p. vannucci uvsq - université de versailles et saint-quentin-en-yvelines the polar method in plane...
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P. VannucciUVSQ - Université de Versailles et Saint-Quentin-en-Yvelines
The polar method in plane anisotropy
Università di PisaFacoltà di Ingegneria
10 luglio 2007
2
Foreword This presentation concerns a series of researches made during the
last years, firstly at ISAT, University of Burgundy and presently at the University of Versailles, by G. Verchery, A. Vincenti, E. Valot and myself.
These researches are based upon an original idea of G. Verchery, the so-called polar method, for the description of plane anisotropy.
The polar method is just an alternative way to the Cartesian one to represent a tensor: in the polar method the whole set of independent invariants are found and they are used to effectively represent the tensor.
In a sense, all these researches turn around a basic question: is there an alternative, effective and advantageous way to represent plane anisotropy?
We only tried to answer to this question…
3
Content
How to describe anisotropy?
The polar method
The mechanical meaning of the polar parameters
Some other results concerning the polar method
Conclusions and perspectives
4
How to describe anisotropy? Anisotropy is the dependence of some property upon the direction;
roughly speaking, that property is not invariant with the direction.
Mathematically, this implies that any orientation-based description of anisotropy is not invariant, i.e. it is based upon some quantities which are not intrinsically representative of the property, being frame dependent: it is the case of the Cartesian representation.
For instance, let us consider the case of elasticity: an isotropic material is described, in any frame, by only two quantities, the Young’s modulus and the Poisson’s ratio, or alternatively the Lamé’s moduli, or C1111 and C1122 in the stiffness tensor or even two other independent combinations of these quantities.
Any of these is an intrinsic quantity describing the elastic behaviour of the material; in a sense, it belongs to the material, and serves to its identification.
5
How to describe anisotropy? Actually, once you know any couple of these numbers, you know
everything about the elastic behaviour of the material, in particular you do not need to specify the frame.
But when you want to make the same with an anisotropic material, for instance a fiber reinforced layer, the situation is much more complicated.
First of all, you need more quantities (how much??? a nice question…), and, more important, these quantities can be frame-dependent, according to the mathematical description of the property you choose.
As a matter of fact, the tensor components and most of their combinations, like engineer constants, are no more invariants.
So, they describe the given property in a given frame and only in this one; when you need the description in another frame, you must use a change-of-frame law (tensor rank dependent and normally rather cumbersome).
6
How to describe anisotropy? For instance, a fiber reinforced layer is orthotropic and described in
its material axes by 4 quantities: E1, E2, 12 and G12 or alternatively by the stiffness matrix:
Nevertheless, none of these quantities is intrinsic: a change of frame transforms these quantities in a rather cumbersome way and the skyline of the stiffness matrix changes: with a Cartesian representation in a general frame, nobody can say in a glance if a layer is orthotropic!
.
00
0
0
1212
22221122
11221111
Q
Q
7
How to describe anisotropy? In fact, an orthotropic layer in a general frame looks like this:
),()22(
,)2()2(
.cos,)2()2(
,sin,)2(2
),()4(
,)2(2
441212
221212112222221111
3121222221122
3121211221111
3121222221122
3121211221111
42222
2212121122
41111
441122
22121222221111
42222
2212121122
41111
csQcsQQQQQ
scQQQcsQQQQ
ccsQQQscQQQQ
scQcsQQsQQ
csQcsQQQQ
sQcsQQcQQ
xyxy
yyyx
xxxy
yyyy
xxyy
xxxx
with,
xyxyyyyxxxxy
yyyxyyyyxxyy
xxxyxxyyxxxx
QQQ
QQQ
QQQ
Q
y
x1
x2
x3 z
x
8
How to describe anisotropy? To remark that in a general frame it is apparent that 6, of which 4
independent, quantities are needed to describe an orthotropic, just like a totally anisotropic, layer.
In a sense, the Cartesian representation is frame dependent and for this reason it is effective whenever frame dependent results are needed (stresses, strains, displacements).
Nevertheless, in some problems the use of an intrinsic representation can be more effective than the Cartesian one.
Another reason to use an alternative representation of the tensor components is to have simpler formulae for the frame rotation; this can be of great importance in inverse problems, where the direction is one of the design variables: it is the case of laminate design.
9
How to describe anisotropy? In other words, alternative tensor representations for anisotropy are
interesting per se and for applications.
In the past, different authors have proposed not-Cartesian tensor representations, mainly in elasticity: in the field of composite mechanics, and hence in two-dimensional elasticity, the most known attempt is that of Tsai and Pagano (1968), who proposed the following formulae:
.4sin4cos
,4sin24cos22sin2cos2
,4sin4cos2sin22cos
,4sin24cos22sin2cos2
,4sin4cos
,4sin4cos2sin22cos
735
3726
73621
3726
734
73621
UUUQ
UUUUQ
UUUUUQ
UUUUQ
UUUQ
UUUUUQ
xyxy
xyyy
yyyy
xxxy
xxyy
xxxx
10
How to describe anisotropy? Here, Ui are the Tsai and Pagano parameters given by:
This representation is not very effective (indeed, it just simplifies a little the rotation formulae) for two reasons: the parameters Ui are not tensor invariants and they do not have a physical meaning.
In addition, 7 parameters are introduced for describing a 6-parameters property.
.2
,2
,8
42
,8
46,
8
42
,2
,8
4323
122211127
122211126
12122222112211115
12122222112211114
12122222112211113
222211112
12122222112211111
QQU
QQU
QQQQU
QQQQU
QQQQU
QQU
QQQQU
11
How to describe anisotropy?
So, the ideal should be the following:
to have an intrinsic representation of an anisotropic property, i.e. making use only of tensor invariants and of a sufficient number of direction parameters fixing the frame, and in addition the tensor invariants should be chosen in such a way that they represent some physical property, if possible linked to the kind of anisotropy of the material.
This is just what has been done in 2D elasticity by Professor G. Verchery in 1979, who has introduced what he called the polar method.
12
The polar method The origin of the polar method is the complex variable
representation of plane quantities, a classical approach in mathematical physics (Klein 1896, Michell 1902, Kolosov 1909, Muskhelishvili 1933, Green et Zerna 1954).
Actually, Verchery introduces a new complex transformation, more effective than that of Green and Zerna, and develops the tensor representation of symmetric tensors of the 2nd rank and of elasticity tensors.
The Verchery’s transformation: for a vector x= (x1, x2) the contravariant components are given by
Some algebraic, a little bit complicate, manipulations give the transformation of symmetric 2nd rank tensors and of elasticity tensors. The results are the following:
),( 21 XXcont X
,1xmX cont .11
11
21
1
ii
iim
13
The polar method Second rank symmetric tensors:
Elasticity tensors:
.
2
101
2
21
22
12
11
22
12
11
T
T
T
ii
ii
T
T
T
.
144241
2002
100201
104201
2002
144241
41
2222
1222
1212
1122
1112
1111
2222
1222
1212
1122
1112
1111
T
T
T
T
T
T
ii
ii
ii
ii
T
T
T
T
T
T
14
The polar method This transformation has a peculiar and fundamental point: it allows
to find the whole set of tensor invariants and so to express the Cartesian components of the tensor by only its invariants plus one parameter (actually, an angle) that fixes the frame.
In the case of 2nd order symmetric tensor the result is very well known: it is the Mohr’s representation:
T, R and are the polar components of T. Indeed, T and R are invariants, and they represent respectively the centre and the radius of the Mohr’s circle (the spherical part and the deviator of the tensor), while is an angle fixing the frame (it gives the direction of the principal components, and depends on the frame where the Cartesian components are known).
converselyand
,2cos
,2sin
,2cos
22
12
11
RTT
RT
RTT
.2
,2
1222112
2211
TiTT
eR
TTT
i
15
The polar method More interesting is the case of elasticity tensors:
Conversely,
.2cos44cos2
,2sin24sin
,4cos
,4cos2
,2sin2 4sin
,2cos44cos2
11 00102222
11 001222
0001212
00101122
11001112
11 00101111
RRTTT
RRT
RTT
RTTT
RRT
RRTTT
. )(2 e 8
, )(4 42e 8
,28
,42 8
12221112222211112
1
1222111222221212112211114
0
2222112211111
2222121211221111 0
1
0
TTiTTR
TTiTTTTR
TTTT
TTTTT
i
i
16
The polar method T0, T1, R0, R1, 0 and 1 are the polar components of T, the last two
being angles and the remaining ones moduli.
It can be shown that T0, T1, R0, R1 and the angular difference0 1 are five independent invariants of T, which can be represented by these intrinsic quantities. The choice of 0 or 1 fixes the frame (normally, 1 =0).
Actually, they describe the elastic properties of a layer, in a different, but intrinsic, way from the usual Cartesian one.
So, now the questions are: is this representation interesting, i.e. useful? have the polar components a mechanical meaning? what are the bounds of polar components?
The answer to the first question is crucial: I will try to convince you that in some problems, but indeed not everytime, the polar representation can be advantageous.
But, to start…
17
The polar method ...what happens for the components in a rotated frame?
The result is rather interesting: it is sufficient to subtract the rotation angle from the polar angles!
This property reveals to be rather useful in laminate design.
).(2cos4)(4cos2
),(2sin2)(4sin
),(4cos
),(4cos2
),(2sin2 )(4sin
),(2cos4)(4cos2
110010
1100
000
0010
1100
110010
RRTTT
RRT
RTT
RTTT
RRT
RRTTT
yyyy
xyyy
xyxy
xxyy
xxxy
xxxx
).(2cos
),(2sin
),(2cos
RTT
RT
RTT
yy
xy
xx
18
The polar method The polar components of the inverse tensor S=T-1 can be given as
function of T0, T1, R0, R1, 0 and 1 :
)].4( cos [16)(8with
,
2
2
2
100021
20
201
4002
12
1
401
4214
0
20
20
1
2110
0
1011
010
RTRRTT
Δ
eRTeR er
,Δ
eRTeR er
,Δ
RTt
,Δ
RTTt
)Φi(ΦiΦi
iΦiΦi
19
The polar method The technical moduli can also be easily calculated, for a given
angle θ:
.)(2cos4)(4cos2
)(2sin2)(4sin2)()(2)(
,)(2cos4)(4cos2
)(2sin2)(4sin2)()(2)(
,)(4cos4
1)(
41
)(
,)(2cos4)(4cos2
)(4cos2)()()(
,)(2cos4)(4cos2)()(
,)(2cos4)(4cos2)()(
110010
1100,
110010
1100,
000
1
110010
0010
1110010
1
1110010
1
rrtt
rrES
rrtt
rrES
rtSG
rrtt
rttES
rrttSE
rrttSE
yyxyyyyxy
xxxxxyxxy
xyxyxy
xxxxyyxy
yyyyyy
xxxxxx
20
The mechanical meaning of the polar parameters A glance at the equations expressing the Cartesian components in a
general frame as functions of the polar parameters; fixing 1=0 it is
shows immediately that T0 and T1 represent the isotropic part of T, while R0, R1 and 0 the anisotropic part.
The necessary and sufficient condition for plane isotropy is then
R0=R1=0.
More generally, the mechanical meaning of the polar components concerns material symmetries and strain energy decomposition.
.2cos4)(4cos2
,2sin2)(4sin
),(4cos
),(4cos2
,2sin2 )(4sin
,2cos4)(4cos2
10010
100
000
0010
100
10010
RRTTT
RRT
RTT
RTTT
RRT
RRTTT
yyyy
xyyy
xyxy
xxyy
xxxy
xxxx
21
The mechanical meaning of the polar parameters Let us first consider the link between elastic symmetries and polar
components.
Actually, the polar method has allowed the first intrinsic definition of plane orthotropy, and the discovery of a special type of orthotropy.
In fact, there are 3 alternative invariant sufficient conditions of plane orthotropy:
general orthotropy
(Vong & Verchery, 1986):
square symmetry (Verchery, 1979):
R0 orthotropy (Vannucci, 2002):
In brackets, the same condition expressed through the Tsai and Pagano parameters: the comparison shows the effectiveness of the polar method.
);044(
,4
236267
227
10 ,
UUUUUUU
KK N
);0(,0 621 UUR
).0(,0 730 UUR
22
The mechanical meaning of the polar parameters The case of general orthotropy is stated by
This means that a layer is generally orthotropic if the harmonics describing the elastic properties are shifted of a multiple of 45°.
If we pose 1=0 (i.e. the frame is fixed so as the x axis is the strong axis) the Cartesian components look like
. ,410 N KK
.2cos44cos)1(2
,2sin24sin)1(
,4cos)1(
,4cos)1(2
,2sin2 4sin)1(
,2cos44cos)1(2
1010
10
00
010
10
1010
RRTTT
RRT
RTT
RTTT
RRT
RRTTT
kyyyy
kxyyy
kxyxy
kxxyy
kxxxy
kxxxx
23
The mechanical meaning of the polar parameters
To be remarked that there are two different types of general orthotropy, K= 0 or 1, for the same invariants T0, T1, R0 and R1 . In other words, two different orthotropic materials share the same values of T0, T1, R0 and R1:
The directional and Cartesian diagram of the Young’s modulus E() for two orthotropic materials sharing the
same values of T0, T1, R0 and R1 but with different K (T0 = 1.3, T1= 0.8, R0=0.7, R1=0.3).
K=0
K=1
K= 1
K= 0
(°)
K=0
K=1
K= 1
K= 0
(°)
24
The mechanical meaning of the polar parameters
The directional and Cartesian diagram of the Poisson's coefficient xy() for two orthotropic materials sharing
the same values of T0, T1, R0 and R1 but with different K (T0 = 1.3, T1= 0.8, R0=0.7, R1=0.3).
(°)
K=0 K=1
K=0
K=1
25
The mechanical meaning of the polar parameters K=0 corresponds to what Pedersen, 1993, called a low shear
modulus orthotropy (U3>0) while the case K=1 corresponds to the case of high shear modulus orthotropy (U3<0) .
It can be shown that materials with K=1 show some peculiar properties, especially for the design with respect to stiffness properties of laminates.
K=1
K=0
(°)
K=0
K=1
The directional and Cartesian diagram of Gxy() for two orthotropic materials sharing the same
values of T0, T1, R0 and R1 but with different K (T0 = 1.3, T1= 0.8, R0=0.7, R1=0.3).
26
The mechanical meaning of the polar parameters The value of K influences also the kind of variation of the
component Txxxx(), hence of the normal stiffness. Three are the possible cases, depending upon the value of K and of the parameter =R0/R1; if once again we choose 1=0, then:
The case of the Young's modulus can be treated in the same way, but using the compliance parameters and remembering that Ex()=1/Sxxxx().
.1
1arccos21
K
K=0 or K=1 and <1: Txxxx() is the highest value and Txxxx(/2) is the lowest;
K=0 and ≥1 Txxxx() is the absolute and Txxxx(/2) the relative maximum; the minimum is at the angle , with
K=1 and ≥1 Txxxx() is the relative and Txxxx(/2) the absolute minimum; the maximum is at the angle .
27
The mechanical meaning of the polar parameters The case of orthotropy stated by R1=0 corresponds to the so-called
square-symmetry. It is the planar equivalent of the cubic syngony.
Though it is still an orthotropic case, algebraically it is different from the previous one. In fact, unlike the previous case, stated by a third-order invariant, this case is ruled by a second-order invariant.
The directional diagrams of the Young’s modulus E() and of Gxy() for two square-symmetric materials sharing the same values of T0, T1, R0 but with
different K (T0 = 1.3, T1= 0.8, R0=0.7).
K=1
K=0
K=1
K=0E() Gxy()
28
The mechanical meaning of the polar parameters Now, the change from K=0 to K=1 corresponds simply to a rotation
of the frame through an angle of 45°.
In fact, the Cartesian components in this case are
The Cartesian conditions for square-symmetry are easily found to be
.
,0
yyyyxxxx
xyyyxxxy
TT
TT
.4cos)1(2
,4sin)1(
,4cos)1(
,4cos)1(2
,4sin)1(
,4cos)1(2
010
0
00
010
0
010
RTTT
RT
RTT
RTTT
RT
RTTT
kyyyy
kxyyy
kxyxy
kxxyy
kxxxy
kxxxx
29
The mechanical meaning of the polar parameters
R0 orthotropy is a special case of orthotropy. Like the previous case of square–symmetry, it is ruled by a second order invariant.
The components of an elasticity R0 orthotropic tensor T behave like those of a 2nd rank symmetric tensor. If 1= 0, then:
Txxyy and Txyxy are isotropic, while Txxxy and Tyyxy are identical .
.2cos42
,2sin2
,
,2
,2sin2
,2cos42
110
1
0
10
1
110
RTTT
RT
TT
TTT
RT
RTTT
yyyy
yyxy
xyxy
xxyy
xxxy
xxxx
30
The mechanical meaning of the polar parameters To be remarked that R0=0 does not imply the same in compliance:
Nevertheless, the independent constants are still 3 also in compliance, as
So, there are also materials r0-orthotropic in compliance but not in stiffness.
.2π
,
,)2(8
,)2(4
,)2(16
,)2(4
1110
2110
112
1100
21
0
2110
012
1100
2110
0
RTT
Rr
RTTT
Rr
RTT
Tt
RTTT
RTTt
.1
21
0 t
rr
31
The mechanical meaning of the polar parameters The Cartesian conditions corresponding to R0-orthotropy can be
easily found:
Once again, two kinds of variations of the Young's modulus E() are possible:
i. T02R1: E(0) is the highest and E(/2) the lowest of E() ;
ii. T0<2R1: E(0) is a local minimum and E(/2) the absolute minimum; the absolute maximum is attained for the orientation
Two examples are shown in the next figures.
.
,42
xyyyxxxy
xyxyxxyyyyyyxxxx
TT
TTTT
.2
arccos21
arccos21
1
0
1
1
R
T
r
t
32
The mechanical meaning of the polar parameters First case: T02R1.
The directional diagrams of E(), Gxy() and xy() for a R0-orthotropic material (T0 = 1.3,
T1= 0.8, R1=0.3).
E() Gxy()
xy()
33
The mechanical meaning of the polar parameters Second case: T0<2R1
The directional diagrams of E(), Gxy() and xy() for a R0-orthotropic material (T0 = 1.3,
T1= 0.8, R1=0.7).
E() Gxy()
xy()
34
The mechanical meaning of the polar parameters A question arise: how R0-orthotropic materials can be obtained?
If we consider a lamina reinforced by fibers disposed in equal amount along two directions, the polar condition to get R0-orthotropy is the following one
So, the solution is
This is only a sufficient condition for R0-orthotropy.
Nevertheless, it is the only possible solution for a lamina reinforced by fibers disposed along only two directions.
In fact, if is the relative volume fraction, the only solution is get for =1 and =12=/4, see the figure.
.021 44 ii ee
.421
35
The mechanical meaning of the polar parameters Some remarks about plane orthotropy: we have seen that:
there is not a unique kind of plane orthotropy, algebraically speaking; the three kinds of plane orthotropy are distinguished by different tensor
conditions: two of them, the R0 and R1 orthotropy are stated by second-order
invariants, the third, the general orthotropy, is stated by a third-order invariant;
in addition, two general orthotropic materials can share the same polar moduli but a different value of the angle 0; so, two are, in principle, the general orthotropic materials sharing the same invariant moduli, and they have qualitative different properties;
R0-orthotropy does not correspond to r0-orthotropy and vice-versa;
R1-orthotropy corresponds to the cubic syngony in 3 dimensions;
the existence of R0-orthropy in 3 dimensions has been recently proved by S. Forte (2005);
in some senses, R0 and R1 orthotropy are stronger forms of orthotropy than the general one: they are conserved in some cases of homogenization.
36
The mechanical meaning of the polar parameters A last consideration about orthotropy.
Usually, it is said that plane orthotropy depends upon 4 constants, and so that 4 independent measures are needed to experimentally characterize such a case.
But, in the stiffness matrix, 6 are the components to be assigned; two of them are zero if the material is orthotropic and if it is described in its material axes: these are two data, numerically equivalent to two zero components.
When we look at the same problem in terms of polar invariants, we find that, once the frame fixed (usually by posing 1=0): general orthotropy is characterized by 5 invariants: T0, T1, R0, R1 and K;
R0 and R1 orthotropy are characterized by 3 invariants: T0, T1 and R0 or R1.
So, it is not so immediate to say how much constants determine plane orthotropy !
37
The mechanical meaning of the polar parameters
Let us now consider the energetic meaning of the polar components.
If then
It can be shown that
T1 is directly responsible of WS, T0, R0 and 0 of WD while R1 couples the two parts.
,,,and,, rtRT εσ
).(2cos,
,21
rRWtTW
WWW
DS
DSεσ
).(2cos4)(4cos22
),(2cos44
1102
02
0
112
1
rtRrRrTW
rtRtTW
D
S
38
The mechanical meaning of the polar parameters And now, let us look at the bounds on the elastic polar components.
It can be shown that the following two conditions are necessary and sufficient to ensure the positive definiteness of W:
For orthotropic materials:
.)(4cos2
,
100021
20
201
00
RTRRTT
RT
.2
)1(
,
1
21
00
00
T
RRT
RT
K
Surface S= 121 /2 TR
a)surface in 3D;
b) level curves of S and existence domain for the two cases of orthotropy.
T1
R1
T1
R1
K= 1
K= 0
existence domain of the case K= 0
existence domain of both the cases
a) b)
39
Some other results concerning the polar method The polar method has been applied to the study of some problems,
mainly concerning laminates but not only.
The problem of assessing the influence of orientation errors on some elastic properties of laminates has been addressed.
For instance, in the case of extension-bending coupling, the degree of coupling
has been introduced to describe the average consequences of angle errors.
It has been shown that, once again, the ratio
is the material parameter that describes the importance of an orientation error.
maxB
B
1
0
R
R
40
Some other results concerning the polar method In particular, if there is only one orientation error , the expression of
is:
is a parameter depending upon the number of layers n:
The maximum of is obtained when the error is located in correspondence of one of the outer layers and when its value is
.1if)2cos1(4)4cos1(1
2
2
,1if)2cos1(4)4cos1(2
22
2
.2mod
12
2 nn
n
41
Some other results concerning the polar method
In both cases, the maximum of is 2. The results are summarized in the figures below, that show a bifurcation-like behaviour with respect to the parameter control , accounting for the anisotropy of the material.
.1if)1
arccos(21
1if2
2
/
/2
0 0.5 1 1.5 2 2.5 3 3.5 4
42
Some other results concerning the polar method The previous figure show that the less sensitive laminates to
orientation errors are those composed by R0-orthotropic materials (=0), while the most sensitive are those composed by R1-orthotropic materials (), i.e. those whose layers are reinforced by balanced fabrics.
A similar analysis has been conducted also for the property of quasi-homogeneity, and numerically also for a random vector of orientation errors for each layer, with similar results.
Another result concerns the classical equation for linear buckling of anisotropic plates
,2
4)2(24
,,,
,,,,,
yyyyxyxyxxxx
yyyyyyyyxyyyxyyyxxyyxyxyxxyyxxxyxxxyxxxxxxxx
wNwNwN
wDwDwDDwDwD
43
If one expresses not only the bending stiffness tensor with polar parameters, but also w, which is a completely index-symmetric tensor, described by 4 polar invariants, indicated with lower case letters, one gets a simpler equation:
At the first member, there are only 3 terms, not 5, accounting for the isotropic and R0- and R1-anisotropic parts of D and w respectively.
At the second member there are only 2 terms, not 3, accounting for the spherical and deviatoric parts of N (capital letters) and of 2w (lower case letters).
This equation is still to be used….
).(2cos22
)(2cos32)(4cos8)2(8 11110000101
rRtT
RrRrTTt
Some other results concerning the polar method
44
Some other results concerning the polar method Others results concerning the mechanics of laminates:
a new kind of tests for angle-ply laminates ;
an experimental study of the strength of in-plane isotropic laminates;
a linear theory of laminates composed by intrinsically coupled layers;
an analytical study of the thermal and piezoelectric expansion coefficients for composite laminates.
The polar method has been recently extended also to the description of other plane properties than elasticity, namely of the tensors describing the phason and the coupled phonon-phason behaviours of quasi-crystals (Lauretti and Vannucci, 2006) and of piezoelectric tensors (Vannucci, 2007).
45
Conclusions and perspectives The polar method is only an alternative method to describe plane
properties based upon the use of tensor invariants.
It reveals to be effective in the qualitative description of some properties, because the polar invariants have a mechanical meaning.
In addition, it is rather useful in some optimal design problems of laminates.
At present, some studies are considering the role played by polar invariants in the minimization of the elastic energy for some structural optimization problems in anisotropic elasticity.
In the future, it should be interesting to apply the same method to the qualitative study of other mechanical properties, like for instance plasticity, damage and strength criteria in anisotropic layers.
Thank you very much for your attention.