composite material & structures 3

8/22/2019 Composite Material & Structures 3

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Composite material & structures

Unit III

Governing differential equation for a

laminated plate, angle ply and cross

ply laminates. Failure criteria forcomposites.


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Analysis of laminates.

A lamina is the fundamental unit of composites.

A lamina ( also called a ply or layer) is a single flat

layer of unidirectional fibres or woven fibresarranged in a matrix.

A laminate is a stack of plies of composites. Each

layer can be laid at various orientations and can be

of different material systems.


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A real structure will consist of laminates

consisting of more than one lamina bonded

together through their thickness.

The reason is that,

1) Lamina thicknesses are of the order of 0.125mm, implying several laminae will be required to

take realistic loads.

(for instance, a glass fibre/epoxy lamina will fail

at a load of 13 kN per m width, along the fibre

direction.)


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2) The mechanical properties of a lamina are

severely limited in the transverse direction. If onestacks several unidirectional layers, it may be an

optimum laminate for unidirectional loads. For

complex loading, this may not be desirable. One can

overcome this by making a laminate with layersstacked at different angles for given loading and

stiffness requirements. This approach increases the

cost and weight of the laminate and hence one need

to optimize ply angles. More over one may use layers

of different composite material systems to develop a

more optimum laminate.


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Laminate code:

A laminate is made up of a groupof single layers bonded to each x

other. Each layer can be identified z

by its location in the element,its material, and its angle of fig. schematic of a

orientation with a reference axis. laminate

Each lamina is represented by the

angle of ply and separated from other

plies by a slash sign. The first ply is the

top ply of the laminate.

y


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Example -1:

[ 0/-45/90/60/30 ] denotes the 0

code of the laminate shown here. -45It consists of five plies, each of which 90

has a different angle with the 60

reference x axis. A slash separates 30each lamina. The above code also

implies that each ply is made of the

same material and is of same thickness.

Sometimes [0/-45/90/60/30]T may also denote

this laminate, where the subscript T stands for

total laminate .


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Example -2:[0/-45/902/60/0] denotes the 0

lamina shown here. It consists of -45

six plies. Since there are two 9090 plies adjacent to each other, 90

902 denotes them, where the 60

subscript 2 is the number of 0

adjacent plies of the same angle.


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Example-3:

[0/-45/60]s denotes the laminate 0shown here. It consists of six plies. -45

Since plies above the mid- surface 60

are of the same orientation, 60

material, and thickness as the -45plies below the mid-surface, 0

it is a symmetric laminate.

The top three plies are

written in the code, while the

subscript s outside the brackets

represents that the three plies are

repeated in the reverse order.


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Example-4:[0/-45/60]s denotes the 0

laminate shown here. -45

It contains five plies. 60Since the number of plies -45

is odd and symmetry exists 0

at the mid- surface, 60 plyis denoted with a bar on the top.


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Example-5:

[0Gr

/45B

]s denotes the graphite/epoxy 0laminate shown here. Boron / epoxy 45

It contains six plies; Boron / epoxy -45

the 0 plies are made Boron / epoxy -45

of graphite, the 45 plies Boron / epoxy 45

are made of boron. Graphite/epoxy 0

Note that 45 notation

indicates that 0 ply befollowed by +45 and

then -45 ply. S indicates symmetry.


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Special cases of laminates:

1. Symmetric laminate:

A laminate is called symmetric if the material,angle, and thicknss of plies are the same above

and below the midplane. An example is :

[0/30/60]sI 0 I 30 I 60 I 30 I 0 I


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2. Cross ply laminate:

A laminate is called cross ply laminate ( alsocalled laminates with specially orthotropic layers),

if only 0 and 90 plies are used to make a

laminate. For example;

I 0 I 90 I 0 I 90 I 90 I 0 I 90 I

3. A laminate is called angle ply laminate, if it has

plies of same material and thickness and orientedonly at + and directions.

For example: I 40 I-40 I 40 I -40 I


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4. Antisymmetric laminates:

A laminate is called antisymmetric if the material andthickness of the plies are the same above and below

the midplane, but the ply orientation at the same

distance above and below of the midplane are

negative of each other. An example is,[45/60/-60/-45].

5. Balanced laminate:

When the laminate consist of pairs of layers of the

same thickness and material, where the angle of plies

are + and. An example is [30/40/-30/30/-30/-40]


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Governing differential equation for a laminate:

As on now, we can assume that the properties

of basic lamina, in terms of elastic constants, are

known. The next logical step in the theoreticalanalysis of composite structures would be to

consider two or more plies or laminae that are

bonded together to form a laminated composite

plate. We have already seen various

combinations of lay up in forming a laminate.

-contd-


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This now becomes a structural rather than a

material problem and we must developmethods by which stresses and strains in each of

the plies and in the interlaminar bond layers can

be predicted for a given loading condition. Thisdevelopment will begin with plates. The classical

laminate theory (CLT) is used to develop the

relations.

-contd-


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Assumptions:

The following assumptions are made in theclassical lamination theory to develop these

relations:

1. Each lamina is orthotropic and homogeneous.2. A line straight and perpendicular to the

middle surface remains straight and

perpendicular to the middle surface during

deformation.(xz=yz=0).

3.A straight line in the z direction remains of

constant length. (z =0)

-contd-


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4.The laminate is thin and the displacements are

continuous and small throughout the laminate

( IuI, IvI, IwI


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Derivation of equations for unidirectional

plane anisotropic plates


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Strain- displacement relation:

If we assume that the classical small deformation

theory is applicable, then we have,

the linear strains,

x = u/x, y= v/y, z= w/z ---- (1)and the angular or shear strains,

xy= yx = (u/y) + (v/x)

xz = zx = (u/z) + (w/x) --------(2)yz = zy = (v/z) + (w/y)

-contd-


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where u,v,w are the displacements in the

x-, y-, z- directions, respectively.

Since these equations are determined from the

geometry of deformations and do not involvethe material properties, they are applicable

regardless of the elastic symmetry of the

material.

-contd-


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Now assume that under the action of a load, the

plate element shown in fig.1, deforms in such a way

that the points 1 and 2 deforms to points 1 and 2.Further, the line element 1-2 perpendicular to the

middle surface of the plate(the x-z plane in the figure)

do not change in length and remain straight and

normal to the deflected middle surface afterdeformation.

Then it can be seen that the deflection of the typical

point 2 becomes,

u = -z(w/x) ---------- (3)

similarly , v = -z (w/y) ----------(4)

-contd-


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2

1

x

x

x

1

2 z

w

x

z

zw/x

u

Fig-1 Deformation of a plate element


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If we substitute these values in eqns (1) and (2) and

neglect the strains in the z- direction,

we get, x = -z2w/x2

y = -z2w/y2

---- (5)

and xy = (-z2w/xy) + (-z2w/xy)= - 2z2w/xy

[Note: u/x = (/x) (zw/x) = -z2w/x2]Eqns (5) represent the strain displacement

relationship, for bending.


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Stress strain relations:

We have the stress- strain relation for unidirectionallamina, when x and y directions coincide with the

material principal directions, as,

x Q11 Q12 0 xy = Q12 Q22 0 y ------------------(6)

xy 0 0 Q66 xy

where the values of Qij are known.


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If the principal directions do not correspondwith x and y directions, these equations willbecome,

x Q11 Q12 Q16 x

y = Q12 Q22 Q26 y ---(7)

xy Q16 Q26 Q66 xy

where the values of [Qij] are known in terms of

material properties.


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For convenience this extensional coefficients are

written as,

Q11= A11, Q12 = A12, Q16= A16,Q22 = A22, Q26 =A26, Q66= A66 --------- (8)

Utilizing this notation and substituting the strain

eqn(5), we have,x A11 A12 A16 (

2w/x2)

y = -z A12 A22 A26 (2w/y2) ---(9)

xy

A16

A26

A66

(2w/xy)

as the matrix form of the stress-displacement

equations for our unidirectional laminate.


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Equilibrium equations:

dx


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dx

h/2

h/2

x

y z

Nx dy

Vxz dy

Ny dxVyz dx

Nyx dx

Nx* dy

Vxz* dy

Ny* dxVyz* dx

Nyx* dx

Fig-2 (a) a laminate plate

Fig-2(b) forces on a laminate plate

q dx dy


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My dx

Myx dx

Mx* dy

My* dx

Mxy dy

Mx dy

Fig-2 (c) Moments acting on a laminate


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Fig-2 depicts a typical element of a plate.

Fig-(a) shows the reference frame and thedimensions. Fig- (b) shows the forces and fig-

the moments.

Note: the stars indicate the larger values of the

loads and moments.

E.g.: Vyz* = Vyz +(Vyz/y)dy

Nx*= Nx+(Nx/X)dx

and My* = My +(My/Y)dyand so on.

-contd-


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Note that the normal loads Nx and Ny and the

shearing loads Nxy, Nyx, Vxz and Vyz of fig-2(b)have units of force per unit length. The pressure

q has units of force per unit area, and the

moments have the units of moments Mx, My,

Mxy and Myx of fig-2(c) have units of moment per

unit length. These units pertain also to the

starred quantities. The total force acting at any

face, for example, on the right face of theelement in fig-2 (b) is shown as the product of

Nx* and the distance dy.


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With the expressions for the stress components as

shown in equations (9), we can determine the

bending and twisting moments of fig-2(c) byintegrating the products of the stress components

and their moment arms over the plate thickness. The

resulting expressions are called the stress couples of

the plate and are given by,

Mx =


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Mx =

zdz

= - [ D11(2w/x2)+D12 (

2w/y2)+ 2D16(2w/xy)]

Where Dij= Aij h3/12


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Similarly,

My =

= -[D12 2w/x2 + D22

2w/y2 +

2D262w/xy]

And,

Mxy = Myx =

= [D16(2w/x2)+D26(

2w/y2)+

2D66(2w/xy)] ---(10)


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Now taking moment about the front edge of fig-2

(b), and equating it to zero, we get,

Vyzdx dy+(Vxz-Vxz*)dy(dy/2)-q dx dy(dy/2)+(My-My*)dx+(Mxy-Mxy*)dy = 0.

When the corresponding expressions for the starred

quantities are inserted and the higher order termsare neglected, this becomes,

Vyzdx dy-(My/y)dx dy(Mxy/x)dx dy=0

or Vzy = (My/y) + (Mxy/x) = 0 ---- (11)


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Similarly, by summing moments about the right

edge of the element, we get,

Vxz = (Mx/x) + (Mxy/y) ------(12)

Finally the summation of the forces in the

z- direction yields,(Vxy/x)dx dy+ (Vyz/y)dx dy + q dx dy=0

from which,(Vxz/x) + (Vyz/y) = -q --------(13)


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When partial derivatives are taken of equations(11) and (12) with respect to y and x respectively

and the values are substituted in equation (13),

we get,(2Mx/x

2)+2(2Mxy/xy)+(2My/y

2)= -q --(14)


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substituting the expressions for Mx, My, and Mxy,as given by eqns (10), we get,

D11(4w/x4)+4D16(

4w/x3y)+2(D12+2D66)4w/x2y2

+4D26(4w/xy3)+D22(4w/y4) = q --(15)

This is the governing differential equation of a

unidirectional laminated plate with arbitrary

orientations of principal material directions.


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If the coordinate planes xz and yz are theplanes of elastic symmetry of the plate, or

equivalently, the x and y axes correspond to the

principal material directions of the plate, then,Q16=Q26= 0, and hence D16=D26= 0 and the

equation (15) reduces to

D11

(4w/x4) +2(D12

+D66

)4w/x2y2

+D22(4w/y4) = q --------(16)


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Governing differential equation for

unidirectional plane anisotropic lamina,

including in-plane loads.

In the above derivation the in plane loads were

not considered. When these forces are included

and the deflection of the plate is noted


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Governing differential equation for

unidirectional plane anisotropiclamina, including in-plane loads.

In the above derivation the in plane loads were notconsidered. When the inplane forces are included and

the deflection of the plate is considered, it can be seen

from fig.3, that they indeed contribute force

components in the Z-direction.With small deflection theory, we can say that the sine

of an angle is approximately equal to the angle in

radians and the cosine of the angle is unity.


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Fig.3

Hence, summing forces in the Z-direction from fig.3,


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Hence, summing forces in the Z direction from fig.3,

we get the following for the unbalanced forces:

-Nxdy(w/x) +[Nx+ (Nx/x) dx] dy [(w/x) +(

2w/x2)]dx

-Nydx(w/y) +

[Ny+(Ny/y)dy]dx[(w/y) +(2w/y2)]dy

-Nxydy(w/y) +

[Nxy

+(Nxy

/x)dx] dy [(w/y)+(2w/xy)]dx

-Nyxdx(w/x) +

[Nyx+(Nyx/Y)dy]dx[(w/x)+(2w/xy)]dy


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On simplification, noting that Nxy=Nyx and

neglecting higher order terms, this expression

reduces to,{ Nx(

2w/x2) + (Nx/x)(w/x)+Ny(2w/y2) +

(Ny/Y)(w/y) + 2Nxy(2w/xy) +

(Nxy/x)(w/y)+(Nxy/y)(w/x)}dydx-------(18)

Now applying the equation of equilibrium for

the summation of forces in the x-direction and

assuming that no body forces are acting, we get,

(Nx/x)dxdy + (Nxy/y)dxdy = 0

or (Nx/x) = -(Nxy/y)


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Simlarly, the summation of forces in the y-

direction gives,

(Ny/y) = -(Nxy/x)

When these expressions are substituted ineqn(18), the expression becomes,

[Nx(2

w/x2

)+Ny(2

w/y2

)+2Nxy(2

w/xy)]dxdy


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If these unbalanced forces in the Z-direction are added to the lateral force q,

equation (14) becomes,

(2Mx/x2) + 2(2Mxy/xy) + (

2My/y2)

= -[q + Nx(2w/x2)+2Nxy(

2w/xy)+

Ny(2w/y2)] ----------(19)

Again substituting the values of Mx My and Mxy


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Again substituting the values of Mx, My, and Mxy,

as given by eqn (10), we Obtain,

D11(4w/x4)+4D16(4w/x3y)+2(D12+2D66)4w/x2y2 +4D26(

4w/xy3)+D22(4w/y4) =

[q+Nx(2w/x2)+2Nxy(

2w/xy)+Ny(2w/y2)]

------------- (20)

This is the governing differential equation to be

satisfied, if the coordinate planes xz and yz are not

the planes of elastic symmetry of the plate. ie. A platewith unidirectional lamina loaded along off-axis. And

the inplane forces are included along with the out of

plane forces.


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Determination of stresses and strains:

Once deflection w has been determined

either in equation as functions of x and y or at

discrete grid or node points throughout theplate, the next problem is the determination of

stresses and strains. We are interested in

determining these values through the thicknessof the plate at any desired location on the plate.


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For a given distance z from the middle surface,

the strains can be computed directly using the

strain-displacement relations (eqns 5). These

equations can be solved by simply taking the

indicated derivatives ifws are expressed inclosed form or by finite difference method.

with the strains known, equations (6) or (7)

can be used to determine the stresses,depending on the conditions.


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Multidirectional plane anisotropic plate.

The great potential of composite materialslies in the fact that, unlike isotropic materials,

the plies of the laminates can be oriented to

meet both load and directions requirements.

Thus the general case is one in which the plies

are oriented in several directions. The uni-

directional laminate discussed so far can be

considered as a special case of the general

classical theory of plates.


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Referring to figure 1 and considering this time

the middle surface stretch as shown in fig.4, due

to inplane forces in addition to bendingdeflections, and also the assumptions are

applicable, then the deflection of the typical

point 1 becomes,u = u0 - z(w/x) and

v = v0 - z(w/y) -------(21)

where u0 and v0 are the displacements along xand y directions respectively of the middle

surface.


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Stress- strain relation for the lamina with mid

plane stretch and bending.u0

w/x

Fig.4


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If the classical small deflection theory is again

considered applicable and the strain in the z-direction

are neglected, we get,

x = (u0/x)z(2w/x2)

y = (v0/y) -z(2w/y2)

xy = [( u0/y)+(v0/x)] -2z(2w/xy)

These are the strain-displacement equations.On the RHS, first set of elements are the midplane

strains and the second set are the midplane

curvatures.

------(22)

ese re at ons can e wr tten n terms o t e ml t i d t i th f ll i f


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plane strains and curvatures in the following form:

x0 u0/x

y0 = v0/y ------------ (23)

xy0 (u0/y)+(v0/x)

where LHS terms are midplane strains.

x -2w/x2

y = -2w/y2

xy -22w/xy

where the LHS terms are the curvatures.

-------------------(24)


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Now the laminate strains can be written as,

x x0 xy = y

0 +z y

xy xy0 xy

----------------(25)


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Strain and stress in a laminate:

If the strains are known at any point in alaminate, the stress strain equation can be used to

calculate the stresses in each lamina.

x

Q11

Q12

Q16

x

y = Q12 Q22 Q26 y ---------(26)

xy Q16 Q26 Q66 xy

The transformed reduced stiffness(TRS) matrix [Q],corresponds to that of the ply located at the point

along the thickness of the laminate.


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Substituting eqn (25) in eqn (26),

x x0 x

y = [Q] y0 + z [Q] yxy xy xy

Where [Q] is the T R S matrix.

From the above equation it can be seen thatthe stresses vary linearly only through the

thickness of each lamina. However, the stresses

may vary from lamina to lamina, since the

transformed reduced stiffness matrix changes

from ply to ply, as [Q] depends on the material

and orientation of the ply.


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fig. Stress and strain variation in a composite laminate.

LAMINATEStrain variation (linear) Stress

variation


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These global stresses can then be

transformed to local stresses through the

transformation equation,1 x

2 = [T] y

12 xy

and globalstrains can be transformed to local

strains,

1 x

2 = [T] y

()12 ()xy


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the local stresses and strains can then be used

in the failure criteria to find when the lamina

fails. Now we should know the midplane strainsand curvatures from the applied loads, to solve

for the local stresses and strains.


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End of 3rd internal

material

(along with previous notes).

All the Best

composite material & structures 3

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