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

Beams and rods made up of composite materials are widely used in mechanical, civil and aerospace engineering. By perfectly bonding various layers with different material properties, one may find a compound with a very high stiffness-to-weight ratio and a composite with a high resistance against a specific load. Even if the single layers are isotropic or orthotropic, the overall behavior of the composite may be anisotropic.

One popular group of these composites is called sandwich structure. These laminates are characterized by a high bending resistance and a very low average density and weight. Usually a relatively soft core is embedded between two stiff skins, which are also called faces. In addition to metallic foam cores, several other possibilities exist to realize soft cores, e.g., truss and lattice cores, honeycomb cores or corrugated cores (see Feng [1] and Ashby [2]). Numerous interesting attributes and further advantages of sandwich structures enlarge the range of engineering applications. Besides the high- strength-to-weight ratio, these advantages include good thermal isolation and protection, acoustic decoupling and energy absorption due to the viscoelastic layer (see Ashby [3]). Many important issues should be considered in the design, analysis, and construction of sandwich structures. Introductions to this field are the well-known books by Zenkert [4], Plantema [5], Marshall [6]. An excellent literature overview on bending, buckling and free vibrations of composite and sandwich beams is presented by Kapania and Racifi [7, 8], Ghugal and Shimpi [9] and Sayyad and Ghugal [10]. Motivated by the remarks in the latter contribution that refined analytical solutions for sandwich structures are rarely available in the scientific literature in comparison with finite element or other numerical solutions, the main focus of this contribution is to close this gap by establishing a modeling procedure that may serve as a starting point for sandwich and laminated beams consisting of orthotropic, anisotropic and piezoelectric properties.

Mead and Markus [11] derived a six-order differential equation for a sandwich beam by neglecting the bending and longitudinal stiffness of the core to calculate resonance frequencies and vibrations of a three-constrained-layer damped beam. Later Mead showed in [12] that Di Taranto [13] had already derived the same outcome for a viscoelastic constraint layer beam. The book by Stamm and Witte [14] gave a very detailed summary of Mead and Markus [11] and Di Taranto [13] model and strongly focused on mathematical modeling of asymmetric sandwich structures. The result is a sixth-order differential equation for the lateral deflection. If the bending stiffness of the skin is not taken into account, one derives a fourth-order differential equation. Only for this case the deflection could be split up into a pure bending deflection of the skin and a shear-dependent deflection.

Heuer [15] derived a fourth-order differential equation for a sandwich beam from d’Alemberts principle. The final equation could be interpreted as the equation of a shear deformable homogeneous beam with effective stiffness if the shear stress is uniformly distributed throughout the laminate depth. In [16], Heuer and Adam investigated a two-layer piezoelectric beam. Interlaminar slip is considered between the layers (the Bernoulli–Euler hypothesis holds for both layers), and it was found that the system is governed by a sixth-order differential equation for the lateral deflection. Adam [17] considered small amplitude vibrations of moderately thick composite plates. Prescribing the continuity of the shear stress according to Hooke’s law a fourth-order differential equation for the plate deflection was derived. In [18], Adam investigated composite plates with eigenstrain, which can be either of thermal, piezoelectric or inelastic nature. It was shown that if Poisson’s ratio and the ratio of mass density and shear modulus of the layers are identical, then the lateral deflection does not depend on the in-plane displacement. Adam and Heuer showed in [19] that the governing equation of a two-layer beam with interfacial slip is similar that of a sandwich beam with thick faces (=anti-sandwich structure). By analogy, it could be shown that the interlaminar slip stiffness is related to the shear modulus and the height of the core. Vo et al.  [20] developed a parabolic shear and normal deformation theory for the bending analysis of functionally graded sandwich beams. Frostig and his co-authors published several works considering the transverse flexibility of the core (see Frostig et al.  [21‐24]). Hence, these models can be used to analyze localized pressure and the relative deflection between the skins. Swanson [25] and Swanson and Kim [26] investigated Frostig’s analytical theory to numberotus sandwich problems of interest. The authors agreed that Frostig’s higher-order sandwich model was able to predict high stress concentrations at locations in the surrounding of concentrated loads and supports by comparing the outcome to finite element results. But it was remarked that numerical difficulties may arise from Frostig’s model. A higher-order analysis model for sandwich plates with flexible core is presented by Tian et al. [27]. A simply supported plate was considered and the results for bending deflections and for predicting the natural eigenfrequencies are compared to the Mindlin theory and ANSYS results. Davalos et al. formulate a one-dimensional beam finite element with layer-wise constant shear. Transverse incompressibility was assumed and the constant shear stress from the constitutive relations is transformed into a parabolic stress distribution in a postprocessing step. A modified zig-zag model for analyzing thick composite beams with rectangular cross sections was developed by Icardi [28], which accurately predicted the displacement and stress fields of composite beams. An efficient one-dimensional static model developed for piezoelectric sandwich beams based on third-order zig-zag functions is presented by Kapuria et al. [29].

This work is mainly motivated by the analytical sandwich model from Mead and Markus [11] and Di Taranto [13]. Anyway, it remains unclear, or at least unanswered according to a literature review, why the bending stiffness of the core is neglected a priori in their contributions and when influences on the deflection of a sandwich beam. A minor aim of this contribution is to show the validity range of some analytical models from the literature. The main objective is to present a sandwich model which consists of three isotropic layers, perfectly bonded together, but with different material properties and thicknesses, which also includes three-layer beams with hard cores. On the one hand, the perfect bonding assumptions require the axial and vertical deflections to be equal, on the other hand the stress continuity for the shear stress and the transverse normal stress has to be fulfilled, too. The underlying equations for the core layer are the Timoshenko equations (TS) and for the external layers, which are usually denoted as faces, the Bernoulli–Euler (BE) equations. The final sandwich model is a sixth-order differential equation for the bending deflections, and hence the complexity and mathematical order are equal to the model from Mead and Markus [11]. Benchmark examples with various boundary conditions and core thicknesses are considered and parameter variations concerning the Young modulus of the core are performed and compared to finite element calculations. The range of applicability of the analytical models is discussed, and it is found that the present three-layer modeling approach yields accurate results for both soft and hard cores. The presented modeling procedure can be a starting point for more advanced composite beam models and different material properties (i.e., orthotropic, anisotropic, piezoelectric).

The basis for the present sandwich model is an accurate beam model taking into account distributed loads that act perpendicular \(q_j\) and parallel \(\tau _j\) to the upper and lower surfaces. Timoshenko assumptions hold for layer j in Fig. 1, and hence the displacement field iswhere \({u}_j (x)\) and \({w}_j (x)\) are the horizontal and the vertical deflection and \({\psi }\) is the rotation angle. The thickness of one layer is \(2 c_j\).

From Eq. (1),it follows for the strainsand for the stressesYoung’s modulus is denoted by \(E_j\), for the shear modulus \(G_j = E_j/(2+2 \nu _j)\) 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 findsHere the axial stiffness, the shear stiffness and the bending stiffness are \(\textrm{EA}_j\), \(\textrm{GA}^*_j\) and \(\textrm{EI}_j\), respectively. The cross-section area is \(A_j = 2 b c_j \) and the geometrical moment of inertia is \(I_j = 2 b c^3_j/3\). The star-symbol \(\textrm{G A}^*_j = \kappa G A_j\) indicates that the shear stiffness also includes a proper shear correction factor \(\kappa \). The equilibrium conditions can be derived from Fig. 1 and readIt 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 \(\tau _{j-1} = - \tau _{j}\)). If unity width is used for the layers \(b=1 \, \textrm{m}\), the expressions for the distributed loads coincide with the stress values (e.g., \(\sigma _{xz} (x,c) = \tau _{j}/b\), \(\sigma _{zz} (x,c) = q_{j}/b\)). Substituting the beam forces (4–6) in Eqs. (7–9), 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 areNote that the shear force is not proportional to the third derivative of the vertical deflection unless for \(\tau _{j-1} = -\tau _{j} \) (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.

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. (17–23) contain 11 unknowns (7 degrees of freedom \({w}_{1}, \, {w}_{2}, \, {w}_{3}\), \({u}_{1}, \, {u}_{2}, \, {u}_{3}\), \({\psi _2}\) and 4 interface stresses \(q_{1},\, q_{2}, \, \tau _{1},\, \tau _{2} \)) which can be computed by 7 differential Eqs. (17–23) and 4 coupling Eqs. (24–26) from the perfect bonding assumptions. The latter readA condensed and more comprehensible form of the differential equations is obtained if a symmetrical setup is considered: \(E_1 = E_3\) and \(t_1 = t_3 \rightarrow EA_f = EA_1 = EA_3 \), \(EI_f = EI_1 = EI_3 \). 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 \(\psi \). In a first step the coupling Eqs. (24–26) are substituted in Eqs. (17–23). In a second step, one solves Eqs. (17–20) for the yet unknown interfacial stresses \(q_{1},\, q_{2}, \, \tau _{1},\, \tau _{2}\) and substitutes the outcome into the remaining three differential Eqs. (21–23)Neglecting the external shear stress \(\tau _{0} = \tau _{3} = 0\) one findsThe solution for the decoupled axial displacement (32) readsThe 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 angleInserting Eq. (34) into Eq. (31) and integrating the resulting equation yields the second-order differential equation for the deflectionwhere \(P,\, Q\) and the inhomogeneous term f(x) readFrom 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. (40). Six integration constants must be satisfied: four constants \(({B_1,\,B_2,\,B_3,\,B_4})\) are already included in f(x) (see Eq. (37)): two additional ones occur from the second-order differential equation (see Eq. (35)).

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 equationwhere 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 ofThe solution for the deflection iswhere \(w_{j}(x)\) is the inhomogeneous solution. The constants \(B_1-B_6\) must be determined from the boundary conditions (see Sect. 3.1).

In this section, the presented refined sandwich theory (RSWT) is compared to both Mead–Markus models: the \(6{\textrm{th}}\)-order model (see Appendix A.1) does not include the core bending stiffness, the simplified \(4{\textrm{th}}\)-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 (\(\textrm{ESL}_\textrm{Timo}\)) taking into account shear deformation (Timoshenko assumption) is computed (Appendix C). For the variations concerning the core elasticity (see Figs. 5, 8 and 10), results are also computed for a structure with totally negligible core stiffness (\(\textrm{BE}_\textrm{face}\), i.e., the structure consists of perfectly slipping faces without interfacial shear stress, and hence for the core \(\textrm{GA}_c^* \approx \textrm{EI}_c \approx 0\) holds, Appendix D). These last two models are limiting cases (\(\textrm{ESL}_\textrm{Timo}\) and \(\textrm{BE}_\textrm{face}\)) 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 \(\textrm{ESL}_\textrm{Timo}\); if the core is very soft, the results from \(\textrm{BE}_\textrm{face}\) should be obtained.

The vertical load on the upper face is uniformly distributed \(q_0 (x) = 1 \, \mathrm {N/m}\) over the length \(l=1 \, {\textrm{m}}\). 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 \(\sigma _{xz}(x,z)\) and the interfacial normal stress (i.e., \(\sigma _{zz}(x,c) = q_2(x)/b\) and \(\sigma _{zz}(x,-c) = q_1(x)/b\) (see Fig. 2)) are computed for the cantilever sandwich. The thickness-to-length ratio is \(\lambda _t = (2c+4t)/l = 1/8\) for all benchmark examples, the relative thickness of the core \(\lambda _c = 2c/(2c+4t)\) (ratio of core thickness to total thickness) is either 0.2 or 0.8.

For the elasticity of the core, the non-dimensional variable \(\mu = E_c / E_f = G_c/ G_f\) (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: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 \(\kappa = 1\) 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 \(\mu = 1/10\).

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.