Formulas and Equations for the Classical Laminate Theory

Formulas and Equations for the Classical Laminate TheoryAuthor: Vincent CalardVersion: 0.6Datum: 19.09.2011

In this document, we prepared a collection of formulas and equations for the classical laminate theory. The firstobjective was to propose a quick reference booklet with the most important formulas to calculate the stiffnessand the compliance matrix of a laminated composite. The document is not yet finished but in a pre-versionrelease. Any comments and/or corrections are also welcome.

Notations

Coordinate axes (xyz) = subscripts (123)

Stress [] and strain [] contracted notations

The stress tensor [] and strain tensor [] are represented by two (3x3) symmetric matrices (denoted by [.]):

[] =

11 12 1322 23Sym. 33

[] =11 12 1322 23

Sym. 33

(1)We chose the following contracted notation in a vector form (denoted by {.}):

{} =

112233223213212

=

123456

{} =

112233231312

=

123456

(2)

From the above relations, we note that the components with subscripts (4,5,6) are associated with the shear andthat the stress and strain state in the (xy) plane is associated with the subscripts (1,2,6).

Hookes law, Compliance matrix {} = [S]{}, 3D21 independent parameters are sufficient to describe the 36 parameters:

123456

=

S11 S12 S13 S14 S15 S16S22 S23 S24 S25 S26

S33 S34 S35 S36S44 S45 S46

Symmetric S55 S56S66

123456

(3)

Hookes law, Stiffness matrix {} = [S]1{} = [C]{}, 3D21 independent parameters are sufficient to determine the 36 components:

1 v0.6/ [email protected]

123456

=

C11 C12 C13 C14 C15 C16C22 C23 C24 C25 C26

C33 C34 C35 C36C44 C45 C46

Symmetric C55 C56C66

123456

(4)

Hookes law in plane stress condition, 3 = 4 = 5 = 0, 2D126

=S11 S12 S16S22 S26

Sym. S66

126

and126

=Q11 Q12 Q16Q22 Q26

Sym. Q66

126

(5)The terms of the stiffness matrix [Q] will not be equal to the corresponding terms of the stiffness matrix [C] in3D. Q11 Q12 Q16Q22 Q26

Sym. Q66

6=C11 C12 C16C22 C26

Sym. C66

(6)The components C16 and C26, respectively S16 and S26, are the (-) coupling components. They are respon-sible of distortions of the material geometry, even if an unidirectional stress or strain condition is applied to thematerial.

Rotation of axe (z) with angle of (xyz) gives (xyz)

The [] = [P ][][P ]1 equation gives the rotation transformation of the stress tensor from the system (xyz)to the system (xyz) and is written as follow with c = cos() and s = sin():1 6 52 4

Sym. 3

= c s 0s c 0

0 0 1

1 6 52 4Sym. 3

c s 0s c 00 0 1

(7)Using the contracted notation, the rotation transformation can be written in a simple form:

123456

=

c2 s2 0 0 0 2css2 c2 0 0 0 2cs0 0 1 0 0 00 0 0 c s 00 0 0 s c 0cs cs 0 0 0 c2 s2

[Jz, ]

123456

(8)

The rotation transformation of the strain tensor with the contracted notation is:


123456

=

123242526

=

c2 s2 0 0 0 122css2 c2 0 0 0 122cs0 0 1 0 0 00 0 0 c s 00 0 0 s c 02cs 2cs 0 0 0 c2 s2

[Jz,]

123242526

(9)

The rotation transformations of the stiffness and the compliance matrices are given by:

[C ] = [Jz,][C][Jz,]1 and [S] = [Jz,][S][Jz,]1 (10)

Rotation in 2D, (xy) plane associated with the subscripts (126)126

= c2 s2 2css2 c2 2cscs cs c2 s2

[J2, ]

126

and126

= c2 s2 122css2 c2 122cs2cs 2cs c2 s2

[J2,]

1226

(11)

with c = cos() and s = sin().The rotation transformation of the stiffness matrix is given by [Q] = [J2,][Q][J2,]1:

Q11Q12Q16Q22Q26Q66

=

c4 2c2s2 4c3s s4 4cs3 4c2s2

c2s2 s4 + c4 2cs3 2c3s c2s2 2c3s 2cs3 4c2s2c3s c3s cs3 c4 3c2s2 cs3 3c2s2 s4 2c3s 2cs3s4 2c2s2 4cs3 c4 4c3s 4c2s2cs3 cs3 c3s 3c2s2 s4 c3s c4 3c2s2 2cs3 2c3sc2s2 2c2s2 2cs3 2c3s c2s2 2c3s 2cs3 (s2 c2)2

Q11Q12Q16Q22Q26Q66

(12)

In case of a stiffness matrix without (-) coupling components (Q16 = Q26 = 0), we have:

Q11Q12Q16Q22Q26Q66

=

c4 s4 2c2s2 4c2s2

c2s2 c2s2 s4 + c4 4c2s2c3s cs3 c3s cs3 2c3s 2cs3s4 c4 2c2s2 4c2s2

cs3 c3s cs3 c3s 2cs3 2c3sc2s2 c2s2 2c2s2 s4 2c2s2 + c4

Q11Q22Q12Q66

(13)

For the compliance matrix, the rotation transformation is:

S11S12S16S22S26S66

=

c4 s4 2c2s2 c2s2

c2s2 c2s2 s4 + c4 c2s22c3s 2cs3 2c3s 2cs3 c3s cs3s4 c4 2c2s2 c2s2

2cs3 2c3s 2cs3 2c3s cs3 c3s4c2s2 4c2s2 8c2s2 s4 2c2s2 + c4

S11S22S12S66

(14)

The stiffness components Q11, Q12, Q22 and Q66, and the compliance components S11, S12, S22 and S66 areeven functions of :

Q11() = Q11() . . . (15)The coupling components S16, S26, Q16 and Q26 are odd functions of :

Q16() = Q16() . . . (16)


Engineering constants

In mechanics, two engineering constants, the Youngs modulus E and the Poissons ratio , are measured fromtensile testing. A third engineering constant, the shear modulus G, is obtained by a pure shear test. In case of apure tensile test in x direction, we have:

{x 6= 0 , y = xy = 0x and y proportionnal to x

x =1Exx

y = xyx = xyEx xxy = 0

The Youngs modulus is defined by the ratio between the applied longitudinal stress and the longitudinal strain,and has the dimension of a stress. The Poissons ratio is defined by the ratio between the longitudinal and thetransversal strain, and is without dimension. In a pure shear test, we have:

{xy 6= 0 , x = y = 0xy proportionnal to xy

{xy =

1Gxy

x = 0 and y = 0

Orthotropic, 3D

The compliance matrix in the symmetry axes of the material is given by:

[Sortho] =

S11 S12 S13 0 0 0S12 S22 S23 0 0 0S13 S23 S33 0 0 00 0 0 S44 0 00 0 0 0 S55 00 0 0 0 0 S66

(17)

9 engineering constants (E1, E2, E3, 12, 23, 13, G23, G13, G12) are necessary to write the matrix in thesymmetry axes:

123456

=

1E1

21E2 31E3 0 0 012E1 1E2 32E3 0 0 013E1 23E2 1E3 0 0 0

0 0 0 1G23 0 0

0 0 0 0 1G13 0

0 0 0 0 0 1G12

123456

(18)

The symmetry relations are:

21E2

=12E1

;31E3

=13E1

;32E3

=23E2

(19)


We have also the following relation 132132 = 122331. The stiffness matrix is obtained by inverting thecompliance matrix which yields the following components:

C11 =(1 2332)

E1;

C12 =(21 + 3123)

E1 =

(12 + 3213)

E2;

C13 =(31 + 2132)

E1 =

(13 + 1223)

E3;

C22 =(1 1331)

E2; (20)

C23 =(32 + 1231)

E2 =

(23 + 1213)

E3;

C33 =(1 1221)

E3;

C44 = G23; C55 = G13; C66 = G12

with the parameter = 1 1221 2332 3113 2213213.We have the condition > 0 to have positiv Cii modulus.

= 1 212E2E1 223

E3E2 13E3

E1 2122313E3

E1(21)

Transversely isotropic, 3D

The compliance matrix in the symmetry axes of the material is given by:

[S] =

S11 S12 S12 0 0 0

S12 S22 S23 0 0 0S12 S23 S22 0 0 00 Plane(yz) 0 2(S22 S23) 0 00 0 0 0 S66 00 0 0 0 0 S66

(22)

The isotropy property in the (yz) plane, subscript (234), yields with equation (14):

[S](yz)plane = [S](yz)plane

S44 = 2(S22 S23)

5 engineer constants (E1, E2, 12, 23, G12) are necessary to write the stiffness matrix:


123456

=

1E1

21E2 21E2 0 0 012E1 1E2 23E2 0 0 012E1 23E2 1E2 0 0 0

0 0 0 2(1+23)E2 0 0

0 0 0 0 1G12 0

0 0 0 0 0 1G12

123456

and the symmetry condition

21E2

= 12E1(23)

The inversion of the compliance matrix yields the stiffness matrix:

[C] =

(123) E1

12 E2

12 E2 0 0 0

(11221)(1+23)

E2(23+1221)(1+23)

E2 0 0 0(11221)(1+23)

E2 0 0 0E2

2(1+23)0 0

Sym. G12 0G12

(24)

with the parameter = (1 23 21221)We can also introduce the 5 engineering constants E1, E2, G12, 12 et 23:

[C] =

(123) E1

12 E2

12 E2 0 0 0

(1212E2/E1)(1+23)

E2(23+212E2/E1)

(1+23)E2 0 0 0

(1212E2/E1)(1+23)

E2 0 0 0E2

2(1+23)0 0

Sym. G12 0G12

(25)

with the parameter = (1 23 2212E2E1 ). We have the condition > 0.

UD Laminate in plane stress and in symmetry axes, 2D

The compliance matrix in the symmetry axes of the material has no (-) coupling components (S16 = S26 = 0)and depends of 4 engineering constants E1, E2, G12 and 12:12

6

= 1E1 21E2 012E1 1E2 0

0 0 1G12

[S]

126

and symmetry condition21E2

= 12E1(26)

The inversion of the compliance matrix yields the stiffness matrix:126

=

1(11221)E1

21(11221)E1 0

12(11221)E2

1(11221)E2 0

0 0 G12

[Q]

126

(27)


We can also introduce the 4 engineering constants E1, E2, G12 and 12:126

=

1(1212E2/E1)

E112

(1212E2/E1)E2 0

1(1212E2/E1)

E2 0

Sym. G12

[Q]

126

(28)

We have the condition (1 212E2/E1) > 0.

UD Laminate in plane stress and out of symmetry axes, 2D

The compliance matrix out of the symmetry axes of the material has two (-) coupling components (S16 6= 0, S26 6= 0) and depends of 6 engineering constants E1, E2, G12, 12, 12, 12:12

6

=

1E121E2

12G12

1E2

12G12

Sym.1G12

[S]

126

(29)

From the components of the laminate in the symmetry axes, the components are calculated with equation (14).The coupling components are equal to zero in the symmetry axes, and they are odd functions of :

12G12

= 2cs(c2

E1 s2E2 + (c2 s2)

(21E2 12G12

))12G12

= 2cs(c2

E1 s2E2 (c2 s2)

(21E2 12G12

)) (30)with c = cos() and s = sin().

Isotropic material, 3D

[S] =

S11 S12 S12 0 0 0S12 S11 S12 0 0 0S12 S12 S11 0 0 00 0 0 2(S11 S12) 0 00 0 0 0 2(S11 S12) 00 0 0 0 0 2(S11 S12)

(31)

Two parameters E and are necessary to describe the compliance and stiffness matrices:


123456

=

1E E E 0 0 0

1E E 0 0 0

1E 0 0 0

1G 0 0

Sym. 1G 01G

123456

and isotropy condition

G = E2(1+)(32)

The stiffness matrix is given by:

123456

=

(1)(1+)(12)E

(1+)(12)E

(1+)(12)E 0 0 0

(1)(1+)(12)E

(1+)(12)E 0 0 0

(1)(1+)(12)E 0 0 0

G 0 0Sym. G 0

G

123456

(33)

We can define a parameter d = (1 + )(1 2) = (1 22).

Considering that the diagonal Cii constants must be positive, We have d > 0, which means < 0.5. A valueof close to 0.5 is characteristic of an incompressible material. The compression dilatation is given by:

V

V= 1 + 2 + 3 =

1 2E

(1 + 2 + 3) (34)

Another representation can be obtained by using the 2 Lame coefficients and .

123456

=

+ 2 0 0 0+ 2 0 0 0

+ 2 0 0 0 0 0

Sym. 0

123456

Lam relations: = E(1+)(12) = E2(1+)

(35)

Isotropic material in plane stress, 2D

Under plane stress condition 33 = 31 = 23 = 0, and the Hookes law takes the form:126

= 1E E 01

E 0Sym.

1G

126

and isotropy conditionG = E2(1+)

(36)

The inversion of the compliance matrix yields:126

=

1(12)E

(12)E 0

1(12)E 0

Sym. G

Isotropy condition:[Q]=[Q]

126

(37)


Summary table of all independent parameters of the [S] and [C]matrices

Number ofIndependentParameters

Parameters

Non zerocomponents

inSym. Axes

Non zerocomponents

outsideSym. Axes

Hooks law 21 Cij 36 36Othotropic 9 E1, E2, E3, 12, 13, 23, G12, G13, G23 12 36Trans. Iso. 5 E1, E2, 12, 23, G12 12 36

UD Laminate 4 E1, E2, 12, G12 5 9Isotropic 2 E, 12 12

Theory of Love-Kirchhoff for thin shell

The following kinematic assumptions are made in Love-Kirchhoff theory:

Straight lines normal to the mid-surface remain straight after deformation. Straight lines normal to the mid-surface remain normal to the mid-surface after deformation. The thickness of the plate does not change during a deformation.

The plane (z = 0) is the mid-plane surface. The displacement of a point (u ,v, w) outside the mid-plane surface(u0, v0, w0) is calculated relatively to the mid-plane displacement and its position z, which allows then thedetermination of the strain tensor on the whole shell (plane condition):

u(x, y, z 6= 0) = u0 z w0xv(x, y, z 6= 0) = u0 z w0yw(x, y, z 6= 0) = w0

1 =

ux =

u0x z

2w0x2

2 =vy =

v0y z

2w0y2

6 =uy +

vx =

u0y +

v0x z2

2w0xy

(38)

Then if we introduce a curvature vector {k0} of the mid-plane and the strain vector {0} of the mid-plane, thevector {} representation of the strain tensor for a point on z coordinate is given by :12

6

=010206

+ zk01k02k06

{ {} = {0}+ z{k0}{} = [Q]{} = [Q]{0}+ z[Q]{k0} (39)


Classical Laminate Theory of thin plate

The classical Laminate Theory is based on the Love-Kirchhoff theorie with the following hypothesis:

The plate is constructed of an arbitrary number of layers of orthotropic sheets bonded together. However,the orthotropic axes of material symmetry of an individual layer need not coincide with the x-y axes ofthe plate. The plate is thin, i.e., the thickness h is much smaller than the other physical dimensions. The displacements u, v, and w are small compared to the plate thickness. Properties of plate are represented by reference plane: the CLT is applicable preferably to symmetric

laminates In-plane strains x , y and xy are small compared to unity. Transverse shear strains xz and yz are negligible. Tangential displacements u and v are linear functions of the z coordinate. The transverse normal strain z is negligible. Each ply obeys Hookes law. No slip between layers The plate has constant thickness.

The forces applied to a small part of the laminate, can be described by 6 components in classical shell theory:3 inplane forces and 3 moments. In a practical notation, the inplane forces and the momnets are calculated perunit length. The inplane forces are denoted by Ni, (i = 1, 2, 6), for all component of the plane stress conditionin i direction. The 3 moments per length are denoted Mi, (i = 1, 2, 6) for the i direction. Mxy is a twistingmoment and can be associated to a distortion of the laminate in the bending mode due to shear stress gradient.Ni and Mi are sometimes called respectively stress resultants and moment resultants.

Ni =

h/2h/2

i dz =

nk=1

hkhk1

ki dz (40)

Mi =

h/2h/2

zi dz =nk=1

hkhk1

zki dz

(41)

The matrix notation is:

N1N2N6M1M2M6

=

A11 A12 A16 B11 B12 B16A12 A22 A26 B12 B22 B26A16 A26 A66 B16 B26 B66B11 B12 B16 D11 D12 D16B12 B22 B26 D12 D22 D26B16 B26 B66 D16 D26 D66

010206k01k02k06

(42)


Aij =

nk=1

Qkij (hk hk1) (43)

Bij =1

2

nk=1

Qkij(h2k h2k1

)(44)

Dij =1

3

nk=1

Qkij(h3k h3k1

)(45)

ABD matrix and mechanical behaviors

Laminate notation

Square Brackets [. . . ] denote a sequence of ply orientations: [0, 90]s = 0/90/90/0.Subscript s [. . . ]s indicates symmetry: [0, 90]s = 0/90/90/0. In this case, each layer is exactly minored aboutthe geometric mid-plane in terms of its properties, thickness and orientation.Repeated plies (. . . )n or repeated sequences [. . . ]n are denoted by a numerical subscript n.

[03, 902]s = 0/0/0/90/90/90/90/0/0/0[0, (+45,-45)2]s = 0/+45/-45/+45/-45/-45/+45/-45/+45/0

[+45,-45]2 = +45/-45/+45/-45

When no Subscript is present, the total description of the laminate lay-up is given by the sequence:

[0,+60,+120] = 0/+60/+120.

Cross-ply laminate: only 0 and 90 plies for the laminate.Angled-ply laminate: only two angles and for the ply orientation.


[A] matrix components in relation to for an UD laminate

The components of the A matrix can be represented in a circular graphic with the variation of the angle ofthe laminate. The principal Youngmodulus, the shear modulus and the Poissons ratio are represented respec-tively by A11, A33 and A12/A11. Calculations have been performed with LamiCens (source: http://www.r-g.de/laminatberechnung.html).

A11 A33 A12/A11

[A] matrix components in relation to for a specially orthotropiclaminate [(0, 90)n]s

A11 A33 A12/A11

[A] matrix components in relation to for a quasi-isotropic laminate[0, 60,60]s

A11 A33 A12/A11


References

P. Bompard, "Matriaux composites", Ecole Centrale Paris, in French, Masters Degree Lecture, (1993). D. Gay and S.V. Hoa, "Composite materials, Design and application", CRC Press, second edition,

ISBN: 978-1-4200-4519-2, (2007). E. Barbero, "Finite element analysis of composite materials", CRC Press, ISBN:978-1-4200-5433-0,

(2008). L. Gornet, "Gnralits sur les matriaux composites", Ecole Centrale Nantes, in French, 43 pages,http:/ /cel.archives-ouvertes.fr, (2008). J.M. Berthelot, "Matriaux composites : Comportement mcanique et analyse des structures", 4me

dition , Paris : Editions Tec& Doc, ISBN: 978-2743007713, (2005). "Faserverbundwerkstoffe Handbuch, Composite materials Handbook", R&G Faserverbundwerkstoffe

GmbH, www.r-g.de, (2009). M.J. Hinton, A.S. Kaddour, P.D. Soden, "Failure Criteria in Fibre Reinforced Polymer Composites: The

World-Wide Failure Exercise (WWFE)", Elsevier, First edition 2004, ISBN: 0-08-044475-X

Softwares

Maxima: Computer algebra system, matrix manipulation, http:/ /maxima.sourceforge.net. Alfalam: Laminate theory, MS Excel, [ABD] matrix, Puck failure criteria, www.klub.tu-darmstadt.de. eLamX: Laminate theory, Java, [ABD] matrix, 3D failure envelope plots, http:/ /tu-dresden.de. LamiCens: Laminate theory, MS Excel, [A] Matrix, Graphics, www.r-g.de. Texgen: Generation of texture, http:/ /texgen.sourceforge.net. Code_Aster: Finite Element Modeling, FEM, www.caelinux.com. MyRTM: RTM simulation, www.iwk.hsr.ch. CADEC: Computer Aided Design Environment for Composites, Laminate theory, [ABDH] matrix,

FEM application, www.mae.wvu.edu/barbero/cadec.html

Author

Dr. Vincent [email protected]

Extended version available soon at www.lulu.com


Documents

Formulas and Equations for the Classical Laminate Theory