In recent engineering applications, structural components of complex shapes are frequently employed in severe environmental conditions [1,2], necessitating refined structural models to predict the bending response under external environmental variables [3,4] with a reduced computational effort. In some applications, high temperature and severe humidity can significantly orient the design process, making it essential to understand their influence on the safety assessment [5–7]. In composite materials, moisture can alter the mechanical properties of both matrix and reinforcing fibers [8–10]. In addition, exposure to high temperature and humidity can lead to additional strains, which alter the mechanical response of the structure [11]. Therefore, multifield formulations are found in literature, which couple various physical problems, including heat conduction, mass diffusion, and electro-mechanical fields [12–15]. For hygro-thermal problems, experimental evidence shows that moisture-induced and thermally-induced deformations are comparable and must be considered. Most theories use classical Fourier and Fick equations to derive temperature and moisture concentration distributions. Then, the corresponding deformations are calculated based on these results [16–18]. The thermal and hygroscopic effects can be evaluated by assuming the analogy between heat conduction and moisture diffusion, as shown in the literature [19,20]. When these phenomena are simultaneously present, a stationary version of the Fourier heat transfer equations can be adopted because thermal equilibrium usually reaches more rapidly than mass equilibrium within the solid [21,22]. However, transient moisture diffusion eventually reaches equilibrium moisture content, which depends on temperature and other environmental variables [23,24]. Research activities by Sih et al. [25–27] highlight the connection between thermal and hygrometric problems from the Onsager reciprocity theorem [28], thus enabling the computation of the Soret and Dufour effects [29–32]. These effects indicate that temperature variation induces moisture to move within the solid, and mass migration slightly alters the temperature. Once the coupled thermo-diffusion theory is established, modeling strategies are introduced to derive the corresponding mechanical response. Many papers address hygro-thermal analysis of laminated structures [33–37] using uncoupled formulations, solving Fick and Fourier equations separately. In this context, a useful paper is Liu et al.’s [37], where the staggered solution of the hygro-thermal problem is derived to update the diffusion coefficient of the constituent material as the temperature varies within the solid. Most modeling strategies consider the infinite body assumption for hygro-thermal analysis [38–40]. They adopt the Classical Plate Theory (CPT) or the First Order Shear Deformation Theory (FSDT) to derive the corresponding mechanical response [41–43]. The solution deviates from that derived from classical approaches for thick and very thick structures, and temperature distribution differs from the typical linear distribution. For example, when granular composite materials are adopted [44–48] with an arbitrary variation of material properties along the panel thickness [49,50], the structural behavior cannot be investigated through classical theories like the CPT and FSDT because they exhibit non-uniform bending [51,52]. For this reason, in mechanical elasticity problems, these structures are mainly studied employing Higher Order Shear Deformation Theories (HSDT) [53–56] or three-dimensional formulations. In addition, recent works [57,58] have shown that when an erroneous temperature and moisture content along the thickness is predicted, inaccurate hygro-thermal deformations within a mechanical model, thus invalidating the design process. Higher-order polynomials can thus be adopted to describe the effective temperature and moisture concentration distribution, which can exhibit abrupt variations in heterogeneous solids, as shown in the literature [59,60]. This aspect can be viewed as a generalization of the zigzag behavior observed in the mechanical case, which was consistently proposed in the literature [61,62]. Consequently, advanced kinematic models should be adopted [63–65], which employ fewer Degrees of Freedom (DOFs). Starting from the pioneering Third Order Shear Deformation Theory (TSDT), refined theories have been derived, namely HSDT, along with zigzag functions in the case of laminated structures, which usually adopt a generalized kinematic model, first proposed in the works by Washizu and Reddy [66,67], based on an arbitrary set of thickness functions. These functions can be assessed following the Equivalent Single Layer (ESL) and the Layer-Wise (LW) approaches [68–71]. In ESL, the configuration variables refer to the entire lamination scheme as they are approximated along the whole thickness of the panel, while LW theories account for unknown variables distributed along the thickness of each layer. In addition to ESL and LW, a hybrid approach called Equivalent Layer-Wise (ELW) has been developed and extensively adopted in References [72,73]. More specifically, the ELW theory allows one to prescribe arbitrary values of the displacement fields at the top and bottom surfaces of the shell. The generalized formulation has been extensively applied to various laminated plate and shell structures with advanced materials [74–76], showing its suitability even in the case of very complicated lamination schemes with softcore, porous anisotropic material, three-dimensional variation of the material properties, and orientation angle. These advanced models, applied to structures of very complex shape and boundary conditions, are tackled numerically with advanced computational techniques like spectral collocation methods since classical approaches usually require a very high computational effort. Among these methods, the Generalized Differential Quadrature (GDQ) [77–79] enables the discretization of arbitrary order derivatives with proper weighting coefficients that depend on the function’s interpolation. Several applications can be found in the literature in which the GDQ weighting coefficients are evaluated with a recursive procedure based on Lagrange interpolating polynomials [80–82]. However, in recent work [83], it has been shown that the accuracy of GDQ-based numerical models using Fourier trigonometric approximating expansions is comparable to that of Lagrange polynomials. For this reason, the Fourier-based GDQ (F-GDQ) numerical technique is introduced. It is also possible to perform integrals after some considerations regarding the GDQ method, leading to the assessment of the Generalized Integral Quadrature (GIQ) numerical technique. The main advantage of the GDQ method is that it enables, from the derivative discretization, the direct numerical solution of the differential equations governing a physical problem in a strong form [84,85], exhibiting a very high convergence rate with a limited number of DOFs. For this reason, it is extensively adopted in many applications that require a very high computational effort when using classical approaches like the Finite Element Method (FEM). Such numerical solutions are validated comparatively for analytical solutions, which can be derived, for instance, following Navier’s method [86,87].

Accordingly, the literature presents some issues regarding hygro-thermal modeling of laminated structures. Above all, most papers deal with subsequent modeling since they account for heat transfer and mass diffusion equations separately and then derive the corresponding mechanical response due to thermal and hygroscopic expansion. This can lead to inaccurate results in some applications since this approach does not model the hygro-thermal coupling. A coupled model was developed in Tornabene [12] to overcome this, where hygro-thermal loads were observed only as sinusoidal distribution of temperature and moisture concentration variation. Therefore, further investigations into more realistic hygro-thermal loading are required. However, the multifield model cited above is suitable only for structures exhibiting complete multifield response. Therefore, a specific HSDT formulation is required to perform hygro-thermal analysis of moderately thick laminated structures with advanced lamination schemes without considering electromagnetic modeling. In this way, unlike Tornabene [12], it is possible to investigate the hygro-thermal coupling even for those materials that cannot exhibit electric and magnetic properties. This study adopts the ELW approach to derive a generalized model for evaluating the fully-coupled hygro-thermo-mechanical response of anisotropic doubly-curved shells under thermodynamic equilibrium conditions to overcome all these limitations. A generalized kinematic model with higher-order theories and zigzag functions is adopted to predict stretching and interlaminar effects. The fundamental relations are derived in curvilinear principal coordinates considering the coupling effects between the physical problems involved in the formulation, and a semi-analytical solution is found for simply supported cross-ply structures with uniform curvature. The formulation in this study focuses on hygro-thermal coupling, while in Tornabene [12], a general formulation is provided where various fields are considered, including electricity and magnetostatics, along with hygro-thermal coupling. The recovery procedure is reported for cross-ply lamination schemes to make the formulation efficient, while in Tornabene [12], it is provided for generally anisotropic materials. The present model is not intended to investigate the possible generation of electric and magnetic fluxes induced by pyroelectricity and pyromagnetic effects, as happens in Tornabene [12] instead. The model is applied in various numerical investigations. Preliminary validating simulations are performed, where the bending response derived from the present formulation is compared to that from a numerical model developed with commercial software based on the three-dimensional FEM (3D-FEM). Then, a systematic investigation is presented, demonstrating the coupling between heat conduction and mass diffusion equations. This analysis is entirely new from those in Tornabene [12], where the sensitivity was not studied. Unlike the numerical examples in Tornabene [12], where only prescribed values of multifield configuration variables with sinusoidal distributions are considered external loads, arbitrary distributions are addressed of hygro-thermo-mechanical secondary variables. The present formulation can be utilized to determine the bending response of laminated structures for various curvatures in a thermal and hygrometric environment with general loads in terms of fluxes and configuration variables. In addition, a semi-analytical procedure is applied to evaluate the coupling effect of hygro-thermal equations on the bending response of the structures, making it a valuable tool for design purposes.

Based on the ELW theory, a doubly-curved shell solid is expressed through the geometric properties of the reference surface, denoted by r(α1,α2). The reference surface is intended to be located in the middle thickness. As a consequence, the position vector R can be described as follows [12]: (1)R(α1,α2,ζ)=r(α1,α2)+h2zn(α1,α2)where z=2ζ/h is a dimensionless parameter that identifies the points distributed along the thickness direction, while n(α1,α2) denotes the normal unit vector calculated in each point of the reference surface. Finally, h is the total thickness of the solid, which is computed as the sum of the thicknesses hk of each k-th layer of the stacking sequence, setting k=1,…,l: (2)h(α1,α2)=∑k=1lhk(α1,α2)=∑k=1l(ζk+1−ζk)

The geometric properties of the shell solid are derived from those of the reference surface r introduced in Eq. (1). To this end, the principal radii of curvatures Ri=R1,R2 and the Lamè parameters Ai=A1,A2 are evaluated for each αi=α1,α2 principal direction: (3)Ri(α1,α2)=−r,i⋅r,ir,ii⋅n,Ai(α1,α2)=r,i⋅r,i

The Lamè parameter Ai is required to compute the curvilinear abscissa, defined as dsi=Aidαi, of an infinitesimal arc of the parametric lines along each principal direction, being dαi the infinitesimal variation of the curvilinear coordinate αi. The integration of dsi along the closed interval [αi0,αi] leads to the definition of the curvilinear abscissa si=s1,s2 defined so that si∈[si0,si1]. On the other hand, the principal radii of curvature Ri, defined in Eq. (3), allows one to define the scaling parameter Hi=H1,H2, which tells how the presence of curvature along αi affects the computation of distances within the three-dimensional shell solid: (4)Hi(α1,α2,ζ)=1+ζRi

Finally, the principal curvature of the shell along α1 and α2 principal directions, denoted as kni=kn1,kn2, are evaluated as kn1=1/R1 and kn2=1/R2, respectively.

The ELW approach and the unified formulation are adopted here to derive a generalized kinematic model coupling the mechanical elasticity problem and the hygro-thermal fundamental relations in stationary conditions. For a three-dimensional solid described with α1,α2,ζ principal coordinates, the mechanical equations are written regarding the displacement field components U1(k),U2(k),U3(k). On the other hand, ΔT(k)=T(k)−T0 accounts for the variation of the absolute temperature of each point of the shell for the reference temperature T0. In the same way, the variation ΔC(k)=C(k)−C0 of the moisture concentration is considered for the reference condition C0, evaluated from the equilibrium moisture content of the material. These quantities are conveniently arranged in the vector Δ(k)(α1,α2,ζ), defined in each point of the three-dimensional solid, which collects the configuration variables of the hygro-thermo-mechanical problem [12]. Employing the unified formulation, the vector Δ(k) is expanded in terms of generalized thickness functions denoted by Fτ(k)αi=Fτ(k)αi(ζ) with i=1,…,5, depending on the thickness coordinate ζ and defined for each τ-th kinematic expansion order with τ=0,…,N+1 [12]: (5)Δ(k)=∑τ=0N+1Fτ(k)δ(τ)⇔[U1(k)U2(k)U3(k)ΔT(k)ΔC(k)]=∑τ=0N+1[Fτ(k)α100000Fτ(k)α200000Fτ(k)α300000Fτ(k)α400000Fτ(k)α5][u1(τ)u2(τ)u3(τ)ξ(τ)κ(τ)]where Fτ(k) is the thickness function matrix, while δ(τ)=[u1(τ)u2(τ)u3(τ)ξ(τ)κ(τ)]T denotes the vector of the generalized configuration variables of the formulation, introduced for each τ=0,…,N+1. The kinematic expansion of Eq. (5) allows one to derive a generalized model with an arbitrary description of the configuration variables, which can be selected based on the lamination scheme object of investigation. Therefore, classical theories like FSDT and TSDT can also be described through Eq. (5). This study considers a higher order axiomatic assumption of the unknown field variable accounting for higher order power functions so that stretching effects and unusual in-plane deformations can be predicted by the two-dimensional model. Hence, the following thickness functions set [12] is adopted from τ=0 to τ=N: (6)Fτ(k)αi={1−z~2forτ=0z~τ+1−z~τ−1forτ=1,..,N−11+z~2forτ=Nwhere the dimensionless quantity z~∈[−1,1] is expressed in terms of the thickness coordinate ζ so that z~=2ζ/h. A representation of the thickness functions of Eq. (6) for N=7 can be found in Fig. 1. The thickness functions associated with τ=0 and τ=N allows for the assessment of kinematic boundary conditions since they are alternatively equal to 0 and 1 at the top and bottom surfaces so that the generalized configuration variable can be arbitrarily enforced. On the other hand, Fτ(k)αi=0 for τ=0,…,N at ζ=±h/2. When τ=N+1, the kinematic expansion of Eq. (5) is performed in terms of the zigzag functions reported below to predict the interlaminar issues:(7)FN+1(k)αi={z1=−ζ−ζ1ζ2−ζ1zk=(−1)k(2z¯k−1),k=2,…,l−1zl=(−1)lζ−ζl+1ζl+1−ζlwhere z¯k=(ζ−ζk)/(ζk+1−ζk). z1,zl, and zk are defined so that at the extreme heights of the laminated shell the condition FN+1(k)αi=0 is satisfied. In this way, the thickness functions of Eq. (6) allows one to enforce the kinematic boundary conditions to the problem. Following the ELW approach, the values assumed at ζ=−h/2 and ζ=h/2 by arbitrary element Δi(k) of vector Δ(k) with i=1,…,5 are equal to that of the corresponding configuration variable δi(τ) of the generalized vector δ(τ) for τ=0 and τ=N+1, respectively: (8)Δi(1)(α1,α2,ζ=−h2)=δi(0)(α1,α2)Δi(l)(α1,α2,ζ=h2)=δi(N)(α1,α2)

The following nomenclature is introduced to identify the thickness function set selected in each simulation based on Tornabene [12]: (9)ELD−NELDZL−Nwhere “ELD” means that the kinematic model is developed according to the ELW approach, while N denotes the maximum expansion order in Eq. (6). Finally, ZL is adopted when the zigzag function (7) is used for τ=N+1. Once the kinematic model is defined, the higher-order two-dimensional definition equations are derived starting from the hygro-thermo-mechanical kinematic relations. These equations are expressed in curvilinear principal coordinates, and relate the displacement field vector U(k), the temperature variation ΔT(k) and the moisture concentration variation ΔC(k) to the strain vector ε(k)=[ε1(k)ε3(k)γ12(k)γ13(k)γ23(k)ε3(k)]T and the temperature and moisture concentration gradients, denoted by θ(k)=[θ1(k)θ2(k)θ3(k)]T and λ(k)=[λ1(k)λ2(k)λ3(k)]T, respectively. These quantities are collected in vector π(k) of the three-dimensional primary variables of the hygro-thermo-mechanical problem. One gets the following relation [12]: (10)π(k)=DΔ(k)⇔[ε(k)ΔT⏜(k)ΔC⏜(k)θ(k)λ(k)]=[D(1)000D(3)000D(3)0D(2)000D(2)][U(k)ΔT(k)ΔC(k)]where D is the kinematic differential operator of the present multifield problem. This operator is built from sub-operators D(1),D(2) and D(3). More specifically, D(1) is the symmetric part of the definition operator for the mechanical case, while D(2) is the gradient of a scalar field. Finally, the operator D(3)=1 is conveniently introduced to derive multifield-induced strain components using the primary variables ΔT⏜(k) and ΔC⏜(k). The definition operator D is expressed as the product of matrices Dζ and DΩ as follows: (11)D=DζDΩwhere Dζ collects the derivatives for the thickness directions and takes the following aspect: (12)Dζ=[Dζ(1)00000Dζ(3)00000Dζ(3)00000Dζ(2)00000Dζ(2)]

The sub-operators Dζ(1),Dζ(2) and Dζ(3) are defined as follows: (13)Dζ(1)=[1H10000000001H20000000001H11H20000000001H10∂∂ζ00000001H20∂∂ζ000000000∂∂ζ],Dζ(2)=[1H10001H2000∂∂ζ],Dζ(3)=1

On the other hand, the differential operator DΩ embeds the derivatives with respect to α1,α2 principal coordinates: (14)DΩ=[DΩ(1)000DΩ(3)000DΩ(3)0DΩ(2)000DΩ(2)]

The quantities DΩ(1),DΩ(2) and DΩ(3) take the following form: (15)DΩ(1)=[D¯Ωα1D¯Ωα2D¯Ωα3]DΩ(2)=[−1A1∂∂α1−1A2∂∂α2−1]TDΩ(3)=1

Finally, operators D¯Ωαi with i=1,2,3 assume the aspect reported below: (16)D¯Ωα1=[1A1∂∂α11A1A2∂A2∂α1−1A1A2∂A1∂α21A2∂∂α2−1R10100],D¯Ωα2=[1A1A2∂A1∂α21A2∂∂α21A1∂∂α1−1A1A2∂A2∂α10−1R2010],D¯Ωα3=[1R11R2001A1∂∂α11A2∂∂α2001]

The operator DΩ of Eq. (14) is now written as the sum of operators DΩαi with i=1,…,5: (17)DΩ=∑i=15DΩαi

The quantities DΩαi introduced in Eq. (17) can be expressed as follows, setting D¯Ω(2)α4=D¯Ω(2)α5=DΩ(2) and D~Ω(3)α4=D~Ω(3)α5=DΩ(3): (18)DΩα1=[D¯Ωα1000000000000000000000000],DΩα2=[0D¯Ωα200000000000000000000000],DΩα3=[00D¯Ωα30000000000000000000000]DΩα4=[00000000D¯Ωα4000000000D~Ωα4000000],DΩα5=[00000000000000D¯Ωα5000000000D~Ωα5]

Substituting the generalized kinematic model of Eq. (5) within Eq. (10), the two-dimensional definition equations with three-dimensional capabilities are provided for the present multifield model. In this way, the vector π(k) of three-dimensional primary variables is expanded up to the τ-th order using the generalized vector π(τ)αi, defined for each τ=0,…,N+1: (19)π(k)=DΔ(k)=Dζ∑i=15DΩαiΔ(k)=∑τ=0N+1∑i=15Z(kτ)αiDΩαiδ(τ)=∑τ=0N+1∑i=15Z(kτ)αiπ(τ)αi

The kinematic relations of Eq. (19) account for the generalized definition operator Z(kτ)αi, which relates the three-dimensional primary variables to the corresponding two-dimensional quantities. This operator is obtained from the assembly of the generalized operators Z1(kτ)αi,Z2(kτ)αi and Z3(kτ)αi belonging to each physical problem involved in the formulation. In a more expanded form, the previous relation can be expressed as follows: (20)[ε(k)ΔT⏜(k)ΔC⏜(k)θ(k)λ(k)]=∑τ=0N+1∑i=15[Z1(kτ)αi00000Z3(kτ)αi00000Z3(kτ)αi00000Z2(kτ)αi00000Z2(kτ)αi][ε(τ)αiΔT⏜(τ)αiΔC⏜(τ)αiθ(τ)αiλ(τ)αi]

The sub-vectors Z1(kτ)αi,Z2(kτ)αi, and Z3(kτ)αi are defined for each τ=0,…,N+1 and i=1,…,5 using the differential operators of Eq. (13) and the thickness function Fτ(k)αi so that Zm(kτ)αi=Dζ(m)Fτ(k)αi with m=1,2,3. The vectors ε(τ)αi,ΔT⏜(τ)αi,ΔC⏜(τ)αi,θ(τ)αi, and λ(τ)αi introduced in Eq. (20), referred to the higher-order kinematic expansion of primary variables, are expressed using an extended notation as follows: (21)ε(τ)αi(α1,α2)=[ε1(τ)αiε2(τ)αiγ1(τ)αiγ2(τ)αiγ13(τ)αiγ23(τ)αiω13(τ)αiω23(τ)αiε3(τ)αi]TΔT⏜(τ)αi(α1,α2)=ΔT⏜(τ)αi,ΔC⏜(τ)αi(α1,α2)=ΔC⏜(τ)αiθ(τ)αi(α1,α2)=[θ1(τ)αiθ2(τ)αiθ3(τ)αi]T,λ(τ)αi(α1,α2)=[λ1(τ)αiλ2(τ)αiλ3(τ)αi]T

This section derives the constitutive relations for the two-dimensional generalized formulation. Each layer of the stacking sequence is modeled as a generally anisotropic elastic continuum, accounting for the full coupling between the mechanical, thermal, and hygrometric physical problems [12]. For an arbitrary k-th layer of the lamination scheme, the following three-dimensional constitutive relation is established: (22)χ(k)=Γ¯(k)π(k)⇔[σ(k)η(k)µ(k)h(k)c(k)]=[Γ¯C(k)−Γ¯z(k)T−Γ¯e(k)T00Γ¯z(k)Γ¯TT(k)Γ¯TC(k)00Γ¯e(k)Γ¯TC(k)Γ¯CC(k)00000Γ¯K(k)Γ¯Y(k)000Γ¯X(k)Γ¯S(k)][ε(k)ΔT⏜(k)ΔC⏜(k)θ(k)λ(k)]where σ(k) is the vector of stress components, while h(k) and c(k) are the thermal flux vector and the mass flux vector, respectively. Finally, η(k) is the specific entropy of the system and µ(k) is the specific chemical potential. Employing an extended notation, Eq. (22) can be expressed as follows:

(23)[σ1(k)σ2(k)τ12(k)τ13(k)τ23(k)σ3(k)η(k)µ(k)h1(k)h2(k)h3(k)c1(k)c2(k)c3(k)]=[C¯11(k)C¯12(k)C¯16(k)C¯14(k)C¯15(k)C¯13(k)−z¯11(k)−e¯11(k)000000C¯12(k)C¯22(k)C¯26(k)C¯24(k)C¯25(k)C¯23(k)−z¯22(k)−e¯22(k)000000C¯16(k)C¯26(k)C¯66(k)C¯46(k)C¯56(k)C¯36(k)−z¯12(k)−e¯12(k)000000C¯14(k)C¯24(k)C¯46(k)C¯44(k)C¯45(k)C¯34(k)−z¯13(k)−e¯13(k)000000C¯15(k)C¯25(k)C¯56(k)C¯45(k)C¯55(k)C¯35(k)−z¯23(k)−e¯23(k)000000C¯13(k)C¯23(k)C¯36(k)C¯34(k)C¯35(k)C¯33(k)−z¯33(k)−z¯33(k)000000z¯11(k)z¯22(k)z¯12(k)z¯13(k)z¯23(k)z¯33(k)ξ¯11(k)ξ¯12(k)000000e¯11(k)e¯22(k)e¯12(k)e¯13(k)e¯23(k)e¯33(k)ξ¯12(k)ξ¯22(k)00000000000000k¯11(k)k¯12(k)k¯13(k)y¯11(k)y¯12(k)y¯13(k)00000000k¯12(k)k¯22(k)k¯23(k)y¯12(k)y¯22(k)y¯23(k)00000000k¯13(k)k¯23(k)k¯33(k)y¯13(k)y¯23(k)y¯33(k)00000000x¯11(k)x¯12(k)x¯13(k)s¯11(k)s¯12(k)s¯13(k)00000000x¯12(k)x¯22(k)x¯23(k)s¯12(k)s¯22(k)s¯23(k)00000000x¯13(k)x¯23(k)x¯33(k)s¯13(k)s¯23(k)s¯33(k)][ε1(k)ε2(k)γ12(k)γ13(k)γ23(k)ε3(k)ΔT⏜(k)ΔC⏜(k)θ1(k)θ2(k)θ3(k)λ1(k)λ2(k)λ3(k)]

Matrix Γ¯(k) is made of sub-matrices that account for a single physical problem. More specifically, Γ¯C(k) is the stiffness matrix of the mechanical elasticity case, while Γ¯K(k) and Γ¯S(k) denote the thermal and moisture conductivity matrix, respectively. Finally, Γ¯Y(k) and Γ¯X(k) are the coupling matrices between the temperature and moisture concentration gradients, which allow one to model the Dufour and Soret effects [12]. Vectors Γ¯z(k) and Γ¯e(k) allow one to compute strains induced by a temperature variation and a moisture concentration variation within the solid. They are calculated from the following expression: (24)Γ¯z(k)T=Γ¯C(k)Γ¯a(k),Γ¯e(k)T=Γ¯C(k)Γ¯b(k)where Γ¯a(k) and Γ¯b(k) contain the rotated thermal and expansion coefficients a¯ij(k) and the rotated hygroscopic expansion coefficients b¯ij(k), with i,j=1,2,3. The constitutive relation (22) is written in the reference system of the problem under consideration. For an arbitrary k-th layer, the elements of the multifield stiffness matrix are provided in the reference system O′α^1(k)α^2(k)ζ^(k), built on the material symmetry axes of the k-th lamina. If π^(k) and χ^(k) are the vectors of the primary and secondary variables indicated in the material reference system, the elastic constitutive relation is written as follows: (25)χ^(k)=Γ(k)π^(k)⇔[σ^(k)η(k)µ(k)h^(k)c^(k)]=[ΓC(k)−Γz(k)T−Γe(k)T00Γz(k)ΓTT(k)ΓTC(k)00Γe(k)ΓTC(k)ΓCC(k)00000ΓK(k)ΓY(k)000ΓX(k)ΓS(k)][ε^(k)ΔT⏜(k)ΔC⏜(k)θ^(k)λ^(k)]