Mechanics Analysis Composite Materials Episode 11 pps

Dependence of an aramid-epoxy composite material temperature on the number of cycles under tensile and compressive loading with frequency IO3 cycles per minute is shown in Fig.. Under cy

336 Mechanics and analysis of composite materials

0 15 30 45 60 75 90 Fig 7.23 Calculated (lines) and experimental (circles) dependencies of dissipation factor on the ply orientation for glass-epoxy(- 0 ) and carbon-epoxy (- -o) unidirectional composites

Dependence of an aramid-epoxy composite material temperature on the number of

cycles under tensile and compressive loading with frequency IO3 cycles per minute is shown in Fig 7.24 (Tamuzh and Protasov, 1986)

Under cyclic loading, structural materials experience a fatigue fracture caused by material damage accumulation As was already noted in Section 3.2.4, heteroge-

neous structure of composite materials provides relatively high resistance of these materials to crack propagation resulting in their specific behavior under cyclic loading As follows from Fig 7.25 showing experimental results obtained by

V.F Kutinov, stress concentration in aluminum specimens practically does not affect material static strength due to plasticity of aluminum but dramatically reduces its fatigue strength Conversely, static strength of carbon-poxy composites that

T " C

r

1 1 0 2 .10

Fig 7.24 Temperature of an aramidxpoxy composite as a function of the number of cycles under

tension (1) and compression (2)

Chapter 7 Environmental, special loading, and manufacturing e f e c f s 337

Fig 7.25 Typical fatigue diagrams for carbon-epoxy composite (solid lines) and aluminum alloy

(broken lines) specimens without (1) and with (2) stress concentration (fatigue strength is normalized to

sLalk strength of specimens without stress concentration)

belong to brittle materials is reduced by stress concentration that practically does not affect the slope of the fatigue curve On average, residual strength of carbon

composites after lo6 loading cycles makes 7&80% of material static strength in comparison with 3WO% for aluminum alloys Qualitatively, this comparative evaluation is true for all fibrous composites that are widely used in structural elements subjected to intensive vibrations such as helicopter rotor blades, airplane propellers, drive shafts, automobile leaf-springs, etc

A typical for composite materials fatigue diagram constructed with experimental results of Apinis et al (1991) is shown in Fig 7.26 Standard fatigue diagrams usually determine material strength for IO3 d N G lo6 and are approximated as

338 Mechanics and analysis of composite materials

Here, N is the number of cycles to failure under stress OR, a and b are experimental

constants depending on frequency of cyclic loading, temperature and other environmental factors, and on the stress ratio R = amitl/amax, where amax and amin

are the maximum and minimum stresses It should be taken into account that results

of fatigue tests are characterized, as a rule with high scatter

Factor R specifies the cycle type The most common bending fatigue test provides the symmetric cycle for which Omin = -a, amnx = a,and R = -1 Tensile load cycle

(amin= 0, omt,,= a) has R = 0, while compressive cycle (amin= -a, ami,,= 0) has

R + -00 Cyclic tension with a , > a m i n > 0 corresponds to 0 < R < 1, while cyclic compression with 0 > a,,, > omin corresponds to 1 < R < 00 Fatigue diagrams for unidirectional aramid-epoxy composite studied by Limonov and Anderson (1991) corresponding to various R-values are presented in Fig 7.27 Analogous results (Anderson et al., 1991) for carbon-epoxy composites are shown

Fabric composites are more sensitive to cyclic loading than materials reinforced with straight fibers This fact is illustrated in Fig 7.29 showing experimental results

of Schulte et al (1987) The foregoing discussion deals with the high-cycle fatigue

Initial interval 1 <N < lo3 corresponding to the so-called low-cycle fatigue is usually studied separately, because the slope of the approximation in Eq (7.57) can

0

*0° t

Fig 7.27 Fatigue diagrams for unidirectional aramid-epoxy composite loaded along the fibers with

various stress ratios

Chapter 7 Environmental, special loading, and manufacturing efficts 339

Fig 7.28 Fatigue diagrams for a unidirectional carbon-epoxy composite loaded along the fibers with

various stress ratios

Fig 7.29 Tensile fatigue diagrams for a cross-ply ( I ) and fabric (2) carbon-epoxy composites

be different for high stresses Typical fatigue diagram for this case is shown in

Fig 7.30 (Tamuzh and Protasov, 1986)

Fatigue has also some effect on the stiffness of composite materials This can be

seen in Fig 7.31 demonstrating reduction of the elastic modulus for a glass

fabric-epoxy-phenolic composite under low-cycle loading (Tamuzh and Protasov, 1986)

This effect should be accounted for in application of composites to the design of

structural members such as automobile leaf-springs which, being subjected to cyclic

loading, are designed under stiffness constraints

Stiffness degradation can be used as an indication of material damage to predict

its fatigue failure The most sensitive characteristic of the stiffness change is the

tangent modulus E, specified by the second equation in Eqs (1 3).Dependence of E,

on the number of cycles, N,normalized to the number of cycles that cause material

Fig 7.31 Dependence of elastic modulus of glass fabric-epoxy phenolic composite on the number of

cycles at stress D = 0.55 (if is the static ultimate stress)

fatigue fracture under the pre-assigned stress is presented in Fig 7.32 corresponding

to f45"angle-ply carbon-epoxy laminate studied by Murakami et al (1 991)

7.3.4 Impact loading

Thin-walled composite laminates possessing high in-plane strength and stiffness are rather sensitive to damage initiated by transverse impact loads that can cause fiber breakage, cracks in the matrix, delamination, and even material penetration by the impactor Depending on the impact energy determined by the impactor mass and velocity and the properties of laminate impact loading can result in considerable reduction in material strength under tension, compression, and shear One of the most dangerous consequences of the impact loading is an internal delamination of

Chapter 7 Environmental special loading, and manufacturing eflects 34 I

for f45" angle-ply carbon-epoxy laminate

laminates that sometimescan be hardly identified by visual examination This type of the defect causesa dramatic reduction in the laminate compressivestrength and results

in unexpected failure of the thin-walled composite structure due to microbucklingo f

fibers or local buckling of plies As follows from Fig 7.33 showing experimental results of Verpoest et al (1989) for unidirectional and fabric composite plates,

impact can reduce material strength in compression by the factor of 5 and more

To study the mechanism of material interlaminar delamination, consider a problem of wave propagation through the thickness of the laminate shown in

Fig 7.34 The motion equation has the following well-known form

(7.58)

Here, u, is the displacement in the z-direction, E, is material modulus in the same direction depending, in the general case on z, and p is the material density For the

laminate in Fig 7.34, the solution of Eq (7.58) should satisfy the following

boundary and initial conditions

342 Mechanics and analysis of composite niateriab

Fig 7.34 Laminate under impact load

Consider first a homogeneous layer such that E, and p do not depend on z Then,

Eq (7.58) acquires the form

where c2 = E,/p Transform this equation introducing new variables, Le., XI =

z + ct and x2 = z -ct Performing traditional transformation we arrive at

Chapter I Environmental, special loading, and manufacturing effects 343

in which the form of function f is governed by the shape of the acting pulse As

can be seen, the stress wave is composed of two components having the opposite signs and moving in the opposite directions with one and the same speed c which is the speed of sound in the material The first term in Eq (7.62) corresponds to the acting pulse that propagates to the free surface z = h (see Fig 7.35 demonstrating the propagation of the rectangular pulse), while the second term corresponds to the pulse reflected from the free surface z = h It is important that for the compressive

direct pulse (which is usually the case), the reflected pulse is tensile and can cause material delamination since the strength of laminated composites under tension across the layers is very low

Fig 7.35 Propagation of direct and reflected pulses through the layer thickness

344 Mechanics and analysis of composite materials

1.5

-1.5

For laminates, such as in Fig 7.34, the boundary conditions, Eqs (7.59) should

be supplemented with the interlaminar conditions u f ) = and cy) = cry-’)

Omitting rather cumbersome solution that can be found elsewhere (Vasiliev and Sibiryakov, 1985) present some numerical results

Consider the two-layered structure the first layer of which has thickness 15 mm

and is made of aramid-epoxy composite material with El’) = 4.2 GPa, p I =

1.4 g/cm3and the second layer is made of boron-epoxy composite material and has

E!2) = 4.55 GPa, p2 = 2g/cm3, h2 = 12mm The duration of a rectangular pulse of

external pressure p acting on the surface of the first layer is tp = 5 x s Dependenceof the interlaminar (z = 15 mm) stress on time is shown in Fig 7.36 As

can be seen, at t M 3tp the tensile interface stress exceeds the intensity of the pulse of pressure by the factor of 1.27 This stress is a result of interaction of the direct stress wave with the waves reflected from the laminate’s inner, outer, and interface surfaces Thus, in a laminate, each interface surface generates elastic waves For laminates consisting of more than two layers, the wave interaction becomes more complicated and, what is more important, can be controlled by the proper

stacking sequence of layers As an example, consider a sandwich structure shown in

Fig 7.37(a) The first (loaded) layer is made of aluminum and has hl = 1 mm,

E!’) = 72 GPa, pI= 2.7g/cm3, the second layer is a foam core with h2 = 10 mm,

E!*) = 0.28 GPa, pz = 0.25 g/cm3, and the third (load-carrying) aramid+poxy

composite layer has h3 = 12 mm, Ei3)= 10 GPa, p 3 = l.4g/cm3 The duration of a rectangular pulse of external pressure is s Maximum tensile stress occurs in the middle plane of the load-carrying layer (plane a -a in Fig 7.37) Normal stress

induced in this plane is presented in Fig 7.38(a) As can be seen, at the moment of

time t equal to about 1.75 x low5s this stress is tensile and can cause delamination

Chapter 7 Environmental, special loading, and manufacturing effects 345

Fig 7.38 Normal stress related to external pressure acting in section a-a of the laminates in

Fig 7.37(a)-(c) respectively

346 Mechanics and analysis of' cornposite materials

Now introduce an additional aluminum layer in the foam core as shown

in Fig 7.37(b) As follows from Fig 7.38(b) this layer suppresses tensile stress

in section a - a Two intermediate aluminum layers (Fig 7.37(c)) working as generators of the compression stress waves eliminate the appearance of tensile stress

in this section Naturally, the effect under discussion can be achieved for a limited period of time But actually, impact tensile stress is dangerous right after the pulse action Damping capacity of real structural materials (it was not taken into account

in the foregoing analysis) dramatically reduces the stress amplitude in time

A flying projectile with relatively high kinetic energy can penetrate through the

laminate As known, composite materials, particularly, high-strength aramid fabrics

are widely used for protection against flying objects To demonstrate the mechanism

of this protection, consider a square composite plate clamped in the steel frame shown in Fig 7.39 and subjected to impact of a rectangular plane projectile (see Fig 7.39) simulating the blade of the turbojet engine compressor The plate consists

of the layers of thin aramid fabric impregnated with epoxy resin at a distance from the window in the frame (see Fig 7.39) and co-cured together as shown in Fig 7.40

The front (loaded) surface of the plate has a 1 mm thick cover sheet made of glass

fabric-epoxy composite Results of ballistic tests are presented in Table 7.2 Front and back views of plate No 2 are shown in Fig 7.39, and the back view of plate

No 3 can be seen in Fig 7.40 Because mechanical properties of the aramid fabric used to make the plates are different in the warp and in the fill directions (see Section

4.6), the plates consist of couples of mutually orthogonal layers of fabric that are

C

(b)

Fig 7.39 Plate no 2 (see Table 7.2) after the impact test: (a) front view; (b) back view

Chapter 7 En~ironmental, special loading and manufacturing effects 347

Fig 7.40 Back view of plate No 3 (see Table 7.2) after the impact test

Table 7.2

Ballistic test of plates made of aramid fabric

by the containment Penetration

further referred to as 0"/90" layers All the plates listed in Table 7.2 have n = 32 of such couples

To calculate the projectile velocity below which it fails to perforate the plate (the so-called ballistic limit) we use the energy conservation law according to which

where K is the projectile striking velocity, V, is its residual velocity, mp = 0.25 kg is

the projectile mass, n = 32 is the number of the 0"/90" layers, Wis the fracture work for the 0"/90" layers, and T i s the kinetic energy of the layer All the other factors and the fiberglass cover of the plate are neglected

Fracture work can be evaluated using the quasi-static test shown in Fig 7.41 A couple of mutually orthogonal fabric layers is fixed along the plate contour and loaded with the projectile The area under the force-deflection curve (solid line in Fig 7.41) can be treated as the work of fracture which for the fabric under study has

been found to be W = 120 Nm

To calculate T, the deformed shape of the fabric membrane has been measured Assuming that the velocities of the membrane points are proportional to deflections

.f and that df,/dt = K kinetic energy of the fabric under study (density of the layer

unit surface is 0.2 kg/m') turn out to be T = 0.0006 v,'

To find the ballistic limit, we should take V, = 0 in Eq (7.63) Substituting the

foregoing results in this equation we get & = 190.5 m/s which is much lower than the experimental result (& = 320 m/s) following from Table 7.2

348 Mechanics and analysis of composite materials

5

Fig 7.41 Fordeflection diagrams for square aramid fabric membranes -couple of layers with orthogonal orientations,- - - -superposition of the diagrams for individually tested layers

Let us change the model of the process and assume that the fabric layers fail one

after another rather than all of them at once, as it is presented in Eq (7.63) The

result is expected to be different because the problem under study is not linear, and the principle of superposition is not valid for it Bearing this in mind, we write

Eq (7.63) in the following incremental form:

(7.64)

Here, 6-1and F$ are the projectile velocities before and after the failure of the kth

couple of fabric layers, W is, as earlier, the fracture work consumed by the kth

couple of layers, G-1 = 0.0006 55, and the last term in the right-hand side of

Eq (7.64) means that we account for the kinetic energy of only those fabric layers

that have been already penetrated by the projectile Solving Eq (7.64) for V, we

Chapter 7 Environmental, special loading, and manufacturing eflects 349

and find that after the failure of the second couple of fabric layers fi = 316.2 m/s This process is repeated until & = 0, and thus found number k determines the

minimum number of 0"/90"layers that can stop the projectile with striking velocity

V, = 320 m/s The result of calculation is presented in Fig 7.42 from which it

follows that k = 32 This is exactly the same number of layers that have been used to

construct the experimental plates

Thus, it can be concluded that the high impact resistance of aramid fabrics is determined by two main factors First, by relatively high work of fracture which is governed not only by the high strength, but also by the interaction of the fabric

layers The broken line in Fig 7.41 shows the fracture process constructed as a result of superposition of experimental diagrams for individual 0" and 90" layers

The solid line corresponds as was noted, to 0" and 90" layers tested together (the

ratio of the fabric strength under tension in the warp and the fill direction is 1.3).As

can be seen, the area under the solid line is much larger that under the broken one which indicates high contribution of the layers interaction to the work of fracture If this conclusion is true, we can expect that for layers with higher anisotropy and for laminates in which the principal material axes of the adjacent layers are not orthogonal, the fracture work can be higher than for the orthotropic laminate under study The second factor increasing the impact resistance of aramid fabrics is associated with a specific process of the failure during which the fabric layers fail one after another but not at once Plates of the same number of layers but consisting

of resin impregnated and co-cured layers that fail at once demonstrate much less impact resistance

350 Mechanics and analysis of coniposite materials

7.4 Manufacturing effects

As was already noted, composite materials are formed in the process of

fabrication of a composite structure, and their properties are strongly dependent on the type and parameters of processing This means that material specimens that are used to determine mechanical properties should be fabricated with the same manufacturing method that is expected to be used to fabricate the structure under study

To demonstrate direct correlation that can exist between processing and material properties, consider the process of circumferential winding on the cylindrical surface

as in Fig 7.43 As a rule, the tapes are wound with some overlap wo shown in

Fig 7.44(a) Introducing dimensionless parameter

Fig 7.43 Winding of a circumferential layer Courtesy of CRISM

Chapter 7 Environmental, special loading,and manufacfuring effects 35I

(7.66)

we can conclude that for the case of complete overlap (Fig 7.44(b)) we have R = 1

Initial position of the tape placed with overlap wo as in Fig 7.44(a) is shown in this Figure with a broken line, while the final position of the tapes is shown with solid lines Assume that after the winding and curing are over, the resulting structure is a unidirectionally reinforced ring which is removed from the mandrel and loaded with internal pressure, so that the ring radius being R before the loading becomes R I

Decompose the resultant force acting in the ring cross-section into two components, i.e

where A = 2w6 is the cross-sectionalarea of the ring made from two tapes as shown

in Fig 7.44 Force F‘corresponds to part BC of the ring (Fig 7.44(a)) and can be

that corresponds to part CD of the ring (Fig 7.44(a)), we should take into account

that the fibers start to take the load only when this part of the tape reaches the position indicated with broken lines, i.e

R1 -(R + S)

R

F’’ = A”E, I

where A” = ( w -wo)6.With due regard to Eqs (7.66), (7.67), and (7.68) we can

write the result of the foregoing analysis in the following form:

352 Mechanics and analysb of composite materials

As follows from Eq (7.69), which is valid for winding without tension, overlap of

the tape results in reduction of material stiffness Because the levels of loading for

the fibers of BC and CD parts of the ring (Fig 7.44(a)) are different, reduction of

material strength can also be expected

Filament winding is usually performed with some initial tension of the tape This

tension improves material properties because it straightens the fibers and densifies

the material However, high tension may result in fiber damage and reduction of

material strength For glass and carbon fibers, preliminary tension usually does not

exceed 5% of the tape strength, while for aramid fibers that are less sensitive to

damage the level of initial tension can reach 20% of the tape strength Preliminary

tension reduces the effect of the tape overlap discussed above and described by

Eq (7.69) However, this effect can show itself in reduction of material strength,

because the initial stresses which are induced by preliminary tension in the fibers can

be different, and some fibers can be overloaded or underloaded under external

forces acting on the structure in operational conditions Strength reduction of

aramid+poxy unidirectional composites on the tape overlap observed in

experi-ments of Rach and Ivanovskii (1986) for winding on a 200 mm diameter mandrel is

demonstrated in Fig 7.45

The absence of the tape preliminary tension or low tension can cause the ply

waviness shown in Fig 7.46 which can occur in the filament wound laminates as a

result of pressure exerted by the overwrapped plies on the undenvrapped plies or in

flat laminates due to material shrinkage in the process of curing

The simplest for analysis is the regular waviness presented in Fig 7.46(a) To

determine the apparent modulus in the x-direction, we can use the expression

similar to one presented in Eqs (4.76), Le

Fig 7.45 Dependence of the normalized longitudinal strength of unidirectional aramid-epoxy

composite on the tape overlap