Chapter 8

Thermal and thermoelastic behaviour in composite materials The behaviour of composite materials is often sensitive to changes in temperature. This arises for two main reasons:

•   the response of the matrix to an applied load is temperature-dependent

•   changes in temperature can cause internal stresses to be set up as a result of differential thermal contraction and expansion of the two constituents.

These stresses affect the thermal expansion coefficient of the composite. Furthermore, significant stresses are normally present in the material at ambient temperatures, since it has in most cases been cooled at the end of the fabrication process. Changes in internal stress state on altering the temperature can be substantial and may strongly influence the response of the material to an applied load. Finally, the thermal conductivity of composite materials is of interest, since many applications and processing procedures involve heat flow of some type. This property can be predicted from the conductivities of the constituents, although the situation may be complicated by poor thermal contact across the interfaces.

 

8.1 Micromechanics: thermal expansion

Figure 8.1: Predicted elastic stress field in a long-fibre composite of polyester/35% glass fibre, showing

the effect of cooling through a temperature interval of 100 K

Let consider a long-fibre composite material. The simplest case is to treat is the axial expansion. It can be seen from Figure 8.1 that the axial stresses are uniform in both constituents, which simplifies the problem. The slab model provides a good basis for this calculation, as it does for the Young's modulus. There are two stages in the calculation, illustrated in Figure 8.2.

Figure 8.2: Schematic representation of the use of the slab model to derive an expression for the axial

thermal expansion coefficient of a long-fibre composite.

Firstly, both constituents are allowed to expand freely. Then, they are constrained to have the same length in the axial direction, by the imposition of appropriate stresses. The final net expansion can then be treated as consisting of two terms; a natural expansion (that of the matrix is usually considered) and an additional contraction or expansion caused by the internal stress. Thus

𝛼"ΔT = 𝛼&ΔT + ϵ& (8.1) where 𝛼" is the thermal expansion coefficient of the composite and ϵ& is the elastic strain in the matrix associated with the internal stress. The value of ϵ& is found by using two simple relationships, based on a strain balance and a force balance. The strain balance is obtained by expressing the sum of the two internal stress-generated strains as the difference between the natural thermal strains

𝛼& − 𝛼* ΔT = ϵ* − ϵ& (8.2)

The force balance is simply a requirement that, since there is no applied stress, the two internal stresses must counterbalance each other when they are converted to forces (multiplied by the relative sectional areas)

1 − 𝑓 σ& + 𝑓σ* = 0

1 − 𝑓 E&ϵ& + 𝑓E*ϵ* = 0 (8.3)

where f is the fraction in volume of fibre. Combining Eq. (8.2) and (8.3) to derive an expression for ϵ&

1 − 𝑓 E&ϵ& + 𝑓E* ϵ& + 𝛼& − 𝛼* ΔT = 0 (8.4)

ϵ& =−𝑓E* 𝛼& − 𝛼* ΔT1 − 𝑓 E& + 𝑓E*

Substitution of this into Eq. (8.1) allows an expression to be obtained for the (axial) coefficient of thermal expansion of the composite

𝛼" = 𝛼& −𝑓E* 𝛼& − 𝛼* ΔT1 − 𝑓 E& + 𝑓E*

=

=𝛼& 1 − 𝑓 E& − 𝛼*𝑓E*

1 − 𝑓 E& + 𝑓E*

(8.5)

Since the axial force balance is reliable for the slab model, this prediction should be quite accurate (it is not entirely rigorous, because differential Poisson contraction strains are neglected.)

The coefficient of thermal expansion in the transverse direction, and the values for short fibre and particulate composites, are more difficult to establish, since the stresses and strains vary with position. Nevertheless, as with the transverse stiffness, some useful approximations can be made. One of the most successful model for the transverse 𝛼 of long-fibre composites is that due to Schapery, who used an energy-based approach to obtain the following expression

𝛼"01 = 𝛼& 1 − 𝑓 1 + 𝜈& + 𝛼*𝑓 1 + 𝜈* − 𝛼"34𝜈56" (8.6)

in which the axial thermal expansion coefficient, 𝛼"34, is that given by Eq. (8.5) and the Poisson ratio 𝜈56" is obtained from a simple rule of mixtures between those of the constituents. The predicted dependence of these two thermal expansion coefficients on fibre content, given by Eq. (8.5) and (8.6), is shown in Figure 8.3 for a polymer matrix composite.

Figure 8.3: Predicted dependence of thermal expansion coefficient on reinforcement content for glass

fibres in an epoxy matrix, according to the force balance (Eq. (8.5)) and Schapery (Eq. (8.6)).

The transverse thermal expansion coefficient of a (long-fibre) composite tends to rise initially as the fibre content increases. This occurs because, on heating the composite, axial expansion of the matrix is strongly inhibited by the presence of the fibres and the resultant axial compression of the matrix generates a Poisson expansion in the transverse direction, which more than compensates for the reduction effected in the normal way by the presence of the fibres, at least for low fibre contents.

8.2 Micromechanics: thermal conductivity

High thermal conductivity is useful in improving the resistance of materials to thermal shock and avoiding the development of hot spots during localised heating. In other situations, a low thermal conductivity can be beneficial in providing thermal insulation. Before considering conductivity levels in fibres, matrices and composites, it is useful to review the mechanisms of thermal conduction.

The conductivity of a composite can be predicted provided suitable assumptions are made about the flow of heat through the constituents, i.e. the shape of the isotherms. For heat flux in long-fibre composites, the slab model can be used. Axial and transverse heat flow are represented schematically in Figure 8.4.

Figure 8.4: Derivation of expressions for axial and transverse conductivity. Schematic depiction of the

slab model for long-fibre composites, axial heat flux (left) and transverse heat flux (right).

For the axial case, the thermal gradient (spacing between the isotherms) is the same in each constituent. The total heat flux is given by the sum of the flows through fibre and matrix

𝑞5" = 𝑓𝑞5* + 1 − 𝑓 𝑞5& (8.7) These can be written in terms of conductivities and thermal gradients

−𝜆5"𝑇: = −𝑓𝜆*𝑇: − 1 − 𝑓 𝜆&𝑇: (8.8) so that the composite conductivity is given by a simple rule of mixtures

𝜆5" = 𝑓𝜆* + 1 − 𝑓 𝜆& (8.9) This expression should give a reliable value for the axial conductivity.

The transverse conductivity, based on the slab model, is obtained by equating the heat fluxes through the two constituents

𝑞6" = −𝜆6"𝑇": = 𝑞6* = −𝜆6*𝑇*: = 𝑞6& = −𝜆6&𝑇&: (8.10) The thermal gradients in the two constituents are related to the overall average value by the expression

𝑇": = 𝑓𝑇*: + 1 − 𝑓 𝑇&: (8.11) leading to the following expression for the transverse conductivity

𝜆6" =𝑓𝜆*+1 − 𝑓𝜆&

;5

(8.12)

As with that derivation, the assumption that the two constituents lie in series with each other leads to this expression being of poor accuracy. In fact, it represents a lower bound. It becomes very unreliable when the fibres have a low conductivity; for insulating fibres, the composite is predicted to have zero conductivity even when the fibre volume fraction is small. More reliable treatments are available for the transverse conductivity. For example, Hatta and Taya have shown that, for the transverse conductivity of a long-fibre composite, this reduces to the expression

𝜆6" = 𝜆& +𝜆& 𝜆* − 𝜆& 𝑓

𝜆& + 1 − 𝑓 𝜆* − 𝜆& 2 (8.12)

Predictions obtained from these equations are shown in Figure 8.5.

Figure 8.5: Predicted plots of composite conductivity (relative to that of the matrix) as a function of

fibre content, for insulating (𝜆𝑓 = 0) and highly conducting (𝜆𝑓 = 10𝜆𝑚) reinforcements.

The rule of mixtures expression gives a reliable prediction for the axial conductivity of long-fibre composites, but it can be seen that the slab model expression for transverse conductivity (Eq. (8.12)), which is completely unreliable for insulating fibres, also gives a large error when the fibres have high conductivity.

8.3 Micromechanics: interfacial thermal resistance

The above calculations are based on the assumption that there is perfect thermal contact between fibre and matrix. In practice, this may be impaired by the presence of an interfacial layer of some sort, or by voids or cracks in the vicinity of the interface. Furthermore, even in the absence of such barriers to heat flow, there may be some loss of heat transfer efficiency across the interface if the carriers are different in the two constituents, as with metal/ceramic systems. Such a thermal resistance is characterised by an interfacial heat transfer coefficient or thermal conductance, h (W/m2 K), defined as the proportionality constant between the heat flux through the boundary and the temperature drop across it

𝑞> = ℎ  Δ𝑇> (8.13) Modelling of the thermal conductivity of composites in the presence of such an interfacial resistance has been addressed by several authors. Hasselman and Johnson derived an analytical expression for the transverse conductivity of a long-fibre composite.

𝜆" = 𝜆&𝑓

𝜆*𝜆&

−𝜆*𝑟ℎ − 1 +

𝜆*𝜆&

+𝜆*𝑟ℎ + 1

𝑓 1 −𝜆*𝜆&

+𝜆*𝑟ℎ +

𝜆*𝜆&

+𝜆*𝑟ℎ + 1

(8.14)

where r is the radius of the fibres. The scale of the structure is now relevant because it determines how frequently interfaces are encountered by the heat flow. This scale effect is represented by

the dimensionless ratio BC1D

.

Figure 8.6: Plots (Gordon et al) of thermal conductivity ratio 𝜆𝑐/𝜆𝑚 against temperature, for a Ti-6A-4V alloy reinforced with 35% of SiC monofilament, with heat flow transverse to the fibre axis.

Figure 8.6 shows measured conductivities over a range of temperature for a Ti-based long-fibre composite, plotted as a ratio to that of the matrix. Also shown are predictions from Eq. (8.14) corresponding to several h values, using the appropriate fibre radius (~50𝜇𝑚). These data are consistent with an h value of around 106 W/m2 K. Although this figure represents relatively poor thermal contact for a fibre/matrix interface, it can be seen that in this system the overall conductivity would not have been much greater had the interface been perfect (ℎ   =  ∞). This illustrates the size effect; the large diameter of the SiC monofilaments means that the magnitude of interfacial resistance is not of much significance (unless it becomes very large indeed).

8.4 Micromechanics: residual stresses

The nature of the interfacial contact is strongly influenced by the presence of residual stresses. These arise from a number of sources, such as plastic deformation of the matrix and phase transformations involving volume changes. There are also volume changes associated with the curing of thermosetting resins. One of the most important sources of residual stress is the thermal contraction which occurs during post-fabrication cooling. Since, for most composite systems, the fibre has a smaller thermal expansion coefficient than the matrix, the resultant stresses are compressive in the fibre and tensile in the matrix. This arises because the matrix contracts onto the fibre and compresses it. The nature of the stress field is illustrated by Figure 8.1. This shows the principal stresses (in axial, hoop and radial directions) for a long glass fibre surrounded by a tube of polyester resin, after cooling through 100 °C. The normal stresses across the interface (radial stresses) are compressive. This is particularly relevant to the interfacial bonding, since this compression ensures that fibre and matrix are kept in close contact and increases the resistance

to debonding and sliding. A relatively high interfacial shear strength results when the residual thermal stresses are large.

Differences between different matrices are highlighted by the data in Table 8.1. This shows, for example, that epoxies can have good resistance to heat distortion and also that they shrink less during curing than polyesters. This is a significant advantage, as is the fact that they can be partially cured, so that pre-pregs can be supplied. In fact, epoxies are superior in most respects to the alternative thermosetting systems, which are sometimes preferred simply on grounds of lower cost.

Table 8.1: Comparison between thermosets and thermoplastics of properties relating to dimensional and

environmental stability.

A simple analysis can be used to estimate the changing axial matrix stress in a long-fibre composite. Assuming no interfacial sliding, fibre yielding, etc., the axial strain of the composite must be equal to that of the fibres, which can be expressed as the sum of their natural thermal expansion and their elastic strain

𝜖5" = 𝜖5* = 𝛼*ΔT +𝜎5* − 𝜈* 𝜎6* + 𝜎M*

𝐸* (8.15)

Since the radial and hoop stresses in the fibre (𝜎6*  and  𝜎M*) are relatively small compared with the axial stress, the contribution from the transverse fibre stresses can be neglected. The axial force balance ( 𝑓𝜎5* − 1 − 𝑓 𝜎5& = 0 ) can then be used to find the axial stress in the matrix

𝜎5& =𝑓

1 − 𝑓 𝐸* 𝛼*ΔT − 𝜖5"   (8.16)

Hence, 𝜎5& is simply proportional to the difference, Δ𝜖, between the natural thermal strain of the fibre and the measured strain of the composite. The initial thermal stress in the matrix can be deduced by taking the composite up to high temperature (~ 0.8 Tm), where the matrix stress will become very small, and running the fibre thermal expansion line back from this region (stretching the fibre only).

8.5 Macromechanics: thermal stresses in a single ply

Consider a rectangular ply of orthotropic material. We define a material reference system (O,1,2) and a global one (O,x,y). In the material reference system, for a single lamina, the linear elastic constitutive model can be represented as follow:

𝜎5𝜎6𝜏56

=1

1 − 𝜈56𝜈65

𝐸5 𝜈65𝐸5 0𝜈56𝐸6 𝐸6 00 0 𝐺56 1 − 𝜈56𝜈65

𝜖5T

𝜖6T

𝛾56T=

                     =𝑄55 𝑄56 0𝑄56 𝑄66 00 0 𝑄WW

𝜖5 − 𝛼5ΔT𝜖6 − 𝛼6ΔT

𝛾56

(8.17)

where 𝐸5and 𝐸6are the elastic modules (for an unidirectional lamina it can be the longitudinal and transversal modules) , 𝜈56 and 𝜈65the Poisson’s ratios and 𝐺56 the shear module. In order to have a symmetric matrix XYZ

[Y= XZY

[Z.

The eq. (8.17) can be rewritten as:

𝜎5𝜎6𝜏56

=𝑄55 𝑄56 0𝑄56 𝑄66 00 0 𝑄WW

𝜖5𝜖6𝛾56

−𝑄55 𝑄56 0𝑄56 𝑄66 00 0 𝑄WW

𝛼5𝛼60

ΔT (8.18)

or

𝜎 56 = 𝑄 56  𝜖 56 − 𝑄 56  𝛼 56 ΔT (8.19)

In order to transform the constitutive relationship from the material to the global reference system it is necessary to apply a rotation of angle 𝜗 (Figure 8.7)

Figure 8.7: Transformation from material to global reference system.

𝜎5𝜎6𝜏56

=cos6 𝜗 sin6 𝜗 2  sin 𝜗 cos 𝜗sin6 𝜗 cos6 𝜗 −2 sin 𝜗 cos 𝜗

−sin 𝜗 cos 𝜗 sin 𝜗 cos 𝜗 cos6 𝜗 −  sin6 𝜗

𝜎4𝜎a𝜏4a

(8.20)

or

𝜎 56 = 𝑇  𝜎 4a (8.21)

Inverse transformation is also possible to be expressed as:

𝜎 4a = 𝑇;5  𝜎 56 (8.22)

We can apply the same transformation over the total strain 𝜖 , the thermal strain 𝜖b and the thermal expansion coefficient 𝛼 vectors.

𝜖 56 = 𝑇  𝜖 4a        ,        𝜖 4a = 𝑇;5  𝜖 456 (8.23)

𝜖 56b = 𝑇  𝜖 4a

b        ,        𝜖 4ab = 𝑇;5  𝜖 456

b (8.24)

𝛼 56 = 𝑇  𝛼 4a        ,        𝛼 4a = 𝑇;5  𝛼 56 (8.25)

In this case we are transforming the strain vector expressed in the form 𝜖5𝜖6𝜖56

and 𝜖4𝜖a𝜖4a

. If we

wont to use the form with the shear strain 𝛾56 and 𝛾4a we have to apply the transformation 𝑅:

𝑅 =1 0 00 1 00 0 2

(8.26)

We will obtain:

𝜖5𝜖6𝛾56

= 𝑅𝜖5𝜖6𝜖56

= 𝑅  𝑇𝜖4𝜖a𝜖4a

= 𝑅  𝑇  𝑅;5𝜖4𝜖a𝛾4a

(8.27)

𝑇e = 𝑅  𝑇  𝑅;5

For the thermal strain we can write:

𝜖 56b = 𝑅  𝛼 56 ΔT

𝜖 4ab = 𝑅  𝛼 4a ΔT = 𝑅  𝑇;5  𝛼 56 ΔT

𝜖4b

𝜖ab

𝛾4ab=

1 0 00 1 00 0 2

cos6 𝜗 sin6 𝜗 −2  sin 𝜗 cos 𝜗sin6 𝜗 cos6 𝜗 2 sin 𝜗 cos 𝜗

sin 𝜗 cos 𝜗 −sin 𝜗 cos 𝜗 cos6 𝜗 −  sin6 𝜗  𝛼5𝛼60

ΔT =

=  𝛼5 cos6 𝜗 + 𝛼6 sin6 𝜗𝛼5 sin6 𝜗 + 𝛼6 cos6 𝜗

2𝛼5 sin 𝜗 cos 𝜗 − 2𝛼6 sin 𝜗 cos 𝜗ΔT =

(8.28)

From Eq. (8.28) it is possible to observe that for an angle 𝜗 ≠ 0°  or  90° there is a shear thermal deformation.

From the above mentioned, it is possible to express the stress in global reference system as:

𝜎 4a = 𝑇;5  𝜎 56 = 𝑇;5  𝑄 56  𝜖 56 − 𝑇;5  𝑄 56  𝛼 56 ΔT (8.29)

𝜎 4a = 𝑇;5  𝑄 56  𝑇e  𝜖 4a − 𝑇;5  𝑄 56  𝛼 56 ΔT (8.30)

𝜎 4a = 𝑄 4a  𝜖 4a − 𝑄 j4a  ΔT (8.31)

where:

𝑄 4a = 𝑇;5  𝑄 56  𝑇e   (8.32)

and

𝑄 j4a = 𝑇;5  𝑄 56  𝛼 56   (8.33)

The matrix 𝑄 4a under the effect of the transformations 𝑇;5and 𝑇e is a full matrix:

𝑄 4a =𝑄55 𝑄56 𝑄5W𝑄65 𝑄66 𝑄6W𝑄W5 𝑄W6 𝑄WW

  (8.34)

The following relations are verified:

𝑄5W −𝜗 = −𝑄5W 𝜗  

𝑄6W −𝜗 = −𝑄6W 𝜗

𝑄>k −𝜗 = 𝑄>k 𝜗        𝑖𝑗 = 11, 22, 12, 66

(8.35)

The vector 𝑄 j4a is:

𝑄 j4a =𝑄j5𝑄j6𝑄j56

  (8.36)

with components:

 𝑄j5 = 𝑄55𝛼5 cos6 𝜗 + 𝑄56𝛼6 cos6 𝜗 + 𝑄56𝛼5 sin6 𝜗 + 𝑄66𝛼6 sin6 𝜗

𝑄j6 = 𝑄55𝛼5 sin6 𝜗 + 𝑄56𝛼6 sin6 𝜗 + 𝑄56𝛼5 cos6 𝜗 + 𝑄66𝛼6 cos6 𝜗

𝑄j56 = 2𝑄55𝛼5 sin 𝜗 cos 𝜗 + 2𝑄56𝛼6 sin 𝜗 cos 𝜗 − 2𝑄56𝛼5 sin 𝜗 cos 𝜗 +− 2𝑄66𝛼6 sin 𝜗 cos 𝜗

(8.37)

The following relations are verified:

𝑄j56 −𝜗 = −𝑄j56 𝜗  

𝑄j> −𝜗 = 𝑄j> 𝜗        𝑖 = 1,2 (8.38)

8.6 Macromechanics: thermal stresses in laminates

Problems associated with thermal stresses can be severe with laminates. Thermal misfit strains now arise, not only between fibre and matrix, but between the individual plies of the laminate. For example, since a lamina usually has a much larger expansion coefficient in the transverse direction than in the axial direction, heating of a cross-ply laminate will lead to the transverse expansion of each ply being strongly inhibited by the presence of the other ply. This is useful in

the sense that it will make the dimensional changes both less pronounced and more isotropic, but it does lead to internal stresses in the laminate. These can cause the laminate to distort, so that it is no longer flat; an illustration of the type of distortion which can occur was shown in Figure 8.8.

Figure 8.8: Schematic illustration of how a cross-ply laminate will tend to distort as a result of through-

thickness coupling stresses when subjected to (a) an external load and (b) a change in temperature.

However, even if such distortions do not occur, the stresses which arise on changing the temperature can cause microstructural damage and impairment of properties.

Consider a rectangular laminated plate composed of N number of orthotropic layers as shown in Figure 8.9. The plate is assumed in a laminate reference system, that we can take, for this discussion, to be coincident with the global reference system (in general it is not: the laminate reference system is a local one). It is convenient to take the xy−plane of the coordinate system to be the undeformed middle plane of the laminate. The z− axis is taken to be positive in a downward direction from the middle plane. The plate thickness is denoted by h while its dimensions, along the x and y directions, are denoted by a and b respectively. Perfect bonding between the orthotropic layers and temperature independent mechanical and thermal properties are assumed. The plate is subjected to a mechanical load q(x, y) and thermal load ΔT 𝑥, 𝑦, 𝑧 .

Figure 8.9: Plate geometry and reference system.

Consider to have a linear distribution of temperature through-the-thickness as in Figure 8.10.

Figure 8.10: Temperature distribution through-the thickness of the lamina.

The temperature load can be expressed as:

ΔT = Δ𝑇3 +𝑧ℎ 2Δ𝑇r (8.39)

where h is the thickness of the laminate.

For the bending analysis, the displacement field of the classical plate theory of Kirchhoff-Love (no shear deformation off-the-plane) at a point in the laminated plate is expressed as:

u 𝑥, 𝑦, 𝑧 = ut 𝑥, 𝑦 − 𝑧𝜕w 𝑥, 𝑦𝜕𝑥 = ut 𝑥, 𝑦 − 𝑧  θa 𝑥, 𝑦

v 𝑥, 𝑦, 𝑧 = vt 𝑥, 𝑦 − 𝑧𝜕w 𝑥, 𝑦𝜕𝑦 = vt 𝑥, 𝑦 − 𝑧  θ4 𝑥, 𝑦

w 𝑥, 𝑦, 𝑧 = w 𝑥, 𝑦, 𝑧

(8.40)

where  θ4 and  θa are the rotations of the section around x and y axes respectively, as in Figure 8.11.

Figure 8.11: displacement field of the classical plate theory of Kirchhoff-Love.

For the small plate deformation, the strain components 𝜖4  , 𝜖a  , 𝛾4a and three displacement components 𝑢  , 𝑣  , 𝑤 are related according to the well-known linear kinematic relations.

𝜖4 =𝜕u𝜕𝑥        ,        𝜖a =

𝜕v𝜕𝑦        ,        𝛾4a =

𝜕u𝜕𝑦 +

𝜕v𝜕𝑥 (8.41)

By applying the strain displacement relations of two-dimensional elasticity to the displacement field given by Eq. (8.40), one obtains the following approximate strain field:

𝜖4 =𝜕ut𝜕𝑥 − 𝑧

𝜕6w𝜕𝑥6 = 𝜖|} + 𝑧  𝜒a

𝜖a =𝜕vt𝜕𝑦 − 𝑧

𝜕6w𝜕𝑦6 = 𝜖|� + 𝑧  𝜒4

𝛾4a =𝜕ut𝜕𝑦 +

𝜕vt𝜕𝑥 − 2𝑧

𝜕6w𝜕𝑥  𝜕𝑦 = 𝛾|}� + 𝑧  𝜒4a

(8.42)

where 𝜒4 = −�Y��aY

       ,        𝜒a = − �Y��4Y

       ,        𝜒4a = −2 �Y��4  �a

are the curvatures.

The stress components of Eq. (8.18), expressed in the laminate reference system (O,x,y) considering thermal effects of Eq. (8.39), for the kth layer can be expressed as:

𝜎4𝜎a𝜏4a

�

=𝑄55 𝑄56 𝑄5W𝑄65 𝑄66 𝑄6W𝑄W5 𝑄W6 𝑄WW

�

 𝜖4𝜖a𝛾4a

−𝑄j5𝑄j6𝑄j56

�

Δ𝑇3 +𝑧ℎ 2Δ𝑇r (8.43)

(total strains have to be the same for all laminae)

If we associate with strains given by Eq. (8.42):

𝜎4𝜎a𝜏4a

�

=𝑄55 𝑄56 𝑄5W𝑄65 𝑄66 𝑄6W𝑄W5 𝑄W6 𝑄WW

�

 𝜖|} + 𝑧  𝜒a𝜖|� + 𝑧  𝜒4𝛾|}� + 𝑧  𝜒4a

−𝑄j5𝑄j6𝑄j56

�

Δ𝑇3 +𝑧ℎ 2Δ𝑇r (8.44)

The resultant forces and moments of a laminated composite plate can be obtained by integrating Eq. (8.44) through thickness, and are written as:

𝑁4𝑁a𝑁4a

=𝜎4𝜎a𝜏4a

�

𝑑𝑧 =D��Z

D�

�

��5

=𝑄55 𝑄56 𝑄5W𝑄65 𝑄66 𝑄6W𝑄W5 𝑄W6 𝑄WW

�𝜖|}𝜖|�𝛾|}�

𝑑𝑧D��Z

D�

�

��5

+𝑄55 𝑄56 𝑄5W𝑄65 𝑄66 𝑄6W𝑄W5 𝑄W6 𝑄WW

�  𝜒a  𝜒4  𝜒4a

𝑧  𝑑𝑧D��Z

D�

�

��5

−𝑄j5𝑄j6𝑄j56

�

Δ𝑇3  𝑑𝑧D��Z

D�

�

��5

−𝑄j5𝑄j6𝑄j56

�

2ℎ  Δ𝑇r  𝑧  𝑑𝑧

D��Z

D�

�

��5

(8.45)

𝑀4𝑀a𝑀4a

=𝜎4𝜎a𝜏4a

�

𝑧  𝑑𝑧 =D��Z

D�

�

��5

=𝑄55 𝑄56 𝑄5W𝑄65 𝑄66 𝑄6W𝑄W5 𝑄W6 𝑄WW

�𝜖|}𝜖|�𝛾|}�

𝑧  𝑑𝑧D��Z

D�

�

��5

+𝑄55 𝑄56 𝑄5W𝑄65 𝑄66 𝑄6W𝑄W5 𝑄W6 𝑄WW

�  𝜒a  𝜒4  𝜒4a

𝑧6  𝑑𝑧D��Z

D�

�

��5

−𝑄j5𝑄j6𝑄j56

�

Δ𝑇3  𝑧  𝑑𝑧D��Z

D�

�

��5

−𝑄j5𝑄j6𝑄j56

�

2ℎ  Δ𝑇r  𝑧

6  𝑑𝑧D��Z

D�

�

��5

(8.46)

If we introduce the coefficients 𝐴>k  , 𝐵>k  , 𝐷>k  , 𝐴j>  , 𝐵j>  , 𝐷j> as:

𝐴>k  , 𝐵>k  , 𝐷>k = 𝑄>k 1  , 𝑧  , 𝑧6 𝑑𝑧D��Z

D�

�

��5

               𝑖, 𝑗 = 1, 2, 6 (8.47)

𝐴j>  , 𝐵j>  , 𝐷j> = 𝑄j> 1  , 𝑧  , 𝑧6 𝑑𝑧D��Z

D�

�

��5

               𝑖, 𝑗 = 1, 2, 12 (8.48)

it is possible to write Eqs. (8.45) and (8.46) in the form:

𝑁𝑀 =

𝐴 𝐵𝐵 𝐷

𝜖t𝜒 −

𝐴j𝐵j

Δ𝑇3 −𝐵j𝐷j

2ℎ  Δ𝑇r (8.49)

from which it is possible to deduce in general the coupling effect between membrane and flexural-torsional behaviour of the laminated plate subjected to mechanical and thermal actions.

For the calculation of the thermal stresses, it is necessary to evaluate the deformation components of the middle plane of the laminate. These can be evaluated taking into account that thermal stresses, in absence of external loads applied to the laminate, makes up an auto-balanced system with force resultant and momentum equal to zero.

00 =

𝐴 𝐵𝐵 𝐷

𝜖t𝜒 −

𝐴j𝐵j

Δ𝑇3 −𝐵j𝐷j

2ℎ  Δ𝑇r

𝐴 𝐵𝐵 𝐷

𝜖t𝜒 =

𝐴j𝐵j

Δ𝑇3 +𝐵j𝐷j

2ℎ  Δ𝑇r

(8.50)

The Eq. (8.50) show how a temperature variation Δ𝑇 = Δ𝑇3 induces in a generic laminate, normal stress and bending stresses, or in terms of deformations, both dilations and curvature and distortions. The curvatures of the middle plane are nil only if the laminate has vector 𝐵j = 0 (and, because 𝛼 is generally ≠ 0, also the matrix 𝐵 = 0), that it means the laminate is symmetrical

in this case. Therefore, in a non-symmetrical laminate, the inevitable variations of the temperature you have during cooling of the treatment temperature (often higher than 100 ° C) at room temperature produce annoying distortions. Symmetry eliminates distortions but not thermal residual stresses, which can be calculated using the constituent equation (8.50). The residual thermal tensions are added to the operating tensions affecting the resistance and the laminate stability. For a proper and accurate design, it is therefore necessary to take into account of residual stresses in the laminate. At the thermal stresses above calculated, due essentially to the fact that the dilations heat of each lamina are partially prevented by the other laminae, it is necessary to add the internal residual thermal stresses, that occurs also in a single free ply due to of different linear thermal expansion coefficient between fibre and matrix.

8.7 Macromechanics: thermal stresses in cross-ply laminates

For the special case of a cross-ply laminate, with two plies, a simple analytical approach can be used. Figure 8.2 is again applicable, but the two constituents are now the two plies (one oriented axially and the other transversely), rather than the matrix and the fibres.

The same equations result, but the meaning of the parameters changes slightly. For example, taking the transverse ply as the matrix and the axial one as the fibres, a modified version of Eqs. (8.4) now gives the elastic strain in the transverse ply on heating through a temperature interval ΔT

ϵ6 =−E5 𝛼6 − 𝛼5 ΔT

E6 + E5 (8.51)

where E5 , E6 are the axial and transverse Young's moduli and 𝛼5, 𝛼6 are the axial and transverse thermal expansion coefficients. The volume fraction, f , of the two constituents does not appear, since it is equal to 0.5 for a cross-ply laminate in which the plies are of equal thickness and therefore cancels out in Eq. (8.4). This strain can be converted directly to a stress, since it arises from the two plies being forced to have the same length (and does not include the free thermal expansion). The stress to which the transverse ply is subjected can therefore be expressed as

σ6 =−E5E6 𝛼6 − 𝛼5 ΔT

E6 + E5 (8.52)

Plots obtained using Eq. (8.52) are presented in Figure 8.8. This shows the stresses in the transverse plies of two cross-ply laminates, as a function of the volume fraction of fibres, after cooling through a temperature interval of 100 deg K. Property data for the fibres and the matrix were taken from Table 8.2.

Table 8.2: Properties of most common fibres and matrices.

Table 8.3: Typical experimental failure data for laminae based on thermoset matrices.

Figure 8.8: Predicted dependence on fibre content of the thermal stress in the transverse direction within

each ply of two cross-ply laminates, arising from a temperature decrease of 100 K.

Several features are of interest in this figure. A peak is observed on increasing the fibre content. This is expected, since the stresses are due to the orthotropy in 𝛼; the difference between 𝛼5and 𝛼6 has a peak at some intermediate value of f (see Figure 8.3). For the carbon fibre composite, the stress does not fall to zero as f approaches 100%. This is because the carbon fibres are inherently orthotropic in thermal expansion (see Table 8.2); they show zero (or negative) expansion axially on heating, but expand in the transverse (radial) directions.

Levels of thermal stress are of interest for prediction of damage development. For a fibre content of ~ 40—70%, it can be seen from Figure 8.8 that stresses of ~ 25—30 MPa are expected. This is approximately the transverse failure stress for typical laminae (see Table 8.3). It follows that cooling through 100 K (from an initially stress-free state) would be likely to cause damage in such a cross-ply laminate. It may be noted that the requirement to avoid such damage is not that the laminae should have a particular transverse strength level, but rather that they should be able to sustain a particular transverse strain without damage.

In general, if a cross-ply laminate is not balanced, thermal stresses induce bending of the plate. The stiffness matrix 𝑄(4a)�  and  the  vector  𝑄 j4a

�   of a cross-ply have components 𝑄5W = 𝑄6W =

𝑄j56 = 0.

𝑄(4a)� =𝑄55 𝑄56 0𝑄65 𝑄66 00 0 𝑄WW

�

       ,        𝑄 j4a� =

𝑄j5𝑄j60

�

then no shear deformation in xy-plane are induced by uniform thermal load.

(8.53)