Mechanics Analysis Composite Materials Episode 12 doc

Optimal composite structures Laminates of uniform strength exist under the following restrictions: For monotropic model of the unidirectional ply considered in the previous section, n =

Chapter 8 Optimal composite structures 37 1

As a rule, helical plies are combined with circumferential plies as in Fig 7.43 For

this case, k = 3, hl = hl = h4/ 2, = -& = 4, h3 = h90, c$3 = 90°, and Eq (8.17) gives

(8.18)

Because the thickness cannot be negative, this equation is valid for 0 6 4 6 40.For

Q 4 6 90", the helical layer should be combined with the axial one, i.e., we

should put k = 3, h l = hZ = h d /2, 4, = = 4 and h3 = ho, 43= 0" Then

(8.19)

Dependencies corresponding to Eqs (8.18) and (8.19) are presented in Fig 8.3 As

an example, consider a filament wound pressure vessel whose parameters are listed

in Table 6.1 Cylindricalpart of the vessel shown in Figs 4.14 and 6.23 consists of a f36" angle-ply helical layer and a circumferentiallayer whose thickness hl = hb and

hZ = h90 are presented in Table 6.1 The ratio hyo/h&for two experimental vessels is 0.97 and 1.01, while Eq (8.18) gives for this case hyo/h&= 0.96 which shows that both vcsscls are close to optimal structures Laminates reinforced with uniformly stressed fibers can exist under some restrictions imposed on the acting forces Ify, F,.,

and iVyy.Such restrictions follow from Eqs (8.13) and (8.14) under the conditions that hi 2 0, 0 6 sin' 4i,cos' 4i6 1 and have the form

Fig 8.3 Optimal thickness ratios for a cylindrical pressure vessel consisting of i$ helical plies combined

with circumferential (90") or axial (0") plies

372 Mechanics and analysis of composite materials

Particularly, Eqs (8.13) and (8.14) do not describe the case of pure shear for which

only shear stress resultant, N,,, is not zero This is quite natural because strength

condition cry) = 51 under which Eqs (8.12)-(8.14) were derived is not valid for shear inducing tension and compression in angle-ply layers

To study in-plane shear of the laminate, we should use both solutions of Eq (8.7) and assume that for some layers, e.g., with i = 1,2,3, , n-1 , of)= 81 while

for the other layers (i = n,n + I , n -t2, ,k),or) = -81 Then, Eqs (8.1) can be reduced to the following forms:

For the case of pure shear (N, =N, = 0), Eqs (8.20) and (8.21) yield h+ =

q5i = -45" for the layers with hi = h i we get from Eq (8.22)

The optimal laminate, as follows from the foregoing derivation, corresponds to

4~45'angle-ply structure shown in Fig 8.2b

8.2 Composite laminates of uniform strength

Consider again the panel in Fig 8.1 and assume that unidirectional plies or fabric layers, that form the panel are orthotropic, i.e., in contrast to the previous section,

we do not neglect now stresses 02 and ~ 1 2in comparison with 01 (see Fig 3.29) Then, constitutive equations for the panel in plane stress state are specified by the first three equations in Eqs (5.35), i.e

Chapter 8 Optimal composite structures 313

of principal strains, or principal stresses because $2 = G I ~ $ for an orthotropic

layer and condition $3 = 0 is equivalent to condition = 0 (see section 2.4) Using the third equation in Eqs (4.69) we can write these conditions as

2(g -E,) sin d icos di +y.rI-cos 24; = 0 (8.25)

This equation can be satisfied for all the layers if we take

Then, Eqs (8.23) yield

Nv = ( B I I+B I ~ ) E ,Nv= (&I +B 2 2 ) ~ , Nry = ( 8 4 1 +B 4 2 ) ~

374 Mechanics and analysis of composite materials

These equations allow us to find strain, i.e

To determine the stresses that act in the optimal laminate, we use Eqs (4.69) and

(8.26) that specify the strains in the principal material coordinates of the layers

as = ~2 = E, y I 2 = 0 Applying constitutive equations, Eqs (4.56), substituting E

from Eq (8.27) and writing the result in the explicit form with the aid of Eqs (8.24)

Chapter 8 Optimal composite structures

Laminates of uniform strength exist under the following restrictions:

For monotropic model of the unidirectional ply considered in the previous section,

n = 0, m = I , and Eqs (8.30) reduce to Eqs (8.9) and (8.10)

To determine the thickness of the optimal laminate, we should use Eqs (8.31)

in conjunction with one of the strength criteria discussed in Chapter 6 For the

simplest case, using the maximum stress criterion in Eqs (6.2), the thickness of

the laminate can be found from the following conditions CTI = or 02 = a,, so that

(8.32)

Obviously, for the optimal structure, we would like to have hl = h2 However, this can happen only if material characteristics meet the following condition:

(8.33)

The results of calculation for typical materials whose properties are listed in Tables

3.5 and 4.4 are presented in Table 8.2 As can be seen, Eq (8.33) is approximately

valid for fabric composites whose stiffness and strength in the warp and fill

directions (see section 4.6) are controlled by the fibers of one and the same nature However for unidirectional polymeric and metal matrix composites, whose longitudinal stiffness and strength are governed by the fibers and transverse characteristics are determined by the matrix properties, a?/al << n In accordance

376 Mechanics and analysis of composite materials

Table 8.2

Parameters of typical advanced composites

Parameter Fabric-epoxy composites Unidirectional-epoxy composites Boron-AI

Glass Carbon Aramid Glass Carbon Aramid Boron

0.83 0.022 0.025 0.012 0.054 0.108

-0~/8l 0.99 0.99

with Eqs (8.32), this means that hl -Kh2, and the ratio h2/h1 varies from 12.7 for

glass-epoxy to 2.04 for boron+poxy composites Now, return to the discussion

presented in section 4.4.2 from which it follows that in laminated composites transverse stresses 02 reaching their ultimate value, &, cause cracks in the matrix which do not result in the failure of the laminate whose strength is controlled by fibers To describe the laminate with cracks in the matrix (naturally, if the cracks are admitted for the structure under design), we can use the monotropic model of the

ply and, hence, results of optimization presented in Section 8.1

Consider again the optimality condition Eq (8.25) As can be seen, this equation can be satisfied not only by strains in Eqs (8.26), but also if we take

Y.VJ

Because the left-hand side of this equation is a periodic function with period 7c,

Eq (8.34) determines two angles, Le

(8.35)

Thus, the optimal laminate consists of two layers, and the fibers in both layers are directed along the lines of principal stresses Assume that the layers are made of the

same composite material and have the same thickness, i.e hl = h2 = h/2, where h is

the thickness of the laminate Then, using Eqs (8.24) and (8.35) we can show that

B I I= B22 and B24 = - B I ~for this laminate After some transformation involving elimination of y.!,, from the first two equations of Eqs (8.23) with the aid of

Eq (8.34) and similar transformation of the third equation from which and E: are

eliminated using again Eq (8.34) we get

Nx = ( B II +8 1 4 tan 2 4 ) ~ :+ (B12 -8 1 4tan %)E!,

N,,= ( ~ 1 2 -B14 tan 2 4 1 4 + ( B I 1 +B14 tan 2 4 ) ~ : ~

Nx?; = (B44 f B14 cot 24)$,

Upon substitution of coefficients B,,, from Eqs (8.24) we arrive at

Chapter 8 Optimal composite structures 377

Introducing average stresses a, = N,/h, 0,.= N,/h, and T.~.~.= N,,/h and solving these equations for strains we have

advanced composites, these constants are listed in Table 8.3 (the properties of unidirectional plies are taken from Table 3.5) Comparing elastic moduli of the optimal laminates with those for quasi-isotropic materials (see Table 5.1) we can see that for polymeric composites the characteristics of the first group of materials are about 40% higher than those for the second group However, it should be emphasized that while the properties of quasi-isotropic laminates are the universal

Table 8.3

Effective elastic constants of an optimal laminate

Property Glass Carbon- Aramid- Boron- Boron- Carbon-

A1203-epoxy epoxy epoxy epoxy AI carbon AI Elastic modulus, E (GPa) 36.9 75.9 50.3 114.8 201.1 95.2 205.4 Poisson’s ratio, Y 0.053 0.039 0.035 0.035 0.21 0.06 0.I76

378 Mechanics and analysis of composite materials

material constants, the optimal laminates demonstrate characteristics shown in

Table 8.3 only if the orientation angles of the fibers are found from Eqs (8.35) or

(8.38) and correspond to a particular distribution of stresses ox, cy,and zxy

As follows from Table 8.3, the modulus of a carbon-epoxy laminate is close to the

modulus of aluminum, while the density of the composite material is less by the factor of 1.7 This is the theoretical weight-saving factor that can be expected if we change aluminum for carbon+poxy composite in a thin-walled structure Because the stiffness of both materials is approximately the same, to find the optimal orientation angles of the structure elements, we can substitute in Eq (8.38) the stresses acting in the aluminum prototype structure Thus designed composite structure will have approximately the same stiffness as the prototype structure and,

as a rule, higher strength because carbon composites are stronger than aluminum alloys

To evaluate the strength of the optimal laminate, we should substitute strains

from Eqs (8.36) into Eqs (4.69) and thus found strains in the principal material coordinates of the layers -into constitutive equations, Eqs (4.56), that specify

stresses ol and o~(z12 = 0)acting in the layers Applying the proper failure criterion

(see Chapter 6 ) we can evaluate the laminate strength

Comparing Tables 1.1 and 8.3 we can see that boron-epoxy optimal laminates

have approximately the same stiffness that titanium (but is lighter by the factor of about 2) and boron-aluminum can be used to substitute steel with a weight-saving

factor of about 3

For preliminary evaluation, we can use a monotropic model of unidirectional

plies neglecting stiffness and load-carrying capacity of the matrix Then, Eqs (8.37)

acquire the following simple forms:

(8.39)

As an example, consider an aluminum shear web with thickness h = 2 mm, elastic constants E, = 72 GPa, v, = 0.3 and density pa = 2.7 g/cm3 The panel is loaded with shear stress z Its shear stiffness is Bg,= 57.6 GPa mm and the mass of a unit

surface is ma= 5.4 kg/m2 For the composite panel, taking ox = = 0 in Eq (8.38)

we get 4 = 45" Thus, the composite panel consists of +45" and -45" unidirectional layers of the same thickness The total thickness of the laminate is h = 2 mm, i.e.,

the same as for an aluminum panel Substituting El = 140 GPa and taking into account that p = 1.55 g/cm3 for a carbon-epoxy composite that is chosen to

substitute aluminum we get B& = 70 GPa mm and m, = 3.1 kg/m3 Stresses acting

in the fiber directions of the composite plies are o;= f 2 z Thus, the composite

panel has 21.5% higher stiffness and its mass makes only 57.4% of the mass of a

metal panel Composite panel has also higher strength because the longitudinal strength of unidirectional carbon-poxy composite under tension and compression

is more than twice higher than the shear strength of aluminum

Possibilities of the composite structure under discussion can be enhanced if we use

different materials in the layers with angles @, and @? specified by Eqs (8.35)

Chapter 8 Optimal composite structures 379

According to the derivation of Obraztsov and Vasiliev (1989), the ratio of the layers’

thicknesses is

and elastic constants in Eqs (8.37) are generalized as

Superscripts 1 and 2 correspond to layers with orientation angles 4I and 4:,

respectively

8.3 Application to pressure vessels

As an example of application of the foregoing results, consider filament wound

membrane shells of revolution, that are widely used as pressure vessels, solid propellant rocket motor cases, tanks for gases and liquids, etc (see Figs 4.14 and

7.43) The shell is loaded with uniform internal pressure p and axial forces T

uniformly distributed along the contour of the shell cross-section Y = ro as in Fig 8.4 Meridional, Nz, and circumferential, Np, stress resultants acting in the shell

Fig 8.4 Axisymmetrically loaded membrane shell of revolution

380 Mechanics and analysis of composite materials

follow from the corresponding free body diagrams of the shell element and can be written as (see, e.g., Vasiliev, 1993)

where z(r)specifiesthe form of the shell meridian, z’ = dz/dr, and

2

Let the shell be made by winding an orthotropic tape a1 angles +4 and -4 with

respect to the shell meridian as in Fig 8.4 Then, N, and Np can be expressed in

terms of stresses 61, 6 2 and 212, referred to the principal material coordinates of the

tape with the aid of Eqs (4.68), i.e

N, = h(ol cos24 + 6 2 sin24 -2 1 2 sin 2&),

where h is the shell thickness Stresses q , ~ ,and 212 are linked with the

corresponding strains by Hooke’s law, Eqs ( 4 2 9 , as

while strains E ~ , E Z ,and y I 2 can be expressed in terms of the meridional, E,, and circumferential, ~ p ,strains of the shell using Eqs (4.69), i.e

(8.44)

Because the right-hand side parts of these three equations include only two strains,

E, and ES,there exists a compatibility equation linking E!,~2 and y12 This equation is

(CI - ~ 2 )sin24 +yI2cos2r$= 0

Writing this equation in terms of stresses with the aid of Eqs (8.43) we get

In conjunction with Eqs (8.42), this equation allows us to determine stresses as

Chapter 8 Optimal composite 38 1

Substituting N, and Np from Eqs (8.40) into Eq (8.46) we arrive at the following

equation for the meridian of the optimal shell:

(8.48)

The first two equations of Eqs (8.45) yield the following expressions for stresses

acting in the tape of the optimal shell:

(8.49)

Taking into account that, in accordance with Eqs (8.45)

382 Mechanics and analysis of composite materials

1

h

61 + a 2 = - ( N , + N p ) ,

we arrive at the following relationships:

which coincide with Eqs (8.31)

Substituting N, from the first equation of Eqs (8.40) into Eq (8.49) we get

Q[l + ( ~ 7 ~ 1’I2

c l h =

Assume that the optimal shell is the structure of uniform stress Differentiating

Eq (8.50) with respect to r and taking into account that according to the foregoing

assumption 01 = constunt, we arrive at the following equation in which 2’ is eliminated with the aid of Eq (8.48):

Consider two particular cases First, assume that a fabric tape of variable width

w ( r ) is laid up on the surface of the mandrel along the meridians of the shell of revolution to be fabricated Then, 4 = 0 and Eq (8.51) acquires the form

where hR = h(r =R) is the shell thickness at the equator r =R (see Fig 8.4)

Assuming that there is no polar opening in the shell (ro = 0)or that it is closed

(T= p r 0 / 2 ) we have from Eq (8.41) Q = $ / 2 Substituting this result in Eqs (8.48) and (8.50) we obtain

(8.53)

(8.54)

Chapter 8 Oprimal composae structures 383 Integrating Eq (8.53) under the condition 1/z‘ = 0 for r = R which means that the tangent line to the shell meridian is parallel to axis z at r = R (see Fig 8.4) we

Here, B, is the a-function (or the Euler integral of the first type) Constant of

integration is found from the condition z ( r = R ) = 0 Meridians corresponding to various n-numbers are presented in Fig 8.5 For n = 1 the optimal shell is a sphere,

while for n = 2 it is a cylinder As follows from Eq (8.52), the thickness of the

spherical (n = 1) and cylindrical (n = 2, r = R ) shells is constant Substituting Eqs (8.52) and (8.55) into Eq (8.54) and taking into account Eqs (8.49) we have

384 Mechanics and analysis of composite materials

This equation allows us to determine the shell thickness at the equator ( r =R), h R ,

matching G Ior rs2 with material strength characteristics

As was already noted, the shells under study can be made laying up fabric tapes of

variable width, ~ ( r ) ,along the shell meridians The tape width can be linked with

the shell thickness, h ( r ) , as

where 4R= &(r=R) It should be noted that this equation is not valid for

r < 1-0+W O ,Le., in the shell area close to the polar opening where tapes are completely overlapped

Substituting h ( r ) from Eq (8.57) into Eq (8.51) we arrive at the following

equation for the tape orientation angle:

d 4 s i n 4 [ n - ( l -n)cos2q5]

d r cos (b[ 1 -( 1 -n) cos2 $1 =

r - -

Chapter 8 Optimal composite StrucIures 385

The solution of this equation that satisfies the boundary condition +(r= R) = 4Ris presented as follows:

As has been already noted in the previous section, the simplest and rather adequate

model of unidirectional fibrous composites for design problems is the monotropic

model ignoring the stiffness of the matrix For this model, we should take n = 0 in the foregoing equations Particularly, Eq (8.58) yields in this case

This is the equation of a geodesic line on the surface of revolution Thus, in the optimal filament wound shell the fibers are directed along the geodesic lines This substantially simplifies the winding process because the tape placed on the surface under tension automatically acquires the form of the geodesic line if thcrc is no friction between the tape and the surface

As follows from Eq (8.59), for 4 = 90”, the tape touches the shell parallel of radius

and the polar opening of this radius is formed in the shell (see Fig 8.4)

Transforming Eq (8.48) with the aid of Eqs (8.59) and (8.60) and taking n = 0

we arrive at the following equation that specifies the meridian of the optimal filament wound shell:

-Integrating Eq (8.61) with due regard for the condition l/z’(R) = 0 which,

as earlier, requires that for r = R the tangent to the meridian be parallel to z-axis,

we get

(8.63)

386 Mechanics and analysis of composite materials

Using this equation to transform Eq (8.50) in which we take n = 0 and substituting

h from Eq (8.57) we get the following equation for the longitudinal stress in the

tape:

p(R2 -4) +2roT

As can be seen, 01 does not depend on Y , and the optimal shell is a structure

reinforced with uniformly stressed fibers

Such fibrous structures are referred to as isotensoids To study the types of

isotensoids corresponding to loading shown in Fig 8.4, factor the expression in the denominator of Eq (8.63) The result can be presented as

As follows from Eq (8.65), quantities R and r-1 are the maximum and minimum

distances from the meridian to the rotation axis Meridians of isotensoids

corresponding to various loading conditions are shown in Fig 8.6 For p = 0,

Chapter 8 Optimal composite structures 387

i.e., under axial tension, a hyperbolic shell is obtained with the meridian determined

as

r' - 2 tan2 4R = R

This meridian corresponds to line 1 in Fig 8.6 For 4R = 0, the hyperbolic shell

degenerates into a cylinder (line 2) Curve 3 corresponds to T = p r 0 / 2 , i.e., to a shell for which the polar opening of radius r0 is closed For the special angle

4R = 4o = 54"44', the shell degenerates into a circular cylindrical shell (line 2)

discussed in section 8.1 For T = 0, Le., in case of an open polar hole, the meridian

has the form corresponding to curve 4 The change in the direction of axial forces T

yields a toroidal shell (line 5) Performing integration of Eq (8.65) and introducing

As an application of the foregoing equations, consider the optimal structure of the

end closure of the pressure vessel shown in Fig, 4.14 The cylindrical part of the

vessel consists of zk4R angle-ply layer with thickness la^, that can be found from

Eq (8.64) in which we should take T = pr0/2, and a circumferential (4 = 90") layer whose thickness is specified by Eq (8.18), i.e

h90 = h&cos2 4 R - 1)

The polar opening of the dome (see Fig 4.14) is closed So T = pr0/2, fj = 0, and the

dome meridian corresponds to curve 3 in Fig 8.6 As has already been noted, upon