Dynamic Stability of Sandwich Beams/Wide Plates Subjected to Axial Impulsive Loads Free

The static critical load gives the limit above which a structure can deform in an alternative way associated with a significant deformation and a possible failure mode when the structure is subjected to a quasi-static load. In contrast, when the load is applied dynamically, the stability limit is no longer as significant as in the static case. It is known that a dynamic impulsive load with a short load duration may safely exceed the static critical load since the structure has no time to deform in comparison to the static case.

Dynamic stability and instability have received considerable attention since the first work by Koning and Taub in 1933 [1], in which a simply supported thin bar subjected to a constant axial load for a specific duration was theoretically studied. Then, Taub [2] studied the same problem of a bar with arbitrary boundary conditions. A literature review of the work related to dynamic stability can be found in the books [3,4] and the survey [5]. Simitses [3] well-summarized various concepts and criteria developed and used by various investigators by the 1980s and also included a few applications to different structures. As dynamic buckling is essentially a dynamic problem, the response is affected by more factors than that in the static buckling case, such as the load can be applied for an infinite duration or a finite duration with various time profiles. Therefore, it is a challenging task to efficiently evaluate the buckling phenomena and many criteria have been proposed [6–10]. In the 1960s, Budiansky and Hutchinson [6,11,12] developed a criterion by determining the applied load under which the maximum response amplitude experiences a large jump. The Budiansky–Roth criterion was developed to study the dynamic stability of structures when loads were applied suddenly with constant magnitude and infinite duration. In 1980, Sheinman [8] studied the dynamic buckling of curved beams with shear deformation, in which the changing of the natural frequency was taken as the criterion. In 1997, Ari-Gur and Simonetta [10] formulated four different buckling criteria by considering the lateral deflection, pulse intensity, deflection shape, force pulse, and displacement pulse. Besides the equations of motion approach [6,8,11,12], Hsu [9] adopted the total energy-phase plane approach to estimate the critical conditions. There are no universal criteria that can apply for all dynamic buckling problems due to the complex nature of this problem, and more criteria were developed recently for different structures subjected to various loads [13,14].

The dynamic response associated with dynamic stability is another aspect that researchers are interested in. The dynamic buckling of basic structural elements, such as beams [15–20], plates [21,22], and shells [23–26], and simply structures, such as mass-spring-beam system [27], and frames [24,28] have been studied. The study is also extended to structures made of elastoplastic material [29] and viscoelastic material [30,31]. With the increasing application of composite materials, the dynamic stability of composite structures has attracted researchers’ interests. Kubiak [32] studied the dynamic buckling behavior of composite plates with varying widthwise material properties. Bisagni [33] studied the dynamic buckling due to impulsive loading of thin-walled carbon fiber reinforced plastic shell structures under axial compression. The laminated hypar and conoid shells were studied by Pradyumna and Bandyopadhyay [34], and a higher order shear deformation theory was considered. Dey and Ramachandra [35] studied both static and dynamic instability of composite cylindrical shells subjected to partial edge loading and another study on the dynamic buckling of composite cylindrical shells study was conducted by Soltanieh et al. [36]. The dynamic snap-through buckling of sandwich panels with initial curvature under transverse loads was studied by Adi Murthy and Alwar [37]. In 2018, Latifi et al. [31] performed the dynamic instability analysis of sandwich beams with integral viscoelastic cores. The model was built on the full-layerwise theory together with the first-order shear deformation theory.

Sandwich composite structures are highly efficient structures of high stiffness and strength with low resultant weight and are widely used in the aerospace, marine, ground transportation, and civil construction. Several sandwich theories have been developed in the last decades with the goal of a convenient structural theory to accurately model the response of sandwich structures. A comparison and discussion on the various sandwich theories can be found in the survey article by Carrera and Brischetto [38]. Due to the usage of soft core materials, sandwich structures exhibit both large amounts of transverse shear and significant compressibility [39]. Therefore, the classical theory (without shear effect) and the first-order shear theory are either inadequate or too conservative [40], and a high-order theory is required. Indeed, recently, the high-order theories are widely used in the theoretical studies on sandwich structures [41–44]. In 1992, Frostig et al. [45] proposed the high-order sandwich panel theory (HSAPT) that accounts for the transverse and shear rigidity of the core (i.e., this theory includes the transverse compressibility of the core). The HSAPT neglects the axial rigidity of the core and results in a constant shear stress distribution through the thickness of the core. In 2012, the extended high-order sandwich panel theory (EHSAPT) was formulated [46], in which the axial, transverse, and shear rigidity of the core are all included. Therefore, the EHSAPT is suitable for sandwich panels with core in a wide range of core stiffnesses, from soft core to medium, even heavy mass density core. The accuracy has been verified by comparisons made with benchmark elasticity solutions for both the static [47] and the dynamic response [48]. Re buckling, the EHSAPT has been successfully adopted to investigate the static buckling and post-buckling behavior of sandwich structures [49–51].

Reviewing the literature, most of the studies on sandwich composite structures are focused on the static, vibration, and static stability behavior. In this work, the dynamic stability of sandwich panels will be studied based on the EHSAPT. The initial geometric imperfections are considered and equations of motions with initial geometric imperfections included are derived. This study focuses on the dynamic response of a sandwich beam/wide plate subjected to edge compressive impulsive loads. Sandwich panels with simply supported edges are investigated semi-analytically in the sense that the dynamic response in the time domain is obtained numerically with the Newmark method (the analytical solution is employed for the spatial domain). Three different impulsive loading profiles are considered, and the effects of oscillation modes, geometries, and materials on the dynamic stability are studied.

A sandwich panel subjected to edge axial loads is schematically shown in Fig. 1. The length is L, and the thicknesses of the top, bottom, and core are f_t, f_b, and 2c, respectively. A unit width is considered to simplify the presentation as a plane problem is considered. The Cartesian coordinate system is set with the origin placed at the left edge of the beam. The x-axis coincides with the middle line of the core and the z-axis is along the thickness direction. The axial displacement field of the top face, core, and bottom face are denoted as u^t(x, z), u^c(x, z), and u^b(x, z), respectively, and the transverse displacement field of the top face, core, and bottom face are w^t(x, z), w^c(x, z), and w^b(x, z). They are measured from the straight configuration, as shown in Fig. 1(b). The initial geometric imperfection of the sandwich panel is also considered. In the top face, core, and bottom face, the components in x-direction are denoted as u^It(x, z), u^Ic(x, z), and u^Ib(x, z), and the components in z-direction are denoted as w^It(x, z), w^Ic(x, z), and w^Ib(x, z).

The displacement field of the sandwich panel is represented by the deformation of the centroid of each layer, ${U¯(x)}=⌊u0tw0tu0bw0bu0cw0cϕ0c⌋T$⁠. In a similar way, the initial geometric imperfections of the sandwich panel, u^It(x, z), w^It(x, z), u^Ic(x, z), w^Ic(x, z), u^Ib(x, z), and w^Ib(x, z), are represented by the values of the initial imperfections at the centroid and are denoted as ${U¯I(x)}=⌊u0Itw0Itu0Ibw0Ibu0Icw0Icϕ0Ic⌋T$⁠. The latter can be obtained by simply replacing the superscript t, b, c with It, Ib, Ic in Eqs. (1)–(2).

Seven differential equations are thus obtained and are given in the  Appendix. Besides, the boundary conditions at both ends are also obtained and are given in the  Appendix.

As a linear elastic problem is considered, and M(·), K(·), $L(⋅,⋅)$⁠, and B(·) are linear differential operators, the original problem can be represented by two subproblems after substituting Eq. (11) into Eq. (10),

• the compressive oscillation due to the axial compressive loads, which is described by

  $M({U¯¨}comp)+K({U¯}comp)+L({N},{U¯}comp)={0}$

  (12a)

  $B({U¯}comp)={N}$

  (12b)
• the bending oscillation due to the initial imperfection, which is described by

  $M({U¯¨}bend)+K({U¯}bend−{U¯I})+L({N},{U¯}bend)={0}$

  (13a)

  $B({U¯}bend−{U¯I})={0}$

  (13b)

Arranging seven differential equations in the same sequence as the sequence of terms appeared in {U_n} yields a symmetric 7 × 7 mass matrix [M_n] and a symmetric 7 × 7 stiffness matrix [K_n].

The static response of the sandwich panel with various magnitudes of initial geometric imperfection, Δ, is given in Fig. 2. It plots the transverse displacement of the top face middle point versus the axial compressive load magnitude P. With the increasing of the axial compressive loads, the static response curve has the Euler critical load line as an asymptote. The smaller initial geometric imperfection, the curve is closer to $P/Pcr1=1$⁠. From Eqs. (15) and (19), it can be seen that both the dynamic response and the static response are proportional to the magnitude of the initial geometric imperfection Δ. Thus, the normalized results will be used in the following.

In this study, the hinged sandwich panel is subjected to equal axial compressive loads at the ends of the top face and bottom face, namely N^t = N^b = P/2. Different impulsive load profiles with load duration τ are considered.

First consider a step impulsive load and the time history is shown in Fig. 3.As a compound structure, the dynamic response of a sandwich panel is affected by multiple factors, such as the material properties, ratio between the faces and core thicknesses, the load profile, also the oscillation shape. The effects caused by these factors will be studied in this part.

When the initial geometric imperfection only has the term of n = 1, the sandwich panel shows the fundamental oscillation only, namely $u0t,b,c=$$U1t,b,ccos(πx/L)$⁠; $w0t,b,c=W1t,b,csin(πx/L)$⁠; $ϕ0c=Φ1ccos(πx/L)$⁠. Consider a sandwich panel made of carbon epoxy faces and glass-phenolic honeycomb core. The material properties are listed in Table 1. The thicknesses of the faces and core are f_t = f_b = 1.0 mm and 2c = 18.0 mm, respectively. The length is L = 40(f_t + f_b + 2c). Different axial compressive load magnitudes and periods are considered. The initial geometric imperfection as the one given by Eq. (21) is applied. The maximum and minimum transverse displacement of the top face during the dynamic response are plotted in Figs. 4 and 5. The displacement plotted as the vertical axis is normalized with the static response, and the load duration plotted as the horizontal axis is normalized with the first free vibration period, T₁ = 2π/ω₁. Here, the “maximum” or “Max” and “minimum” or “Min” are defined as the highest value and lowest value occurring in the dynamic response when subjected to a particular axial compressive impulsive load for the specific duration. Since both the dynamic response and the static response have the same distribution over the x coordinate, sin (πx/L), Figs. 4 and 5 are independent of the axial coordinate x. The response when the step impulsive load is lower than the first critical load is shown in these figures.

Figure 4 shows that the ratio of the maximum transverse displacement to the static response is small if the impulsive load is only applied for a very short duration. The ratio between the maximum dynamic response and static response has a steady value of 2, which is also the highest value when the step impulsive load is applied for a long duration and is lower than the first critical load, $Pcr1$⁠. Before the ratio reaches the steady value, the lower axial impulsive load leads to a higher dynamic/static response ratio for the same load duration. The lower axial compressive impulsive load also requires a shorter load duration to reach the highest value of 2.

Figure 5 plots the ratio of the minimum transverse displacement of the top face and the static response versus τ/T₁, for various axial step impulsive loads. The lowest value is −2. However, the minimum transverse displacement does not have a steady value when the impulsive load is applied for a longer duration. Instead, it is a periodical function in terms of the impulsive load duration τ, with a highest value of 0 and a lowest value of −2. The dynamic response of the transverse displacements of the top and bottom faces and the core under an impulsive load $P=0.6Pcr1$ with durations of τ = 0.35 T₁, τ = 1.4 T₁, τ = 1.58 T₁, and τ = 2.37 T₁ are plotted in Fig. 6. The displacements are normalized with the magnitude of initial imperfection Δ. These four states, given by Figs. 6(a)–6(d), correspond to the dots marked as “a,” “b,” “c,” and “d” in Figs. 4 and 5. In Fig. 6, the dynamic response curves of the transverse displacement of the faces and the core are almost identical to each other. The deformation of the top face, bottom face, and core are synchronized. The sandwich panel undergoes forced vibration due to the impulsive load in the region before the dash line shown in Fig. 6. After that, the load is removed and the sandwich panel undergoes free vibration. The maximum transverse displacement may happen during the free vibration, as shown by Fig. 6(a), or during the forced vibration, as shown by Figs. 6(b) and 6(c), or both, as shown by Fig. 6(d). During the forced vibration, the transverse displacement $w0t−w0It$ is always positive, which means the top face never moves to the position below the initial geometric imperfection. After the axial impulsive load is removed, the sandwich panel may move to the position below than the initial geometric imperfection, and it depends on the impulsive load duration. Only when the impulsive load is removed, when $Max((w0t−w0It)/(w0St−w0It))=2$ (Fig. 6d), the minimum dynamic transverse displacement to the static transverse displacement, $Min((w0t−w0It)/(w0St−w0It))$ can reach the lowest value of −2.

Since the displacement fields are expanded as Fourier series, the strain fields have a similar distribution along the axial direction. When the fundamental oscillation is considered, in which the displacements only contain terms of n = 1, the strains and stresses also have a sinusoidal distribution. For example, the axial strain of the top face can be written as $ϵxxt(x)=Exxtsin(πx/L)$⁠, and the shear strain at the interface of the top face and core can be written as $γxzc(x,z=c)=Γxzc(z=c)cos(πx/L)$⁠. The $Exxt/Δ$⁠, axial strain per unit geometric imperfection, is plotted on a logarithmic scale in Fig. 7. It includes eight cases with impulsive loads lower than, equal to, and higher than the first critical load. When the impulsive load is applied for a short duration, the maximum axial strain increases at the same rate for all curves. In other words, the slope is independent of the magnitude of the impulsive load. With longer load duration, the maximum axial strain during the dynamic response is bounded and the curve has a plateau when the impulsive load is lower than the critical load, whereas the maximum axial strain is unbounded when the axial impulsive equals or is higher than the corresponding critical load. Same tendency is observed in the dynamic response of other strains, stresses, and displacements, as shown in Figs. 8 and 9. Only the scales of the vertical axis are different in Figs. 7–9. Thus, it can be concluded that all measurements change at the same rate with the change of the external load. Therefore, a scaling factor plot can be obtained via one of the measurements, such as the transverse displacement of the top face, given by Fig. 9. The scaling factor is a function of the duration and the magnitude of the impulsive load. Then, one can easily get the maximum displacements, strains, and stresses for an arbitrary step impulsive load magnitude and duration with the scaling factor plot and the value of the measurement of one known case.

Besides, Figs. 7–9 show that it is possible that a sandwich panel can bear an axial impulsive load with a magnitude higher than the static critical load and have a small deformation associated with a low stress state, if the impulsive load duration is short enough. For a given sandwich panel, the maximum allowable stress or deformation can be plotted as a horizontal line in Figs. 7–9. Any points below the line give the permissible load magnitudes and allowable load durations.

When the initial geometric imperfection has a higher order term (n ≥ 2), it will initiate the corresponding higher order oscillation. It is best to evaluate the dynamic response with the nth critical load $Pcrn$ and the nth free vibration period T_n, whose mode shapes are represented by $u0t,b,c=Unt,b,ccos(nπx/L)$⁠; $w0t,b,c=Wnt,b,csin(nπx/L)$⁠; $ϕ0c=Φnccos(nπx/L)$⁠.

When the axial step impulsive load is lower than the mode 2 buckling load $Pcr2$⁠, the maximum dynamic response has a steady value when the step impulsive load is applied for a long duration as well, and it is twice that of the static response. It has same curves as the ones given by Figs. 4–6, and the only difference is that $Pcr1$ is now replaced with $Pcr2$ and T₁ is now replaced with T₂. With the initial geometric imperfection given by Eq. (23), the axial strain in the top face can be written as $ϵxxt(x)=Exxtsin(2πx/L)$⁠, and the shear strain at the interface of the top face and core is $γxzc(x,z=c)=Γxzc(z=c)cos(2πx/L)$⁠. $Exxt$ and $Γxzc(z=c)$ have exactly same tendency given by Figs. 7 and 8, except that the scales on the vertical axis are different, as shown in Figs. 10 and 11.

Same phenomena are observed for other higher order oscillations. For the sandwich panel considered (made of carbon epoxy face and glass-phenolic honeycomb core with f_t = f_b = 1.0 mm, 2c = 18.0 mm, and L = 800 mm), the buckling loads and natural vibration periods corresponding to the first five modes (n = 1, 2, 3, 4, 5) are listed in Table 2. The higher mode, which has more half sine waves, has higher buckling load and shorter free vibration period. So, it can be concluded that although it takes same normalized impulsive load duration for a sandwich panel to have a dynamic response increasing at the same rate when the sandwich panel experiences different mode shapes, the required absolute load duration is shorter when a higher mode is active. In addition to the global buckling mode, sandwich panels may also exhibit wrinkling. With appropriate initial geometric imperfections, the present approach can be extended to wrinkling modes. When a sandwich panel exhibits a mode that consists of multiple oscillation mode components, each components can be studied independently. In this paper, antisymmetric wrinkling modes with uniform wavelengths can be obtained as the superposition of multiple modes with different wavelengths. However, symmetric wrinkling is beyond the scope of this paper, and thus, it is not included.

Let us consider three different sandwich panels, a sandwich panel made of carbon epoxy faces and glass-phenolic honeycomb core with the geometry of case 1, a sandwich panel made of carbon epoxy faces and glass-phenolic honeycomb core with the geometry of case 2, and a sandwich panel made of E-glass polyester faces and balsa wood core with the geometry of case 1. The material properties and geometries are listed in Tables 1 and 3. The ratios $f(P,τ,0.2Pcr1)$ of these three cases are plotted in Fig. 12. It is seen that the rate of change when subjected to different normalized step impulsive loads with different normalized durations is the same and independent of material and geometry. When the impulsive load is lower than the buckling load, the dynamic response is bounded, the curve has a plateau, and the lower impulsive load requires shorter duration to reach the constant value. When the impulsive load equals to or is higher than the buckling load, the dynamic response is unbounded. But it is possible to have a dynamic response with small magnitude if the impulsive load is applied for a short duration.

The buckling load and period of free vibration (or natural frequency) are two fundamental characteristics of a sandwich panel, and they should be used to normalize the impulsive load magnitude and duration. These two items change with changing of material and geometry. For the dynamic response of a sandwich panel subjected to an impulsive load, normalizing the impulsive load magnitude and duration with the buckling load and free vibration period can lead to a consistent normalized response. The normalized response curves are independent of the material and geometry and can simplify the analysis effort.

Consider an impulsive load with a magnitude of P at t = 0 that linearly decays to 0 when t = τ. The time history of the load profile is plotted in Fig. 13. The sandwich panel is made of carbon epoxy faces and glass-phenolic honeycomb core. Geometry is taken as the case 1 listed in Table 3, which has thicknesses of the faces f_t = f_b = 1.0 mm, thickness of the core 2c = 18.0 mm, and length L = 40(f_t + f_b + 2c).

The fundamental oscillation is considered. Thus, the load magnitude and time are normalized with the first mode buckling load and natural period. The ratios of the maximum and minimum dynamic transverse displacement to the static transverse displacement of the top face, $Max((w0t−w0It)/(w0St−w0It))$ and $Min((w0t−w0It)/(w0St−w0It))$⁠, when subjected to various linear decay impulsive loads are given in Fig. 14. A sandwich panel subjected to a higher axial impulsive load requires a longer load duration to have the same ratio. However, a sandwich panel subjected to a linear decay load with a long duration has a different behavior when compared with the one when a step impulsive load is applied, i.e., although the ratio of the maximum dynamic to static response increases with the longer linear decay impulsive load duration, the curve does not reach a plateau even when the load duration is 10 times that of the natural period. In contrast, when a sandwich panel subjected to a step impulsive load lower than 0.8 of the buckling load, the maximum dynamic to static ratio curve reaches a plateau when the load duration is less than one natural period, as shown in Fig. 4.

Figure 14(b) gives the ratio of the minimum dynamic response to the static one. The curves are no longer periodic, which is different from the ones when subjected to step impulsive load. Since the impulsive load decays linearly, the sandwich panel will not rest at the initial position with zero velocity and zero acceleration at the end of the impulsive load, which may happen when subjected to a step impulsive load, shown by Fig. 6(c). Therefore, $Min((w0t−w0It)/(w0St−w0It))$ cannot have a zero value.

The plot of maximum transverse displacement per unit imperfection of the top face is given in Fig. 15. The maximum dynamic displacement increases with the load duration and also with the load magnitude. The dynamic response is bounded when the impulsive load is lower than the static critical load and unbounded when the load equals or is higher than the static critical load. It is noted that other measurements change at the same rate as the change of the external load. In other words, when plotting the curves of the maximum of other displacements, strains, or stresses, similar plots are observed. The only difference between these plots and Fig. 15 is that the scales of the vertical axes are different.

The dynamic response when the sandwich panel experiences higher order oscillations and the dynamic response of sandwich panels for different materials and geometries is also studied. When normalizing the time with the natural period of the corresponding mode and load duration with the buckling load of the corresponding mode, a plot, similar to Fig. 15, is always obtained for any measurement. These plots are omitted for conciseness. The conclusion reached for the case of a step impulsive load example still holds. All measurements change at the same rate as the change of the linear decay impulsive load magnitude and duration, and the rate is independent of the oscillation mode number, material, and geometry.

In this part, an impulsive load with a triangular profile, shown in Fig. 16, is considered. The impulsive load reaches the highest magnitude P at t = τ/2, and equals zero at t = 0 and t = τ. The sandwich panel studied is made out of carbon epoxy faces and glass-phenolic honeycomb core. The geometry is taken as the case 1 listed in Table 3.

For the fundamental oscillation, the ratios of the maximum and minimum dynamic transverse displacement to the static transverse displacement of the top face, $Max((w0t−w0It)/(w0St−w0It))$ and $Min((w0t−w0It)/(w0St−w0It))$ are plotted in Fig. 17. As shown in Fig. 17(a), the $Max((w0t−w0It)/(w0St−w0It))$ curve does not always have a monotonic increase with the load duration τ, which is different from the behavior observed in the cases of step impulsive load and linear decay impulsive load. When the sandwich panel is subjected to a small magnitude triangular impulsive load, such as $0.2Pcr1$⁠, $0.4Pcr1$⁠, $0.6Pcr1$⁠, the highest value occurs at the first peak.

The highest value of the maximum dynamic response versus static response ratio is always less than 2. The ratio of the minimum dynamic transverse displacement to the static transverse displacement does not change monotonically with the load duration, no matter if a step load, a linear decay load, or a triangular shape load is applied to a sandwich panel. When a step load is applied, $Min((w0t−w0It)/(w0St−w0It))$ versus load duration τ curve is a periodic function. When a linear decay load is applied, the curve oscillates around a value close to the lowest value. When a triangular shape load is applied, the curve oscillates and decays.

The dynamic response of the top face transverse displacement when subjected to a triangular impulsive load of $P=0.2Pcr1$ with durations of τ = 0.35 T₁, τ = 1.03 T₁, τ = 2.29 T₁, and τ = 3.25 T₁ is plotted in Fig. 18. These four states are marked with dots and “a,” “b,” “c,” “d” in Fig. 17. As the impulsive load has a triangular profile in time, the forced vibration is not a periodic function.

The plot of maximum transverse displacement per unit imperfection of the top face in logarithmic scale is given in Fig. 19. It includes the loads lower than, equal to, and higher than the buckling load. The curve shows oscillation when the load magnitude is small, and the oscillation vanishes with the increasing of the load magnitude. When the load equals to and is higher than the static buckling load, the curve is unbounded. Other measurements, e.g., displacements, strains, and stresses, all change at the same rate as the change of the external loads. In other words, when plotting the curves of maximum displacement or other measurements, similar plots to Fig. 19 are obtained. The only difference between these plots and Fig. 19 is that the scales of the vertical axes are different.