A refined couple stress model for size-dependent thermo-mechanical analysis of laminated composite microplates

Abstract

The present work develops a novel size-dependent analytical framework for analyzing the thermo-mechanical behavior of laminated microplates by combining a refined couple stress model with an inverse-hyperbolic plate theory. The formulation captures anisotropic material characteristics while incorporating size dependency through an intrinsic length-scaling parameter aligned with the fiber orientation angle of the laminate. The use of the inverse-hyperbolic shear deformation function provides an improved representation of transverse shear behavior without requiring shear correction factors, offering a distinct advantage over conventional higher-order shear deformation theories. Using the principle of virtual work, the governing equations are established and solved using Navier’s closed-form technique under simply supported boundary conditions. An extensive parametric investigation is conducted to examine the effects of micro-scale shear deformation and thermal loads on the bending characteristics of micro-laminates. The results show that the inclusion of couple stress effects leads to a reduction in deflection of up to approximately 40%, while thermal loading further contributes to reductions of about 12–63% depending on the loading and geometric parameters. Additionally, the model converges to classical plate theory predictions at small length-scale values, validating its accuracy. The inverse-hyperbolic shear deformation model shows strong agreement with higher-order shear deformation theories without requiring shear correction factors. Overall, the comparative results validate the model’s accuracy and efficiency, establishing it as a comprehensive analytical framework for assessing the performance of numerical methods in the analysis and design of advanced micro-scale systems, including micro-electromechanical systems (MEMS), where precise prediction of thermo-mechanical deformation is critical for the reliability of sensors, actuators, and micro-structural components, where conventional plate theories fail to capture size-dependent effects.

Keywords

refined couple stress theory thermo-mechanical response shear deformation theory composite plates laminated microplates

Introduction

In recent decades, the growing use of micro-scale structural elements has played a crucial role in advancing next-generation engineering systems, especially because of their significance in micro-electromechanical systems (MEMS) and other miniaturized technologies. Among the diverse class of engineered materials, laminated composite plates have emerged as indispensable constituents in critical domains such as aerospace, automotive, civil infrastructure, and marine engineering. This significance is largely due to their exceptional specific stiffness, high thermal resistance, and the ability to achieve customized anisotropic properties through controlled lamination.

The escalating demand for miniaturized high-performance devices, coupled with rapid progress in precision micro-manufacturing, has facilitated the fabrication of micro-scale laminated composites through sophisticated processes such as micro-rolling, micro-milling, micro-deep drawing, micro-drilling, micro-extrusion, and micro-grinding.¹ These advanced manufacturing routes offer fine-scale control over geometrical and material characteristics, which is essential for achieving functional reliability and dimensional accuracy at the micro-scale. Consequently, micro-laminated composite plates have become fundamental structural components not only in MEMS, but also in high-precision biomedical devices, implantable sensors, and environmentally responsive microsystems.^1,2

Despite their expanding role in micro-scale engineering systems, the mechanical characterization of laminated microplates presents substantial theoretical challenges. At reduced length scales, conventional continuum theories fail to capture the pronounced size-dependent behaviors that arise due to the material’s micro-structural characteristics. These limitations are further exacerbated when structures are subjected to multiphysics environments, particularly combined thermal and mechanical fields, which are common in applications such as aerospace, biomedical, and marine structures.

Thermo-mechanical loading induces complex deformation mechanisms and stress redistributions that significantly influence the structural response of laminated composites. The differential expansion or contraction due to temperature gradients can lead to inter-laminar stresses, warping, and premature failure. Thermo-mechanical behavior of laminated microstructures has attracted significant attention due to its critical role in the performance of advanced engineering systems such as micro-electromechanical systems (MEMS), flexible electronics, and biomedical micro-devices. In these applications, temperature variations can induce coupled thermal stresses and deformations that significantly influence structural reliability and functional accuracy. In particular, heat shrinkage and thermal expansion variations in composite materials can alter the effective stiffness and deformation characteristics of laminated structures. Recent studies have highlighted that thermal shrinkage effects in composite and sandwich structures can strongly influence their thermo-mechanical response and stability characteristics.^3,4 Accurate modeling of these phenomena is vital for the functional performance of micro-laminated plates deployed in advanced applications, such as flexible electronics, biomedical implants, aerospace insulation systems, and environmental sensing platforms.^5,6 Recent studies have also highlighted the importance of tensile expansion and interfacial effects in laminated composites, particularly for MEMS applications, where tailored microstructures can significantly enhance strength, ductility, and thermo-mechanical performance.^7,8

Consequently, the development of advanced theoretical frameworks that incorporate both scale-dependent elasticity and thermo-mechanical coupling has become a focal point in the mechanics of laminated composite microstructures.

Experimental investigations^9–11 have consistently demonstrated that materials exhibit enhanced stiffness and strength when scaled down to micro- and sub-micrometer dimensions, a phenomenon widely recognized as the scale effect. This behavior stems from micro-structural constraints that become significant at small scales, leading to deviations from classical elasticity predictions. Traditional continuum theories, which assume scale-invariant material behavior and lack intrinsic length-scaling parameters, are inadequate for capturing such phenomena.

To overcome the limitations of classical continuum theories, higher-order models have been developed. The couple stress elasticity theory (CSET), originally formulated by Mindlin and Tiersten,¹² Toupin,¹³ and Koiter,¹⁴ incorporates couple stresses and a material length-scaling parameter into the governing equations, thereby enabling the accurate representation of size-dependent behavior in isotropic media. In parallel, Eringen’s nonlocal elasticity theory¹⁵ offers an alternative framework by accounting for long-range interactions via nonlocal integral formulations, which further improves predictions of mechanical response at small length scales.

Subsequent developments in strain gradient elasticity^16,17 introduced higher-order strain gradients to account for micro-structural interactions, while micropolar elasticity¹⁸ extended the kinematics to include microrotations. Among these, the modified couple stress theory (MCST) by Yang et al.¹⁹ has gained particular prominence for engineering applications due to its simplicity. It requires only one length-scaling parameter and adopts a symmetric couple stress tensor and augments classical equilibrium equations with additional moment equilibrium relations, offering a computationally efficient means to capture size effects.

Further refinement was introduced by Lam et al.²⁰ through the modified strain gradient theory (MSGT). These higher-order theories have been successfully employed in the modeling and analysis of micro-structural components using both analytical and numerical techniques.^21–26 Specifically, MCST has been extensively applied to model the static and dynamic responses of micro-beams^27–31 and microplates.^32–39

While the modified couple stress theory (MCST) effectively models the size-dependent response of isotropic microplates, its application is confined to isotropic materials despite its computational efficiency. To address this shortcoming and extend the theory to anisotropic systems, Chen et al.⁴⁰ adapted the couple stress framework for the analysis of laminated micro-beams. Subsequently, Chen and Li⁴¹ integrated CST with Kirchhoff’s plate theory to study laminated microplates. Nevertheless, the classical Kirchhoff formulation disregards transverse shear deformation, a factor that plays a critical role in the mechanical analysis of moderately thick and thick laminated plates.

To overcome the limitations of Kirchhoff’s plate theory, researchers have proposed several CST-based formulations that incorporate either the First-Order Shear Deformation Theory (FSDT)^42–44 or the Higher-Order Shear Deformation Theory (HSDT).^42,43 Among these methods, the results obtained from HSDT are more accurate in laminated composite plates due to its consideration of the actual distribution of transverse shear strains in the thickness direction. Moreover, in HSDT, it is not necessary to use empirical shear correction factors because the shear stress-free conditions in the top and bottom surfaces are automatically satisfied.

Chen and Wang⁴⁵ developed a global–local HSDT formulation within the CST framework to analyze the bending response of laminated microplates. Yang and He⁴⁶ explored the behavior of composite micro-laminates by integrating CST with zigzag theory, which enhanced the representation of inter-laminar continuity. Thanh et al.¹ proposed a CST-based model using Reddy’s HSDT for bending, vibration, and buckling analyses through an isogeometric approach. In a similar direction, Ma et al.⁴⁷ performed a finite element investigation of the size-dependent free vibration characteristics of composite microplates within a CST–HSDT framework. More recently, Joshan et al.⁴⁸ addressed the size-dependent static, free vibration, and buckling responses of composite microplates using non-polynomial HSDTs.

The thermo-mechanical analysis of micro-laminates has gained significant attention due to the increasing use of such materials in micro-scale devices operating under thermal environments. Traditional numerical methods often face challenges in accurately capturing size effects and maintaining geometric exactness. To address these issues, researchers have employed isogeometric analysis (IGA) in conjunction with non-classical theories.

Thanh et al.⁴⁹ formulated a refined HSDT within the CST framework to study the thermo-mechanical bending and buckling behavior of composite microplates, employing isogeometric analysis (IGA) for continuity and accurate geometry representation. Their approach effectively captured both size-dependent effects and thermal influences. Building on this, Ma et al.⁴⁷ combined CST with Reddy’s HSDT and applied the finite element method to investigate the free vibration characteristics of microplates in thermal environments.

A comprehensive review of existing studies indicates that CST-based higher-order plate models have been widely applied to analyze the size-dependent behavior of laminated microplates. In contrast, relatively limited attention has been given to their thermo-mechanical response, even though temperature variations are unavoidable in practical applications such as aerospace, marine, and biomedical structures. Moreover, various non-polynomial shear deformation theories (NPSDTs) such as trigonometric,^50–52 exponential,^53–55 inverse trigonometric,^56,57 inverse hyperbolic,^58–60 and hyperbolic^61,62 shear deformation theories are presented for laminated structures in the last three decades. NPSDTs are recognized due to their high accuracy in modeling the effects of transverse shear without any correction factors. The precision of these theories has also been established in the literature for composite and functionally graded (FG) microplates.^48,63–70 However, these theories have yet to be employed for the analytical thermo-mechanical modeling of laminated microplates within the CST framework. Integrating NPSDT with refined couple stress theory (RCST), therefore, holds significant potential for advancing robust and precise models for predicting the thermo-mechanical performance of anisotropic laminates at the microscale.

In this paper, a generalized size-dependent non-polynomial shear deformation model has been developed within the context of the couple stress theory to carry out thermo-mechanical analyses of laminated composite microplates. An inverse hyperbolic function⁶⁰ has been adopted to depict the variation of the displacement field through the thickness of the laminate, which offers an accurate description of the deformation of plates. To account for size effects, CST has been utilized by considering a material length-scale parameter along the fiber direction. The governing equations have been derived employing the virtual work principle, which have been solved analytically employing Navier’s method to validate the results for simply supported boundary conditions. Thermo-mechanical responses of micro-laminates have been analyzed, and results have been compared with those available in the literature employing other HSDT models. This comparison indicates the capability, accuracy, and efficiency of the proposed model to evaluate laminated composite microplates subjected to thermal loading.

Mathematical formulation

In this study, a Cartesian coordinate system (x₁, x₂, x₃) defines a rectangular multilayered micro-composite micro-laminate. The laminate consists of N individual laminae with in-plane dimensions of a and b along the x₁ and x₂ axes, respectively, while the total thickness is denoted by h along the x₃ direction. The geometric mid-surface of the laminate is assumed to lie at x₃ = 0. The following assumptions are considered in this formulation:

(1) The strains are taken to be infinitesimal. The material is assumed to be linear elastic. Additionally, the displacements and rotations are assumed to be sufficiently small such that the influence of geometric nonlinearity can be ignored.

(2) The normal strain along the thickness direction is considered to be negligible.

(3) The material properties in the microplate do not change with the change in temperature.

Since the variables are defined at the middle layer of the plate in the equivalent single-layer framework, the present work does not predict inter-laminar stresses accurately, which limits the scope of applicability. However, this can be achieved with the use of equilibrium equations or by defining the variables in zigzag/layerwise approach.

Principle of virtual work

The formulation of laminated composite microplates that incorporate couple stress theory is established using the principle of virtual work. This fundamental principle serves as the foundation for deriving the governing equations and can be expressed in its generalized form as follows⁷¹:

\int_{0}^{t} (δ U - δ W) d t = 0

(1)

where the first variation in the virtual internal energy is denoted by δU, which is defined as¹:

δ U = \int_{V} (σ \cdot δ ε + η \cdot δ χ) d x_{1} d x_{2} d x_{3}

(2)

Here, σ denotes the Cauchy stress, and ɛ indicates the classical strain tensor. The tensor η corresponds to the couple stress tensor, while χ represents the curvature tensor related to size-dependent effects.⁴⁸ In the couple stress theory, the curvature tensor describes the gradient of material rotation and is the conjugate of the couple stress tensor in the strain energy function. In physical terms, the curvature tensor represents the bending and twisting of the micro-elements of the material due to the non-uniform rotation of the material.

The components of the curvature tensor χ_ij are defined as the spatial gradients of the rotation vector $ϕ_{i}^{c}$ , and can be expressed as⁴⁸:

χ_{i j} = ϕ_{i, j}^{c}

(3)

where

ϕ_{i}^{c}

represents the components of the infinitesimal rotation vector and the subscript comma signifies the partial differentiation with respect to the spatial coordinate x_j. The above definition indicates that the curvature tensor quantifies the rate of rotation with respect to the spatial directions, which assumes a significant role in micro-scale structures to account for the size-dependent effects. The components

ϕ_{i}^{c} (i = 1,2,3)

are given as follows in terms of displacement components⁶⁹:

ϕ_{1}^{c} = \frac{1}{2} (\frac{\partial u_{3}}{\partial x_{2}} - \frac{\partial u_{2}}{\partial x_{3}})

(4)

ϕ_{2}^{c} = \frac{1}{2} (\frac{\partial u_{1}}{\partial x_{3}} - \frac{\partial u_{3}}{\partial x_{1}})

(5)

ϕ_{3}^{c} = \frac{1}{2} (\frac{\partial u_{2}}{\partial x_{1}} - \frac{\partial u_{1}}{\partial x_{2}})

(6)

Here, u₁, u₂, and u₃ denote the displacement components along the x₁, x₂, and x₃ coordinate directions, respectively. The virtual work δW associated with the externally applied forces, as referenced in equation (1), is defined as⁷²:

δ W = - \int_{Ω_{0}} (q δ u_{3}) d x_{1} d x_{2}

(7)

Here, q is the transversely distributed load applied on the microplate.

Displacement function

The displacements of a material point (x₁, x₂, x₃) are described in terms of mid-surface displacements and rotations, based on a shear strain function-driven shear deformation plate theory. Accordingly, the displacement field within this theoretical framework is given by⁶⁰:

u_{1} = u_{0} (x_{1}, x_{2}) - x_{3} (\frac{\partial w_{0} (x_{1}, x_{2})}{\partial x_{1}}) + f (x_{3}) θ_{1} (x_{1}, x_{2})

(8)

u_{2} = v_{0} (x_{1}, x_{2}) - x_{3} (\frac{\partial w_{0} (x_{1}, x_{2})}{\partial x_{2}}) + f (x_{3}) θ_{2} (x_{1}, x_{2})

(9)

u_{3} = w_{0} (x_{1}, x_{2})

(10)

Here, u₀, v₀, and w₀ correspond to the displacements of the mid-surface in the x₁, x₂, and x₃ directions, respectively, while θ₁ and θ₂ denote the shear rotations about the x₂ and x₁ axes. The shear strain distribution is governed by the function f(x₃), and specifically selected to satisfy the following conditions⁷³:

f^{'} (\pm \frac{h}{2}) = 0, \int_{- h / 2}^{h / 2} f (x_{3}) d x_{3} = 0

(11)

The shear strain functions used from the literature, as well as the one used in the current formulation, are given in Table 1. The shear strain functions capture the thickness variation of the transverse shear strain and do not require any shear correction factors, which are usually needed in the first-order shear deformation theory (FSDT). The modeling of the transverse shear deformation has a prominent role in moderately thick laminated plates and sandwich plates, where the shear deformation has a dominant influence on the overall mechanical response.

Table 1.

Shear strain functions used in various shear deformation theories. A comparison of transverse shear strain variation functions f(x₃) used in various higher-order shear deformation theories (HSDTs). These functions describe how transverse shear strain varies with the plate thickness coordinate x₃, where h is the total thickness of the plate.

Theory	f(x₃)
Touratier⁵⁰	$(h / π) s i n (π x_{3} / h)$
Mantari et al.⁷⁴	$t a n (r x_{3}) - x_{3} r {s e c}^{2} (r h / 2); [r = 0.2]$
Sarangan and Singh⁵⁵	$t a n h (r x_{3} / h) - x_{3} r / h {s e c h}^{2} (r / 2); [r = 2.5]$
Soldatos⁶¹	$x_{3} c o s h (0.5) - h s i n h (x_{3} / h)$
Sarangan and Singh⁵⁵	${(r x_{3} / h)}^{3} e^{(2 \sqrt{2 π})} - x_{3} e^{(2 \sqrt{2 π})} (3 r^{3} / 4 h); [r = 0.95]$
Grover et al.⁷⁵	${c o t}^{- 1} (h r / x_{3}) - x_{3} 4 r / h (r^{2} + 1); [r = 0.42]$
Joshan et al.⁵⁹	${t a n h}^{- 1} (r x_{3} / h) - x_{3} r / h (1 - (r^{2} / 4)); [r = 0.088]$
Aydogdu⁵⁴	$(1 - 4 x_{3}^{2} / h^{2}) r^{- 2 x_{3}^{2} / h^{2} / I n (r)}; [r = 3]$
Joshan et al.⁶⁰ (present)	$1 / h π c o s e c h^{- 1} (h / x_{3} π) - x_{3} 2 / h^{2} \sqrt{(4 + π^{2})}$

Some authors have proposed different expressions for the distribution of shear strain, such as trigonometric,⁵⁰ trigonometric and secant,⁷⁴ hyperbolic,^55,61 inverse trigonometric,⁷⁵ and exponential functions,⁵⁴ in order to accurately capture the effects of transverse shear deformations and satisfy the traction-free conditions.

The current formulation employs an inverse hyperbolic function, and it can be seen that several advantages are incorporated in the analysis of the laminated microstructure by employing such a function. The inverse hyperbolic functions employed in the present formulation inherently have a smooth nonlinear distribution similar to the actual shear stress profile predicted by three-dimensional elasticity theory. This allows the function to accurately capture the rapid changes in transverse shear strains and predict the mechanical response of thick plates with greater precision compared to other theories considered in this study.

The infinitesimal strain components, that is, ɛ₁₁, ɛ₂₂, ɛ₁₂, ɛ₁₃, and ɛ₂₃, are now determined using equations (8)–(10) as follows:

ε_{11} = \frac{\partial u_{1}}{\partial x_{1}} = \frac{\partial u_{0}}{\partial x_{1}} - x_{3} \frac{\partial^{2} w_{0}}{\partial x_{1}^{2}} + f (x_{3}) \frac{\partial θ_{1}}{\partial x_{1}}

(12)

ε_{22} = \frac{\partial u_{2}}{\partial x_{2}} = \frac{\partial v_{0}}{\partial x_{2}} - x_{3} \frac{\partial^{2} w_{0}}{\partial x_{2}^{2}} + f (x_{3}) \frac{\partial θ_{2}}{\partial x_{2}}

(13)

ε_{12} = \frac{\partial u_{1}}{\partial x_{2}} + \frac{\partial u_{2}}{\partial x_{1}} = \frac{\partial u_{0}}{\partial x_{2}} + \frac{\partial v_{0}}{\partial x_{1}} - 2 x_{3} \frac{\partial^{2} w_{0}}{\partial x_{1} \partial x_{2}} + f (x_{3}) (\frac{\partial θ_{1}}{\partial x_{2}} + \frac{\partial θ_{2}}{\partial x_{1}})

(14)

ε_{13} = \frac{\partial u_{3}}{\partial x_{1}} + \frac{\partial u_{1}}{\partial x_{3}} = \frac{\partial w_{0}}{\partial x_{1}} - \frac{\partial w_{0}}{\partial x_{1}} + f^{'} (x_{3}) θ_{1} = f^{'} (x_{3}) θ_{1}

(15)

ε_{23} = \frac{\partial u_{3}}{\partial x_{2}} + \frac{\partial u_{2}}{\partial x_{3}} = \frac{\partial w_{0}}{\partial x_{2}} - \frac{\partial w_{0}}{\partial x_{2}} + f^{'} (x_{3}) θ_{2} = f^{'} (x_{3}) θ_{2}

(16)

The rotation vector $ϕ_{i}^{c}$ is now obtained as:

ϕ_{1}^{c} = \frac{1}{2} (\frac{\partial u_{3}}{\partial x_{2}} - \frac{\partial u_{2}}{\partial x_{3}}) = \frac{\partial w_{0}}{\partial x_{2}} - \frac{1}{2} f^{'} (x_{3}) θ_{2}

(17)

ϕ_{2}^{c} = \frac{1}{2} (\frac{\partial u_{1}}{\partial x_{3}} - \frac{\partial u_{3}}{\partial x_{1}}) = - \frac{\partial w_{0}}{\partial x_{1}} + \frac{1}{2} f^{'} (x_{3}) θ_{1}

(18)

ϕ_{3}^{c} = \frac{1}{2} (\frac{\partial u_{2}}{\partial x_{1}} - \frac{\partial u_{1}}{\partial x_{2}}) = \frac{1}{2} (\frac{\partial v_{0}}{\partial x_{1}} - \frac{\partial u_{0}}{\partial x_{2}} + f (x_{3}) (\frac{\partial θ_{2}}{\partial x_{1}} - \frac{\partial θ_{1}}{\partial x_{2}}))

(19)

Furthermore, the components of the curvature tensor χ_ij are found by using equation (3) as follows:

χ_{11} = \frac{\partial ϕ_{1}^{c}}{\partial x_{1}} = \frac{\partial^{2} w_{0}}{\partial x_{2} \partial x_{1}} - \frac{1}{2} f^{'} (x_{3}) \frac{\partial θ_{2}}{\partial x_{1}}

(20)

χ_{12} = \frac{\partial ϕ_{1}^{c}}{\partial x_{2}} = \frac{\partial^{2} w_{0}}{\partial x_{2}^{2}} - \frac{1}{2} f^{'} (x_{3}) \frac{\partial θ_{2}}{\partial x_{2}}

(21)

χ_{13} = \frac{\partial ϕ_{1}^{c}}{\partial x_{3}} = - \frac{1}{2} f^{″} (x_{3}) θ_{2}

(22)

χ_{21} = \frac{\partial ϕ_{2}^{c}}{\partial x_{1}} = - \frac{\partial^{2} w_{0}}{\partial x_{1}^{2}} + \frac{1}{2} f^{'} (x_{3}) \frac{\partial θ_{1}}{\partial x_{1}}

(23)

χ_{22} = \frac{\partial ϕ_{2}^{c}}{\partial x_{2}} = - \frac{\partial^{2} w_{0}}{\partial x_{1} \partial x_{2}} + \frac{1}{2} f^{'} (x_{3}) \frac{\partial θ_{1}}{\partial x_{2}}

(24)

χ_{23} = \frac{\partial ϕ_{2}^{c}}{\partial x_{3}} = \frac{1}{2} f^{″} (x_{3}) θ_{1}

(25)

χ_{31} = \frac{\partial ϕ_{3}^{c}}{\partial x_{1}} = \frac{1}{2} (\frac{\partial^{2} v_{0}}{\partial x_{1}^{2}} - \frac{\partial^{2} u_{0}}{\partial x_{2} \partial x_{1}} + f (x_{3}) (\frac{\partial θ_{2}}{\partial^{2} x_{1}^{2}} - \frac{\partial^{2} θ_{1}}{\partial x_{2} \partial x_{1}}))

(26)

χ_{32} = \frac{\partial ϕ_{3}^{c}}{\partial x_{2}} = \frac{1}{2} (\frac{\partial^{2} v_{0}}{\partial x_{1} \partial x_{2}} - \frac{\partial^{2} u_{0}}{\partial x_{2}^{2}} + f (x_{3}) (\frac{\partial θ_{2}}{\partial^{2} x_{1} \partial x_{2}} - \frac{\partial^{2} θ_{1}}{\partial x_{2}^{2}}))

(27)

χ_{33} = \frac{\partial ϕ_{3}^{c}}{\partial x_{3}} = \frac{1}{2} f^{'} (x_{3}) (\frac{\partial θ_{2}}{\partial x_{1}} - \frac{\partial θ_{1}}{\partial x_{2}})

(28)

Constitutive relations

The equations establishing relationship between the stress components {σ₁₁, σ₂₂, σ₁₂, σ₂₃, σ₁₃} (expressed in Voigt notation) and the corresponding strain components {ɛ₁₁, ɛ₂₂, ɛ₁₂, ɛ₂₃, ɛ₁₃}, including the strain contributions induced by the applied thermal field, are expressed as follows⁷⁶:

\{\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{12} \\ σ_{23} \\ σ_{13} \end{matrix}\} = [\begin{matrix} Q_{11}^{t} & Q_{12}^{t} & Q_{16}^{t} & 0 & 0 \\ Q_{12}^{t} & Q_{22}^{t} & Q_{26}^{t} & 0 & 0 \\ Q_{16}^{t} & Q_{26}^{t} & Q_{66}^{t} & 0 & 0 \\ 0 & 0 & 0 & Q_{44}^{t} & Q_{45}^{t} \\ 0 & 0 & 0 & Q_{45}^{t} & Q_{55}^{t} \end{matrix}] \{\begin{matrix} ε_{11} - α_{11} Δ T \\ ε_{22} - α_{2} Δ T \\ ε_{12} - α_{12} Δ T \\ ε_{23} \\ ε_{13} \end{matrix}\}

(29)

The parameters $Q_{i j}^{t}$ denote the scalar components of the transformed stiffness matrix, α_ij (i, j = 1, 2) are in-plane thermal expansion characteristics of the laminate, and ΔT is the applied change in temperature. The temperature variation through the thickness is assumed following the distribution reported in earlier studies^59,76,77:

Δ T = T_{0} + \frac{x_{3}}{h} T_{1} + \frac{f (x_{3})}{h} T_{2}

(30)

Here, T₀, T₁, and T₂ represent the constant, linear, and nonlinear components of the temperature field, respectively, across the thickness of the microplate.

For an orthotropic lamina, the coefficients of thermal expansion in the global coordinate system depend on the fiber orientation angle Φ. The thermal expansion coefficients in the global coordinate system can be expressed in terms of the material principal coefficients as follows⁷¹:

\begin{align} α_{11} & = α_{1} \cos^{2} Φ + α_{2} \sin^{2} Φ, \end{align}

(31)

\begin{align} α_{22} & = α_{1} \sin^{2} Φ + α_{2} \cos^{2} Φ, \end{align}

(32)

\begin{align} α_{12} & = (α_{1} - α_{2}) \sin Φ \cos Φ . \end{align}

(33)

Here, α₁ and α₂ represent the thermal expansion coefficients along the fiber direction and the transverse direction of the lamina, respectively.

Constitutive relations: Couple stress-curvature tensor relationship

Based on the refined CST presented by Chen and Li,⁴¹ the constitutive relations specifically tailored for anisotropic materials are given by:

η_{i j} = l_{i}^{2} G_{j} χ_{i j} + l_{j}^{2} G_{i} χ_{j i}

(34)

where l_i denotes the length-scaling parameters in the respective directions, while G_i represents the shear moduli in the plane perpendicular to l_i (i.e., G₁ = G₁₃, G₂ = G₂₃, and G₃ = G₁₂). The formulation can be specialized to isotropic FG composites by considering G₁ = G₂ = G₃. Due to the anisotropic nature of the material, the couple stress theory considers three distinct length-scaling parameters. Accordingly, equation (34) can be expressed as follows¹:

\begin{align} \{\begin{matrix} η_{11} \\ η_{22} \\ η_{33} \\ η_{12} \\ η_{21} \\ η_{23} \\ η_{32} \\ η_{13} \\ η_{31} \end{matrix}\} = [\begin{matrix} 2 G_{13} l_{1}^{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 2 G_{23} l_{2}^{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 G_{12} l_{3}^{2} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & G_{13} l_{1}^{2} & G_{23} l_{2}^{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & G_{13} l_{1}^{2} & G_{23} l_{2}^{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & G_{23} l_{2}^{2} & G_{12} l_{3}^{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & G_{23} l_{2}^{2} & G_{12} l_{3}^{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & G_{13} l_{1}^{2} & G_{12} l_{3}^{2} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & G_{13} l_{1}^{2} & G_{12} l_{3}^{2} \end{matrix}] \{\begin{matrix} χ_{11} \\ χ_{22} \\ χ_{33} \\ χ_{12} \\ χ_{21} \\ χ_{23} \\ χ_{32} \\ χ_{13} \\ χ_{31} \end{matrix}\} \end{align}

(35)

Such quantitative examinations by Thanh et al.¹ established that the contributions of the length-scaling parameters l₂ and l₃ on the structural behavior of composite micro-laminates are negligible.

The maximum percentage variance is found to be 0.136%, and practically no difference is observed between the cases l₂/l₁ = l₃/l₁ = 0 and l₃ = l₂ = l₁ = 1.⁷⁸ A similar trend is also observed in the thermo-mechanical analysis, where both thermal and mechanical loadings are simultaneously applied. In this case, the maximum percentage variance is found to be 0.139%, further confirming the minimal influence of these parameters on the predicted response. It is observed that the size-dependent effects are mainly influenced by the length-scaling parameter in the reference direction, l₁. Thus, in simplifying the current formulation here, we keep only the fiber-direction length-scaling parameter (l₁). In this manner, the transformed couple stress constitutive relations are given by¹:

{\{\begin{matrix} η_{11} \\ η_{22} \\ η_{12} \\ η_{21} \\ η_{23} \\ η_{32} \\ η_{13} \\ η_{31} \end{matrix}\}}^{(k)} = {[\begin{matrix} Q_{11}^{b c} & 0 & 0 & Q_{15}^{b c} & 0 & 0 & 0 & 0 \\ 0 & Q_{22}^{b c} & Q_{24}^{b c} & 0 & 0 & 0 & 0 & 0 \\ Q_{41}^{b c} & Q_{42}^{b c} & Q_{44}^{b c} & Q_{45}^{b c} & 0 & 0 & 0 & 0 \\ Q_{41}^{b c} & Q_{42}^{b c} & Q_{44}^{b c} & Q_{45}^{b c} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & Q_{11}^{s c} & 0 & Q_{13}^{s c} & 0 \\ 0 & 0 & 0 & 0 & Q_{11}^{s c} & 0 & Q_{13}^{s c} & 0 \\ 0 & 0 & 0 & 0 & Q_{31}^{s c} & 0 & Q_{33}^{s c} & 0 \\ 0 & 0 & 0 & 0 & Q_{31}^{s c} & 0 & Q_{33}^{s c} & 0 \end{matrix}]}^{(k)} {\{\begin{matrix} χ_{11} \\ χ_{22} \\ χ_{12} \\ χ_{21} \\ χ_{23} \\ χ_{32} \\ χ_{13} \\ χ_{31} \end{matrix}\}}^{(k)}

(36)

η^{(k)} = Q^{c (k)} χ^{(k)}

(37)

where the components of Q^c(k) are given by:

\begin{align} Q_{11}^{b c (k)} = 2 G_{13}^{(k)} l_{1}^{2} α_{m}^{4} + 2 G_{13}^{(k)} l_{1}^{2} α_{m}^{2} α_{n}^{2} \\ Q_{15}^{b c (k)} = Q_{24}^{b c (k)} = 2 G_{13}^{(k)} l_{1}^{2} α_{m} α_{n} (α_{m}^{2} + α_{n}^{2}) \\ Q_{22}^{b c (k)} = 2 G_{13}^{(k)} l_{1}^{2} α_{n}^{4} + 2 G_{13}^{(k)} l_{1}^{2} α_{m}^{2} α_{n}^{2} \\ Q_{41}^{b c (k)} = Q_{42}^{b c (k)} = G_{13}^{(k)} l_{1}^{2} α_{m} α_{n} (α_{m}^{2} + α_{n}^{2}) \\ Q_{44}^{b c (k)} = G_{13}^{(k)} l_{1}^{2} α_{m}^{2} (α_{m}^{2} + α_{n}^{2}) \\ Q_{45}^{b c (k)} = G_{13}^{(k)} l_{1}^{2} α_{n}^{2} (α_{m}^{2} + α_{n}^{2}) \\ Q_{11}^{s c (k)} = G_{13}^{(k)} l_{1}^{2} α_{n}^{2} \\ Q_{33}^{s c (k)} = G_{13}^{(k)} l_{1}^{2} α_{m}^{2} \\ Q_{13}^{s c (k)} = Q_{31}^{s c (k)} = G_{13}^{(k)} l_{1}^{2} α_{m} α_{n} \\ α_{m} = c o s Φ, α_{n} = s i n Φ \end{align}

(38)

Derivation of governing equations

Based on the NPSDT framework, governing equations for the microplate are now obtained by substituting curvature tensor components (20)–(28) and strain components (12)–(16) into internal energy equation (2), the following expression is obtained:

\begin{align} δ U & = \int_{Ω_{0}} \int_{x_{3}} [σ_{11} δ (\frac{\partial u_{0}}{\partial x_{1}} - x_{3} \frac{\partial^{2} w_{0}}{\partial x_{1}^{2}} + f (x_{3}) \frac{\partial θ_{1}}{\partial x_{1}}) + σ_{22} δ (\frac{\partial v_{0}}{\partial x_{2}} - x_{3} \frac{\partial^{2} w_{0}}{\partial x_{2}^{2}} + f (x_{3}) \frac{\partial θ_{2}}{\partial x_{2}}) \\ + σ_{12} δ (\frac{\partial u_{0}}{\partial x_{2}} + \frac{\partial v_{0}}{\partial x_{1}} - 2 x_{3} \frac{\partial^{2} w_{0}}{\partial x_{1} \partial x_{2}} + f (x_{3}) (\frac{\partial θ_{1}}{\partial x_{2}} + \frac{\partial θ_{2}}{\partial x_{1}})) + σ_{13} f^{'} (x_{3}) δ θ_{1} \\ + σ_{23} f^{'} (x_{3}) δ θ_{2} + η_{11} δ (\frac{\partial^{2} w_{0}}{\partial x_{2} \partial x_{1}} - \frac{1}{2} f^{'} (x_{3}) \frac{\partial θ_{2}}{\partial x_{1}}) + η_{12} δ (\frac{\partial^{2} w_{0}}{\partial x_{2}^{2}} - \frac{1}{2} f^{'} (x_{3}) \frac{\partial θ_{2}}{\partial x_{2}}) \\ + η_{13} δ (- \frac{1}{2} f^{''} (x_{3}) θ_{2}) + η_{21} δ (- \frac{\partial^{2} w_{0}}{\partial x_{1}^{2}} + \frac{1}{2} f^{'} (x_{3}) \frac{\partial θ_{1}}{\partial x_{1}}) \\ + η_{22} δ (- \frac{\partial^{2} w_{0}}{\partial x_{1} \partial x_{2}} + \frac{1}{2} f^{'} (x_{3}) \frac{\partial θ_{1}}{\partial x_{2}}) + η_{23} δ (\frac{1}{2} f^{''} (x_{3}) θ_{1}) \\ + η_{31} δ (\frac{1}{2} (\frac{\partial^{2} v_{0}}{\partial x_{1}^{2}} - \frac{\partial^{2} u_{0}}{\partial x_{2} \partial x_{1}} + f (x_{3}) (\frac{\partial^{2} θ_{2}}{\partial^{2} x_{1}^{2}} - \frac{\partial^{2} θ_{1}}{\partial x_{2} \partial x_{1}}))) \\ + η_{32} δ (\frac{1}{2} (\frac{\partial^{2} v_{0}}{\partial x_{1} \partial x_{2}} - \frac{\partial^{2} u_{0}}{\partial x_{2}^{2}} + f (x_{3}) (\frac{\partial^{2} θ_{2}}{\partial^{2} x_{1} \partial x_{2}} - \frac{\partial^{2} θ_{1}}{\partial x_{2}^{2}}))) \\ + η_{33} δ (\frac{1}{2} f^{'} (x_{3}) (\frac{\partial θ_{2}}{\partial x_{1}} - \frac{\partial θ_{1}}{\partial x_{2}}))] d x_{3} d x_{1} d x_{2} \end{align}

(39)

Upon integrating with respect to x₃ through the thickness of the plate, the corresponding two-dimensional variational form of the governing equation is obtained as:

\begin{align} δ U & = \int_{Ω_{0}} [N_{11} \frac{\partial δ u_{0}}{\partial x_{1}} - M_{11} \frac{\partial^{2} δ w_{0}}{\partial x_{1}^{2}} + P_{11} \frac{\partial δ θ_{1}}{\partial x_{1}} + N_{22} \frac{\partial δ v_{0}}{\partial x_{2}} - M_{22} \frac{\partial^{2} δ w_{0}}{\partial x_{2}^{2}} + P_{22} \frac{\partial δ θ_{2}}{\partial x_{2}} \\ + N_{12} \frac{\partial δ u_{0}}{\partial x_{2}} + N_{12} \frac{\partial δ v_{0}}{\partial x_{1}} - 2 M_{12} \frac{\partial^{2} δ w_{0}}{\partial x_{1} \partial x_{2}} + P_{12} (\frac{\partial δ θ_{1}}{\partial x_{2}} + \frac{\partial δ θ_{2}}{\partial x_{1}}) + K_{13} δ θ_{1} \\ + K_{23} δ θ_{2} + N_{11}^{c} \frac{\partial^{2} δ w_{0}}{\partial x_{2} \partial x_{1}} - \frac{1}{2} K_{11}^{c} \frac{\partial δ θ_{2}}{\partial x_{1}} + N_{12}^{c} \frac{\partial^{2} δ w_{0}}{\partial x_{2}^{2}} - \frac{1}{2} K_{12}^{c} \frac{\partial δ θ_{2}}{\partial x_{2}} \\ - \frac{1}{2} L_{13}^{c} δ θ_{2} - N_{21}^{c} \frac{\partial^{2} δ w_{0}}{\partial x_{1}^{2}} + \frac{1}{2} K_{21}^{c} \frac{\partial δ θ_{1}}{\partial x_{1}} - N_{22}^{c} \frac{\partial^{2} δ w_{0}}{\partial x_{1} \partial x_{2}} + \frac{1}{2} K_{22}^{c} \frac{\partial δ θ_{1}}{\partial x_{2}} \\ + \frac{1}{2} L_{23}^{c} δ θ_{1} + \frac{1}{2} (N_{31}^{c} \frac{\partial^{2} δ v_{0}}{\partial x_{1}^{2}} - N_{31}^{c} \frac{\partial^{2} δ u_{0}}{\partial x_{2} \partial x_{1}} + P_{31}^{c} (\frac{\partial δ θ_{2}}{\partial^{2} x_{1}^{2}} - \frac{\partial^{2} δ θ_{1}}{\partial x_{2} \partial x_{1}})) \\ + \frac{1}{2} (N_{32}^{c} \frac{\partial^{2} δ v_{0}}{\partial x_{1} \partial x_{2}} - N_{32}^{c} \frac{\partial^{2} δ u_{0}}{\partial x_{2}^{2}} + P_{32}^{c} (\frac{\partial δ θ_{2}}{\partial^{2} x_{1} \partial x_{2}} - \frac{\partial^{2} δ θ_{1}}{\partial x_{2}^{2}})) \\ + \frac{1}{2} K_{33}^{c} (\frac{\partial δ θ_{2}}{\partial x_{1}} - \frac{\partial δ θ_{1}}{\partial x_{2}}))] d x_{1} d x_{2} \end{align}

(40)

where

[\begin{matrix} N_{11} & M_{11} & P_{11} \\ N_{22} & M_{22} & P_{22} \\ N_{12} & M_{12} & P_{12} \end{matrix}] = \int_{x_{3}} \{\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{12} \end{matrix}\} [\begin{matrix} 1 & x_{3} & f (x_{3}) \end{matrix}] d x_{3}

(41)

[\begin{matrix} K_{23} \\ K_{13} \end{matrix}] = \int_{x_{3}} \{\begin{matrix} σ_{23} \\ σ_{13} \end{matrix}\} [f^{'} (x_{3})] d x_{3}

(42)

[\begin{matrix} N_{i j}^{c} \\ P_{i j}^{c} \\ K_{i j}^{c} \\ L_{i j}^{c} \end{matrix}] = \int_{x_{3}} \{\begin{matrix} η_{i j} \end{matrix}\} [\begin{matrix} 1 \\ f (x_{3}) \\ f^{'} (x_{3}) \\ f^{″} (x_{3}) \end{matrix}] d x_{3}

(43)

The governing equations can be obtained from equations (7) and (40) in the context of the principle of virtual work. These equations can thus be written as:

u_{0} : \frac{\partial N_{11}}{\partial x_{1}} + \frac{\partial N_{12}}{\partial x_{2}} + \frac{1}{2} (\frac{\partial^{2} N_{31}^{c}}{\partial x_{1} \partial x_{2}} + \frac{\partial^{2} N_{32}^{c}}{\partial x_{2}^{2}}) = 0

(44)

v_{0} : \frac{\partial N_{12}}{\partial x_{1}} + \frac{\partial N_{22}}{\partial x_{2}} - \frac{1}{2} (\frac{\partial^{2} N_{31}^{c}}{\partial x_{1}^{2}} + \frac{\partial^{2} N_{32}^{c}}{\partial x_{2} \partial x_{1}}) = 0

(45)

w_{0} : \frac{\partial^{2} M_{11}}{\partial x_{1}^{2}} + \frac{\partial^{2} M_{22}}{\partial x_{2}^{2}} + 2 \frac{\partial^{2} M_{12}}{\partial x_{1} \partial x_{2}} - \frac{\partial^{2} N_{11}^{c}}{\partial x_{2} \partial x_{1}} + \frac{\partial^{2} N_{22}^{c}}{\partial x_{1} \partial x_{2}} - \frac{\partial^{2} N_{12}^{c}}{\partial x_{2}^{2}} + \frac{\partial^{2} N_{21}^{c}}{\partial x_{1}^{2}} = - q

(46)

θ_{1} : \frac{\partial P_{11}}{\partial x_{1}} + \frac{\partial P_{12}}{\partial x_{2}} - K_{13} + \frac{1}{2} (\frac{\partial K_{21}^{c}}{\partial x_{1}} + \frac{\partial K_{22}^{c}}{\partial x_{2}} + \frac{\partial^{2} P_{31}^{c}}{\partial x_{2} \partial x_{1}} + \frac{\partial^{2} P_{32}^{c}}{\partial x_{2}^{2}} - \frac{\partial K_{33}^{c}}{\partial x_{2}} - L_{23}^{c}) = 0

(47)

θ_{2} : \frac{\partial P_{12}}{\partial x_{1}} + \frac{\partial P_{22}}{\partial x_{2}} - K_{23} + \frac{1}{2} (- \frac{\partial K_{11}^{c}}{\partial x_{1}} - \frac{\partial K_{12}^{c}}{\partial x_{2}} - \frac{\partial^{2} P_{31}^{c}}{\partial x_{1}^{2}} - \frac{\partial^{2} P_{32}^{c}}{\partial x_{1} \partial x_{2}} + \frac{\partial K_{33}^{c}}{\partial x_{1}} + L_{13}^{c}) = 0

(48)

The corresponding boundary conditions in terms of natural and essential variables are given by:

\begin{align} O n e d g e s x_{1} = 0 a n d x_{1} = a \\ e i t h e r u_{0} = 0 o r N_{11} - \frac{1}{2} \frac{\partial N_{31}^{c}}{\partial x_{2}} = 0, e i t h e r \frac{\partial w_{0}}{\partial x_{1}} = 0 o r M_{11} + N_{21}^{c} = 0, \\ e i t h e r w_{0} = 0 o r \frac{\partial M_{11}}{\partial x_{1}} + 2 \frac{\partial M_{12}}{\partial x_{2}} - \frac{\partial N_{11}^{c}}{\partial x_{2}} + \frac{\partial N_{21}^{c}}{\partial x_{1}} + \frac{\partial N_{22}^{c}}{\partial x_{2}} = 0, \\ e i t h e r θ_{1} = 0 o r P_{11} + \frac{1}{2} K_{21}^{c} - \frac{1}{2} \frac{\partial P_{31}^{c}}{\partial x_{2}} = 0, \\ e i t h e r θ_{2} = 0 o r P_{12} - \frac{1}{2} (K_{11}^{c} - K_{33}^{c}) - \frac{1}{2} \frac{\partial P_{31}^{c}}{\partial x_{1}} - \frac{1}{2} \frac{\partial P_{32}^{c}}{\partial x_{1}} = 0, \\ e i t h e r v_{0} = 0 o r N_{12} - \frac{1}{2} \frac{\partial N_{31}^{c}}{\partial x_{1}} - \frac{1}{2} \frac{\partial N_{32}^{c}}{\partial x_{2}} = 0, \\ e i t h e r \frac{\partial v_{0}}{\partial x_{1}} = 0 o r N_{31}^{c} = 0, e i t h e r \frac{\partial θ_{2}}{\partial x_{1}} = 0 o r P_{31}^{c} = 0 \end{align}

(49)

\begin{align} O n e d g e s x_{2} = 0 a n d x_{2} = b \\ e i t h e r v_{0} = 0 o r N_{22} + \frac{1}{2} \frac{\partial N_{32}^{c}}{\partial x_{1}} = 0, e i t h e r \frac{\partial w_{0}}{\partial x_{2}} = 0 o r M_{22} - N_{12}^{c} = 0, \\ e i t h e r w_{0} = 0 o r \frac{\partial M_{22}}{\partial x_{2}} + 2 \frac{\partial M_{12}}{\partial x_{1}} - \frac{\partial N_{11}^{c}}{\partial x_{1}} - \frac{\partial N_{12}^{c}}{\partial x_{2}} + \frac{\partial N_{22}^{c}}{\partial x_{1}} = 0, \\ e i t h e r θ_{2} = 0 o r P_{22} - \frac{1}{2} K_{12}^{c} + \frac{1}{2} \frac{\partial P_{32}^{c}}{\partial x_{1}} = 0, \\ e i t h e r θ_{1} = 0 o r P_{12} + \frac{1}{2} (K_{22}^{c} - K_{33}^{c}) + \frac{1}{2} \frac{\partial P_{31}^{c}}{\partial x_{1}} + \frac{1}{2} \frac{\partial P_{32}^{c}}{\partial x_{2}} = 0, \\ e i t h e r u_{0} = 0 o r N_{12} + \frac{1}{2} \frac{\partial N_{31}^{c}}{\partial x_{1}} + \frac{1}{2} \frac{\partial N_{32}^{c}}{\partial x_{2}} = 0, \\ e i t h e r \frac{\partial u_{0}}{\partial x_{2}} = 0 o r N_{32}^{c} = 0, e i t h e r \frac{\partial θ_{1}}{\partial x_{2}} = 0 o r P_{32}^{c} = 0 \end{align}

(50)

\begin{align} A t a l l c o r n e r s \\ e i t h e r w_{0} = 0 o r 2 M_{12} - N_{11}^{c} + N_{22}^{c} = 0, e i t h e r u_{0} = 0 o r N_{31}^{c} = 0, \\ e i t h e r θ_{1} = 0 o r P_{31}^{c} = 0, e i t h e r v_{0} = 0 o r N_{32}^{c} = 0, \\ e i t h e r θ_{2} = 0 o r P_{32}^{c} = 0 \end{align}

(51)

Additionally, the stiffness constants are introduced to describe the elastic stiffness response of the composite laminates described by⁴⁸:

\begin{align} [\begin{matrix} A_{i j} & B_{i j} & D_{i j} & E_{i j} & F_{i j} & H_{i j} & S_{i j} & L_{i j} \end{matrix}] \\ = \sum_{k = 1}^{N} \int_{x_{3}} [Q_{i j}^{t (k)}] [\begin{matrix} 1 & x_{3} & x_{3}^{2} & f & x_{3} f & f^{2} & f^{'} & f^{' 2} \end{matrix}] d x_{3} \end{align}

(52)

\begin{align} [\begin{matrix} A_{i}^{T} & B_{i}^{T} & D_{i}^{T} & E_{i}^{T} & F_{i j}^{T} & H_{i j}^{T} \end{matrix}] = \sum_{k = 1}^{N} \int_{x_{3}} [Q_{i j}^{t (k)}] \{\begin{matrix} α_{11} \\ α_{22} \\ α_{12} \\ 0 \\ 0 \end{matrix}\} [\begin{matrix} 1 & x_{3} & x_{3}^{2} & f & x_{3} f & f^{2} \end{matrix}] d x_{3} \end{align}

(53)

[\begin{matrix} A_{11}^{b} & E_{11}^{b} & F_{11}^{b} & H_{11}^{b} & K_{11}^{b} & L_{11}^{b} \end{matrix}] = \sum_{k = 1}^{N} \int_{x_{3}} [Q_{11}^{b c (k)}] [\begin{matrix} 1 & f^{''} & f^{''} f & f^{'' 2} & f^{'} & f^{' 2} \end{matrix}] d x_{3}

(54)

[\begin{matrix} A_{22}^{b} & E_{22}^{b} & F_{22}^{b} & H_{22}^{b} & K_{22}^{b} & L_{22}^{b} \end{matrix}] = \sum_{k = 1}^{N} \int_{x_{3}} [Q_{22}^{b c (k)}] [\begin{matrix} 1 & f^{''} & f^{''} f & f^{'' 2} & f^{'} & f^{' 2} \end{matrix}] d x_{3}

(55)

[\begin{matrix} A_{44}^{b} & E_{44}^{b} & F_{44}^{b} & H_{44}^{b} & K_{44}^{b} & L_{44}^{b} \end{matrix}] = \sum_{k = 1}^{N} \int_{x_{3}} [Q_{44}^{b c (k)}] [\begin{matrix} 1 & f^{''} & f^{''} f & f^{'' 2} & f^{'} & f^{' 2} \end{matrix}] d x_{3}

(56)

[\begin{matrix} A_{45}^{b} & E_{45}^{b} & F_{45}^{b} & H_{45}^{b} & K_{45}^{b} & L_{45}^{b} \end{matrix}] = \sum_{k = 1}^{N} \int_{x_{3}} [Q_{45}^{b c (k)}] [\begin{matrix} 1 & f^{''} & f^{''} f & f^{'' 2} & f^{'} & f^{' 2} \end{matrix}] d x_{3}

(57)

[\begin{matrix} A_{11}^{s} & E_{11}^{s} & F_{11}^{s} & H_{11}^{s} & K_{11}^{s} & L_{11}^{s} \end{matrix}] = \sum_{k = 1}^{N} \int_{x_{3}} [Q_{11}^{s c (k)}] [\begin{matrix} 1 & f^{''} & f^{''} f & f^{'' 2} & f^{'} & f^{' 2} \end{matrix}] d x_{3}

(58)

[\begin{matrix} A_{33}^{s} & E_{33}^{s} & F_{33}^{s} & H_{33}^{s} & K_{33}^{s} & L_{33}^{s} \end{matrix}] = \sum_{k = 1}^{N} \int_{x_{3}} [Q_{33}^{s c (k)}] [\begin{matrix} 1 & f^{''} & f^{''} f & f^{'' 2} & f^{'} & f^{' 2} \end{matrix}] d x_{3}

(59)

The use of these coefficients in the governing equations (44)–(48) gives rise to the determination of the governing equations (in explicit form) used for cross-ply micro-laminates given by:

\begin{align} \frac{\partial}{\partial x_{1}} (A_{11} \frac{\partial u_{0}}{\partial x_{1}} - B_{11} (\frac{\partial^{2} w_{0}}{\partial x_{1}^{2}}) + E_{11} \frac{\partial θ_{1}}{\partial x_{1}}) + \frac{\partial}{\partial x_{1}} (A_{12} \frac{\partial v_{0}}{\partial x_{2}} - B_{12} (\frac{\partial^{2} w_{0}}{\partial x_{2}^{2}}) + E_{12} \frac{\partial θ_{2}}{\partial x_{2}}) + \\ \frac{\partial}{\partial x_{2}} (A_{66} \frac{\partial u_{0}}{\partial x_{2}} + A_{66} \frac{\partial v_{0}}{\partial x_{1}} - 2 B_{66} (\frac{\partial^{2} w_{0}}{\partial x_{1} \partial x_{2}}) + E_{66} (\frac{\partial θ_{1}}{\partial x_{2}} + \frac{\partial θ_{2}}{\partial x_{1}})) \\ + \frac{1}{4} [\frac{\partial^{2}}{\partial x_{1} \partial x_{2}} (- E_{33}^{s} θ_{2}) + \frac{\partial^{2}}{\partial x_{2}^{2}} (E_{11}^{s} θ_{1})] = \frac{\partial}{\partial x_{1}} (A_{1}^{T} T_{0} + B_{1}^{T} T_{1} + E_{1}^{T} T_{2}) \end{align}

(60)

\begin{align} \frac{\partial}{\partial x_{2}} (A_{12} \frac{\partial u_{0}}{\partial x_{1}} - B_{12} (\frac{\partial^{2} w_{0}}{\partial x_{1}^{2}}) + E_{12} \frac{\partial θ_{1}}{\partial x_{1}}) + \frac{\partial}{\partial x_{2}} (A_{22} \frac{\partial v_{0}}{\partial x_{2}} - B_{22} (\frac{\partial^{2} w_{0}}{\partial x_{2}^{2}}) + E_{22} \frac{\partial θ_{2}}{\partial x_{2}}) + \\ \frac{\partial}{\partial x_{1}} (A_{66} \frac{\partial u_{0}}{\partial x_{2}} + A_{66} \frac{\partial v_{0}}{\partial x_{1}} - 2 B_{66} (\frac{\partial^{2} w_{0}}{\partial x_{1} \partial x_{2}}) + E_{66} (\frac{\partial θ_{1}}{\partial x_{2}} + \frac{\partial θ_{2}}{\partial x_{1}})) \\ - \frac{1}{4} [\frac{\partial^{2}}{\partial x_{1}^{2}} (- E_{33}^{s} θ_{2}) + \frac{\partial^{2}}{\partial x_{1} \partial x_{2}} (E_{11}^{s} θ_{1})] = \frac{\partial}{\partial x_{2}} (A_{2}^{T} T_{0} + B_{2}^{T} T_{1} + E_{2}^{T} T_{2}) \end{align}

(61)

\begin{align} \frac{\partial^{2}}{\partial x_{1}^{2}} (B_{11} \frac{\partial u_{0}}{\partial x_{1}} - D_{11} (\frac{\partial^{2} w_{0}}{\partial x_{1}^{2}}) + F_{11} \frac{\partial θ_{1}}{\partial x_{1}}) + \frac{\partial^{2}}{\partial x_{1}^{2}} (B_{12} \frac{\partial v_{0}}{\partial x_{2}} - D_{12} (\frac{\partial^{2} w_{0}}{\partial x_{2}^{2}}) \\ + F_{12} \frac{\partial θ_{2}}{\partial x_{2}}) + \frac{\partial^{2}}{\partial x_{2}^{2}} (B_{12} \frac{\partial u_{0}}{\partial x_{1}} - D_{12} (\frac{\partial^{2} w_{0}}{\partial x_{1}^{2}}) + F_{12} \frac{\partial θ_{1}}{\partial x_{1}}) + \frac{\partial^{2}}{\partial x_{2}^{2}} (B_{22} \frac{\partial v_{0}}{\partial x_{2}} \\ - D_{22} (\frac{\partial^{2} w_{0}}{\partial x_{2}^{2}}) + F_{22} \frac{\partial θ_{2}}{\partial x_{2}}) + 2 \frac{\partial^{2}}{\partial x_{1} \partial x_{2}} (B_{66} (\frac{\partial u_{0}}{\partial x_{2}} + \frac{\partial v_{0}}{\partial x_{1}}) - 2 D_{66} (\frac{\partial^{2} w_{0}}{\partial x_{1} \partial x_{2}}) \\ + F_{66} (\frac{\partial θ_{1}}{\partial x_{2}} + \frac{\partial θ_{2}}{\partial x_{1}})) - \frac{\partial^{2}}{\partial x_{2} \partial x_{1}} (A_{11}^{b} \frac{\partial^{2} w_{0}}{\partial x_{2} \partial x_{1}} - \frac{1}{2} K_{11}^{b} \frac{\partial θ_{2}}{\partial x_{1}}) \\ + \frac{\partial^{2}}{\partial x_{1} \partial x_{2}} (- A_{22}^{b} \frac{\partial^{2} w_{0}}{\partial x_{1} \partial x_{2}} + \frac{1}{2} K_{22}^{b} \frac{\partial θ_{1}}{\partial x_{2}}) - \frac{\partial^{2}}{\partial x_{2}^{2}} (A_{44}^{b} \frac{\partial^{2} w_{0}}{\partial x_{2}^{2}} - \frac{1}{2} K_{44}^{b} \frac{\partial θ_{2}}{\partial x_{2}} - A_{45}^{b} \frac{\partial^{2} w_{0}}{\partial x_{1}^{2}} \\ + \frac{1}{2} K_{45}^{b} \frac{\partial θ_{1}}{\partial x_{1}}) + \frac{\partial^{2}}{\partial x_{1}^{2}} (A_{44}^{b} \frac{\partial^{2} w_{0}}{\partial x_{2}^{2}} - \frac{1}{2} K_{44}^{b} \frac{\partial θ_{2}}{\partial x_{2}} - A_{45}^{b} \frac{\partial^{2} w_{0}}{\partial x_{1}^{2}} + \frac{1}{2} K_{45}^{b} \frac{\partial θ_{1}}{\partial x_{1}}) = - q \\ + \frac{\partial^{2}}{\partial x_{1}^{2}} (B_{1}^{T} T_{0} + D_{1}^{T} T_{1} + F_{1}^{T} T_{2}) + \frac{\partial^{2}}{\partial x_{2}^{2}} (B_{2}^{T} T_{0} + D_{2}^{T} T_{1} + F_{2}^{T} T_{2}) \end{align}

(62)

\begin{align} \frac{\partial}{\partial x_{1}} (E_{11} \frac{\partial u_{0}}{\partial x_{1}} - F_{11} (\frac{\partial^{2} w_{0}}{\partial x_{1}^{2}}) + H_{11} \frac{\partial θ_{1}}{\partial x_{1}}) + \frac{\partial}{\partial x_{1}} (E_{12} \frac{\partial v_{0}}{\partial x_{2}} - F_{12} (\frac{\partial^{2} w_{0}}{\partial x_{2}^{2}}) + H_{12} \frac{\partial θ_{2}}{\partial x_{2}}) \\ + \frac{\partial}{\partial x_{2}} (E_{66} \frac{\partial u_{0}}{\partial x_{2}} + E_{66} \frac{\partial v_{0}}{\partial x_{1}} - 2 F_{66} (\frac{\partial^{2} w_{0}}{\partial x_{1} \partial x_{2}}) + H_{66} (\frac{\partial θ_{1}}{\partial x_{2}} + \frac{\partial θ_{2}}{\partial x_{1}})) - L_{55} θ_{1} \\ + \frac{1}{2} [\frac{\partial}{\partial x_{1}} (K_{44}^{b} \frac{\partial^{2} w_{0}}{\partial x_{2}^{2}} - \frac{1}{2} L_{44}^{b} \frac{\partial θ_{2}}{\partial x_{2}} - K_{45}^{b} \frac{\partial^{2} w_{0}}{\partial x_{1}^{2}} + \frac{1}{2} L_{45}^{b} \frac{\partial θ_{1}}{\partial x_{1}}) \\ + \frac{\partial}{\partial x_{2}} (- K_{22}^{b} \frac{\partial^{2} w_{0}}{\partial x_{1} \partial x_{2}} + \frac{1}{2} L_{22}^{b} \frac{\partial θ_{1}}{\partial x_{2}}) + \frac{\partial^{2}}{\partial x_{2} \partial x_{1}} (- \frac{1}{2} F_{33}^{s} θ_{2}) + \frac{\partial^{2}}{\partial x_{2}^{2}} (\frac{1}{2} F_{11}^{s} θ_{1}) - \frac{1}{2} H_{11}^{s} θ_{1}] \\ = \frac{\partial}{\partial x_{1}} (E_{1}^{T} T_{0} + F_{1}^{T} T_{1} + H_{1}^{T} T_{2}) \end{align}

(63)

\begin{align} \frac{\partial}{\partial x_{2}} (E_{12} \frac{\partial u_{0}}{\partial x_{1}} - D_{12} (\frac{\partial^{2} w_{0}}{\partial x_{1}^{2}}) + H_{12} \frac{\partial θ_{1}}{\partial x_{1}}) + \frac{\partial}{\partial x_{2}} (E_{22} \frac{\partial v_{0}}{\partial x_{2}} - F_{22} (\frac{\partial^{2} w_{0}}{\partial x_{2}^{2}}) + H_{22} \frac{\partial θ_{2}}{\partial x_{2}}) \\ + \frac{\partial}{\partial x_{1}} (E_{66} \frac{\partial u_{0}}{\partial x_{2}} + E_{66} \frac{\partial v_{0}}{\partial x_{1}} - 2 D_{66} (\frac{\partial^{2} w_{0}}{\partial x_{1} \partial x_{2}}) + H_{66} (\frac{\partial θ_{1}}{\partial x_{2}} + \frac{\partial θ_{2}}{\partial x_{1}})) - L_{44} θ_{2} \\ + \frac{1}{2} [- \frac{\partial}{\partial x_{1}} (K_{11}^{b} \frac{\partial^{2} w_{0}}{\partial x_{2} \partial x_{1}} - \frac{1}{2} L_{11}^{b} \frac{\partial θ_{2}}{\partial x_{1}}) - \frac{\partial}{\partial x_{2}} (K_{44}^{b} \frac{\partial^{2} w_{0}}{\partial x_{2}^{2}} - \frac{1}{2} L_{44}^{b} \frac{\partial θ_{2}}{\partial x_{2}} \\ - K_{45}^{b} \frac{\partial^{2} w_{0}}{\partial x_{1}^{2}} + \frac{1}{2} L_{45}^{b} \frac{\partial θ_{1}}{\partial x_{1}}) - \frac{\partial^{2}}{\partial x_{1}^{2}} (- \frac{1}{2} F_{33}^{s} θ_{2}) - \frac{\partial^{2}}{\partial x_{1} \partial x_{2}} (\frac{1}{2} F_{11}^{s} θ_{1}) - \frac{1}{2} H_{33}^{s} θ_{2}] \\ = \frac{\partial}{\partial x_{2}} (E_{2}^{T} T_{0} + F_{2}^{T} T_{1} + H_{1}^{T} T_{2}) \end{align}

(64)

Solution methodology

The governing equations provided in their explicit forms (60)–(64) are solved using Navier’s solution procedure. This solution approach is applicable to plates with simply supported boundary conditions. For other boundary conditions such as clamped or free edges, alternative analytical approaches (e.g., the Lévy-type solution⁷¹) or numerical methods such as the finite element method may be employed to obtain the solution. The present formulation can be extended to such boundary conditions by adopting appropriate admissible displacement functions or numerical discretization schemes. The present solution assumes a simply supported arrangement, and the boundary conditions that apply on every edge of the microplate, are given as:⁴⁸

\begin{align} A t x_{1} = 0, a \\ v_{0} = w_{0} = θ_{2} = N_{11} - \frac{1}{2} \frac{\partial N_{31}^{c}}{\partial x_{2}} = M_{11} + N_{21}^{c} = P_{11} + \frac{1}{2} K_{21}^{c} - \frac{1}{2} \frac{\partial P_{31}^{c}}{\partial x_{2}} = 0 \\ N_{31}^{c} = P_{31}^{c} = 0 \\ A t x_{2} = 0, b \\ u_{0} = w_{0} = θ_{1} = N_{22} + \frac{1}{2} \frac{\partial N_{32}^{c}}{\partial x_{1}} = M_{22} - N_{12}^{c} = P_{22} - \frac{1}{2} K_{12}^{c} + \frac{1}{2} \frac{\partial P_{32}^{c}}{\partial x_{1}} = 0 \\ N_{32}^{c} = P_{32}^{c} = 0 \end{align}

(65)

To satisfy the simply supported boundary conditions, the series solution for the displacement variables is given by⁷⁷:

u_{0} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} U_{m n} c o s (p x_{1}) s i n (l x_{2})

(66)

v_{0} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} V_{m n} s i n (p x_{1}) c o s (l x_{2})

(67)

w_{0} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} W_{m n} s i n (p x_{1}) s i n (l x_{2})

(68)

θ_{1} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} X_{m n} c o s (p x_{1}) s i n (l x_{2})

(69)

θ_{2} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} Y_{m n} s i n (p x_{1}) c o s (l x_{2})

(70)

where p = πm/a and l = πn/b. The Navier solution coefficients U_mn, V_mn, W_mn, X_mn, and Y_mn are unknowns which are obtained using the closed-form solution. The mechanical load q(x₁, x₂) and temperature field are further assumed as:

q = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} q_{m n} s i n (p x_{1}) s i n (l x_{2})

(71)

where

q_{m n} = \frac{4}{a b} \int_{x_{1} = 0}^{a} \int_{x_{2} = 0}^{b} q (x_{1}, x_{2}) s i n (p x_{1}) s i n (l x_{2}) d x_{1} d x_{2}

(72)

T_{0} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} {\bar{T}}_{0} s i n (p x_{1}) s i n (l x_{2})

(73)

T_{1} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} {\bar{T}}_{1} s i n (p x_{1}) s i n (l x_{2})

(74)

T_{2} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} {\bar{T}}_{2} s i n (p x_{1}) s i n (l x_{2})

(75)

The series solutions given by equations (66)–(75) are substituted into the governing equations to transform them into an algebraic system. The resulting equations can be expressed in a compact matrix form as follows:

R U = F,

(76)

where U is the displacement vector given by:

U = {U_{m n} V_{m n} W_{m n} X_{m n} Y_{m n}}^{T},

(77)

F is the resultant force vector; R is the resultant stiffness matrix. For an invertible matrix R equation (76) can be solved to obtain U.

Results and discussion

In this section, a comprehensive investigation of bending response of laminated microplates under thermo-mechanical loading, with explicit consideration of the material length-scale effects. Numerical simulations are carried out for various laminate configurations, including single-layer, two-layer, three-layer, four-layer, and six-layer microplates, featuring cross-ply stacking sequences such as [0/90], [0/90/0], [0/90/0/90], [0/90/90/0], and [0/90/0/90/0/90].

The results are expressed as dimensionless transverse deflections under both sinusoidal and uniformly distributed loading conditions. Several examples are focused on the influence of size effects, that is, (l/h) ratio, which plays a critical role in capturing micro-scale size-dependent behavior. A systematic comparison with existing higher-order models is also carried out, demonstrating superior predictive capability of the present model. The findings highlight the necessity of incorporating scale-dependent effects for accurately modeling the mechanical response of micro-structured laminated plates.

For consistency and to facilitate direct comparison with previously published results, the present analysis employs a benchmark orthotropic material model, hereafter referred to as MM1. The constitutive behavior of each lamina is characterized by its orthotropic elastic constants, namely the Young’s moduli (Y₁₁, Y₂₂), shear moduli (G₁₂, G₁₃, G₂₃), and Poisson’s ratio (ν₁₂), which are specified as follows^71,78:

M M 1 : \frac{Y_{11}}{Y_{22}} = 25, Y_{22} = 6.9 GPa,, ν_{12} = 0.25 G_{23} = 0.2 Y_{22}, G_{12} = G_{13} = 0.5 Y_{22} .

(78)

This formulation can also be used for isotropic FG materials if (Y₁₁ = Y₂₂) is considered, taking into account the gradation of material properties through thickness. The coefficients of stiffness (equations (52)–(59)) can be defined according to the gradation law used in FG materials. Moreover, the modified couple stress theory has been used in various works^34,35,70 to study the structural response of FG structures. Unless otherwise stated, the thickness of the laminated composite microplate is assumed to be h = 25 µm for all numerical case studies presented in this section.¹ This thickness is consistent with typical dimensions of micro-electromechanical system (MEMS) components, ensuring that the selected configuration realistically reflects practical micro-scale applications.

Bending response of laminated microplates under pure mechanical loading

This subsection examines the bending behavior of laminated microplates subjected to a uniform transverse mechanical load q₀. For this representative case, the dimensionless central deflection is introduced as⁵⁸:

{\bar{w}}_{1} = w_{0} (\frac{100 Y_{22} h^{3}}{q_{0} a^{4}}),

(79)

and evaluated for a three-layered [0/90/0] and four-layered [0/90/90/0] laminate under sinusoidal loading (SSL) and uniformly distributed load (UDL).

In the first example, the numerical values for the non-dimensional deflection of the macro [0/90/0] composite plate (l/h = 0) subjected to UDL are shown in Table 2. In the present example, the results are compared with the results obtained using the 3D elasticity solution,⁷⁹ Tauratier,⁵⁰ Mantari et al.,⁷⁴ Grover et al.,⁷⁵ HSDT-1 by Grover et al.,⁵⁸ HSDT-2 by Joshan et al.,⁵⁹ zigzag (ZZ) theory,⁸⁰ HSDT-3 by Reddy,⁷² and FSDT by Reddy.⁷² In the present example, absolute percentage error for the above plate theories is also evaluated for the given plate. The table shows that the average percentage error for the present theory (1.54%) is lower than that for the above plate theories. Hence, the present plate theory can be used to solve the plate problems. The table shows that the absolute percentage error for the plate model increases as the span-to-thickness ratio decreases.

Table 2.

Dimensionless deflection ${\bar{w}}_{1}$ for [0/90/0] laminate under UDL (MM1, l/h = 0).

Theory	a/h = 100	% err.	a/h = 10	% err.	a/h = 4	% err.	Avg. err.
Present	0.6709	0.04	1.1238	2.56	2.9800	2.03	1.54
3D Elasticity⁷⁹	0.6712	-	1.1533	-	3.0416	-	-
Tauratier⁵⁰	0.6706	0.09	1.0988	4.73	2.9332	3.56	2.79
Mantari et al.⁷⁴	0.6705	0.10	1.0898	5.51	2.9085	4.38	3.33
Grover et al.⁷⁵	0.6708	0.06	1.1189	2.98	2.9730	2.26	1.77
ZZ⁸⁰	0.6709	0.04	1.1237	2.57	2.9737	2.23	1.61
HSDT-1⁵⁸	0.6709	0.04	1.1224	2.68	2.9787	2.07	1.60
HSDT-2⁵⁹	0.6705	0.10	1.0899	5.50	2.9090	4.36	3.32
HSDT-3⁷²	0.6705	0.10	1.0900	5.49	2.9091	4.36	3.32
FSDT⁷²	0.6697	0.22	1.0219	11.39	2.6596	12.56	8.06

As a next example, the considered laminated plate is subjected to SSL, and the results are presented in Table 3. The normalized stresses are expressed as⁷²:

({\bar{σ}}_{11}, {\bar{σ}}_{22}, {\bar{σ}}_{12}) = (σ_{11}, σ_{22},, σ_{12}) (\frac{h^{2}}{q_{0} a^{2}}),

(80)

({\bar{σ}}_{13}, {\bar{σ}}_{23}) = (σ_{13}, σ_{23}) (\frac{h}{q_{0} a}) .

(81)

Table 3.

Dimensionless stresses and deflection for [0/90/0] laminated plate for different a/h values (MM1, l/h = 0, SSL).

a/h	Theory	$\bar{w}$	${\bar{σ}}_{x x}$	${\bar{σ}}_{y y}$	${\bar{σ}}_{x y}$	${\bar{σ}}_{y z}$	${\bar{σ}}_{x z}$	Avg. err. (%)
a/h	Position	(a/2, b/2, 0)	(a/2, b/2, h/2)	(a/2, b/2, h/6)	(0, 0, h/2)	(a/2, 0, 0)	(0, b/2, 0)
10	Present	0.7337	0.5853	0.2760	0.0286	0.1155	0.3134	4.34
	3D Elasticity⁷⁹	0.7405	0.59	0.288	0.0289	0.123	0.357	–
	Sarangan and Singh⁵⁵	0.7321	0.5847	0.2755	0.0286	0.1140	0.3059	4.86
	Karama et al.⁵³	0.723	0.576	0.272	0.0281	0.108	0.272	8.51
	Grover et al.⁵⁸	0.7329	0.5845	0.2757	0.0286	0.1148	0.3091	4.56
	Reddy⁷²	0.7125	0.568	0.2689	0.0277	0.103	0.2447	9.11
20	Present	0.5104	0.5505	0.2066	0.0233	0.0908	0.3293	3.07
	3D Elasticity⁷⁹	-	0.552	0.21	0.0234	0.094	0.385	–
	Karama et al.⁵³	0.508	0.548	0.205	0.0231	0.086	0.285	6.31
	Grover et al.⁵⁸	0.5102	0.5503	0.2065	0.0233	0.0903	0.3252	3.15
	Reddy⁷²	0.5041	0.546	0.2043	0.0230	0.0825	0.2549	7.04
50	Present	0.4441	0.5406	0.1840	0.0216	0.0829	0.3345	2.56
	3D Elasticity⁷⁹	-	0.541	0.185	0.0216	0.084	0.393	–
	Karama et al.⁵³	0.444	0.54	0.183	0.0216	0.079	0.289	5.28
	Grover et al.⁵⁸	0.4441	0.5406	0.184	0.0216	0.0825	0.3302	2.74
100	Present	0.4345	0.5392	0.1807	0.0214	0.0817	0.3352	2.40
	3D Elasticity⁷⁹	-	0.539	0.181	0.0213	0.083	0.395	–
	Sarangan and Singh⁵⁵	0.4344	0.5392	0.1807	0.0214	0.0808	0.3275	2.56
	Karama et al.⁵³	0.435	0.538	0.18	0.0213	0.078	0.289	3.61
	Grover et al.⁵⁸	0.4344	0.5392	0.1807	0.0214	0.0813	0.3309	2.46
	Reddy⁷²	0.4342	0.539	0.1806	0.0214	0.075	0.2586	5.12

The results are compared with 3D elasticity solution,⁷⁹ Sarangan and Singh,⁵⁵ Grover et al.,⁵⁸ Karama et al.,⁵³ and Reddy’s HSDT.⁷² It is observed that the present theory performs well even for thick plates (a/h ≤ 10). For the moderately thick plate case (a/h = 10), the present formulation predicts the response with an average error of approximately (4.34%) relative to the three-dimensional elasticity solution. This error level is slightly lower than that obtained from the zigzag theory of Sarangan and Singh⁵⁵ ((4.86%)) and is also marginally better than the higher-order theory of Grover et al.⁵⁸ (4.56%). In contrast, the formulations proposed by Karama et al.⁵³ and Reddy⁷² exhibit significantly larger deviations, with average errors of approximately (8.51%) and (9.11%), respectively. This indicates that the present model provides improved accuracy compared with several classical higher-order shear deformation theories for moderately thick laminated plates.

For the thick plate case (a/h = 20), the present formulation demonstrates an average error of about (3.07%), which is comparable to the predictions of Grover et al.⁵⁸ (3.15%). The theories of Karama et al.⁵³ and Reddy⁷² show larger deviations of (6.31%) and (7.04%), respectively. These results suggest that the present formulation maintains good predictive capability for thick plates where transverse shear deformation effects are significant.

For the intermediate thickness ratio (a/h = 50), the present formulation yields an average error of approximately (2.56%), which is slightly lower than the corresponding error obtained using the formulation of Grover et al.⁵⁸ (2.74%). The theory proposed by Karama et al.⁵³ shows a comparatively larger deviation of about (5.28%). This demonstrates that the present model retains good accuracy even for moderately thin laminated plates.

For the thin-plate case (a/h = 100), the present formulation provides highly accurate predictions with an average error of about (2.40%). Comparable results are obtained using the models of Grover et al.⁵⁸ (2.46%) and Sarangan and Singh⁵⁵ (2.56%). In contrast, the classical higher-order theories of Karama et al.⁵³ and Reddy⁷² exhibit larger deviations of approximately (3.61%) and (5.12%), respectively.

Overall, the present formulation consistently demonstrates lower or comparable errors relative to several established higher-order and zigzag plate theories across a wide range of thickness ratios. The results indicate that the proposed theory is capable of accurately predicting both deflection and stress components of laminated composite plates under sinusoidal loading conditions.

Further, the size-dependent bending behavior of laminated microplates is analyzed. Tables 4 and 5 summarize the numerically computed central deflections for the two considered laminate configurations, evaluated for span-to-thickness ratios of a/h = 10 and a/h = 100, under systematic variations of l/h. The present results are benchmarked against the exact three-dimensional elasticity solution (ES)⁸¹ in the classical limit (l/h = 0) and further validated through comparison with several refined plate models, including the inverse-trigonometric shear deformation theory (ITSDT),⁴⁸ sinusoidal plate theory (SPT),⁴⁸ and Reddy’s higher-order shear deformation theory (HSDT).¹

Table 4.

Dimensionless central deflection for a three-layered micro-laminate [0/90/0] under SSL.

a/h	l/h	ES⁸¹	Present	ITSDT⁴⁸	SPT⁴⁸	HSDT¹
100	0	0.4390	0.4345	0.4344	0.4343	0.4342
	0.2		0.4271	0.4271	0.4270	0.4270
	0.25		0.4231	0.4231	0.4231	0.4230
	0.33		0.4147	0.4148	0.4148	0.4147
	0.5		0.3925	0.3926	0.3927	0.3927
	1		0.3047	0.3048	0.3051	0.3052
10	0	0.7405	0.7337	0.7307	0.7180	0.7125
	0.2		0.7061	0.7060	0.6987	0.6950
	0.25		0.6916	0.6929	0.6882	0.6856
	0.33		0.6626	0.6665	0.6669	0.6661
	0.5		0.5935	0.6021	0.6128	0.6160
	1		0.3917	0.4029	0.4276	0.4384

Table 5.

Dimensionless central deflection for a four-layered composite microplate [0/90/90/0] under sinusoidal loading.

a/h	l/h	ES⁸¹	Present	ITSDT⁴⁸	SPT⁴⁸	HSDT¹
100	0	0.439	0.4345	0.4345	0.4344	0.4343
	0.2		0.4270	0.4270	0.4271	0.4270
	0.25		0.4229	0.4230	0.4231	0.4231
	0.33		0.4144	0.4145	0.4147	0.4147
	0.5		0.3920	0.3921	0.3925	0.3926
	1		0.3043	0.3044	0.3048	0.3050
10	0	0.743	0.7286	0.7280	0.7198	0.7147
	0.2		0.6884	0.6924	0.6956	0.6945
	0.25		0.6686	0.6745	0.6829	0.6839
	0.33		0.6312	0.6399	0.6572	0.6614
	0.5		0.5517	0.5637	0.5950	0.6060
	1		0.3614	0.3699	0.4034	0.4217

The present model exhibits superior agreement with the ES in the classical regime. For finite l/h, the proposed formulation consistently captures the stiffening behavior induced by micro-structural size effects while remaining in close correspondence with other refined theories. For example, at a/h = 10 and l/h = 1, results illustrate prominent size effects considered by present model in comparison to those from ITSDT, SPT, and HSDT. Moreover, as the plate becomes slender (a/h = 100), the discrepancies among various theories diminish, and all approaches tend toward the classical response. Notably, the present formulation accurately reproduces both the conventional plate behavior (l/h → 0) and the enhanced stiffness observed at higher l/h values.

Furthermore, the through thickness distributions of the dimensionless in-plane normal stresses and transverse shear stresses are investigated for the four-layered [0/90/90/0] laminate with a span-to-thickness ratio of a/h = 10 subjected to sinusoidal surface loading (SSL). The variation of in-plane stresses $({\bar{σ}}_{11}, {\bar{σ}}_{22})$ at the plate center and transverse shear stresses $({\bar{σ}}_{23}, {\bar{σ}}_{13})$ along the edges is plotted in Figure 1 for l/h = 0, 0.5, and 1. A noticeable reduction in stress magnitudes is observed with increasing l/h, reflecting the size-dependent stiffening effect.

Figure 1.

Through thickness plots for the dimensionless transverse shear stresses $({\bar{σ}}_{23}, {\bar{σ}}_{13})$ , and in-plane normal stresses $({\bar{σ}}_{11}, {\bar{σ}}_{22})$ for [0/90/90/0] microplate (MM1, a/h = 10). A significant reduction is observed in stress magnitudes, with increasing l/h, reflecting the size-dependent stiffening effect.

Next, the bending response under uniformly distributed load (UDL) is studied. Tables 6 and 7 report the central deflections for the [0/90/0] and [0/90/90/0] laminates, respectively, for different l/h values and a/h ratios. Comparisons with ITSDT, SPT, HSDT, and ES (where available) are presented. For slender plates (a/h = 100), all models converge, while for a/h = 10, the influence of size effects is more pronounced. In both SSL and UDL cases, deflections decrease monotonically with increasing l/h, confirming the stiffening trend associated with micro-scale effects.

Table 6.

Dimensionless central deflection for [0/90/0] micro-laminate under UDL.

a/h	l/h	ES⁸¹	Present	ITSDT⁴⁸	SPT⁴⁸	HSDT¹
100	0	0.6712	0.6709	0.6708	0.6706	0.6705
	0.2		0.6612	0.6612	0.6610	0.6609
	0.25		0.6558	0.6558	0.6557	0.6556
	0.33		0.6443	0.6444	0.6443	0.6443
	0.5		0.6128	0.6130	0.6132	0.6132
	1		0.4809	0.4811	0.4816	0.4817
10	0	1.1533	1.1238	1.1189	1.0988	1.0892
	0.2		1.0851	1.0848	1.0730	1.0650
	0.25		1.0644	1.0664	1.0587	1.0516
	0.33		1.0226	1.0287	1.0288	1.0233
	0.5		0.9208	0.9341	0.9503	0.9486
	1		0.6136	0.6308	0.6683	0.6747

Table 7.

Dimensionless central deflection for four-layered micro-laminate [0/90/90/0] under UDL.

a/h	l/h	Present	ITSDT⁴⁸	SPT⁴⁸	HSDT¹
100	0	0.6845	0.6845	0.6844	0.6843
	0.2	0.6730	0.6731	0.6731	0.6731
	0.25	0.6667	0.6668	0.6669	0.6669
	0.33	0.6535	0.6536	0.6540	0.6540
	0.5	0.6187	0.6190	0.6196	0.6197
	1	0.4816	0.4817	0.4824	0.4826
10	0	1.1270	1.126	1.1134	1.1049
	0.2	1.0662	1.0724	1.0776	1.0734
	0.25	1.0362	1.0454	1.0586	1.0566
	0.33	0.9796	0.9932	1.0201	1.0222
	0.5	0.8590	0.8775	0.9261	0.9364
	1	0.5673	0.5804	0.6318	0.6516

Finally, Figure 2 illustrates the influence of stacking sequence by comparing multilayered anti-symmetric laminates [0/90]_n (n = 1, 2, 3) with a/h = 10 under SSL. A decreasing trend in deflection with l/h is evident, with the two-layered laminate showing the steepest reduction compared to the other cases.

Figure 2.

Dimensionless deflection for anti-symmetric [0/90]_n microplates.

In addition, the effect of the modular ratio (Y₁₁/Y₂₂) is examined on the deflection response for [0/90/90/0] micro-laminate. The results thus obtained are given in Figure 3 for l/h ratios varying from 0 to 1 and a/h = 10. The ratio (Y₁₁/Y₂₂) has an inverse relationship with the deflection when it comes to dimensionless form.

Figure 3.

Dimensionless deflection with varying l/h for different (Y₁₁/Y₂₂) ratios for [0/90/90/0] microplates.

Bending response of laminated microplates under thermal loading

This subsection addresses the static bending behavior of cross-ply microplates under a pure thermal load. The dimensionless central deflection is defined as⁷⁶:

{\bar{w}}_{2} = w_{0} (\frac{10 h}{T_{2} α_{11} a^{2}}),

(82)

Table 8 presents the maximum dimensionless central deflections of orthotropic and three-layered cross-ply laminates subjected to a linearly varying sinusoidal thermal field. The predictions from the present theory are compared with those of ITSDT,⁸² HSDT,⁷⁶ and SPT⁷⁶ for a/h = 10, 20, 100. It is observed that the present results are in excellent agreement with HSDTs across all cases. This highlights the capability of the proposed formulation in accurately capturing thermal bending responses, while maintaining computational efficiency. Furthermore, the results are very close to those of ITSDT, HSDT, and SPT, with only marginal deviations (less than 0.5%). For slender plates (a/h = 100), all theories converge to nearly identical predictions, as expected due to the reduced influence of transverse shear deformation. In contrast, for a/h = 10 and 20, the present theory demonstrates consistency with refined higher-order models while effectively representing the thermal bending behavior of both single-layer and multilayer laminates.

Table 8.

Dimensionless deflection of orthotropic and three-layer cross-ply plate [0/90/0] under linearly varying sinusoidal thermal field (T₁ = 100, T₀ = T₂ = 0, l/h = 0).

Theory	[0]			[0/90/0]
Theory	a/h = 10	a/h = 20	a/h = 100	a/h = 10	a/h = 20	a/h = 100
HSDT⁷⁶	1.0439	1.0346	1.0313	1.1463	1.1087	1.0950
SPT⁷⁶	1.0438	1.0346	1.0313	1.1475	1.1090	1.0950
ITSDT⁸²	1.0436	1.0345	1.0313	1.1498	1.1098	1.0950
Present	1.0435	1.0345	1.0313	1.1509	1.1101	1.0950

In addition, the effect of the aspect ratio (a/b) of the plate on the thermo-mechanical response of the laminated composite plate is also considered. The results given in Table 9 show the effects of (a/h) and (a/b) on the maximum value of the non-dimensional transverse displacement of a three-layer symmetric laminate subjected to the sinusoidally varying temperature distribution along its surface. The results obtained indicate that the current approach yields a close correlation with existing theories, such as HSDT and SPT.⁷⁶ As the span to thickness ratio increases from a/h = 10 to a/h = 100, the dimensionless deflection is found to decrease gradually. This implies that the plate is becoming a thin plate. As the aspect ratio a/b increases for a given a/h, the dimensionless deflection is found to decrease. This implies that the plate is becoming stiffer. The good agreement between the results of the proposed formulation and the results of the existing solution in the literature proves the reliability of the proposed formulation in the analysis of the thermoelastic bending of a rectangular laminated composite plate.

Table 9.

Effect of aspect ratio on the dimensionless deflection for [0/90/0] plate (T₁ = 100, T₀ = T₂ = 0, l/h = 0, α₂₂/α₁₁ = 3).

a/h	Model	a/b = 0.33	a/b = 0.5	a/b = 1.5	a/b = 2
10	Present	1.0734	1.1031	1.0006	0.7431
	HSDT⁷⁶	1.0724	1.1008	0.9997	0.7455
	SPT⁷⁶	1.0726	1.1014	0.9999	0.7449
20	Present	1.0628	1.0815	0.9897	0.7574
	HSDT⁷⁶	1.0625	1.0808	0.9892	0.7583
	SPT⁷⁶	1.0626	1.0810	0.9893	0.7581
100	Present	1.0593	1.0741	0.9847	0.7642
	HSDT⁷⁶	1.0593	1.0741	0.9847	0.7642
	SPT⁷⁶	1.0593	1.0741	0.9847	0.7642

Tables 10 and 11 summarize the maximum dimensionless central deflections of orthotropic [0] and symmetric cross-ply [0/90/0] microplates subjected to sinusoidal and uniform temperature fields, respectively, for varying l/h and a/h. It is evident that in both loading cases, the consideration of the length scale significantly reduces the predicted deflections, with a more pronounced reduction observed as l/h increases. For example, in the sinusoidal thermal load case, the central deflection of the [0] laminate decreases from 1.0435 at l/h = 0 to 0.6389 at l/h = 1 for a/h = 10, representing a reduction of nearly 38.8%. Similarly, for the cross-ply laminate, the deflection decreases from 1.1509 to 0.7346, corresponding to a reduction of approximately 36.2%. This decrease in deflection with increasing l/h clearly demonstrates the stiffening effect of the material length-scale parameter. The results indicate that the size-dependent behavior becomes increasingly significant at larger l/h values, leading to enhanced resistance to thermo-mechanical deformation in microplate structures.

Table 10.

Maximum dimensionless deflection ${\bar{w}}_{2}$ for orthotropic [0] and symmetric cross-ply microplate [0/90/0] under sinusoidal thermal field (T₁ = 100, T₀ = T₂ = 0, α₂₂/α₁₁ = 3).

l/h	[0]			[0/90/0]
l/h	a/h = 10	a/h = 20	a/h = 100	a/h = 10	a/h = 20	a/h = 100
0	1.0435	1.0345	1.0313	1.1509	1.1101	1.0950
0.2	1.0186	1.0154	1.0142	1.1219	1.0891	1.0768
0.25	1.0049	1.0049	1.0048	1.1064	1.0777	1.0669
0.33	0.9764	0.9829	0.9851	1.0749	1.0540	1.0459
0.5	0.9024	0.9250	0.9328	0.9964	0.9924	0.9905
1	0.6389	0.7010	0.7251	0.7346	0.7610	0.7703

Table 11.

Dimensionless deflection for orthotropic [0] and symmetric cross-ply microplate [0/90/0] under uniform temperature field (T₁ = 100, T₀ = T₂ = 0, α₂₂/α₁₁ = 3).

l/h	[0]			[0/90/0]
l/h	a/h = 10	a/h = 20	a/h = 100	a/h = 10	a/h = 20	a/h = 100
0	1.4606	1.4419	1.4347	1.6530	1.5825	1.5547
0.2	1.4529	1.4395	1.4340	1.6220	1.5619	1.5378
0.25	1.4445	1.4352	1.4309	1.6046	1.5501	1.5280
0.33	1.4210	1.4213	1.4202	1.5675	1.5243	1.5064
0.5	1.3382	1.3650	1.3733	1.4682	1.4514	1.4432
1	0.9635	1.0582	1.0949	1.1009	1.1360	1.1480

Moreover, the cross-ply [0/90/0] laminate consistently exhibits larger deflections than the single-layer [0] laminate across all a/h and l/h values. The effect is most significant for a/h = 10 and gradually diminishes for slender plates (a/h = 100), where the influence of transverse shear deformation and scale effects becomes negligible, and the responses approach those predicted by classical plate theory.

The influence of (a/b) on the non-dimensional transverse displacement of a three-layered [0/90/0] rectangular microplate under sinusoidal temperature distribution is shown in Table 12. The deflection reduces from 1.0734 to 0.9439 (12.1%) for a/b = 1/3 and from 0.7431 to 0.2762 (62.8%) for a/b = 2 as (l/h) increases from 0 to 1, indicating enhanced stiffness due to size effects. Additionally, for a fixed (l/h), deflection decreases with increasing (a/b), confirming that plates with higher aspect ratios exhibit lower thermoelastic deformation.

Table 12.

Dimensionless deflection of [0/90/0] rectangular microplate with varying l/h and a/b (T₁ = 100, T₀ = T₂ = 0, α₂₂/α₁₁ = 3).

l/h	a/b = 0.33	a/b = 0.5	a/b = 1.5	a/b = 2
0	1.0734	1.1031	1.0006	0.7431
0.2	1.0667	1.0931	0.9514	0.6903
0.25	1.0631	1.0877	0.9261	0.6642
0.33	1.0554	1.0763	0.8766	0.6149
0.5	1.0346	1.0463	0.7634	0.5104
1	0.9439	0.9219	0.4649	0.2762

In addition to this, the study is further carried out for anti-symmetric micro-laminates having the stacking sequence [0/90]_n (n = 1, 2, 3), which are exposed to sinusoidal thermal loading (T₁ = 100, T₀ = T₂ = 0, α₂₂/α₁₁ = 3). The plot between ${\bar{w}}_{2}$ and (l/h) for micro-laminates having a/h = 10 is shown in Figure 4. From this graph, it is clear that that ${\bar{w}}_{2}$ decreases as l/h increases, demonstrating the stiffening effect imparted by the size-dependent parameter.

Figure 4.

Dimensional deflection ${\bar{w}}_{2}$ for [0/90]_n microplates under sinusoidal thermal load.

Furthermore, the effect of the thermal expansion coefficient ratio (α₂₂/α₁₁) on the dimensionless deflection ${\bar{w}}_{2}$ of a four-layered micro-laminate [0/90/90/0] is investigated under a sinusoidal thermal field. The corresponding results, presented in Figure 5 for a/h = 10 and different l/h values, reveal a nearly linear increase in ${\bar{w}}_{2}$ as the ratio α₂₂/α₁₁ increases. This trend is consistently observed across all considered values l/h, indicating that the anisotropy in thermal expansion significantly governs the thermo-mechanical response of the micro-laminates.

Figure 5.

Dimensionless deflection for varying l/h and different thermal coefficient ratios (α₂₂/α₁₁) for four-layered microplate [0/90/90/0].

Figure 6 illustrates the variation of ${\bar{w}}_{2}$ with size parameter (l/h) for [0/90/0] microplate. Two different types of through thickness temperature distributions are considered, namely the purely linear temperature field (T₁ = 100, T₀ = T₂ = 0), and the combined linear–nonlinear temperature field (T₁ = T₂ = 100, T₀ = 0). It can be observed that ${\bar{w}}_{2}$ decreases consistently (34.5%) with an increase in the size parameter. This trend highlights the significant influence of the length-scale parameter on the bending response of laminated microplates, indicating that the structural stiffness effectively increases as the size-dependent effects become more pronounced. Furthermore, among the considered thermal loading cases, the higher deflection values are obtained for the combined linear–nonlinear temperature distribution. The results clearly demonstrate the importance of considering both the temperature gradient distribution and the material length-scale parameter when analyzing the thermoelastic response of laminated micro-scale structures.

Figure 6.

Variation in dimensionless deflection with l/h for different types of through thickness temperature variations for a three-layered microplate [0/90/0].

Bending response of laminated microplates under combined thermal and mechanical loads

This subsection addresses the static bending behavior of cross-ply microplates under combined thermo-mechanical load. For this case, the dimensionless central deflection is defined as⁷⁶:

{\bar{w}}_{3} = w_{0} {[(\frac{q_{0} a^{4}}{h^{3} λ}) + (\frac{α_{11} T_{2} a^{2}}{10 h})]}^{- 1}

(83)

where

λ = (\frac{π^{4}}{12}) [4 G_{12} + \frac{Y_{11} + (1 + ν_{12}) Y_{22}}{1 - ν_{12} ν_{21}}]

(84)

Table 13 presents the dimensionless deflection

{\bar{w}}_{3}

of orthotropic [0] and symmetric cross-ply plate [0°/90°/0°] subjected to a combined sinusoidal thermal field and transverse loading l/h = 0, T₁ = 100, q = 100. The results are computed for a/h = 10, 20, and 100. It can be observed that the present model provides results in excellent agreement with ITSDT,⁸² HSDT,⁷⁶ and SPT,⁷⁶ thereby validating its accuracy. For both lamination schemes, the deflection decreases with increasing a/h, as expected due to the increased plate stiffness in thinner configurations. The cross-ply [0°/90°/0°] plate exhibits slightly higher deflections than the orthotropic [0] plate, reflecting the relative reduction in bending stiffness due to the inclusion of transverse plies. The differences among the various higher-order theories remain marginal, particularly for large a/h, where all models tend to converge toward the thin-plate limit. Overall, the present formulation captures both the laminate configuration effects and loading influences with high fidelity, further demonstrating its robustness.

Table 13.

Dimensionless deflection ${\bar{w}}_{3}$ of orthotropic [0] and symmetric cross-ply plate [0/90/0] subjected to sinusoidal thermal and mechanical loading (l/h = 0, T₁ = 100, T₀ = T₂ = 0, q = 100, α₂₂/α₁₁ = 3, α₁₁ = 10⁻⁶).

Theory	[0°]			[0°/90°/0°]
Theory	a/h = 10	a/h = 20	a/h = 100	a/h = 10	a/h = 20	a/h = 100
Present	1.4488	1.1079	0.9959	1.6865	1.1732	0.9986
ITSDT⁸²	1.4553	1.1094	0.9960	1.6754	1.1698	0.9985
HSDT⁷⁶	1.4634	1.1113	0.9961	1.6366	1.1587	0.9980
SPT⁷⁶	1.4621	1.1111	0.9960	1.6493	1.1624	0.9982

The influence of thermal and transverse loadings (sinusoidal and uniformly distributed) on the size-dependent deflection of a single-ply [0] and symmetric cross-ply [0°/90°/0°] microplate is discussed through Tables 14 and 15. The deflection decreases monotonically with increasing (l/h) ratio for both cases, highlighting the stiffening effect introduced by the couple stress theory at the micro-scale. Similarly, an increase in a/h ratio results in reduced deflection, consistent with the trend observed in slender plates. The cross-ply configuration consistently exhibits higher deflections compared to the orthotropic plate, which can be attributed to its lower bending rigidity due to the inclusion of transverse plies. Furthermore, the uniform loading case yields significantly higher deflections than the sinusoidal loading case under the same thermal and mechanical conditions, indicating the stronger influence of uniform load distribution on the overall plate response. At very large values of a/h, the deflection values tend to converge toward the thin-plate predictions, showing the asymptotic consistency of the present model with classical plate theory in the absence of size effects.

Table 14.

Size-dependent dimensionless deflection of single-ply [0] and [0/90/0] microplates under sinusoidal thermo-mechanical load (T₁ = 100, T₀ = T₂ = 0, q = 100, α₂₂/α₁₁ = 3, α₁₁ = 10⁻⁶).

l/h	[0]			[0/90/0]
l/h	a/h = 10	a/h = 20	a/h = 100	a/h = 10	a/h = 20	a/h = 100
0	1.4488	1.1079	0.9959	1.6865	1.1732	0.9986
0.2	1.4117	1.0870	0.9794	1.6229	1.1453	0.9818
0.25	1.3918	1.0755	0.9703	1.5896	1.1303	0.9726
0.33	1.3508	1.0517	0.9513	1.5229	1.0995	0.9532
0.5	1.2464	0.9893	0.9008	1.3641	1.0218	0.9021
1	0.8805	0.7494	0.7002	0.9003	0.7530	0.7003

Table 15.

Size-dependent dimensionless deflection of single-ply [0] and symmetric three-layer [0/90/0] microplates under uniform temperature field and uniform transverse field (T₁ = 100, T₀ = T₂ = 0, q = 100, α₂₂/α₁₁ = 3, α₁₁ = 10⁻⁶).

l/h	[0]			[0/90/0]
l/h	a/h = 10	a/h = 20	a/h = 100	a/h = 10	a/h = 20	a/h = 100
0	2.1632	1.6641	1.5003	2.5831	1.8062	1.5421
0.2	2.1340	1.6491	1.4888	2.4940	1.7681	1.5198
0.25	2.1143	1.6389	1.4809	2.4466	1.7473	1.5074
0.33	2.0680	1.6142	1.4619	2.3505	1.7040	1.4809
0.5	1.9304	1.5366	1.4010	2.1164	1.5920	1.4086
1	1.3796	1.1822	1.1080	1.4104	1.1857	1.1053

In addition, the response of the anti-symmetric micro-laminates [0/90]_n (n = 1, 2, 3) is investigated when subjected to sinusoidal thermo-mechanical loading. Figure 7 shows the effect of (l/h) on ${\bar{w}}_{3}$ for plates with a/h = 10. As can be seen, ${\bar{w}}_{3}$ tends to decrease as l/h increases, which is attributed to the stiffness enhancement effect of size-dependent behavior. In addition, the rate of decrease is greater for the two-layer micro-laminate than for other arrangements, implying that the effect of scale becomes less significant with an increase in the number of layers because of increased bending stiffness.

Figure 7.

Dimensionless deflection for [0/90]_n microplate under sinusoidal thermo-mechanical load.

Finally, the influence of the material anisotropy ratio (Y₁₁/Y₂₂) on the dimensionless deflection ${\bar{w}}_{3}$ of a four-layered micro-laminate [0/90/90/0] is examined under a sinusoidal thermo-mechanical loading condition. The results, illustrated in Figure 8 for different values of l/h with a/h = 10, indicate a consistent reduction in ${\bar{w}}_{2}$ as the ratio Y₁₁/Y₂₂ increases. This decreasing trend is observed for all considered l/h values, signifying that higher stiffness along the principal material direction effectively suppresses the overall deflection response of the micro-laminate.

Figure 8.

Variation in dimensionless deflection ${\bar{w}}_{3}$ with l/h for different material anisotropy ratios (Y₁₁/Y₂₂) for four-layered microplate [0/90/90/0] under thermo-mechanical load.

Conclusion

Laminated composite microplates have been analyzed within the framework of a scale-dependent higher-order shear deformation theory integrated with the refined couple stress theory. The proposed model incorporates material anisotropy and accounts for thermo-mechanical loading effects. Analytical solutions are obtained using Navier’s method for simply supported boundary conditions, and the results are benchmarked against established higher-order theories.

The influence of length-scaling effects, laminate configuration, and thermal loading on the structural response is systematically investigated, leading to the following key observations:

• Deflection behavior: The inclusion of scale effects leads to a noticeable reduction in transverse deflections. The stiffening influence is more significant in moderately thick plates, while slender plates show negligible deviation from classical models.

• Stress distribution: In-plane stresses and transverse shear stresses decrease with increasing length-scale parameter.

• Thermal loading: Both uniform and sinusoidal temperature fields highlight the stiffening role of length-scale effects. Under sinusoidal thermal loading, the central deflection decreases significantly with increasing size parameter, with reductions of about 38.8% for the [0] laminate and 36.2% for the cross-ply laminate as l/h increases from 0 to 1, highlighting the pronounced size-dependent stiffening effect.

• Comparative accuracy: Results show excellent agreement with inverse-hyperbolic shear deformation theory and other refined plate theories, confirming the accuracy and computational efficiency of the present formulation.

These findings confirm the effectiveness of the proposed scale-dependent model in capturing the thermo-mechanical response of laminated composite microplates. The proposed formulation serves as an analytical benchmark for assessing the accuracy of numerical methods in the analysis and design of advanced micro-scale structures, particularly in regimes where conventional plate theories fail to capture size-dependent effects. Furthermore, the model holds strong practical relevance for applications in micro-electromechanical systems (MEMS), flexible electronics, biomedical devices, and aerospace microstructures, where accurate prediction of thermo-mechanical behavior is essential for ensuring structural reliability and performance. Future studies might focus on experimental testing the proposed model for micro-scale structures to confirm its size-dependent predictions. Additionally, extending the current formulation to include dynamic, vibration, and stability analyses under varying thermo-mechanical loading conditions would provide a more comprehensive understanding of the behavior and performance of laminated microstructures.

Footnotes

ORCID iDs

Manu Dev Sharma

Yadwinder Singh Joshan

Neeraj Grover

Lakshay Taneja

Kishore Khanna

Author contributions

Manu Dev Sharma: Formal analysis, investigation, methodology, software, validation, and writing—original draft.

Yadwinder Singh Joshan: Conceptualization, methodology, supervision, visualization, and writing—review and editing.

Neeraj Grover: Conceptualization, supervision, visualization, and writing—review and editing.

Lakshay: Solution methodology, supervision, visualization, and writing—review and editing.

Kishore Khanna: Numerical verification, supervision, and writing—review and editing.

All authors have read and approved the final version of the manuscript and agree to be accountable for all aspects of the work.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

Data Availability Statement

No data was used for the research described in the article.*

Appendix

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