Dynamic response of monolithic and sandwich structures subjected to impulsive and impact loadings

Abstract

Monolithic and sandwich structures have been widely used as energy absorption structures in military and civil engineering. This article reviews theoretical analyses of monolithic beams and plates subjected to static loading, impulsive loading and impact by a mass systematically. Experimental data collected from the literatures are compared with these theoretical results. In addition, the critical impulses for the failure of the monolithic structures are also reviewed. Furthermore, sandwich structures under quasi-static, low-velocity impact, high-velocity impact and blast loading, as well as their failure modes, are also summarized. The research methodology involves experimental investigations, theoretical analyses and numerical simulations.

Keywords

impact loading impulsive loading large deformation monolithic structure sandwich structure

Introduction

Monolithic beams and plates have been widely researched since 1950s using experimental, theoretical and numerical methods. Studies of the dynamic behaviour of monolithic structures were important for understanding their protection ability. Fully clamped monolithic beams made of different materials (mild steel, aluminium, epoxy bonding of carbon fibre to metal, titanium, etc.) with different cross sections (rectangular, circular, I-shaped, T-shaped, etc.) have been investigated under mass impact loading, uniformly distributed impulsive loading and localized loading.

Experiments for monolithic plates made of different materials (aluminium alloy and H.R. steel, hot-rolled mild steel, etc.) were conducted under uniformly distributed impulses (Jones et al., 1970; Wierzbicki and Florence, 1970), striking by large masses (Wen and Jones, 1993) and underwater explosion (Rajendran and Narasimhan, 2001). The failure modes were observed in the experiments: Mode I – large inelastic deformation, Mode II – tearing at the support and Mode III – shear failure at the support (Neuberger et al., 2007). Different shaped plates were investigated by analytical methods. The values predicted by the bending theory of rigid, perfectly plastic plates were higher than those attained from analysis with the effects of large deflections and strain rate sensitivity, when comparing the theoretical solutions with the experimental permanent displacements of the strain-rate insensitive aluminium and the strain rate–sensitive steel plates (Florence, 1966; Wang and Hopkins, 1954). When the loadings were large enough, geometry changes could not be disregarded. In large deflections, the effect of stretching could not be disregarded either, and when the deflection exceeded the beam thickness, normal resultant forces of bending moment and stretching were introduced (Qin et al., 2009; Yu and Stronge, 1990).

In recent years, sandwich structures have been widely used in military aircraft and ships, or in civil engineering structures that may suffer air or water blast (Lu and Yu, 2003). Sandwich structures outperform monolithic structures with the same mass within a certain amount of loadings. According to the geometries of the cores, sandwich structures can be classified as those with metallic foam, polymer foam, lattice, truss and thin-walled tubes. The face sheets of sandwich structures include metal, carbon or glass-reinforced composite (fibre metal laminates (FMLs)). Sandwich structures subjected to quasi-static, impact, ballistic and blast loadings have been investigated by many researchers. Failure modes of sandwich structure were classified as follows: face yielding/micro buckling, face wrinkling, core yielding and indentation.

In this article, static analysis of monolithic beams applied with a concentrated force and uniformly distributed pressure are employed, which is easy for a beginner to grasp the basic concepts. Theoretical solutions of monolithic structures subjected to uniformly impulsive loading and impact loading are reviewed. Experimental data have been collected to examine the validity of the theoretical methods. It is evidenced that the theoretical solutions could be used for design, safety assessments and security studies. The critical impulses for the failure of the monolithic structures are also reviewed. Sandwich structures under quasi-static, low-velocity impact, high-velocity impact and blast loading, as well as their failure modes, are summarized.

Monolithic beams

Static analysis of a monolithic beam

Monolithic beam loaded with a concentrated force

A clamped beam with length L applied with a concentrated force loading P (see Figure 1(a)) was analysed (Haythornthwaite, 1957). It is assumed that the beam is made of perfectly plastic material with yield stress $σ_{0}$ , and it has rectangular cross section with width b and depth H (see Figure 2(a)). The static yield load $P_{Y}$ is

P_{Y} = 2 M_{0} L / (l_{1} l_{2})

(1)

where $M_{0} = σ_{0} b H^{2} / 4$ is the fully plastic bending moment of a beam with rectangular cross section, and $l_{1}$ and $l_{2}$ are the distance between the loading point and the left and right clamped ends, respectively. The dimensionless force is

P = P / P_{Y} = m + δ N / (2 M_{0})

(2)

where $m = M / M_{0}$ , and $δ$ is the displacement of the loading point (see Figure 1(c)).

Figure 1.

A clamped beam with a concentrated force: (a) sketch of the clamped beam and applied force, (b) principle of virtual work - a small deflection at the middle of the beam and the changed angles at two ends and (c) force and bending moment for the left half of the beam. (Haythornthwaite, 1957).

Figure 2.

Relationship between the bending moment and axial force: (a) stress distribution on the rectangular cross section and (b) yield conditions.

As shown in Figure 2(a), the neutral fibres are at a distance $(2 η - 1) H / 2$ above the centroid. The stress distribution can be divided into two cases: pure bending moment $M$ and pure axial force $N$

M = σ_{0} b (H - η H) η H

(3)

N = σ_{0} b [H - 2 (H - η H)]

(4)

The dimensionless bending moment and axial force are $m = M / M_{0} = 4 η (1 - η)$ and $n = N / N_{0} = 2 η - 1$ , respectively; where $N_{0}$ is the fully plastic axial force $σ_{0} b H$ . The distance between neutral fibres and the centroid axis $y = (η - (1 / 2)) H = δ / 2$ . At the rectangular cross section, the axial force N and the bending moment M should satisfy the yield condition as follows (see solid line in Figure 2(b))

n^{2} + | m | = 1

(5)

where $\bar{δ}$ is the dimensionless displacement $(\bar{δ} = δ / H)$ , the equation of dimensionless force p can be transferred into a non-dimensional form

p = {\begin{matrix} 1 + {\bar{δ}}^{2} & \bar{δ} \leq 1 \\ 2 \bar{δ} & \bar{δ} \geq 1 \end{matrix}

(6)

Monolithic beam loaded with uniformly distributed pressure

It is assumed that the beam is made of rigid, perfectly plastic material. A uniformly distributed pressure $p$ is applied on the simply supported beam (see Figure 3(a)) and it leads to a plastic hinge at the middle of the beam (see Figure 3(b)). Based on the first order of classical theory, governing equations were developed without considering the change of geometry (Jones, 1989). Jones (1989) developed general equations of motion based on the idea that the external work rate equates to the internal energy dissipated at plastic hinges and within plastic zones

2 (p^{u} L) (L \overset{\cdot}{θ} / 2) = M_{0} 2 \overset{\cdot}{θ}

(7)

where L is the semi-length of the beam. From equation (7), the upper limit load $p^{u}$ can be obtained as follows

p^{u} = 2 M_{0} / L^{2}

(8)

Figure 3.

Simply supported monolithic beam subjected to uniformly distributed transverse pressure: (a) pressure distribution, (b) transverse displacement field.

The distribution of the bending moment in the region $0 \leq x \leq L$ of the beam is

M = p (L^{2} - x^{2}) / 2

(9)

which has the largest value, $M_{max} = p L^{2} / 2$ , at the centre of the beam. So, the bending moment distribution is statically admissible for the lower limit load $p^{l}$

p^{l} = 2 M_{0} / L^{2}

(10)

Obviously, the exact static collapse pressure is

p_{c} = 2 M_{0} / L^{2}

(11)

which simultaneously satisfies the requirements of both upper- and lower-bound theorems of plasticity. For a clamped beam, $M_{0}$ should be replaced by $2 M_{0}$ , so the exact collapse pressure is

{\bar{p}}_{c} = 4 M_{0} / L^{2}

(12)

Dynamic plastic behaviour of monolithic beams

Monolithic beam under uniformly distributed impulsive loading

Experiments of fully clamped beams loaded with uniformly distributed impulsive loadings were conducted by many researchers. The dynamic behaviours of fully clamped beams made of mild steel and aluminium 6061-T6 were investigated under uniformly distributed impulsive loads by Jones et al. (1971). The permanent transverse deflections at the centre of the specimen were more dependent on geometry changes than the strain rate sensitivity of the material. When the beam obtains uniform velocity initially, the non-dimensional kinetic energy is defined as follows

λ = \frac{{m V}_{0}^{2} L^{2}}{4 M_{0} H} = \frac{K_{0} L}{σ_{0} b H^{3}}

(13)

where $V_{0}$ is the initial velocity of the beam, and $K_{0} = {m L V}_{0}^{2}$ is the initial kinetic energy of the beam.

Several analytical methods were employed under the same assumption of the deformation mode. In the first phase, two plastic hinges, B and C, exist between the two supported points (see Figure 4(a)). Both B and C travel to the middle of the beam, and the segment BC translates with a constant velocity $V_{0}$ , while the segment AB rotates around the clamped point A. In the second phase, the two plastic hinges coalesce, and only one plastic hinge exists, which moves downwards until it stops (see Figure 4(b)).

Figure 4.

(a) First phase, (b) second phase and (c) inner forces analysis.

The expression of the continuity of velocity at the rigid segment AB is

ω ξ l = V_{0}

(14)

where $V_{0}$ is the initial velocity, $ξ l$ denotes the location of the plastic hinge B and $ω$ is the angular velocity of segment AB.

As described in section ‘Monolithic beam loaded with a concentrated force’, at the rectangular cross section, the axial force N and the bending moment M satisfy the yield condition

N^{2} / {N_{0}}^{2} + M / M_{0} - 1 = 0

(15)

To simplify the analysis, the square yield condition (circumscribing or inscribing yield curve, see dotted lines in Figure 2(b)) is adopted. With the effects of the axial force N, which prevents motion at the ends, the final displacement is much smaller compared with those without considering the axial constraint. It is assumed that the slope of segment AB $d w / d x$ would be so small that $(d w / d x)^{2} < < 1$ . The flow rule was discussed by Onat and Prager (1953)

\frac{N}{N_{0}} = \frac{N_{0} W}{4 M_{0}}

(16)

In the first phase, the equation for the angular acceleration of segment AB is as follows (Symonds and Mentel, 1958)

\frac{1}{3} m {(ξ l)}^{3} \overset{\cdot}{ω} (t) = - 2 M - N W

(17)

Together with the equations (14) to (17), the deflection in terms of the beam theory with axial constraint effects was given by Symonds and Mentel (1958) as follows

\frac{W_{f}}{H} + \frac{1}{4} {(\frac{W_{f}}{H})}^{3} = \frac{2 λ}{3}

(18)

and it is plotted as solid line in Figure 5(a). When the deflection at the centre of the beam exceeds the height of the beam H, the axial force N is a constant value $N_{0}$ , while the bending moment $M = 0$ extends throughout the beam. Symonds and Mentel (1958) proposed the plastic string theory: the beam behaved as a plastic string with tension $N_{0}$ ; therefore, the centre line should undergo extension rather than contraction. The beam had finite thickness, which means that the ratio of curvature to centre-line strain must be within a certain range. Any points of the cross section were assumed to be in tension at the yield stress. Symonds and Mentel (1958) also gave the range of the final deflection as follows

3^{\frac{1}{4}} \sqrt{\frac{λ}{6}} - \frac{2}{\sqrt{3}} < \frac{W_{f}}{H} < 3^{\frac{1}{4}} \sqrt{\frac{λ}{6}} - \frac{1}{\sqrt{3}}

(19)

Figure 5.

The relationship between $λ$ and $W_{f} / H$ for monolithic beams: (a) uniformly distributed impulsive loading and (b) struck by a mass.

The lower bound is plotted as dotted line in Figure 5(a).

In order to simplify the analysis and calculation, Chen and Yu (1993) introduced a membrane factor $f_{n}$ to substitute for the result of bending moment M and stretching force N with $f_{n} M_{0}$ in large deflections. For beams with a rectangular cross section

f_{n} = {\begin{matrix} 1 + {(\frac{W}{H})}^{2} & \frac{W}{H} \leq 1 \\ 2 (\frac{W}{H}) & \frac{W}{H} \geq 1 \end{matrix}

(20)

In the first phase of motion for a clamped beam

\int_{0}^{ξ L} ω x^{2} m d x + \int_{ξ L}^{L} V_{0} x m d x - \int_{0}^{L} V_{0} x m d x = - 2 M_{0} \int_{0}^{t} f_{n} (t) d t

(21)

In the second phase of motion

\frac{1}{3} m l^{3} \overset{\cdot}{ω} (t) = - 2 M_{0} f_{n} (t)

(22)

The relationship between non-dimensional initial energy and final displacement at the centre of the beam is as follows

{\begin{matrix} \frac{W_{f}}{H} + \frac{1}{3} {(\frac{W_{f}}{H})}^{3} = \frac{2}{3} λ & λ \leq 2 \\ \frac{W_{f}}{H} = \frac{1}{\sqrt{6}} \sqrt{4 λ - 2} & λ \geq 2 \end{matrix}

(23)

and it is plotted as dashed line in Figure 5(a). The lines in Figure 5(a) give the theoretical predictions according to equations for beams made from rigid, perfectly plastic material. The maximum permanent transverse displacements of monolithic beams subjected to blast loading experiments were reported in the three references (see the points in Figure 5(a)). When the initial energy is small, all the three theoretical methods match well with the experimental data. When the initial energy is large, the beam theory with axial constraint effects gives good estimates for beams made from Al strengthened with carbon fibre–reinforced polymer (CFRP), while the membrane factor method gives quite reasonable estimates for beams made from Al 6061-T6. However, the plastic string model overestimates the permanent transverse displacements.

Monolithic beam struck by a mass

Clamped beams struck transversely by a mass G at the mid-span were briefly illustrated by previous researchers. Non-dimensional initial energy is given by

λ = \frac{{G V}_{0}^{2} L}{2 σ_{0} b H^{3}} = \frac{K_{0} L}{σ_{0} b H^{3}}

(24)

where G is the striker mass, $V_{0}$ is the impact velocity and $K_{0} = {G V}_{0}^{2} / 2$ is the initial kinetic energy of the striker. Liu and Jones (1988) built a model to examine the transverse shear and bending response, and they presented detailed procedures to calculate the final displacement. They assumed that the square yield curve (dotted lines in Figure 2(b)) controlled the plastic yielding of the beam. Their theoretical method included the influence of geometry changes, axial restraints and bending moments, but it disregarded the transverse shear force in the yield condition (Liu and Jones, 1988; see the solid line in Figure 5(b)). They gave the final displacement as follows

\frac{W_{f}}{H} = \frac{1}{2} (- 1 + \sqrt{1 + 4 λ})

(25)

Chen and Yu (1993) also analysed a beam struck by a mass using the membrane factor method. Both bending moment and axial force joined the energy dissipation (Yu and Qiu, 2011), and the final dimensionless displacement is as follows

{\begin{matrix} \frac{W_{f}}{H} + \frac{1}{3} {(\frac{W_{f}}{H})}^{3} = \frac{2}{3} λ & λ \leq 2 \\ \frac{W_{f}}{H} = \frac{\sqrt{2 λ - 1}}{\sqrt{3}} & λ \geq 2 \end{matrix}

(26)

and it is plotted as dashed line in Figure 5(b). Experimental data (see the points) of the mass impact loading on beams made from steel and alloy (Alves and Jones, 2002; Liu and Jones, 1987; Yu and Jones, 1991) were also plotted in Figure 5(b). Liu and Jones’s (1987) prediction matches well with the experimental results of the flat Al alloy from the reference. Chen and Yu’s solution matches well with the experimental results of the flat steel and mild steel.

Failures of monolithic beams

Three distinctly different damage modes of beams subjected to uniformly distributed velocities have been noted by Menkes and Opat (1973): Mode I – large inelastic deformation; Mode II – tearing in outer fibres, at or over the support; Mode III – transverse shear failure at the support (Jones, 1976). The threshold for tearing (Mode II) is taken at first as the impulse increases intensity, and as the impulse increases, tearing and share failure overlap. They also found that the Mode II and Mode III thresholds do not depend on the length of the beam, but only on the thickness. The elementary failure criteria was proposed by Jones (1976). When the maximum total strain at the supports reaches to the fracture strain of the material under uniaxial tensile tests, tensile failure (Mode II) happens; when the maximum shear displacement reaches to the thickness of the beam, shear deformation happens. As described in the previous section, the theoretical equations were reasonably successful in predicting the dynamic response of fully clamped beams which exhibited large plastic deformation (Mode I).

Plastic hinges involves both axial strain $ε$ and curvature k. Each plastic hinge at the two ends of the rigid segment (AB and BD in Figure 3(b)) are assumed to have a length l, and the total hinge length at the centre of the beam is assumed to be 2l. Thus, the maximum total strain $ε_{m}$ at the supports is

ε_{m} = ε + H k / 2

(27)

where $ε \approx W^{2} / 4 L l$ and $k \approx W / L l$ , provided the angle is small. When the displacement is small, the plastic hinges are located near the positions $M = \pm M_{0}$ , $N = 0$ ; while as the displacement increases, the plasticity conditions of the hinges move around the yield curve from the positions $M = \pm M_{0}$ , $N = 0$ to the position $M = 0$ , $N = N_{0}$ when $W_{f} / H \geq 1$ (Jones, 1976). In order to evaluate the maximum stain $ε_{m}$ , the value length of the plastic hinge should be estimated. When the dimensionless final permanent displacement $W_{f} / H \leq 1$ , the plastic hinges at the two supports are assumed to have the length of the thickness of the beam, $l = H$ . Then, the membrane force N increases with the increasing displacement. The hinge length is simplified as an average value, $l = 3 H / 2$ for $0 \leq W / H \leq 1$ , provided the dimensionless final permanent displacement $W_{f} / H \geq 1$ . In this condition, the maximum strain at the beam supports is

ε_{m 1} = \frac{H}{2 L}

(28)

when $W / H = 1$ .

After that, the fully clamped beam behaves as a string with the membrane force $N = N_{0}$ . For $1 \leq W / H \leq W_{f} / H$ , it is assumed that the axial stretching occurs over the entire span 2L, which means that the length of the plastic hinge $l = L / 2$ , together with $ε \approx W^{2} / 4 L l$ , the maximum strain accumulated between $W / H = 1$ and $W_{f} / H$

ε_{m 2} = \frac{W_{f}^{2}}{2 L^{2}} - \frac{H^{2}}{2 L^{2}}

(29)

The total tensile strain at the supports when $W = W_{f}$ is the sum of equations (28) and (29)

ε_{m} = {(\frac{H}{2 L})}^{2} {2 {(\frac{W_{f}}{H})}^{2} + \frac{2 L}{H} - 2}

(30)

When the tensile failure strain is given, $W_{f} / H$ when failure occurs at the supports can be obtained. It can also be expressed in terms of the dimensionless initial kinetic energy $λ$ (see equations (13) and (24)). Thus, $λ$ required for the onset of failure Mode II can be estimated, which depends on the beam with given values of $L / H$ . The experimental data (Menkes and Opat, 1973) of the relationship between the dimensionless kinetic energy $λ$ and the dimensionless final displacement are plotted in Figure 6 (hollow scatters). The relationship between the threshold dimensionless kinetic energy and the corresponding dimensionless displacement for onset of failure Mode II is also plotted in Figure 6 (solid scatters). The values of $L / H$ are 21.4, 16, 10.7, 8 and 5.3, respectively, and the tensile failure strain was taken to be 0.25 for the aluminium beam.

Figure 6.

Failure estimation of monolithic beams under uniformly distributed impulsive loading.

Monolithic plates

Sufficiently large loadings can cause finite displacements, and the geometry changes must be considered. Based on the theory that the external work rate is equal to the internal energy dissipated at plastic hinges and plastic zones, the following governing equation was given by Jones (2013) for plates neglecting transverse shear effects

\begin{matrix} - G \overset{\cdot\cdot}{W} \overset{\cdot}{W} - \int_{A} μ \overset{\cdot\cdot}{w} \overset{\cdot}{w} d A + \int_{A} p \overset{\cdot}{w} d A \\ = \int_{A} {(M_{r} + w N_{r}) {\overset{\cdot}{k}}_{r} + (M_{θ} + w N_{θ}) {\overset{\cdot}{k}}_{θ}} d A \\ + \sum_{i = 1}^{n} \int_{C_{i}} (M_{r} + w N_{r}) (\partial \overset{\cdot}{w} / \partial r) d C_{i} + \sum_{j = 1}^{m} \int_{C_{j}} Q_{r} {(\overset{\cdot}{w})}_{j} d C_{j} \end{matrix}

(31)

where G is an impact mass and $μ$ is the mass per unit surface area, A is the surface area. $w$ , $\overset{\cdot}{w}$ and $\overset{\cdot\cdot}{w}$ are the transversal displacement, the associated velocity and acceleration, respectively. The first two terms on the left are the work rate due to the inertia forces of the impact mass and the plate mass, and the third term is the work rate due to the pressure pulse. On the right side, the first term is the energy dissipated in the continuous deformation fields, and the last two terms are the energy dissipated in n plastic hinges with an angular velocity $(\partial \overset{\cdot}{w} / \partial r)_{i}$ across a hinge of length $C_{i}$ and in m transversal shear hinges with a velocity discontinuity $(\overset{\cdot}{w})_{j}$ across a hinge length $C_{j}$ .

A kinematically admissible velocity profile should be assumed first. Combining with the normality requirements of plasticity and the appropriate yield condition, equation (31) derives a differential equation, which is linear under circumscribing or inscribing yield conditions (Jones, 2013). When the initial conditions are given, the solution of the governing equation provides the theoretical solution for the permanent transverse displacement. The experimental data from the references were compared with the analytical predictions. The difference between the analytical methods and experimental results might be due to the non-uniform loading, non-ideal clamped boundary condition around the plate boundary or radial slippage of the plates within the clamped supports (Jones, 2013).

Rectangular plates

Rectangular plates under uniform impulsive loading

Fully clamped rectangular plates subjected to uniformly distributed impulsive loading were investigated experimentally by many researchers. The theoretical solution is obtained by the governing equation (equation (31)). The non-dimensional initial kinetic energy is defined as follows

λ = \frac{μ V_{0}^{2} L^{2}}{σ_{0} H^{3}}

(32)

For a rectangular plate with the dimensions $2 B \times 2 L$ (2B < 2L) and an aspect ratio $β = B / L$ , Jones (2013) predicted maximum permanent transverse displacements without considering strain rate–sensitive effect

\frac{W_{f}}{H} = \frac{(3 - ζ_{0}) [{1 + 2 λ ζ_{0}^{2} (1 - ζ_{0} + 1 / (2 - ζ_{0})) / 3}^{1 / 2} - 1]}{2 {1 + (ζ_{0} - 1) (ζ_{0} - 2)}}

(33)

where $ζ_{0} = β (3 + β^{2})^{1 / 2} - β^{2}$ .

For strain rate–sensitive material, the Cowper–Symonds constitutive equation is as follows (Jones, 2013)

{σ'}_{0} / σ_{0} = 1 + {({\overset{\cdot}{ε}}_{e} / D)}^{1 / q}

(34)

where $σ'_{0}$ is the dynamic flow stress, ${\overset{\cdot}{ε}}_{e} = (λ (3 - ζ_{0})^{2} / 3 π (2 - ζ_{0})) (H / L)^{2} \sqrt{(2 σ_{0} (ζ_{0}^{2} - 3 ζ_{0} + 3) / ρ B^{2} (2 - ζ_{0}) (3 - 2 ζ_{0}))}$ is the equivalent strain rate and D and q are constants for a particular material. The value of $σ'_{0}$ is used instead of $σ_{0}$ in the dimensionless initial kinetic energy $λ$ in equation (32). For the different materials, D and q are given in Jones (1989).

Experimental data for rectangular plates applied with blast loading in Rajendran and Narasimhan (2001) and Jones et al. (1970) are plotted in Figure 7(a). Jones et al. (1970) reported experimental results of rectangular plates, which were made from mild steel or Al 6064-T6 with $β = 0.593$ , subjected to blast loadings. Rajendran and Narasimhan (2001) obtained experimental data on high-strength low-alloy (HSLA) steel rectangular plates with $β = 0.833$ . The permanent transverse displacements, without considering strain rate–sensitivity effects, are plotted, and the values are higher than the experimental data for rectangular plates with $β = 0.593$ in Figure 7(a).

Figure 7.

The relationship between $λ$ and $W_{f} / H$ for rectangular plates: (a) uniformly distributed impulsive loading and (b) struck by a mass.

Rectangular plates struck by a mass

The governing equation (equation (31)) also predicts the response of rectangular plates subjected to mass impact. The non-dimensional initial kinetic energy is defined as follows

λ = \frac{{G V}_{0}^{2}}{2 σ_{0} H^{3}} = \frac{K_{0}}{σ_{0} H^{3}}

(35)

For large impact masses (ratio of G to the plate mass >> 1), the final displacement at the centre of the plate was predicted by Jones (2013) as follows

\frac{W_{f}}{H} = \frac{(1 + n)}{2} {{[1 + \frac{4 β λ}{(1 + β^{2}) {(1 + n)}^{2}}]}^{1 / 2} - 1}

(36)

where n is a constant value for a certain boundary condition, and $0 \leq n \leq 1$ nM₀ is the bending resistance around the plate boundary. For the extreme case, the fully clamped case, $n = 1$ , and for the simply supported case, $n = 0$ . Clamped rectangular plates struck by a mass were also investigated experimentally by previous researchers. The experimental data were collected, and the analytical methods are compared in Figure 7(b). Experimental data for low-velocity impact loading on square plates made from mild steel and for high-velocity impact loading on rectangular plates made from Al alloy have been reported in Zhu et al. (1994) and Jones et al. (2008), respectively. For low-velocity impact loading case, the experimental values lie near to the prediction from equation (36). For high-velocity impact loading on rectangular plates made from Al alloy, the prediction agrees well with the experimental data, while for plates made from mild steel, the prediction overestimates the experimental data (Zhu et al., 1994).

Circular plates

Circular plates under uniform impulsive loading

For the circular plates subjected to uniformly distributed impulsive loading, the non-dimensional initial energy is as follows

λ = \frac{μ V_{0}^{2} R^{2}}{σ_{0} H^{3}}

(37)

For rigid, perfectly plastic material, Jones (2013) gave the following theoretical solution

\frac{W_{f}}{H} = \frac{(1 + n)}{2} {{[1 + \frac{8 λ}{3 {(1 + n)}^{2}}]}^{1 / 2} - 1}

(38)

For strain rate–sensitive material, the Cowper–Symonds constitutive equation (equation (34)) is employed, and for circular plates, the equivalent strain rate is ${\overset{\cdot}{ε}}_{e} = (4 \sqrt{2} V_{0}^{2} / \sqrt{3} π R) \sqrt{(ρ / σ_{0})}$ . The expression of the final displacement with the effect of strain rate sensitivity is as follows

\frac{W_{f}}{H} = \frac{(1 + n)}{2} {{[1 + \frac{8 λ}{3 {(1 + n)}^{2} {1 + {(4 \sqrt{2} V_{0}^{2} / \sqrt{3} π R) {(ρ / σ_{0})}^{1 / 2}} / D}^{\frac{1}{q}}}]}^{1 / 2} - 1}

(39)

The permanent transverse displacements of circular plates made from alloy steel, Al 6061-T6 and H.R. Steel A285, which were subjected to uniformly distributed impulsive loading, were reported in Wierzbicki and Florence (1970) and Rajendran and Narasimhan (2001). The theoretical methods are compared with experimental data in Figure 8(a). The solid line gives the analytical method result based on equation (38) for a rigid, perfectly plastic material, while the dashed line is the prediction based on equation (39) with the effect of strain rate sensitivity. For lower impulsive loading, the experimental values lie near to the prediction of rigid, perfectly plastic material, while the prediction of strain rate–sensitivity material fits the experimental data better when the input of impulsive loading is large.

Figure 8.

The relationship between $λ$ and $W_{f} / H$ for circular plates: (a) uniformly distributed impulsive loading and (b) struck by a mass.

Circular plates struck by a mass

For the rigid, perfectly plastic material, the permanent transverse displacement of a circular plate stuck at the centre by a rigid mass G is obtained (Jones, 2013) as follows

\frac{W_{f}}{H} = \frac{(1 + n)}{2} {{[1 + \frac{8 λ}{π {(1 + n)}^{2}}]}^{1 / 2} - 1}

(40)

where the non-dimensional initial kinetic energy, $λ$ , is defined as equation (35). Clamped circular plates struck by a mass were also investigated experimentally by previous researchers and the results are plotted in Figure 8(b). For the low-velocity impact case (Wen and Jones, 1993), the middle velocity impact case (Wen and Jones, 1993) and the high-velocity impact case (Tian et al., 1996), the theoretical solution perfectly fits the experimental data.

Failure of monolithic plates

The failure of mild steel, armour steel, aluminium alloy and fibre-reinforced polymer composite square plates subjected to impulsive loading at small stand-off distances were investigated experimentally by Langdon et al. (2015). They found that in their experimental scenario, the non-dimensional rupture impulse increases with the increasing specific energy to tensile fracture obtained from quasi-static tensile tests (Langdon et al., 2015). The three modes of beams subjected to uniformly distributed velocities have been modified by Wen et al. (1995) to solve the problem of a clamped circular plate subjected to uniformly distributed impulsive loading. They gave the expressions of the dimensionless initial energy required for causing the tensile tearing and transverse shear failure

λ_{t} = \frac{\sqrt{2} ε_{m}^{\frac{1}{2}}}{(\frac{H}{R})} [\frac{ε_{m}^{\frac{1}{2}}}{\sqrt{2} (\frac{H}{R})} + 1] + \frac{γ_{c}}{4 (1 + n)} [\frac{\sqrt{6} ε_{m}^{\frac{1}{2}}}{(\frac{H}{R})} + \sqrt{3}] {[\frac{\sqrt{6} ε_{m}^{\frac{1}{2}}}{2} + \frac{\sqrt{3}}{2} (\frac{H}{R})]}^{\frac{1}{n}}

(41)

where $λ_{t}$ is the dimensionless initial energy required for causing the tensile tearing failure; $ε_{m}$ and $γ_{c}$ are the tensile and shear failure strain, respectively; and n is the work-hardening index of the material

\sqrt{\frac{2}{3}} - \frac{\sqrt{2}}{2} \frac{λ^{*}}{λ_{t}} \frac{H}{R} = ε_{m}^{\frac{1}{2}}

(42)

where $λ^{*}$ is the dimensionless critical total input impulse, which causes a change in failure mode from tensile tearing to transverse shear. Comparison of theoretically predicted failure boundaries with experimental observations (Teeling-Smith and Nurick, 1991) for clamped circular mild steel plates was also investigated by Wen et al. (1995). The results are plotted in Figure 9 when $ε_{m} = 0.285$ ; $γ_{c}$ and n are taken to be the typical values of 0.8 and 0.25, respectively, for mild steel. The solid line is the boundary between the Mode I deformation and Mode II failure, and the dashed line is the boundary between the Mode II failure and Mode III failure. The solid square scatter denotes the dimensional critical input impulse that causes plate tensile tearing failure (Mode II), while the hollow circle scatter denotes the threshold for Mode III failure. The theoretical results are smaller than the experimental data, and this is possibly because the strain rate effect is not considered and the boundary conditions in the experiments conducted are not fully clamped, as noted in Teeling-Smith and Nurick (1991).

Figure 9.

Comparison of theoretically predicted failure boundaries with experimental observations for clamped circular mild steel plates subjected to uniformly distributed impulsive loading.

Sandwich structures

Quasi-static and low-velocity loading

Quasi-static loadings, such as three-point or four-point bending tests, on sandwich beams have been reviewed by Zhu et al. (2010) to investigate the quasi-static deformation mechanism. The structural failure modes for initial plastic collapse were considered. Failure modes, including face sheet compressive failure, face sheet de-bonding, indentation failure, core failure and face sheet wrinkling, depend on loading methods, material properties and geometrical shapes (Crupi and Montanini, 2007; Daniel et al., 2002). Typical failure modes of a sandwich beam under three-point bending are shown in Figure 10(a) and (b) (McCormack et al., 2001; Steeves and Fleck, 2004), respectively. The damage evolution of a typical sandwich panel with composite face sheets and an aluminium honeycomb core is shown in Figure 10(c) (Raju et al., 2008).

Figure 10.

Typical failure modes of sandwich structures: (a) sandwich beam with composite skins (Steeves and Fleck, 2004), (b) sandwich beam with metallic skins (McCormack et al., 2001) and (c) damage evolution of sandwich panels with different core thicknesses (Raju et al., 2008).

The low-velocity impact responses of sandwich structures were investigated by many researchers (Foo et al., 2008; Hazizan and Cantwell, 2003; Meo et al., 2005). Andrey et al. (2003) studied the effect of low-velocity impact on sandwich beams with foam core subjected to compression, shear and bending loadings. Serious damage occurred in the core, and there was no significant damage to the face laminates. For energy absorption use, such damage should be avoided (Andrey et al., 2003).

High-velocity loading

Ballistic tests are normally conducted by a gas gun or a drop hammer with different shapes of strikers. Ballistic limit and perforation energy are two critical parameters to evaluate the perforation resistant behaviour and capacity of energy absorption. Failure behaviour of sandwich structures under impact loading is complicated and it depends on the structural configuration, material of the face sheets and the core, the impact velocity and the shape of the indenters. The ballistic performances of the monolithic and sandwich panels with a pyramidal lattice core were almost identical in Yungwirth et al. (2008); however, their failure behaviours were different. Under high impact loading, the monolithic panels failed by ductile hole enlargement. However, only the centre of face sheet of the sandwich panel underwent this type of failure, and the distal face sheet failed by a petalling mode in their entire investigated velocity range (Yungwirth et al., 2008).

Split-Hopkinson pressure bar (SHPB) technique is usually used to investigate the behaviours of materials and structures under high strain rate. Crushing at high strain rates could significantly raise the energy absorption and resistance of the cellular cores of sandwich structures (Zhu et al., 2010). The dynamic behaviour of these cores at high strain rates were studied by many researchers. Yu et al. (2003) investigated the response of sandwich beams with an open-cell Al–foam core using the SHPB technique. A series of tests of localized, penetrating impact on sandwich panels were conducted and the results showed that membrane state fibre stretching, core compression and friction between the core and the impactor absorbed more energy than delamination, core fracture and de-bonding (Kepler, 2004).

Blast loading

A research group led by Lu (Shen et al., 2010; Zhu et al., 2008a, 2008b) have investigated different configurations of metallic sandwich panels under air blast loading by experiments, finite element (FE) analysis and theoretical method. The experimental results show that panels with thicker face sheets, a higher density core and loaded by larger charges tended to deform locally on the front face, while those with thinner skins and a low-density core and subjected to lower level shocks were prone to have global deformation (Zhu et al., 2008a). At the central area of the front face sheet, indenting or pitting occurred, and at different regions, folding damage took place in the honeycomb core. Air blast tests were also applied on curved sandwich panels with two aluminium face sheets and an aluminium foam core (Shen et al., 2010). The curvature had significant influence on panels’ failure behaviour, because wrinkling of the back sheet of the panels occurred, which was not observed in the flat sandwich panels (Shen et al., 2010). Vaziri et al. (2007) assessed the failure behaviours of steel sandwich beams with square honeycomb core and folded plate core by numerical simulations. They found that under intense impulsive loads, core would undergo extensive plastic buckling under compression or shear, and extensive buckling could promote failure in either the core or the faces. Core failures occur as core tearing or as shear fracture at welded joints; however, core failures did not necessarily imply ultimate failure since a sandwich panel could have substantial residual strength afterward (Vaziri et al., 2007).

Conclusion

Plastic hinges were employed to predict the energy absorption by monolithic structures. Based on the theory that the external work rate is equal to the internal energy dissipated at plastic hinges and plastic zones, the governing equation for the exact solution was developed. However, plastic hinges can only be used in rigid, perfectly plastic material, because for strain rate–sensitive material, infinitely large stresses are yielded due to the infinitely large strain rate at the plastic hinges. The general characteristics of the modal deformation profile should be assumed first and then the development of the equations was obtained.

For monolithic beams, many researchers have provided good predictions of the maximum permanent transverse deflections subjected to uniformly distributed impulsive loading and impact mass loading. Empirical solutions of the maximum permanent transverse displacements for monolithic plates have also been given. Based on the theory that the external work rate is equal to the internal energy dissipated at plastic hinges and plastic zones, Jones (2013) gave a governing equation for arbitrarily shaped monolithic plates with different boundary conditions and with a variety of dynamic transverse loading applied, such as rectangular impulse loading, uniformly distributed impulsive loading and impact mass loading. Parameters for strain rate–sensitive materials were also introduced to the analytical procedures. It gives credible estimates for the final permanent displacement when compared with the experimental data for steel and aluminium alloy.

Sandwich structures under quasi-static, low-velocity impact, high-velocity impact and blast loading, as well as their failure modes, were summarized. The research methodology involves experimental investigations, theoretical analyses and numerical simulations. Failure behaviour of sandwich structures under impact loading is complicated, and it depends on the structural configuration, material of the face sheets and the core and the loading conditions. Failure modes include face sheet compressive failure, face sheet de-bonding, indentation failure, core failure and face sheet wrinkling.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This work was supported by the National Natural Science Foundation of China under grant number 51578361, the National Key Research and Development Plan under grant number 2016YFC0701100 and the Tianjin Basic Research Program under grant number 15JCZDJC39900. The first two authors thank the Australian Research Council for the financial support through a Discovery Grant (grant number DP160102612).

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