A new one-dimensional high-order sandwich panel theory for curved panels is presented and compared with the theory of elasticity. The theory accounts for the sandwich core compressibility in the radial direction as well as the core circumferential rigidity. Two distinct core displacement fields are proposed and investigated. One is a logarithmic (it includes terms that are linear, inverse, and logarithmic functions of the radial coordinate). The other is a polynomial (it consists of second and third-order polynomials of the radial coordinate), and it is an extension of the corresponding field for the flat panel. In both formulations, the two thin curved face sheets are assumed to be perfectly bonded to the core and follow the classical Euler–Bernoulli beam assumptions. The relative merits of these two approaches are assessed by comparing the results to an elasticity solution. The case examined is a simply supported curved sandwich panel subjected to a distributed transverse load, for which a closed-form elasticity solution can be formulated. It is shown that the logarithmic formulation is more accurate than the polynomial especially for the stiffer cores and for curved panels of smaller radius.

## Introduction

In aerospace or naval construction, structural shapes are often not flat (e.g., ship hulls or airplane fuselages). Thus, although the majority of the structural theories are formulated and studied on flat panels, there is a definite need to formulate such theories for the geometry of curved panels and properly address the effect of curvature. When it comes to sandwich structures, which consist of two thin high-stiffness face sheets, usually metallic or laminated composites, bonded to a core made of low-density and low-stiffness materials such as honeycomb, polymer foam, or balsa wood, most of the research has been done on flat panels, see book by Carlsson and Kardomateas [1].

In sandwich panels, the core is supposed to provide the shear resistance/stiffness and including transverse shear has been long recognized as a necessary characteristic of sandwich analysis [1]. Thus, the simplest sandwich structural theories assume that the core is incompressible in the transverse direction and with negligible in-plane rigidity in the longitudinal direction. The most popular theory that includes the transverse shear effect is the first-order shear deformation theory (FOSDT) that replaces the layered panel with an equivalent single layer and assumes that the core is incompressible, see, for example, the books by Allen [2] or Carlsson and Kardomateas [1]. A more advanced approach is based on high-order models, see, for example, Ref. [3], in which the core displacement field is derived from the elasticity equations in a closed-form or Carrera and Brischetto [4] that presumed the displacement fields of the core. In such models, the overall response is a combination of the responses of the face sheets and the core through equilibrium and compatibility. A recent advanced high-order theory, the extended high-order sandwich panel theory (EHSAPT), which includes the effect of the in-plane core rigidity has been formulated for flat panels, Ref. [5] for the static and Ref. [6] for the dynamic case. This theory was shown to be very close to the elasticity prediction in both the static and the dynamic cases.

The literature is rather limited when it comes to sandwich curved panels. Noor et al. [7,8] outlined models that assume that the core is incompressible following the first-order shear deformable (FOSD) theory, and Vaswani et al. [9] studied the vibration and damping analysis of curved sandwich beams by using the Flugge shell theory, while assuming that the face sheets are membranes only and the core is incompressible. Yin-Jiang [10] studied the stability of shallow cylindrical sandwich panels with orthotropic layers by use of the FOSD theory but with membrane and flexural rigidities of the face sheets. Furthermore, Di Sciuva [11] developed a model that takes into account the shear deformation but assumes that the core is incompressible and linear. The compressibility of the core was included in the high-order sandwich panel theory introduced by Frostig et al. [3] and was adopted for the curved sandwich panel configuration by Frostig [12] for the linear and Bozhevolnaya and Frostig [13] for the nonlinear case. In addition, Bozhevolnaya and Frostig [14] studied the free vibration and Frostig and Thomsen [15] the thermal effects that induce deformation as well as degradation of properties in curved sandwich panels.

As already mentioned, the effect of the in-plane core rigidity has been included in the EHSAPT, already formulated for flat panels in Ref. [5]. In this paper, we extend this formulation to the configuration of a curved sandwich panel. The theory will be developed based on the following assumptions: The face sheets have in-plane (circumferential) and bending rigidities with negligible shear deformations, see Refs. [16,17]. The core is considered as a 2D linear elastic continuum obeying small deformation kinematic relations and where the core height may change (compressible core) and the section plane does not remain plane after deformation. The core is assumed to possess shear, radial normal, and circumferential stiffness; full bonding between the face sheets and the core is assumed, and the interfacial layers can resist shear as well as radial normal stresses. Subsequently, the linear response of the sandwich curved panel from the two versions of the high-order theory is studied and compared to the predictions from a recently derived elasticity solution [18].

## Formulation

### Kinematic Description.

In the following, we consider a curved sandwich panel of unit width, and consisting of two thin top/bottom face sheets of thickness ft and fb, respectively, separated by a thick core of thickness 2c. The panel is bounded in the plane by two concentric circles of radii R1 and R2 and two radial segments forming an arbitrary angle $α≤π$ (Fig. 1). The mean radii for the top face, core, and bottom face are denoted by Rt, Rc, and Rb, respectively. In addition, we denote by rtc =  $Rc+c$, the radius at the top face/core interface, and by rbc =  $Rc−c$, the corresponding radius at the bottom face/core interface. A polar coordinate system ($r,θ$) is used to describe the sandwich geometry and kinematics. The polar angle θ is measured from the left end. Furthermore, local normal-tangential coordinate systems are defined at mid top/bottom face sheets and mid core to help simplify the formulation, which are ($zt,θ$), ($zc,θ$), and ($zb,θ$). Radial and circumferential displacements are represented by w and u, respectively, and the superscripts t, b, and c denote top face sheet, bottom face sheet, and core.

Fig. 1
Fig. 1
Close modal
The two face sheets are assumed to follow the assumptions of the Euler–Bernoulli beam theory. Therefore, the displacement field for the top face sheet, $−ft/2≤zt≤ft/2$, is
$wt(zt,θ)=w0t(θ) ; ut(zt,θ)=u0t(θ)+[u0t(θ)−w0t ′(θ)]ztRt$
(1a)
and for the bottom face sheet, $−fb/2≤zb≤fb/2$, is
$wb(zb,θ)=w0b(θ) ; ub(zb,θ)=u0b(θ)+[u0b(θ)−w0b ′(θ)]zbRb$
(1b)
The only nonzero corresponding linear-strain is
$ϵθθt,b=ϵθθ0t,b+zt,bκθθ0t,b$
(1c)
where
$ϵθθ0t,b(θ)=1Rt,b[u0t,b ′(θ)+w0t,b(θ)] , κθθ0t,b(θ)=1Rt,b[u0t,b ′(θ)+w0t,b(θ)]$
(1d)

The extended high-order sandwich panel theory allows for core compressibility and circumferential rigidity effects. Two distinct presumed core displacement fields are proposed

#### Logarithmic Core Displacement Field.

One is a closed-form displacement function involving a logarithmic term. The rational for adopting this displacement profile is as follows: In general, the core in sandwich structures has axial rigidity significantly less than that of the faces. Accordingly, if we neglect the core circumferential stress, i.e., we assume that $σθθ=0$, then the elasticity equilibrium equations become
$∂σrr∂r+1r∂τrθ∂θ+σrrr=0 ; ∂τrθ∂r+2τrθr=0$
(2a)
These can be rewritten as
$∂(rσrr)∂r+∂τrθ∂θ=0 ; ∂(r2τrθ)∂r=0$
(2b)
Integrating the second of Eq. (2b) for $τrθ$ and then substituting in the first of Eq. (2b) and integrating for σrr result in
$τrθ=f1(θ)r2 ; σrr=f1′(θ)r2+f2(θ)r$
(2c)
Using the constitutive and strain–displacement relations gives
$σrr=Ecϵrr=Ec∂w∂r=f1′(θ)r2+f2(θ)r$
(2d)
which would integrate to
$w(r,θ)=1Ec[f2(θ)lnr−f1′(θ)r+f3(θ)]$
(2e)
Similarly, the shear stress relation becomes
$τrθ=Gc(1r∂w∂θ+∂u∂r−ur)=Gcr[1r∂w∂θ+∂∂r(ur)]=f1(θ)r2$
(2f)
Substituting w from Eq. (2e) and integrating give
$u(r,θ)=−f1(θ)2Gcr−1Ec[f2′(θ)2rln2r+f1″(θ)+f3′(θ)rlnr]+rf4(θ)$
(2g)

Therefore, it can be seen that in this case the displacements would contain terms $lnr$, r, and $1/r$.

Although our high-order theory includes the axial rigidity of the core, it is reasonable to assume, based on the above discussion, that the curvature of the panel would induce a logarithmic dependence, thus the first displacement profile, termed “logarithmic,” is as follows:
$uc(r,θ)=u0c(θ)+ru1c(θ)+u2c(θ)r+u3c(θ)ln(r)$
(3a)
$wc(r,θ)=w0c(θ)+w1c(θ)r+w2c(θ)ln(r)$
(3b)
Assuming perfect bonding of the two face sheets with the core, displacement continuity is imposed at the two interfaces resulting in four compatibility equations
$wt(zt=−ft2,θ)=wc(r=rtc,θ) , ut(zt=−ft2,θ)=uc(r=rtc,θ)$
(4a)
$wb(zb=+fb2,θ)=wc(r=rbc,θ) , ub(zb=+fb2,θ)=uc(r=rbc,θ)$
(4b)
The four compatibility equations are then solved for the four dependent variables $u2c(θ), u3c(θ), w1c(θ)$, and $w2c(θ)$. These are obtained in terms of
$D=rtclnrtc−rbclnrbc$
(5a)
as follows:
$u2c(θ)=rtcrbcD{u0b(θ)lnrtc−u0t(θ)lnrbc+u0c(θ)(lnrbc−lnrtc)+u1c(θ)(rtclnrbc−rbclnrtc)+(fb/2)lnrtcrb[u0b(θ)−w0b ′(θ)]+(ft/2)lnrbcrt[u0t(θ)−w0t ′(θ)]}$
(5b)
$u3c(θ)=1D[rtcu0t(θ)−rbcu0b(θ)+(rbc−rtc)u0c(θ)+(rbc2−rtc2)u1c(θ)−(fb/2)rbcrb[u0b(θ)−w0b ′(θ)]−(ft/2)rtcrt[u0t(θ)−w0t ′(θ)]$
(5c)
$w1c(θ)=rtcrbcD[w0b(θ)lnrtc−w0t(θ)lnrbc+w0c(θ)lnrbcrtc]$
(5d)
$w2c(θ)=1D[rtcw0t(θ)−rbcw0b(θ)+(rbc−rtc)w0c(θ)]$
(5e)
The corresponding linear-strains in polar coordinates are
$ϵrrc(r,θ)=∂wc(r,θ)∂r$
(6a)
$ϵθθc(r,θ)=1r∂uc(r,θ)∂θ+wc(r,θ)r$
(6b)
$γrθc(r,θ)=∂uc(r,θ)∂r+1r∂wc(r,θ)∂θ−uc(r,θ)r$
(6c)

#### Polynomial Core Displacement Field.

The other presumed core displacement field takes the form of a high-order polynomial, and it is a direct extension of the displacement field for sandwich flat panels [5]. The polynomial core kinematics description is simple when expressed in the local coordinate $(zc,θ)$ as follows:
$uc(zc,θ)=(1−zc2c2)u0c(θ)+zc(1−zc2c2)u1c(θ)+[(fb/2)+Rb]zc22c2Rb(1−zcc)u0b(θ)+[(−ft/2)+Rt]zc22c2Rt(1+zcc)u0t(θ)+fbzc24c2Rb(−1+zcc)w0b ′(θ)+ftzc24c2Rt(1+zcc)w0t ′(θ)$
(7a)
$wc(zc,θ)=(−zc2c+zc22c2)w0b(θ)+(1−zc22c2)w0c(θ)+(zc2c+zc22c2)w0t(θ)$
(7b)

The polynomial displacement field in Eq. (7) has been defined in such a way that satisfies the four interfacial displacement compatibility conditions, Eqs. (4a) and 4(b). Thus, in contrast with Eq. (2), the polynomial functions do not contain $u2c(θ), u3c(θ), w1c(θ)$, and $w2c(θ)$ that need to be determined through interfacial compatibilities.

The corresponding linear-strain in the local coordinate $(zc,ϕ)$ is the same as in Eq. (6), with interchanging variable, $r=zc+Rc$.

Thus, the extended high-order theory formulation for sandwich curved panels is in terms of seven dependent variables as a function of θ: two for the top face sheet, $w0t, u0t$, two for the bottom face sheet, $w0b, u0b$, and three for the core, $w0c, u0c$, and $u1c$.

##### Constitutive relations.
In the following, $cijt,b,c$ denote the orthotropic material stiffness constants where $i,j=1,3,5$ and $1≡θ, 3≡r$, and $5≡rθ$. The orthotropic stress–strain relations for the core read
$[σrrcσθθcτrθc]=[c11cc13c0c13cc33c000c55c] [ϵrrcϵθθcγrθc]$
(8a)
For the face sheets, from the kinematic assumptions (1), the only nonzero strain is the $ϵθθ$, and as a consequence, the two nonzero resulting stresses are
$σθθt,b=c11t,bϵθθt,b , σrrt,b=c13t,bϵθθt,b$
(8b)

However, the $σrrt,b(r,θ)$ will not be included in the subsequent principle of minimum total potential energy, because the corresponding strain, $ϵrrt,b(r,θ)$, is zero.

##### Principle of minimum total potential energy.
Governing equations and associated boundary conditions are derived from the principle of minimum total potential energy
$δ(U+V)=0$
(9a)

where U is the strain-energy of the sandwich panel, and V is the external potential due to the applied loads.

The first variation of the strain-energy of the sandwich beam is
$δU=∫0α[∫−fb/2fb/2σθθbδϵθθbRbdzb+∫rbcrtc(σrrcδϵrrc+σθθcδϵθθc+τrθcδγrθc)rdr+∫−ft/2ft/2σθθtδϵθθtRtdzt]dθ$
(9b)

Recall that, in this paper, two distinct core displacement fields are presented: one is described in polar coordinates $(r,θ$), and the other is described in the local tangential coordinates $(zc,θ)$. In Eq. (8b), the core strain-energy integral, denoted by the superscript c, is in the polar coordinates. If the core strain-energy is expressed in the local tangential coordinates $(zc,θ)$, the integral can be easily converted to the local coordinate as $r=zc+Rc$, dr = dzc, and the integration limits is changed from $∫rbcrtcto$$∫−cc$. Also, please notice that due to the small thickness of the faces, in the first integral in Eq. (9b) we carry the integration with $Rbdzb$ instead of $(Rb+zb)dzb$; same with the last integral in Eq. (9b).

The sandwich panel is subjected to various loadings on both face sheets, and the first variation of the external potential is
$δV=−∫0α{[nθθt(θ)δu0t(θ)+qt(θ)δw0t(θ)+mt(θ)δβt(θ)]Rt+[nθθb(θ)δu0b(θ)+qb(θ)δw0b(θ)+mb(θ)δβb(θ)]Rb}dθ−Ntδu0t(θe)−Ptδw0t(θe)+Mtδβt(θe)−Nbδu0b(θe)−Pbδw0b(θe)+Mbδβb(θe)$
(9c)

where $qt,b(θ)$ is the distributed normal force (along the radius), $nt,b(θ)$ is the distributed tangential force (along θ), and $mt,b(θ)$ is the distributed moment on the top and bottom face sheets, respectively. In addition, $Pt,b$ is the concentrated normal (along the radius) force, $Nt,b$ is the concentrated tangential (along θ) force, and $Mt,b$ is the concentrated moment applied at the end θe on the top and bottom face sheets, respectively.

In the foregoing equation, we denote by θe the boundary points, commonly $θe=0$ or $θe=α$. The procedure below will make this assumption. If concentrated external forces/moments are applied at a θe between 0 and α, then the boundary conditions can be treated in separate ranges, i.e., $0≤θ≤θe$ and $θe≤θ≤α$, with continuity conditions applied at $θ=θe$.

Then, the governing equations are obtained by substituting the stress–strain relations, Eq. (8), into the strain-energy, Eq. (9b); this way the expressions are in terms of strains. Next, by substituting the strain–displacement relations, Eq. (6), into the strain-energy Eq. (9b), the expressions are written in terms of displacements. Once all the terms in the variational expressions are in terms of displacements, integration by parts is carried in order to obtain the governing differential equations and the associated boundary conditions. As a result, we obtain seven linear ordinary differential equations in terms of the seven generalized coordinates: $w0t, u0t, w0b, u0b, w0c, u0c$, and $ϕ0c$.

The resulting governing equations for $0≤θ≤α$ and associated boundary conditions are as follows:

##### Governing equations and associated boundary conditions.

Differential equations

Top face sheet

$δw0t:$
$[A3c+c11tbftRt+(A3a−A7b+A9c)∂2∂θ2+(A9a+c11tbft312Rt3)∂4∂θ4]w0t+[A1c+(A1a−A5b+A8c)∂2∂ϕ2+A8a∂4∂θ4]w0b+[A2c+(A2a−A6b)∂2∂θ2]w0c+[(A6c−A3b+c11tbftRt)∂∂θ+(A6a+c11tbft312Rt3)∂3∂θ3]u0t+[(A4c−A1b)∂∂θ+A4a∂3∂θ3]u0b+[(A5c−A2b)∂∂θ+A5a∂3∂θ3]u0c+[(A7c−A4b)∂∂θ+A7a∂3∂θ3]u1c=Rtqt+∂∂θmt$
(10a)
$δu0t:$
$[(D7c−D3b−c11tbftRt)∂∂θ+(c11tbft312Rt3−D9b)∂3∂θ3]w0t+[(D5c−D1b)∂∂θ−D8b∂3∂θ3]w0b+(D6c−D2b)∂∂θw0c+[D3c−(D6b+c11tbftRt+c11tbft312Rt3)∂2∂θ2]u0t+(D1c−D4b∂2∂θ2)u0b+(D2c−D5b∂2∂θ2)u0c+(D4c−D7b∂2∂θ2)u1c=Rtnθθt+mt$
(10b)

Bottom face sheet

$δw0b:$
$[B3c+(B3a−B7b+B9c)∂2∂θ2+B9a∂4∂θ4]w0t+[B1c+c11bbfbRb+(B1a−B5b+B8c)∂2∂θ2+(B8a+c11bbfb312Rb3)∂4∂θ4]w0b+[B2c+(B2a−B6b)∂2∂θ2]w0c+[(B6c−B3b)∂∂θ+B6a∂3∂θ3]u0t+[(B4c−B1b+c11bbfbRb)∂∂θ+(B4a−c11bbfb312Rb3)∂3∂θ3]u0b+[(−B2b+B5c)∂∂θ+B5a∂3∂θ3]u0c+[(−B4b+B7c)∂∂θ+B7a∂3∂θ3]u1c=−Rbqb−∂∂θmb$
(10c)
$δu0b:$
$[(E7c−E3b)∂∂θ−E9b∂3∂θ3]w0t+[(E5c−E1b−c11bbfbRb)∂∂θ+(c11bbfb312Rb3−E8b)∂3∂θ3]w0b+(E6c−E2b)∂∂θw0c+(E3c−E6b∂2∂θ2)u0t+[E1c−(E4b+c11bbfbRb+c11bbfb312Rb3)∂2∂θ2]u0b+(E2c−E5b∂2∂θ2)u0c+(E4c−E7b∂2∂θ2)u1c=−Rbnss,b−mb$
(10d)

Core

$δw0c:$
$[C3c+(C9c−C7b)∂2∂θ2]w0t+[C1c+(C8c−C5b)∂2∂θ2]w0b+(C2c−C6b∂2∂θ2)w0c+(C6c−C3b)∂∂θu0t+(C4c−C1b)∂∂θu0b+(C5c−C2b)∂∂θu0c+(C7c−C4b)∂∂θu1c=0$
(10e)
$δu0c:$
$[(F7c−F3b)∂∂θ−F9b∂3∂θ3]w0t+[(F5c−F1b)∂∂θ−F8b∂3∂θ3]w0b+(F6c−F2b)∂∂θw0c+(F3c−F6b∂2∂θ2)u0t+(F1c−F4b∂2∂θ2)u0b+(F2c−F5b∂2∂θ2)u0c+(F4c−F7b∂2∂θ2)u1c=0$
(10f)
$δu1c:$
$[(G7c−G3b)∂∂θ−G9b∂3∂θ3]w0t+[(G5c−G1b)∂∂θ−G8b∂3∂θ3]w0b+(G6c−G2b)∂∂θw0c++(G3c−G6b∂2∂θ2)u0t+(G1c−G4b∂2∂θ2)u0b+(G2c−G5b∂2∂θ2)u0c+(G4c−G7b∂2∂θ2)u1c=0$
(10g)

The corresponding boundary conditions are at θ = 0 and $θ=α$, read as follows (at each end, there are nine boundary conditions, three for each of the two face sheets and three for the core):

Top face sheet

Either $δw0t=0$ or
$[(A7b−A3a)∂∂θ−(A9a+c11tbft312Rt3)∂3∂θ3]w0t+[(A5b−A1a)∂∂θ−A8a∂3∂θ3]w0b+(A6b−A2a)∂∂θw0c+[A3b+(c11tbft312Rt3−A6a)∂2∂θ2]u0t+(A1b−A4a∂2∂θ2)u0b+(A2b−A5a∂2∂θ2)u0c+(A4b−A7a∂2∂θ2)u1c=Pt−mt$
(11a)
Either $δw0t ′=0$ or
$[A3a+(A9a+c11tbft312Rt3)∂2∂θ2]w0t+(A1a+A8a∂2∂θ2)w0b+A2aw0c+[A6a∂∂θ−c11tbft312Rt3∂3∂θ3]u0t+A4a∂∂θu0b+A5a∂∂θu0c+A7a∂∂θu1c=−MtRt$
(11b)
Either $δu0t=0$ or
$[D3b+c11tbftRt+(D9b−c11tbft312Rt3)∂2∂θ2]w0t+(D1b+D8b∂2∂θ2)w0b+D2bw0c+[D6b+c11tbft312Rt3+c11tbftRt]∂∂θu0t+D4b∂∂θu0b+D5b∂∂θu0c+D7b∂∂θu1c=Nt+MtRt$
(11c)

Bottom face sheet

Either $δw0b=0$ or
$[(B7b−B3a)∂∂θ−B9a∂3∂θ3]w0t+[(B5b−B1a)∂∂θ−(c11bbfb312Rb3+B8a)∂3∂θ3]w0b+(B6b−B2a)∂∂θw0c+(B3b−B6a∂2∂θ2)u0t+[B1b+(c11bbfb312Rb3−B4a)∂2∂θ2]u0b+(B2b−B5a∂2∂θ2)u0c+(B4b−B7a∂2∂θ2)u1c=Pt+mb$
(11d)
Either $δw0b ′=0$ or
$(B3a+B9a∂2∂θ2)w0t+[B1a+(B8a+c11bbfb312Rb3)∂2∂θ2]w0b+B2aw0c+B4a∂∂θu0t+(B4a∂∂θ−c11bbfb312Rb3∂3∂θ3)u0b+B5a∂∂θu0c+B7a∂∂θu1c=−MbRb$
(11e)
Either $δu0b=0$ or
$(E3b+E9b∂2∂θ2)w0t+(E1b+c11bbfbRb+(E8b−c11bbfb312Rb3)∂2∂θ2)w0b+E2bw0c+E5b∂∂θu0t(E4b+c11bbfb312Rb3+c11bbfbRb)∂∂θu0b+E5b∂∂θu0c+E7b∂∂θu1c=Nb+MbRb$
(11f)

Core

Either $δw0c=0$ or
$C7b∂∂θw0t+C5b∂∂θw0b+C6b∂∂θw0c+C3butc+C1bu0b+C2bu0c+C4bu1c=0$
(11g)
Either $δu0c=0$ or
$(F3b+F9b∂2∂θ2)w0t+(F1b+F8b∂2∂θ2)w0b+F2bw0c+F6b∂∂θu0t+F4b∂∂θu0b+F5b∂∂θu0c+F7b∂∂θu1c=0$
(11h)
Either $δu1c=0$ or
$(G3b+G9b∂2∂θ2)w0t+(G1b+G8b∂2∂θ2)w0b+G2bw0c+G6b∂∂θu0t+G4b∂∂θu0b+G5b∂∂θu0c+G7b∂∂θu1c=0$
(11i)

where $Aia,b,c, Bia,b,c, Cib,c, Dib,c, Eib,c, Fib,c, Gib,c$ are constants which include both geometric and material properties and are defined in Appendix  A for the logarithmic EHSAPT and in Appendix  B for the Polynomial EHSAPT.

## Solution Procedure

In the following, we outline the solution procedure for a simply supported curved panel subjected to a distributed load on the top face sheet $qt(θ)$, which can be expressed as a Fourier series
$qt(θ)=∑n=1∞qn sin(nπθα)$
(12a)
The solution that satisfies the simply supported boundary conditions is
$w0t,b,c(θ)=∑n=1∞W0nt,b,c sin(nπθα) ; u0t,b,c(θ)=∑n=1∞U0nt,b,c cos(nπθα)$
(12b)
$u1c(θ)=∑n=1∞U1nc cos(nπθα)$
(12c)

Substituting the foregoing equations (12) into the governing differential equations (10) results for each n in a system of linear algebraic equations, $[Kn]{Xn}={Fn}$, where $[Kn]$ is a 7 × 7 stiffness matrix, ${Fn}$ is a 1 × 7 force matrix, and ${Xn}$ is a 1 × 7 unknown displacement matrix, namely: ${W0,nt,W0,nb,W0,nc,U0,nt,U0,nb,U0,nc,U1,nc}$. Each individual n linear algebraic equations system can be easily solved obtaining ${Xn}$, then the analytical solution is obtained from the series (12b) and (12c). In Eq. (12), $∑n=1∞$ is replaced by $∑n=1N$, where N is the total number of terms included in the Fourier series equation (12a) and corresponding solutions equations (12b) and (12c); the no of terms, N, is determined from a study of the series convergence.

## Results

Simply supported curved sandwich panels subjected to a half sine distributed load
$qt(θ)=q0 sin πθα$
(13)

are studied. The solutions from four different theories are presented. The first two versions of the EHSAPT with logarithmic and polynomial core displacement functions are formulated and presented in this paper. And the other two are the theory of elasticity and the FOSDT with equivalent shear modulus, which are presented in Refs. [18] and [19], respectively. The elasticity solution serves as the benchmark to assess the accuracy of all theories, whereas the first-order shear deformation theory is widely used in the sandwich structures community due to its simplicity. These four different solutions are then compared for various sandwich geometries and core materials in order to validate and assess the relative merits of the two versions of the EHSAPTs.

In the subsequent paragraphs, various geometries of the sandwich curved panel in Fig. 1 are analyzed. In particular, we consider a symmetric construction with thin faces of thickness $ft,b=1$ mm, made out of isotropic aluminium (2024-T3) with modulus $Et,b=69.13$ GPa and a thick core of thickness 2c = 25 mm. In addition, the (out-of-plane) width is 30 mm.

Cases 1 and 2 consist of core made out of the relatively stiffer Balsawood (Gurit Balsaflex) with Ec = 5199 MPa, Gc = 206 MPa, and νc = 0.30. The angular span in both cases is $α=π$. Case 1 has a large radius of the top face midline Rt = 813 mm and a per unit width load of q0 = 2.40 × 10−6 N/m applied, whereas case 2 has a relatively small radius Rt = 82.25 mm and a per unit width load of q0 = 2.40 × 10−2 N/m applied.

Cases 3 and 4 consist of core made out of the relatively flexible foam Divinycell H160, with Ec = 170 MPa, Gc = 66 MPa, and νc = 0.30. Case 3 has a small radius of the top face midline Rt = 41.11 mm and an angular span $α=π$ and a per unit width load of q0 = 4.00 × 10−2 N/m applied, whereas case 4 has a large radius Rt = 813 mm and a much smaller angular span $α=0.1012π$ and a per unit width load of q0 = 1.20 × 104 N/m is applied.

In the following, we shall normalize the stresses with q0, the circumferential (hoop) displacement with $Rtα$ (i.e., the top face arc length), and the transverse displacement with the quantities that scale the maximum transverse displacement of a flat plate of length $Rtα$ and bending rigidity $EI=Et(ft+fb)c2$ under distributed load q0, i.e., the
$w̃(r,θ)=w(r,θ)wnorm ; wnorm=q0Rt4α4Et(ft+fb)c2$
(14a)
Radial through-thickness and angular spanwise coordinates are presented in the dimensionless quantities
$r̃=r−R1R2−R1; θ̃=θα$
(14b)

i.e., the sandwich top panel surface is at $r̃=1$, and bottom surface is at $r̃=0$.

In addition, we define the stiffness core constants by
$c11c=c33c=Ec1−νc2; c13c=Ecνc1−νc2; c55c=Gc$
Fig. 2
Fig. 2
Close modal
Fig. 3
Fig. 3
Close modal

Figures 2(a)2(d) present the core through-thickness radial stress $σrr̃$ at $θ=α/2$ for the four cases considered. These figures show clearly that the logarithmic EHSAPT (Eq. (3)) has a superior accuracy over the polynomial EHSAPT (Eq. (7)), especially for the stiffer cores (cases 1 and 2). When the core is very flexible and the panel radius is small, the difference between the two EHSPATs is negligible (case 4). Accurate determination of the radial stresses is needed because significant compressive stresses can develop within the core, for example, under impact loading, which can result in core crushing failure modes. Note also that the FOSDT is an incompressible theory; therefore, there is no information on $σ̃rr$.

The shear stiffness of a sandwich structure is dictated by its core. And in a typical sandwich structure, core material stiffness and strength are weak compared to the face sheets. Thus, it is necessary to determine the shear stress correctly in order to prevent core failure. Figure 3 shows the through-thickness core shear stress $τrθ$ at θ = 0, where the maximum takes place. Both EHSAPTs predict nonlinear results in agreement to the elasticity, while the FOSDT cannot capture the profile and underestimates the maximum shear stress. The logarithmic EHSAPT seems to have a slight edge over the polynomial EHSPAT.

In sandwich structures, the circumferential stress of interest is the one on the stiff face sheets, while it is often neglected in the flexible core. Figure 4 presents the through thickness $σθθ$ within the top/bottom face sheets. The FOSDT predicts that compression occurs in the top face, while tension occurs in the bottom face. However, both versions of the EHSAPT and elasticity show that within each face sheet, both tension and compression coexist; moreover, the magnitudes are much greater than the FOSDT prediction (ten times greater for tension in the top face sheet and nine times greater in the bottom face sheet). Thus, the FOSDT is inadequate in calculating the bending stress. On the contrary, the EHSAPT allows for the appropriate degrees-of-freedom in the two face sheets and the core, resulting in a very accurate solution. Both the logarithmic and the polynomial versions of the EHSAPT are in excellent agreement with the elasticity.

Fig. 4
Fig. 4
Close modal

The transverse displacement at the top face, wt, along the span of the curved panel is shown in Fig. 5(a), it can be seen that both the EHSAPT-logarithmic and the EHSAPT-polynomial predict a displacement profile in very close agreement with the elasticity, unlike the FOSDT, which seems to be underestimating the displacement. However, a closer look shows that the EHSAPT-logarithmic is more accurate. In particular, Figs. 5(b) and 5(c) show the radial (transverse) displacement profiles at $θ=α/2$ through the thickness for cases 3 and 4; it can be seen that the EHSAPT logarithmic is more accurate than the EHSAPT polynomial, especially for the smaller radius case 3. For larger radius, the difference is negligible (case 4). It also shows again that the FOSDT is inaccurate.

Fig. 5
Fig. 5
Close modal

Finally, Fig. 6 shows the profile of the circumferential (hoop) displacement at θ = 0 through the thickness, showing again that the logarithmic EHSAPT is more accurate than the polynomial EHSAPT.

Fig. 6
Fig. 6
Close modal

## Conclusions

In summary, curved sandwich panels are studied, and two variants of the EHSAPT are formulated. One is based on a logarithmic displacement field for the core, while the other is based on polynomial functions of the thicknesswise coordinate. The system of governing differential equations and the associated boundary conditions are derived via the principle of minimum total potential energy. Closed-form analytical solution and its procedure are outlined for a simply supported curved sandwich panel subjected to a distributed load acting on the top face. Various geometries and core materials are studied, and the corresponding results are shown and compared among the two variants of the EHSAPT, the elasticity solution and the FOSDT. The elasticity solution serves as a benchmark. Results show that in general both EHSAPTs show very good accuracy (unlike the FOSDT) but the logarithmic EHSAPT has an edge over the polynomial EHSAPT especially with regard to the radial normal stresses σrr and for configurations involving stiffer cores.

## Acknowledgment

The financial support of the Office of Naval Research, Grant No. N00014-16-1-2831, and the interest and encouragement of the Grant Monitor, Dr. Y. D. S. Rajapakse, are both gratefully acknowledged.

### Appendix A: Logarithmic EHSAPT Constants

In terms of
$k0=b(rbclnrbc−rtclnrtc)2$
(A1)
and the $H1ja,b,c,d, j=1⋯9$ constants defined as
$H11a=1rbcrtc[2rbc(c11c+c13c)(rbc−rtc)−2rbcrtcc11clnrbc+lnrtc[rbc2(c11c+c13c)+rtc2(c11c−c13c)]]$
(A2a)
$H12a=(c11c+c13c)(rbc−rtc)rbcrtc[(rbc+rtc)ln(rbcrtc)−2rbc+2rtc]$
(A2b)
$H13a=1rbcrtc[−2rtc(c11c+c13c)(rbc−rtc)−2rbcrtcc11clnrtc+lnrbc[rbc2(c11c−c13c)+rtc2(c11c+c13c)]]$
(A2c)
$H14a=c11c(fb+2Rb)2Rbrbcrtc[2rbc(−rtclnrbc+rbc−rtc)+(rbc2+rtc2)lnrtc]$
(A2d)
$H15a=c11c(rbc−rtc)rbcrtc[(rbc+rtc)ln(rbcrtc)−2rbc+2rtc]$
(A2e)
$H16a=c11c(ft−2Rt)2Rtrbcrtc[−2rtc(−rbclnrtc−rbc+rtc)−(rbc2+rtc2)lnrbc]$
(A2f)
$H17a=c11crbcrtc[2rbcrtcln(rbcrtc)(rbclnrbc−rtclnrtc)−(rbc−rtc)(rbc+rtc)[rbc(lnrtc+2)−rtc(lnrbc+2)]]$
(A2g)
$H18a=c11cfb2Rbrbcrtc[2rbc(rtclnrbc−rbc+rtc)−(rbc2+rtc2)lnrtc]$
(A2h)
$H19a=c11cft2Rtrbcrtc[2rtc(−rbclnrtc−rbc+rtc)+(rbc2+rtc2)lnrbc]$
(A2i)
$H11b=c11c[rbc(ln2rbc−ln2rtc−2lnrtc)+2rtclnrtc]+2c13c(rbclnrbc−rtclnrtc)$
(A3a)
$H12b=c11cln(rbcrtc)[(rbc+rtc)ln(rbcrtc)−2rbc+2rtc]$
(A3b)
$H13b=c11c[rtc(ln2rtc−ln2rbc−2lnrbc)+2rbclnrbc]+2c13c(rtclnrtc−rbclnrbc)$
(A3c)
$H14b=c11c(fb+2Rb)2Rb[rbc(ln2rbc−ln2rtc−2lnrtc)+2rtclnrtc]$
(A3d)
$H15b=c11cln(rbcrtc)[(rbc+rtc)ln(rbcrtc)−2rbc+2rtc]$
(A3e)
$H16b=c11c(ft−2Rt)2Rt[rtc(ln2rbc+2lnrbc−ln2rtc)−2rbclnrbc]$
(A3f)
$H17b=c11c(rtc−rbc)[(rbc+rtc)ln(rbcrtc)−2rbc+2rtc]ln(rbcrtc)$
(A3g)
$H18b=c11cfb2Rb[rbc(ln2rtc−ln2rbc)+2(rbc−rtc)lnrtc]$
(A3h)
$H19b=c11cft2Rt[rtc(ln2rtc−ln2rbc)+2(rbc−rtc)lnrbc]$
(A3i)
$H11c=12Rb{−(c11c−c13c)(rbc−rtc)+c11crbclnrbc+[−c11c+(c11c−c13c)ln(rtcrbc)]rtclnrtc}$
(A4a)
$H12c=2(c11c−c13c)[(rbc−rtc)2−rbcrtcln(rbcrtc)2]$
(A4b)
$H13c=12Rt{(c11c−c13c)(rbc−rtc)+c11crtclnrtc−[c11c+(c11c−c13c)ln(rtcrbc)]rbclnrbc}$
(A4c)
$H14c=c11crbc(fb+2Rb)Rb[−rbc+rtc+rbclnrbc+(−1−lnrbc+lnrtc)rtclnrtc]$
(A4d)
$H15c=2c11c[(rbc−rtc)2−rbcrtcln(rbcrtc)2]$
(A4e)
$H16c=c11crtc(ft−2Rt)Rt[−rbc+rtc+rbclnrbc+(1+lnrbc−lnrtc)rtclnrtc]$
(A4f)
$H17c=c11c[2rbcrtcln(rbcrtc)(rbclnrtc−rtclnrbc)+(rbc−rtc)(rbc+rtc)[rbc(2−lnrbc)−rtc(2−lnrtc)]]$
(A4g)
$H18c=c11cfbrbcRb[rbc−rtc−rbclnrbc+(1+lnrbc−lnrtc)rtclnrtc]$
(A4h)
$H19c=c11cftrtcRt[rbc−rtc+rtclnrtc−(1−lnrbc+lnrtc)rbclnrbc]$
(A4i)
$H11d=−23rbcc11cln3rtc+rbc(c13c−2c11c)ln2rtc+2(c13c−c11c)(−rtclnrbc+rbc−rtc)lnrtc+13rbc(2c11clnrbc+3c13c)ln2rbc$
(A5a)
$H12d=13ln(rbcrtc)[−6(c11c−c13c)(rbc−rtc)+c11c(rbc+2rtc)ln2rbc+c11c[(rbc−rtc)lnrtc+6rtc]lnrbc−c11c[(2rbc+rtc)lnrtc+6rbc]lnrtc+3c13c(rbc+rtc)ln(rtcrbc)]$
(A5b)
$H13d=−23rtcc11cln3rbc+rtc(c13c−2c11c)ln2rbc+2(c11c−c13c)(rbclnrtc+rbc−rtc)lnrbc+13rtcln2rtc(2c11clnrtc+3c13c)$
(A5c)
$H14d=c11c(fb+2Rb)3Rb[3rtclnrbclnrtc−[3(rbc−rtc)+(lnrtc+3)rbclnrtc]lnrtc+rbcln3rbc]$
(A5d)
$H15d=13c11cln(rbcrtc)[(rbc+2rtc)ln2rbc+[(rbc−rtc)lnrtc+6rtc]lnrbc−[(2rbc+rtc)lnrtc+6rbc]lnrtc−6rbc+6rtc]$
(A5e)
$H16d=c11c(ft−2Rt)3Rt[rtcln3rbc+3rtcln2rbc−3(rbclnrtc+rbc−rtc)lnrbc−rtcln3rtc]$
(A5f)
$H17d=23c11c[(rtc2−rbc2)ln3rbc+3(rbc2+rtc2)ln2rbc−3(4rbcrtclnrtc+rbc2−rtc2)lnrbc+[rbc2(ln2rtc+3lnrtc+3)−rtc2(ln2rtc−3lnrtc+3)]lnrtc]$
(A5g)
$H18d=c11cfb3Rb[rbc(ln3rtc−ln3rbc)+3(rbclnrtc−rtclnrbc+rbc−rtc)lnrtc]$
(A5h)
$H19d=c11cft3Rt[3rbclnrbclnrtc−[rtc(lnrbc+3)lnrbc−3rbc+3rtc]lnrbc+rtcln3rtc]$
(A5i)
and the $H2ja,b, j=1⋯9$ constants defined as
$H21a=1rbcrtc[2rbc(c13c+c33c)(rbc−rtc)−2rbcrtcc13clnrbc+[rbc2(c13c+c33c)+rtc2(c13c−c33c)]lnrtc]$
(A6a)
$H22a=(c13c+c33c)(rbc−rtc)rbcrtc[(rbc+rtc)ln(rbcrtc)−2rbc+2rtc]$
(A6b)
$H23a=1rbcrtc[−2rtc(c13c+c33c)(rbc−rtc)−2rbcrtcc13clnrtc+[rbc2(c13c−c33c)+rtc2(c13c+c33c)]lnrbc]$
(A6c)
$H24a=c13c(fb+2Rb)2Rbrbcrtc[2rbc(−rtclnrbc+rbc−rtc)+(rbc2+rtc2)lnrtc]$
(A6d)
$H25a=c13c(rbc−rtc)rbcrtc[(rbc+rtc)ln(rbcrtc)−2rbc+2rtc]$
(A6e)
$H26a=−c13c(ft−2Rt)2Rtrbcrtc[2rtc(rtc−rbc−rbclnrtc)+(rbc2+rtc2)lnrbc]$
(A6f)
$H27a=c13crbcrtc{−2(rbc+rtc)(rbc−rtc)2+[2rbc2ln(rbcrtc)+rbc2−rtc2]rtclnrbc−[2rtc2ln(rbcrtc)+rbc2−rtc2]rbclnrtc}$
(A6g)
$H28a=c13cfb2Rbrbcrtc[2rbc(rtclnrbc−rbc+rtc)−(rbc2+rtc2)lnrtc]$
(A6h)
$H29a=c13cft2Rtrbcrtc[2rtc(−rbclnrtc−rbc+rtc)+(rbc2+rtc2)lnrbc]$
(A6i)
$H21b=−[rbcc13c(lnrtc+2)+2rtc(c33c−c13c)]lnrtc+rbcc13cln2rbc+2rbcc33clnrbc$
(A7a)
$H22b=c13cln(rbcrtc)[(rbc+rtc)ln(rbcrtc)−2rbc+2rtc]$
(A7b)
$H23b=−[rtcc13c(lnrbc+2)lnrbc−2rbc(c13c−c33c)]+rtcc13cln2rtc+2rtcc33clnrtc$
(A7c)
$H24b=c13c(fb+2Rb)2Rb[rbcln2rbc−(rbclnrtc+2rbc−2rtc)lnrtc]$
(A7d)
$H25b=c13cln(rbcrtc)[(rbc+rtc)ln(rbcrtc)−2rbc+2rtc]$
(A7e)
$H26b=c13c(ft−2Rt)2Rt[rtcln2rbc+2(rtc−rbc)lnrbc−rtcln2rtc]$
(A7f)
$H27b=−c13c(rbc−rtc)[(rbc+rtc)ln(rbcrtc)−2rbc+2rtc]ln(rbcrtc)$
(A7g)
$H28b=c13cfb2Rb[(rbclnrtc+2rbc−2rtc)lnrtc−rbcln2rbc]$
(A7h)
$H29b=c13cft2Rt[rtcln2rtc−(rtclnrbc−2rbc+2rtc)lnrbc]$
(A7i)
and the $H3ja,b,c, j=1⋯7$ constants defined as
$H31a=1rbcRt(fb+2Rb)(rbclnrbc−rtclnrtc)$
(A8a)
$H32a=0$
(A8b)
$H33a=1rtcRb(ft−2Rt)(rbclnrbc−rtclnrtc)$
(A8c)
$H34a=0$
(A8d)
$H35a=Rtrbcrtc[−rbcrtc(fb+2Rb)lnrbc+[Rbrbc2+rtc2(fb+Rb)]lnrtc+2Rbrbc(rbc−rtc)]$
(A8e)
$H36a=RbRt(rbc−rtc)rbcrtc[(rbc+rtc)ln(rbcrtc)−2rbc+2rtc]$
(A8f)
$H37a=Rbrbcrtc[rbcrtc(ft−2Rt)lnrtc+[Rt(rbc2+rtc2)−rbc2ft]lnrbc+2Rtrtc(rtc−rbc)]$
(A8g)
$H31b=rbcRt(fb+2Rb)[12(−4rtcrbc+lnrtc+2)lnrtc−12ln2rbc+lnrbc]$
(A9a)
$H32b=RbRtln(rbcrtc)[2(rbc−rtc)+(rbc+rtc)ln(rtcrbc)]$
(A9b)
$H33b=12Rb(ft−2Rt)[−rtcln2rbc+rtc[ln2rtc−2ln(rbcrtc)]+4rbclnrbc]$
(A9c)
$H34b=RbRt(rtc−rbc)[2(rbc−rtc)+(rbc+rtc)ln(rtcrbc)]ln(rbcrtc)$
(A9d)
$H35b=12Rt[−[rbc(fb+2Rb)lnrtc+2rbc(fb+2Rb)−4rtc(fb+Rb)]lnrtc+rbc(fb+2Rb)ln2rbc−2fbrbclnrbc]$
(A9e)
$H36b=RbRtln(rbcrtc)[(rbc+rtc)ln(rbcrtc)−2rbc+2rtc]$
(A9f)
$H37b=12Rb[[2ft(rtc−2rbc)+(ft−2Rt)rtclnrbc+4Rt(rbc−rtc)]lnrbc−(ft−2Rt)rtcln2rtc+2ftrtclnrtc]$
(A9g)
$H31c=16Rt(fb+2Rb)[2rbcln3rtc+9rbcln2rtc+12(−rtclnrbc+rbc−rtc)lnrtc+rbc(3−2lnrbc)ln2rbc]$
(A10a)
$H32c=13RbRtln(rbcrtc)[12(rbc−rtc)+(2rbc+rtc)ln2rtc+[3(3rbc+rtc)+(rtc−rbc)lnrbc]lnrtc−[3(rbc+3rtc)+(rbc+2rtc)lnrbc]lnrbc]$
(A10b)
$H33c=16Rb(ft−2Rt)[−2rtcln3rbc−9rtcln2rbc+12(rbclnrtc+rbc−rtc)lnrbc+rtcln2rtc(2lnrtc−3)] ,$
(A10c)
$H34c=−13RbRt[2(rtc2−rbc2)ln3rbc+3(rbc2+3rtc2)ln2rbc+12rtc(−2rbclnrtc−rbc+rtc)lnrbc+[12rbc(rbc−rtc)+2(rbc−rtc)(rbc+rtc)ln2rtc+3(3rbc2+rtc2)lnrtc]lnrtc]$
(A10d)
$H35c=16Rt[−2rbc(fb+2Rb)ln3rtc−3rbc(3fb+4Rb)ln2rtc−12(fb+Rb)(−rtclnrbc+rbc−rtc)lnrtc+rbc[2(fb+2Rb)lnrbc−3fb]ln2rbc]$
(A10e)
$H36c=13RbRtln(rbcrtc)[(rbc+2rtc)ln2rbc+[(rbc−rtc)lnrtc+6rtc]lnrbc−[(2rbc+rtc)lnrtc+6rbc]lnrtc−6rbc+6rtc]$
(A10f)
$H37c=16Rb[2rtcln3rbc(ft−2Rt)+3rtcln2rbc(3ft−4Rt)−12(ft−Rt)(rbclnrtc+rbc−rtc)lnrbc+rtc[3ft−2(ft−2Rt)lnrtc]ln2rtc]$
(A10g)
the constants $Aia,b,c$ and $Bia,b,c$ in the logarithmic EHSAPT are defined as follows:
$Aia=k0ftrtc4Rt(H1id−H1iarbclnrbc)$
(A11a)
$Aib=k0c55crtc4RbRt2[ftH3ib−(ft−2Rt)H3ic+2H3ia(ft−Rt)rbclnrbc]$
(A11b)
$Aic=k0rtc2(H2ib+H1id+H2ia−H1ia)rbclnrbc$
(A11c)
$Bia=k0fbrbc4Rb(H1id−H1iartclnrtc)$
(A12a)
$Bib=k0c55crbc4Rb2Rt[fbH3ib−(fb+2Rb)H3ic+2H3ia(fb+Rb)rtclnrtc]$
(A12b)
$Bic=−k0rbc2[H2ib+H1id+(H2ia−H1ia)rtclnrtc]$
(A12c)
and the constants $Cib,c, Dib,c, Eib,c, Fib,c$, and $Gib,c$ in the logarithmic EHSAPT are defined as follows:
$Cib=k0c55c2RbRt[H3ic(rbc−rtc)+H3iarbcrtclnrbcrtc+H3ib(−rbclnrbc+rtclnrtc)]$
(A13a)
$Cic=k012[(H2ib+H1id)(rbc−rtc)−H1ibrbclnrbc+(H1ia−H2ia)rbcrtclnrbcrtc+H1ibrtclnrtc]$
(A13b)
$Dib=k0(ft−2Rt)rtc4Rt(−H1id+H1iarbclnrbc)$
(A14a)
$Dic=−k0c55c(ft−2Rt)rtc4RbRt2(H3ib−H3ic+2H3iarbclnrbc)$
(A14b)
$Eib=k0(fb+2Rb)rbc4Rb(−H1id+H1iartclnrtc)$
(A15a)
$Eic=−k0c55c(fb+2Rb)rbc4Rb2Rt(H3ib−H3ic+2H3iartclnrtc)$
(A15b)
$Fib=k012[H1id(rbc−rtc)+H1iarbcrtclnrbcrtc+H1ib(−rbclnrbc+rtclnrtc)]$
(A16a)
$Fic=k0c55c2RbRt[(H3ib+H3ic)(rbc−rtc)+H3ibrbclnrbc−H3ibrtclnrtc+2H3iarbcrtclnrtcrbc]$
(A16b)
$Gib=k012[H1id(rbc−rtc)(rbc+rtc)+rbc(−H1ic+H1iartc2)lnrbc+(H1ic−H1iarbc2)rtclnrtc]$
(A17a)
$Gic=k0c55c2RbRt[(H3ib−H3ic)(rbc−rtc)(rbc+rtc)+2H3iarbcrtc(−rtclnrbc+rbclnrtc)]$
(A17b)

### Appendix B: Polynomial EHSAPT Constants

In terms of the $H1ja,b,c,d, j=1⋯9$ defined as
$H11a=23c2[c(c11c+2c13c)−3Rcc13c] ; H12a=83c3(c11c−c13c)$
(B1a)
$H13a=23c2[3Rcc13c+c(c11c+2c13c)] ; H14a=c3c11c(fb+2Rb)3Rb$
(B1b)
$H15a=83c3c11c ; H16a=−c3c11c(ft−2Rt)3Rt$
(B1c)
$H17a=0 ; H18a=−c3fbc11c3Rb ; H19a=c3ftc11c3Rt$
(B1d)
$H11b=−2c2c13c−(Rc+c)c11c[Rcln(Rc+cRc−c)+2c]$
(B2a)
$H12b=4c11c[12(c2−Rc2)ln(Rc−cRc+c)+cRc]$
(B2b)
$H13b=2c2c13c+(c−Rc)c11c[Rcln(Rc+cRc−c)+2c]$
(B2c)
$H14b=c11c(fb+2Rb)3cRb[32Rc2(Rc+c)ln(Rc−cRc+c)−c(c2+3cRc+3Rc2)]$
(B2d)
$H15b=4c11c[12(c2−Rc2)ln(Rc−cRc+c)+cRc]$
(B2e)
$H16b=c11c(ft−2Rt)3cRt(ft−2Rt)[32(c−Rc)Rc2ln(Rc+cRc−c)−c(c2−3cRc+3Rc2)]$
(B2f)
$H17b=43c11c[2c3+32Rc(c2−Rc2)ln(Rc+cRc−c)−3cRc2]$
(B2g)
$H18b=fbc11c3cRb[c(c2+3cRc+3Rc2)+32(Rc+c)Rc2ln(Rc+cRc−c)]$
(B2h)
$H19b=ftc11c3cRt[c(c2−3cRc+3Rc2)+32(c−Rc)Rc2ln(Rc−cRc+c)]$
(B2i)
$H11c=−23c3[(c−2Rc)c13c+cc11c] ; H12c=−83c3Rcc13c$
(B3a)
$H13c=23c3[(2Rc+c)c13c+cc11c] ; H14c=−c4c11c(fb+2Rb)5Rb$
(B3b)
$H15c=0 ; H16c=−c4c11c(ft−2Rt)5Rt ; H17c=815c5c11c$
(B3c)
$H18c=c4fbc11c5Rb ; H19c=c4ftc11c5Rt$
(B3d)
$H11d=215c4[(6c−5Rc)c13c+3cc11c] ; H12d=815c5(c11c−3c13c)$
(B4a)
$H13d=215c4[(5Rc+6c)c13c+3cc11c] ; H14d=c5c11c(fb+2Rb)5Rb$
(B4b)
$H15d=815c5c11c ; H16d=−c5c11c(ft−2Rt)5Rt$
(B4c)
$H17d=0 ; H18d=−c5fbc11c5Rb ; H19d=c5ftc11c5Rt$
(B4d)
and the $H2ja,b, j=1⋯9$ defined as
$H21a=43[(2c−3Rc)c33c+cc13c] ; H22a=163c(c13c−c33c)$
(B5a)
$H23a=43[(3Rc+2c)c33c+cc13c] ; H24a=2cc13c(fb+2Rb)3Rb$
(B5b)
$H25a=163cc13c ; H26a=−2cc13c(ft−2Rt)3Rt ; H27a=0$
(B5c)
$H28a=−2cfbc13c3Rb ; H29a=2cftc13c3Rt$
(B5d)
$H21b=−43c[(c−2Rc)c33c+cc13c] ; H22b=−163cRcc33c$
(B6a)
$H23b=43c[(2Rc+c)c33c+cc13c] ; H24b=−2c2c13c(fb+2Rb)5Rb$
(B6b)
$H25b=0 ; H26b=−2c2c13c(ft−2Rt)5Rt ; H27b=1615c3c13c$
(B6c)
$H28b=2c2fbc13c5Rb ; H29b=2c2ftc13c5Rt$
(B6d)
and the $H3ja,b,c,d, j=1⋯7$ defined as
$H31a=23c3(c−3Rc)Rt(fb+2Rb) ; H32a=−323c4RbRt$
(B7a)
$H33a=−23c3Rb(3Rc+c)(ft−2Rt) ; H34a=0$
(B7b)
$H35a=−23c3Rt[fb(c−3Rc)−2cRb] ; H36a=163c4RbRt$
(B7c)
$H37a=23c3Rb[(3Rc+c)ft+2cRt]$
(B7d)
$H31b=Rt(fb+2Rb)[−4c33+2c2Rc+2cRc2+Rc2(Rc+c)ln(Rc+cRc−c)]$
(B8a)
$H32b=−4cRbRt[2cRc+(c−Rc)(Rc+c)ln(Rc−cRc+c)]$
(B8b)
$H33b=−23Rb[c(2c2+3cRc−3Rc2)+32(Rc−c)Rc2ln(Rc−cRc+c)](ft−2Rt)$
(B8c)
$H34b=−83cRbRt[2c3+32Rc(c2−Rc2)ln(Rc+cRc−c)−3cRc2]$
(B8d)
$H35b=13Rt[4c3(fb−3Rb)−6c2Rc(fb+2Rb)−6cfbRc2+3Rc(Rc+c)ln(Rc−cRc+c)(fbRc+2cRb)]$
(B8e)
$H36b=8cRbRt[12(c2−Rc2)ln(Rc−cRc+c)+cRc]$
(B8f)
$H37b=23Rb[c[(2c2+3cRc−3Rc2)ft+6c(c−Rc)Rt]+32(c−Rc)Rcln(Rc+cRc−c)(Rcft+2cRt)]$
(B8g)
$H31c=−415c4(3c−5Rc)Rt(fb+2Rb) ; H32c=−163c4RbRcRt$
(B9a)
$H33c=−415c4Rb(5Rc+3c)(ft−2Rt) ; H34c=−165c6RbRt$
(B9b)
$H35c=415c4Rt[fb(3c−5Rc)−5cRb] ; H36c=0$
(B9c)
$H37c=415c4Rb[(5Rc+3c)ft+5cRt]$
(B9d)
$H31d=25c5(c−3Rc)Rt(fb+2Rb) ; H32d=−6415c6RbRt$
(B10a)
$H33d=−25c5Rb(3Rc+c)(ft−2Rt) ; H34d=−3215c6RbRcRt$
(B10b)
$H35d=−25c5Rt[fb(c−3Rc)−2cRb] ; H36d=1615c6RbRt$
(B10c)
$H37d=25c5Rb[(3Rc+c)ft+2cRt]$
(B10d)
the constants $Aia,b,c$ and $Bia,b,c$ are defined as follows:
$Aia=bft8c5Rt[(c−Rc)[Rc(RcH1ib−H1ia)+H1ic]+H1id]$
(B11a)
$Aib=bc55c16c6RbRt2[ft[−Rc2(H3ia+cH3ib)+Rc(cH3ia+H3ic)+Rc3H3ib+cH3ic+2H3id]+2cRt[(c−Rc)(H3ia−RcH3ib)+H3ic]]$
(B11b)
$Aic=b8c4[(c−Rc)(−RcH1ia+Rc2H1ib+H1ic)+H1id]$
(B11c)
$Bia=bfb8c5Rb[H1id−(Rc+c)[Rc(RcH1ib−H1ia)+H1ic]]$
(B12a)
$Bib=bc55c16c6Rb2Rt[−2c2Rb(H3ia−RcH3ib)+fb[Rc(−RcH3ia+Rc2H3ib+H3ic)+2H3id]+c[2Rb(−RcH3ia+Rc2H3ib+H3ic)−fb(Rc(H3ia−RcH3ib)+H3ic)]]$
(B12b)
$Bic=b8c4[c2(2H2ib−cH2ia)−2(Rc+c)H1ia+2Rc(Rc+c)H1ib+2H1ic]$
(B12c)
and the constants $Cib,c, Dib,c, Eib,c, Fib,c$, and $Gib,c$ are defined as follows:
$Cib=bc55c4c5RbRt[RcH3ia+(c2−Rc2)H3ib−H3ic]$
(B13a)
$Cic=b2c4[RcH1ia+(c2−Rc2)H1ib−c2H2ib−H1ic]$
(B13b)
$Dib=−b(ft−2Rt)8c5Rt[(c−Rc)(−RcH1ia+Rc2H1ib+H1ic)+H1id]$
(B14a)
$Dic=−bc55c(ft−2Rt)16c6RbRt2[(c−Rc)RcH3ia−(c−Rc)Rc2H3ib+(Rc+c)H3ic+2H3id]$
(B14b)
$Eib=b(fb+2Rb)8c5Rb[(Rc+c)(−RcH1ia+Rc2H1ib+H1ic)−H1id]$
(B15a)
$Eic=bc55c(fb+2Rb)16c6Rb2Rt[Rc(Rc+c)H3ia−Rc2(Rc+c)H3ib+(c−Rc)H3ic−2H3id]$
(B15b)
$Fib=b2c4[RcH1ia+(c2−Rc2)H1ib−H1ic]$
(B16a)
$Fic=−bc55c4c5RbRt[RcH3ia+(c2−Rc2)H3ib+H3ic]$
(B16b)
$Gib=b2c4[(c2−Rc2)H1ia+(Rc3−c2Rc)H1ib+RcH1ic−H1id]$
(B17a)
$Gic=bc55c4c5RbRt[Rc[RcH3ia+(c2−Rc2)H3ib−H3ic]−2H3id]$
(B17b)

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