Formula Collection for Laminates

Formulas and Equations for the Classical Laminate TheoryAuthor: Vincent CalardVersion: 0.5Datum: 24.05.2011

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 [σ] and strain [ε] contracted notations

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

[ε] =

ε11 ε12 ε13ε22 ε23

Sym. ε33

[σ] =

σ11 σ12 σ13σ22 σ23

Sym. σ33

(1)

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

{ε} =

ε11ε22ε332ε232ε132ε12

=

ε1ε2ε3ε4ε5ε6

{σ} =

σ11σ22σ33σ23σ13σ12

=

σ1σ2σ3σ4σ5σ6

(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).

Hooke’s law, Compliance matrix {ε} = [S]{σ}, 3D

21 independent parameters are sufficient to describe the 36 parameters:

ε1ε2ε3ε4ε5ε6

=

S11 S12 S13 S14 S15 S16S22 S23 S24 S25 S26

S33 S34 S35 S36S44 S45 S46

Symmetric S55 S56S66

σ1σ2σ3σ4σ5σ6

(3)

Hooke’s law, Stiffness matrix {σ} = [S]−1{ε} = [C]{ε}, 3D

21 independent parameters are sufficient to determine the 36 components:

1 v0.5/ [email protected]

σ1σ2σ3σ4σ5σ6

=

C11 C12 C13 C14 C15 C16

C22 C23 C24 C25 C26

C33 C34 C35 C36

C44 C45 C46

Symmetric C55 C56

C66

ε1ε2ε3ε4ε5ε6

(4)

Hooke’s law in plane stress condition, σ3 = σ4 = σ5 = 0, 2Dε1ε2ε6

=

S11 S12 S16S22 S26

Sym. S66

σ1σ2σ6

and

σ1σ2σ6

=

Q11 Q12 Q16

Q22 Q26

Sym. Q66

ε1ε2ε6

(5)

The terms of the stiffness matrix [Q] will not be equal to the corresponding terms of the stiffness matrix [C] in3D. Q11 Q12 Q16

Q22 Q26

Sym. Q66

6=C11 C12 C16

C22 C26

Sym. C66

(6)

The components C16 and C26, respectively S16 and S26, are the (σ-γ) 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 θ of (xyz) gives (x′y′z′)

The [σ′] = [P ][σ][P ]−1 equation gives the rotation transformation of the stress tensor from the system (xyz)to the system (x′y′z′) and is written as follow with c = cos(θ) and s = sin(θ):σ′1 σ′6 σ′5

σ′2 σ′4Sym. σ′3

=

c s 0−s c 00 0 1

σ1 σ6 σ5σ2 σ4

Sym. σ3

c −s 0s c 00 0 1

(7)

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

σ′1σ′2σ′3σ′4σ′5σ′6

=

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 0−cs cs 0 0 0 c2 − s2

︸︷︷︸

[Jz,σ ]

σ1σ2σ3σ4σ5σ6

(8)

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


ε′1ε′2ε′3ε′4ε′5ε′6

=

ε′1ε′2ε′32ε′42ε′52ε′6

=

c2 s2 0 0 0 122cs

s2 c2 0 0 0 −122cs

0 0 1 0 0 00 0 0 c s 00 0 0 −s c 0−2cs 2cs 0 0 0 c2 − s2

︸︷︷︸

[Jz,ε]

ε1ε2ε32ε42ε52ε6

(9)

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

[C ′] = [Jz,σ][C][Jz,ε]−1 and [S′] = [Jz,ε][S][Jz,σ]−1 (10)

Rotation θ in 2D, (xy) plane associated with the subscripts (126)σ′1σ′2σ′6

=

c2 s2 2css2 c2 −2cs−cs cs c2 − s2

︸︷︷︸

[J2,σ ]

σ1σ2σ6

and

ε′1ε′2ε′6

=

c2 s2 122cs

s2 c2 −122cs

−2cs 2cs c2 − s2

︸︷︷︸

[J2,ε]

ε1ε22ε6

(11)

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

−1:

Q′11Q′12Q′16Q′22Q′26Q′66

=

c4 2c2s2 4c3s s4 4cs3 4c2s2

c2s2 s4 + c4 2cs3 − 2c3s c2s2 2c3s− 2cs3 −4c2s2

−c3s c3s− cs3 c4 − 3c2s2 cs3 3c2s2 − s4 2c3s− 2cs3

s4 2c2s2 −4cs3 c4 −4c3s 4c2s2

−cs3 cs3 − c3s 3c2s2 − s4 c3s c4 − 3c2s2 2cs3 − 2c3sc2s2 −2c2s2 2cs3 − 2c3s c2s2 2c3s− 2cs3 (s2 − c2)2

Q11

Q12

Q16

Q22

Q26

Q66

(12)

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

Q′11Q′12Q′16Q′22Q′26Q′66

=

c4 s4 2c2s2 4c2s2

c2s2 c2s2 s4 + c4 −4c2s2

−c3s cs3 c3s− cs3 2c3s− 2cs3

s4 c4 2c2s2 4c2s2

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

Q11

Q22

Q12

Q66

(13)

For the compliance matrix, the rotation transformation is:

S′11S′12S′16S′22S′26S′66

=

c4 s4 2c2s2 c2s2

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

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

S11S22S12S66

(14)

The stiffness components Q′11, Q′12, Q′22 and Q′66, and the compliance components S′11, S′12, S′22 and S′66 areeven functions of θ:

Q′11(−θ) = Q′11(θ) . . . (15)

The coupling components S′16, S′26, Q′16 and Q′26 are odd functions of θ:

Q′16(−θ) = −Q′16(θ) . . . (16)


Engineering constants

In mechanics, two engineering constants, the Young’s modulus E and the Poisson’s ratio ν, 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:

{σx 6= 0 , σy = τxy = 0εx and εy proportionnal to σx

⇒

εx = 1

Exσx

εy = −νxyεx = −νxyExσx

γxy = 0

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

{τxy 6= 0 , σx = σy = 0γxy proportionnal to τxy

⇒{γxy = 1

Gτxyεx = 0 and εy = 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, ν12, ν23, ν13, G23, G13, G12) are necessary to write the matrix in thesymmetry axes:

ε1ε2ε3ε4ε5ε6

=

1E1

−ν21E2

−ν31E3

0 0 0

−ν12E1

1E2

−ν32E3

0 0 0

−ν13E1

−ν23E2

1E3

0 0 0

0 0 0 1G23

0 0

0 0 0 0 1G13

0

0 0 0 0 0 1G12

σ1σ2σ3σ4σ5σ6

(18)

The symmetry relations are:

−ν21E2

=−ν12E1

;−ν31E3

=−ν13E1

;−ν32E3

=−ν23E2

(19)


We have also the following relation ν13ν21ν32 = ν12ν23ν31. The stiffness matrix is obtained by inverting thecompliance matrix which yields the following components:

C11 =(1− ν23ν32)

∆E1;

C12 =(ν21 + ν31ν23)

∆E1 =

(ν12 + ν32ν13)

∆E2;

C13 =(ν31 + ν21ν32)

∆E1 =

(ν13 + ν12ν23)

∆E3;

C22 =(1− ν13ν31)

∆E2; (20)

C23 =(ν32 + ν12ν31)

∆E2 =

(ν23 + ν12ν13)

∆E3;

C33 =(1− ν12ν21)

∆E3;

C44 = G23; C55 = G13; C66 = G12

with the parameter ∆ = 1− ν12ν21 − ν23ν32 − ν31ν13 − 2ν21ν32ν13.We have the condition ∆ > 0 to have positiv Cii modulus.

∆ = 1− ν212E2

E1− ν223

E3

E2− ν13

E3

E1− 2ν12ν23ν13

E3

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 0

0 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, ν12, ν23, G12) are necessary to write the stiffness matrix:


ε1ε2ε3ε4ε5ε6

=

1E1

−ν21E2

−ν21E2

0 0 0

−ν12E1

1E2

−ν23E2

0 0 0

−ν12E1

−ν23E2

1E2

0 0 0

0 0 0 2(1+ν23)E2

0 0

0 0 0 0 1G12

0

0 0 0 0 0 1G12

σ1σ2σ3σ4σ5σ6

and the symmetry condition

−ν21E2

= −ν12E1

(23)

The inversion of the compliance matrix yields the stiffness matrix:

[C] =

(1−ν23)δ E1

ν12δ E2

ν12δ E2 0 0 0

(1−ν12ν21)(1+ν23)δ

E2(ν23+ν12ν21)(1+ν23)δ

E2 0 0 0(1−ν12ν21)(1+ν23)δ

E2 0 0 0E2

2(1+ν23)0 0

Sym. G12 0G12

(24)

with the parameter δ = (1− ν23 − 2ν12ν21)We can also introduce the 5 engineering constants E1, E2, G12, ν12 et ν23:

[C] =

(1−ν23)δ E1

ν12δ E2

ν12δ E2 0 0 0

(1−ν212E2/E1)(1+ν23)δ

E2(ν23+ν212E2/E1)

(1+ν23)δE2 0 0 0

(1−ν212E2/E1)(1+ν23)δ

E2 0 0 0E2

2(1+ν23)0 0

Sym. G12 0G12

(25)

with the parameter δ = (1− ν23 − 2ν212E2E1

). We have the condition δ > 0.

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

The compliance matrix in the symmetry axes of the material has no (σ-γ) coupling components (S16 = S26 = 0)and depends of 4 engineering constants E1, E2, G12 and ν12:ε1ε2

ε6

=

1E1

−ν21E2

0

−ν12E1

1E2

0

0 0 1G12

︸︷︷︸

[S]

σ1σ2σ6

and symmetry condition−ν21E2

= −ν12E1

(26)

The inversion of the compliance matrix yields the stiffness matrix:σ1σ2σ6

=

1

(1−ν12ν21)E1ν21

(1−ν12ν21)E1 0ν12

(1−ν12ν21)E21

(1−ν12ν21)E2 0

0 0 G12

︸︷︷︸

[Q]

ε1ε2ε6

(27)


We can also introduce the 4 engineering constants E1, E2, G12 and ν12:σ1σ2σ6

=

1

(1−ν212E2/E1)E1

ν12(1−ν212E2/E1)

E2 01

(1−ν212E2/E1)E2 0

Sym. G12

︸︷︷︸

[Q]

ε1ε2ε6

(28)

We have the condition (1− ν212E2/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 (σ-γ) coupling components (S′16 6= 0, S′26 6= 0) and depends of 6 engineering constants E′1, E′2, G′12, ν ′12, µ′12, η′12:

ε′1ε′2ε′6

=

1E′1−ν′21E′2

η′12G′12

1E′2

µ′12G′12

Sym.1G′12

︸︷︷︸

[S′]

σ′1σ′2σ′6

(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 θ:

η′12G′12

= −2cs(c2

E1− s2

E2+ (c2 − s2)

(ν21E2− 1

2G12

))µ′12G′12

= −2cs(c2

E1− s2

E2− (c2 − s2)

(ν21E2− 1

2G12

)) (30)

with c = cos(θ) and s = sin(θ).

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 ν are necessary to describe the compliance and stiffness matrices:


ε1ε2ε3ε4ε5ε6

=

1E − ν

E − νE 0 0 0

1E − ν

E 0 0 01E 0 0 0

1G 0 0

Sym. 1G 0

1G

σ1σ2σ3σ4σ5σ6

and isotropy condition

G = E2(1+ν)

(32)

The stiffness matrix is given by:

σ1σ2σ3σ4σ5σ6

=

(1−ν)(1+ν)(1−2ν)E

ν(1+ν)(1−2ν)E

ν(1+ν)(1−2ν)E 0 0 0

(1−ν)(1+ν)(1−2ν)E

ν(1+ν)(1−2ν)E 0 0 0

(1−ν)(1+ν)(1−2ν)E 0 0 0

G 0 0Sym. G 0

G

ε1ε2ε3ε4ε5ε6

(33)

We can define a parameter d = (1 + ν)(1− 2ν) = (1− ν − 2ν2).

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

∆V

V= ε1 + ε2 + ε3 =

1− 2ν

E(σ1 + σ2 + σ3) (34)

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

σ1σ2σ3σ4σ5σ6

=

λ+ 2µ λ λ 0 0 0λ+ 2µ λ 0 0 0

λ+ 2µ 0 0 0µ 0 0

Sym. µ 0µ

ε1ε2ε3ε4ε5ε6

Lamé relations:λ = νE

(1+ν)(1−2ν)µ = E

2(1+ν)

(35)

Isotropic material in plane stress, 2D

Under plane stress condition σ33 = σ31 = σ23 = 0, and the Hooke’s law takes the form:ε1ε2ε6

=

1E − ν

E 01E 0

Sym.1G

σ1σ2σ6

and isotropy conditionG = E

2(1+ν)(36)

The inversion of the compliance matrix yields:σ1σ2σ6

=

1

(1−ν2)Eν

(1−ν2)E 01

(1−ν2)E 0

Sym. G

︸︷︷︸Isotropy condition:[Q]=[Q′]

ε1ε2ε6

(37)


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

Number ofIndependentParameters

Parameters

Non zerocomponents

inSym. Axes

Non zerocomponents

outsideSym. Axes

Hook’s law 21 Cij 36 36Othotropic 9 E1, E2, E3, ν12, ν13, ν23, G12, G13, G23 12 36Trans. Iso. 5 E1, E2, ν12, ν23, G12 12 36

UD Laminate 4 E1, E2, ν12, G12 5 9Isotropic 2 E, ν 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 ∂w0∂x

v(x, y, z 6= 0) = u0 − z ∂w0∂y

w(x, y, z 6= 0) = w0

⇒

ε1 = ∂u

∂x = ∂u0∂x − z

∂2w0∂x2

ε2 = ∂v∂y = ∂v0

∂y − z∂2w0∂y2

ε6 = ∂u∂y + ∂v

∂x = ∂u0∂y + ∂v0

∂x − z2∂2w0∂x∂y

(38)

Then if we introduce a curvature vector {k0} of the mid-plane and the strain vector {ε0} of the mid-plane, thevector {ε} representation of the strain tensor for a point on z coordinate is given by :ε1ε2

ε6

=

ε01ε02ε06

+ z

k01k02k06

⇒ {{ε} = {ε0}+ z{k0}{σ} = [Q]{ε} = [Q]{ε0}+ 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 σx , σy and σxy are small compared to unity.• Transverse shear strains σxz and σyz are negligible.• Tangential displacements u and v are linear functions of the z coordinate.• The transverse normal strain σz is negligible.• Each ply obeys Hooke’s 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 =

ˆ h/2

−h/2σi dz =

n∑k=1

ˆ hk

hk−1

σki dz (40)

Mi =

ˆ h/2

−h/2zσi dz =

n∑k=1

ˆ hk

hk−1

zσki dz

(41)

The matrix notation is:

N1

N2

N6

M1

M2

M6

=

A11 A12 A16 B11 B12 B16

A12 A22 A26 B12 B22 B26

A16 A26 A66 B16 B26 B66

B11 B12 B16 D11 D12 D16

B12 B22 B26 D12 D22 D26

B16 B26 B66 D16 D26 D66

ε01ε02ε06k01k02k06

(42)


Aij =

n∑k=1

Qkij (hk − hk−1) (43)

Bij =1

2

n∑k=1

Qkij(h2k − h2k−1

)(44)

Dij =1

3

n∑k=1

Qkij(h3k − h3k−1

)(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 θ and −θ for the ply orientation.


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

The components of the A matrix can be represented in a circular graphic with the variation of the θ angle ofthe laminate. The principal Young’modulus, the shear modulus and the Poisson’s 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 θ for a specially orthotropiclaminate [(0, 90)n]s

A11 A33 A12/A11

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

A11 A33 A12/A11


References

• P. Bompard, "Matériaux composites", Ecole Centrale Paris, in French, Master’s 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, "Généralités sur les matériaux composites", Ecole Centrale Nantes, in French, 43 pages,

http:/ /cel.archives-ouvertes.fr, (2008).• J.M. Berthelot, "Matériaux composites : Comportement mécanique et analyse des structures", 4ème

édition , 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 CalardAerospace & Advanced Composites GmbHc/o Forschungszentrum Seibersdorf2444 Seibersdorf, AustriaT: +43 (0)2622 90550 610F: +43 (0)2622 90550 [email protected]: / / www.aac-research.at


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