ADVANCED MECHANICS OF COMPOSITE MATERIALS Episode 12 ppsx

Then, the temperature change will cause a thermal strain associated with material expansion, and the force P0, being constant, also induces additional strain because the material stiffne

laminate element with the aid of Eqs (5.3) and (5.14), i.e.,

It follows from Eqs (7.23) that in the general case, uniform heating of laminates induces,

in contrast to homogeneous materials, not only in-plane strains but also changes to thelaminate curvatures and twist Indeed, assume that the laminate is free from edge andsurface loads so that forces and moments in the left-hand sides of Eqs (7.23) are equal to

zero Since the CTE of the layers, in the general case, are different, the thermal terms N T and M T in the right-hand sides of Eqs (7.23) are not equal to zero even for a uniform

temperature field, and these equations enable us to find ε T , γ T , and κ T specifying thelaminate in-plane and out-of-plane deformation Moreover, using the approach described

in Section 5.11, we can conclude that uniform heating of the laminate is accompanied, inthe general case, by stresses acting in the layers and between the layers

As an example, consider the four-layered structure of the space telescope described inSection 7.1.1

First, we calculate the stiffness coefficients of the layers, i.e.,

• for the internal layer of aluminum foil,

sin2φcos2φ = 1.91 GPa

• for the external skin,

A ( 4) = E1ecos4φe+ E2esin4φe+ 2 E1eνe + 2Ge !

sin2φecos2φe= 99.05 GPa

A (224) = E1esin4φe + E2ecos4φe+ 2 E1eν12e + 2Ge

12

!sin2φecos2φe = 13.39 GPa

A (124) = E1eν12e +E1e+ Ee2− 2 E1eν12e + 2Ge

12

!

sin2φecos2φe = 13.96 GPa

Using Eqs (7.18), we find the thermal coefficients of the layers (the temperature isuniformly distributed over the laminate thickness)

"

α2e+ νe

21αe1#sin2φ

2

"

α2e+ νe

21αe1#cos2φ



T

= 317.61 · 10−6T GPa/◦C

Since the layers are orthotropic, A T

12 = 0 for all of them Specifying the coordinates ofthe layers (see Fig 5.10) i.e.,

t0= 0 mm, t1= 0.02 mm, t2= 1.02 mm, t3= 10.02 mm, t4= 13.52 mm and applying Eq (7.27), we calculate the parameters J mn (r)for the laminate

!

2 t22− t2

1

!+ A T11

!

3 t32− t2

2

!+ A T11

To determine M mn T , we need to specify the reference surface of the laminate Assumethat this surface coincides with the middle surface, i.e., that e= h/2 = 6.76 mm Then,

so that κ xT = 0 and κ yT = 0 Since there are no external loads, the free body diagram

enables us to conclude that N x = 0 and N y = 0 As a result, the first two equations ofEqs (7.23) for the structure under study become

B11ε0xT + B12ε0yT = N T

11

B21ε xT0 + B22ε0yT = N T

22Solving these equations for thermal strains and taking into account Eqs (7.20), we get

12 For the laminate under study, calculation yields

α x = −0.94 · 10−61/◦C, α

y = 14.7 · 10−61/◦C

Return to Eqs (7.13) and (7.20) based on the assumption that the coefficients of thermalexpansion do not depend on temperature For moderate temperatures, this is a reasonableapproximation This conclusion follows from Fig 7.6, in which the experimental results

of Sukhanov et al (1990) (shown with solid lines) are compared with Eqs (7.20), in which

T = T − 20◦C (dashed lines) represent carbon–epoxy angle-ply laminates However,

for relatively high temperatures, some deviation from linear behavior can be observed

In this case, Eqs (7.13) and (7.20) for thermal strains can be generalized as

ε T =

T

T

α(T )dT

−20

−10

10 20 30

Fig 7.6 Experimental dependencies of thermal strains on temperature (solid lines) for±φ angle-ply carbon–

epoxy composite and the corresponding linear approximations (dashed lines).

Temperature variations can also result in a change in material mechanical properties

As follows from Fig 7.7, in which the circles correspond to the experimental data of

Ha and Springer (1987), elevated temperatures result in either higher or lower reduction

of material strength and stiffness characteristics, depending on whether the correspondingmaterial characteristic is controlled mainly by the fibers or by the matrix The curvespresented in Fig 7.7 correspond to a carbon–epoxy composite, but they are typicalfor polymeric unidirectional composites The longitudinal modulus and tensile strength,being controlled by the fibers, are less sensitive to temperature than longitudinal com-pressive strength, and transverse and shear characteristics Analogous results for a moretemperature-sensitive thermoplastic composite studied by Soutis and Turkmen (1993) arepresented in Fig 7.8 Metal matrix composites demonstrate much higher thermal resis-tance, whereas ceramic and carbon–carbon composites have been specially developed towithstand high temperatures For example, carbon–carbon fabric composite under heat-ing up to 2500◦C demonstrates only a 7% reduction in tensile strength and about 30%

reduction in compressive strength without significant change of stiffness

Analysis of thermoelastic deformation for materials whose stiffness characteristicsdepend on temperature presents substantial difficulties because thermal strains are causednot only by material thermal expansion, but also by external forces Consider, for example,

a structural element under temperature T0 loaded with some external force P0, and assume that the temperature is increased to a value T1 Then, the temperature change will cause a thermal strain associated with material expansion, and the force P0, being constant, also induces additional strain because the material stiffness at temperature T1 is less than its

stiffness at temperature T0 To determine the final stress and strain state of the structure,

Fig 7.7 Experimental dependencies of normalized stiffness (solid lines) and strength (dashed lines)

character-istics of unidirectional carbon–epoxy composite on temperature.

0 0.2 0.4 0.6 0.8 1

s+ 2

s+ 1

Fig 7.8 Experimental dependencies of normalized stiffness (solid lines) and strength (dashed lines)

character-istics of unidirectional glass–polypropylene composite on temperature.

we should describe the process of loading and heating using, e.g., the method of successiveloading (and heating) presented in Section 4.1.2.

7.2 Hygrothermal effects and aging

Effects that are similar to temperature variations, i.e., expansion and degradation ofproperties, can also be caused by moisture Moisture absorption is governed by Fick’slaw, which is analogous to Fourier’s law, Eq (7.1), for thermal conductivity, i.e.,

Consider a laminated composite material shown in Fig 7.9 for which n coincides with the

zaxis Despite the formal correspondence between Eq (7.2) for thermal conductivity and

Eq (7.32) for moisture diffusion, there is a difference in principle between these problems

This difference is associated with the diffusivity coefficient D, which is much lower than

x x

z

(b)(a)

Fig 7.9 Composite material exposed to moisture on both surfaces z = 0 and z = h (a), and on the surface

z= 0 only (b).

the thermal conductivity λ of the same material As is known, there are materials, e.g., metals, with relatively high λ and practically zero D coefficients Low D-value means

that moisture diffusion is a rather slow process As shown by Shen and Springer (1976),the temperature increase in time inside a surface-heated composite material reaches asteady (equilibrium) state temperature about 106 times faster than the moisture contentapproaching the corresponding stable state This means that, in contrast to Section 7.1.1 inwhich the steady (time-independent) temperature distribution is studied, we must considerthe time-dependent process of moisture diffusion To simplify the problem, we can neglect

the possible variation of the mass diffusion coefficient D over the laminate thickness, taking D= constant for polymeric composites Then, Eq (7.33) reduces to

D∂2W

∂z2 = ∂W

Consider the laminate in Fig 7.9a Introduce the maximum moisture content W m that

can exist in the material under the preassigned environmental conditions Naturally, W m

depends on the material nature and structure, temperature, relative humidity (RH) of the

gas (e.g., humid air), or on the nature of the liquid (distilled water, salted water, fuel,lubricating oil, etc.) to the action of which the material is exposed Introduce also thenormalized moisture concentration as

The integration constants can be found from the boundary conditions on the surfaces z= 0

and z = h (see Fig 7.9a) Assume that on these surfaces W = W m or w = 1 Then, inaccordance with Eq (7.36), we get

The first of these conditions yields C2n = 0, whereas from the second condition we have sin r n h = 0, which yields



(7.40)

where z = z/h.

For the structure in Fig 7.9b, the surface z = h is not exposed to moisture, and hence

q W (z = h) = 0 So, in accordance with Eq (7.31), the second boundary condition in Eqs (7.37) must be changed to w(h, t ) = 0 Then, instead of Eq (7.38), we must use

r n h= π

2(2n − 1)

Comparing this result with Eq (7.38), we can conclude that for the laminate in Fig 7.9b,

w(z, t ) is specified by the solution in Eq (7.40) in which we must change h to 2h The mass increase of the material with thickness h is

M = A

h

0

mdz where A is the surface area Using Eqs (7.32) and (7.35), we get

M = AmW m

h

wdz

Switching to a dimensionless variable z = z/h and taking the total moisture content as

of Con t found in accordance with Eq (7.42) is presented in Fig 7.11.

An interesting interpretation of the curve in Fig 7.11 can be noted if we change the

100

50

10

1

Fig 7.10 Distribution of the normalized moisture concentration w over the thickness of 1-mm-thick carbon–

epoxy composite for various exposure times t.

t, hours

0 0.2

hour

t,

Fig 7.12 Dependence of the normalized moisture concentration on √

t

part of the curve is close to a straight line whose slope can be used to determine the

diffusion coefficient of the material matching the theoretical dependence C(t) with the

experimental one Note that experimental methods usually result in rather approximate

evaluation of the material diffusivity D with possible variations up to 100% (Tsai, 1987) The maximum value of the function C(t) to which it tends to approach determines the maximum moisture content C = W

Thus, the material behavior under the action of moisture is specified by two

experimen-tal parameters – D and C m– which can depend on the ambient media, its moisture content,

and temperature The experimental dependencies of C in Eq (7.41) on t for 0.6-mm-thick carbon–epoxy composite exposed to humid air with various relative humidity (RH ) levels

are shown in Fig 7.13 (Survey, 1984) As can be seen, the moisture content is mately proportional to the air humidity The gradients of the curves in Fig 7.13 depend

approxi-on the laminate thickness (Fig 7.14, Survey, 1984)

Fig 7.14 Dependencies of the moisture content on time for a carbon–epoxy composite with thickness

3.6 mm (1), 1.2 mm (2), and 0.6 mm (3) exposed to humid air with 75% RH.

Among polymeric composites, the highest capacity for moisture absorption under room

temperature is demonstrated by aramid composites (7 ± 0.25% by weight) in which both

the polymeric matrix and fibers are susceptible to moisture Glass and carbon polymeric

composites are characterized with moisture content 3.5 ±0.2% and 2±0.75%, respectively.

In real aramid–epoxy and carbon–epoxy composite structures, the moisture content isusually about 2% and 1%, respectively The lowest susceptibility to moisture is demon-strated by boron composites Metal matrix, ceramic, and carbon–carbon composites arenot affected by moisture

The material diffusivity coefficient D depends on temperature in accordance with the

Arrhenius relationship (Tsai, 1987)

D(T a )= D0

e k/T a

in which D0 and k are some material constants and T ais the absolute temperature imental dependencies of the moisture content on time in a 1.2-mm-thick carbon–epoxy

Exper-composite exposed to humid air with 95% RH at various temperatures are presented

in Fig 7.15 (Survey, 1984) The most pronounced effect of temperature is observedfor aramid–epoxy composites The corresponding experimental results of Milyutin et al.(1989) are shown in Fig 7.16

When a material absorbs moisture, it expands, demonstrating effects that are similar tothermal effects, which can be modeled using the equations presented in Section 7.1.2, if

we treat α1 , α2and α x , α y as coefficients of moisture expansion and change T for C.

Similar to temperature, increase in moisture reduces material strength and stiffness Forcarbon–epoxy composites, this reduction is about 12%, for aramid–epoxy composites,about 25%, and for glass–epoxy materials, about 35% After drying out, the effect ofmoisture usually disappears

Fig 7.15 Dependencies of the moisture content on time for 1.2-mm-thick carbon–epoxy composite exposed to

humid air with 95% RH under temperatures 25◦C (1), 50◦C (2), and 80◦C (3).

0 4 8 12 16

Fig 7.16 Moisture content as a function of time and temperature for aramid–epoxy composites.

The cyclic action of temperature, moisture, or sun radiation results in material aging, i.e.,

in degradation of the material properties during the process of material or structure storage.For some polymeric composites, exposure to elevated temperature, which can reach 70◦C,

and radiation, whose intensity can be as high as 1 kW/m2, can cause more complete curing

of the resin and some increase of material strength in compression, shear, or bending.However, under long-term action of the aforementioned factors, the material strengthand stiffness decrease To evaluate the effect of aging, testing under transverse bending(see Fig 4.98) is usually performed The flexural strength obtained

σf = 3P l

2bh2

allows for both fiber and matrix material degradation in the process of aging Experimentalresults from G.M Gunyaev et al showing the dependence of the normalized flexuralstrength on time for advanced composites are presented in Fig 7.17 The most dramatic

is the effect of aging on the ultimate transverse tensile deformation ε2 of unidirectionalcomposites: the low value of which results in cracking of the matrix as discussed inSections 4.4.2 and 6.4 After accelerated aging, i.e., long-term moisture conditioning attemperature 70◦C, a 0.75% moisture content in carbon–epoxy composites results in about

20% reduction of ε2, whereas a 1.15% moisture content causes about 45% reduction.

Environmental effects on composite materials are discussed in detail elsewhere(Tsai, 1987; Springer, 1981, 1984, 1988)

0 0.2 0.4 0.6 0.8 1

1

2

4 3

s f

Fig 7.17 Dependence of the normalized flexural strength on the time of aging for boron (1), carbon (2),

aramid (3), and glass (4) epoxy composites.

7.3 Time and time-dependent loading effects

7.3.1 Viscoelastisity

Polymeric matrices are characterized with pronounced viscoelastic properties ing in time-dependent behavior of polymeric composites that manifests itself in creep(see Section 1.1), stress relaxation, and dependence of the stress–strain diagram on therate of loading It should be emphasized that in composite materials, viscoelastic defor-mation of the polymeric matrix is restricted by the fibers that are usually linear elasticand do not demonstrate time-dependent behavior The one exception to existing fibers

result-is represented by aramid fibers that are actually polymeric themselves by their nature.The properties of metal matrix, ceramic, and carbon–carbon composites under normalconditions do not depend on time Rheological (time-dependent) characteristics of struc-tural materials are revealed in creep tests allowing us to plot the dependence of strain ontime under constant stress Such diagrams are shown in Fig 7.18 for the aramid–epoxycomposite described by Skudra et al (1989) An important characteristic of the mate-rial can be established if we plot the so-called isochrone stress–strain diagrams shown in

Fig 7.19 Three curves in this figure are plotted for t = 0, t = 100, and t = 1000 days,

and the points on these curves correspond to points 1, 2, 3 in Fig 7.18 As can be seen,the initial parts of the isochrone diagrams are linear, which means that under moderatestress, the material under study can be classified as a linear-viscoelastic material To char-acterize such a material, we need to have only one creep diagram, whereby the othercurves can be plotted, increasing strains in proportion to stress For example, the creep

curve corresponding to σ1= 450 MPa in Fig 7.18 can be obtained if we multiply strains

corresponding to σ1= 300 MPa by 1.5

Fig 7.18 Creep strain response of unidirectional aramid–epoxy composite under tension in longitudinal

direction with three constant stresses.

0 200 400 600

t = 1000

1 1

2 2

3 3

e1,%

Fig 7.19 Isochrone stress–strain diagrams corresponding to creep curves in Fig 7.18.

Linear-viscoelastic material behavior is described with reasonable accuracy by thehereditary theory, according to which the dependence of strain on time is expressed as

t

t dt

t

Fig 7.20 Geometric interpretation of the hereditary constitutive theory.

strain ε v depending on the loading process as if the material ‘remembers’ this process

Within the framework of this interpretation, the creep compliance C(θ ), where θ = t − τ

can be treated as some ‘memory function’ that should, obviously, be infinitely high at

θ = 0 and tend to zero for θ → ∞, as in Fig 7.21.

The inverse form of Eq (7.43) is

Here, R(t − τ) is the relaxation modulus or the relaxation kernel that can be expressed,

as shown below, in terms of C(t − τ).

The creep compliance is determined using experimental creep diagrams Transforming

to a new variable θ = t − τ, we can write Eq (7.43) in the following form

q = t – t

C(q)

Fig 7.21 Typical form of the creep compliance function.

For a creep test, the stress is constant, so σ = σ0,and Eq (7.44) yields

ε(t ) = ε0



1+

t0