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Chapter 2. Laminate theory 9

Chapter 2

Laminate Theory

This chapter aims to give a brief description of the type of mechanical analysis applied to

determine the behaviour of the proposed structure which is made up of laminate composite

panels. The analysis includes the fundamentals required to understand the mechanical

behaviour of a deformable solid through the application of the theory of elasticity. From here

the elastic behaviour of the composite material is described through stress-strain relations and

visa versa, in terms of its engineering constants from a three dimensional state to the more

representative two-dimensional state of the composite plate. The effects of ply orientation are

examined with corresponding transformations between principal axis and orientated

coordinates outlined and their subsequent application and relevance in the project mentioned.

The analysis determines the conditions required to be met by the laminas so as to constitute a

laminate. If the laminas meet the conditions required, the classical theory outlined can be

appropriately applied to the laminate. The theory attempts to find effective and realistic

simplifying assumptions that reduces the three dimensional elastic problem to a two

dimensional one. It determines the response of the laminate to forces and moments acting on

the laminate by applying the hypothesis of thin laminates where a number of deformed

geometrical occurrences are assumed. Finally, other types of mechanical behaviour are

outlined in this chapter. These behaviours are considered worthy in presenting for discussion as

they are directly related to the project in terms of geometry and service conditions. These

topical mechanical behaviours include the presence of holes in laminates, vibration and fatigue.

2.1. Elastic Theory

Composite materials, as with all deformable solids, change shape at different points of the

material when a system of external loading is applied on the solid in equilibrium, giving rise to a

new geometric or deformed configuration. Figure 2.1 shows a deformable solid subjected to

the application of a system of forces indirectly where external loading is applied on some

arbitrary zones of the solid’s boundary but with limited displacement in another zone which

generates forces necessary to equilibrate the external applied system.


The physical magnitudes that are incurred in the deformation of a solid are the external

loading: applied in the body Xi and/or on the boundary ti. The second type of physical

magnitudes is the displacements ui of the body. The objective of the mechanical behavioural

analysis of a deformable solid is to determine its displacement when external loading is applied.

However, the solid’s displacement cannot be determined directly from the applied external

loading. It is therefore necessary to define internal variables that are related to the physical

magnitudes in equilibrium, these include the stresses σij and strains εij of the deformable solid.

Figure 2.1 represents the elastic problem in terms of its forces, displacements, stresses and

strains. Given that the stresses are related to the external loading, the same as the strains are

related with displacements and given the relation between the displacements and the loads, it

must exist a material relation between the stresses and the strains. This material relation is

known as the Behavioural Law or the constitutive equations of the material.

Figure 2.1: The problem of deformable solids

Analysing the elastic problem in the above figure, the relation between the exterior loads Xi and

ti (i =1,2,3) and stresses σij (i, j = 1,2,3) are the equations of internal equilibrium.

0, =+ ijij Xσ (2.1)

( )tijij Dtn ∂=σ (2.2)

Between displacements εij (i, j = 1,2,3) and strains are the equations of compatibility.

( )ijjiij uu ,,21 +=ε

(2.3)


( )uii Duu ∂= (2.4)

Between the stresses and strains are the constitutive equations or the Behavioural Law.

ijkkijij G δλεεσ += 2 (2.5)

ijkkijij E

v

E

v δσσε −+= 1 (2.6)

The elastic problem is therefore made up of a system of 15 differential equations which include

three equilibrium equations, six strain-displacement relations, and six constitutive equations. In

total, there are 15 unknowns, made up of six components from the stress tensor, six from the

strain tensor and three displacements [4].

2.2. Elastic Material Behaviour In Composite Materials

2.2.1. Stiffness Matrix C

The generalised Hooke's Law relating stresses to strains can be written as the following

expression

jiji C εσ .=

(2.7)

Where σi are the stress components, Cij is the stiffness matrix, and εj are the strain components.

The stress-strain relationship and the corresponding stiffness matrix for the anisotropic or

triclinic (no planes of symmetry for the material properties) linear elastic case are shown below.

=

12

13

23

33

22

11

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

12

13

23

33

22

11

2

2

2

ε

ε

ε

ε

ε

ε

σ

σ

σ

σ

σ

σ

CCCCCC

CCCCCC

CCCCCC

CCCCCC

CCCCCC

CCCCCC

(2.8)

where the stiffness matrix itself is symmetric, implying that only 21 of the 36 are independent

elastic constants. According to the material type, different extents of symmetry of material

properties occur and subsequent reduction in the number of elastic constants in the stiffness

matrix is observed. One of such is the stiffness matrix shown below which describes the case of


the stress-strain relations in coordinates aligned with the principal material directions i.e., the

directions that are parallel to the intersections of the three orthogonal planes of the material

property symmetry. This matrix defines an orthotropic material which is fundamental in the

composite analysis in this project. It is important to note also that orthotropic materials can

exhibit apparent anisotropy when stressed in non-principal material coordinates [3].

=

12

13

23

33

22

11

66

55

44

333231

232221

131211

12

13

23

33

22

11

2

2

2

00000

00000

00000

000

000

000

ε

ε

ε

ε

ε

ε

σ

σ

σ

σ

σ

σ

C

C

C

CCC

CCC

CCC

(2.9)

2.2.2. Compliance Matrix S

For ease of resolving the elastic material behaviour we define the inverse of the previous stress-

strain relation such that

jiji S σε .= (2.10)

where Sij is the compliance matrix which contains more reduced expressions of the elastic

constants. The complete 6x6 compliance matrix is given as

=

12

13

23

33

22

11

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

12

13

23

33

22

11

σ

σ

σ

σ

σ

σ

γ

γ

γ

ε

ε

ε

SSSSSS

SSSSSS

SSSSSS

SSSSSS

SSSSSS

SSSSSS

(2.11)

For an anisotropic material, there exists a significant coupling effect between the applied stress

and the resulting deformation. The types of coupling for above the strain-stress expression are

shown in figure 2.2. S11, S22 and S33 represent the coupling due to the individual applied stresses

σ1, σ2 and σ3, respectively, in the same direction. S44, S55 and S66 represent the shear strain

response due to the applied shear stress in the same plane. S12, S13 and S23 represent the


extension-extension coupling or coupling between the distinct normal stresses and normal

strains, also known as the Poisson effect. S15, S16, S24, S25, S26, S34, S35 and S36 represent the

shear-extension coupling or a more complex coupling of the normal strain response to applied

shear stress than for the preceding compliances. S45, S46 and S56 represent shear-shear coupling

or the shear strain response to shear stress applied in another plane. The remaining terms of

compliance matrix are a result of symmetry [3].

=

12

13

23

33

22

11

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

12

13

23

33

22

11

σ

σ

σ

σ

σ

σ

γ

γ

γ

ε

ε

ε

SSSSSS

SSSSSS

SSSSSS

SSSSSS

SSSSSS

SSSSSS

Figure 2.2: Physical significance of anisotropic stress-strain relations

For an anisotropic material, the compliance matrix components in terms of the engineering

constants are shown in equation (2.12), using the reduced index notation of Voigt (1910). The

values of the compliance matrix can be physically measured by specimen testing. The elastic

constants that can be physically measured include Young’s Modulus E, Poisson’s ratio v, shear

modulus G, and analytically measured constants include shear-extension coupling or mutual

influence coefficients η (Lekhnitskii), and shear-shear coupling coefficients μ (Chenstov).

−−

−−

−−

=

12

13

23

33

22

11

65

56

4

46

3

36

2

26

1

16

6

65

54

45

3

35

2

25

1

15

6

64

5

54

43

34

2

24

1

14

6

63

5

53

4

43

32

23

1

13

6

62

5

52

4

42

3

32

21

12

6

61

5

51

4

41

3

31

2

21

1

12

13

23

33

22

11

.

1

1

1

1

1

1

σ

σ

σ

σ

σ

σ

µµηηη

µµηηη

µµηηη

ηηηνν

ηηηνν

ηηηνν

γ

γ

γ

ε

ε

ε

GGGEEE

GGGEEE

GGGEEE

GGGEEE

GGGEEE

GGGEEE

(2.12)

Shear-Extension

Coupling

Shear-Shear

Coupling

Shear

Extension

Extension-Extension Coupling


In relation to more realistic cases of engineering problems of thin plate elements which include

panel-type composite structures, the 2-D case of plane stress of the lamina in principal axes is

characterised by the reductions below and is shown in figure 2.3.

031233 === σσσ

2231133 σσε SS +=

023 =ε

013 =ε

(2.13)

This idealisation is physically achieved as the lamina can only resist significant stresses in the

fibre direction, any stresses out of the 1-2 plane, such as σi3, would subject the lamina to

unnatural stresses.

Figure 2.3: Coordinates of unidirectional reinforced lamina [3]

This simplification reduces the 6x6 stiffness matrix to a 3x3 one and implies the following

reduction of the strain-stress relation as

=

6

2

1

662616

262212

162111

6

2

1

.

σ

σ

σ

γ

ε

ε

SSS

SSS

SSS

(2.14)

Following that, the engineering constants of the compliance matrix of the above relation are

shown in equation (2.15). It must be noted that Young’s Modulus and Poisson’s ratio can be

measured relatively efficiently through testing specimens with it is principal coordinates

coinciding with the orientated coordinates. However, the degree of accuracy of the measured

value of the shear modulus depends on the type of test procedure adopted where there are a

number of proposed procedures that include direct and indirect methods [5].


−

−

=

6

2

1

62

26

1

16

6

62

21

12

6

61

2

21

1

6

2

1

.

1

1

1

σ

σ

σ

ηη

ην

ην

ε

ε

ε

GEE

GEE

GEE

(2.15)

2.2.3. Orthotropic Lamina

For the orthotropic lamina, the stiffness matrix can be further reduced to the following:

=

12

2

1

66

2221

1211

12

2

1

.

00

0

0

σ

σ

σ

γ

ε

ε

S

SS

SS

(2.16)

Where there are only five constants of which only four are independent. The orthotropic

compliances in terms of the elastic constants are

111

1E

S = ;

222

1E

S = ;

1

12

2

212112 EE

SSνν −=−== ;

1266

1G

S =

(2.17)

The inverted strain-stress relation reduces to

=

12

2

1

66

2221

1211

12

2

1

2

.

00

0

0

ε

ε

ε

σ

σ

σ

Q

QQ

QQ

(2.18)

Where Qij are the reduced stiffnesses of the lamina that are related to the compliance matrix

components and elastic constants by


2112

12

122211

2211 1 νν−

=−

= E

SSS

SQ

2112

2122

122211

122112 1 νν

ν−

=−

== E

SSS

SQQ

2112

22

122211

1122 1 νν−

=−

= E

SSS

SQ

12

6666

1G

SQ ==

(2.19)

2.2.4. Ply Orientation

It is often necessary to move between the principal coordinates and the orientated coordinates

of the lamina. The first coordinate transformation considered below is utilised in the area of

design so as to determine the effect of the lamina properties when a load is applied in non-

principal material coordinates. The second transformation considered below is employed in the

area of material property testing, specifically in the area of testing where the principal

coordinates are not parallel to the applied loading. This transformation procedure is applied in

this project in the area of failure criteria written in parametric script language (APDL) described

in Section 4.4. The resulting stresses in global coordinates are extracted from the model in the

post processing stage and are transformed into their principal axes equivalents and subjected

to the applied failure criteria. Figure2. 4 represents the variation in terms of angle θ between

the off-axis coordinates (x,y) and the principal axes (1,2).

Figure 2.4: Rotation to principal material coordinates from off-axis coordinates [3]


Principal Axis (1,2) Off-Axis Coordinates (x,y)

The transformation equations in principal axes of the material to the off-axis coordinates for

the stress tensor are given by the expression below where θ is the angle from the x-axis to the

1-axis as demonstrated in the above figure.

−−

−

=

12

2

1

22

22

22

.

sincossin.cossin.cos

sin.cos2cossin

sin.cos2sincos

σ

σ

σ

θθθθθθ

θθθ

θθθθ

σ

σ

σ

xy

y

x

(2.20)

Similarly, the same transformation matrix is applied to the strain tensor, the expressions for

both transformed stress and strain tensors are written in short as

[ ]

=

−

12

2

1

1.

σ

σ

σ

σ

σ

σ

T

xy

y

x

[ ]

=

−

12

2

1

1.

ε

ε

ε

ε

ε

ε

T

xy

y

x

(2.21)

Where the inverse of the transformation matrix in short is written as

[ ]

−−

−

=−

22

22

22

1

..

.2

.2

scscsc

sccs

scsc

T

(2.22)

Off-Axis Coordinates (x, y) Principal Axis (1, 2)

The transformation of the equations of the off-axis coordinates to the principal axis of the

material stress tensor is

−−

−=

xy

y

x

σ

σ

σ

θθθθθθ

θθθθ

θθθθ

σ

σ

σ

.

sincossin.cossin.cos

sin.cos2cossin

sin.cos2sincos

22

22

22

12

2

1

(2.23)

And as before, the same transformation matrix is applied to the strain tensor, the expressions

for both transformed stress and strain tensors are written in short as


[ ]

=

xy

y

x

T

ε

ε

ε

ε

ε

ε

.

12

2

1

[ ]

=

xy

y

x

T

σ

σ

σ

σ

σ

σ

.

12

2

1

(2.24)

Where the transformation matrix in short is written as

−−

−=22

22

22

..

.2

.2

scscsc

sccs

scsc

T

(2.25)

Resolving the transformation in the equations in (2.23), the stress in principal axis in plane

stress are the following

θθσθσθσσ sin.cos2sincos 221 xyyx ++=

θθσθσθσσ sin.cos2cossin 22

2 xyyx −+=

( )θθσθθσθθσσ 22

12 sincossin.cossin.cos −++−= xyyx

(2.26)

And the strains in the principal axis in the plane stress state are

θθεθεθεε sin.cos2sincos 22

1 xyyx ++=

θθεθεθεε sin.cos2cossin 22

2 xyyx −+=

( )θθεθθεθθεε 22

12 sincossin.cossin.cos −++−= xyyx

(2.27)

2.2.5. Transformed Stiffness and Compliance Matrices

It is possible to substitute the transformation in (2.21) into the stress-strain relations in the

principal material coordinates in (2.18) in order to obtain the stress-strain relations in

orientated or off-axis coordinates which are expressed in the following relation

[ ] [ ]

=

−

xy

y

x

xy

y

x

T

Q

QQ

QQ

T

ε

ε

ε

σ

σ

σ

2

.

00

0

0

66

2221

1211

1

(2.28)

Resolving the matrices in (2.28), the stress-strain relation in xy coordinates is


=

xy

y

x

xy

y

x

QQQ

QQQ

QQQ

γ

ε

ε

σ

σ

σ

.

662616

262221

161211

(2.29)

In which [ ] [ ][ ]TQTQ 1−= is the component of the stiffness matrix of the transformed lamina

and is defined as

( ) θθθθ 226612

422

41111 cossin22sincos QQQQQ +++=

( ) ( )θθθθ 44

1222

6622112112 sincoscossin4 ++−+== QQQQQQ

( ) θθθθ 22

66124

224

1122 cossin22cossin QQQQQ +++=

( ) ( ) θθθθ 3

6612223

6612116116 sincos2sincos2 QQQQQQQQ −−−−−==

( ) ( ) θθθθ sincos2sincos2 3

6612223

6612116226 QQQQQQQQ −−−−−==

( ) ( )θθθθ 44

6622

6612221166 cossincossin22 ++−−+= QQQQQQ

(2.30)

Where the ijQ matrix denotes that we are dealing with the transformed reduced stiffness

instead of the reduced stiffness ijQ . It is worth noting that the transformed reduced stiffness

matrix contains terms in all nine positions of the matrix while the reduced stiffness matrix

contains a number of zero terms. Alternatively to the above the procedure, the compliance

matrix in strain-stress relations in orientated coordinates is given as

=

xy

y

x

xy

y

x

SSS

SSS

SSS

σ

σ

σ

γ

ε

ε

.

662616

262212

161211

(2.31)

where the transformed orthotropic compliances ijS are

( ) 4

2222

66124

1111 sincossin2cos SSSSS +++= θθθ

( ) ( ) 22

66221144

1212 cossincossin θθθ SSSSS −+++=

( ) 4

2222

66124

1122 coscossin2sin SSSSS +++= θθθ

( ) ( ) θθθθ cossin22cossin22 3

6612223

66121116 SSSSSSS −−−−−=

( ) ( ) θθθθ 3

6612223

66121126 cossin22cossin22 SSSSSSS −−−−−=

(2.32)


( ) ( )θθθθ 44

6622

6612221166 cossincossin4222 ++−−+= SSSSSS

where the anisotropic compliances in terms of engineering constants are

111

1

ES = ;

222

1

ES = ;

1

12

2

212112 EE

SSνν

−=−== ;

1266

1

GS =

6

16

1

6116 GE

Sηη == ;

6

26

2

6226 GE

Sηη ==

(2.33)

where the new engineering constants are called the coefficients of mutual influence η by

Lekhnitskii which were presented in the compliance matrix of the strain-stress relations in

(2.14) and are defined as

ij

iiji γ

εη =, ;

i

ijiij ε

γη =,

(2.34)

where iji,η is the coefficient of mutual influence that characterises the stretching in the i-direction

caused by shear stress in the ij-plane and iij ,η

is the coefficient of mutual influence that

characterises shearing in the ij-plane caused by normal stress in the i-direction. Note that the

mutual influences given in (2.34) are expressed in Voigt notation.

The presence of the 16Q and 26Q , and 16S

and 26S in the stiffness and compliance matrices,

respectively, creates a more complex problem solution of the generally orthotropic laminas

than that of the specially orthotropic laminas. The presence of the mutual influence coefficients

causes shear-extension coupling which complicates the solution of practical problems [3], [5].

2.3. Mechanical Behaviour of Laminate

A laminate is two or more laminas or plies bonded together to act as a unique structural

element. The laminas are required to meet certain conditions so as to constitute a laminate,

also the laminate response as a result of imposed boundary conditions including support

conditions and loading. The mechanical behaviour of the laminate is presented in this project

on a macromechanical scale in which the individual components of the lamina such as the fibre

and matrix are not considered individually but the entire lamina and its response in the

laminate. The conditions required by two laminas of different orientations perfectly bonded in

a laminate include deformation compatibility: the laminas in the laminate must deform alike

along the interface between those laminas in the direction of the applied force and the stresses

in the transversal direction must be self-equilibrating so as to comply with the deformation

compatibility. The other two conditions include stress-strain relations and equilibrium.


Difficulties arise when more than two laminas of arbitrary angles are contained in the laminate

and thus a different approach namely the Classical Lamination Theory (CLT) is required to

satisfy the required conditions already mentioned. The CLT approach attempts to find effective

and realistic simplifying assumptions that reduces the three dimensional elastic problem to a

two dimensional one. The process includes a review of the stress-strain behaviour of an

individual lamina which is expressed as the kth

lamina in the laminate. Secondly, the stress and

strain variations through the thickness of the laminate are determined. Finally, the relation of

the laminate forces and moments to the strains and the curvatures are characterised [3].

2.3.1. Formulation of the Laminate (Constitutive Equations)

Considering the first part of the process for the CLT approach which includes the stress-strain

behaviour of an individual lamina, the stress-strain relations in principal axis for a lamina of an

orthotropic material under plane stress are given in (2.18) and for ease of demonstrating the

approach are shown again below.

=

12

2

1

66

2221

1211

12

2

1

.

00

0

0

γ

ε

ε

σ

σ

σ

Q

QQ

QQ

(2.35)

As a result of the arbitrary orientation of the laminas, the stresses and strains of the laminas are

resolved into the in-plane orientated coordinates so as to define the laminate stiffness.

Similarly, these stress-strain relations and the transformed reduced stiffness matrix are given in

(2.29) and again, are shown below.

=

xy

y

x

xy

y

x

QQQ

QQQ

QQQ

γ

ε

ε

σ

σ

σ

.

662616

262221

161211

(2.36)

In general for a lamina that occupies the kth

position in the laminate, the previous expression

can be written as

{ } [ ] { }kkk Q εσ =

(2.37)

The CLT approach assumes that the complete laminate acts as a single layer where there is

perfect bonding between the laminas enabling continuous displacement between the laminas

so that no lamina can slip relative to the other. The Hypothesis of Kirchhoff assumes that, if the

laminate is thin, a line that is originally straight and perpendicular to the middle surface of the


laminate before deformation is assumed to remain straight and perpendicular to the middle

surface when the laminate is deformed. Figure 2.5 shows from left to right the thin laminate

and its orientation, a sectional view (xz-plane) of the laminate in both the undeformed and

deformed state.

Figure 2.5: Laminate axis orientation, laminate section before and after deformation [3]

The Hypothesis of Kirchhoff in which the normal to the middle surface remains straight is

depicted in the figure above. This assumption thereby ignores the shearing strains in planes

perpendicular to the middle surface, that is

0== yzxz γγ (2.38)

In addition, the lines perpendicular to the middle surface are presumed to have a constant

length so that the strain perpendicular to the middle surface is ignored

0=zε (2.38)

The laminate cross section derives the Hypothesis of Kirchoff in which the displacement in the

x-direction of the point B (middle surface) from the undeformed to deformed state is u0.

Because the line ABCD remains straight after deformation, the displacement of point C in the x-

direction is

β.0 zuu −= (2.39)

From the Hypothesis of Kirchoff-Love for shells where under deformation, the line ABCD

remains perpendicular to the middle surface, β is the slope of the middle laminate surface in

the x-direction and is

x

w

∂∂= 0β (2.40)


Then, the displacement at any point through the laminate thickness is

x

wzuu

∂∂−= 0

0 (2.41)

Similarly, for the displacement in the y-direction is

y

wzvv

∂∂−= 0

0 (2.42)

As a consequence of the Hypothesis of Kirchoff, the remaining laminate strains are defined in

terms of displacements as

20

20

x

wz

x

u

x

ux ∂

∂−∂∂=

∂∂=ε

20

20

y

wz

y

v

y

vy ∂

∂−∂∂=

∂∂=ε

yx

wz

x

v

y

u

x

v

y

uxy ∂∂

∂−∂∂+

∂∂=

∂∂+

∂∂= 0

200γ

(2.43)

or they can be expressed in vector form as

+

=

∂∂∂

−

∂∂

−

∂∂

−

+

∂∂

+∂∂

∂∂∂

∂

=

0

0

0

0

0

0

02

20

2

20

2

00

0

0

2 xy

y

x

xy

y

x

xy

y

x

k

k

k

z

yx

wy

wx

w

z

x

v

y

uy

vx

u

γ

ε

ε

γ

ε

ε

(2.44)

where 0xε , 0

yε and 0xyγ are the three middle strains (elongations and distortions) and 0

xk , 0yk and

0xyk are the three middle-surface curvatures (bending curvatures and torsion). The stress-strain

relations given in (2.36) can be modified by the substitution of the strain variation through the

thickness given above in (2.44). The stresses for the kth

layer are expressed in terms of the

laminate middle-surface strains and curvatures as


+

=

0

0

0

0

0

0

662616

262221

161211

xy

y

x

xy

y

x

kk

xy

y

x

k

k

k

z

QQQ

QQQ

QQQ

γ

ε

ε

σ

σ

σ

(2.45)

where z corresponds with the coordinates of the kth

lamina. The component of the stiffness

matrix ijQ can be different for the each layer of the laminate. That implies that the stresses at

the interface are not continuous even though the strain variation is linear through the lamina

interface. Figure 2.6 demonstrates the distribution of strain ε, characteristic stiffness moduli Q

and stress σ distribution for a four layer laminate. While the stress variation is discontinuous at

the interface it does vary linearly within each of the laminas [5].

Figure 2.6: Strain and stress distribution [5]

The final stage of the CLT approach includes the characterisation of the relation of the laminate

forces and moments to the strains and the curvatures. The loading includes Nx which is a force

per unit width (in-plane) of the cross section of the laminate and Mx which is a moment per unit

width and is shown acting on the laminate in figure 2.7.

Figure 2.7: In-plane forces and moments on a laminate [5]


The resultant forces and moments acting on a laminate, as shown in the above figure, are

obtained by integration of the stresses in each layer or lamina through the laminate thickness

and are defined as

dzdz

N

N

Nk

N

k

z

z

xy

y

x

t

t

xy

y

x

xy

y

x

k

k∑∫∫

=−

−

=

=

1

2

2 1

σ

σ

σ

σ

σ

σ

(2.46)

dzzdzz

M

M

Mk

N

k

z

z

xy

y

x

t

t

xy

y

x

xy

y

x

k

k

..1

2

2 1∑∫∫

=−

−

=

=

σ

σ

σ

σ

σ

σ

(2.47)

where zk and zk-1 are the laminate geometry and the configurations of the laminas are shown in

figure 2.8 in which z is positive downwards.

Figure 2.8: Lamina configurations [5]

The stress-strain relations in (2.45) can be substituted into the forces and moments equations

in (2.46) and (2.47), respectively, and the results of these substitutions are shown below in

(2.48) and (2.49). If there does not exist temperature dependent or moisture dependent

properties and a temperature gradient or a moisture gradient in the lamina, the stiffness matrix

can be taken outside the integration over each layer but remains within the summation of the

force and moments resultants for each layer. If an elevated temperature or moisture exists

throughout the layers the stiffness matrix remains constant but its value is altered due to

degradation. In cases where the stiffness matrix is not constant throughout the layers, it


remains within the integration over each layer thereby leading to a more complicated

numerical solution [3], [5].

∑ ∫ ∫=

+

=

− −

N

k

z

z

z

z

xy

y

x

xy

y

x

k

xy

y

x

zdz

k

k

k

dz

QQQ

QQQ

QQQ

N

N

N

k

k

k

k10

0

0

0

0

0

662616

262221

161211

1 1

γ

ε

ε

(2.48)

∑ ∫ ∫=

+

=

− −

N

k

z

z

z

z

xy

y

x

xy

y

x

k

xy

y

x

dzz

k

k

k

zdz

QQQ

QQQ

QQQ

M

M

M

k

k

k

k1

2

0

0

0

0

0

0

662616

262221

161211

1 1

γ

ε

ε

(2.49)

Given that the three middle strains ( 0xε , 0

yε , 0xyγ ) and the three middle-surface curvatures ( 0

xk ,

0yk , 0

xyk ) are independent of z, and are instead middle surface values, they can be removed

from within the summation signs. The equations in (2.50) and (2.51) can be written as

+

=

oxy

oy

ox

oxy

oy

ox

xy

y

x

k

k

k

BBB

BBB

BBB

AAA

AAA

AAA

N

N

N

662616

262212

161211

662616

262212

161211

γ

ε

ε

(2.50)

+

=

oxy

oy

ox

oxy

oy

ox

xy

y

x

k

k

k

DDD

DDD

DDD

BBB

BBB

BBB

M

M

M

662616

262212

161211

662616

262212

161211

γ

ε

ε

(2.51)

where

( )∑=

−−=N

kkk

kijij zzQA

11

( )∑=

−−=N

kkk

kijij zzQB

1

21

2

2

1

( )∑=

−−=N

kkk

kijij zzQD

1

31

3

3

1

(2.52)


The Aij are extensional stiffnesses with A16 and A26 representing shear-extension coupling, the

Bij are bending-extension coupling stiffnesses, and the Dij are bending stiffnesses with D16 and

D26 representing bend-twist coupling. The presence of Bij implies coupling between bending and

extension of a laminate. This in physical terms causes not only extensional deformations but

bending and/or twisting of the laminate when only an in-plane force, e.g. Nx is applied on the

laminate [3].

2.4. Other Analysis and Behavioural Topics

The complexity of the composite model requires a number of mechanical behavioural topics to

be analysed. The ones felt most relevant to this project are included below. Those presented

are a direct consequence of the model’s profile (holes in laminates) and its subjected

environment (vibration and fatigue in laminates).

2.4.1. Holes in Laminates

The existence of holes in composite laminate structures is a result of numerous service and

mechanical requirements including weight and surface area reduction, bolt accommodation,

and access through the structure. In isotropic materials, the main influence for failure with

holes is due to the magnitude of the stress concentration factor from which the maximum

stress is obtained. However, for orthotropic materials, a combined stress failure criterion is

required. It includes stress concentration factors at the hole’s edge and an appropriate failure

criterion for composite materials as described in Section 4.3. Many isotropic materials such as

aluminium or steel are, in terms of deformation before failure, more ductile than composite

materials thereby allowing localised yielding to accommodate stress concentrations in these

critical zones whereas the majority of composite materials contain higher stress concentrations

and a lesser ability to yield than isotropic materials.

The stress concentration factor around the circumference of the hole is caused by the

combination of the principal material direction and secondly the load direction in which the

material is subjected. Where the principal material direction does not coincide with the loading,

the lamina is considered effectively as being anisotropic or generally orthotropic. Figure 2.9

shows a lamina with its fibre direction at an arbitrary angle α from the x-direction of loading.

The angle θ represents the circumferential stress at the edge of the circular hole and thus its

magnitude varies in accordance with the fibre direction. As α approaches 90o, the peak stress

concentration factor decreases and shifts its location θ around the hole. As a result, stress

concentrations around the hole circumference are quite intrinsic. Its complexity increases with

the analysis of a laminate with laminas of various orientations where each layer and their

stresses must be determined by the use of the Classic Laminate Theory approach and applied to

an appropriate strength criterion for failure analysis.


Figure 2.9: Loading and principal material direction of composite lamina

Stress concentration around holes in composite laminates can be reduced by a method known

as the Stiffening Strip Concept. This process includes the addition of stiffer composite material

in the zones located on either side of the hole but away from its boundary. The concept of the

stiffener is to remove loading from around the hole boundary by transferring the loading

through the stiffener itself. A second method is the addition of a more flexible strip situated

right at the edge of the hole so as to reduce the load concentration at the holes edge and

transferring it to some other unknown region of the laminate.

2.4.2. Vibration of Laminates

The main objective of this type of analysis is to determine the response of the laminate due to

vibration in terms of its magnitude of deflection and its modes shapes. Vibration is a transverse

load which causes deflection of the laminate due to bending and is generally larger than in-

plane deflections, because flexural stiffnesses are lower than extensional stiffnesses. The

general equilibrium equations governing transverse deflections include both in-plane and out-

of-plane forces. The analysis of laminate or plate deflections is based on the CLT outlined in

Section 2.3 and in the differential equations of equilibrium. For clarity of representation, the

differential equations are developed more conveniently through the use of a planar element

dimensioned dx by dy. Figure 2.10 shows the in-plane stress resultants (a), the moment

resultants (b) and the transverse shear resultants (c). Because the plate does not remain flat

during vibration, the analysis cannot be derived from equilibrium of the differential element

and it is therefore assumed that the transverse deflections remain small, so that the out-of-

plane components of the in-plane resultants Nx, Ny, and Nxy are negligible [6].


Figure 2.10: Stress, moments, and transverse shear resultants of laminate [6]

The equilibrium differential equations for vibration of a composite laminate with arbitrary ply

orientations are presented below beginning with the summation of forces along the x-direction

as

2

02

0 t

udxdydxNdyNdxdy

y

NdxNdxdy

x

NdyN xyx

xyxy

xx ∂

∂=−−∂

∂++

∂∂+ ρ (2.53)

where ρ0 is the mass per unit area of laminate and u0(x, y, t) is the middle surface displacement

in the x-direction. The previous equation can be simplified to

2

02

0 t

u

y

N

x

N xyx

∂∂=

∂∂

+∂

∂ ρ (2.54)

Similarly, the summation of forces along the y-direction gives

2

02

0 t

vdxdydyNdxNdxdy

x

NdyNdxdy

y

NdxN xyy

xyxy

xx ∂

∂=−−∂

∂++

∂∂+ ρ (2.55)

and simplifies to

2

02

0 t

v

x

N

y

N xyy

∂∂=

∂∂

+∂

∂ρ (2.56)


where v0(x, y, t) is the middle surface displacement in the y-direction. The summation of the

forces along the z-direction yields

( )2

2

0,t

wyxqdxQdyQdxdy

y

QdxQdxdy

x

QdyQ yy X

yX

X ∂∂=+−−

∂∂

++∂

∂+ ρ (2.57)

where

dzQt

t xzx ∫−=2

2σ dzQ

t

t yzy ∫−=2

2σ

(2.58)

and simplifies to

( )2

2

0,t

wyxq

y

Q

x

QyX

∂∂=+

∂∂

+∂

∂ρ (2.59)

where w(x, y, t) is the displacement in the z-direction.

For the moment equilibrium, the moments are considered about the x-axis and y-axis but

rotary inertia is neglected. The summation of the moments about the x-axis simplifies to

yxyy Q

x

M

y

M=

∂∂

+∂

∂ (2.60)

And similarly, the summation of moments about the y-axis yields

xxyx Q

y

M

x

M =∂

∂+

∂∂

(2.61)

Substitution of the two moments in equations (2.60) and (2.61) in equation of (2.59) produces

( )2

2

02

22

2

2

,2t

wyxq

y

M

yx

M

x

M xyxyx

∂∂=+

∂∂

+∂∂

∂+

∂∂ ρ (2.62)

The laminate force-deformation equations in (2.48) and the strain and curvatures relations in

terms of displacement in (2.43) are substituted into differential equations of motion (2.54),

(2.56), and (2.62) to produce the corresponding equations of motion in terms of displacements.

( )3

3

112

02

26

02

66122

02

162

02

66

02

162

02

11 2x

wB

y

vA

xdy

vAA

x

vA

dy

uA

xdy

uA

x

uA

∂∂−

∂∂+

∂∂++

∂∂+∂+

∂∂+

∂∂

( ) 023 3

3

262

3

66122

3

16 =∂∂−

∂∂∂+−

∂∂∂−

y

wB

yx

wBB

yx

wB

(2.63)


( ) 3

3

162

02

22

02

262

02

662

02

26

02

66122

02

16 2x

wB

y

vA

xdy

vA

dx

vA

dy

uA

xdy

uAA

x

uA

∂∂−

∂∂+

∂∂+∂+∂+

∂∂++

∂∂

( ) 032 3

3

222

3

262

3

6612 =∂∂−

∂∂∂−

∂∂∂+−

y

wB

yx

wB

yx

wBB

(2.64)

( )3

03

114

4

223

4

2622

4

66123

4

164

4

11 4224x

uB

y

wD

yx

wD

yx

wDD

dyx

wD

x

wD

∂∂−

∂∂+

∂∂∂+

∂∂∂++

∂∂+

∂∂

( ) ( )yx

vBB

x

vB

y

uB

yx

uBB

yx

uB

∂∂∂+−

∂∂−

∂∂−

∂∂∂+−

∂∂∂−

2

03

66123

02

163

03

262

03

66122

03

16 223

( )yxqy

vB

yx

vB ,3

3

03

222

03

26 =∂∂−

∂∂∂−

(2.65)

The various coupling stiffnesses such as A16 and A26 (shear-extension coupling), Bij (bending-

extension coupling), and D16 and D26 (bend-twist coupling) are present in the above equilibrium

equations analysis and must be considered in their effect on the vibration behaviour of the

laminate plate. It is important to recognise the effect of the lamina configuration within the

laminate on the various coupling stiffnesses. If the laminate is symmetric about the middle

surface (as is intended to be the case for all the modelled composite structures in this project)

the bending-extension coupling Bij is reduced to zero [3]. Furthermore, if the laminate is

specially orthotropic i.e. the principal material directions coincide with the loading direction,

the shear-extension coupling and the bend-twist coupling simplifying equation (2.65) of

transverse displacements to

( ) ( )yxqy

wD

yx

wDD

x

wD ,22 4

4

2222

4

66124

4

11 =∂∂+

∂∂∂++

∂∂

(2.66)

2.4.3. Fatigue

The vast majority of service failures in materials are due to fatigue of the material. Fatigue of

isotropic materials has been investigated for many years and its process is quite well

documented. However, fatigue of orthotropic and anisotropic composite materials is relatively

new in comparison. Fatigue of unidirectional composites is generally controlled by the lamina

with orientation 0o even with the laminate in question containing laminas of various

orientations. Due to the importance of the effects of fatigue in service life, testing of

representative laminate specimens of the structure for an appropriate load history is required

to determine the life of the structure or the number of load cycles before failure.


Fatigue is controlled by a number of methods including displacement, energy and load

controlled tests with the ultimate considered the most appropriated to represent actual fatigue

life in service conditions. The S-N diagram describes the applied global stress level with respect

to the number of cycles to failure. For composite materials, the diagram is more readably

interpreted if it is replotted with the maximum strain attained in the first load cycle against the

number of cycles (log) to failure. The maximum strain recorded in the first load cycle can be

described as the damage state reached in the initial stage which is seen to contribute to any

progression of the damage after the initial cycles and during the course of the fatigue life.

The fatigue life diagram consists of three distinct regions as depicted in figure 2.11 and

represents regions of different damage mechanisms incurred by the composite material. These

failure mechanisms are associated damage of the fibre and matrix components [5].

Figure 2.11: Fatigue life diagram of longitudinal composites in tension-tension fatigue [7]

Region I, known also as the static region, is the zone in which the strain level coincides with the

maximum strain level of the static test. The mechanism in this region is evidently breakage of

the fibre in the 0o direction which is similar to that of static testing where fibre breakage in the

composite is random.

Region II or the progressive region is the zone consisting of a downward slope that is a

consequence of the decrease in the strain level and an increase in life. The mechanisms

attributed to failure in this region include fibre bridged cracking, and debond propagation.

Region II can be further subdivided in terms of macroscopic fatigue damage mechanisms which

include fibre breakage as being the prevalent mechanism at high load levels (high portion of

region II) or known also as initiation triggered mechanisms. At low load levels the, the main

mechanism is matrix or interface crack propagation.


Region III is the fatigue limit of the composite. Below this limit failure does not occur prior a

large number of cycles of typically 106 or 10

7 cycles. In this region, the damage is constrained

and obstructed from further growth by the fibres. Crack arrest and subsequent inhibition of

damage accumulation is believed to be caused by the fact that the strain level is too low and

the threshold value for propagation is not reached and secondly, the fibre-matrix debonding

and crack arresting by proximate fibres prevent damage accumulation and subsequent failure.

It is important to note that composites with high fibre mechanical resistance and less ductile

matrices have an adverse effect on the fatigue performance. Graphically, this resembles a

steeper slope in the scatter band of region II and an increased fatigue limit as shown in figure

2.12.

Figure 2.12: Fatigue life influenced by fibre stiffness and matrix toughness [7]

It has been observed from numerous investigations that multidirectional composites are more

sensitive to fatigue in tension-compression loading than in tension-tension loading. This

occurrence can be attributed to the greater number of transverse cracks that appear in cross-

ply laminates (e.g. 90o

plies) under tension-compression loading than that of the same laminate

under tension-tension loading. Observations show that the rate of debond propagation is

higher in tension-compression loading ply which subsequently causes an accelerated initiation

of transverse cracks and a reduction in fatigue life [7].

chapter 2 laminate theory - universidad de...

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