B
x
2B
x
1A
x
2A
cos sin
sin cos
x
1B
x
1A
x
2B
x
2A
. (30)
For an afne deformation, by substituting (24) and (25) into (8) and setting
Fe
1
= f
11
e
1
+ f
21
e
2
, (31)
Fe
2
= f
12
e
1
+ f
22
e
2
, (32)
we obtain the following coordinate description
x
1B
x
2B
x
1A
x
2A
f
11
f
12
f
21
f
22
x
1B
x
1A
x
2B
x
2A
. (33)
5 Parameterizations
Given a Cartesian coordinate system in a Euclidean space E of dimension two, the position of a
body point A in the placement p is described by (24). If we set s
1A
:= x
1A
, s
2A
:= x
2A
, then (24)
becomes
p
A
= o + s
1A
e
1
+ s
2A
e
2
. (34)
Hence, for each position there exists a couple of coordinates
p
A
(s
1A
, s
2A
) (35)
DISAT, University of LAquila, April 23, 2011 (1432) A. Tatone Mechanics of Solids and Materials.
PLACEMENTS AND DEFORMATIONS 5
and viceversa any couple of coordinates denes a position
(s
1A
, s
2A
) p
A
. (36)
Denoting by such a function, called parameterization of the body shape
R := im p, we get
p
A
= (s
1A
, s
2A
). (37)
In general the parameterization of a body shape is independent of the coordinate system. If the
dimension of the Euclidean space is 2, we assume that the domain of the parameterization is the
closure of an open set of R
2
. Since the deformation gradient F is never singular the dimension of
the shape does not change.
A coordinate description of a deformation can be given through two scalar functions
1
and
2
such that
((s
1
, s
2
)) = o +
1
(s
1
, s
2
)e
1
+
2
(s
1
, s
2
)e
2
. (38)
If the dimension of the Euclidean space is 3, we assume that the domain of the parameterization
is the closure of an open set of R
3
. In this case we need three scalar functions
1
,
2
and
3
in
order to give a coordinate description of the deformation.
6 Deformation gradient
Let us consider a placement p and the line through p
O
= (s
1
, s
2
)
c
1
(h) := p
O
+ h e
1
. (39)
By using a coordinate system as in (34) and the parameterization (37) we obtain the following
description
c
1
(h) := p
O
+ h e
1
= (s
1
, s
2
) + h e
1
= o + (s
1
+ h)e
1
+ s
2
e
2
= (s
1
+ h, s
2
). (40)
This line is transformed by into the curve
c
1
(h) := ( c
1
(h)) = ((s
1
+ h, s
2
)). (41)
If the deformation is not afne then this curve is not a straight line in general. By using the
coordinate expression (38) for , we get the following coordinate expression for the curve (41)
c
1
(h) = o +
1
(s
1
+ h, s
2
)e
1
+
2
(s
1
+ h, s
2
)e
2
, (42)
c
1
(0) = o +
1
(s
1
, s
2
)e
1
+
2
(s
1
, s
2
)e
2
. (43)
Hence the tangent vector at ( p
O
) is given by
c
1
= lim
h0
1
h
(c
1
(h) c
1
(0)) =
1
1
e
1
+
1
2
e
2
, (44)
where
1
is the derivative with respect to the rst argument. The line
c
2
(h) := p
O
+ h e
2
= (s
1
, s
2
) + h e
2
= o + s
1
e
1
+ (s
2
+ h)e
2
= (s
1
, s
2
+ h) (45)
is transformed into the curve
c
2
(h) := ( c
2
(h)) = ((s
1
, s
2
+ h)). (46)
DISAT, University of LAquila, April 23, 2011 (1432) A. Tatone Mechanics of Solids and Materials.
6 PLACEMENTS AND DEFORMATIONS
By using (38) we get
c
2
(h) = o +
1
(s
1
, s
2
+ h)e
1
+
2
(s
1
, s
2
+ h)e
2
, (47)
c
2
(0) = o +
1
(s
1
, s
2
)e
1
+
2
(s
1
, s
2
)e
2
. (48)
hence the tangent vector at ( p
O
) is given by
c
2
= lim
h0
1
h
(c
2
(h) c
2
(0)) =
2
1
e
1
+
2
2
e
2
, (49)
where
2
is the derivative with respect to the second argument. Finally, the line
c(h) := p
O
+ h a = (s
1
, s
2
) + h a = (s
1
, s
2
) + h(
1
e
1
+
2
e
2
)
= o + (s
1
+ h
1
)e
1
+ (s
2
+ h
2
)e
2
= (s
1
+ h
1
, s
2
+ h
2
)
(50)
is transformed into the curve
c(h) := ( c(h)) = ((s
1
+ h
1
, s
2
+ h
2
)). (51)
Again, from (38) we obtain
c(h) = o +
1
(s
1
+ h
1
, s
2
+ h
2
)e
1
+
2
(s
1
+ h
1
, s
2
+ h
2
)e
2
, (52)
c(0) = o +
1
(s
1
, s
2
)e
1
+
2
(s
1
, s
2
)e
2
. (53)
Thus the tangent vector at ( p
O
) turns out to be
c
= lim
h0
1
h
(c(h) c(0)) = (
1
1
1
+
2
2
1
) e
1
+ (
1
1
2
+
2
2
2
) e
2
=
1
(
1
1
e
1
+
1
2
e
2
) +
2
(
2
1
e
1
+
2
2
e
2
) .
(54)
Note that
c
1
= lim
h0
1
h
( c
1
(h) c
1
(0)) = e
1
, (55)
c
2
= lim
h0
1
h
( c
2
(h) c
2
(0)) = e
2
, (56)
c
= lim
h0
1
h
( c(h) c(0)) = a =
1
e
1
+
2
e
2
. (57)
It follows that tangent vectors are such that
c
=
1
c
1
+
2
c
2
, (58)
c
=
1
c
1
+
2
c
2
, (59)
Then there exists a linear map, called the deformation gradient F( p
O
) transforming tangent vectors
to lines at p
O
into tangent vectors to corresponding curves at ( p
O
):
F( p
O
) : c
1
c
1
, (60)
F( p
O
) : c
2
c
2
, (61)
F( p
O
) :
1
c
1
+
2
c
2
1
c
1
+
2
c
2
. (62)
DISAT, University of LAquila, April 23, 2011 (1432) A. Tatone Mechanics of Solids and Materials.
PLACEMENTS AND DEFORMATIONS 7
e
1
e
2
Fe
1
Fe
2
Fe
1
Fe
2
Fe
1
Fe
2
(a)
(b)
(c)
Figure 1: Deformation: (a) rigid, (b) afne, (c) generic.
From the expressions for c
1
and c
2
we obtain
F( p
O
)e
1
=
1
1
e
1
+
1
2
e
2
, (63)
F( p
O
)e
2
=
2
1
e
1
+
2
2
e
2
. (64)
Hence the matrix of F( p
O
) is
[F( p
O
)] =
1
s
1
1
s
2
2
s
1
2
s
2
. (65)
It can be proved that F( p
O
) is independent of the parameterization.
Further it can be proved that if F( p
O
) = F( p
A
) p
O
, i.e. F is a uniform tensor eld, then is
afne.
We assume that every deformation is such that
det F( p
O
) > 0 p
O
. (66)
7 Displacement gradient
From the denition (6) of displacement eld we get
u( p
O
) = ( p
O
) p
O
, (67)
DISAT, University of LAquila, April 23, 2011 (1432) A. Tatone Mechanics of Solids and Materials.
8 PLACEMENTS AND DEFORMATIONS
which when evaluated along the curve (50) through p
O
becomes
u( c(h)) = ( c(h)) c(h) = c(h) c(h), (68)
u( c(0)) = ( c(0)) c(0) = c(0) c(0). (69)
By using the denition of gradient of a vector eld together with the deniton of deformation
gradient, we obtain
u( p
O
) c
= c
= F( p
O
) c
= (F( p
O
) I) c
. (70)
Since this relation holds true for the derivative along any curve, then
u( p
O
) = F( p
O
) I. (71)
A component description of the displacement eld u can be given by dening on the parameter-
ization domain two scalar functions u
1
e u
2
such that
u((s
1
, s
2
)) = u
1
(s
1
, s
2
)e
1
+ u
2
(s
1
, s
2
)e
2
. (72)
Since the displacement eld on curves (40) and (45) is
u( c
1
) = u
1
(s
1
+ h, s
2
)e
1
+ u
2
(s
1
+ h, s
2
)e
2
, (73)
u( c
2
) = u
1
(s
1
, s
2
+ h)e
1
+ u
2
(s
1
, s
2
+ h)e
2
, (74)
from the dention of gradient of a vector eld we get
u( p
O
) c
1
=
1
u
1
e
1
+
1
u
2
e
2
, (75)
u( p
O
) c
2
=
2
u
1
e
1
+
2
u
2
e
2
. (76)
Hence the matrix of u( p
O
) turns out to be
[u( p
O
)] =
u
1
s
1
u
1
s
2
u
2
s
1
u
2
s
2
. (77)
8 Local deformation
Note that if the deformation is afne then the line through p
O
(39) is transformed into the line
c
1
(h) = ( p
O
) + h F( p
O
)e
1
. (78)
If the deformation is not afne then (78) is not true in general and we can dene the difference
o(h) := c
1
(h)
( p
O
) + h F( p
O
)e
1
c
1
(h) c
1
(0)
h F( p
O
)e
1
, (79)
which, since by (60)
lim
h0
1
h
c
1
(h) c
1
(0)
= c
1
= F( p
O
)e
1
, (80)
has the property
lim
h0
o(h)
|h|
= 0. (81)
Hence for a generic deformation, by using the denition (79), the expression (78) is replaced by
c
1
(h) = ( p
O
) + h F( p
O
)e
1
+ o(h). (82)
Note that, because of (81), o(h) goes to zero faster than h: we can say that sufciently close to
p
O
every deformation is an afne deformation.
DISAT, University of LAquila, April 23, 2011 (1432) A. Tatone Mechanics of Solids and Materials.
PLACEMENTS AND DEFORMATIONS 9
8.1 Stretch and shear
The stretch U describes how lines through any position are transformed by an afne deformation:
they are stretched while the angles between them are changed. Difference vectors between any
two positions along the principal directions are stretched, while keeping their direction, by a
factor equal to the corresponding eigenvalue of U.
In general, a non afne deformation transforms lines into curves, but the stretch U( p
O
), given
by the polar decomposition of F( p
O
), still describes how tangent vectors to curves through p
O
are transformed: they are stretched while the angles between them are changed. Tangent vectors
which are eigenvectors of U( p
O
) are stretched, while keeping their direction, by a factor equal to
the corresponding eigenvalue of U.
Since F( p
O
) depends on the position p
O
also the eigenvalues and the eigenvectors of U depend
on p
O
.
For any position p
O
and any vector a we call stretch and elongation along the direction a, re-
spectively
:=
Ua
a
, :=
Ua a
a
= 1. (83)
Along each principal direction, since Ua
i
=
i
a
i
the stretch is =
i
. The stretch along a principal
directions is called principal stretch.
Two orthogonal tangent vectors a
1
e a
2
at p
O
are transformed by U( p
O
) into linearly indepen-
dent vectors which are not orthogonal in general. We call shear strain between a
1
e a
2
the angle
2
< <
2
such that
cos (
2
) =
Ua
1
Ua
2
Ua
1
Ua
2
. (84)
Note that if a
1
e a
2
eigenvectors of U then = 0.
DISAT, University of LAquila, April 23, 2011 (1432) A. Tatone Mechanics of Solids and Materials.
