CHAPTER  6
MACROMECHANICAL BEHAVIOUR
6.1 INTRODUCTION
6.2 THREEDIMENSIONAL MATERIAL ANISOTROPY
6.4 ELASTIC CONSTANTS AND COMPLIANCES IN TERMS OF ENGINEERING CONSTANTS
6.6 TWODIMENSIONAL CASE: PLANE STRESS
6.7 TRANSFORMATION OF ELASTIC CONSTANTS AND COMPLIANCES
6.7.1 ThreeDimensional Case
6.7.2 TwoDimensional Case
6.8 PARTICULATE AND SHORT FIBRE COMPOSITES
6.9 MULTIDIRECTIONAL FIBRE REINFORCED COMPOSITES
6.11 BIDIRECTIONAL LAMINA
6.12 GENERAL LAMINATES
6.13 LAMINATE HYGROTHERMAL STRAINS
6.14 STRENGTH CRITERIA FOR ORTHOTROPIC LAMINA
6.15 BIBLIOGTAPHY
6.16 EXERCISES
The heterogeneity in a composite material is introduced due to not only its biphase or in some cases multiphase composition, but also laminations. This leads to a distinctly different stress strain behaviour in the case of laminates. The anisotropy caused due to fibre orientations and the resulting extensionshear and bendingtwisting coupling as well as the extensionbending coupling developed due to unsymmetric lamination add to the complexities. A clear understanding of the constitutive equations of a composite laminate is thus desirable before these are used in analysis and design of composite structures. In this chapter, we first introduce to the readers the basic constitutive equations for a general threedimensional anisotropic material with and without material symmetry, elastic constants and compliances and their relations to engineering constants, as well as transformation laws for elastic constants and compliances for both three and twodimensional cases. We also discuss constitutive relations for several composite materials–particulate and short fibre composites, multidirectional fibre reinforced composites, unidirectional lamina and general laminates as well as lamina strength criteria.
6.2 THREEDIMENSIONAL MATERIAL ANISOTROPY
For a threedimensional elastic anisotropic body (Fig. 6.1), the generalized Hook's law is expressed as
(i, j = 1,2,3) (6.1)
where and are the stress and strain tensors, respectively, and are the elastic constants. Here the indices i, j, k and l can assume values of 1, 2 and 3. This implies that there may exist 3^{4} = 81 independent elastic constants. However, it is known from the theory of elasticity, that both stress tensor and strain tensor are symmetric. As = , =
and as = , = (6.2)
Thus, = = = (6.3)
This results in reduction of possible independent elastic constants to thirtysix.
Further, if there exists a strain energy U such that
(6.4)
with the property that , then
= (6.5)
Equation 6.5 in conjunction with Eq. 6.3 finally reduce the total number of independent elastic constants from thirtysix to twentyone only. Such an anisotropic material with twentyone independent elastic constants is termed as triclinic. Now, using the following contracted single index notations
(6.6)
the constitutive relations for the general case of material anisotropy are expressed as
(6.7)
or, ; i, j = 1, 2,….,6 (6.8)
Here, [ ] is the elastic constant matrix.
Conversely, { } = [S_{ij} ] {} ; i, j =1, 2,…..,6 (6.9)
where [S_{ij}] is the compliance matrix.
Note that
[S_{ij}] = [C_{ij }]^{1} (6.10)
Also, [ ] =[ ] and [S_{ij}] = [S_{ji}] due to symmetry.
There may exist several situations when the distribution and orientation of reinforcements may give rise to special cases of material property symmetry. When there is one plane of material property symmetry (say, the plane of symmetry is x_{3 = }0, i.e., the rotation of 180 degree around the x_{3} axis yields an equivalent material), the elastic constant matrix [ ] is modified as
(6.11)
Thus there are thirteen independent elastic constants, and the material is monoclinic. The compliance matrix [S_{ij}] for a monoclinic material may accordingly be written from Eq. 6.11 by replacing 'C ' with 'S '.
If there are three mutually orthogonal planes of symmetry, the material behaviour is orthotropic. The elastic constant matrix is then expressed as
_{orthotropic}_{ }= (6.12)
Thus there are nine independent elastic constants. Correspondingly there exist nine independent compliances.
Two special cases of symmetry, square symmetry and hexagonal symmetry, may arise due to packing of fibres in some regular fashion. This results in further reduction of independent elastic constant. For instance, if the fibres are packed in a square array (Fig. 6.2) in the X_{2}X_{3} plane. Then
[]
_{ square array} = (6.13)
There exist now six independent elastic constants. Similarly, when the fibres are packed in hexagonal array (Fig. 6.3),
(6.14)
In the case of hexagonal symmetry, the number of independent elastic constants is reduced to five only. The material symmetry equivalent to the hexagonal symmetry, is also achieved, if the fibres are packed in a random fashion (Fig. 6.4) in the X_{2}X_{3} plane. This form of symmetry is usually termed as transverse isotropy. The [ ] matrix due to the transverse isotropy is the same as that given in Eq. 6.14. The compliance matrices corresponding to Eqs. 6.12 through 6.14 can be accordingly written down. However,it may be noted that in the case of rectangular array (Fig. 6.5), C_{12 }≠ C_{13}, C_{22 } ≠ C_{33} and C_{55 }≠ C_{66 }(Eq. 6.13).
Material Isotropy
The material properties remain independent of directional change for an isotropic material. The elastic constant matrix [ ] for a three dimensional isotropic material are expressed as
(6.15)
The compliance matrix [S_{ij}] for an isotropic material can be accordingly derived.
6.4 ELASTIC CONSTANTS AND COMPLIANCES IN TERMS OF
The elastic constants or compliances are essentially material constants. Incidentally, the determination of all these elastic constants or compliances is not easy to accomplish by simple tests. The material constants that are normally determined through characterization experiments (see chapter 4) are termed as engineering constants. They can also be evaluated using the micromechanics material models (chapter 5).
All nine independent compliances and therefore elastic constants listed in Eq. 6.12 are now expressed in terms of nine independent engineering constants. The stressstrain relations for a threedimensional orthotropic material, in terms of engineering constants, can be written as follows:
(6.16)
We know that, in terms of compliances, the stressstrain relations are
(6.17)
Comparing Eqs. 6.16 and 6.17, we can express the compliances in terms of engineering constants.
(6.18)
The elastic constants can then be derived by inversion of the compliance matrix i.e. [] = [S_{ij}]^{1} and are given as follows:
(6.19)
where
(6.20)
In terms of engineering constants, the elastic constants and compliances for an isotropic material are given by
(6.21)
and
Consider cylindrical coordinates r, θ, z as illustrated in Fig. 6.6. Here the zaxis is assumed to coincide with the X_{3}axis. The stress and strain components are represented as
and (6.22)
The stressstrain relations, in terms of compliances, become
(6.23)
where
(6.24)
The elastic constant matrix [] is obtained by inversion of the compliance matrix [S_{ij}] i.e., [] = [S_{ij}]^{1} or from Eq. 6.19 by replacing the indices 1,2,3 with r, θ, z respectively.
6.6 TWODIMENSIONAL CASE: PLANE STRESS
For the case of plane stress (Fig. 6.7)
σ_{3 }= σ_{4 }= σ_{5 }= 0 (6.25)
The stressstrain relations, with twodimensional anisotropy, are
(6.26)
or, i, j = 1,2,6 (6.27)
Where [Q_{ij}] are the reduced stiffnesses (elastic constants) for plane stress.
Similarly, in terms of compliances, the stressstrain relations are
(6.28)
or, i, j =1,2,6 (6.29)
For the case of twodimensional orthotropy (Fig. 6.8) the stressstrain relations are
(6.30)
and
(6.31)
with
(6.32)
and
(6.33)
Note that the engineering constants , , , (or ) and G'_{12 }are referred to the orthotropic axis system X'_{1}X'_{2} (i.e., the material axes).
6.7 TRANSFORMATION OF ELASTIC CONSTANTS AND COMPLIANCES
If the elastic constants and compliances of a material are known with respect to a given coordinate system, then the corresponding values with respect to any other mutually perpendicular coordinates can be determined using laws of transformation. These are explained in Appendix A.
The transformation of elastic constants from the X'_{1}X'_{2}X'_{3 }coordinates to mutually orthogonal X_{1}X_{2} X_{3 }coordinates (Refer Fig. A.1 and Eq. A. 22) is given as follows:
(6.34)
where transformation matrix is given by Eq. A.8.
Note that the elements of and correspond to the X_{1}X_{2} X_{3 } and X'_{1}X'_{2}X'_{3 }coordinates, respectively.
One can use Eq. 6.34 in the following form
= ^{T} ^{1}
= [T_{σ}] [T_{σ}]^{T} (6.35)
if the transformation is required from X_{1}X_{2} X_{3 } coordinates to theX'_{1}X'_{2}X'_{3 }coordinates. Note that [T_{σ}] is defined by Eq. A.13.
Simalarly,
(6.36)
(6.37)
The corresponding elastic constants and compliances [S_{ij}] due to special cases of material symmetry and transformation matrices and [T_{σ}] due to specific orientation of axes are to be reduced from the general three dimensional cases, before transformation is sought from one axis system to the other.
If the elements of and refer to the X_{1}X_{2} and X'_{1}X'_{2} coordinates (see Eqs. 6.26 and 6.30 and Figs. 6.7 and 6.8), respectively, then transformation laws for reduced elastic constants are obtained as follows:
(6.38)
(6.39)
where and are defined by Eqs. A.18 and A.19. The compliance matrices are accordingly transformed using Eqs. 6.36 and 6.37.
Accordingly, from Eqs. 6.38 and A.18 it can be shown that
(6.40)
In a similar way from Eqs. 6,36 and A.19 one obtains
(6.41)
Note that m = cos and n = sin and and are defined by Eqs. 6.32 and 6.33 respectively, in terms of engineering constants , , (or ) and G'_{12 }corresponding to principal material directions.
If transformation is required from one anisotropic material axis system (say X_{1}X_{2} X_{3}) to another anisotropic material axis system (say, ), then from Eqs. 6.38 and A. 18 we can
or,
or,
(6.42)
Similarly, using Eqs. 6.36 and A.19 one can write
or,
(6.43)
6.8 PARTICULATE AND SHORT FIBRE COMPOSITES
Particulate composites, where reinforcements are in the form of particles, platelets and flakes, and short fibre composites may exhibit a wide range of elastic material behaviour depending on the shapes, sizes, orientations and distributions of reinforcements in the matrix phase as well as elastic properties of the constituent materials. The matrix behaviour is normally isotropic. The composition of these composites are first established by examining their morphology and then proper stressstrain relations can be obtained from the equations developed in the preceding sections. It is also to be noted whether the composite body under consideration is threedimensional or two dimensional in character.
For example, the behaviour of a threedimensional composite with a typical reinforcement packing shown in Fig. 6.9a is anisotropic in nature. Here the reinforcements are oriented in some regular fashion with respect to the reference axes X_{1}X_{2} X_{3}. The stress strain relations = for this type of composites are given by Eq 6.8 with elements of listed in Eq. 6.7. When the reinforcements are arranged parallel to the axes (Fig. 6.9b), the composite behaviour is orthotropic and Eq. 6.12 defines the corresponding . If the orientation and distribution of reinforcements are found to be random in the matrix phase, as shown in Fig. 6.9c, the composite is assumed to behave like an isotropic material. Consequently, the elastic constant matrix is reduced to that given in Eq. 6.15.
For twodimensional anisotropic, orthotropic and isotropic cases, some possible reinforcement arrangements are illustrated in Fig. 6.10. The stressstrain relations, as presented in section 6.6 can be accordingly used for these cases.
If transformations of elastic constants and compliances are required from one axes system to another, then one can use the transformation rules discussed in section 6.7.
Fig. 6.10
6.9 MULTIDIRECTIONAL FIBRE REINFORCED COMPOSITES
Composites exhibit strong directional properties, when reinforcements are in the form of continuous fibres. In a multidirectional composite, fibres can be placed in any desired direction in a threedimensional space, along which better stiffness (or strength) is desired. The shear properties can be greatly improved by providing diagonal reinforcements. Carboncarbon composites form an important class of multidirectional composites due to several variations in weave design and perform construction. Similar multidirectional composite systems can also be designed and developed with both metalmatrix and ceramicmatrix composites. A typical multidirectional (5D) composite is shown in Fig. 6.11a. There are three bundles of orthogonal fibres f_{1}, f_{2}, f_{3} and two bundles of diagonal fibres f_{4}, f_{5}. We consider here an integrated multidirectional fibre reinforced composite moder which contains n number of unidirectional fibre composite blocks that are oriented in n arbitrary directions with respect to a threedimensional reference axes X_{1}X_{2} X_{3}. Each unit block may have different fobre volume fractions. This arrangement makes n number of material axis systems, and therefore yield n sets of direction cosines between n material axis systems and the reference axes X_{1}X_{2} X_{3}. For example, Figure 6.11b represents the orientation of the material axis system for the ith block. The corresponding transformation matrices and can then be written down using Eqs. A.8 and A.13, respectively.
The material behaviour for each block with respect to its axes is orthotropic. The elastic constants for the ith block are then given as
(6.44)
The effective elastic constants for the ndirectional fibre reinforced composite are then determined by averaging the transformed properties as follows:
(6.45)
Note that the overall fibre volume fraction is given as
(6.46)
A unidirectional lamina is a thin layer (ply) of composite and is normally treated as a twodimensional problem. It contains parallel, continuous fibres and provides extremely high directional properties. It is the basic building unit of a laminate and finds very wide applications in composite structures specially in the form of laminates. Therefore, the knowledge of its elastic macromechanical behaviour is of utmost importance to composite structural designers.
Figure 6.12a depicts a unidirectional lamina where parallel, continuous fibres, are aligned along the X'_{1} axis (fibre axis or longitudinal direction). The X'_{2 }axis (transverse direction) is normal the fibre axis. The axes X'_{1}X'_{2} are referred as material axes. The material axes are oriented counter clockwise by angle with respect to the reference axes X_{1}X_{2}. The angle (also referred as fibre angle) is considered positive when measured counterclockwise from the X_{1} axis. This type of unidirectional lamina is termed as “offaxis” lamina. An offaxis lamina behaves like an anisotropic twodimensional body, and the stressstrain relations, given by Eqs. 6.26 through 6.29, can be used for the present case.
When the material axes coincide with the reference axes (i.e., =0), as shown in Fig. 6.12b, the lamina is termed as “onaxis” lamina and its behaviour is orthotropic in nature. The stressstrain relations are defined by Eqs. 6.30 and 6.31.
The engineering constants , , , (or ) and G'_{12} are usually known, as these can be determined either by using micromechanics theories (chapter 4) or by characterization tests (chapter 5). Using these engineering constants, the reduced stiffnesses and compliances are then determined for an orthotropic lamina with the help of Eqs. 6.32 and 6.33. The transformed reduced stiffnesses and can now be evaluated employing Eqs. 6.40 and 6.41. The stiffness and compliances for three composite systems are computed for various fibre orientations and are listed in Tables 6.1 and 6.2. Typical variations of transformed properties and with change in the fibre angle are illustrated in Figs. 6.13 and 6.14. Such plots aid to the basic understanding of the stiffness behaviour of an offaxis lamina with different fibre orientations. Note that the case =0 corresponds to an onaxis lamina.
A bidirectional lamina is one which contains parallel, continuous fibres aligned along mutually perpendicular directions, as shown in Fig. 6.15. A lamina reinforced with woven fabrics that have fibres in the mutually orthogonal warp and fill directions can also be treated as a bidirectional lamina. The effects of undulation (crimp) and other problems associated with different weaving patterns are however, neglected. In Fig. 6.15 the X_{1}' X_{2}' is referred as material axes. The amount of fibres in both directions need not necessarily be the same. In a hybrid lamina, even the fibres in two directions may vary, but when the material axes X_{1}' X_{2}' coincide with the reference axes X_{1}X_{2} (Fig. 6.15a), the material behaviour is orthotropic and the lamina may be termed as “onaxis” bidirectional lamina. If the X_{1}' X_{2}' plane rotates by an angle with respect to the X_{1}X_{2} axes (Fig. 6.15a), then the oriented lamina behaves as an anisotropic material and it can be identified as an “offaxis” bidirectional lamina can also be treated as a twodimensional problem and its elastic properties can be determined in an usual manner as discussed in sections 6.6 and 6.10. It may be mentioned that the anisotropy and stiffness behaviour of a bidirectional lamina can be greatly controlled by varying the types of fibres (say, carbon fibre along the X_{1}' direction and glass fibre along the X_{2}' direction) and volume fractions of fibres (V_{f}) in both directions. When the fibres and V_{f } are same in both directions, then
E'_{11} = E'_{22 }and the material behaviour is square symmetric. Note that a square symmetric material is different from an isotropic material.
We consider here a general thin laminate of thickness h (Fig. 6.16). The X_{3 }axis is replaced here by the z axis for convenience. The laminate consists of n number of unidirectional and/or bidirectional laminae, where each lamina may be of different materials and thicknesses and have different fibre orientations (). A thin general laminate is essentially a twodimensional problem, but cannot be treated as a twodimensional plane stress problem as has been done for a unidirectional lamina. The existence of extension –bending couling causes bending, even if the laminate is subjected to inplane loads only. Therefore, thin plate bending theories are employed in derivation of constitutive relations. We assume that Kirchhoff 's assumptions related to the thin plate bending theory are applicable in the present case.
Let u_{1}^{0}, u_{2}^{0 }and w are the midplane displacements, and w is constant through the thickness of the lamina. Then the midplane strains are given by
^{ }(6.47)
and the curvatures, which are constant through the thickness of the laminate, are
(6.48)
The strains at any distance z are then given as
(6.49)
Now from Eq. 6.26, we have at any distance z
(6.50)
The stress and moment resultants (Fig. 6.17) are evaluated per unit length of the laminate as follows:
and (6.51)
Thus,
where and
where
Proceeding in a similar manner, all stress and moment resultants can be expressed as listed below:
(6.52)
with (A_{ij}, B_{ij}, D_{ij}) = _{ij} (1, z, z^{2}) dz; i, j = 1, 2, 6 (6.53)
Equation 6.52 represents the constitutive relations for a general laminate, and A_{ij}, B_{ij}, and D_{ij} are the inplane, extension bending coupling and bending stiffnesses, respectively. Note that all these stiffnesses are derived for a unit length of the laminate. The elastic properties of each lamina are generally assumed to be constant through its thickness, as these laminae are considered to be thin. Then A_{ij}, B_{ij}, and D_{ij }are approximated as
(6.54)
From Eq. 6.52, it is seen that there exist several types of mechanical coupling in a general laminate. These are grouped together as follows:
Extension – Shear : A_{16}, A_{26}
Extension – Bending : B_{11}, B_{12}, B_{22}
_{ }Extension – Twisting : B_{16 }, B_{26}
Shear – Bending : B_{16 }, B_{26}
_{ }Shear – Twisting : B_{66}
Bending – Twisting : D_{16 }, D_{26}
Biaxial – Extension : A_{12}
Biaxial – Bending : D_{12}
As stated earlier, the coupling terms B_{ij} occur due to unsymmetry about the middle surface of a laminate. However, all terms containing suffices '16 ' and '26 ' are resulted due to anisotropy caused by the fibre orientation other than 0^{0 }and 90^{0}. Those containing suffices '12 ' are due to Poisson's effect. Although a heneral unsymmetric laminate contains all coupling terms, there are several laminates where some of these may vanish. These are listed in Table 6.3. There are several important points that are to be noted here. The first two laminates (serial nos. 1 and 2) which are christened as “offaxis laminate” and “onaxis laminate”; respectively are essentially paralles ply laminates where all laminae in a laminate have the same fibre orientation and therefore are stacked parallel to each other. These are, in fact, similar to unidirectional laminae. For a symmetric balanced angleply laminate D_{16} and D_{26} do not vanish, although A_{16} = A_{26} = 0. The only coupling effect that appears in an antisymmetric crossply laminate is the extensionbending coupling due to presence of B_{11}and B_{22} and note that B_{22} =  B_{11}. But the existence of B_{16} and B_{26} cause an antisymmetric angleply laminate to experience extensiontwisting coupling. Note also that extensionbending coupling is predominant for an unsymmetric crossply laminate.
The mechanical coupling, as discussed above, influences the deformation behaviour of a laminate to a great extent. This can be better understood by examining the deformed shapes of a couple of laminates as illustrated in Figs. 6.18 through 6.20. Here the dotted lines represent the undeformed shape and the firm lines, deformed shapes. Consider first a simple offaxis laminate (or unidirectional lamina), subjected to an inplane stress resultant N_{1} (Fig.6.18a) and an outofplane moment resultant M_{1} (Fig. 6.18b). We know from Eq. 6.52 and Table 6.1 (B_{ij}=0) that
(6.55)
Thus, as illustrated in Fig. 6.18a, it is noted that a simple tension causes not only extension and contraction, but also shearing of the laminate. While the extension and contraction are due to A_{11 }and A_{12}, respectively and the inplane shear deformation is due to presence of A_{16}. This characteristic behaviour is seen especially in an anisotropic (offaxis) laminate. The shear deformation vanishes, if A_{16 }= 0, as in the case of an orthotropic (onaxis) laminate (serial no.2 of Table 6.1). Similarly, as can be seen in Fig. 6.18b, a simple bending due to M_{1} has resulted not only longitudinal bending (due to D_{11}) and transverse bending (due to D_{12}), but also twisting (due to D_{16}).
Figure 6.19 describes the deformation behaviour of an antisymmetric crossply laminate. The extensionbending coupling due to B_{11} and B_{22} can be clearly observed. In Fig. 6.19a a simple inplane tension is found to introduce bending in the laminate. Conversely, a simple bending causes extension of the laminate, a shown in Fig. 6.19b.
Figure 6.20 depicts the deformed shape of an antisymmetric angleply laminate. Here the extensionbending and bendingshear coupling effects due to B_{16} and B_{26 }are presented. In a similar manner, the deformation characteristics of other types of laminates can be illustrated. The most important point that is to be focused here is that fibre orientation and lamina stacking sequence affect laminate stiffness properties, which, in turn, control the deformation behaviour of a laminate.
Table 6.4 provides the stiffnesses [A_{ij}], [B_{ij}]and [D_{ij}] for various stacking sequences of carbon/epoxy composites. The [Q_{ij}] values given in Table 6.1 have been used to compute the above stiffnesses.
6.13 LAMINATE HYGROTHERMAL STRAINS
The changes in moisture concentration and temperature introduce expansional strains in each lamina. The stressstrain relation of an offaxis lamina (Eq. 6.28) is then modified as follows
(6.56)
with
(6.57)
and
and (6.58)
where the superscripts e, H, T refer to expansion, moisture and temperature, respectively, ΔC and ΔT are the change in specific moisture concentration and temperature, respectively, and β's and α's are coefficients of moisture expansion and thermal expansion respectively.
Note that the spatial distributions of moisture concentration and temperature are determined from solution of moisture diffusion and heat transfer problems.
Expansional strains transform like mechanical strains (Appendix A) i.e., .
Inversion of Eq. 6.56 yields (see also Eq. 6.26), at any distance z (Fig. 6.16), (6.59)
Thus, for a general laminate Eq. 6.52 will be modified as
(6.60)
where the expansional force resultants are
(6.61)
and the expansional moments are
(6.62)
These expansional force resultants and moments may considerably influence the deformation behaviour of a laminate.
6.14 STRENGTH CRITERIA FOR ORTHOTROPIC LAMINA
Isotropic materials do not have any preferential direction and in most cases tensile strength and compressive strength are equal. The shear strength is also dependent on the tensile strength. A strength criterion for an isotropic lamina is, therefore, based on stress components, σ_{1}, σ_{2 }and σ_{6 }for a twodimensional problem and a single strength constant i.e., ultimate strength X. An orthotropic lamina (Fig. 6.8), on the other hand, exhibits five independent strength constants e.g., tensile strength X'_{11}^{t} a dcompressive strength X'_{11}^{c } along the X'_{1} direction; tensile strength X'_{22}^{t } and compressive strength X'_{22}^{c }along the X'_{2} direction and inplane shear strength X'_{12}. Hence a strength criterion for a twodimensional orthotropic lamina should involve the stress components σ'_{1}, σ'_{2 }and σ'_{6 }and strength constants X'_{11}^{t}, X'_{11}^{c}, X'_{22}^{t} X'_{22}^{c} and X^{'}_{12}. We present here a few important strength criteria that are commonly used to evaluate the failure of an orthotropic lamina. Maximum Stress Criterian
A lamina is assumed to fail, if any of the following relations is satisfied
, when and are tensile
, when and are compressive. (6.63)
It is assumed that inplane shear strengths are equal under positive or negative shear load.
Maximum Strain Criterian
A lamina fails, if any of the following is satisfied
when and are tensile
when and are compressive. (6.64)
Note that the addition of suffix 'u' in strain components indicates the corresponding ultimate strains. The ultimate shear strains are also assumed to be equal under positive or negative shear load. If a material behaves linearly elastic till failure, the ultimate strains can be related to ultimate strength constants as follows:
(6.65)
TsaiHill Criterion
The general threedimensional orthotropic strength criterion is given by
(6.66)
Assuming that normal stresses , and an dshear stress act independently and substituting = X'_{11}, = X'_{22}, and in the above strength criterion, we obtain
; ; _{ }(6.67)
Combining Eqs. 6.67 we get
(6.68)
Assuming transverse symmetry X'_{22 }= X'_{33 } and twodimensional plane stress case (σ_{3 }= σ_{4 }= σ_{5 } =0), Eq. 6.66 reduces to
or,
(6.69)
When , or both are tensile or compressive, Eq. 6.69 can be used by substituting the corresponding tensile or compressive strength constants in it. Thus, if is tensile, X'_{11 }= X'_{11}^{t }, and if is compressive, X'_{22}= X'_{22}^{c} and so on.
TsaiHill/Hoffman criterion accounts for unequal tensile and compressive strengths. For a threedimensional state of stress in an orthotropic material, this criterion is given as
(6.70)
If tensile acts only and =X'_{11}^{t }, then from Eq. 6.70
(C_{2} + C_{3}) X'_{11}^{t} + C_{4} = 1/X'_{11}^{t} (6.71)
If compressive acts only and = X'_{11}^{c}, then
(C_{2} + C_{3}) X'_{11}^{c} – C_{4} = 1/X'_{11}^{c} (6.72)
From Eqs. 6.71 and 6.72, we obtain
(6.73)
Similarly, consideration of and yields
(6.74)
(6.75)
Now, assuming , we derive from the above the following relations for C_{1}, C_{2} and C_{3}:
(6.76)
(6.77)
Further, applying only and yields
(6.78)
Now, considering a two dimensional state of plane stress condition and substituting the values of C_{1}, C_{2}, C_{3}, C_{4}, C_{5 }and C_{9 }from the above relations, the strength criterion takes the following form:
(6.79)
For an orthotropic material under a twodimensional state of plane stress condition, this criterion assumes the form
(6.80)
Considering that the positive or negative inplane shear stress should not affect the results, the terms F_{16 } , F_{26} and F_{6} should vanish. Hence Eq. 6.80 reduces to
(6.81)
Now applying independently tensile and compressive normal stresses and , and inplane shear stresses , and substitution of =X'_{11}^{t } =X'_{11}^{c}, =X'_{22}^{t },  = X'_{22}^{c }and =X'_{12 }in Eq. 6.81 yields
(6.82)
Employing the von Mises plane stress analogy, the remaining interaction coefficient F_{12} can be defined
(6.83)
Combining Eqs. 6.816.83, the TsaiWu criterian takes the following form:
(6.84)
It is to be mentioned that the TsaiWu criterion (Eq. 6.84) accounts for interaction of stress components as well as both tensile and compressive strength constants and shear strength and is considered as a reasonably accurate and consistent representation of failure of an orthotropic lamina under biaxial stresses. The TsaiHill criterion (Eq. 6.69) is also very popular with composite structural designers.
Table 6.1: Stiffnessesand for three unidirectional composites (GPa)
Material 





Kelvar/Epoxy 
91.87 
4.03 
1.41 
2.26 

Carbon/Epoxy 
133.94 
8.32 
2.16 
3.81 

Boron/Polyimide 
242.39 
14.93 
3.88 
5.53 

Material 
(degree) 







Kelvar/ Epoxy 
0 
91.87 
4.03 
1.41 
2.26 
0.00 
0.00 

30 
54.15 
10.23 
17.17 
18.02 
28.12 
9.92 

45 
26.93 
26.93 
22.42 
23.27 
21.96 
21.96 

60 
10.23 
54.15 
17.17 
18.02 
9.92 
28.12 

90 
4.03 
91.87 
1.41 
2.26 
0.00 
0.00 

Carbon/ Epoxy 
0 
133.94 
8.32 
2.16 
3.81 
0.00 
0.00 

30 
79.53 
16.72 
25.17 
26.82 
40.48 
13.92 

45 
40.46 
40.46 
32.84 
34.48 
31.40 
31.40 

60 
16.72 
79.53 
25.17 
26.82 
13.92 
40.48 

90 
8.32 
133.94 
2.16 
3.81 
0.00 
0.00 

Boron/ Plyimide

0 
242.39 
14.93 
3.88 
5.53 
0.00 
0.00 

30 
142.88 
29.15 
46.53 
48.17 
73.87 
24.62 

45 
71.80 
71.80 
60.74 
62.39 
56.87 
56.87 

60 
29.15 
142.88 
46.53 
48.17 
24.62 
73.87 

90 
14.93 
242.39 
3.88 
5.53 
0.00 
0.00 

Table 6.2: Compliance and for three unidirectional composites (TPa)^{1}
Material 





Kelvar/Epoxy 
10.94 
249.75 
3.83 
443.49 

Carbon/Epoxy 
7.50 
120.66 
1.95 
262.47 

Boron/Polyimide 
4.14 
67.27 
1.08 
180.96 

Material 
(degree) 







Kelvar/ Epoxy 
0 
10.94 
249.75 
3.83 
443.46 
0.00 
0.00 

30 
103.48 
222.88 
36.66 
312.13 
141.32 
65.50 

45 
174.12 
174.12 
47.61 
268.35 
119.40 
119.40 

60 
222.88 
103.48 
36.66 
312.13 
65.50 
141.32 

90 
249.75 
10.94 
3.83 
443.46 
0.00 
0.00 

Carbon/ Epoxy 
0 
7.50 
120.66 
1.95 
262.47 
0.00 
0.00 

30 
60.24 
116.82 
26.40 
164.66 
77.23 
20.76 

45 
96.68 
96.68 
34.55 
132.05 
56.58 
56.58 

60 
116.82 
60.24 
26.40 
164.66 
20.76 
77.23 

90 
120.66 
7.50 
1.95 
262.47 
0.00 
0.00 

Boron/ Plyimide

0 
4.14 
67.27 
1.08 
180.96 
0.00 
0.00 

30 
40.06 
71.63 
21.21 
100.42 
50.59 
4.08 

45 
62.56 
62.56 
27.93 
73.57 
31.56 
31.56 

60 
71.63 
40.06 
21.21 
100.42 
4.08 
50.59 

90 
67.27 
4.14 
1.08 
180.96 
0.00 
0.00 

Table 6.3 : Stiffnesses for various types of laminates
Case Laminate type Elastic behaviour Stiffnesses
1. Offaxis laminate anisotropic all B_{ij}=0; A_{ij}= h Q_{ij}
_{ }(all plies oriented and uncoupled D_{ij} = (h^{3}/12) Q_{ij}
at )
2. Onaxis laminate orthotropic all B_{ij}=0; A_{ij}=h Q_{ij}
(all plies oriented either and uncoupled and D_{ij} = h^{3}/12) Q_{ij}
0^{0} or 90^{0}) with Q_{16} = Q_{26} = 0
3. Symmetric crossply specially all B_{ij}=0; A_{16}= A_{26}=
(odd number of orthropic and D_{16}= D_{26}=0; rest of
0^{0 }/ 90^{0 }/ 0^{0}, etc. plies) uncoupled A_{ij} and D_{ij} are finite
4. Symmetric angleply anisotropic and all B_{ij}=0; all A_{ij }and D_{ij}
(odd number of //, uncoupled are finite
etc. plies)
5. Symmetric balanced angle anisotropic and all B_{ij}=0; A_{16}= A_{26}=0 rest
ply (///, etc. plies) uncoupled of A_{ij} and D_{ij }are finite.
6. Antisymmetric crossply orthotropic and A_{16}= A_{26}= B_{16}= B_{26}=
(even number of partly coupled B_{12}= B_{66}= D_{16}= D_{26}=0
0^{0 }/ 90^{0 }/ 0^{0}/90^{0}, etc. plies) rest of A_{ij} ,B_{ij} and D_{ij }are
finite with B_{22}=B_{11};
D_{22}=D_{11}
7. Antisymmetric angleply anisotropic and A_{16}= A_{26}=B_{11}=B_{22}
(even number of partly coupled B_{12}= B_{66}= D_{16}= D_{26}=0
(///, etc. plies) rest of A_{ij}, B_{ij} and D_{ij} are
finite.
8. Unsymmetric crossply orthotropic but A_{16}= A_{26}= B_{16}= B_{26}=
(irregular stacking of coupled D_{16}= D_{26}=0; rest of
0^{0 }or 90^{0} plies) A_{ij}, B_{ij} and D_{ij} are
finite.
9. General unsymmetric anisotropic and all A_{ij}, B_{ij} and D_{ij} are
laminate strongly coupled finite.
Table 6.4: Stiffneses for carbon/epoxy composite laminates
Laminate Thickness : 4mm
Units : [A_{ij}], GPamm; [B_{ij}], GPamm^{2}; [D_{ij}], GPamm^{3}
1. 0^{0 }/ 90^{0 }/ 0^{0} laminates
2. 45^{0 }/ 45^{0 }/ 45^{0 } laminate
3. 45^{0} /45^{0 }/45^{0 }/ 45^{0 } laminate
4. 0^{0} /90^{0 }/ 0^{0 }/ 90^{0 } laminate
5. 45^{0} /45^{0 }/45^{0 }/ 45^{0 } laminate
6. 0^{0} /90^{0 }/0^{0 }/ 0^{0 } laminate
1. S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, McGraw Hill, N.Y., 1970.
2. Y.C. Fung, Foundations of Solid Mechanics, Englewood Cliffs, N.J., 1965.
3. S.G. Lekhnitskii, Theory of Elasticity of an Anisotropic Body, MIR Publ. Moscow, 1981.
4. J.C, Halpin, Primer or Composite Materials: Analysis, Technomic Publ. Co., Inc. Lancaste, 1984.
5. R.M. Christensen, Mechanics of Composite Materials, Wiley Interscience, N.Y., 1979.
6. Z. Hashin and C.T. Herakovich (Eds.), Mechanics of Composite MaterialsRecent Advances, Pergamon Press, N.Y.,1983.
7. S.W. Tsai and H.T. Hahn, Introduction to Composite Materials Technomic Publ. Co., Inc., Lancaster,1980.
8. J.M. Whitney, Structural Analysis of Laminted Composites, Technomic Publ. Co., Inc.,Lancaster, 1987.
9. J.R. Vinson and R.L. Sierakowski, The Behaviour of Structures Composed of Composite Materials, Kluwar Academic Publ., MA,1985.
10. S.W. Tsai, J.C. Halpin and N.J. Pangano (Eds.) Composite Materials Workshop, Technomic Publ. Co., Inc., Lancaster, 1968.
1. State the generalized Hooke's law for a threedimensional elastic anisotropic material and show that there are twentyone independent elastic constants for a triclinic material.
2. Write down the elastic constant matrix for threedimensional orthtropic, square symmetric, hexagonal symmetric and isotropic materials.
3. Distinguish between elastic constants and engineering constants.
4. For a twodimensional orthotropic case, express and in terms of engineering constants.
5. Derive expressions for and in terms of angle and show that
and
6. Assume properties given in Table 4.4 for Kevlar/epoxy and carbon/epoxy/composites and determine [A_{ij}], [B_{ij}] and [D_{ij}] for a hybrid laminate (thickness 4 mm).
7. Make a critical assessment of various lamina failure theories.
8. Derive expressions for TsaiHill and TsaiWu strength criteria.