2.4 Equilibrium equations - Invariant form
3.1.3 First-order plate theory
In this case φ(x)i = φ(y)i = 0, λ(x)i = λ(y)i = 0 in Eq.(2.1). Eqs.(3.1)-(3.2) apply, however Eqs.(3.3)-(3.4) are omitted again (shear strain continuity at the interface plane is not possible to ensure because of the piecewise constant distributions, refer to Figure 3.1b). Thus, we can eliminate only four parameters from Eq.(2.1), the secondary parameters are: u0i,v0i for i= 1,2. The vector of primary parameters is:
ψ(p)=
θ(p)1 θ(p)2 T
, p=xory. (3.18)
The elements of the matrices Kij(0) and Kij(1) are defined in Appendix A.3. Apparently Kij(2) = Kij(3) = 0 in this case. The in-plane equilibrium is defined by Eq.(3.7). The The equation for δw is equivalent to Eq.(3.15). The equivalent moments can be obtained by taking back the Kij constants given by Appendix A.3 into Eq.(3.19) and subtracting the shear forces: that have an important role in the assignment of the boundary and continuity conditions.
3.2 Delaminated region
In the delaminated region (refer to Figures 3.1b and 3.2b) the top and bottom plates are captured by independent ESLs, and thus these plates are modeled by traditional first-, second-order plates and third-order Reddy plates. The displacement field can be given by:
u(i) =u0b+
viz., the global membrane displacement components are zero in this case andj is a summation index. It is important to note that in accordance with Eq.(3.21) the transverse deflections of the top and bottom plates of the delaminated region are identical (constrained mode model, (Szekr´enyes (2014c))). In other words, the crack opening is eliminated in the plate, and the problem provides essentially mixed-mode II/III fracture without the presence of mode-I (refer to Figure 1.3). The aim is to predict the mechanical fields in the plate as accurately as possible compared to FE calculations. If this is done for mixed-mode II/III, then the model can be extended to general mixed-mode I/II/III case in the course of further research work.
3.2.1 Reddy’s third-order plate theory
In this theory the top and bottom plates are traction-free at the top and bottom boundaries (Reddy (2004), refer to Figures 3.1b and 3.2b), thus, from Eq.(3.4) we have:
(γxz(1), γyz(1))
z(1)=±t1/2 = (γxz(2), γyz(2))
z(2)=±t2/2 = 0, (3.22)
meaning eight conditions. The secondary parameters are: φ(x)i, φ(y)i,λ(x)i,λ(y)i fori= 1..2.
The modified displacement field has the same form as that given by Eq.(3.21), the coefficients denoted by Kij are placed in Appendix A.1. The vector of primary parameters becomes:
ψ(p) =
θ(p)1 θ(p)2 ∂w∂p T
, p=xory, (3.23)
where θ(p)2 is an autocontinuity parameter. The equilibrium equations with respect to the membrane displacements are: The equilibrium equations for the bending and twisting moments are:
δψ(x)j : ∂Mˆx(j) Finally, the equation with respect to δw in accordance with Reddy (2004) is:
δw : where the equivalent shear forces are defined as (Reddy (2004)):
Qˆx(i) =Qx(i)+ 3Ki3(3)Sx(i), Qˆy(i)=Qy(i)+ 3Ki3(3)Sy(i), i= 1,2. (3.27) Finally, the equivalent bending moments become:
⎛
CHAPTER 3. THE METHOD OF TWO EQUIVALENT SINGLE LAYERS
3.2.2 Second-order plate theory
In this case λ(x)i = 0 and λ(y)i = 0 in Eq.(2.1). No conditions are imposed against the displacement field, and so there are only primary parameters in Eq.(3.21), the vector of primary parameters takes the form below:
ψ(p)=
θ(p)1 φ(p)1 θ(p)2 φ(p)2 T
, p=xory. (3.29)
The elements of the matrices Kij(0), Kij(1) and Kij(2) are defined in Appendix A.2. Apparently Kij(3) = 0 in this case. The membrane force equilibrium is given by Eq.(3.24), the other equilibrium equations are:
δθ(x)i : ∂Mx(i)
∂x +∂Mxy(i)
∂y −Qx(i) = 0, i= 1,2, δθ(y)i : ∂Mxy(i)
∂x +∂My(i)
∂y −Qy(i)= 0, i= 1,2,
(3.30)
δφ(x)i : ∂Lx(i)
∂x +∂Lxy(i)
∂y −2Rx(i) = 0, i= 1,2, δφ(y)i : ∂Lxy(i)
∂x +∂Ly(i)
∂y −2Ry(i) = 0, i= 1,2.
(3.31)
Finally, the shear force equilibrium involves:
δw: 2
i=1
∂Qx(i)
∂x + ∂Qy(i)
∂y
+q= 0. (3.32)
3.2.3 First-order plate theory
For the FSDT φ(x)i = 0, φ(y)i = 0, λ(x)i = 0 and λ(y)i = 0 in Eq.(2.1). Obviously, there are only primary parameters in Eq.(3.21): u0i,v0i, θ(x)i, θ(y)i for i= 1,2. The vector of primary parameters is:
ψ(p)=
θ(p)1 θ(p)2 T
, p=xory. (3.33)
The elements of the matrices Kij(0) and Kij(1) are defined in Appendix A.3. In this case Kij(2) =Kij(3) = 0. The in-plane force equilibrium involves Eq.(3.24), the moment and shear force equilibrium leads to:
δθ(x)i : ∂Mx(i)
∂x +∂Mxy(i)
∂y −Qx(i) = 0, i= 1,2, δθ(y)i : ∂Mxy(i)
∂x +∂My(i)
∂y −Qy(i)= 0, i= 1,2,
(3.34)
dx
Figure 3.3: Equilibrium of stress resultants of the FSDT (a), SSDT (b) and TSDT (c) for an equivalent single layer.
The equilibrium equations of ESL theories can be obtained by using differential plate elements assuming differential changes of the stress resultants going from one boundary to the other. This scheme is very simple in the case of FSDT as it is shown by Figure 3.3a.
For the SSDT Figure 3.3a (moment and shear force equilibrium) should be complemented with Figure 3.3b showing the possible interpretation of the higher-order stress resultants (L and R). Finally, Figure 3.3a, b and c should be considered together if the general TSDT is applied (L, P, R and S), although the equilibrium equations of this theory were not presented, these are documented by Szekr´enyes (2014d).
The method of four equivalent single layers 4
The concept of the method of 4ESLs (developed recently bySzekr´enyes (2016a,b)) is shown in Figures 4.1a and 4.2a. The bottom and top plates are modeled by two ESLs, and thus it is a refinement compared to the method of 2ESLs. It will be shown later that in certain cases it is not enough to apply 2ESLs. The SEKC is applied to the problem shown in Figures 4.1a and 4.2a. Using the conditions defined by Eqs.(2.2)-(2.6) we can eliminate certain parameters from Eq.(2.1), which (for 4 ESLs) involves 34 parameters altogether plus the deflections (w(i)(x, y) = w(x, y)). The parameter elimination is carried out similarly to that of the method of 2ESLs. In the subsequent sections the undelaminated and delaminated regions are discussed separately. In the first step, the general TSDT displacement field is presented, then in the subsequent steps the SSDT and FSDT fields are obtained by the reduction of the TSDT equations. The meaning of the transverse splitting in Figure 4.1a
Figure 4.1: Cross sections and deformation of the top (ESL3 and ESL4) and bottom (ESL1 and ESL2) plate elements of a delaminated plate in the X-Z plane (a). Distribution of the transverse shear strains by FSDT, SSDT and TSDT (b).
is that different mathematical models are applied in the undelaminated and delaminated regions.
Figure 4.2: Cross sections and deformation of the top (ESL3 and ESL4) and bottom (ESL1 and ESL2) plate elements of a delaminated plate in the Y-Z plane (a). Distribution of the transverse shear strains by FSDT, SSDT and TSDT (b).
4.1 Undelaminated region
The transition zone around the delamination front in the X −Z plane of the composite plate is shown in Figure 4.1a. The distribution of the in-plane displacement functions is piecewise linear by FSDT, quadratic in the case of the SSDT and cubic for the TSDT. The corresponding shear strain distributions are shown in Figure 4.1b: it is piecewise constant by FSDT, piecewise linear by SSDT and piecewise quadratic by TSDT with continuous derivatives and curvatures in the latter case at the perturbation planes. The Y −Z plane is shown in Figure 4.2. In accordance with Figures 4.1a and 4.2a and Eq.(2.2), the following conditions are formulated between the four ESLs (continuity of in-plane displacements at the interface planes):
(u(1), v(1), w(1))
z(1)=t1/2 = (u(2), v(2), w(2))
z(2)=−t2/2, (u(2), v(2), w(2))
z(2)=t2/2 = (u(3), v(3), w(3))
z(3)=−t3/2, (u(3), v(3), w(3))
z(3)=t3/2 = (u(4), v(4), w(4))
z(4)=−t4/2.
(4.1)
The reference plane belongs to the second ESL (see Figures 4.1a and 4.2b), therefore, the following condition is imposed (refer to Eq.(2.3)):
(u(2), v(2))
z(2)=zR(2) = (u0(x, y), v0(x, y)), (4.2)
CHAPTER 4. THE METHOD OF FOUR EQUIVALENT SINGLE LAYERS
wherez(2)R = 1/2(t3+t4−t1) in accordance with Figure 4.1a and it gives the position of the global midplane of the model with respect to ESL2. The next set of conditions imposes the continuous shear strains at the interface planes using Eq.(2.4):
(γxz(1), γyz(1))
As discussed previously, the oscillations in the shear strain distribution can be reduced by ensuring continuous shear strain derivatives at interface planes 1-2 and 3-4 (Eq.(2.5)):
∂γxz(1)
furthermore, by imposing continuous second derivatives of shear strains in the same planes by using the conditions below (Eq.(2.6)):
∂2γxz(1)
To further reduce the number of parameters in the displacement field and to obtain more accurate results, the SSCC is applied at the top and bottom boundaries (Eq.(2.7)):
(γxz(1), γyz(1))
z(1)=−t1/2 = (γxz(4), γyz(4))
z(4)=t4/2. (4.6)
The concept of the shear strain control condition (SSCC) is shown in Figure 4.3, where in the undelaminated portion at two points in each cross section the shear strain is identical.
Although the SSCC is applicable even in the case of the TSDT it leads to large oscillations in the transverse shear strains (Szekr´enyes (2016b)), therefore it is applied only to the SSDT solution. It is also important to note that the large oscillations in the shear strain distribution take place even if the SSDT solution without the SSCC is applied (Szekr´enyes (2016a)).
In Eq.(2.1) the displacement functions are modified in order to satisfy Eqs.(4.1)-(4.6).
In the general sense, by applying the FSDT, SSDT and TSDT theories the displacement functions can be written as:
u(i) =u0+
where the matrices denoted by Kij are related to the geometry (ESL thicknesses),irefers to the ESL number, the summation index j defines the component in ψ, which is the vector of primary parameters, finally w(x, y) is the transverse deflection and identical for each ESL.
Figure 4.3: The concept of controlled shear strain distribution on theX-Z plane (a) andY-Z plane (b) in the undelaminated and delaminated regions of the SSDT and TSDT solutions.