Abstract

This paper presents an analysis for the dynamic stability of sandwich beams/wide plates subjected to axial impulsive loads. The formulation and solution of the problem is done by use of the extended high-order sandwich panel theory (EHSAPT). With the initial geometric imperfection included, the equations of motion in terms of seven generalized displacements are derived. The dynamic response of sandwich panels subjected to three different types of impulsive loads, namely, step, linear decay, and triangular impulse, is studied. Furthermore, the effects of the oscillation mode number, face/core materials, and geometries are investigated. It is observed that all measurements of the dynamic response, such as the maximum displacements, strains, and stresses, change at the same rate as the change of the impulse load magnitude and duration, for a specific impulse load profile. When the impulse load is lower than the static buckling load, the dynamic response is bounded no matter how long the load is applied. A step impulsive axial load with magnitude lower than the static buckling load can lead a sandwich panel to have a dynamic response as high as twice the static response. When the impulse load is higher than the static critical load, the dynamic response is unbounded with increasing load duration. However, it is possible that the dynamic response can be controlled at a low level if the duration of the impulse load is short enough, and thus, in this case, the load can safely exceed the static critical load.

1 Introduction

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 [610]. 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 [1520], plates [21,22], and shells [2326], 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 [4144]. 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 [4951].

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.

2 Formulation

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 ft, fb, 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 ut(x, z), uc(x, z), and ub(x, z), respectively, and the transverse displacement field of the top face, core, and bottom face are wt(x, z), wc(x, z), and wb(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 uIt(x, z), uIc(x, z), and uIb(x, z), and the components in z-direction are denoted as wIt(x, z), wIc(x, z), and wIb(x, z).

Fig. 1
Sandwich panel geometry
Fig. 1
Sandwich panel geometry
Close modal
The EHSAPT [46] is considered to model the sandwich panel. The thin high stiffness faces adopt the Euler–Bernoulli assumptions, and the displacement field in the top face (c < zc + ft) is
wt(x,z)=w0t(x);ut(x,z)=u0t(x)(zcft2)w0,xt(x)
(1a)
and in the bottom face (− cfbz < −c) is
wb(x,z)=w0b(x),ub(x,z)=u0b(x)(z+c+fb2)w0,xb(x)
(1b)
where the superscripts t and b stand for the top face and the bottom face, and the subscript 0 refers to the value at the centroid.
In the core (−czc), the deformation in the axial direction and thickness direction are both considered. The axial displacement uc(x, z) and transverse displacement wc(x, z) are cubic and quadratic polynomials, in terms of the z coordinate, respectively [46]. Note that these displacements satisfy the displacement continuity conditions along the interfaces between the faces and core (z = ±c): one has
wc(x,z)=(z2c+z22c2)w0t(x)+(1z2c2)w0c(x)+(z2c+z22c2)w0b(x)
(2a)
uc(x,z)=z22c2(1+zc)u0t(x)+ftz24c2(1+zc)w0,xt(x)+(1z2c2)u0c(x)+z(1z2c2)ϕ0c(x)+z22c2(1zc)u0b(x)+fbz24c2(1+zc)w0,xb(x)
(2b)
where u0c, w0c, and ϕ0c are the axial displacement, transverse displacement, and the slope of the centroid of the core.

The displacement field of the sandwich panel is represented by the deformation of the centroid of each layer, {U¯(x)}=u0tw0tu0bw0bu0cw0cϕ0cT. In a similar way, the initial geometric imperfections of the sandwich panel, uIt(x, z), wIt(x, z), uIc(x, z), wIc(x, z), uIb(x, z), and wIb(x, z), are represented by the values of the initial imperfections at the centroid and are denoted as {U¯I(x)}=u0Itw0Itu0Ibw0Ibu0Icw0Icϕ0IcT. The latter can be obtained by simply replacing the superscript t, b, c with It, Ib, Ic in Eqs. (1)(2).

As a result of the Euler–Bernoulli assumption, the axial normal strain is the only nonzero strain in the top and bottom face and is given by
ϵxxt=ϵxx(ut,wt)ϵxx(uIt,wIt)
(3a)
ϵxxb=ϵxx(ub,wb)ϵxx(uIb,wIb)
(3b)
In the core, the axial normal strain, transverse normal strain, and shear strain all exist and are given by
ϵxxc=ϵxx(uc,wc)ϵxx(uIc,wIc)
(4a)
ϵzzc=ϵzz(uc,wc)ϵzz(uIc,wIc)
(4b)
γxzc=γxz(uc,wc)γxz(uIc,wIc)
(4c)
where
ϵxx=ux,ϵzz=wz,γxz=uz+wx
(4d)
The normal axial stress of the faces are σxxt,b=E1t,bϵxxt,b. In the orthotropic core, there are three stress components:
[σxxcσzzcτxzc]=[C11cC13c0C13cC33c000C55c][ϵxxcϵzzcγxzc]=[1E1cν31cE3c0ν31cE3c1E3c0001G31c]1[ϵxxcϵzzcγxzc]
(5)
where Cijc are the stiffness constants of the core, which are C11c=E1cE3c/(E3cE1cν31c2), C13c=ν31cE1cE3c/(E3cE1cν31c2), C33c=E3c2/(E3cE1cν31c2), and C55c=G31c.
The strain energy of the sandwich panel is
U=120Lcc+ftσxxtϵxxtdzdx+120Lcc(σxxcϵxxc+σzzcϵzzc+τxzcγxzc)dzdx+120Lcfbcσxxbϵxxbdzdx
(6a)
and the kinetic energy is
T=120Lcc+ftρt{[(utuIt)t]2+[(wtwIt)t]2}dzdx+120Lccρc{[(ucuIc)t]2+[(wcwIc)t]2}dzdx+120Lcfbcρb{[(ubuIb)t]2+[(wbwIb)t]2}dzdx
(6b)
where ρt,b,c is the mass density of the top, bottom, and core, respectively. Since the initial geometric imperfection is independent of time t, the kinetic energy given in Eq. (6b) is further reduced to
T=120Lcc+ftρt[(utt)2+(wtt)2]dzdx+120Lccρc[(uct)2+(wct)2]dzdx+120Lcfbcρb[(ubt)2+(wbt)2]dzdx
(6c)
The work done by the external forces is
9
W=Nt{u0t|x=0x=Lu0It|x=0x=L[120L(w0tx)2dx120L(w0Itx)2dx]}Nb{u0b|x=0x=Lu0Ib|x=0x=L[120L(w0bx)2dx120L(w0Ibx)2dx]}
(7)
where the positive values of Nt and Nb stand for compressive loads, and the terms in the curly brackets are the displacements of their points of application.
The equations of motion are derived from Hamilton’s principle:
t1t2δ(UWT)dt=0
(8)

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 simply supported sandwich panel is considered, the displacement boundary conditions are
Atx=0andx=L:w0tw0It=w0bw0Ib=w0cw0Ic=0
(9)
Therefore, the force boundary conditions that should be considered are the ones given by Eqs. (A3a), (A3d), (A3e), (A3g), (A3h), and (A3k). In a compact form, the governing equations can be written as follows:
M({U¯¨})+K({U¯}{U¯I})+L({N},{U¯})={0}
(10a)
with boundary conditions
B({U¯}{U¯I})={N}
(10b)
where {U¯}=[u0tw0tu0bw0bu0cw0cϕ0c]T is the displacement, {U¯I}=[u0Itw0Itu0Ibw0Ibu0Icw0Icϕ0Ic]T is the initial geometric imperfection, {N} is the external load vector, and {0} is the zero vector. Also, M(·), K(·), L(,), and B(·) are linear differential operators.
The dynamic response of the sandwich panel is the combination of two components. One is the compressive oscillation due to the axial compressive loads, and the other is the bending oscillation caused by the coupling effect between the transverse deformation and the axial deformation and initialed by the initial geometric imperfection. Hence, {U¯} can be written as follows:
{U¯}={U¯}comp+{U¯}bend
(11)

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)
Here, we are interested in the dynamic stability behavior of sandwich panels, which is associated with the transverse deformation. Therefore, the equations of motion given by (II), Eq. (13), are considered.

The initial geometric imperfection can be expended into the Fourier series. By considering the simply supported displacement boundary conditions, Eq. (9), the initial geometric imperfection is expressed as
u0It=n=1UnItcosnxLπ,u0Ic=n=1UnIccosnxLπ,u0Ib=n=1UnIbcosnxLπ
(14a)
w0It=n=1WnItsinnxLπ;w0Ic=n=1WnIcsinnxLπ;w0Ib=n=1WnIbsinnxLπ
(14b)
ϕ0Ic=n=1ΦnIccosnxLπ
(14c)
After substituting Eq. (14) into Eq. (13a), it is seen that the solution of the nth term has the terms Untcos(nπx/L), Wntsin(nπx/L), Unbcos(nπx/L), Wnbsin(nπx/L), Unccos(nπx/L), Wncsin(nπx/L) and Φnccos(nπx/L), which also satisfy both the displacement boundary conditions and force boundary conditions, given by Eq. (9) and Eqs. (A3a), (A3d), (A3e), (A3g), (A3h), and (A3k). After substituting the nth term into the equations of motion, Eq. (13a) becomes
[Mn]{U¨n}+[Kn]{Un}+[Ln]{Un}=Kn{UnI}
(15)
where {Un}=[UntWntUnbWnbUncWncΦnc]T and {UnI}=[UnItWnItUnIbWnIbUnIcWnIcΦnIc]T.

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

When the sandwich panel has an arbitrary initial geometric imperfection, the general solution of Eq. (13a) equals to the summation of all Fourier series terms, which is,
u0t=n=1UntcosnxLπ;u0c=n=1UnccosnxLπ;u0b=n=1UnbcosnxLπ
(16a)
w0t=n=1WntsinnxLπ;w0c=n=1WncsinnxLπ;w0b=n=1WnbsinnxLπ
(16b)
ϕ0c=n=1ΦnccosnxLπ
(16c)
The dynamic response is obtained by the numerical integration method. In the present work, the implicit Newmark method is adopted, which has the following scheme:
{U˙}k+1={U˙}k+Δt[(1α){U¨}k+α{U¨}k+1]
(17a)
{U}k+1={U}k+Δt{U˙}k+(Δt)2[(12β){U¨}k+β{U¨}k+1]
(17b)
where α and β are coefficients and are taken as α = 1/2 and β = 1/4 herein. By this choice of the constants, it leads to an implicit unconditionally stable scheme.

2.1 Behavior Under Static Load.

The dynamic response of sandwich panels subjected to axial compressive loads is evaluated by comparing it to its behavior under the corresponding static loads. The static response can be obtained by neglecting the time related terms in Eq. (13a). Thus, the governing equations together with the boundary conditions are
K({U¯}bend{U¯I})+L({N},{U¯}bend)={0}
(18a)
B({U¯}bend{U¯I})={0}
(18b)
Similar to the dynamic analysis, by considering the Fourier expansion of the initial geometric imperfection, the solution corresponding to the nth term is determined by
[Kn]{Un}+[Ln]{Un}=[Kn]{UnI}
(19)
Solving these seven linear algebra equations yields {Un} directly.

2.2 Euler Critical Load and Natural Frequency.

The linear static critical load and natural frequency are also determined in order to evaluate the magnitude of the impulsive loads and the load duration. The Euler critical load of the nth mode shape [ut,b,c=Unt,b,ccos(nπx/L); wt,b,c=Wnt,b,csin(nπx/L); ϕc=Φnccos(nπx/L)] is obtained by solving the following eigen-value problem:
det([Kn]+Pcrn[Ln])=0
(20)
where Pcrn is the nth buckling load if unit load is considered in the [Ln] matrix.
Let us consider a sandwich panel has faces with thicknesses ft = fb = 1.0 mm, and core with thickness 2c = 18.0 mm. The length is L = 40(ft + fb + 2c). The faces are made of carbon epoxy, and the core is made out of glass-phenolic honeycomb. The material properties are listed in Table 1. Solving the eigen-value problem yields the first Euler critical load, Pcr1=19.88kN. Same sandwich panel with initial geometric imperfection subjected to static loading is also studied. It can be easily proved that only the out-of-plane imperfections can induce the out-of-plane deformation. When sandwich panels possess imperfections that are only in the in-plane direction of the faces and core or rotations of the core, the out-of-plane deformations are zero. Therefore, only the out-of-plane imperfections, namely, w0It, w0Ib, and w0Ic, are imposed, and all the other terms are taken as null. The following first mode initial geometric imperfection is considered,
u0It=u0Ib=u0Ic=ϕ0Ic=0
(21a)
w0It=w0Ib=w0Ic=ΔsinπxL
(21b)
Table 1

Material properties

Carbon epoxy faceE-glass polyester faceBalsa wood coreGlass-phenolic honeycomb core
E1181.040.00.6710.032
E210.310.00.1580.032
E310.310.07.720.300
G235.963.50.3120.048
G317.174.50.3120.048
G127.174.50.2000.013
ν320.400.400.490.25
ν310.0160.260.230.25
ν120.2770.0650.660.25
ρ1632200025064
Carbon epoxy faceE-glass polyester faceBalsa wood coreGlass-phenolic honeycomb core
E1181.040.00.6710.032
E210.310.00.1580.032
E310.310.07.720.300
G235.963.50.3120.048
G317.174.50.3120.048
G127.174.50.2000.013
ν320.400.400.490.25
ν310.0160.260.230.25
ν120.2770.0650.660.25
ρ1632200025064

Note: Moduli data are in GPa. Densities are in kg/m3.

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.

Fig. 2
Top face mid-point transverse displacement under static axial compressive loads with various initial imperfections
Fig. 2
Top face mid-point transverse displacement under static axial compressive loads with various initial imperfections
Close modal
The nth free vibration mode shape also has the same shape as the nth Euler buckling mode, u0t,b,c=Unt,b,ccos(nπx/L), w0t,b,c=Wnt,b,csin(nπx/L), ϕ0c=Φnccos(nπx/L). The natural frequency is obtained by solving the following eigen-value problem,
det([Kn]ωn2[Mn])=0
(22)
The period of the nth free vibration mode is Tn = 2π/ωn.

3 Results and Discussions

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 Nt = Nb = P/2. Different impulsive load profiles with load duration τ are considered.

3.1 Response When Subjected to a Step Impulsive Load.

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.

Fig. 3
Step impulsive load time profile
Fig. 3
Step impulsive load time profile
Close modal

3.1.1 The Fundamental Oscillation.

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 ft = fb = 1.0 mm and 2c = 18.0 mm, respectively. The length is L = 40(ft + fb + 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.

Fig. 4
Ratio of maximum dynamic response to static response versus ratio τ/T1 of impulsive duration to first free vibration period for various axial loads
Fig. 4
Ratio of maximum dynamic response to static response versus ratio τ/T1 of impulsive duration to first free vibration period for various axial loads
Close modal
Fig. 5
Ratio of minimum dynamic response to static response versus ratio τ/T1 of impulsive duration to first free vibration period for various axial loads
Fig. 5
Ratio of minimum dynamic response to static response versus ratio τ/T1 of impulsive duration to first free vibration period for various axial loads
Close modal

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 w0tw0It 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((w0tw0It)/(w0Stw0It))=2 (Fig. 6d), the minimum dynamic transverse displacement to the static transverse displacement, Min((w0tw0It)/(w0Stw0It)) can reach the lowest value of −2.

Fig. 6
Dynamic response of the top face, bottom face and core transverse displacement when P=0.6Pcr1 with various durations: (a) τ/T1 = 0.35, (b) τ/T1 = 1.4, (c) τ/T1 = 1.58, and (d) τ/T1 = 2.37
Fig. 6
Dynamic response of the top face, bottom face and core transverse displacement when P=0.6Pcr1 with various durations: (a) τ/T1 = 0.35, (b) τ/T1 = 1.4, (c) τ/T1 = 1.58, and (d) τ/T1 = 2.37
Close modal

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. 79. 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.

Fig. 7
Maximum axial strain per unit imperfection of the top face, Exxt/Δ, plotted against impulse duration, τ/T1, for various axial compressive loads (logarithmic scale)
Fig. 7
Maximum axial strain per unit imperfection of the top face, Exxt/Δ, plotted against impulse duration, τ/T1, for various axial compressive loads (logarithmic scale)
Close modal
Fig. 8
Maximum shear strain per unit imperfection at the top face/core interface, Γxzc(z=c)/Δ, plotted against impulse duration, τ/T1, for various axial compressive loads (logarithmic scale)
Fig. 8
Maximum shear strain per unit imperfection at the top face/core interface, Γxzc(z=c)/Δ, plotted against impulse duration, τ/T1, for various axial compressive loads (logarithmic scale)
Close modal
Fig. 9
Maximum transverse displacement per unit imperfection of the top face, (W1t−W1It)/Δ, plotted against impulse duration, τ/T1, for various axial compressive loads (logarithmic scale)
Fig. 9
Maximum transverse displacement per unit imperfection of the top face, (W1t−W1It)/Δ, plotted against impulse duration, τ/T1, for various axial compressive loads (logarithmic scale)
Close modal

Besides, Figs. 79 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. 79. Any points below the line give the permissible load magnitudes and allowable load durations.

3.1.2 The nth higher order oscillation.

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 Tn, 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).

Let us consider the following initial geometric imperfection,
u0It=u0Ib=u0Ic=ϕ0Ic=0
(23a)
w0It=w0Ib=w0Ic=Δsin2πxL
(23b)
which leads to a dynamic response that only contains the second mode oscillation with n = 2 term.

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. 46, 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.

Fig. 10
Maximum axial strain per unit imperfection of the top face, Exxt/Δ, plotted against impulse duration, τ/T2, for various axial compressive loads (mode 2 geometric imperfection, logarithmic scale)
Fig. 10
Maximum axial strain per unit imperfection of the top face, Exxt/Δ, plotted against impulse duration, τ/T2, for various axial compressive loads (mode 2 geometric imperfection, logarithmic scale)
Close modal
Fig. 11
Maximum axial strain per unit imperfection of the top face, Γxzc/Δ, plotted against impulse duration, τ/T2, for various axial compressive loads (mode 2 geometric imperfection, logarithmic scale)
Fig. 11
Maximum axial strain per unit imperfection of the top face, Γxzc/Δ, plotted against impulse duration, τ/T2, for various axial compressive loads (mode 2 geometric imperfection, logarithmic scale)
Close modal

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 ft = fb = 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.

Table 2

Buckling load and natural vibration period of the nth mode shape carbon epoxy face/honeycomb core, case 1

nMode 1Mode 2Mode 3Mode 4Mode 5
Pcrn (N)331.2893653.3081797.9326866.6279904.5836
Tn (ms)5.84302.080321.25470.90280.7068
nMode 1Mode 2Mode 3Mode 4Mode 5
Pcrn (N)331.2893653.3081797.9326866.6279904.5836
Tn (ms)5.84302.080321.25470.90280.7068

3.1.3 Effects of the Geometric and Material Properties.

As demonstrated in the above section, for a specific sandwich panel, all measurements change at the same rate when the magnitude and duration of the step impulsive load changes. Since sandwich composite structures are compound structures, we are interested in the dynamic response when sandwich structures are made of faces and core with different ratios in materials and geometries. As vertical axis scales are different when compared with different measurements and different sandwich panels, the following ratio is introduced to compare the changing rate of different measurements uniformly. It is defined as
f(P,τ,0.2Pcr1)=Max[F(P,τ)]Max[F(0.2Pcr1,τ)]
(24)
where F(P, τ) is a measurement during the dynamic response when subjected to a load P with duration τ. It can be the displacement, e.g., W1tW1It, strain, e.g., Exxt, Γxzc(z=c), or stress. Since f(P,τ,0.2Pcr1) is the ratio between the response when subjected to a load P and 0.2Pcr1, it remains the same no matter which measurement is used as F(P, τ). Here, only the response of the fundamental oscillation needs to be considered since the higher oscillation has the same changing rate.

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.

Fig. 12
Ratio of maximum dynamic response under step impulse load P to the corresponding dynamic response under step impulse 0.2Pcr1 for various ratios P/Pcr1: (a) carbon epoxy face/honeycomb core and case 1 geometry, (b) carbon epoxy face/honeycomb core and case 2 geometry, and (c) E-glass polyester face/balsa wood core and case 1 geometry
Fig. 12
Ratio of maximum dynamic response under step impulse load P to the corresponding dynamic response under step impulse 0.2Pcr1 for various ratios P/Pcr1: (a) carbon epoxy face/honeycomb core and case 1 geometry, (b) carbon epoxy face/honeycomb core and case 2 geometry, and (c) E-glass polyester face/balsa wood core and case 1 geometry
Close modal
Table 3

Geometry properties (mm)

ftfb2cL
Case 11.01.018.0800
Case 22.02.016.0800
ftfb2cL
Case 11.01.018.0800
Case 22.02.016.0800

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.

3.2 Response When Subjected to a Linear Decay Impulsive Load.

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 ft = fb = 1.0 mm, thickness of the core 2c = 18.0 mm, and length L = 40(ft + fb + 2c).

Fig. 13
Linear decay impulsive load profile
Fig. 13
Linear decay impulsive load profile
Close modal

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((w0tw0It)/(w0Stw0It)) and Min((w0tw0It)/(w0Stw0It)), 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.

Fig. 14
Ratio of dynamic response to static response under linear decay impulse load with duration τ/T1 and magnitude P/Pcr1: (a) maximum transverse displacement and (b) minimum transverse displacement
Fig. 14
Ratio of dynamic response to static response under linear decay impulse load with duration τ/T1 and magnitude P/Pcr1: (a) maximum transverse displacement and (b) minimum transverse displacement
Close modal

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((w0tw0It)/(w0Stw0It)) 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.

Fig. 15
Maximum transverse displacement per unit imperfection of the top face (W1t−W1It)/Δ under linear decay impulse load with duration τ/T1 and magnitude P/Pcr1 (logarithmic scale)
Fig. 15
Maximum transverse displacement per unit imperfection of the top face (W1t−W1It)/Δ under linear decay impulse load with duration τ/T1 and magnitude P/Pcr1 (logarithmic scale)
Close modal

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.

3.3 Response When Subjected to a Triangular Impulsive Load.

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.

Fig. 16
Triangular impulse load profile
Fig. 16
Triangular impulse load profile
Close modal

For the fundamental oscillation, the ratios of the maximum and minimum dynamic transverse displacement to the static transverse displacement of the top face, Max((w0tw0It)/(w0Stw0It)) and Min((w0tw0It)/(w0Stw0It)) are plotted in Fig. 17. As shown in Fig. 17(a), the Max((w0tw0It)/(w0Stw0It)) 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.

Fig. 17
Ratio of dynamic response to static response under triangular impulse load with duration τ/T1 and magnitude P/Pcr1: (a) maximum transverse displacement and (b) minimum transverse displacement
Fig. 17
Ratio of dynamic response to static response under triangular impulse load with duration τ/T1 and magnitude P/Pcr1: (a) maximum transverse displacement and (b) minimum transverse displacement
Close modal

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((w0tw0It)/(w0Stw0It)) 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.

Fig. 18
Top face transverse displacement, under a triangular impulse load with P=0.2Pcr1 and various durations: (a) τ/T1 = 0.35, (b) τ/T1 = 1.03, (c) τ/T1 = 2.29, and (d) τ/T1 = 3.25
Fig. 18
Top face transverse displacement, under a triangular impulse load with P=0.2Pcr1 and various durations: (a) τ/T1 = 0.35, (b) τ/T1 = 1.03, (c) τ/T1 = 2.29, and (d) τ/T1 = 3.25
Close modal

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.

Fig. 19
Maximum transverse displacement per unit imperfection of the top face (W1t−W1It)/Δ under triangular impulse load with duration τ/T1 and magnitude P/Pcr1 (logarithmic scale)
Fig. 19
Maximum transverse displacement per unit imperfection of the top face (W1t−W1It)/Δ under triangular impulse load with duration τ/T1 and magnitude P/Pcr1 (logarithmic scale)
Close modal

The dynamic response when the sandwich panel experiences higher order oscillations and the dynamic response of sandwich panels of different materials and geometries are also studied. When normalizing the time with the natural period of the corresponding mode and the load duration with the buckling load of the corresponding mode, a plot, similar to Fig. 19, is always obtained for any measurement. These plots are omitted for conciseness. The conclusion given in the previous examples still holds. All measurements change at the same rate as the change of the load magnitude and duration, and the rate is independent of the oscillation mode number, material, and geometry.

4 Conclusions

The dynamic stability of sandwich panels subjected to axial compressive impulsive loads is studied based on the EHSAPT. The initial geometric imperfections are considered. Equations of motion are derived and are in terms of seven generalized displacement coordinates. With the numerical studies, it is observed that the corresponding static response, static buckling load, and free vibration period are three good parameters to evaluate the dynamic stability of sandwich panels. In the present work, three different impulsive loading profiles, namely, step load, linear decay load, and triangular load are considered. It is seen that, for a given impulsive load profile, all measurements during the dynamic response, such as the maximum displacement, maximum strain, and maximum stress, change at the same rate with the change of the impulsive load magnitude and duration. Furthermore, the rate of change is independent of the oscillation mode number, material, and geometry. Therefore, one scaling factor plot can be used for a specific impulsive load profile. With the scaling factor plot and one dynamic response under this specific impulsive load profile, one can easily get the dynamic response when subjected to the same load profile but with different impulsive load magnitude and duration. The rate of change is quite different when a sandwich panel is subjected to different types of impulsive loads. For the load lower than the static buckling load, the dynamic response is bounded: (1) when the step impulsive load is applied, the ratio of the maximum dynamic response and static response increases to the highest value of 2 when the load duration is long enough; (2) when the linear decay impulsive load is applied, the ratio approaches 2 but increases much more slowly; (3) when the triangular impulsive load is applied, the ratio oscillates with the change of load duration, and the highest ratio does not exceed the value of 2. For the load higher than the static buckling load, the dynamic response is unbounded. However, the dynamic response can be controlled under a small level as long as the load duration is short enough. Thus, it is possible that a sandwich panel can survive an impulsive load which is much higher than the static buckling load.

Acknowledgment

The financial support of the Office of Naval Research, Grant N00014-20-1-2605, and the interest and encouragement of the Grant Monitor, Dr. Y. D. S. Rajapakse, are both gratefully acknowledged.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.

Appendix

Seven partial differential equations in t and x are obtained from the variational principle, Eq. (8), as:

δu0t:
(6cρc35+ftρt)u¨0t+(3cftρc35x)w¨0t+(cρc35)u¨0b(cfbρc70x)w¨0b+(2cρc15)u¨0c+(2c2ρc35)ϕ¨0c+(4730cC55cα1t2x2)(u0tu0It)+(α3tx3cft35C11c3x3)(w0tw0It)(730cC55c+c35C11c2x2)(u0bu0Ib)(α2bxcfb70C11c3x3)(w0bw0Ib)(43cC55c+2c15C11c2x2)(u0cu0Ic)+(β1x)(w0cw0Ic)(45C55c+2c235C11c2x2)(ϕ0cϕ0Ic)=0
(A1a)
δw0t:
(3cftρc35x)u¨0t+[4cρc15+ftρt(3cft2ρc70+ft3ρt12)2x2]w¨0t(cftρc70x)u¨0b+(cρc15+cftfbρc1402x2)w¨0b(cftρc15x)u¨0c+(2cρc15)w¨0c(c2ftρc35x)ϕ¨0c+(α3tx+3cft35C11c3x3)(u0tu0It)+(76cC33c+α8t2x2+α9t4x4)(w0tw0It)+(α5tx+cft70C11c3x3)(u0bu0Ib)+(16cC33c+β22x2cfbft140C11c4x4)(w0bw0Ib)+(α6tx+cft15C11c3x3)(u0cu0Ic)+(43cC33c+α7t2x2)(w0cw0Ic)+(α4tx+c2ft35C11c3x3)(ϕ0cϕ0Ic)+Nt2x2w0t=0
(A1b)
δu0b:
(cρc35)u¨0t+(cftρc70x)w¨0t+(6cρc35+fbρb)u¨0b(3cfbρc35x)w¨0b+(2cρc15)u¨0c(2c2ρc35)ϕ¨0c(730cC55c+c35C11c2x2)(u0tu0It)+(α2txcft70C11c3x3)(w0tw0It)+(4730cC55cα1b2x2)(u0bu0Ib)+(α3bx+3cfb35C11c3x3)(w0bw0Ib)(43cC55c+2c15C11c2x2)(u0cu0Ic)(β1x)(w0cw0Ic)+(45C55c+2c235C11c2x2)(ϕ0cϕ0Ic)=0
(A1c)
δw0b:
(cfbρc70x)u¨0t+(cρc15+cftfbρc1402x2)w¨0t+(3cfbρc35x)u¨0b+[4cρc15+fbρb(3cfb2ρc70+ft3ρt12)2x2]w¨0b+(cfbρc15x)u¨0c+(2cρc15)w¨0c(c2fbρc35x)ϕ¨0c(α5bx+cfb70C11c3x3)(u0tu0It)+(16cC33c+β22x2cfbft140C11c4x4)(w0tw0It)+(α3bx3cfb35C11c3x3)(u0bu0Ib)+(76cC33c+α8b2x2+α9b4x4)(w0bw0Ib)(α6bx+cfb15C11c3x3)(u0cu0Ic)+(43cC33c+α7b2x2)(w0cw0Ic)+(α4bx+c2fb35C11c3x3)(ϕ0cϕ0Ic)+Nb2x2w0b=0
(A1d)
δu0c:
(2cρc15)u¨0t+(cftρc15x)w¨0t+(2cρc15)u¨0b(cfbρc15x)w¨0t(43cC55c+2c15C11c2x2)(u0tu0It)(α6tx+cft15C11c3x3)(w0tw0It)(43cC55c+2c15C11c2x2)(u0bu0Ib)+(α6bx+cfb15C11c3x3)(w0bw0Ib)+(83cC55c16c15C11c2x2)(u0cu0Ic)=0
(A1e)
δw0c:
(2cρc15)w¨0t+(2cρc15)w¨0b+(16cρc15)w¨0c(β1x)(u0tu0It)+(43cC33c+α7t2x2)(w0tw0It)+(β1x)(u0bu0Ib)+(43cC33c+α7b2x2)(w0bw0Ib)+(83cC33c16c15C55c2x2)(w0cw0Ic)(β3x)(ϕ0cϕ0Ic)=0
(A1f)
δϕ0c:
(2c2ρc35)u¨0t+(c2ftρc35x)w¨0t(2c2ρc35)u¨0b+(c2fbρc35x)w¨0b+(16c3ρc105)ϕ¨0c(45C55c+2c235C11c2x2)(u0tu0It)(α4tx+c2ft35C11c3x3)(w0tw0It)+(45C55c+2c235C11c2x2)(u0bu0Ib)(α4bx+c2fb35C11c3x3)(w0bw0Ib)+(β3x)(w0cw0Ic)+(8c5C55c16c3105C11c2x2)(ϕ0cϕ0Ic)=0
(A1g)
In the above equations, the α and β constants are defined as:
α1i=6c35C11c+fiC11i;α2i=130C13c+(1307fi60c)C55c
(A2a)
α3i=1130C13c+(1930+47fi60c)C55c;α4i=4c15C13c+(4c15+2fi5)C55c
(A2b)
α5i=130C13c+(130+7fi60c)C55c;α6i=23C13c+(23+2fi3c)C55c
(A2c)
α7i=fi5C13c(2c15+fi5)C55c;α8i=11fi30C13c(4c15+19fi30+47fi2120c)C55c
(A2d)
α9i=fi312C11i+3cfi270C11c;i=tori=b
(A2e)
and
β1=25(C13c+C55c);β2=fb+ft60C13c+(c15+fb+ft607fbft120c)C55c
(A2f)
β3=8c15C13c+C55c
(A2g)

The over double dot represents the second order derivative with respect to time t.

In addition, nine boundary conditions at either end are also obtained from the variational principle, Eq. (8):

  1. at x = 0 and x = L, either δu0t=0 or,
    [(6c35C11c+ftC11t)x](u0tu0It)+(1130C13c+3cft35C11c2x2)(w0tw0It)+(c35C11cx)(u0bu0Ib)+(130C13ccfb70C11c2x2)(w0bw0Ib)+(2c15C11cx)(u0cu0Ic)(25C13c)(w0cw0Ic)+(2c235C11cx)(ϕ0cϕ0Ic)=Nt
    (A3a)
  2. at x = 0 and x = L, either δw0t=0 or,
    [(38c+47ft)60cC55c3cft35C11c2x2](u0tu0It)+[(11ft60C13cα8t)xα9t3x3](w0tw0It)+[(2c7ft)60cC55ccft70C11c2x2](u0bu0Ib)+[(fb60C13cβ2)x+cfbft140C11c3x3](w0bw0Ib)[2(c+ft)3cC55c+cft15C11c2x2](u0cu0Ic)(α7tx)(w0cw0Ic)[2(2c+3ft)15C55c+c2ft35C11c2x2](ϕ0cϕ0Ic)+Lwt=Ntxw0t
    (A3b)
    where Lwt is the inertial term:
    Lwt=ft420[35ft2ρtxw¨0t+cρc(36u¨0t+18ftxw¨0t+6u¨0b3fbxw¨0b+28u¨0c+12cϕ¨0c)]
    (A3c)
  3. at x = 0 and x = L, either δw0,xt = 0, or,
    (3cft35C11cx)(u0tu0It)+(11ft60C13c+α9t2x2)(w0tw0It)+(cft70C11cx)(u0bu0Ib)+(ft60C13ccfbft140C11c2x2)(w0bw0Ib)+(cft15C11cx)(u0cu0Ic)(ft5C13c)(w0cw0Ic)+(c2ft35C11cx)(ϕ0cϕ0Ic)=0
    (A3d)
  4. at x = 0 and x = L, either δu0c=0 or,
    (2c15C11cx)(u0tu0It)+(23C13c+cft15C11c2x2)(w0tw0It)+(2c15C11cx)(u0bu0Ib)(23C13c+cfb15C11c2x2)(w0bw0Ib)+(16c15C11cx)(u0cu0Ic)=0
    (A3e)
  5. at x = 0 and x = L, either δw0c=0 or,
    C55c[25(u0tu0It)+(2c+3ft)15x(w0tw0It)25(u0bu0Ib)+(2c+3fb)15x(w0bw0Ib)+16c15x(w0cw0Ic)+8c15(ϕ0cϕ0Ic)]=0
    (A3f)
  6. at x = 0 and x = L, either δϕ0c=0 or,
    (2c235C11cx)(u0tu0It)+(4c15C13c+c2ft35C11c2x2)(w0tw0It)(2c235C11cx)(u0bu0Ib)+(4c15C13c+c2fb35C11c2x2)(w0bw0Ib)(8c15C13c)(w0cw0Ic)+(16c3105C11cx)(ϕ0cϕ0Ic)=0
    (A3g)
  7. at x = 0 and x = L, either δu0b=0 or,
    (c35C11cx)(u0tu0It)+(130C13c+cft70C11c2x2)(w0tw0It)+[(6c35C11c+fbC11b)x](u0bu0Ib)(1130C13c+3cfb35C11c2x2)(w0bw0Ib)+(2c15C11cx)(u0cu0Ic)+(25C13c)(w0cw0Ic)(2c235C11cx)(ϕ0cϕ0Ic)=Nb
    (A3h)
  8. at x = 0 and x = L, either δw0b=0 or
    [(2c+7fb)60cC55c+cfb70C11c2x2](u0tu0It)+[(ft60C13cβ2)x+cfbft140C11c3x3](w0tw0It)+[(38c47fb)60cC55c+3cfb35C11c2x2](u0bu0Ib)+[(11fb60C13cα8b)xα9b3x3](w0bw0Ib)+[2(c+fb)3cC55c+cfb15C11c2x2](u0cu0Ic)(α7bx)(w0cw0Ic)[2(2c+3fb)15C55c+c2fb35C11c2x2](ϕ0cϕ0Ic)+Lwb=Nbxw0b
    (A3i)
    where Lwb is the inertial term:
    Lwb=fb420[35fb2ρbxw¨0b+cρc(6u¨0t3ftxw¨0t36u¨0b+18fbxw¨0b28u¨0c+12cϕ¨0c)]
    (A3j)
    and
  9. at x = 0 and x = L, either δw0,xb = 0, or,
    (cfb70C11cx)(u0tu0It)+(fb60C13ccfbft140C11c2x2)(w0tw0It)(3cfb35C11cx)(u0bu0Ib)+(11fb60C13c+α9b2x2)(w0bw0Ib)(cfb15C11cx)(u0cu0Ic)(fb5C13c)(w0cw0Ic)+(c2fb35C11cx)(ϕ0cϕ0Ic)=0
    (A3k)

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