Sandwich theory describes the behaviour of a beam, plate, or shell which consists of three layers—two facesheets and one core. The most commonly used sandwich theory is linear and is an extension of first-order beam theory. The linear sandwich theory is of importance for the design and analysis of sandwich panels, which are of use in building construction, vehicle construction, airplane construction, and refrigeration engineering. Some advantages of sandwich construction are:
Sandwich cross-sections are composite. They usually consist of a low- to moderate-stiffness core which is connected with two stiff exterior facesheets. The composite has a considerably higher ratio of shear stiffness to weight than an equivalent beam made of only the core material or the facesheet material. The composite also has a high ratio of tensile strength to weight. The high stiffness of the facesheets leads to a high ratio of bending stiffness to weight for the composite. The behavior of a beam with sandwich cross-section under a load differs from a beam with a constant elastic cross section. If the radius of curvature during bending is large compared to the thickness of the sandwich beam and the strains in the component materials are small, then the deformation of a sandwich composite beam can be separated into two parts:
deformations due to bending moments or bending deformation, and deformations due to transverse forces, also called shear deformation. Sandwich beam, plate, and shell theories usually assume that the reference stress state is one of zero stress. However, during curing, differences of temperature between the facesheets persist because of the thermal separation by the core material. These temperature differences, coupled with different linear expansions of the facesheets, can lead to a bending of the sandwich beam in the direction of the warmer facesheet. If the bending is constrained during the manufacturing process, then residual stresses can develop in the components of a sandwich composite. The superposition of a reference stress state on the solutions provided by sandwich theory is possible when the problem is linear. However, when large elastic deformations and rotations are expected, the initial stress state has to be incorporated directly into the sandwich theory.
Engineering sandwich beam theory
In the engineering theory of sandwich beams, the axial strain is assumed to vary linearly over the cross-section of the beam as in Euler-Bernoulli theory:
ε x x ( x , z ) = − z d 2 w d x 2 {\displaystyle \varepsilon _{xx}(x,z)=-z~{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}}
Therefore, the axial stress in the sandwich beam is given by
σ x x ( x , z ) = − z E ( z ) d 2 w d x 2 , {\displaystyle \sigma _{xx}(x,z)=-z~E(z)~{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}},}
where E ( z ) {\displaystyle E(z)} is the Young's modulus, which is a function of the location along the thickness of the beam. The bending moment in the beam is then given by
M x ( x ) = ∬ z σ x x d z d y = − ( ∬ z 2 E ( z ) d z d y ) d 2 w d x 2 =: − D d 2 w d x 2 {\displaystyle M_{x}(x)=\iint z~\sigma _{xx}~\mathrm {d} z\,\mathrm {d} y=-\left(\iint z^{2}E(z)~\mathrm {d} z\,\mathrm {d} y\right)~{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}=:-D~{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}}
The quantity D {\displaystyle D} is called the flexural stiffness of the sandwich beam. The shear force Q x {\displaystyle Q_{x}} is defined as
Q x = d M x d x . {\displaystyle Q_{x}={\frac {\mathrm {d} M_{x}}{\mathrm {d} x}}.}
Using these relations, we can show that the stresses in a sandwich beam with a core of thickness 2 h {\displaystyle 2h} and modulus E c {\displaystyle E^{c}} and two facesheets each of thickness f {\displaystyle f} and modulus E f {\displaystyle E^{f}} are given by
σ x x f = z E f M x D ; σ x x c = z E c M x D τ x z f = Q x E f 2 D [ ( h + f ) 2 − z 2 ] ; τ x z c = Q x 2 D [ E c ( h 2 − z 2 ) + E f f ( f + 2 h ) ] {\displaystyle {\begin{aligned}\sigma _{xx}^{\mathrm {f} }&={\cfrac {zE^{\mathrm {f} }M_{x}}{D}}~;~~&\sigma _{xx}^{\mathrm {c} }&={\cfrac {zE^{\mathrm {c} }M_{x}}{D}}\\\tau _{xz}^{\mathrm {f} }&={\cfrac {Q_{x}E^{\mathrm {f} }}{2D}}\left[(h+f)^{2}-z^{2}\right]~;~~&\tau _{xz}^{\mathrm {c} }&={\cfrac {Q_{x}}{2D}}\left[E^{\mathrm {c} }\left(h^{2}-z^{2}\right)+E^{\mathrm {f} }f(f+2h)\right]\end{aligned}}}
For a sandwich beam with identical facesheets and unit width, the value of D {\displaystyle D} is
D = E f ∫ w ∫ − h − f − h z 2 d z d y + E c ∫ w ∫ − h h z 2 d z d y + E f ∫ w ∫ h h + f z 2 d z d y = 2 3 E f f 3 + 2 3 E c h 3 + 2 E f f h ( f + h ) . {\displaystyle {\begin{aligned}D&=E^{f}\int _{w}\int _{-h-f}^{-h}z^{2}~\mathrm {d} z\,\mathrm {d} y+E^{c}\int _{w}\int _{-h}^{h}z^{2}~\mathrm {d} z\,\mathrm {d} y+E^{f}\int _{w}\int _{h}^{h+f}z^{2}~\mathrm {d} z\,\mathrm {d} y\\&={\frac {2}{3}}E^{f}f^{3}+{\frac {2}{3}}E^{c}h^{3}+2E^{f}fh(f+h)~.\end{aligned}}}
If E f ≫ E c {\displaystyle E^{f}\gg E^{c}} , then D {\displaystyle D} can be approximated as
D ≈ 2 3 E f f 3 + 2 E f f h ( f + h ) = 2 f E f ( 1 3 f 2 + h ( f + h ) ) {\displaystyle D\approx {\frac {2}{3}}E^{f}f^{3}+2E^{f}fh(f+h)=2fE^{f}\left({\frac {1}{3}}f^{2}+h(f+h)\right)}
and the stresses in the sandwich beam can be approximated as
σ x x f ≈ z M x 2 3 f 3 + 2 f h ( f + h ) ; σ x x c ≈ 0 τ x z f ≈ Q x 4 3 f 3 + 4 f h ( f + h ) [ ( h + f ) 2 − z 2 ] ; τ x z c ≈ Q x ( f + 2 h ) 2 3 f 2 + h ( f + h ) {\displaystyle {\begin{aligned}\sigma _{xx}^{\mathrm {f} }&\approx {\cfrac {zM_{x}}{{\frac {2}{3}}f^{3}+2fh(f+h)}}~;~~&\sigma _{xx}^{\mathrm {c} }&\approx 0\\\tau _{xz}^{\mathrm {f} }&\approx {\cfrac {Q_{x}}{{\frac {4}{3}}f^{3}+4fh(f+h)}}\left[(h+f)^{2}-z^{2}\right]~;~~&\tau _{xz}^{\mathrm {c} }&\approx {\cfrac {Q_{x}(f+2h)}{{\frac {2}{3}}f^{2}+h(f+h)}}\end{aligned}}}
If, in addition, f ≪ 2 h {\displaystyle f\ll 2h} , then
D ≈ 2 E f f h ( f + h ) {\displaystyle D\approx 2E^{f}fh(f+h)}
and the approximate stresses in the beam are
σ x x f ≈ z M x 2 f h ( f + h ) ; σ x x c ≈ 0 τ x z f ≈ Q x 4 f h ( f + h ) [ ( h + f ) 2 − z 2 ] ; τ x z c ≈ Q x ( f + 2 h ) 4 h ( f + h ) ≈ Q x 2 h {\displaystyle {\begin{aligned}\sigma _{xx}^{\mathrm {f} }&\approx {\cfrac {zM_{x}}{2fh(f+h)}}~;~~&\sigma _{xx}^{\mathrm {c} }&\approx 0\\\tau _{xz}^{\mathrm {f} }&\approx {\cfrac {Q_{x}}{4fh(f+h)}}\left[(h+f)^{2}-z^{2}\right]~;~~&\tau _{xz}^{\mathrm {c} }&\approx {\cfrac {Q_{x}(f+2h)}{4h(f+h)}}\approx {\cfrac {Q_{x}}{2h}}\end{aligned}}}
If we assume that the facesheets are thin enough that the stresses may be assumed to be constant through the thickness, then we have the approximation
σ x x f ≈ ± M x 2 f h ; σ x x c ≈ 0 τ x z f ≈ 0 ; τ x z c ≈ Q x 2 h {\displaystyle {\begin{aligned}\sigma _{xx}^{\mathrm {f} }&\approx \pm {\cfrac {M_{x}}{2fh}}~;~~&\sigma _{xx}^{\mathrm {c} }&\approx 0\\\tau _{xz}^{\mathrm {f} }&\approx 0~;~~&\tau _{xz}^{\mathrm {c} }&\approx {\cfrac {Q_{x}}{2h}}\end{aligned}}}
Hence the problem can be split into two parts, one involving only core shear and the other involving only bending stresses in the facesheets.
Linear sandwich theory
Bending of a sandwich beam with thin facesheets
The main assumptions of linear sandwich theories of beams with thin facesheets are:
the transverse normal stiffness of the core is infinite; that is, the core thickness in the z-direction does not change during bending the in-plane normal stiffness of the core is small compared to that of the facesheets; that is, the core does not lengthen or compress in the x-direction the facesheets behave according to the Euler-Bernoulli assumptions; that is, there is no xz-shear in the facesheets and the z-direction thickness of the facesheets does not change However, the xz shear-stresses in the core are not neglected.
Constitutive assumptions The constitutive relations for two-dimensional orthotropic linear elastic materials are
[ σ x x σ z z σ z x ] = [ C 11 C 13 0 C 13