Analysis of non-skew and skew laminated composite and sandwich plates under hygro-thermo-mechanical conditions including transverse stress variations

Abstract

A new fourth-order in-plane displacement field based refined higher-order zigzag theory is proposed for analysis of laminated sandwich plate (for both skew and non-skew shaped) subjected to hygro-thermal conditions. In order to predict behavior of thick laminated sandwich plates, third-order transverse displacement field is taken. Zig-zag effects are introduced using linear unit Heaviside step function. The theory satisfies zeros transverse normal and shear stress condition at the bottom and top surface of the plate along with continuity condition at interface. The proposed model is free from any kind of C-1 or penalty function requirements. Nine-noded isoparametric finite element having twelve degrees of freedom per node is used during analysis. Since, the present theory incorporates transverse displacement field along with continuity conditions, is able to predict the behavior of thick sandwich plates more efficiently. In literature no results are present for skew laminated composite and sandwich plates therefore, present results for skew plates are entirely new and will serve as benchmark for future studies.

Keywords

Laminated sandwich plate hygro-thermal finite element stress continuity zig-zag theory

Introduction

Laminated composite and sandwich structures are widely used in various fields of engineering such as aerospace, automobile, civil, naval etc. Since these structures exist in form of layers and hence, are difficult to analyse. Along with this problem, these structures are subjected to wide range of temperature and moisture variation during their service life period. Therefore, the stresses arising due to such variations must be included during analysis stage prior to construction. A detailed review on the influence of such conditions on the behavior of the laminated composite and sandwich structures is given by [1 –3] and recently by the authors Garg and Chalak [4].

From the review work it is seen that no work is carried out till date on the hygro-thermo-mechanical based analysis of laminated composite and sandwich plates using C-0 higher-order zigzag theory (HOZT). Khandelwal et al. [5] and Garg et al. [6] recently carried out analysis of the same using finite element (FE) third-order zigzag theory (ZZT) and trigonometric based ZZT respectively. However, both these theories assumed constant transverse displacement function and at the same time requires penalty function or C-1 continuity requirement conditions. Because of these limitations, the theory is not able to predict the behavior of laminated composite and sandwich plates more efficiently especially for thick plates. In case of limitation of applicability of C-1 FEs for practical applications, Chalak et al. [7] stated, “Moreover, most of the available FEs are not attractive for practical application due to the presence of higher order derivative in the field variables and some other disadvantages”. Introduction of transverse displacement function instead of assuming it constant, helps in predicting the behavior of sandwich plates more effectively as seen in the review work of Garg and Chalak [4]. In the work of Khandelwal et al. [5] it is noticed that in order to predict the transverse stresses more effectively, post-processing technique was implemented. Also, the work is carried out on very limited scale for understanding the behavior of sandwiches when combined hygro-thermo-mechanical loading is acting.

Research gaps

After the detailed review analysis on the bending analysis of laminated composite and sandwich structures under hygrothermal conditions following gaps are identified:

Most of the available theories assumes constant transverse displacement field which are not able to predict the behavior of thick laminated composite and sandwich structures efficiently

In order to predict the behavior of thick laminates more efficiently, some researchers used post processing techniques which adds to the computational efforts

Transverse normal stresses are neglected by most of the researchers during analysis because of the assumption of constant transverse displacement field which fails to predict the accurate behavior of the sandwich structures

Very few works are concentrated on using C-0 finite element for the bending analysis of laminated composite and sandwich structures under hygrothermal conditions

No work is reported in literature on the bending analysis of skew laminated composite and sandwich plates under hygrothermal conditions to the authors’ best knowledge.

After the careful investigation, a new nine noded FE based fourth-order ZZT is proposed for the analysis of laminated composite and sandwich plates for hygro-thermo-mechanical analysis of laminated composite plates. Third-order transverse displacement field is assumed along with linear unit step function. The present formulation satisfies zero condition of transverse normal and shear stresses at top and bottom surfaces. Also, the formulation satisfies inter-laminar transverse normal and shear stress continuity at interfaces. Along with these, the theory is also free from any kind of penalty function or C-1 requirements. In case of FE, twelve degrees of freedom per node is considered. After solving number of problems, it is seen that present model is able to predict the behavior of laminated composite and sandwich plates efficiently even for thick plates.

Mathematical modelling

Considering a laminated plate of thickness ‘h’ having ‘N^(u)’ number of upper layers and ‘N^(l)’ number of lower layers with respect to the mid-plane of the plate which is considered as the reference plane during analysis.

The in-plane displacement field is chosen as (Figure 1):

U_{(x)} = u^{(0)} + z φ^{(x)} + z^{2} μ^{(x)} + z^{3} ξ^{(x)} + z^{4} ψ^{(x)} + \sum_{i = 1}^{N^{(u)} - 1} (z - z_{i}^{(u)}) H (z - z_{i}^{(u)}) Φ_{i}^{(x_{u})} + \sum_{j = 1}^{N^{(l)} - 1} (z - z_{j}^{(l)}) H (- z + z_{j}^{(l)}) Φ_{j}^{(x_{l})}

(1)

U_{(y)} = v^{(0)} + z φ^{(y)} + z^{2} μ^{(y)} + z^{3} ξ^{(y)} + z^{4} ψ^{(y)} + \sum_{i = 1}^{N^{(u)} - 1} (z - z_{i}^{(u)}) H (z - z_{i}^{(u)}) Φ_{i}^{(y_{u})} + \sum_{j = 1}^{N^{(l)} - 1} (z - z_{j}^{(l)}) H (- z + z_{j}^{(l)}) Φ_{j}^{(y_{l})}

(2)

The transverse displacement field is taken as:

U_{(z)} = w^{(0)} + z φ^{(z)} + z^{2} μ^{(z)} + z^{3} ξ^{(z)} + \sum_{i = 1}^{N^{(u)} - 1} (z - z_{i}^{(u)}) H (z - z_{i}^{(u)}) Φ_{i}^{(z_{u})} + \sum_{j = 1}^{N^{(l)} - 1} (z - z_{j}^{(l)}) H (- z + z_{j}^{(l)}) Φ_{j}^{(z_{l})}

(3)

where,

u^{(0)}, v^{(0)}, w^{(0)}

are the displacement of any point on the mid-plane about x- y- and z-axis,

φ^{(x)}, φ^{(y)}, φ^{(z)}

are the rotation of the mid-plane about y-, x- and z-axis respectively.

μ^{(x)}, μ^{(y)}, μ^{(z)}, ξ^{(x)}, ξ^{(y)}, ξ^{(z)}, ψ^{(x)}, ψ^{(y)}

represents higher order unknowns.

Φ

denotes the slope of the ith/jth layer for upper and lower layers.

H (z - z_{i}^{(u)}), H (z - z_{i}^{(u)})

represents Heaviside unit step function.

The stress-strain relationship for an orthotropic material along the fibre-axis (local coordinate axis) for k^th lamina can be written as:

{\{σ\}}_{(l)} = {[C]}_{k} {\{ε - α Δ T - β Δ c\}}_{(l)}

(4)

{\{σ\}}_{(l)} = {[C]}_{k} {\{ε_{net}\}}_{(l)}

where,

\{σ\}

and

\{ε_{net}\} = \{ε\} - \{ε_{hygro t hermal}\}

are the stress and strain vector for the k^th lamina along local axis.

The above constitutive relationship must be transformed to the global axis system which is done by multiplying the compliance matrix [C] with the transformation matrix [T] and can be re-written as:

\{σ\} = {[\bar{C}]}_{k} \{ε_{net}\}

(5)

where,

\{σ\}, \{ε_{net}\}

are the transformed stress and strain vector while

[\bar{C}]

is the transformed compliance matrix for the k^th lamina.

Now utilising the following conditions:

Transverse normal and shear stress-free condition at the top and bottom surface of the plate

Condition of transverse normal and shear-stress continuity at interface

Additional conditions: $u = u^{(u)}, v = v^{(u)}, w = w^{(u)}$ at top of the plate and $u = u^{(l)}, v = v^{(l)}, w = w^{(l)}$ at bottom of the plate.

With the help of above discussed conditions, the following unknowns

μ^{(x)}, μ^{(y)}, μ^{(z)}, ξ^{(x)}, ξ^{(y)}, ξ^{(z)}, ψ^{(x)}, ψ^{(y)}, Φ_{i}^{{[x}^{(u)}]}, Φ_{j}^{{[x}^{(l)}]}, Φ_{i}^{{[y}^{(u)}]}, Φ_{j}^{{[y}^{(l)}]}, Φ_{i}^{{[z}^{(u)}]}, Φ_{j}^{{[z}^{(l)}]}, \frac{\partial w^{(u)}}{\partial x}, \frac{\partial w^{(l)}}{\partial x}, \frac{\partial w^{(u)}}{\partial y}, \frac{\partial w^{(l)}}{\partial y}

can be written in terms of displacements

u^{(0)}, v^{(0)}, w^{(0)}, φ^{(x)}, φ^{(y)}, φ^{(z)}, u^{(u)}, w^{(u)}, u^{(l)}, w^{(l)}

as:

\{M\} = [N] \{Θ\}

(6)

where,

\{Θ\} = {\{u^{(0)}, v^{(0)}, w^{(0)}, φ^{(x)}, φ^{(y)}, φ^{(z)}, u^{(u)}, v^{(u)}, w^{(u)}, u^{(l)}, v^{(l)}, w^{(l)}\}}^{T}

\begin{array}{l} \{M\} = μ^{(x)}, μ^{(y)}, μ^{(z)}, ξ^{(x)}, ξ^{(y)}, ξ^{(z)}, ψ^{(x)}, ψ^{(y)}, Φ_{1}^{{[x}^{(u)}]} Φ_{2}^{{[x}^{(u)}]} \dots Φ_{N^{(u)} - 1}^{{[x}^{(u)}]} Φ_{1}^{{[x}^{(l)}]} Φ_{2}^{{[x}^{(l)}]} \dots Φ_{N^{(l)} - 1}^{{[x}^{(l)}]} \\ Φ_{1}^{{[y}^{(u)}]} Φ_{2}^{{[y}^{(u)}]} \dots Φ_{N^{(u)} - 1}^{{[y}^{(u)}]} Φ_{1}^{{[y}^{(l)}]} Φ_{2}^{{[y}^{(l)}]} \dots Φ_{N^{(l)} - 1}^{{[y}^{(l)}]} Φ_{1}^{{[z}^{(u)}]} Φ_{2}^{{[z}^{(u)}]} \dots Φ_{N^{(l)} - 1}^{{[z}^{(u)}]} Φ_{1}^{{[z}^{(l)}]} Φ_{2}^{{[z}^{(l)}]} \dots Φ_{N^{(l)} - 1}^{{[z}^{(l)}]} \\ \frac{\partial w^{(u)}}{\partial x}, \frac{\partial w^{(l)}}{\partial x}, \frac{\partial w^{(u)}}{\partial y}, \frac{\partial w^{(l)}}{\partial y}}^{T} \end{array}

and the elements of [N] depends upon the property of the material of the lamina.

Since, the last four terms in the {M} are the derivative terms and can be expressed easily in terms of displacement components $\{Θ\}$ , hence the problem associated with the C-1 continuity requirements can be avoided easily using the proposed model.

With the help of equation (6), equations (1), (2) and (3) can be re-arranged as:

U_{(x)} = a^{(1)} u^{(0)} + a^{(2)} v^{(0)} + a^{(3)} w^{(0)} + a^{(4)} φ^{(x)} + a^{(5)} φ^{(y)} + a^{(6)} φ^{(z)} + a^{(7)} u^{(u)} + a^{(8)} v^{(u)} + a^{(9)} w^{(u)} + a^{(10)} u^{(l)} + a^{(11)} v^{(l)} + a^{(12)} w^{(l)}

(7)

U_{(y)} = b^{(1)} u^{(0)} + b^{(2)} v^{(0)} + b^{(3)} w^{(0)} + b^{(4)} φ^{(x)} + b^{(5)} φ^{(y)} + b^{(6)} φ^{(z)} + b^{(7)} u^{(u)} + b^{(8)} v^{(u)} + b^{(9)} w^{(u)} + b^{(10)} u^{(l)} + b^{(11)} v^{(l)} + b^{(12)} w^{(l)}

(8)

U_{(z)} = c^{(1)} u^{(0)} + c^{(2)} v^{(0)} + c^{(3)} w^{(0)} + c^{(4)} φ^{(x)} + c^{(5)} φ^{(y)} + c^{(6)} φ^{(z)} + c^{(7)} u^{(u)} + c^{(8)} v^{(u)} + c^{(9)} w^{(u)} + c^{(10)} u^{(l)} + c^{(11)} v^{(l)} + c^{(12)} w^{(l)}

(9)

Here, the coefficients a’s, b’s and c’s are the function of material properties, thickness coordinates and the unit step function.

By imposing the additional condition as stated above, the derivative terms appearing in equation (6) are replaced with nodal variables. Hence, above three equations (1), (2) and (3) are free from any kind of C-1 continuity requirements without defining any new variable [6] or without using any penalty function [5].

In the present study, Lagrangian nine-noded C-0 isoparametric FE having twelve degrees of freedom per node is used.

The associated dof per node are

\{u^{(0)}, v^{(0)}, w^{(0)}, φ^{(x)}, φ^{(y)}, φ^{(z)}, u^{(u)}, v^{(u)}, w^{(u)}, u^{(l)}, v^{(l)}, w^{(l)}\}

The generalised displacement for an element can be written as:

\{δ\} = [N_{i}] \{δ_{i}\}

(10)

where,

N_{i}

is the shape function associated with the node and are as follows:

N_{1} = (rs (r - 1) (s - 1)) / 4; N_{2} = - ((r - 1) (s^{2} - 1)) / 2; N_{3} = (r s (r - 1) (s + 1)) / 4; N_{4} = - ((r^{2} - 1) (s + 1) s) / 2, N_{5} = (r s (r + 1) (s + 1)) / 4; N_{6} = - (r (r + 1) (s^{2} - 1)) / 2; N_{7} = (r s (r + 1) (s - 1)) / 4; N_{8} = - ((r^{2} - 1) (s - 1) s) / 2; N_{9} = ((r^{2} - 1) (s^{2} - 1))

Writing strain-displacement relationship in linear range, and with the help of equations (1) to (6), the strains can be written in form of unknowns as:

{\{\bar{ε}\}}_{6 \times 1} = {[H]}_{6 \times 36} {\{ε\}}_{36 \times 1}

(11)

where,

\{ε\} = \begin{matrix} {u^{(0)} v^{(0)} w^{(0)} φ^{(x)} φ^{(y)} φ^{(z)} u^{(u)} v^{(u)} w^{(u)} u^{(l)} v^{(l)} w^{(l)} \frac{\partial u^{(0)}}{\partial x} \frac{\partial u^{(0)}}{\partial y} \frac{\partial v^{(0)}}{\partial x} \frac{\partial v^{(0)}}{\partial y} \\ \frac{\partial w^{(0)}}{\partial x} \frac{\partial w^{(0)}}{\partial y} \frac{\partial φ^{(x)}}{\partial x} \frac{\partial φ^{(x)}}{\partial y} \frac{\partial φ^{(y)}}{\partial x} \frac{\partial φ^{(y)}}{\partial y} \frac{\partial φ^{(z)}}{\partial x} \frac{\partial φ^{(z)}}{\partial y} \frac{\partial u^{(u)}}{\partial x} \frac{\partial u^{(u)}}{\partial y} \frac{\partial v^{(u)}}{\partial x} \frac{\partial v^{(u)}}{\partial y} \frac{\partial w^{(u)}}{\partial x} \frac{\partial w^{(u)}}{\partial y} \\ \frac{\partial u^{(l)}}{\partial x} \frac{\partial u^{(l)}}{\partial y} \frac{\partial v^{(l)}}{\partial x} \frac{\partial v^{(l)}}{\partial y} \frac{\partial w^{(l)}}{\partial x} \frac{\partial w^{(l)}}{\partial y}} \end{matrix}

[H]

Using equation (10), in above equation $\{ε\}$ can be written with the help of unknowns (structural deformation)

{\{ε\}}_{6 \times 1} = {[B]}_{6 \times 108} {\{δ\}}_{108 \times 1}

(12)

where, [B] is the strain-displacement relationship on the Cartesian coordinates.

The total potential energy of the plate under hygrothermal and transverse load may can be written as:

Π_{e} = U_{s} - W_{ext}

(13)

where, U_S is the plate’s strain energy and W_ext is the energy due to external loading.

U_{s} = \frac{1}{2} \sum_{k = 1}^{n} \int \int \int {\{{\bar{ε}}_{net}\}}^{T} {[\bar{C}]}_{k} \{{\bar{ε}}_{net}\} d x d y d z

(14)

U_{s} = \frac{1}{2} \int \int [{\{ε\}}^{T} [D] \{ε\}] d x d y - \int \int {\{ε\}}^{T} [f^{hygrothermal}] d x d y + \frac{1}{2} \sum_{k = 1}^{n} \int \int \int {\{ε_{hygrothermal}\}}^{T} {[\bar{Q}]}_{k} \{ε_{hygrothermal}\} d x d y d z

(15)

where,

[D] = \sum_{k = 1}^{n} \int {[H]}^{T} {[\bar{Q}]}_{k} [H] d z

and

[f^{hygrothermal}] = \sum_{k = 1}^{n} \int {[H]}^{T} {[\bar{Q}]}_{k} [ε_{hygrothermal}] d z

While applying the principle of minimum potential energy, the last term appearing in equation (15) will vanish. The energy due to external load of intensity q = f (x, y) is

$W_{ext} = \iint wq d x d y or W_{ext} = \iint {\{δ\}}^{T} {[N_{t d}]}^{T} q d x d y$ (16)

Now, calculating the elemental potential energy by combining equations (12), (14) and (16) as

Π_{e} = \frac{1}{2} \iint {\{δ\}}^{T} {[B]}^{T} [D] [B] \{δ\} d x d y - \frac{1}{2} \iint {\{δ\}}^{T} {[B]}^{T} [f^{hygrothermal}] d x d y - \frac{1}{2} \iint {\{δ\}}^{T} {[N_{t d}]}^{T} q d x d y Π_{e} = \frac{1}{2} {\{δ\}}^{T} [K_{e}] \{δ\} - \frac{1}{2} {\{δ\}}^{T} \{P_{e}^{hygrothermal}\} - \frac{1}{2} {\{δ\}}^{T} [P_{e}]

(17)

where,

[K_{e}] = \iint {[B]}^{T} [D] [B] d x d y

is the elemental stiffness matrix,

\{P_{e}^{hygrothermal}\} = \iint {[B]}^{T} [f^{hygrothermal}] d x d y

is hygrothermally induced elemental load vector, and

[P_{e}] = \iint {[N_{t d}]}^{T} q d x d y

is elemental mechanical load vector in which

[N_{t d}]

is the shape function like matrix in which only transverse displacement at node associated terms occurs.

By minimising equation (17) with respect to $\{δ\}$ , we get

[K_{e}] \{δ\} = \{P_{e}^{hygrothermal}\} + [P_{e}]

(18)

For skew plates, the displacement components $\{δ\}$ along the skew edges (parallel to y-axis) has to be transformed to the global axis to corresponding local axis i.e. from X-Y-Z axis to X’-Y’-Z’ axis (Figure 1(b)). Hence, equation 10 has to be modified and can be re-written as:

{\{δ\}}^{T} = [N_{i}] {\{δ_{i}\}}^{T}

(19)

The above discussed code is written in MATLAB and is validated by comparing the present results with those available in literature. Some new results are also given for thick sandwich plates which will be served as benchmark for future works on thick laminated plates.

Results and discussion

Example 1: Convergence and validation studies on laminated composite plate: At first convergence and validation studies are carried on three-layered simply supported (0°/90°/0°) laminated composite square plate subjected to sinusoidal thermal loading with equal rise and fall on top and bottom surface $(Δ T (x, y, h / 2) = - Δ T (x, y, - h / 2) = T_{0} \sin \frac{π x}{a} \sin \frac{π y}{b})$ . The material properties are taken as E₁ = 172.4 GPa, E₂ = E₃ = 6.89 GPa, G₁₂ = G₁₃ = 3.45 GPa, G₂₃ = 1.378 GPa, ν₁₂ = ν₁₃ = ν₂₃ = 0.25, α₁ = 1 × 10⁻⁶ K⁻¹, α₂ = 1125 × 10⁻⁶ K⁻¹. All layers are of same thickness. Results for non-dimensional deflection $(\bar{w} = \frac{w h^{3}}{T_{0} α_{1} a^{4}})$ and stresses $(({\bar{σ}}^{x x}, {\bar{σ}}^{y y}, {\bar{σ}}^{x y}) = \frac{h^{2} (σ^{x x}, σ^{y y}, σ^{x y})}{E_{2} T_{0} α_{1} a^{2}}; ({\bar{σ}}^{x z}, {\bar{σ}}^{y z}) = \frac{h (σ^{x z}, σ^{y z})}{E_{2} T_{0} α_{1} a}; ({\bar{σ}}^{z z} = \frac{σ^{z z}}{E_{2} T_{0} α_{1}}))$ are presented in Table 1. It can be seen that $\bar{w}$ converges at mesh size of 10 × 10 but in order to predict stresses more correctly, mesh size of 12 × 12 is used in further studies.

Same problem is already solved by Khandelwal et al. [5] using C-1 HOZT (9-noded FE with 7-degrees of freedom per node) and Kant et al. [8] using semi-analytical solution. Earlier, Bhaskar et al. [9] published elasticity solutions for the same problem. Singh and Chakrabarti [10] analysed similar plate with higher-order shear deformation theory (HOSDT) with 9-noded FE having 7 degrees of freedom per node while Patel et al. [11] used 8-noded FE having 13 degrees of freedom per node. Along with these references, FE based (nine noded FE having eleven degrees of freedom per node) C-0 HOZT developed by Chalak et al. [7] is modified for application under hygro-thermo-mechanical conditions by present authors and is also used for comparison. C-0 theory proposed by Chalak et al. [7] included transverse displacement variation is able to predict more accurate results though Khandelwal et al. [5] also used HOZT assuming constant transverse displacement field. Hence, it is recommended to introduce appropriate transverse displacement field during formulation of HOZT. Incorporation of zigzag field in same helps in predicting transverse stresses more effectively.

It can be seen that the results predicted using present model are in good agreement when compared with elasticity results given by Bhaskar et al. [9]. Incorporation of transverse displacement field instead of assuming it constant helps in predicting results with more accuracy. Along with the same, introduction of zigzag effects in same further improves the performance of model especially for thick plates. Without using any post-processing technique, present model is able to predict inter-laminar transverse normal and shear-stress continuity at interface (Figure 6) as used by Khandelwal et al. [5] in their work. Figures 3 to 6 shows variation of stress $({\bar{σ}}^{x x}, {\bar{σ}}^{y y}, {\bar{σ}}^{x y}, {\bar{σ}}^{y z})$ for same plate with h/a = 0.02, 0.05 and 0.10. The results obtained by HOZT reported by Chalak et al. [5] also predicts good results but not able to predict transverse stresses more correctly as predicted by the present theory. In all types of stresses, it can be seen that large amount of stresses borne by top and bottom layers. In-plane normal stresses exhibit almost linear variation in all the layers with zero at mid of the plate while parabolic or higher order variation is observed for shear stress variation along the thickness of the plate for all h/a values. However, in case of in-plane shear stress it is seen that with increase in thickness of the plate, the non-linearity in stress distribution along the thickness increases. Hence, the present model is able to predict the highly non-linear behavior effectively. Symmetry in all types of variations with reference to mid plane is observed as expected.

Table 2 shows the non-dimensional deflection and stresses for the same plate subjected to three different loadings namely thermo-mechanical (TM), hygro-thermal (HT) and hygro-thermo-mechanical (HTM) loading. The value for β₂ is taken as 0.44 (wt. % water)⁻¹. Results are reported for two different end conditions: all sides simply supported (SSSS) and all sides clamped (CCCC). Temperature distribution is taken as $Δ T (x, y, h / 2) = - Δ T (x, y, - h / 2) = T_{0} \sin \frac{π x}{a} \sin \frac{π y}{b}$ , moisture distribution as $Δ c (x, y, \pm h / 2) = C_{0} \sin \frac{π x}{a} \sin \frac{π y}{b}$ and mechanical loading as $q (x, y, 0) = q_{0} \sin \frac{π x}{a} \sin \frac{π y}{b}$ . For converting dimensional values to non-dimensional form, following relations are used:

Table 1.

Variation of non-dimensional deflection and stresses for three-layered simply supported laminated composite plate subjected to thermal loading.^a

h/a	Source	$\bar{w}$ (a/2, b/2, 0)	± ${\bar{σ}}^{x x}$ (a/2, b/2, ±h/2)	± ${\bar{σ}}^{y y}$ (a/2, b/2, $\mp$ h/2)	± ${\bar{σ}}^{x y}$ (0, 0, $\mp$ h/2)	${\bar{σ}}^{x z}$ (0, b/2, $\mp$ h/6)	– ${\bar{σ}}^{y z}$ (a/2, 0, $\mp$ h/6)
0.01	Present (4 × 4)	0.001042	0.09821	0.10833	0.00509	0.06632	0.06733
	Present (8 × 8)	0.001031	0.09734	0.10725	0.00505	0.06795	0.06823
	Present (12 × 12)	0.001024	0.09689	0.10673	0.00503	0.06831	0.06915
	Present (16 × 16)	0.001024	0.09657	0.10654	0.00502	0.06973	0.06966
	Singh and Chakrabarti [10]	0.001030	0.09716	0.10750	0.00507	–	–
	Chalak et al. [7]	0.001024	0.09701	0.10690	0.00506	0.06971	0.06963
	Khandelwal et al. [5]	0.001023	0.09688	$0$ .10644	0.00506	–	–
	Bhaskar et al. [9]	0.001026	0.09654	0.10650	0.00505	0.07073	0.0708
	% Error	–0.194	0.031	0.037	–0.594	–1.41	–1.61
0.02	Present	0.004194	0.38730	0.42556	0.02047	0.27802	0.2785
	Patel et al. [11]	0.004200	–	–	–	–	–
	Singh and Chakrabarti [10]	0.004180	0.38984	0.42960	0.02063	–	–
	Chalak et al. [7]	0.004194	0.38728	0.42546	0.02041	0.27793	0.2776
	Khandelwal et al. [5]	0.004155	0.38894	0.42513	0.02063	–	–
	Bhaskar et al. [9]	0.004200	0.38700	0.42520	0.02056	0.28140	0.2826
	% Error	–0.142	0.077	0.084	–0.437	–1.20	–1.45
0.05	Present	0.03027	2.4573	2.6302	0.1429	1.6828	1.7209
	Patel et al. [11]	0.03027	–	–	–	–	–
	Singh and Chakrabarti [10]	0.02825	2.4127	2.6475	0.1386	–	–
	Chalak et al. [7]	0.03022	2.4554	2.6270	0.1426	1.6800	1.7103
	Khandelwal et al. [5]	0.02857	2.4388	2.6299	0.1422	–	–
	Kant et al. [8]	0.03030	2.4550	2.6281	0.1438	1.6992	1.7381
	Bhaskar et al. [9]	0.03030	2.4550	2.6275	0.1437	1.6990	1.7380
	% Error	–0.099	0.093	0.102	–0.556	–0.953	–0.983
0.10	Present	0.1738	10.2700	10.1521	0.7575	6.0019	6.5528
	Patel et al. [11]	0.1737	–	–	–	–	–
	Singh and Chakrabarti [10]	0.1413	09.4560	10.2900	0.6843	–	–
	Chalak et al. [7]	0.1731	10.1027	10.0312	0.7437	6.1234	6.6464
	Khandelwal et al. [5]	0.1463	09.6775	10.1876	0.7323	–	–
	Kant et al. [8]	0.1739	10.2630	10.1436	0.7629	6.0541	6.6008
	Bhaskar et al. [9]	0.1739	10.2600	10.1400	0.7629	6.0510	6.6010
	% Error	–0.057	0.097	0.119	–0.707	–0.811	–0.730
0.25	Present	2.6688	74.0117	53.5754	9.7221	21.0858	32.0269
	Patel et al. [11]	2.6456	–	–	–	–	–
	Chalak et al. [7]	2.4132	70.8591	54.9267	9.0145	21.1234	32.1047
	Khandelwal et al. [5]	1.6045	52.4943	55.7223	8.3921	–	–
	Kant et al. [8]	2.6679	73.9496	53.5062	9.8113	21.2030	32.1750
	Bhaskar et al. [9]	2.6680	73.9375	53.5059	9.8113	21.2025	32.1750
	% Error	0.0299	0.100	0.130	–0.909	–0.550	–0.461

$% Error = \frac{Present results - Bhaskar e t a l . [9] results}{Bhaskar e t a l . [9] results} \times 100$ .

Table 2.

Variation of non-dimensional deflection and stresses for three layered laminated composite plate with different end conditions (T₀ = 100, C₀ = 1, q₀ = 100).

End condition		SSSS				CCCC
h/a	Source	$\bar{w}$ (a/2, b/2, 0)	± ${\bar{σ}}^{x x}$ (a/2, b/2, ±h/2)	± ${\bar{σ}}^{x y}$ (0, 0, $\mp$ h/2)	${\bar{σ}}^{x z}$ (0, b/2, $\mp$ h/6)	$\bar{w}$ (a/2, b/2, 0)	± ${\bar{σ}}^{x x}$ (a/2, b/2, ±h/2)	± ${\bar{σ}}^{x y}$ (0, 0, $\mp$ h/2)	${\bar{σ}}^{x z}$ (0, b/2, $\mp$ h/6)
Thermo-mechanical loading
0.01	Present	0.0102	1.2101	0.0516	1.0836	0.0026	0.4733	0	0.7630
0.01	Khandelwal et al. [5]	0.0107	1.2758	0.0530	–	–	–	–	–
0.02	Present	0.0171	1.8425	0.0832	1.5942	0.0047	0.5844	0	0.7177
0.02	Khandelwal et al. [5]	0.0172	1.8662	0.0848	–	–	–	–	–
0.05	Present	0.0707	5.9855	0.4092	5.2689	0.0243	1.4352	0	1.1906
0.05	Khandelwal et al. [5]	0.0675	5.9795	5.1985	–	–	–	–	–
0.10	Present	0.3241	20.8440	1.9756	14.6892	0.1378	3.5608	0	1.7973
0.10	Khandelwal et al. [5]	0.3077	19.4211	1.4681	–	–	–	–	–
0.25	Present	3.7668	109.2762	29.1537	12.1525	1.4254	82.6900	0	2.4698
0.25	Khandelwal et al. [5]	3.2495	104.9886	16.7842	–	–	–	–	–
Hygro-thermal loading
0.01	Present	0.0411	0.0371	0.0020	0.0332	0.0162	0.0075	0	0.0228
0.02	Present	0.2087	0.1706	0.0095	0.1469	0.0473	0.0386	0	0.0687
0.05	Present	1.2759	1.2152	0.0617	0.8106	0.4287	0.2057	0	0.1415
0.10	Present	6.0033	4.0120	0.3684	2.5567	2.4360	0.6531	0	0.1959
0.25	Present	66.9250	23.9953	6.3982	9.1575	25.4481	2.2453	0	0.6264
Hygro-thermo-mechanical loading
0.01	Present	0.0083	0.0751	0.0041	0.0632	0.0021	0.0152	0	0.0464
0.02	Present	0.0338	0.3022	0.0165	0.2549	0.0096	0.0605	0	0.1181
0.05	Present	0.2352	1.9315	0.1188	1.5621	0.0797	0.3721	0	0.2631
0.10	Present	1.2007	8.7450	1.0737	5.4618	0.6109	1.3070	0	0.3419
0.25	Present	12.9865	41.9912	11.2662	15.7820	4.6900	9.5966	0	9.5295

Table 3.

Variation of non-dimensional deflection and stresses for four layered (0°/90°/0°/90°) laminated composite plate under thermal loading.

h/a	Source	$\bar{w}$ (a/2, b/2, 0)	– ${\bar{σ}}^{x x}$ (a/2, b/2, h/2)	${\bar{σ}}^{y y}$ (a/2, b/2, h/2)	$\mp {\bar{σ}}^{x y}$ (0, 0, ±h/2)	${\bar{σ}}^{x z}$ (0, b/2, 0.2h)	${\bar{σ}}^{y z}$ (a/2, 0, 0.2h)
0.01	Present	0.0131	0.0058⁺ 0.0036^–	0.0036⁺ 0.0058^–	0.0064	–0.0013⁺ 0.0014^–	0.0014⁺ –0.0013^–
0.01	Khandelwal et al. [5]	0.0131	0.0056⁺ 0.0036^–	0.0036⁺ 0.0056^–	0.0065	–	–
0.02	Present	0.0525	0.0230⁺ 0.0143^–	0.0143⁺ 0.0230^–	0.0258	–0.0196⁺ 0.0135^–	0.0135⁺ –0.0196^–
0.02	Khandelwal et al. [5]	0.0524	0.0222⁺ 0.0145^–	0.0145⁺ 0.0222^–	0.0260	–	–
0.05	Present	0.3392	0.1391⁺ 0.0917^–	0.0917⁺ 0.1391^–	0.1631	–0.0756⁺ 0.0509^–	0.0509⁺ –0.0756^–
	Khandelwal et al. [5]	0.3286	0.1388⁺ 0.0910^–	0.0910⁺ 0.1338^–	0.1630	–	–
	Kant et al. [8]	0.3422	0.1385⁺ 0.0903^–	0.0903⁺ 0.1385^–	0.1643	–0.0714⁺ 0.0476^–	0.0476⁺ –0.0714^–
0.10	Present	1.5106	0.5503⁺ 0.3723^–	0.3723⁺ 0.5503^–	0.6713	–0.2967⁺ 0.1893^–	0.1893⁺ –0.2967^–
	Khandelwal et al. [5]	1.3297	0.5542⁺ 0.3714^–	0.5542⁺ 0.3714^–	0.6587	–	–
	Kant et al. [8]	1.5447	0.5479⁺ 0.3719^–	0.3719⁺ 0.5479^–	0.6894	–0.2818⁺ 0.1870^–	0.1870⁺ –0.2818^–
0.25	Present	8.9106	2.1035⁺ 1.6512^–	1.6512⁺ 2.1035^–	3.5195	–1.0745⁺ 0.6993^–	0.6993⁺ –1.0745^–
	Khandelwal et al. [5]	8.9133	3.4326⁺ 2.6136^–	2.6136⁺ 3.4326^–	4.3811	–	–
	Kant et al. [8]	8.9098	2.1008⁺ 1.6496^–	1.6496⁺ 2.1008^–	3.2408	–1.0726⁺ 0.6986^–	0.6986⁺ –1.0726^–

For TM loading:

\bar{w} = \frac{w h^{3}}{T_{0} α_{1} a^{4}} + \frac{w E_{2} h^{3}}{q_{0} a^{4}}; ({\bar{σ}}^{x x}, {\bar{σ}}^{y y}, {\bar{σ}}^{x y}) = \frac{h^{2} (σ^{x x}, σ^{y y}, σ^{x y})}{E_{2} T_{0} α_{1} a^{2}} + \frac{h^{2} (σ^{x x}, σ^{y y}, σ^{x y})}{q_{0} a^{2}}; ({\bar{σ}}^{x z}, {\bar{σ}}^{y z}) = \frac{h (σ^{x z}, σ^{y z})}{E_{2} T_{0} α_{1} a} + \frac{h (σ^{x z}, σ^{y z})}{q_{0} a} .

For HT loading: $\bar{w} = \frac{w h^{3}}{10 T_{0} α_{1} a^{4}}; ({\bar{σ}}^{x x}, {\bar{σ}}^{y y}, {\bar{σ}}^{x y}) = \frac{h^{2} (σ^{x x}, σ^{y y}, σ^{x y})}{10^{3} E_{2} T_{0} α_{1} a^{2}}; ({\bar{σ}}^{x z}, {\bar{σ}}^{y z}) = \frac{h (σ^{x z}, σ^{y z})}{10^{3} E_{2} T_{0} α_{1} a}$ .

For HTM loading:

\bar{w} = \frac{1}{10^{2}} (\frac{w h^{3}}{T_{0} α_{1} a^{4}} + \frac{w E_{2} h^{3}}{q_{0} a^{4}}); ({\bar{σ}}^{x x}, {\bar{σ}}^{y y}, {\bar{σ}}^{x y}) = \frac{1}{10^{3}} (\frac{h^{2} (σ^{x x}, σ^{y y}, σ^{x y})}{E_{2} T_{0} α_{1} a^{2}} + \frac{h^{2} (σ^{x x}, σ^{y y}, σ^{x y})}{q_{0} a^{2}}); ({\bar{σ}}^{x z}, {\bar{σ}}^{y z}) = \frac{1}{10^{3}} (\frac{h (σ^{x z}, σ^{y z})}{E_{2} T_{0} α_{1} a} + \frac{h (σ^{x z}, σ^{y z})}{q_{0} a}) .

The present results for TM loading are in good agreement when compared with the results reported by Khandelwal et al. [5]. It can be seen that the maximum deflection and stresses are seen for combined HTM loading and then in decreasing order for HT, TM while minimum for thermal loading only. With increase in h/a value non-dimensional deflection and stresses increases. In-plane shear stresses are zero in case of CCCC plate as expected. However, non-dimensional deflection and stresses in case of CCCC plate are much lesser than SSSS plate.

Example 2: Unsymmetric laminated composite plate: In this example, 4-layered (0°/90°/0°/90°) unsymmetric cross-ply laminated composite SSSS plate is analysed under thermal loading. Material properties are taken as: E₁ = 150 GPa, E₂ = E₃ = 10 GPa, G₁₂ = G₁₃ = 5 GPa, G₂₃ = 3.378 GPa, ν₁₂ = ν₁₃ = ν₂₃ = 0.3, α₁ = 0.139e-6 K⁻¹, α₂ = 9e-6 K⁻¹. All layers are of same thickness. Results for non-dimensional deflection $(\bar{w} = \frac{100 \times w h^{3}}{T_{0} α_{1} a^{4}})$ and stresses $(({\bar{σ}}^{x x}, {\bar{σ}}^{y y}, {\bar{σ}}^{x y}) = \frac{h^{2} (σ^{x x}, σ^{y y}, 10 \times σ^{x y})}{E_{2} T_{0} α_{1} a^{2}}; ({\bar{σ}}^{x z}, {\bar{σ}}^{y z}) = \frac{h (σ^{x z}, σ^{y z})}{E_{2} T_{0} α_{1} a})$ are presented in Table 3. Superscript ‘+’ denotes stress at respective height above reference plane while superscript ‘−’ represents the value of stress at respective height below the reference plane. Present results are compared with the results reported by Kant et al. [8] and Khandelwal et al. [5] and are found in good agreement. The stresses predicted by present model especially the transverse stresses are more accurate than those predicted by Khandelwal et al. [5].

Example 3: 3-layered skew laminated composite plate: A 3-layered (0°/90°/0°) rhombic shaped SSSS skew laminated composite plate (a = b) made up of material A is analysed under thermal, HT and HTM conditions. Since, no results are available in the literature on the analysis of skew laminates under thermal, HT and HTM loading hence, C-0 HOZT given by Chalak et al. [12] is modified for predicting behavior of skew laminates under respective loading. Results for non-dimensional deflection and stresses are reported in Table 4 for different values of skew angle (δ) and thickness ratio (h/a) of the plate under different loading conditions. Relations used for converting dimensional parameters to non-dimensional form are same as used in example 1 for respective loading case. It can be seen that the present results are in good agreement with those predicted by using HOZT given by Chalak et al. [12]. The behavior of skew plate widely depends upon the skew angle of the plate. It is noticed that with increase in δ of the plate, non-dimensional deflection and stresses decreases. In case of transverse shear stresses, nature changes. ${\bar{σ}}^{x z}$ changes its nature when δ increases from 15° to 45° while the nature of ${\bar{σ}}^{y z}$ changes when δ increases from 30° to 45° in all the three types of loading. Therefore, if skew laminated composite plates used in structure expected to be acted by environmental loads, transverse shear stresses must be analysed accurately.

Table 4.

Variation of non-dimensional deflection and stresses for three layered (0°/90°/0°) rhombic shaped SSSS skew laminated composite plate under different loading.

h/a	δ	Source	$\bar{w}$ (a/2, b/2, 0)	$\pm {\bar{σ}}^{x x}$ (a/2, b/2, ±h/2)	$\mp {\bar{σ}}^{y y}$ (a/2, b/2, ±h/2)	$\mp {\bar{σ}}^{x y}$ (0, 0, ±h/2)	${\bar{σ}}^{x z}$ (0, b/2, ±h/6)	– ${\bar{σ}}^{y z}$ (a/2, 0, ±h/6)
Thermal loading
0.01	15°	Present	9.88e-4	0.0875	0.1048	0.0030	0.0428	0.0855
	15°	Chalak et al. [13]	1.31e-3	0.0901	0.1432	0.0042	0.0597	0.0993
	30°	Present	7.83e-4	0.0689	0.0851	1.85e-3	–0.0786	0.0186
	30°	Chalak et al. [13]	8.21e-4	0.0728	0.0903	2.13e-3	–0.0906	0.0229
	45°	Present	4.99e-4	0.0472	0.0604	4.12e-4	–0.1897	–0.0705
	45°	Chalak et al. [13]	5.40e-4	0.0505	0.0642	4.96e-4	–0.2611	–0.0820
0.05	15°	Present	0.0283	2.2756	2.5984	0.0756	0.4592	1.4203
	15°	Chalak et al. [13]	0.0305	2.3961	2.7163	0.0839	0.5160	1.9962
	30°	Present	0.0213	1.8292	2.0078	0.0109	–7.3730	0.9985
	30°	Chalak et al. [13]	0.0256	2.0866	2.3191	0.0251	–8.1059	1.5361
	45°	Present	0.0201	1.1638	1.8091	0.0042	–25.067	–1.0642
	45°	Chalak et al. [13]	0.0326	1.3960	2.1103	0.0036	–27.361	–1.6867
0.10	15°	Present	0.1432	9.1407	10.0138	0.0459	2.6638	6.0108
	15°	Chalak et al. [13]	0.1547	9.1523	10.0241	0.0574	3.7686	6.3235
	30°	Present	0.1101	7.6393	8.1168	0.0950	–18.478	2.6638
	30°	Chalak et al. [13]	0.1216	7.6424	8.1251	0.1081	–19.590	3.0254
	45°	Present	0.0624	3.9918	5.7724	0.0600	–62.193	–3.9461
	45°	Chalak et al. [13]	0.0673	4.0154	5.7905	0.0715	–63.859	–4.1866
Hygro-thermal loading
0.01	15°	Present	0.0381	0.0345	0.0407	0.0013	0.0173	0.0347
	15°	Chalak et al. [13]	0.0506	0.0472	0.0541	0.0025	0.0291	0.0468
	30°	Present	0.0303	0.0279	0.0330	7.09e-4	–0.0302	0.0077
	30°	Chalak et al. [13]	0.0432	0.0395	0.0499	8.25e-4	–0.0427	0.0090
	45°	Present	0.0193	0.0193	0.0235	1.53e-4	–0.4271	–0.0283
	45°	Chalak et al. [13]	0.0318	0.0261	0.0400	2.81e-4	–0.5613	–0.0426
0.05	15°	Present	1.0952	0.9028	1.0065	0.0291	0.1765	0.8127
	15°	Chalak et al. [13]	1.2392	1.0195	1.2560	0.0400	0.2991	1.3370
	30°	Present	0.8648	0.7221	0.8189	0.0083	–3.2907	0.3380
	30°	Chalak et al. [13]	0.9991	0.8518	0.9443	0.0095	–3.5718	0.9064
	45°	Present	0.5302	0.4626	0.5849	0.0011	–10.563	–0.4260
	45°	Chalak et al. [13]	0.6719	0.5939	0.7003	0.0024	–11.981	–1.0057
0.10	15°	Present	5.5583	3.5698	3.9706	0.1533	1.0274	3.0089
	15°	Chalak et al. [13]	5.6072	3.7215	4.1863	0.1697	1.5591	3.6304
	30°	Present	4.2771	2.8331	3.1546	0.0362	–7.2569	1.0580
	30°	Chalak et al. [13]	4.3815	3.1007	3.3790	0.0511	–7.8814	1.7133
	45°	Present	2.4272	1.6620	2.2441	0.0281	–24.147	–1.5589
	45°	Chalak et al. [13]	2.5560	1.9034	2.3867	0.0407	–25.908	–2.0917
Hygro-thermo-mechanical loading
0.01	15°	Present	0.0772	0.0699	0.0814	0.0026	0.0338	0.0695
	15°	Chalak et al. [13]	0.0651	0.0785	0.0917	0.0031	0.0359	0.0742
	30°	Present	0.0610	0.0564	0.0661	0.0014	–0.0610	0.0155
	30°	Chalak et al. [13]	0.0573	0.0622	0.0700	0.0019	–0.0532	0.0273
	45°	Present	0.0038	0.0389	0.0487	3.09e-4	–1.4850	–0.0567
	45°	Chalak et al. [13]	0.0019	0.0427	0.0518	3.08e-4	–1.3182	–0.0422
0.05	15°	Present	0.2191	1.8067	2.0130	0.0582	0.3532	1.6256
	15°	Chalak et al. [13]	0.3054	1.9930	2.1292	0.0661	0.4464	1.7997
	30°	Present	0.1730	1.4449	1.6377	0.0166	–0.6580	0.6761
	30°	Chalak et al. [13]	0.2008	1.5636	1.7462	0.0249	–0.5323	0.7004
	45°	Present	0.1060	9.2565	1.1699	0.0023	–21.125	–0.8521
	45°	Chalak et al. [13]	0.2099	9.3612	1.2593	0.0034	–19.199	–0.7306
0.10	15°	Present	1.1178	7.2446	7.7814	0.3067	2.0554	6.0178
	15°	Chalak et al. [13]	1.3442	7.4301	7.9645	0.4122	3.5513	6.4618
	30°	Present	0.8555	5.6670	6.3092	0.0725	–14.513	2.1160
	30°	Chalak et al. [13]	0.9867	5.7924	6.4117	0.0843	–13.040	2.9781
	45°	Present	0.4854	3.3244	4.4882	0.0456	–48.834	–3.1178
	45°	Chalak et al. [13]	0.6003	3.5007	4.6010	0.0549	–47.681	–2.3419

Table 5 shows the results for non-dimensional deflection and stresses for same plate of parallelogram shape under HTM load for two values of aspect ratio (a/b) 0.50 and 2.0. From Table 5, it can be seen that along with δ, amount of non-dimensional deflection and stress along with its nature also depends upon value of a/b. From Tables 4 and 5, it can be noticed that with increase in a/b value, non-dimensional deflection increases. In-plane stress along the larger edge of the plate is more than the in-plane stress along shorted edge at centre of the plate. Decrease in value of stress for plate with a/b = 0.50 is more than the amount fall for plate with a/b = 2.0. For plate with aa/b = 0.50, the plane of deflection of plate reverses at δ = 45°. However, such reversal in the direction of the bending of plate is not seen when a/b = 2.0.

Table 5.

Variation of non-dimensional deflection and stresses for three layered (0°/90°/0°) parallelogram shaped SSSS skew laminated composite plate under HTM loading.

a/b	h/a	δ	$\bar{w}$ (a/2, b/2, 0)	$\pm {\bar{σ}}^{x x}$ (a/2, b/2, ±h/2)	$\mp {\bar{σ}}^{y y}$ (a/2, b/2, ±h/2)	$\mp {\bar{σ}}^{x y}$ (0, 0, ±h/2)	${\bar{σ}}^{x z}$ (0, b/2, ±h/6)	− ${\bar{σ}}^{y z}$ (a/2, 0, ±h/6)
0.50	0.01	0°	0.0045	0.0273	0.0922	0.0011	0.0117	–0.0351
		15°	0.0033	0.0195	0.0698	9.60e-4	–0.2250	–0.0041
		30°	5.91e-4	0.0023	0.0182	8.82e-4	–1.1635	0.0596
		45°	–0.0017	–0.0082	0.0277	6.36e-4	–1.8076	0.0596
	0.10	0°	0.5610	2.7552	9.1408	0.2185	1.0866	–1.4150
		15°	0.4123	1.9588	6.9272	0.0936	–3.9826	–0.7808
		30°	0.0676	0.1273	2.1479	0.0318	–29.052	3.8360
		45°	–0.1078	–0.5086	–3.0894	0.0373	–57.374	2.8120
2.0	0.01	0°	0.0093	0.0922	0.0692	0.0087	0.0813	–0.0467
		15°	0.0090	0.0908	0.0674	0.0052	0.0802	–0.1095
		30°	0.0078	0.0848	0.0628	0.0024	–0.1922	–0.1185
		45°	0.0059	0.0777	0.0555	2.91e-4	–0.7515	–0.1074
	0.10	0°	1.3370	8.4620	6.0606	1.4337	5.8490	–8.3954
		15°	1.2752	8.2546	5.8324	0.6729	27.268	–10.713
		30°	1.0734	7.1402	5.3231	0.2234	–4.7425	–10.320
		45°	0.7606	5.1314	4.6438	0.1465	–26.188	–8.2608

Example 4: 3-layered laminated sandwich plate: In this example, analysis of 3-layered (0°/core/0°) SSSS square laminated sandwich plate is studied under thermal loading same as used in example 1. Thickness of core is taken as 0.8 times the thickness of plate while the face sheets are of same thickness. Material properties of the plate are taken as: for face layer: E₁ = 172.4 GPa, E₂ = E₃ = 6.89 GPa, G₁₂ = G₁₃ = 3.45 GPa, G₂₃ = 1.378 GPa, ν₁₂ = ν₁₃ = ν₂₃ = 0.25, α₁ = α₃ = 0.1e-5 K⁻¹, α₂ = 2e-5 K⁻¹; for core: E₁ = E₂ = 0.276 GPa, E₃ = 3.45 GPa, G₁₂ = 0.1104 GPa, G₁₃ = G₂₃ = 0.414 GPa, ν₁₂ = ν₁₃ = ν₂₃ = 0.25, α₁ = α₃ = 0.1e-6 K⁻¹, α₂ = 0.2e-5 K⁻¹. The results for non-dimensional deflection and stresses are reported in Table 6. Predicted results are compared with those given by Khandelwal et al. [5] and Kant et al. [8] and are found in good agreement. Present model is able to predict the results especially for transverse stresses more efficiently because of incorporation of transverse displacement field.

Table 6.

Variation of non-dimensional deflection and stresses for three layered (0°/core/0°) square laminated sandwich plate under thermal loading.

h/a	Source	$100 \times \bar{w}$ (a/2, b/2, 0)	$\pm {\bar{σ}}^{x x}$ (a/2, b/2, ±h/2)	$\mp {\bar{σ}}^{y y}$ (a/2, b/2, ±h/2)	$\mp {10 \times \bar{σ}}^{x y}$ (0, 0, ±h/2)	${\bar{σ}}^{x z}$ (0, b/2, ±h/6)	− ${\bar{σ}}^{y z}$ (a/2, 0, ±h/6)
0.01	Present	0.0034	0.0015	0.0017	0.0002	0.0004	0.0005
0.01	Khandelwal et al. [5]	0.0035	0.0015	0.0018	0.0002	–	–
0.02	Present	0.0146	0.0057	0.0072	0.0007	0.0009	0.0009
0.02	Khandelwal et al. [5]	0.0145	0.0058	0.0072	0.0007	–	–
0.05	Present	0.1001	0.0356	0.0042	0.0050	0.0108	0.0111
	Khandelwal et al. [5]	0.0998	0.0357	0.0441	0.0050	–	–
	Kant et al. [8]	0.1002	0.0351	0.0441	0.0049	0.0109	0.0111
0.10	Present	0.5128	0.1326	0.1650	0.0253	0.0378	0.0387
	Khandelwal et al. [5]	0.5094	0.1339	0.1649	0.0254	–	–
	Kant et al. [8]	0.5139	0.1317	0.1651	0.0253	0.0382	0.0394
0.25	Present	5.5733	0.6305	0.7899	0.2747	0.1256	0.1420
	Khandelwal et al. [5]	5.4775	0.6332	0.7890	0.2756	–	–
	Kant et al. [8]	5.6024	0.6271	0.7904	0.2744	0.1259	0.1425

Table 7 shows the variation of non-dimensional deflection and stresses for same plate with different skew angles (δ) under HTM loading. It can be seen that with increase in value of δ, non-dimensional deflection and stresses decreases. But for very thin skew sandwich plate (h/a = 0.01), ${\bar{σ}}^{x z}$ increases with increase in skew angle while for remaining values of h/a, same decreases with increase in skew angle of the plate.

Table 7.

Variation of non-dimensional deflection and stresses for three layered (0°/core/0°) rhombic laminated sandwich plate under HTM loading.

h/a	δ	$\bar{w}$ (a/2, b/2, 0)	$\pm {\bar{σ}}^{x x}$ (a/2, b/2, ±h/2)	$\mp {\bar{σ}}^{y y}$ (a/2, b/2, ±h/2)	$\mp {\bar{σ}}^{x y}$ (0, 0, ±h/2)	${\bar{σ}}^{x z}$ (0, b/2, ±h/6)	− ${\bar{σ}}^{y z}$ (a/2, 0, ±h/6)
0.01	0°	0.0088	0.0823	0.0873	0.0043	0.0105	0.0241
	10°	0.0086	0.0800	0.0859	0.0034	0.0435	0.0236
	20°	0.0080	0.0742	0.0771	0.0271	0.0793	0.0182
	30°	0.0069	0.0647	0.0642	0.0020	0.0967	0.0094
0.05	0°	0.2749	1.9951	2.1268	0.1329	0.4974	0.5201
	10°	0.2680	1.9524	2.0501	0.0923	0.5720	0.5032
	20°	0.2486	1.8260	1.8388	0.0606	0.5120	0.3945
	30°	0.2173	1.6259	1.5302	0.0362	0.2249	0.1925
0.10	0°	1.6480	7.3822	7.8948	0.9151	1.7532	1.7328
	10°	1.6063	7.2258	7.5944	0.6029	1.6442	1.4348
	20°	1.4785	6.7272	6.7640	0.3727	1.4304	0.8773
	30°	1.2680	5.8984	5.5532	0.1964	0.9919	0.3150
0.25	0°	21.4112	35.1650	37.2296	18.2329	5.7115	6.6399
	10°	20.7084	34.2671	35.3018	12.6690	5.8771	6.5528
	20°	18.4528	30.9560	30.5896	7.6993	4.3847	4.6684
	30°	14.9531	25.2685	24.2543	3.9898	1.3780	1.5883

Conclusion

A new C-0 HOZT is proposed in this work including transverse deformation under hygro-thermo-mechanical loading for laminated composite and sandwich plates. The theory satisfies transverse stress continuity condition at interfaces along with zero value of transverse stresses at top and bottom surfaces of plate. Proposed theory is free from any penalty function or C-1 requirements. Nine-noded C-0 FE having 12 d.o.f. per node is used. During the analysis, following conclusion are drawn:

Inclusion of transverse displacement filed during analysis of laminated composite and sandwich plates helps to predict behavior of the same more efficiently especially for thick one.

Introduction of zigzag field also helps in predicting behavior of laminate composite and sandwich structures more efficiently especially for thicker one.

With increase in thickness ratio of the plate, non-dimensional deflection and stresses increases for both laminated composite and sandwich plates.

Minimum non-dimensional deflection and stresses are observed when only thermal load acts on the plate while maximum when all three loads i.e., hygro, thermal and mechanical loads act on plate. Increasing order of non-dimensional deflection and stresses with respect to loading is: thermal < hygro-thermal < hygro-thermo-mechanical load.

The behavior of skew plates depends widely on its skew angle as the nature of transverse shear stresses changes their nature with increase in skew angle of the plate.

Along with skew angle, nature and amount of stresses and deflection in plate also depends upon the geometry of plate.

Therefore, the present model can be applied for various research, industrial applications for laminated composite and sandwich plates during analysis stage.

Scope of future research and social application

The present research can be extended for carrying out the following studies:

Present study can be extended for the analysis of laminated composite and sandwich plates containing de-laminations, imperfections or micro-failures

Present formulation may be extended for the free vibration and buckling analysis of laminated composite and sandwich plates under hygrothermal conditions

Crack initiation and propagation along with failure mechanism can also be studied.

In terms of social application, present model can be used in industries dealing with laminates where the structures are expected to be served under changing environmental conditions. Potential fields for the same are: aeronautical, aerospace, marine, automobile etc. where the structures serve under wide range of thermal and/or moisture concentration.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is financially supported jointly by MHRD, GoI and NIT Kurukshetra through PhD scholarship grant (2K17/NITK/PHD/6170004).

ORCID iD

Aman Garg

References

Noor

Burton

WS.

Computational models for high-temperature multilayered composite plates and shells. Appl Mech Rev 1992; 45: 419–446.

Noor

Burton

WS.

Steady-state heat conduction in multilayered composite plates and shells. Comput Struct 1991; 39: 185–193.

Tauchert

TR.

Thermally induced flexure, buckling, and vibration of plates. Appl Mech Rev 1991; 44: 347–360.

Garg

Chalak

HD.

A review on analysis of laminated composite and sandwich structures under hygrothermal conditions. Thin-Walled Struct 2019; 142: 205–226.

Khandelwal

Chakrabarti

Bhargava

Efficient 2D thermo-mechanical analysis of composites and sandwich laminates. Mech Adv Mater Struct 2018; 26: 526–538.

Garg

Karkhanis

Sahoo

, et al. Trigonometric zigzag theory for static analysis of laminated composite and sandwich plates under hygro-thermo-mechanical loading. Compos Struct 2019; 209: 460–471.

Chalak

Chakrabarti

Iqbal

, et al. An improved C0 FE model for the analysis of laminated sandwich plate with soft core. Finite Elem Anal Des 2012; 56: 20–31.

Kant

Pendhari

Desai

YM.

An efficient semi-analytical model for composite and sandwich plates subjected to thermal load. J Thermal Stresses 2007; 31: 77–103.

Bhaskar

Varadan

Ali

Thermoelastic solutions for orthotropic and anisotropic composite laminates. Compos Part B Eng 1996; 27: 415–420.

10.

Singh

Chakrabarti

Hygrothermal analysis of laminated composite plates by using efficient higher order shear deformation theory. J Solid Mech 2011; 3: 85–95.

11.

Patel

Ganapathi

Makhecha

DP.

Hygrothermal effects on the structural behaviour of thick composite laminates using higher-order theory. Compos Struct 2002; 56: 25–34.

12.

Chalak

Chakrabarti

Iqbal

, et al. C0 FE model based on HOZT for the analysis of laminated soft core skew sandwich plates: Bending and vibration. Appl Math Model 2014; 38: 1211–1223.