1 Introduction

Different types of sandwich structure are well known in mechanical and civil engineering. These structures are fabricated from two thin but stiff layers named faces and a thick core. Very popular are composite structures. The most commonly used materials for the core are honeycombs, synthetic, or metal foams sometimes wood. The type of used materials for the core depend on application of whole structure. Advantages of several layer structures are lower weight, novel physical and mechanical properties, a high level of impact energy absorption, heat dissipation, electrical conductivity, and acoustic shielding. Such properties are possible to achieve through the development of manufacturing technology of porous material among other metal foams. Popularity of such structures is also evident in the rising number of publications on this topic. The authors describe the study of such structures in many directions. In the literature, one might find the statement that the first mathematician who described the low relating to mentione structures was Archimedes. He described the way how to calculate density. Considering the last century, the foundations of theory of that kind of structures after War World II were presented in the literature by Libove and Butdorf^[1], Plantema^[2], and Reissner^[3]. They concerned on modelling of the three-layered structure taking into account the change in the shape of a normal straight to the middle surface after deformation. Manufacture, characterization, and application of cellular metals and metal foams were described by Banhart^[4]. Magnucki and Szyc^[5] and Magnucki et al.^[6] investigated the strength, stability, and deflection of sandwich structures with an aluminum foam core. Dynamic stability of a sandwich plate was studied by Dębowski et al.^[7]. Cao and Zhong^[8] investigated dynamic response of an infinite beam placed on a Pasternak foundation when the system was subjected to a moving load. The similar problem was investigated by Pradhan et al.^[9]. They analyzed the static and dynamic stability of an asymmetric sandwich beam with viscoelastic core lying on a variable Pasternak foundation under the action of a pulsating axial load subjected to one-dimensional thermal gradient under three different boundary conditions by the computational method. Global and local buckling of sandwich beams was investigated by Douville and Le Grognec^[10], Huang and Kardomateas^[11], Jasion et al.^[12-13], and Jasion and Magnucki^[14], Javidinejad^[15], Magnucki and Stasiewicz^[16], and Magnucki^[17]. Grygorowicz et al.^[18] described elastic buckling of a sandwich beam with variable mechanical properties of the core. They also presented analytical and finite element method (FEM) studies on buckling of sandwich beams^[19].

Dynamic buckling of foam stabilized composite skin was described by Rivallant et al.^[20]. Tagarielli et al.^[21] investigated the dynamic response of composite sandwich beams to transverse impact. Wang et al.^[22] led research on structural response of clamped sandwich beams subjected to impact loading. A similar investigation was conducted by Tan et al.^[23]. They presented dynamic response of clamped sandwich beam with aluminium alloy foam core subjected to impact loading. The behavior of structural fibre composite sandwich beams made up of glass fibre composite skins and phenolic core material was investigated by Manalo^[24] under three-point short beam and asymmetrical beam shear tests. Grygorowicz and Magnucka-Blandzi^[25] studied stability of sandwich beams with variable properties of the core with dynamic loads. Aly et al.^[26] described parametric models for design optimization and material selection of sandwich panels with the objective of weight and cost minimization subject to structural integrity constraints such as strength, stiffness, and buckling resistance. Ashjari and Khoshravan^[27] developed a method for the single-objective optimization of material distribution of simply supported functionally graded isotropic plates. Brighenti^[28] presented smart behavior of layered plates through the use of auxetic materials. Zingoni^[29] described the group-theoretic insights on the vibration of symmetric structures in engineering. Hadji et al.^[30] used the four-variable refined plate theory for the free vibration analysis of functionally graded material sandwich rectangular plates. The purpose of this paper is a mathematical description of the behavior of three-layered beam with the metal foam core dynamically loaded. The discussion is adopted variable mechanical properties of the core, which is a new approach.

The object of investigations is a three-layered beam with the metal foam core and a rectangular cross-section (see Fig. 1(a)). The beam is simply supported and axially compressed. The scheme of a considered sandwich beam, its geometry, the load, and supports are shown in Fig. 1(b). The beam is compressed by the load $F_0$. The buckling problem is considered in the elastic range. The influence of the core porosity (the parameter $k_0$) on the critical load, static and dynamic paths (if the load is changed linearly in time), and two unstable regions (if the load is pulsating) is studied. The analytical results are verified numerically with the FEM and the ANSYS system.

2 Mechanical properties of beam core

The considered beam is three-layered. The isotropic facings are made of metal sheets, and their thickness equals $h_{\rm f}$. Young's modulus $E_{\rm f}$ and the density $\rho_{\rm f}$ of facings are constant. The isotropic but not homogeneous core is made of metal alloy foam. The density $\rho_{\rm c}$ in the middle surface of the core is ten times smaller than that in the facings. Poisson's ratio $\nu_{\rm c}$ is kept constant. The thickness of the core is $h_{\rm c}$, and the length of the beam is $L$. The mechanical properties are variable through the thickness of the core (along the $z$-direction). The biggest value of Young's modulus occurs in the middle plane of the beam (see Fig. 2) in its symmetry plane.

The minimal values occur at the top and bottom surfaces of the core. Young's modulus decreases in the direction to the facings according to the function

where $k_0$ is the parameter, and $k_0\geq 0$.

If the parameter $k_0$ equals 0, then Young's modulus of the core is constant and equals $E_{\rm c0}$. Otherwise, if this parameter increases ($0＜k_0$), the mechanical properties vary. Its nonlinear characteristic is presented in Fig. 3. The density of the core is also variable through its thickness, i.e.,

The density of the beam core is changing with Young's modulus. However, the densities of the facings $\rho_{\rm f}$ are constant.

3 Displacements, stresses, and strains

The stability problem of three-layered beams with variable mechanical properties of the core is studied with the use of two different hypotheses of deformation of the flat cross-section. The first one is the classical broken line hypothesis (see Fig. 4(a)). The second one is the nonlinear hypothesis (see Fig. 4(b)), which is a generalization of the classical one. Magnucka-Blandzi^[31] presented detailed description of this hypothesis.

The field of displacements for each layer according to the assumed hypotheses (see Fig. 4) is formulated as follows.

(i) For the upper and lower sheet-faces $-(\frac{1}{2}+x_1)\leq\zeta\leq -\frac{1}{2}$ and $ \frac{1}{2}\leq \zeta\leq \frac{1}{2}+x_1$,

where "-" denotes the upper sheet-face and "+" denotes the lower sheet-face.

(ii) For the metal foam core $-\frac{1}{2}\leq \zeta\leq \frac{1}{2}$,

where $x$ and $z$ are the coordinates, $t$ is the time, $\zeta=z/h_{\rm c}$ is the dimensionless coordinate, $w$ is the deflection, $\psi_1(x, t)=u_{\rm f}(x, t)/h_{\rm c}$ and $\psi_2$ are the dimensionless functions, and $x_1=h_{\rm f}/h_{\rm c}$ is the dimensionless parameter.

The classical broken line hypothesis is a particular case of the above nonlinear hypothesis (when $\psi_2(x)\equiv 0$).

The relations between the strains and the displacements are nonlinear.

(i) For the upper and lower sheet-faces $-(\frac{1}{2}+x_1)\leq\zeta\leq -\frac{1}{2}$ and $\frac{1}{2}\leq \zeta\leq \frac{1}{2}+x_1$,

(ii) For the metal foam core $-\frac{1}{2}\leq \zeta\leq \frac{1}{2}$,

The stresses in the facings are defined according to Hooke's law.

(i) For the upper and lower sheet-faces $-(\frac{1}{2}+x_1)\leq\zeta\leq -\frac{1}{2}$ and $\frac{1}{2}\leq \zeta\leq \frac{1}{2}+x_1$,

(ii) For the metal foam core $-\frac{1}{2}\leq \zeta\leq \frac{1}{2}$,

4 Equations of motion

On the basis of the Hamilton principle

the system of equations of motion is derived for the considered problem. If the time is not considered, the equations of static equilibrium are obtained. The limits of integration $t_1$ and $t_2$ indicate the start and end of time.

The kinetic energy of the beam is

where f1 and f2 denote the upper and lower facings of the sandwich beam, respectively.

The kinetic energy of the upper sheet is

The kinetic energy of the core is

The kinetic energy of the lower sheet is

The elastic strain energy of the beam is

where

Therefore, the elastic strain energy is

where

The work of the external load is

Based on the principle (9), three partial differential equations are obtained as follows:

where $\rho_{\rm c0}$ is the mass density in the middle surface of the beam (for $\zeta=0$), $\rho_{\rm f}$ is the mass density of the facings, and $\rho_{\rm b}=2\rho_{\rm f}x_1+\rho_{\rm c0}J_4.$

5 Static and dynamic equilibrium paths

The system of static equilibrium is obtained, if the dependency on time in the system (18) is omitted. Then, the system can be written as follows:

The equations for unknown functions are assumed as follows:

which identically satisfy the boundary conditions

The parameters $\psi_{\rm a1}$ and $\psi_{\rm a2}$ are calculated from the second and third equations of the system (19),

where

and

After substituting the functions of $\psi_{\rm a1}$ and $\psi_{\rm a2}$ into the first equation of the system (19) and using the Bubnov-Galerkin method, the static equilibrium path is obtained,

The static critical load $F_{\rm cr}$ follows from the above equation (if $w_{\rm a}=0$ which means that the deflection equals zero)

Therefore, the static equilibrium path is

If the dependency of time is taken into account for the unknown functions in Eq.(20), i.e.,

then the system (18) may be reduced to one ordinary differential equation of second-order,

For the sandwich beam subjected to the load linearly variable in time,

where $t_0$ is the base time. The dynamic equilibrium path is obtained. Equation (27) is solved with the use of Runge-Kutta method of fourth-order.

6 Unstable regions

In the case of linear analysis, i.e., if the relationships between the strains and the displacements are linear, the system (18) is

In this case, the external load is assumed as the pulsating force,

where $F_{\rm c}$ is an average value of the load, $F_{\rm a}$ is an amplitude of the load, and $\Theta$ is a frequency of the load. Assuming Eq.(26) for the unknown functions, the system (29) may be reduced to the Mathieu equation,

where

and the angular frequency

Therefore, the natural frequency (in Herz) is

and two unstable regions, according to the monograph^[32], may be written as follows.

(i) The first unstable region is

(ii) The second unstable region is

7 Numerical calculations

Numerical calculations are performed for the family of the sandwich beams with the dimensions and mechanical properties given in Table 1.

The total hight of the beam equals 20 mm. Using the formula for the critical loads (24), their values for the family of beams are determined (see Table 2).

Moreover, to verify analytical results, the finite element model of the beam is performed. The numerical results are also given in Table 2. The numerical calculations are done with use of the ANSYS software version 14.5. Because of symmetry of the problem, only a half of the beam is considered in the FEM models with proper boundary conditions imposed on the symmetry planes (see Fig. 5). The ANSYS analysis is performed using static structural analysis and linear buckling and dynamic modal analysis. The beam is supported at the ends by displacement. Support is defined by the coordinates $x$ (along the neutral axis), $y$, and $z$. The displacement is possible only along the $x$-axis, and other coordinates are constant. The force is loaded to the edge of the beam. The facings are modeled using shell element (SHELL 181) with 4-node with six degrees of freedom at each node: translations in the $x$-, $y$-, and $z$-directions, and rotations about the $x$-, $y$-, and $z$-axes. The core is modeled using solid element (SOLID 186), a higher order three-dimensional 20-node solid element that exhibits quadratic displacement behavior. The element is defined by 20 nodes with three degrees of freedom per node: translations in the nodal $x$-, $y$-, and $z$-directions. The ANSYS FEM model of the sandwich beam consists of 20.010 rectangular elements. Bigger number of elements has no influence on the results of calculations. The buckled beam is shown in Fig. 6.

The influence of the parameter $k_0$ on the critical load is shown in Fig. 7. With the increase of the parameter $k_0$, the value of critical load decreases.

According to Eqs.(34) and (35), two unstable regions are determined (see Fig. 8). The pulsating load is assumed as follows:

Therefore, the maximal value of the load does not exceed the critical load. The natural frequencies are also verified numerically using the FEM. For example, for $k_0=1$, the natural frequency obtained analytically according to Eq.(33) is equal to $54.80$ Hz and numerically $51.21$ Hz.

To determine static and dynamic paths of equilibrium, the load linear variable in time is assumed as it is written in Eq.(28). These paths are presented in Fig. 9 for two different values of the parameter $k_0$.

The paths of dynamic equilibrium (continuous lines) oscillate around the paths of static equilibrium (dashed lines).

8 Conclusions

In the design process, the structures are adopt for convenience and safety of man and not vice versa. For this reason, new solutions are sought. This is why the popularity of sandwich structure with the metal foam core increased last decades. Sandwich constructions are not only beams or plates used in mechanical engineering. They are also parts used in medicine or dentistry as spinal vertebrae or other implants. The exact mathematical description, supported by numerical and experimental studies of behavior mentioned structures dynamically loaded, is very important. Dynamic stability of a family of sandwich aluminum alloy beams with the foam core is investigated in the paper. The structure is loaded with an axial pulsating force. The variable mechanical properties of the core are taken to calculations. The deformation of flat cross-section is analyzed with use of two hypotheses: the classical broken line and the proposed nonlinear hypothesis. Strains and stresses of each layer of the beam are described. The Hamilton principle is used to derive the system of equations of motion. The bending of the beam is assumed as a function of time. The use of Galerkin method conducts to the second-order differential equation of motion for dimensionless bending. Solving this equation numerically with use of the Runge-Kutta method, the dynamic equilibrium path is obtained. Finally, the system of equations of motion is reduced to one equation known as the Mathieu equation. At last, numerical calculations for family of beams are done. Firstly, the critical load $F_\mathrm {cr}$ is calculated for selected values of $k_0$ parameter. As a reminder, this parameter is responsible for mechanical properties of the core. For the beams with a length of $1~000$ mm, the critical load changes from 5.81 kN (for $k_0=1$) to 3.16 kN (for $k_0=20$). Secondly, two unstable regions for the pulsating load $F_0(t)$ for selected $k_0$ are also calculated. It should be mentioned here that the pulsating load does not exceed the critical load. Finally, the dynamic equilibrium path is achieved in the shape of the curve oscillating around the parabola-equilibrium path for the static load.