A verified analytical sandwich beam model for soft and hard cores: comparison to existing analytical models and finite element calculations

This paper presents a novel approach of modeling of three-layer beam. Such composites are usually known as sandwich structures if the modulus of elasticity of the core is much smaller than those of the faces. In the present approach, the faces are modeled as Bernoulli–Euler beams, the core as a Timoshenko beam. Taking into account the kinematic and dynamic interface conditions, which means that the perfect bonding assumptions hold for the displacement and each layer is subjected to continuous traction stresses across the interface, a sixth-order differential equation is derived for the bending deflection, and a second-order system for the axial displacement. No restrictions are imposed on the elastic properties of the middle layer, and hence the developed theory also yields accurate results for hard cores. The presented refined theory is compared to analytical models from the literature and to finite element calculations for various benchmark examples. Special focus is laid the boundary conditions and the core stiffness. A parametric study varying the Young modulus of the core shows that the present sandwich model agrees very well with the target solutions obtained from finite element calculations under plane stress assumptions, in particular concerning the transverse deflection, the shear stress distribution and the interfacial normal stress.

Free body diagram of a beam layer j (surface tractions are denoted by q j−1 , q j and τ j−1 , τ j , the beam forces are Q j , N j and moment M j )

Recapitulation: equations of motion of a single isotropic layer
The basis for the present sandwich model is an accurate beam model taking into account distributed loads that act perpendicular q j and parallel τ j to the upper and lower surfaces. Timoshenko assumptions hold for layer j in Fig. 1, and hence the displacement field is where u j (x) and w j (x) are the horizontal and the vertical deflection and ψ is the rotation angle. The thickness of one layer is 2c j . From Eq. (1),it follows for the strains ε x x =ū j,x = u j,x γ xz = 2ε xz =ū j,z +w j,x = ψ j + w j,x (2) and for the stresses Young's modulus is denoted by E j , for the shear modulus G j = E j /(2 + 2ν j ) holds if an isotropic material is assumed. The forces and the bending moment on beam level follow by integration of the stresses. Substituting Eqs. (1) and (2) in Eq. (3), one finds Here the axial stiffness, the shear stiffness and the bending stiffness are EA j , GA * j and EI j , respectively. The cross-section area is A j = 2bc j and the geometrical moment of inertia is I j = 2bc 3 j /3. The star-symbol GA * j = κG A j indicates that the shear stiffness also includes a proper shear correction factor κ. The equilibrium conditions can be derived from Fig. 1 and read M j,x − Q j + τ j + τ j−1 c j = 0 ( 8 ) Q j,x + q j − q j−1 = 0 ( 9 ) It is noted that the common relation M j,x = Q j is violated if shear loads over the surfaces are taken into account (see Eq. (8) unless for τ j−1 = −τ j ). If unity width is used for the layers b = 1 m, the expressions for Substituting the beam forces (4)(5)(6) in Eqs. (7)(8)(9), one finds three differential equations In case of thin layers the influence of shear on the deflection can be neglected. The Bernoulli-Euler equations follow by differentiating Eq. (11) with respect to x and adding to Eq. (12) The beam forces and the moment are Note that the shear force is not proportional to the third derivative of the vertical deflection unless for τ j−1 = −τ j (see Eq. (15)). This is important to note because the derivations presented by Mead and Markus [11], Di Taranto [13] and Stamm and Witte [14] are erroneous where M j,x = Q j holds despite the presence of shear tractions.

Equations of motion of a three-layer beam with soft or hard core
In general, the faces (or skins) of a sandwich construction are relatively thin compared to the core thickness, and hence shear effects play a minor rule. Modeling the faces as two BE-beams as in Eq. (13), one finds for the upper (subscript 1) and the lower face (subscript 3) (see also Fig. 2) The main focus here is to establish a mathematical model for a three-layer beam whose core bending stiffness is not negligible. Considering Timoshenko assumptions, one finds three differential equations for the core from (10-12) These Eqs. (17)(18)(19)(20)(21)(22)(23) contain 11 unknowns (7 degrees of freedom w 1 , w 2 , w 3 , u 1 , u 2 , u 3 , ψ 2 and 4 interface stresses q 1 , q 2 , τ 1 , τ 2 ) which can be computed by 7 differential Eqs. (17)(18)(19)(20)(21)(22)(23) and 4 coupling Eqs. (24)(25)(26) from the perfect bonding assumptions. The latter read A condensed and more comprehensible form of the differential equations is obtained if a symmetrical setup is considered: The subscripts 1 and 3 are replaced by the subscript f (=face), the subscript 2 is replaced by c (=core). In the following, the vertical and horizontal deflections and the rotation angle of the core are denoted by w, u and ψ. In a first step the coupling Eqs. (24)(25)(26) are substituted in Eqs. (17)(18)(19)(20)(21)(22)(23). In a second step, one solves Eqs. (17)(18)(19)(20) for the yet unknown interfacial stresses q 1 , q 2 , τ 1 , τ 2 and substitutes the outcome into the remaining three differential Eqs. (21-23) Neglecting the external shear stress τ 0 = τ 3 = 0 one finds The solution for the decoupled axial displacement (32) reads The two unknowns A 0 , A 1 can be determined by corresponding kinematic or dynamic boundary conditions. One observes that the axial displacement of the middle axis of the core is zero (unless normal forces for the boundary conditions are considered). Integrating Eq. (30) and adding the results to Eq. (31) one finds for the rotation angle Inserting Eq. (34) into Eq. (31) and integrating the resulting equation yields the second-order differential equation for the deflection where P, Q and the inhomogeneous term f (x) read From a mathematical point of view, the three-layer sandwich model is reduced to a second-order differential Eq. (36), which holds for an arbitrary distributed load q 0 (x). The eigenvalues of the system are given by Eq. In order to provide a more detailed relation to other sandwich models (see sections A.1 and A.2) and for reasons of an easy implementation in symbolic computational software like MATHEMATICA, it is more appropriate to convert Eq. (35) into a sixth-order differential equation where the constants R and S read Equation (38) is a sixth-order differential equation for the deflection w where four eigenvalues are zero, the remaining nontrivial eigenvalues are the root of The solution for the deflection is where w j (x) is the inhomogeneous solution. The constants B 1 − B 6 must be determined from the boundary conditions (see Sect. 3.1).

Boundary conditions
As opposed to the classical beam theory, where four boundary conditions must be prescribed in order to calculate the constants from integration, the sixth-order differential equation for the three-layer beam requires three boundary conditions on the left and on the right side of the beam (note: the boundary conditions for the axial deflection u(x) (see Eq. (33)), are not discussed here, because external shear tractions and normal forces are not considered, hence A 0 = A 1 = 0 holds). In the following, a clamped end, a hinge and a free end are discussed. For each boundary, two possibilities exist depending if a rigid end plate is attached or not (see also the extensive discussion in Stamm and Witte [14] for Mead and Markus' sandwich model [11]). It is clear from Fig. 3 that an end plate is an additional kinematic constraint causing the core and face angles to be equal ψ = −w ,x . According to Eqs. (17), (19), (24) and (25) the interfacial shear stresses read in case of a a symmetrical sandwich panel and a vanishing axial displacement of the core u = 0 The following relations hold between the kinematic degrees of freedom and the beam forces of the core and the face: • normal force: • shear force: • bending moment: Considering an end plate means that none of the kinematic degrees of freedom can move at x * = 0 Another possibility is if the upper and the lower face are fixed by two hinges. The bending moment of the face will vanish M f (x * ) = 0 and from Fig. 3a it becomes clear that the rotation angle of the core will be nonzero, i.e., ψ(x * ) = 0, Eqs. (24) and (25) describe the kinematic constraints between deflection w and ψ

Hinged boundary: with and without end plate
In case of a hard hinged support (with rigid end plate), the face bending moment condition must be replaced by the kinematic constraint Eq. (24)

Free end: with and without end plate
If all of the vertical faces of the layers are free, the dynamic boundaries are free If an end plate is present, then hold.

Examples: verification of the presented sandwich model
In this section, the presented refined sandwich theory (RSWT) is compared to both Mead-Markus models: the 6th-order model (see Appendix A.1) does not include the core bending stiffness, the simplified 4th-order model (see Appendix A.2) also disregards the face bending stiffness. It becomes clear that the latter one cannot satisfy all boundary conditions (see also Stamm and Witte [14] by comparing p.25-p.90). The results from the two-dimensional finite element calculations under plane stress assumptions serve as target solutions. Additionally, the outcome of an equivalent single layer model (ESL Timo ) taking into account shear deformation (Timoshenko assumption) is computed (Appendix C). For the variations concerning the core elasticity (see Table 1 Parameters for the numerical examples Poisson ratio of core and face κ = 1 (−) 1 Shear correction factor of core (varies) Young's modulus of core Cross section area of core , results are also computed for a structure with totally negligible core stiffness (BE face , i.e., the structure consists of perfectly slipping faces without interfacial shear stress, and hence for the core GA * c ≈ EI c ≈ 0 holds, Appendix D). These last two models are limiting cases (ESL Timo and BE face ) and should not be understood as specific sandwich models. The only intention of these models is to show their very limited validity range: If the Young modulus of the core and of the faces are of the same order, the results should be very close to ESL Timo ; if the core is very soft, the results from BE face should be obtained.
The vertical load on the upper face is uniformly distributed q 0 (x) = 1 N/m over the length l = 1 m. As benchmark examples a cantilever and a clamped-hinged sandwich beam are considered and the lateral deflections are shown. The shear stress distribution in thickness direction σ xz (x, z) and the interfacial normal stress (i.e., σ zz (x, c) = q 2 (x)/b and σ zz (x, −c) = q 1 (x)/b (see Fig. 2)) are computed for the cantilever sandwich. The thickness-to-length ratio is λ t = (2c + 4t)/l = 1/8 for all benchmark examples, the relative thickness of the core λ c = 2c/(2c + 4t) (ratio of core thickness to total thickness) is either 0.2 or 0.8.
For the elasticity of the core, the non-dimensional variable μ = E c /E f = G c /G f (ratio of Young's and shear moduli) is introduced. From a practical point of view, parameter variations of the relative core elasticity immediately show the validity range and accuracy of each analytical model under investigation. The following abbreviations are used for the analytical models: • RSWT (refined sandwich theory): governed by the 6th-order differential equations (30) and (31) taking into account the core bending stiffness (see Sect. 3. • MM 6th : Mead-Markus [11] or Di Taranto [13] sandwich model disregarding the core bending stiffness, see Appendix A.1 and Stamm and Witte [14] p.86. • MM 4th : Mead-Markus [11] or Di Taranto [13] sandwich model disregarding the core and the face bending stiffness, see Appendix A.2 and Stamm and Witte [14] p.22. • ESL Timo : equivalent single layer theory with Timoshenko assumption, see Appendix C • BE face : the core bending and shear stiffness is totally neglected. The structure is reduced to a sliding two-layer beam with vanishing interfacial shear stress τ 1 = τ 2 = 0, see Appendix D.
The geometrical and material parameters for the core and the faces of the sandwich beam are summarized in Table 1. Special attention must be paid to the choice of the shear correction factor for the core layer. For typical sandwich structures, when the elasticity of the core is small, the shear stress distribution of the core will be constant or at least very close to a constant distribution, and hence setting κ = 1 is justified. As the results from Fig. 6 and the comparison to FE results show this is also a reasonable choice for moderately soft cores, i.e., if μ = 1/10.

Clamped (with rigid end plate): free (without rigid end plate)
The first example is a cantilever sandwich beam with a thickness-to-length ratio λ t = 1/8. The core thickness is 0.08 m, the thickness of each face is 0.01 m, and hence λ c = 0.8 follows from Table 1.  (Fig. 4a). But the zoom figure reveals that the deflection from the FE (gray) and the present sandwich theory (blue) is −2.21 × 10 −9 m at x = l/2. The 4th (magenta) and 6th (red)-order models of Mead-Markus overestimate the outcome −2.41×10 −9 m and −2.42 × 10 −9 m, respectively, because the bending stiffness of the core is neglected. The ESL-theory (black) is a little too stiff and underestimates the result −1.99 × 10 −9 m.
If the core is weaker (Fig. 4b), RSW T (blue) and M M 6th (red) almost coincide −2.49 × 10 −8 m and both results are close to the the numerical outcome −2.50 × 10 −8 m (gray) at x = l/2. M M 4th overestimates the outcome: the clamped boundary is inaccurately modeled because the bending stiffness of the faces are neglected. As expected the ESL (black) yields much too stiff results for the soft core example.
A good overview of the validity range for all the analytical theories shows Fig. 5. The relative tip deflection is shown (normalized by the RSWT outcome w RSW T (l)) over the variation of the Young and the shear modulus of the core. The non-dimensional variable is μ = E c /E f = G c /G f . The Timoshenko E SL-beam (black) yields reliable results for μ > 1/10 when the error is below 5%. When the core bending stiffness becomes less important (i.e., for μ < 1/50) the results from both Mead-Markus models are accurate. It can be seen that the 6th-order model converges to the presented RSWT results. Contrary, the 4th-order MM model should be used only for 1/200 > μ > 1/10. It is noted that compressibility effects of the core cannot be considered by any of the analytical models. The finite element results includes such effects, of course, but they play a minor role even for extremely soft cores: at μ = 10 −7 the thickness deformation is only 0.98% of the vertical deflection. Figure 6a, b shows the shear stress distribution at x = l/2 for μ = 1/10 and μ = 1/1000. One can recalculate the shear stress of each layer in some simple post-processing steps: assuming a quadratic shear stress of the form σ xz = C 0 + C 1 z + C 2 z 2 for the core and computing τ 1 , τ 2 from Eq. (42) and the shear force Q c , the three unknowns C 0 , C 1 , C 2 are determined. For the faces the same procedure can be done in an analogous manner.
One observes a perfect fit between the finite element and the RSWT results: The parabolic part of the core vanishes for the M M model by definition because normal stress is neglected. The shear stress distribution in the faces is almost linear: According to the balance of linear momentum σ x x,x + σ xz,z = 0, the linear shear stress contribution is caused by the constant axial stress due to the axial force N f , the quadratic part is caused The interfacial normal stresses σ zz (x, ±c), which equal q 1 and q 2 for beam of unit width, is shown in Fig. 6c, d. At the upper face the normal stress is close to one (F E: 0.950 Nm −2 , RSW T : 0.948 Nm −2 at x = l/2, Fig. 6c). Only close to the clamped and the free end where some end effects occur the F E and RSW T differ.
In case of a softer core (Fig. 6d), the presented theory also agrees with the numerical result. It is noted that the original paper of Mead-Markus [11] does not give any hints how to calculate the interfacial normal stress, so it is omitted here.

Clamped (with rigid end plate): soft hinged support
Next a clamped-hinged sandwich panel is investigated. Again the deflection curves with moderately hard and soft core are plotted and the thickness-to-length ratio is λ t = 1/8. Parameter variations of the core stiffness show the relative difference to the outcome from RSWT. Section 4.2.1 shows the results for a thick core λ c = 0.8 (sandwich); Sect. 4.2.2 shows the results for a thin core λ c = 0.2 (anti-sandwich).

Thick core (λ c = 0.8)
The core thickness is 0.08 m; the thickness of each face is 0.01 m. The deflection is shown in Fig. 7. For μ = 1/10 (Fig. 7a), one observes that the FE outcome (gray) matches with the present theory (blue). At For soft cores (Fig. 7b), M M 6th (red) is close to RSWT (blue), and the deflections are −8.17 × 10 −10 m at x = l/2. The target solution (FE gray) is −8.19 × 10 −10 m. As for the cantilever beam (Fig. 4) M M 4th (magenta) overestimates the outcome mainly due to improper modeling of the clamped boundary condition at x = 0. The relative difference of the deflection at x = l/2 over the core elasticity is shown in Fig. 8. The tendency and validity ranges of the various analytical theories are very similar as in the previous section for the cantilever: The ESL should be used only for hard cores, i.e., μ > 1/5, while the assumption of a vanishing axial core stress (M M 6th ) is accurate for μ < 1/10. Neglecting both the bending stiffness of the core and of the faces  (M M 4th ) yield unrealistic results unless for the short interval 1/100 < μ < 1/10 when the error is below 5%. It is noted that the incompressibility assumption holds for the whole range: the relative thickness deformation is 0.3% at μ = 10 −5 in the FE model.

Thin core (λ c = 0.2)
Now the core thickness of the clamped-hinged anti-sandwich beam is 0.02 m, but the thickness of each face is 0.04 m.
The bending stiffness of the core is small, and hence the M M 6th (red) and the RSWT outcome (blue) are almost identical: −1.66 × 10 −10 m at x = l/2 see Fig. 9a for μ = 1/10. The target solution from FE (gray, −1.74 × 10 −10 m) is underestimated because both theories assume infinite shear stiffness of the faces due to the Bernoulli-Euler assumption. This can be explained as follows: The beam length is l = 0.8 m, so the face thickness is 0.04 m which corresponds to a thickness-to-length ratio 1/20 and the influence of shear on the deflection becomes more dominant than in Sect. 4.2.1. Without going into detail, M M 4th (magenta) overshoots and E SL (black) undershoots the outcome. If the core is less stiff (Fig. 9b, μ = 1/1000), both 6th-order theories of course match again, but the difference to the FE results remains due to the infinite shear rigidity of the faces as before.
One observes from Fig. 10 that FE (gray), RSWT (blue) and M M 6th (red) are in good agreement. If the core is harder, the no-bending-stiffness assumption from the latter theory compensates the Bernoulli-Euler assumption for the faces. For softer cores μ < 1 its results converge to the RSWT as expected. For μ = 1/10 both analytical results underestimate the FE outcome by 5.3%. For μ < 1/10 5 the difference is 1.6% and remains constant because of the infinite shear rigidity of the faces. The 4th-order Mead-Markus theory yields inaccurate results independent of the core's Young modulus and should be avoided for anti-sandwich beams.

Conclusion
In this contribution, a novel approach of modeling a three-layer beam is presented. This type of composite includes a sandwich beam as representative example. The middle layer is modeled as a Timoshenko beam subjected to traction forces parallel and perpendicular to its upper and lower surfaces. The faces, also denoted as constraining layers, are modeled as Bernoulli-Euler beams. Eliminating the interfacial stresses between the middle and its attached layers and considering the perfect bonding assumptions, a sixth-order differential equation for the bending deflections is derived. The outcome is an extension of the sixth-order sandwich theory from Di Taranto and Mead-Markus but it includes also the bending stiffness of the core layer although the mathematical complexity remains the same. Hence, the presented analytical theory also holds for both soft and hard cores and can be extended to find suitable models for arbitrary composite layouts with different layer properties. Finally, the presented refined theory is compared to analytical models from the literature and to finite element calculations for various benchmark examples. Different boundary conditions and core stiffness (by varying its thickness and Young's and shear moduli) are considered showing that the derived theory agrees best with the target results from finite element analysis.
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Appendix A Sandwich theory according to Mead-Markus [11] and Di Taranto [13]
In the following, the sandwich equations from Mead-Markus [11] and Di Taranto [13] are summarized. Their theory is limited to relatively soft cores because of E c ≈ 0 (i.e., the longitudinal and the bending stiffness of the core are neglected). A.1 Sixth-order sandwich theory Stamm and Witte [14] present a more comprehensive derivation and the find the following differential equations It can be shown that the variable F is related to the shear stiffness of the core, it holds F = G A c (1 + t/c) 2 (note: κ = 1 holds since the shear stress is uniformly distributed). The angle γ * may not be confused with the rotation angle of the core ψ, it holds γ * = γ c/(c + t), see Fig. 11. Eliminating γ * in (A1), one finds the sixthorder differential equations derived by Mead-Markus [11] and Di Taranto [13] (note: that the nomenclature is different in their original works) Eqs. (A1) and (A2) are equivalent, but Stamm and Witte's notation is more convenient for implementation in symbolic software programs like MATHEMATICA, when boundary conditions are considered in terms of w and γ * . Furthermore, it is noted that the differential Eq. (A2) is also obtained if one neglects the bending stiffness of the core in Eq. (38). This demonstrates that the presented sandwich theory is a consistent generalization of the works by Mead-Markus and Di Taranto, who both neglect the longitudinal stiffness of the core from the beginning. Solving the eigenvalue problem, one finds 4 trivial eigenvalues and 2 nontrivial ones The term 2E A f (c + t) 2 (Steiner's parallel axis theorem) can be interpreted as bending stiffness. Introducing w = w B + w S one is able to split the deflection into two terms This is also called the method of partial deflections, where w B is the deflection due to bending and w S is the deflection due to shear.