Sandwich theory

Bending of a sandwich beam after incorporating shear of the core into the deformation.

The main assumptions of linear sandwich theories of beams with thin facesheets are:

• the transverse normal stiffness of the core is infinite, i.e., 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, i.e., the core does not lengthen or compress in the x-direction
• the facesheets behave according to the Euler-Bernoulli assumptions, i.e., 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.

The constitutive relations for two-dimensional orthotropic linear elastic materials are

${\displaystyle {\begin{bmatrix}\sigma _{xx}\\\sigma _{zz}\\\sigma _{zx}\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{13}&0\\C_{13}&C_{33}&0\\0&0&C_{55}\end{bmatrix}}{\begin{bmatrix}\varepsilon _{xx}\\\varepsilon _{zz}\\\varepsilon _{zx}\end{bmatrix}}}$

The assumptions of sandwich theory lead to the simplified relations

${\displaystyle \sigma _{xx}^{\mathrm {face} }=C_{11}^{\mathrm {face} }~\varepsilon _{xx}^{\mathrm {face} }~;~~\sigma _{zx}^{\mathrm {core} }=C_{55}^{\mathrm {core} }~\varepsilon _{zx}^{\mathrm {core} }~;~~\sigma _{zz}^{\mathrm {face} }=\sigma _{xz}^{\mathrm {face} }=0~;~~\sigma _{zz}^{\mathrm {core} }=\sigma _{xx}^{\mathrm {core} }=0}$

and

${\displaystyle \varepsilon _{zz}^{\mathrm {face} }=\varepsilon _{xz}^{\mathrm {face} }=0~;~~\varepsilon _{zz}^{\mathrm {core} }=\varepsilon _{xx}^{\mathrm {core} }=0}$

The equilibrium equations in two dimensions are

${\displaystyle {\cfrac {\partial \sigma _{xx}}{\partial x}}+{\cfrac {\partial \sigma _{zx}}{\partial z}}=0~;~~{\cfrac {\partial \sigma _{zx}}{\partial x}}+{\cfrac {\partial \sigma _{zz}}{\partial z}}=0}$

The assumptions for a sandwich beam and the equilibrium equation imply that

${\displaystyle \sigma _{xx}^{\mathrm {face} }\equiv \sigma _{xx}^{\mathrm {face} }(z)~;~~\sigma _{zx}^{\mathrm {core} }=\mathrm {constant} }$

Therefore, for homogeneous facesheets and core, the strains also have the form

${\displaystyle \varepsilon _{xx}^{\mathrm {face} }\equiv \varepsilon _{xx}^{\mathrm {face} }(z)~;~~\varepsilon _{zx}^{\mathrm {core} }=\mathrm {constant} }$

Bending of a sandwich beam. The total deflection is the sum of a bending part w_b and a shear part w_s

Shear strains during the bending of a sandwich beam.

Let the sandwich beam be subjected to a bending moment ${\displaystyle M}$ and a shear force ${\displaystyle Q}$. Let the total deflection of the beam due to these loads be ${\displaystyle w}$. The adjacent figure shows that, for small displacements, the total deflection of the mid-surface of the beam can be expressed as the sum of two deflections, a pure bending deflection ${\displaystyle w_{b}}$ and a pure shear deflection ${\displaystyle w_{s}}$, i.e.,

${\displaystyle w(x)=w_{b}(x)+w_{s}(x)}$

From the geometry of the deformation we observe that the engineering shear strain (${\displaystyle \gamma }$) in the core is related the effective shear strain in the composite by the relation

${\displaystyle \gamma _{zx}^{\mathrm {core} }={\tfrac {2h+f}{2h}}~\gamma _{zx}^{\mathrm {beam} }}$

Note the shear strain in the core is larger than the effective shear strain in the composite and that small deformations (${\displaystyle \tan \gamma =\gamma }$) are assumed in deriving the above relation. The effective shear strain in the beam is related to the shear displacement by the relation

${\displaystyle \gamma _{zx}^{\mathrm {beam} }={\cfrac {\mathrm {d} w_{s}}{\mathrm {d} x}}}$

The facesheets are assumed to deform in accordance with the assumptions of Euler-Bernoulli beam theory. The total deflection of the facesheets is assumed to be the superposition of the deflections due to bending and that due to core shear. The ${\displaystyle x}$-direction displacements of the facesheets due to bending are given by

${\displaystyle u_{b}^{\mathrm {face} }(x,z)=-z~{\cfrac {\mathrm {d} w_{b}}{\mathrm {d} x}}}$

The displacement of the top facesheet due to shear in the core is

${\displaystyle u_{s}^{\mathrm {topface} }(x,z)=-\left(z-h-{\tfrac {f}{2}}\right)~{\cfrac {\mathrm {d} w_{s}}{\mathrm {d} x}}}$

and that of the bottom facesheet is

${\displaystyle u_{s}^{\mathrm {botface} }(x,z)=-\left(z+h+{\tfrac {f}{2}}\right)~{\cfrac {\mathrm {d} w_{s}}{\mathrm {d} x}}}$

The normal strains in the two facesheets are given by

${\displaystyle \varepsilon _{xx}={\cfrac {\partial u_{b}}{\partial x}}+{\cfrac {\partial u_{s}}{\partial x}}}$

Therefore,

${\displaystyle \varepsilon _{xx}^{\mathrm {topface} }=-z~{\cfrac {\mathrm {d} ^{2}w_{b}}{\mathrm {d} x^{2}}}-\left(z-h-{\tfrac {f}{2}}\right)~{\cfrac {\mathrm {d} ^{2}w_{s}}{\mathrm {d} x^{2}}}~;~~\varepsilon _{xx}^{\mathrm {botface} }=-z~{\cfrac {\mathrm {d} ^{2}w_{b}}{\mathrm {d} x^{2}}}-\left(z+h+{\tfrac {f}{2}}\right)~{\cfrac {\mathrm {d} ^{2}w_{s}}{\mathrm {d} x^{2}}}}$

The shear stress in the core is given by

${\displaystyle \sigma _{zx}^{\mathrm {core} }=C_{55}^{\mathrm {core} }~\varepsilon _{zx}^{\mathrm {core} }={\cfrac {C_{55}^{\mathrm {core} }}{2}}~\gamma _{zx}^{\mathrm {core} }={\tfrac {2h+f}{4h}}~C_{55}^{\mathrm {core} }~\gamma _{zx}^{\mathrm {beam} }}$

or,

${\displaystyle \sigma _{zx}^{\mathrm {core} }={\tfrac {2h+f}{4h}}~C_{55}^{\mathrm {core} }~{\cfrac {\mathrm {d} w_{s}}{\mathrm {d} x}}}$

The normal stresses in the facesheets are given by

${\displaystyle \sigma _{xx}^{\mathrm {face} }=C_{11}^{\mathrm {face} }~\varepsilon _{xx}^{\mathrm {face} }}$

Hence,

${\displaystyle {\begin{aligned}\sigma _{xx}^{\mathrm {topface} }&=-z~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w_{b}}{\mathrm {d} x^{2}}}-\left(z-h-{\tfrac {f}{2}}\right)~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w_{s}}{\mathrm {d} x^{2}}}&=&-z~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}+\left({\tfrac {2h+f}{2}}\right)~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w_{s}}{\mathrm {d} x^{2}}}\\\sigma _{xx}^{\mathrm {botface} }&=-z~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w_{b}}{\mathrm {d} x^{2}}}-\left(z+h+{\tfrac {f}{2}}\right)~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w_{s}}{\mathrm {d} x^{2}}}&=&-z~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}-\left({\tfrac {2h+f}{2}}\right)~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w_{s}}{\mathrm {d} x^{2}}}\end{aligned}}}$

The resultant normal force in a face sheet is defined as

${\displaystyle N_{xx}^{\mathrm {face} }:=\int _{-f/2}^{f/2}\sigma _{xx}^{\mathrm {face} }~\mathrm {d} z_{f}}$

and the resultant moments are defined as

${\displaystyle M_{xx}^{\mathrm {face} }:=\int _{-f/2}^{f/2}z_{f}~\sigma _{xx}^{\mathrm {face} }~\mathrm {d} z_{f}}$

where

${\displaystyle z_{f}^{\mathrm {topface} }:=z-h-{\tfrac {f}{2}}~;~~z_{f}^{\mathrm {botface} }:=z+h+{\tfrac {f}{2}}}$

Using the expressions for the normal stress in the two facesheets gives

${\displaystyle {\begin{aligned}N_{xx}^{\mathrm {topface} }&=-f\left(h+{\tfrac {f}{2}}\right)~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w_{b}}{\mathrm {d} x^{2}}}=-N_{xx}^{\mathrm {botface} }\\M_{xx}^{\mathrm {topface} }&=-{\cfrac {f^{3}~C_{11}^{\mathrm {face} }}{12}}\left({\cfrac {\mathrm {d} ^{2}w_{b}}{\mathrm {d} x^{2}}}+{\cfrac {\mathrm {d} ^{2}w_{s}}{\mathrm {d} x^{2}}}\right)=-{\cfrac {f^{3}~C_{11}^{\mathrm {face} }}{12}}~{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}=M_{xx}^{\mathrm {botface} }\end{aligned}}}$

In the core, the resultant moment is

${\displaystyle M_{xx}^{\mathrm {core} }:=\int _{-h}^{h}z~\sigma _{xx}^{\mathrm {core} }~\mathrm {d} z=0}$

The total bending moment in the beam is

${\displaystyle M=N_{xx}^{\mathrm {topface} }~(2h+f)+2~M_{xx}^{\mathrm {topface} }}$

or,

${\displaystyle M=-{\cfrac {f(2h+f)^{2}}{2}}~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w_{b}}{\mathrm {d} x^{2}}}-{\cfrac {f^{3}}{6}}~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}}$

The shear force ${\displaystyle Q_{x}}$ in the core is defined as

${\displaystyle Q_{x}^{\mathrm {core} }=\kappa \int _{-h}^{h}\sigma _{xz}~dz={\tfrac {\kappa (2h+f)}{2}}~C_{55}^{\mathrm {core} }~{\cfrac {\mathrm {d} w_{s}}{\mathrm {d} x}}}$

where ${\displaystyle \kappa }$ is a shear correction coefficient. The shear force in the facesheets can be computed from the bending moments using the relation

${\displaystyle Q_{x}^{\mathrm {face} }={\cfrac {\mathrm {d} M_{xx}^{\mathrm {face} }}{\mathrm {d} x}}}$

or,

${\displaystyle Q_{x}^{\mathrm {face} }=-{\cfrac {f^{3}~C_{11}^{\mathrm {face} }}{12}}~{\cfrac {\mathrm {d} ^{3}w}{\mathrm {d} x^{3}}}}$

For thin facesheets, the shear force in the facesheets is usually ignored.[2]

The bending stiffness of the sandwich beam is given by

${\displaystyle D^{\mathrm {beam} }=-M/{\tfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}}$

From the expression for the total bending moment in the beam, we have

${\displaystyle M=-{\cfrac {f(2h+f)^{2}}{2}}~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w_{b}}{\mathrm {d} x^{2}}}-{\cfrac {f^{3}}{6}}~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}}$

For small shear deformations, the above expression can be written as

${\displaystyle M\approx -{\cfrac {f[3(2h+f)^{2}+f^{2}]}{6}}~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}}$

Therefore, the bending stiffness of the sandwich beam (with ${\displaystyle f\ll 2h}$) is given by

${\displaystyle D^{\mathrm {beam} }\approx {\cfrac {f[3(2h+f)^{2}+f^{2}]}{6}}~C_{11}^{\mathrm {face} }\approx {\cfrac {f(2h+f)^{2}}{2}}~C_{11}^{\mathrm {face} }}$

and that of the facesheets is

${\displaystyle D^{\mathrm {face} }={\cfrac {f^{3}}{12}}~C_{11}^{\mathrm {face} }}$

The shear stiffness of the beam is given by

${\displaystyle S^{\mathrm {beam} }=Q_{x}/{\tfrac {\mathrm {d} w_{s}}{\mathrm {d} x}}}$

Therefore, the shear stiffness of the beam, which is equal to the shear stiffness of the core, is

${\displaystyle S^{\mathrm {beam} }=S^{\mathrm {core} }={\cfrac {\kappa (2h+f)}{2}}~C_{55}^{\mathrm {core} }}$

A relation can be obtained between the bending and shear deflections by using the continuity of tractions between the core and the facesheets. If we equate the tractions directly we get

${\displaystyle n_{x}~\sigma _{xx}^{\mathrm {face} }=n_{z}~\sigma _{zx}^{\mathrm {core} }}$

At both the facesheet-core interfaces ${\displaystyle n_{x}=1}$ but at the top of the core ${\displaystyle n_{z}=1}$ and at the bottom of the core ${\displaystyle n_{z}=-1}$. Therefore, traction continuity at ${\displaystyle z=\pm h}$ leads to

${\displaystyle 2fh~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w_{s}}{\mathrm {d} x^{2}}}-(2h+f)~C_{55}^{\mathrm {core} }~{\cfrac {\mathrm {d} w_{s}}{\mathrm {d} x}}=4h^{2}~C_{11}^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{2}w_{b}}{\mathrm {d} x^{2}}}}$

The above relation is rarely used because of the presence of second derivatives of the shear deflection. Instead it is assumed that

${\displaystyle n_{z}~\sigma _{zx}^{\mathrm {core} }={\cfrac {\mathrm {d} N_{xx}^{\mathrm {face} }}{\mathrm {d} x}}}$

which implies that

${\displaystyle {\cfrac {\mathrm {d} w_{s}}{\mathrm {d} x}}=-2fh~\left({\cfrac {C_{11}^{\mathrm {face} }}{C_{55}^{\mathrm {core} }}}\right)~{\cfrac {\mathrm {d} ^{3}w_{b}}{\mathrm {d} x^{3}}}}$

Using the above definitions, the governing balance equations for the bending moment and shear force are

${\displaystyle {\begin{aligned}M&=D^{\mathrm {beam} }~{\cfrac {\mathrm {d} ^{2}w_{s}}{\mathrm {d} x^{2}}}-\left(D^{\mathrm {beam} }+2D^{\mathrm {face} }\right)~{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}\\Q&=S^{\mathrm {core} }~{\cfrac {\mathrm {d} w_{s}}{\mathrm {d} x}}-2D^{\mathrm {face} }~{\cfrac {\mathrm {d} ^{3}w}{\mathrm {d} x^{3}}}\end{aligned}}}$

We can alternatively express the above as two equations that can be solved for ${\displaystyle w}$ and ${\displaystyle w_{s}}$ as

${\displaystyle {\begin{aligned}&\left({\frac {2D^{\mathrm {face} }}{S^{\mathrm {core} }}}\right){\cfrac {\mathrm {d} ^{4}w}{\mathrm {d} x^{4}}}-\left(1+{\frac {2D^{\mathrm {face} }}{D^{\mathrm {beam} }}}\right){\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}+\left({\cfrac {1}{S^{\mathrm {core} }}}\right)~{\cfrac {\mathrm {d} Q}{\mathrm {d} x}}={\frac {M}{D^{\mathrm {beam} }}}\\&\left({\frac {D^{\mathrm {beam} }}{S^{\mathrm {core} }}}\right){\cfrac {\mathrm {d} ^{3}w_{s}}{\mathrm {d} x^{3}}}-\left(1+{\frac {D^{\mathrm {beam} }}{2D^{\mathrm {face} }}}\right){\cfrac {\mathrm {d} w_{s}}{\mathrm {d} x}}-{\cfrac {1}{S^{\mathrm {core} }}}~{\cfrac {\mathrm {d} M}{\mathrm {d} x}}=-\left(1+{\cfrac {D^{\mathrm {beam} }}{2D^{\mathrm {face} }}}\right){\frac {Q}{S^{\mathrm {core} }}}\,\end{aligned}}}$

Using the approximations

${\displaystyle Q\approx {\cfrac {\mathrm {d} M}{\mathrm {d} x}}~;~~q\approx {\cfrac {\mathrm {d} Q}{\mathrm {d} x}}}$

where ${\displaystyle q}$ is the intensity of the applied load on the beam, we have

${\displaystyle {\begin{aligned}&\left({\frac {2D^{\mathrm {face} }}{S^{\mathrm {core} }}}\right){\cfrac {\mathrm {d} ^{4}w}{\mathrm {d} x^{4}}}-\left(1+{\frac {2D^{\mathrm {face} }}{D^{\mathrm {beam} }}}\right){\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}={\frac {M}{D^{\mathrm {beam} }}}-{\cfrac {q}{S^{\mathrm {core} }}}\\&\left({\frac {D^{\mathrm {beam} }}{S^{\mathrm {core} }}}\right){\cfrac {\mathrm {d} ^{3}w_{s}}{\mathrm {d} x^{3}}}-\left(1+{\frac {D^{\mathrm {beam} }}{2D^{\mathrm {face} }}}\right){\cfrac {\mathrm {d} w_{s}}{\mathrm {d} x}}=-\left({\cfrac {D^{\mathrm {beam} }}{2D^{\mathrm {face} }}}\right){\frac {Q}{S^{\mathrm {core} }}}\,\end{aligned}}}$

Several techniques may be used to solve this system of two coupled ordinary differential equations given the applied load and the applied bending moment and displacement boundary conditions.

Assuming that each partial cross section fulfills Bernoulli's hypothesis, the balance of forces and moments on the deformed sandwich beam element can be used to deduce the bending equation for the sandwich beam.

Figure 1 - Equilibration of a deflected sandwich beam under temperature load and burden in comparison with the undeflected cross section

The stress resultants and the corresponding deformations of the beam and of the cross section can be seen in Figure 1. The following relationships can be derived using the theory of linear elasticity:[3][4]

${\displaystyle {\begin{aligned}M^{\mathrm {core} }&=D^{\mathrm {beam} }\left({\cfrac {\mathrm {d} \gamma _{2}}{\mathrm {d} x}}+\vartheta \right)=D^{\mathrm {beam} }\left({\cfrac {\mathrm {d} \gamma }{\mathrm {d} x}}-{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}+\vartheta \right)\\M^{\mathrm {face} }&=-D^{\mathrm {face} }{\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}\\Q^{\mathrm {core} }&=S^{\mathrm {core} }\gamma \\Q^{\mathrm {face} }&=-D^{\mathrm {face} }{\cfrac {\mathrm {d} ^{3}w}{\mathrm {d} x^{3}}}\end{aligned}}\,}$

where

${\displaystyle w\,}$	transverse displacement of the beam
${\displaystyle \gamma \,}$	Average shear strain in the sandwich	${\displaystyle \gamma =\gamma _{1}+\gamma _{2}\,}$
${\displaystyle \gamma _{1}\,}$	Rotation of the facesheets	${\displaystyle \gamma _{1}={\cfrac {\mathrm {d} w}{\mathrm {d} x}}\,}$
${\displaystyle \gamma _{2}\,}$	Shear strain in the core
${\displaystyle M^{\mathrm {core} }\,}$	Bending moment in the core
${\displaystyle D^{\mathrm {beam} }\,}$	Bending stiffness of the sandwich beam
${\displaystyle M^{\mathrm {face} }\,}$	Bending moment in the facesheets
${\displaystyle D^{\mathrm {face} }\,}$	Bending stiffness of the facesheets
${\displaystyle Q^{\mathrm {core} }\,}$	Shear force in the core
${\displaystyle Q^{\mathrm {face} }\,}$	Shear force in the facesheets
${\displaystyle S^{\mathrm {core} }\,}$	Shear stiffness of the core
${\displaystyle \vartheta \,}$	Additional bending as a consequence of a temperature drop	${\displaystyle \vartheta ={\frac {\alpha (T_{o}-T_{u})}{e}}\,}$
${\displaystyle \alpha \,}$	Temperature coefficient of expansion of the converings

Superposition of the equations for the facesheets and the core leads to the following equations for the total shear force ${\displaystyle Q}$ and the total bending moment ${\displaystyle M}$:

${\displaystyle {\begin{alignedat}{3}&S^{\mathrm {core} }\gamma -D^{\mathrm {face} }{\cfrac {\mathrm {d} ^{3}w}{\mathrm {d} x^{3}}}=Q&\quad \quad &(1)\\&D^{\mathrm {beam} }\left({\cfrac {\mathrm {d} \gamma }{\mathrm {d} x}}+\vartheta \right)-\left(D^{\mathrm {beam} }+D^{\mathrm {face} }\right){\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}=M&\quad \quad &(2)\,\end{alignedat}}}$

We can alternatively express the above as two equations that can be solved for ${\displaystyle w}$ and ${\displaystyle \gamma }$, i.e.,

${\displaystyle {\begin{aligned}&\left({\frac {D^{\mathrm {face} }}{S^{\mathrm {core} }}}\right){\cfrac {\mathrm {d} ^{4}w}{\mathrm {d} x^{4}}}-\left(1+{\frac {D^{\mathrm {face} }}{D^{\mathrm {beam} }}}\right){\cfrac {\mathrm {d} ^{2}w}{\mathrm {d} x^{2}}}={\frac {M}{D^{\mathrm {beam} }}}-{\cfrac {q}{S^{\mathrm {core} }}}-\vartheta \\&\left({\frac {D^{\mathrm {beam} }}{S^{\mathrm {core} }}}\right){\cfrac {\mathrm {d} ^{2}\gamma }{\mathrm {d} x^{2}}}-\left(1+{\frac {D^{\mathrm {beam} }}{D^{\mathrm {face} }}}\right)\gamma =-\left({\cfrac {D^{\mathrm {beam} }}{D^{\mathrm {face} }}}\right){\frac {Q}{S^{\mathrm {core} }}}\,\end{aligned}}}$