Chapter 2. Laminate theory 9
Chapter 2 Laminate Theory
This chapter aims to give a brief description of the type of mechanical analysis applied to determine the behaviour of the proposed structure which is made up of laminate composite panels. The analysis includes the fundamentals required to understand the mechanical behaviour of a deformable solid through the application of the theory of elasticity. From here the elastic behaviour of the composite material is described through stress-strain relations and visa versa , in terms of its engineering constants from a three dimensional state to the more representative two-dimensional state of the composite plate. The effects of ply orientation are examined with corresponding transformations between principal axis and orientated coordinates outlined and their subsequent application and relevance in the project mentioned. The analysis determines the conditions required to be met by the laminas so as to constitute a laminate. If the laminas meet the conditions required, the classical theory outlined can be appropriately applied to the laminate. The theory attempts to find effective and realistic simplifying assumptions that reduces the three dimensional elastic problem to a two dimensional one. It determines the response of the laminate to forces and moments acting on the laminate by applying the hypothesis of thin laminates where a number of deformed geometrical occurrences are assumed. Finally, other types of mechanical behaviour are outlined in this chapter. These behaviours are considered worthy in presenting for discussion as they are directly related to the project in terms of geometry and service conditions. These topical mechanical behaviours include the presence of holes in laminates, vibration and fatigue.
2.1. Elastic Theory Composite materials, as with all deformable solids, change shape at different points of the material when a system of external loading is applied on the solid in equilibrium, giving rise to a new geometric or deformed configuration. Figure 2.1 shows a deformable solid subjected to the application of a system of forces indirectly where external loading is applied on some arbitrary zones of the solid’s boundary but with limited displacement in another zone which generates forces necessary to equilibrate the external applied system.
Chapter 2. Laminate theory 10
The physical magnitudes that are incurred in the deformation of a solid are the external loading: applied in the body Xi and/or on the boundary ti. The second type of physical magnitudes is the displacements ui of the body. The objective of the mechanical behavioural analysis of a deformable solid is to determine its displacement when external loading is applied. However, the solid’s displacement cannot be determined directly from the applied external loading. It is therefore necessary to define internal variables that are related to the physical magnitudes in equilibrium, these include the stresses σij and strains εij of the deformable solid. Figure 2.1 represents the elastic problem in terms of its forces, displacements, stresses and strains. Given that the stresses are related to the external loading, the same as the strains are related with displacements and given the relation between the displacements and the loads, it must exist a material relation between the stresses and the strains. This material relation is known as the Behavioural Law or the constitutive equations of the material.
Figure 2.1: The problem of deformable solids
Analysing the elastic problem in the above figure, the relation between the exterior loads Xi and ti (i =1,2,3) and stresses σij (i, j = 1,2,3) are the equations of internal equilibrium.
σ + = (2. 1) , jij X i 0
σ = (∂ ) (2. 2) n jij ti Dt
Between displacements εij (i, j = 1,2,3) and strains are the equations of compatibility.
1 (2. 3) ε = ()u + u ij 2 , ji ,ij
Chapter 2. Laminate theory 11
= (∂ ) (2. 4) ui ui Du
Between the stresses and strains are the constitutive equations or the Behavioural Law.
σ = ε + λε δ (2. 5) ij 2G ij ijkk
+ ε = 1 vσ − v σ δ ij ij ijkk (2.6) E E
The elastic problem is therefore made up of a system of 15 differential equations which include three equilibrium equations, six strain-displacement relations, and six constitutive equations. In total, there are 15 unknowns, made up of six components from the stress tensor, six from the strain tensor and three displacements [4].
2.2. Elastic Material Behaviour In Composite Materials 2.2.1. Stiffness Matrix C The generalised Hooke's Law relating stresses to strains can be written as the following expression
σ = C .ε (2.7 ) i ij j
Where σi are the stress components, Cij is the stiffness matrix, and εj are the strain components. The stress-strain relationship and the corresponding stiffness matrix for the anisotropic or triclinic (no planes of symmetry for the material properties) linear elastic case are shown below.
σ ε 11 C11 C12 C13 C14 C15 C16 11 σ ε 22 C21 C22 C23 C24 C25 C26 22 σ C C C C C C ε 33 31 32 33 34 35 36 33 = (2.8) σ C C C C C C 2ε 23 41 42 43 44 45 46 23 σ C C C C C C 2ε 13 51 52 53 54 55 56 13 σ ε 12 C61 C62 C63 C64 C65 C66 2 12 where the stiffness matrix itself is symmetric, implying that only 21 of the 36 are independent elastic constants. According to the material type, different extents of symmetry of material properties occur and subsequent reduction in the number of elastic constants in the stiffness matrix is observed. One of such is the stiffness matrix shown below which describes the case of
Chapter 2. Laminate theory 12
the stress-strain relations in coordinates aligned with the principal material directions i.e., the directions that are parallel to the intersections of the three orthogonal planes of the material property symmetry. This matrix defines an orthotropic material which is fundamental in the composite analysis in this project. It is important to note also that orthotropic materials can exhibit apparent anisotropy when stressed in non-principal material coordinates [3].
σ ε 11 C11 C12 C13 0 0 0 11 σ ε 22 C21 C22 C23 0 0 0 22 σ C C C 0 0 0 ε 33 31 32 33 33 = (2.9) σ 0 0 0 C 0 0 2ε 23 44 23 σ 0 0 0 0 C 0 2ε 13 55 13 σ 0 0 0 0 0 C 2ε 12 66 12
2.2.2. Compliance Matrix S For ease of resolving the elastic material behaviour we define the inverse of the previous stress- strain relation such that
ε = σ (2.10 ) i S ij . j where Sij is the compliance matrix which contains more reduced expressions of the elastic constants. The complete 6x6 compliance matrix is given as
ε σ 11 S11 S12 S13 S14 S15 S16 11 ε σ 22 S21 S22 S23 S24 S25 S26 22 ε σ 33 S31 S32 S33 S34 S35 S36 33 = (2.11) γ S S S S S S σ 23 41 42 43 44 45 46 23 γ S S S S S S σ 13 51 52 53 54 55 56 13 γ σ 12 S61 S62 S63 S64 S65 S66 12
For an anisotropic material, there exists a significant coupling effect between the applied stress and the resulting deformation. The types of coupling for above the strain-stress expression are shown in figure 2.2. S11 , S22 and S33 represent the coupling due to the individual applied stresses
σ1, σ2 and σ3, respectively, in the same direction. S44 , S55 and S66 represent the shear strain response due to the applied shear stress in the same plane. S12 , S13 and S23 represent the
Chapter 2. Laminate theory 13
extension-extension coupling or coupling between the distinct normal stresses and normal strains, also known as the Poisson effect. S 15 , S16 , S24 , S25 , S26 , S34 , S35 and S36 represent the shear-extension coupling or a more complex coupling of the normal strain response to applied shear stress than for the preceding compliances. S45 , S46 and S56 represent shear-shear coupling or the shear strain response to shear stress applied in another plane. The remaining terms of compliance matrix are a result of symmetry [3].
Extension -Extension Coupling
ε σ 11 S11 S12 S13 S14 S15 S16 11 Extension Shear -Extension ε S S S S S S σ Coupling 22 21 22 23 24 25 26 22 ε σ 33 S31 S32 S33 S34 S35 S36 33 = γ σ 23 S41 S42 S43 S44 S45 S46 23 Shear -Shear γ S S S S S S σ 13 51 52 53 54 55 56 13 Coupling γ σ 12 S61 S62 S63 S64 S65 S66 12
Shear Figure 2.2: Physical significance of anisotropic stress-strain relations
For an anisotropic material, the compliance matrix components in terms of the engineering constants are shown in equation (2.12), using the reduced index notation of Voigt (1910). The values of the compliance matrix can be physically measured by specimen testing. The elastic constants that can be physically measured include Young’s Modulus E, Poisson’s ratio v, shear modulus G, and analytically measured constants include shear-extension coupling or mutual influence coefficients η (Lekhnitskii), and shear-shear coupling coefficients μ (Chenstov).
ν ν η η η 1 − 21 − 31 41 51 61 E E E G G G ε 1 2 3 4 5 6 σ 11 ν 1 ν η η η 11 − 12 − 32 42 52 62 ε E1 E2 E3 G4 G5 G6 σ 22 ν ν η η η 22 13 23 1 43 53 63 ε − − σ 33 33 (2.12) = E1 E2 E3 G4 G5 G6 η η η µ µ . γ 14 24 34 1 54 64 σ 23 23 E1 E2 E3 G4 G5 G6 γ η η η µ µ σ 13 15 25 35 45 1 65 13 γ E1 E2 E3 G4 G5 G6 σ 12 η η η µ µ 12 16 26 36 46 56 1 E1 E2 E3 G4 G5 G6
Chapter 2. Laminate theory 14
In relation to more realistic cases of engineering problems of thin plate elements which include panel-type composite structures, the 2-D case of plane stress of the lamina in principal axes is characterised by the reductions below and is shown in figure 2.3.
σ = σ = σ = 0 3 23 31 (2.13) ε = S σ + S σ ε = 0 ε = 0 3 13 1 23 2 23 13 This idealisation is physically achieved as the lamina can only resist significant stresses in the fibre direction, any stresses out of the 1-2 plane, such as σi3 , would subject the lamina to unnatural stresses.
Figure 2.3: Coordinates of unidirectional reinforced lamina [3]
This simplification reduces the 6x6 stiffness matrix to a 3x3 one and implies the following reduction of the strain-stress relation as
ε σ 1 S11 S21 S16 1 ε = σ 2 S12 S22 S26 . 2 (2.14) γ σ 6 S16 S26 S66 6
Following that, the engineering constants of the compliance matrix of the above relation are shown in equation (2.15). It must be noted that Young’s Modulus and Poisson’s ratio can be measured relatively efficiently through testing specimens with it is principal coordinates coinciding with the orientated coordinates. However, the degree of accuracy of the measured value of the shear modulus depends on the type of test procedure adopted where there are a number of proposed procedures that include direct and indirect methods [5].
Chapter 2. Laminate theory 15
ν η 1 − 21 61 ε σ 1 E1 E2 G6 1 ν η ε = − 12 1 62 σ (2.15) 2 . 2 E E G 1 2 6 ε η η 1 σ 6 16 26 6
E1 E2 G6
2.2.3. Orthotropic Lamina For the orthotropic lamina, the stiffness matrix can be further reduced to the following:
ε σ 1 S11 S12 0 1 ε = σ 2 S21 S22 0 . 2 (2.16) γ σ 12 0 0 S66 12
Where there are only five constants of which only four are independent. The orthotropic compliances in terms of the elastic constants are
1 1 ν ν 1 S = ; S = ; S = S −= 21 −= 12 ; S = 11 22 12 21 66 (2.17) E1 E2 E2 E1 G12
The inverted strain-stress relation reduces to
σ ε 1 Q11 Q12 0 1 σ = ε 2 Q21 Q22 0 . 2 (2.18) σ ε 12 0 0 Q66 2 12
Where Qij are the reduced stiffnesses of the lamina that are related to the compliance matrix components and elastic constants by
Chapter 2. Laminate theory 16
S E Q = 22 = 1 11 S S − S 2 1 −ν ν 11 22 12 12 21 S ν E Q = Q = 12 = 12 2 12 21 − 2 1−ν ν (2.19) S11 S22 S12 2112
S E Q = 11 = 2 22 − 2 1−ν ν S11 S22 S12 2112
1 Q = = G 66 S 12 66
2.2.4. Ply Orientation It is often necessary to move between the principal coordinates and the orientated coordinates of the lamina. The first coordinate transformation considered below is utilised in the area of design so as to determine the effect of the lamina properties when a load is applied in non- principal material coordinates. The second transformation considered below is employed in the area of material property testing, specifically in the area of testing where the principal coordinates are not parallel to the applied loading. This transformation procedure is applied in this project in the area of failure criteria written in parametric script language (APDL) described in Section 4.4. The resulting stresses in global coordinates are extracted from the model in the post processing stage and are transformed into their principal axes equivalents and subjected to the applied failure criteria. Figure2. 4 represents the variation in terms of angle θ between the off-axis coordinates (x,y) and the principal axes (1,2).
Figure 2.4: Rotation to principal material coordinates from off-axis coordinates [3]
Chapter 2. Laminate theory 17
Principal Axis (1,2) Off-Axis Coordinates (x,y) The transformation equations in principal axes of the material to the off-axis coordinates for the stress tensor are given by the expression below where θ is the angle from the x-axis to the 1-axis as demonstrated in the above figure.
σ 2 θ 2 θ − θ θ σ x cos sin 2cos sin. 1 σ = 2 θ 2 θ θ σ (2.20) y sin cos 2cos sin. . 2 σ cos θ sin. θ − cos θ sin. θ cos 2 θ − sin 2 θ σ xy 12 Similarly, the same transformation matrix is applied to the strain tensor, the expressions for both transformed stress and strain tensors are written in short as
σ σ ε ε x 1 x 1 σ = []−1 σ ε = []−1 ε (2.21) y T . 2 y T . 2 σ σ ε ε xy 12 xy 12
Where the inverse of the transformation matrix in short is written as
c2 s2 − .2 sc − []T 1 = s2 c2 .2 sc (2.22) .sc − .sc c2 − s2
Off-Axis Coordinates (x, y) Principal Axis ( 1, 2)
The transformation of the equations of the off-axis coordinates to the principal axis of the material stress tensor is
σ 2 θ 2 θ θ θ σ 1 cos sin 2cos sin. x σ = 2 θ 2 θ − θ θ σ (2.23) 2 sin cos 2cos sin. . y σ − cos θ sin. θ cos θ sin. θ cos 2 θ − sin 2 θ σ 12 xy
And as before, the same transformation matrix is applied to the strain tensor, the expressions for both transformed stress and strain tensors are written in short as
Chapter 2. Laminate theory 18
ε ε σ σ 1 x 1 x ε = []ε σ = []σ (2.24) 2 T . y 2 T . y ε ε σ σ 12 xy 12 xy
Where the transformation matrix in short is written as
c2 s2 .2 sc T = s2 c2 − .2 sc (2.25) − 2 − 2 .sc .sc c s
Resolving the transformation in the equations in (2.23), the stress in principal axis in plane stress are the following
σ = σ 2 θ + σ 2 θ + σ θ θ 1 x cos y sin 2 xy cos sin.
σ = σ sin 2 θ + σ cos 2 θ − 2σ cos θ sin. θ 2 x y xy (2.26) σ −= σ cos θ sin. θ + σ cos θ sin. θ + σ (cos 2 θ − sin 2 θ ) 12 x y xy
And the strains in the principal axis in the plane stress state are
ε = ε 2 θ + ε 2 θ + ε θ θ 1 x cos y sin 2 xy cos sin.
ε = ε sin 2 θ + ε cos 2 θ − 2ε cos θ sin. θ 2 x y xy (2.27) ε −= ε cos θ sin. θ + ε cos θ sin. θ + ε (cos 2 θ − sin 2 θ ) 12 x y xy
2.2.5. Transformed Stiffness and Compliance Matrices It is possible to substitute the transformation in (2.21) into the stress-strain relations in the principal material coordinates in (2.18) in order to obtain the stress-strain relations in orientated or off-axis coordinates which are expressed in the following relation
σ ε x Q11 Q12 0 x σ = [][]−1 ε (2.28) y T Q21 Q22 0 T . y σ ε xy 0 0 Q66 2 xy
Resolving the matrices in (2.28), the stress-strain relation in xy coordinates is
Chapter 2. Laminate theory 19
σ Q Q Q ε x 11 12 16 x σ = ε (2.29) y Q21 Q22 Q26 . y σ γ xy Q16 Q26 Q66 xy
− In which Q = [T ] 1[Q][T ] is the component of the stiffness matrix of the transformed lamina and is defined as
= 4 θ + 4 θ + ( + ) 2 θ 2 θ Q11 Q11 cos Q22 sin 2 Q12 2Q66 sin cos
Q = Q = (Q + Q − 4Q )sin 2 θ cos 2 θ + Q (cos 4 θ + sin 4 θ ) 12 21 11 22 66 12
Q = Q sin 4 θ + Q cos 4 θ + 2(Q + 2Q )sin 2 θ cos 2 θ (2.30) 22 11 22 12 66
Q = Q = (Q − Q − 2Q )cos 3 θ sin θ − (Q − Q − 2Q )cos θ sin 3 θ 16 61 11 12 66 22 12 66 Q = Q = (Q − Q − 2Q )cos θ sin 3 θ − (Q − Q − 2Q )cos 3 θ sin θ 26 62 11 12 66 22 12 66 Q = (Q + Q − 2Q − 2Q )sin 2 θ cos 2 θ + Q (sin 4 θ + cos 4 θ ) 66 11 22 12 66 66
Where the Q matrix denotes that we are dealing with the transformed reduced stiffness ij instead of the reduced stiffness Qij . It is worth noting that the transformed reduced stiffness matrix contains terms in all nine positions of the matrix while the reduced stiffness matrix contains a number of zero terms. Alternatively to the above the procedure, the compliance matrix in strain-stress relations in orientated coordinates is given as
ε S S S σ x 11 12 16 x ε = σ (2.31) y S12 S22 S26 . y γ S S S σ xy 16 26 66 xy where the transformed orthotropic compliances Sij are
= 4 θ + ( + ) 2 θ 2 θ + 4 S11 S11 cos 2S12 S66 sin cos S22 sin
S = S (sin 4 θ + cos 4 θ )+ (S + S − S )sin 2 θ cos 2 12 12 11 22 66
S = S sin 4 θ + (2S + S )sin 2 θ cos 2 θ + S cos 4 (2.32) 22 11 12 66 22
S = (2S − 2S − S )sin θ cos 3 θ − (2S − 2S − S )sin 3 θ cos θ 16 11 12 66 22 12 66 S = (2S − 2S − S )sin 3 θ cos θ − (2S − 2S − S )sin θ cos 3 θ 26 11 12 66 22 12 66
Chapter 2. Laminate theory 20
S = (22 S + 2S − 4S − S )sin 2 θ cos 2 θ + S (sin 4 θ + cos 4 θ ) 66 11 22 12 66 66
where the anisotropic compliances in terms of engineering constants are
1 1 ν ν 1 = ; = ; = −= 21 −= 12 ; = S11 S22 S12 S 21 S66 E E E E G 1 2 2 1 12 (2.33) η η η η S = 61 = 16 ; S = 62 = 26 16 E G 26 E G 1 6 2 6 where the new engineering constants are called the coefficients of mutual influence η by Lekhnitskii which were presented in the compliance matrix of the strain-stress relations in (2.14) and are defined as ε γ η = i ; η = ij ,iji γ ,iij ε (2.34) ij i where η is the coefficient of mutual influence that characterises the stretching in the i-direction ,iji caused by shear stress in the ij -plane and η is the coefficient of mutual influence that ,iij characterises shearing in the ij -plane caused by normal stress in the i-direction. Note that the mutual influences given in (2.34) are expressed in Voigt notation.
The presence of the Q and Q , and S and S in the stiffness and compliance matrices, 16 26 16 26 respectively, creates a more complex problem solution of the generally orthotropic laminas than that of the specially orthotropic laminas. The presence of the mutual influence coefficients causes shear-extension coupling which complicates the solution of practical problems [3], [5].
2.3. Mechanical Behaviour of Laminate
A laminate is two or more laminas or plies bonded together to act as a unique structural element. The laminas are required to meet certain conditions so as to constitute a laminate, also the laminate response as a result of imposed boundary conditions including support conditions and loading. The mechanical behaviour of the laminate is presented in this project on a macromechanical scale in which the individual components of the lamina such as the fibre and matrix are not considered individually but the entire lamina and its response in the laminate. The conditions required by two laminas of different orientations perfectly bonded in a laminate include deformation compatibility: the laminas in the laminate must deform alike along the interface between those laminas in the direction of the applied force and the stresses in the transversal direction must be self-equilibrating so as to comply with the deformation compatibility. The other two conditions include stress-strain relations and equilibrium.
Chapter 2. Laminate theory 21
Difficulties arise when more than two laminas of arbitrary angles are contained in the laminate and thus a different approach namely the Classical Lamination Theory (CLT) is required to satisfy the required conditions already mentioned. The CLT approach attempts to find effective and realistic simplifying assumptions that reduces the three dimensional elastic problem to a two dimensional one. The process includes a review of the stress-strain behaviour of an individual lamina which is expressed as the kth lamina in the laminate. Secondly, the stress and strain variations through the thickness of the laminate are determined. Finally, the relation of the laminate forces and moments to the strains and the curvatures are characterised [3].
2.3.1. Formulation of the Laminate (Constitutive Equations) Considering the first part of the process for the CLT approach which includes the stress-strain behaviour of an individual lamina, the stress-strain relations in principal axis for a lamina of an orthotropic material under plane stress are given in (2.18) and for ease of demonstrating the approach are shown again below.
σ ε 1 Q11 Q12 0 1 σ = ε 2 Q21 Q22 0 . 2 (2.35) σ γ 12 0 0 Q66 12
As a result of the arbitrary orientation of the laminas, the stresses and strains of the laminas are resolved into the in-plane orientated coordinates so as to define the laminate stiffness. Similarly, these stress-strain relations and the transformed reduced stiffness matrix are given in (2.29) and again, are shown below.
σ Q Q Q ε x 11 12 16 x σ = ε (2.36) y Q21 Q22 Q26 . y σ γ xy Q16 Q26 Q66 xy
In general for a lamina that occupies the kth position in the laminate, the previous expression can be written as {σ }k = [Q ]k {ε}k (2.37) The CLT approach assumes that the complete laminate acts as a single layer where there is perfect bonding between the laminas enabling continuous displacement between the laminas so that no lamina can slip relative to the other. The Hypothesis of Kirchhoff assumes that, if the laminate is thin, a line that is originally straight and perpendicular to the middle surface of the
Chapter 2. Laminate theory 22
laminate before deformation is assumed to remain straight and perpendicular to the middle surface when the laminate is deformed. Figure 2.5 shows from left to right the thin laminate and its orientation, a sectional view (xz -plane) of the laminate in both the undeformed and deformed state.
Figure 2.5: Laminate axis orientation, laminate section before and after deformation [3]
The Hypothesis of Kirchhoff in which the normal to the middle surface remains straight is depicted in the figure above. This assumption thereby ignores the shearing strains in planes perpendicular to the middle surface, that is
γ = γ = (2.38 ) xz yz 0
In addition, the lines perpendicular to the middle surface are presumed to have a constant length so that the strain perpendicular to the middle surface is ignored
ε = (2.38 ) z 0 The laminate cross section derives the Hypothesis of Kirchoff in which the displacement in the x-direction of the point B (middle surface) from the undeformed to deformed state is u 0. Because the line ABCD remains straight after deformation, the displacement of point C in the x- direction is
= − β (2.39 ) u u0 z.
From the Hypothesis of Kirchoff-Love for shells where under deformation, the line ABCD remains perpendicular to the middle surface, β is the slope of the middle laminate surface in the x-direction and is
∂ w0 β = (2.40) ∂x
Chapter 2. Laminate theory 23
Then, the displacement at any point through the laminate thickness is
∂ = − w0 u u0 z (2.41) ∂x
Similarly, for the displacement in the y-direction is
∂w v = v − z 0 0 ∂y (2.42)
As a consequence of the Hypothesis of Kirchoff, the remaining laminate strains are defined in terms of displacements as
∂u ∂u ∂ 2w ε = = 0 − 0 x z 2 ∂x ∂x ∂x
∂v ∂v ∂2w ε = = 0 − z 0 y ∂y ∂y ∂y 2 (2.43)
∂u ∂v ∂u ∂v ∂ 2w γ = + = 0 + 0 − z 0 xy ∂y ∂x ∂y ∂x ∂ ∂yx or they can be expressed in vector form as
2 ∂u ∂ w 0 − 0 ε 2 ε 0 0 x ∂x ∂x x k x 2 ∂v ∂ w ε = 0 + z − 0 = ε 0 + zk 0 y ∂y ∂y 2 y y (2.44) γ ∂u ∂v ∂ 2 w γ 0 k 0 xy 0 + 0 − 0 xy xy 2 ∂y ∂x ∂ ∂yx
ε 0 ε 0 γ 0 0 0 where x , y and xy are the three middle strains (elongations and distortions) and kx , k y and 0 k xy are the three middle-surface curvatures (bending curvatures and torsion). The stress-strain relations given in (2.36) can be modified by the substitution of the strain variation through the thickness given above in (2.44). The stresses for the kth layer are expressed in terms of the laminate middle-surface strains and curvatures as
Chapter 2. Laminate theory 24
k k σ Q Q Q ε 0 k 0 x 11 12 16 x x σ = ε 0 + 0 (2.45) y Q21 Q22 Q26 y z k y σ γ 0 0 xy Q16 Q26 Q66 xy kxy where z corresponds with the coordinates of the kth lamina. The component of the stiffness matrix Qij can be different for the each layer of the laminate. That implies that the stresses at the interface are not continuous even though the strain variation is linear through the lamina interface. Figure 2.6 demonstrates the distribution of strain ε, characteristic stiffness moduli Q and stress σ distribution for a four layer laminate. While the stress variation is discontinuous at the interface it does vary linearly within each of the laminas [5].
Figure 2.6: Strain and stress distribution [5]
The final stage of the CLT approach includes the characterisation of the relation of the laminate forces and moments to the strains and the curvatures. The loading includes Nx which is a force per unit width (in-plane) of the cross section of the laminate and Mx which is a moment per unit width and is shown acting on the laminate in figure 2.7.
Figure 2.7: In-plane forces and moments on a laminate [5]
Chapter 2. Laminate theory 25
The resultant forces and moments acting on a laminate, as shown in the above figure, are obtained by integration of the stresses in each layer or lamina through the laminate thickness and are defined as
k N σ σ x x x t N z N = 2 σ dz = k σ dz (2.46) y ∫−t y ∑ ∫ y z − 2 k =1 k 1 σ σ Nxy xy xy
k M σ σ x x x t N z M = 2 σ .dzz = k σ .dzz (2.47) y ∫−t y ∑∫ y z − 2 k =1 k 1 σ σ M xy xy xy
where z k and z k-1 are the laminate geometry and the configurations of the laminas are shown in figure 2.8 in which z is positive downwards.
Figure 2.8: Lamina configurations [5]
The stress-strain relations in (2.45) can be substituted into the forces and moments equations in (2.46) and (2.47), respectively, and the results of these substitutions are shown below in (2.48) and (2.49). If there does not exist temperature dependent or moisture dependent properties and a temperature gradient or a moisture gradient in the lamina, the stiffness matrix can be taken outside the integration over each layer but remains within the summation of the force and moments resultants for each layer. If an elevated temperature or moisture exists throughout the layers the stiffness matrix remains constant but its value is altered due to degradation. In cases where the stiffness matrix is not constant throughout the layers, it
Chapter 2. Laminate theory 26
remains within the integration over each layer thereby leading to a more complicated numerical solution [3], [5].
k N Q Q Q ε 0 k 0 x 11 12 16 x x N zk zk = ε 0 + 0 (2.48) N y ∑ Q21 Q22 Q26 ∫y dz ∫ ky zdz = zk−1 zk − 1 k 1 γ 0 0 Nxy Q16 Q26 Q66 xy kxy
k M Q Q Q ε 0 k 0 x 11 12 16 x x N z z = k ε 0 + k 0 2 (2.49) M y ∑ Q21 Q22 Q26 ∫y zdz ∫ ky dzz = zk−1 zk − 1 k 1 γ 0 0 M xy Q16 Q26 Q66 xy kxy
ε 0 ε 0 γ 0 0 Given that the three middle strains ( x , y , xy ) and the three middle-surface curvatures ( kx , 0 0 k y , k xy ) are independent of z, and are instead middle surface values, they can be removed from within the summation signs. The equations in (2.50) and (2.51) can be written as
ε o o Nx A11 A12 A16 x B11 B12 B16 kx = ε o + o N y A12 A22 A26 y B12 B22 B26 ky (2.50) γ o o Nxy A16 A26 A66 xy B16 B26 B66 kxy
ε o o M x B11 B12 B16 x D11 D12 D16 kx = ε o + o M y B12 B22 B26 y D12 D22 D26 k y (2.51) γ o o M xy B16 B26 B66 xy D16 D26 D66 k xy where N = k ()− Aij ∑ Qij zk zk −1 k =1
N = 1 k ()2 − 2 Bij ∑Qij zk zk −1 (2.52) 2 k =1
N = 1 k ()3 − 3 Dij ∑Qij zk zk −1 3 k =1
Chapter 2. Laminate theory 27
The Aij are extensional stiffnesses with A16 and A26 representing shear-extension coupling, the
Bij are bending-extension coupling stiffnesses, and the Dij are bending stiffnesses with D16 and
D26 representing bend-twist coupling. The presence of Bij implies coupling between bending and extension of a laminate. This in physical terms causes not only extensional deformations but bending and/or twisting of the laminate when only an in-plane force, e.g. Nx is applied on the laminate [3].
2.4. Other Analysis and Behavioural Topics The complexity of the composite model requires a number of mechanical behavioural topics to be analysed. The ones felt most relevant to this project are included below. Those presented are a direct consequence of the model’s profile (holes in laminates) and its subjected environment (vibration and fatigue in laminates).
2.4.1. Holes in Laminates The existence of holes in composite laminate structures is a result of numerous service and mechanical requirements including weight and surface area reduction, bolt accommodation, and access through the structure. In isotropic materials, the main influence for failure with holes is due to the magnitude of the stress concentration factor from which the maximum stress is obtained. However, for orthotropic materials, a combined stress failure criterion is required. It includes stress concentration factors at the hole’s edge and an appropriate failure criterion for composite materials as described in Section 4.3. Many isotropic materials such as aluminium or steel are, in terms of deformation before failure, more ductile than composite materials thereby allowing localised yielding to accommodate stress concentrations in these critical zones whereas the majority of composite materials contain higher stress concentrations and a lesser ability to yield than isotropic materials.
The stress concentration factor around the circumference of the hole is caused by the combination of the principal material direction and secondly the load direction in which the material is subjected. Where the principal material direction does not coincide with the loading, the lamina is considered effectively as being anisotropic or generally orthotropic. Figure 2.9 shows a lamina with its fibre direction at an arbitrary angle α from the x-direction of loading. The angle θ represents the circumferential stress at the edge of the circular hole and thus its magnitude varies in accordance with the fibre direction. As α approaches 90 o, the peak stress concentration factor decreases and shifts its location θ around the hole. As a result, stress concentrations around the hole circumference are quite intrinsic. Its complexity increases with the analysis of a laminate with laminas of various orientations where each layer and their stresses must be determined by the use of the Classic Laminate Theory approach and applied to an appropriate strength criterion for failure analysis.
Chapter 2. Laminate theory 28
Figure 2.9: Loading and principal material direction of composite lamina
Stress concentration around holes in composite laminates can be reduced by a method known as the Stiffening Strip Concept. This process includes the addition of stiffer composite material in the zones located on either side of the hole but away from its boundary. The concept of the stiffener is to remove loading from around the hole boundary by transferring the loading through the stiffener itself. A second method is the addition of a more flexible strip situated right at the edge of the hole so as to reduce the load concentration at the holes edge and transferring it to some other unknown region of the laminate.
2.4.2. Vibration of Laminates The main objective of this type of analysis is to determine the response of the laminate due to vibration in terms of its magnitude of deflection and its modes shapes. Vibration is a transverse load which causes deflection of the laminate due to bending and is generally larger than in- plane deflections, because flexural stiffnesses are lower than extensional stiffnesses. The general equilibrium equations governing transverse deflections include both in-plane and out- of-plane forces. The analysis of laminate or plate deflections is based on the CLT outlined in Section 2.3 and in the differential equations of equilibrium. For clarity of representation, the differential equations are developed more conveniently through the use of a planar element dimensioned dx by dy . Figure 2.10 shows the in-plane stress resultants (a), the moment resultants (b) and the transverse shear resultants (c). Because the plate does not remain flat during vibration, the analysis cannot be derived from equilibrium of the differential element and it is therefore assumed that the transverse deflections remain small, so that the out-of- plane components of the in-plane resultants Nx, Ny, and Nxy are negligible [6].
Chapter 2. Laminate theory 29
Figure 2.10: Stress, moments, and transverse shear resultants of laminate [6]
The equilibrium differential equations for vibration of a composite laminate with arbitrary ply orientations are presented below beginning with the summation of forces along the x-direction as
∂ 02 ∂N N xy ∂ u N dy + x dxdy + N dx + dxdy − N dy − N dx = ρ dxdy (2.53) x ∂x xy ∂y x xy 0 ∂t 2
0 where ρ0 is the mass per unit area of laminate and u (x, y, t) is the middle surface displacement in the x-direction. The previous equation can be simplified to
∂ 02 ∂N N xy ∂ u x + = ρ (2.54) ∂x ∂y 0 ∂t 2
Similarly, the summation of forces along the y-direction gives
∂ 02 ∂N N xy ∂ v N dx + x dxdy + N dy + dxdy − N dx − N dy = ρ dxdy (2.55) x ∂y xy ∂x y xy 0 ∂t 2 and simplifies to
∂ ∂ 02 N y N xy ∂ v + = ρ (2.56) ∂y ∂x 0 ∂t 2
Chapter 2. Laminate theory 30
where v0(x, y, t) is the middle surface displacement in the y-direction. The summation of the forces along the z-direction yields
∂Q ∂Q ∂2w Q dy + X dxdy +Q dx + y dxdy −Q dy −Q dx + (), yxq = ρ (2.57) X ∂x y ∂y X y 0 ∂t 2 where
= t 2 σ = t 2 σ Qx xz dz Qy yz dz ∫−t 2 ∫−t 2 (2.58) and simplifies to
∂Q ∂Q ∂2w X + y + (), yxq = ρ (2.59) ∂x ∂y 0 ∂t 2 where w(x, y, t) is the displacement in the z-direction.
For the moment equilibrium, the moments are considered about the x-axis and y-axis but rotary inertia is neglected. The summation of the moments about the x-axis simplifies to
∂M ∂M y + xy = Q (2.60) ∂y ∂x y And similarly, the summation of moments about the y-axis yields
∂M ∂M x + xy = Q (2.61) ∂x ∂y x
Substitution of the two moments in equations (2.60) and (2.61) in equation of (2.59) produces
∂2M ∂2M ∂2M ∂2w x + 2 xy + y + (), yxq = ρ x (2.62) ∂x2 ∂ ∂yx ∂y2 0 ∂t 2
The laminate force-deformation equations in (2.48) and the strain and curvatures relations in terms of displacement in (2.43) are substituted into differential equations of motion (2.54), (2.56), and (2.62) to produce the corresponding equations of motion in terms of displacements.
∂ u02 ∂ u02 ∂ u02 ∂ v02 ∂ v02 ∂ v02 ∂3w A + 2A + A + A + ()A + A + A − B (2.63) 11 ∂x2 16 ∂xdy 66 dy 2 16 ∂x2 12 66 ∂xdy 26 ∂y2 11 ∂x3 ∂3w ∂3w ∂3w − 3B − ()B + 2B − B = 0 16 ∂x2∂y 12 66 ∂ ∂yx 2 26 ∂y3
Chapter 2. Laminate theory 31
∂ u02 ∂ u02 ∂ u02 ∂ v02 ∂ v02 ∂ v02 ∂3w A + ()A + A + A + A + 2A + A − B (2.64) 16 ∂x2 12 66 ∂xdy 26 dy 2 66 dx 2 26 ∂xdy 22 ∂y2 16 ∂x3 ∂3w ∂3w ∂3w − ()B + 2B − 3B − B = 0 12 66 ∂x2∂y 26 ∂ ∂yx 2 22 ∂y3
∂4w ∂4w ∂4w ∂4w ∂4w ∂ u03 D + 4D + 2()D + 2D + 4D + D − B 11 ∂ 4 16 ∂ 3 12 66 ∂ 2∂ 2 26 ∂ ∂ 3 22 ∂ 4 11 ∂ 3 x dyx x y yx y x ∂ u03 ∂ u03 ∂ u03 ∂ v02 ∂ v03 (2.65) − 3B − ()B + 2B − B − B − ()B + 2B 16 ∂x2∂y 12 66 ∂ ∂yx 2 26 ∂y3 16 ∂x3 12 66 ∂x2∂y ∂ v03 ∂ v03 − 3B − B = (), yxq 26 ∂ ∂yx 2 22 ∂y3
The various coupling stiffnesses such as A16 and A26 (shear-extension coupling), Bij (bending- extension coupling), and D16 and D26 (bend-twist coupling) are present in the above equilibrium equations analysis and must be considered in their effect on the vibration behaviour of the laminate plate. It is important to recognise the effect of the lamina configuration within the laminate on the various coupling stiffnesses. If the laminate is symmetric about the middle surface (as is intended to be the case for all the modelled composite structures in this project) the bending-extension coupling Bij is reduced to zero [3]. Furthermore, if the laminate is specially orthotropic i.e. the principal material directions coincide with the loading direction, the shear-extension coupling and the bend-twist coupling simplifying equation (2.65) of transverse displacements to
∂ 4w ∂4w ∂4w D + 2()D + 2D + D = (), yxq (2.66) 11 ∂x4 12 66 ∂x2∂y 2 22 ∂y 4
2.4.3. Fatigue The vast majority of service failures in materials are due to fatigue of the material. Fatigue of isotropic materials has been investigated for many years and its process is quite well documented. However, fatigue of orthotropic and anisotropic composite materials is relatively new in comparison. Fatigue of unidirectional composites is generally controlled by the lamina with orientation 0 o even with the laminate in question containing laminas of various orientations. Due to the importance of the effects of fatigue in service life, testing of representative laminate specimens of the structure for an appropriate load history is required to determine the life of the structure or the number of load cycles before failure.
Chapter 2. Laminate theory 32
Fatigue is controlled by a number of methods including displacement, energy and load controlled tests with the ultimate considered the most appropriated to represent actual fatigue life in service conditions. The S-N diagram describes the applied global stress level with respect to the number of cycles to failure. For composite materials, the diagram is more readably interpreted if it is replotted with the maximum strain attained in the first load cycle against the number of cycles (log) to failure. The maximum strain recorded in the first load cycle can be described as the damage state reached in the initial stage which is seen to contribute to any progression of the damage after the initial cycles and during the course of the fatigue life.
The fatigue life diagram consists of three distinct regions as depicted in figure 2.11 and represents regions of different damage mechanisms incurred by the composite material. These failure mechanisms are associated damage of the fibre and matrix components [5].
Figure 2.11: Fatigue life diagram of longitudinal composites in tension-tension fatigue [7]
Region I, known also as the static region, is the zone in which the strain level coincides with the maximum strain level of the static test. The mechanism in this region is evidently breakage of the fibre in the 0 o direction which is similar to that of static testing where fibre breakage in the composite is random.
Region II or the progressive region is the zone consisting of a downward slope that is a consequence of the decrease in the strain level and an increase in life. The mechanisms attributed to failure in this region include fibre bridged cracking, and debond propagation. Region II can be further subdivided in terms of macroscopic fatigue damage mechanisms which include fibre breakage as being the prevalent mechanism at high load levels (high portion of region II) or known also as initiation triggered mechanisms. At low load levels the, the main mechanism is matrix or interface crack propagation.
Chapter 2. Laminate theory 33
Region III is the fatigue limit of the composite. Below this limit failure does not occur prior a large number of cycles of typically 10 6 or 10 7 cycles. In this region, the damage is constrained and obstructed from further growth by the fibres. Crack arrest and subsequent inhibition of damage accumulation is believed to be caused by the fact that the strain level is too low and the threshold value for propagation is not reached and secondly, the fibre-matrix debonding and crack arresting by proximate fibres prevent damage accumulation and subsequent failure.
It is important to note that composites with high fibre mechanical resistance and less ductile matrices have an adverse effect on the fatigue performance. Graphically, this resembles a steeper slope in the scatter band of region II and an increased fatigue limit as shown in figure 2.12.
Figure 2.12: Fatigue life influenced by fibre stiffness and matrix toughness [7]
It has been observed from numerous investigations that multidirectional composites are more sensitive to fatigue in tension-compression loading than in tension-tension loading. This occurrence can be attributed to the greater number of transverse cracks that appear in cross- ply laminates (e.g. 90 o plies) under tension-compression loading than that of the same laminate under tension-tension loading. Observations show that the rate of debond propagation is higher in tension-compression loading ply which subsequently causes an accelerated initiation of transverse cracks and a reduction in fatigue life [7].
Chapter 2. Laminate theory 34