A general non-dimensionless analysis framework on the plastic deformation of typical plates under repeated blast loading

Abstract

This study presents a novel and general dimensionless modeling framework for calculating the midpoint displacement of monolithic/double-layered/triple-layered circular plates and welded rectangular plates subjected to repeated blast loading, addressing a critical gap in existing blast response analyses. Unlike conventional approaches that rely on the measured impulsive load, the proposed model framework explicitly incorporates the charge mass, stand-off distance, material strain effect (via Cowper–Symonds and Johnson–Cook constitutive laws), and localized blast effect. By employing singular value decomposition to calibrate empirical coefficients against experimental data, the derived empirical models from the general modeling framework achieve high calculation accuracy across diverse structural configurations and repeated blast scenarios. The comparison with other methods also validates the effectiveness and the high accuracy of the derived empirical models. This validated general model framework offers an empirically grounded tool for structural response assessment of blast-resistant structures, significantly advancing the field by bridging empirical and theoretical methodologies.

Introduction

In recent decades, various blast conditions, including air blasts [1, 2–3], underwater explosions [4, 5], internal blasts [6, 7], and confined blasts [8], have caused considerable damage to structures. Dynamic plastic response and failure modes of thin-walled structures such as steel cylinders, pipes, and shells subjected to blast loads have become the hottest because of their numerous applications in actual engineering structures like aircraft, pressure vessels, and ships [9, 10, 11–12].

As one of the critical components of modern structures, the plate has attracted the attention of several researchers who have been analyzing its behaviors and failure modes [13]. Various plate structures like unstiffened plates [14], stiffened plates [15], circular plates [16], welded plates [17], double-layered plates [18, 19], circular sandwich plates [20, 21], triangular plates [22], armor-grade steel plates, and fiber metal laminates [23] have been explored by experiments or numerical simulation techniques. For example, Aune et al. [24] studied the structural response of thin aluminum and steel plates under blast loads by numerical simulation. Granum et al. [25] explored the effect of heat treatment on the structural response of aluminum plates with pre-cut slits. Xu et al. [26] combined the experiments and numerical techniques to analyze the failure mode of welded plates subjected to air blast. Mostofi et al. [27] studied the structural behavior of double-layered rectangular plates under a gas mixture detonation load.

Regarding the behavior of plates under internal blast, Li et al. [28] explored the effects of stand-off distance and stiffener on the deflection and failure modes of stiffened plates by experimental and numerical methods. It has been found that the deflection of the stiffened plate first decreased and then increased with the increase of stand-off distance. This is different from the deflection of the plates under air blast. The plate displacement gradually decreases with the stand-off distance increasing [16]. Li et al. [2] studied the deformation of fully clamped square plates under enclosed explosions at different stand-off distances and proposed a failure mode prediction method based on the scaled distance. Besides, a conclusion similar to the research of Li et al. [28] has been drawn. Besides, Yao's research group has conducted a couple of research works to study the dynamic response of box-shaped structures [6], the dynamic response and damage of multi-box structures [7].

In addition to the experimental and numerical simulation studies mentioned above, dimensional analysis is another commonly used method to study the structural response and failure modes for thin-walled structures under blast load. As a pioneer, Jones proposed a dimensionless number and an analytical solution to predict the large plastic deformation of fully clamped circular plates [29]. Based on Johnson's damage number, Nurick and Martin [30] proposed modified dimensionless numbers for plates subjected to uniform and local blast loads. Jacob [16] introduced a stand-off distance parameter to the dimensional number based on the research work of Nurick and Martin [30]. Mojtaba et al. [31] developed a non-dimensional analysis method on the large plastic deformation triangular plates under impulsive loading. Babaei and Mostofi [32] proposed new dimensionless numbers for the large plastic deformation prediction of circular plates under localized and uniform impulsive loading. Further, a dimensionless form for the structural response of double-layered circular plates has also been developed [18]. Regarding rectangular plates, Jacob et al. [33] proposed a dimensionless number by considering the ratio of the charge area to the target plate area. Babaei et al. [34] introduced new dimensionless numbers for the dynamic plastic response of quadrangular plates by considering the strain rate effect, the plate size, and the load types. Yao et al. [35] developed a dimensionless number by considering the charge mass and structural geometry. For the structural response prediction of steel box structures under internal blast loading, a couple of dimensionless numbers are developed by considering the explosive energy, static yield strength, and the effect of membrane forces [36, 37, 38–39].

The above-mentioned studies [30, 31, 32, 33, 34, 35, 36, 37, 38–39] mainly focus on a single blast load. Repeated blast loads are equally deserving of attention. Structural elements, such as plates and shells, are commonly seen in ships, vehicles, and building structures and may suffer from multiple explosions when traffic accidents occur. In the general form, the blast energy is dissipated by large plastic deformation and failure of structures around the contact zones. During repeated blast loading, plastic deformation can accumulate, and the permanent deflections will be characterized by the number of blasts and the severity of the loadings.

There are only a few studies about the large plastic deformation and the damage behavior of structures under repeated blast loading. Henchie et al. [40] conducted an experimental and numerical study to investigate the response of single-layered circular plates to repeated uniform blast loading up to five blasts at a given stand-off distance. The experimental and numerical results show that the midpoint displacement increment decreased as the blast number increased. Shin et al. [41] studied the accumulation of deformation of plates under repeated slamming loads. Zhu et al. [42] investigated the dynamic response of stiffened plates under repeated impacts. Rezasefat et al. [3] numerically investigated the behavior of monolithic and multi-layered circular plate configurations subjected to repeated localized impulsive loading. Ziya-Shamami et al. [43] conducted an experimental study to investigate the structural response of monolithic and multi-layered circular metallic plates subjected to repeated distributed impulsive loading up to five times. Behtaj et al. [44] conducted a combined experimental and numerical assessment of the deformation mode and failure mechanism of welded rectangular plates subjected to three successive blast loads. Regarding the dimensionless analysis, there is minimal work on the structural response of structures under repeated blast loading. In the reference [43], empirical models based on the tested impulsive loads were proposed to predict the structural response of different plate configurations. Tian et al. [45] proposed a theoretical method to predict the structural response of clamped circular plates. Further, Tian et al. [46] proposed a deep-learning method to predict damage and conduct sensitivity analysis in rectangular welded plates subjected to repeated blast loads.

Several existing studies adopted the tested impulsive loads to analyze the structural response of the plates. However, the impulsive loads need to be measured by special equipment. Some structural response analysis models also deal with the intact plates. The structural response analysis model for "imperfect" plates like welded plates has not been fully explored. Besides, the existing studies mainly consider the structural response analysis by the Cowper–Symonds constitutive equation. If the material parameter for this constitutive equation is unknown, it may cause difficulties in calculating the structural response of the target plates by only the Cowper–Symonds constitutive equation. Therefore, it is significant to propose a structural response analysis model by considering both the Cowper–Symonds and the John–Cook constitutive equations so that users can choose which one to use according to the known material parameters.

To bridge the above-mentioned gaps, the present study aims to propose a novel and general dimensionless modeling framework for calculating the midpoint displacement of monolithic/double-layered/triple-layered circular plates and welded rectangular plates subjected to repeated blast loading, addressing a critical gap in existing blast response analyses. Unlike conventional approaches that rely on the measured impulsive load, the proposed model framework explicitly incorporates the charge mass, stand-off distance, material strain effect (via Cowper–Symonds and Johnson–Cook constitutive laws), and localized blast effects. This general model framework aims to offer an empirically grounded tool for structural response assessment of blast-resistant structures. The organization of this study is as follows: Sect. 2 briefly introduces the existing experimental data used in this study. The structural response analysis models are built in Sect. 3. Section 4 presents the results of the analysis and discussion. Finally, conclusions are drawn in Sect. 5.

Experiment study review

There are a few experimental studies [40, 43, 44] about the large plastic deformation of structures under repeated blast loading. This section briefly introduces the experimental results from the existing research works. These experimental results are used in Sect. 4 to solve the unknown coefficients of the proposed structural response analysis models.

In the reference [40], clamped specimens consisting of a 2-mm- or 3-mm-thick test plate made from DOMEX 700 of yield strength 750 MPa are used. The experimental setup is schematically shown in Fig. 1. The test plate is placed between two 20-mm-thick mild steel plates, and the diameter of the exposed circular area is 106mm. These two thick steel plates are assumed to be rigid bodies, considering that no significant permanent deformation is observed during the experiment. The impulsive load is measured by the clamping rig. The explosive (PE4) varying from 5 to 40 g was donated at a given stand-off distance of 150 mm to generate the blast loading. The radius of the explosive is 17 mm and is spread evenly onto a polystyrene foam pad. The radius of the circular plate exposed to the explosive is 53 mm. For repeated blasts, the plate that experienced the blast was left in the clamping rig and reloaded with the same mass of explosive detonated at the same location.

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Fig. 1

Diagram of a horizontal ballistic pendulum with an illustration of a clamping rig [40]

In this study, 44 series of experimental results in the reference [40] are used to solve the coefficients of the structural response analysis models in Sect. 3, and they are presented in Table 3 of Appendix.

In the reference [43], experiments were conducted to investigate the response of monolithic and multi-layered circular plates under blast loading. The experimental setup is schematically shown in Fig. 2. Two types of plates are used, including aluminum (Al-1050) and steel plates (St-12). The explosives varying from 3 to 13.5 g were donated at two types of stand-off distances of 200 mm and 300 mm to generate the blast loading. The radius of the explosive is 17.5 mm. The 250 mm × 250 mm square test plates are placed between two 25-mm-thick steel clamping frames, leaving a circular exposed area with a radius equaling 50 mm. Steel plates and aluminum alloy plates with different thicknesses are used as the test plates. This study uses 73 experimental results, including 31 experimental results for double-layered plates and 42 for triple-layered plates. These experimental results are summarized in Tables 4 and 5 of Appendix.

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Fig. 2

Two-dimensional schematic diagram of the experimental test rig [43]

In the reference [44], the deformation of welded rectangular plates exposed to three successive blast loads was investigated by experiment. Three different categories of 0 W (a plate without any weld seam), 1 W (a plate with a single-weld seam), and 2 W (a plate with double-weld seams) were compared. The experimental setup is schematically shown in Fig. 3. Two types of plates are used, including aluminum and steel plates. The explosives varying from 25 to 50 g were donated at a given stand-off distance of 200 mm to generate the blast loading. The radius of the explosive is 15 mm. The 2-mm-thick rectangular plate is placed between two 25-mm-thick rear and front clamping support plates. And the exposed dimension for each test plate is $250\text{mm}\times 150\text{mm}$. These two clamping plates are assumed to be rigid members because they do not undergo any significant permanent deformation during the experiment. The explosive charge (PE4) with a radius of 15 mm is used, and the stand-off distance is 200 mm for each experimental case. In this study, 34 series of experimental results from Table 4 in the reference [44] are used. These experimental results are summarized in Table 6 of Appendix.

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Fig. 3

Schematic of the experimental setup [44]

A general dimensionless modeling framework for repeated blast loading

The general dimensionless framework

Previous empirical models are based on the tested impulsive load and the Cowper–Symonds constitutive equation [43]. Inspired by their work, this section proposes a general framework for the midpoint deflection analysis models of single-layered, double-layered, and welded rectangular plates. In our general form, Cowper–Symonds and Johnson–Cook constitutive equations are considered, respectively. Therefore, users can choose which one to use according to the known material parameters. Besides, the impulsive load is replaced by the charge mass, as the charge mass can be directly acquired. This charge-mass-based form makes it convenient to calculate the structural response without knowing the impulsive load.

The proposed general empirical form can be stated as follows.

1

$$\frac{W}{H}=f\left(\overline{\lambda },\overline{\theta },\overline{R},\overline{S},\overline{N}\right)$$where $\frac{W}{H}$ is the ratio of the displacement to thickness. $\overline{\lambda }$ represents the dimensionless number for circular or rectangular plates. For the Cowper–Symonds constitutive equation,$\overline{\theta }={\overline{\theta }}_1$ and it denotes the strain rate effect. For the Johnson–Cook constitutive equation, $\overline{\theta }=\left({\overline{\theta }}_1,{\overline{\theta }}_2\right)$, where ${\overline{\theta }}_1$ denotes the strain rate effect and ${\overline{\theta }}_2$ denotes the strain-hardening effect. $\overline{R}$ is the dimensionless load parameter. $\overline{S}$ is the dimensionless stand-off distance parameter. $\overline{N}$ is related to the number of weld seams.

For a given single-layered circular or rectangular plate subjected to single blast loading [16, 33], the original dimensionless number $\lambda$ can be expressed as follows.

2

$$\lambda =\left\{\begin{array}{cc}\frac{4I_{}^2}{{\pi }^2{R}^2{H}^4\rho {\sigma }_0}\hfill & \text{for}\text{circular}\text{plate}\\ \frac{I}{4H^2\sqrt{bl\rho {\sigma }_0}}\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$where I is the impulsive load. H is the plate thickness. $\sigma$ is the static yield strength.$\rho$ is the density of the plate. R is the radius of the circular plate. b and l are, respectively, the half-width and half-length of the rectangular plate.

As shown in Fig. 4, the previous studies [40, 43, 44] show that the impulsive loading grows linearly with the charge mass increasing. Here, for simplicity, we assume that

3

$$I=\alpha M$$where M is the charge mass, $\alpha$ is the coefficient.

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Fig. 4

The graph of impulse versus charge mass

For repeated blast loading, I in Eq. (2) is replaced by the total impulsive loads, i.e.,

4

$$I_t=I_1+I_2+\cdots +I_{n_1}\propto \alpha (M_1+M_2+\cdots +M_{n_1})=\alpha M_t={\overline{M}}_t$$where $I_t$ and $M_t$ are, respectively, the total impulsive and the total mass charge.$I_{i_1}$ and $M_{i_1}$ are, respectively, the impulsive load and the charge mass of the $i_1$ th blast. n₁ is the blast number. $\alpha$ has the same unit of $\frac{I}{M_t^{}}$.

The Cowper–Symonds constitutive equation can be stated as follows [47]:

5

$${\sigma }_d={\sigma }_0\left(1+{\left(\frac{\dot{\epsilon }}{D}\right)}^{\frac{1}{q}}\right)$$where ${\sigma }_0$ and ${\sigma }_d$ are, respectively, the static and dynamic yield strength.$D$ and $q$ are material constants. The Cowper–Symonds constitutive equation is a classical model used to describe the plastic behavior of materials under high strain rates. The model captures the strain rate-dependent strengthening effect on yield stress through a power-law formulation, where higher strain rates result in increased dynamic yield stress. It applies to moderate-to-high strain rate regimes, such as impact deformation of metals. This model neglects the influence of temperature and strain hardening.

Refer to the references [32, 34], the strain rate effect for circular and rectangular plates can be, respectively, shown as follows:

6

$${\theta }_1=\left\{\begin{array}{cc}{\left(\frac{I}{3\sqrt{2}\pi \rho R^{4}D}\right)}^{\frac{1}{q}}\hfill & \text{for}\text{circular}\text{plate}\\ {\left(\frac{I}{12\sqrt{2}\rho b^2{l}^{2}D}\right)}^{\frac{1}{q}}\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$where $\rho$ is the density of the plate. R is the radius of the circular plate. b and l are, respectively, the half width and half length of the rectangular plate.

The Johnson–Cook constitutive equation can be expressed as follows [48]:

7

$${\sigma }_d=\left({\sigma }_0+B{\epsilon }^{\text{n}}\right)\left(1+Cln\frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}\right)\left[1-{\left(\frac{T-T_r}{T_m-T_r}\right)}^m\right]$$where ${\sigma }_0$ and ${\sigma }_d$ are, respectively, the static and dynamic yield strength. n is the material parameter. B is the work-hardening modulus. C is the strain rate constant. m is the thermal softening constant.$\epsilon$ and $\dot{\epsilon }$ are, respectively, the equivalent plastic strain and strain rate. ${\dot{\epsilon }}_0$ is the reference strain rate and is a constant. $T_r$ and $T_m$ are, respectively, the reference temperature and the melting temperature. The Johnson–Cook constitutive equation incorporates strain-hardening effects, strain rate strengthening effects, and thermal softening effects. It assumes that these three mechanisms are mutually independent and express their combined influence through a multiplicative formulation, which significantly simplifies the calibration of material parameters.

According to the references [29, 45] and Taylor's formula, when the midpoint deflection reaches its maximum value, the strain can be approximately expressed as follows:

8

$$\epsilon =\left\{\begin{array}{cc}\frac{\sqrt{R^2+W^2}-R}{R}=\sqrt{1+{\frac{W}{R^2}}^2}-1\approx \frac{W^2}{2R^2}\hfill & \text{for}\text{circular}\text{plate}\\ \frac{\sqrt{l^2+b^2+W^2}-\sqrt{l^2+b^2}}{\sqrt{l^2+b^2}}=\sqrt{1+{\frac{W}{l^2+b^2}}^2}-1\approx \frac{W^2}{2\left(l^2+b^2\right)}\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$

Here, according to the reference [29, 45], the results of the first-order Taylor expansion are preserved.

Based on the treatments in the references [31, 32, 34], the strain rate effect and the strain-hardening effect for the Johnson–Cook constitutive equation can be expressed as follows:

9

$${\theta }_1=\left\{\begin{array}{cc}{\left(\frac{I}{3\sqrt{2}\pi \rho R^{4}D}\right)}^{\text{C}}\hfill & \text{for}\text{circular}\text{plate}\\ {\left(\frac{I}{12\sqrt{2}\rho b^2{l}^{2}D}\right)}^{\text{C}}\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$

10

$${\theta }_2=\left\{\begin{array}{cc}\frac{B}{{\sigma }_0}{\left(\frac{H^2}{2R^2}\right)}^n\hfill & \text{for}\text{circular}\text{plate}\\ \frac{B}{{\sigma }_0}{\left(\frac{H^2}{2\left(l^2+b^2\right)}\right)}^n\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$

Based on the treatments in the reference [43] and considering the effect of different stand-off distances, more generalized forms of the dimensionless load parameter $\overline{R}$ and the dimensionless stand-off distance parameter $\overline{S}$ for repeated blast loading are given here. Considering the physical significance of these two parameters, the plate area and structural size are adopted to replace the plate thickness in the denominator, which is used in the reference [43]. As shown in Fig. 5, $\overline{R}$ and $\overline{S}$ reflect the localized property of the blast loading. Referring to the treatments in the study [43], the sum of the ratio of the stand-off distance to the half-length of the plate and the ratio of the area of the explosive to the area of the plate is used to describe the cumulative effect of repeated blasts. This measure is empirical, and its effectiveness is verified in the reference [43]. $\overline{R}$ and $\overline{S}$ can be, respectively, expressed as follows:

11

$$\overline{R}=\left\{\begin{array}{cc}\frac{R_1^{}+R_2^{}+\cdots +R_{n_1}^{}}{R}\hfill & \text{for}\text{circular}\text{plate}\\ \frac{\pi \left(R_1^2+R_2^2+\cdots +R_{n_1}^2\right)}{4bl}\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$

12

$$\overline{S}=\left\{\begin{array}{cc}\frac{S_1+S_2+\cdots +S_{n_1}}{R}\hfill & \text{for}\text{circular}\text{plate}\\ \frac{S_1+S_2+\cdots +S_{n_1}}{l}\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$

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Fig. 5

Schematic of the load parameter and stand-off distance parameter

Finally, referring to the study in [43], a simple expression for Eq. (1) can be written as follows:

13

$$\frac{W}{H}=\left\{\begin{array}{cc}{a}_0{\overline{\lambda }}^{a_1}{\overline{\theta }}_1^{a_2}{\overline{R}}^{a_3}{\overline{S}}^{a_4}{\overline{N}}^{a_5}\hfill & \text{for}\text{Cowper - Symonds}\text{constitutive}\text{equation}\\ a_0{\overline{\lambda }}^{a_1}{\overline{\theta }}_1^{a_2}{\overline{\theta }}_2^{a_3}{\overline{R}}^{a_4}{\overline{S}}^{a_5}{\overline{N}}^{a_6}\hfill & \text{for}\text{Johnson - Cook}\text{constitutive}\text{equation}\end{array}\right)$$where $a_0,a_1,a_{2,}\dots ,a_6$ are the unknown coefficients. Due to the exponential form, $\alpha$ can be separated from $\lambda$ and $\theta _1^{}$ can be separated and be classified into $a_0$. An additional solution for $\alpha$ is unnecessary so that the total charge mass $M_t$ is directly used in the dimensionless number $\overline{\lambda }$.

$\overline{\lambda }$ is expressed as follows.

14

$$\overline{\lambda }=\left\{\begin{array}{cc}\frac{4M_t^2}{{\pi }^2{R}^2{H}^4\rho {\sigma }_0}\hfill & \text{for}\text{circular}\text{plate}\\ \frac{M_t^{}}{4H^2\sqrt{bl\rho {\sigma }_0}}\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$

In order to avoid the multiplication result being zero in Eq. (13), $\overline{N}$ is expressed as follows:

15

$$\overline{N}=N+1$$where N is the number of weld seams. Especially, N = 0 for an intact plate.

For Cowper–Symonds constitutive equation, ${\overline{\theta }}_1$ is expressed as follows.

16

$${\overline{\theta }}_1=\left\{\begin{array}{cc}{\left(\frac{M_t}{3\sqrt{2}\pi \rho R^{4}D}\right)}^{\frac{1}{q}}\hfill & \text{for}\text{circular}\text{plate}\\ {\left(\frac{M_t}{12\sqrt{2}\rho b^2{l}^{2}D}\right)}^{\frac{1}{q}}\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$

For Johnson–Cook constitutive equation, ${\overline{\theta }}_1$ is expressed as follows.

17

$${\overline{\theta }}_1=\left\{\begin{array}{cc}{\left(\frac{M_t}{3\sqrt{2}\pi \rho R^{4}D}\right)}^{\text{C}}\hfill & \text{for}\text{circular}\text{plate}\\ {\left(\frac{M_t}{12\sqrt{2}\rho b^2{l}^{2}D}\right)}^{\text{C}}\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$

If each blast in the successive blast cases has the same charge mass, Eqs. (11), (12), (14), (16), and (17) can be simplified as follows:

18

$$\overline{R}=\left\{\begin{array}{cc}{n}_1\frac{R_0}{R}\hfill & \text{for}\text{circular}\text{plate}\\ n_1\frac{\pi R_0^2}{4bl}\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$

19

$$\overline{S}=\left\{\begin{array}{cc}{n}_1\frac{S_{}^{}}{R}\hfill & \text{for}\text{circular}\text{plate}\\ n_1\frac{S_{}^{}}{l}\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$

20

$$\overline{\lambda }=\left\{\begin{array}{cc}\frac{4n_1^{2}M_{}^2}{{\pi }^2{R}^2{H}^4\rho {\sigma }_0}\hfill & \text{for}\text{circular}\text{plate}\\ \frac{n_{1}M_{}^{}}{2H^2\sqrt{bl\rho {\sigma }_0}}\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$

21

$${\overline{\theta }}_1=\left\{\begin{array}{cc}\left(\frac{n_{1}M_{}^{}}{3\sqrt{2}\pi \rho R^{4}D}\right)_{}^{\frac{1}{q}}\hfill & \text{for}\text{circular}\text{plate}\\ {\left(\frac{n_{1}M}{12\sqrt{2}\rho b^2{l}^{2}D}\right)}^{\frac{1}{q}}\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$

22

$${\overline{\theta }}_1=\left\{\begin{array}{cc}{\left(\frac{n_{1}M}{3\sqrt{2}\pi \rho R^{4}D}\right)}^{\text{C}}\hfill & \text{for}\text{circular}\text{plate}\\ {\left(\frac{n_{1}M}{12\sqrt{2}\rho b^2{l}^{2}D}\right)}^{\text{C}}\hfill & \text{for}\text{rectangular}\text{plate}\end{array}\right)$$

For multi-layered plates, $\overline{R}$,$\overline{S}$, and $\overline{N}$ remain the same as those for single-layered plates, while $\overline{\lambda }$,${\overline{\theta }}_1$,and $\theta _2^{}$ are modified as follows.

23

$$\left\{\begin{array}{c}\overline{\lambda }={\overline{\lambda }}_1{\overline{\lambda }}_2\cdots {\overline{\lambda }}_{n_2}\hfill \\ {\overline{\theta }}_1={\overline{\theta }}_{1,1}{\overline{\theta }}_{1,2}\cdots {\overline{\theta }}_{1,n_2}\hfill \\ {\overline{\theta }}_2={\overline{\theta }}_{2,1}{\overline{\theta }}_{2,2}\cdots {\overline{\theta }}_{2,n_2}\hfill \end{array}\right)$$where n₂ denotes the layers of the multi-layered plate.

Empirical models derived from the general framework

This subsection introduces the empirical models derived from the proposed general dimensionless framework in Sect. 3.1. Based on the Buckingham $\pi$-theorem [34], Eq. (13) can be rewritten as follows.

24

$${\pi }_0=a_0{\left({\pi }_1\right)}^{a_1}{\left({\pi }_2\right)}^{a_2}{\left({\pi }_3\right)}^{a_3}{\left({\pi }_4\right)}^{a_4}{\left({\pi }_5\right)}^{a_5}{\left({\pi }_6\right)}^{a_6}$$where $a_i(i=1,2,3,4,5)$ are the unknown coefficients. ${\pi }_i(i=1,2,3,4,5)$ are $\frac{W}{H}$,$\overline{\lambda }$, ${\overline{\theta }}_1$,$\theta _2^{}$,$\overline{R}$,$\overline{S}$, and $\overline{N}$ as shown in Eq. (13).

According to Eq. (24), a group of empirical models for single-layered and multi-layered circular plates and welded rectangular plates under repeated blast loading are derived and summarized in Table 1. For the double-layered and triple-layered plates, because only the material parameters for the Cowper–Symonds constitutive equation are given in the reference [43], the empirical model based on the Cowper–Symonds constitutive equation is used here. The empirical model based on the Johnson–Cook constitutive equation is built for the welded plates because only the material parameters for the Johnson–Cook constitutive equation are given in the reference [44].

Table 1. The empirical models for different plate configurations

Plate configuration		Output ${\pi }_0$	Input ${\pi }_1$	Input ${\pi }_2$	Input ${\pi }_3$	Input ${\pi }_4$	Input ${\pi }_5$	Input ${\pi }_6$
Circular plate	Single-layered	$\frac{W}{H}$	$\overline{\lambda }$	${\overline{\theta }}_1$	–	$\overline{R}$	$\overline{S}$	–
		$\frac{W}{H}$	$\overline{\lambda }$	${\overline{\theta }}_1$	${\overline{\theta }}_2$	$\overline{R}$	$\overline{S}$	–
	Double-layered	$\frac{W}{H}$	$\overline{\lambda }$	${\overline{\theta }}_1$	–	$\overline{R}$	$\overline{S}$	–
	Triple-layered	$\frac{W}{H}$	$\overline{\lambda }$	${\overline{\theta }}_1$	–	$\overline{R}$	$\overline{S}$	–
Rectangular plate	Welded	$\frac{W}{H}$	$\overline{\lambda }$	${\overline{\theta }}_1$	${\overline{\theta }}_2$	$\overline{R}$	$\overline{S}$	$\overline{N}$

Unknown coefficient solution method

This subsection introduces the unknown coefficient solution method for the empirical models derived from the proposed general dimensionless framework in Sect. 3.2.

To acquire the coefficients in Eq. (24), a natural logarithm is used [32]. Equation (24) can be rewritten as follows:

25

$$\begin{array}{cc}\hfill ln\left({\pi }_0\right)& =ln\left(a_0\right)+a_{1}ln\left({\pi }_1\right)+a_{2}ln\left({\pi }_2\right)+a_{3}ln\left({\pi }_3\right)\hfill \\ \hfill & +a_{4}ln\left({\pi }_4\right)+a_{5}ln\left({\pi }_5\right)+a_{6}ln\left({\pi }_6\right)\hfill \\ \hfill \end{array}$$

If there are m experimental results, a system of linear equations can be obtained as follows:

26

$$\mathbf{Y}=\mathbf{AX}$$where $\mathbf{A}$,$\mathbf{X}$, and $\mathbf{Y}$ can be stated as follows:

27

$$\mathbf{A}=\left[\begin{array}{ccccccc}1& ln\left({\pi }_{11}\right)& ln\left({\pi }_{12}\right)& ln\left({\pi }_{13}\right)& ln\left({\pi }_{14}\right)& ln\left({\pi }_{15}\right)& ln\left({\pi }_{16}\right)\\ 1& ln\left({\pi }_{21}\right)& ln\left({\pi }_{22}\right)& ln\left({\pi }_{23}\right)& ln\left({\pi }_{24}\right)& ln\left({\pi }_{25}\right)& ln\left({\pi }_{26}\right)\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ 1& ln\left({\pi }_{m1}\right)& ln\left({\pi }_{m2}\right)& ln\left({\pi }_{m3}\right)& ln\left({\pi }_{m4}\right)& ln\left({\pi }_{m5}\right)& ln\left({\pi }_{m6}\right)\end{array}\right]$$

28

$$\mathbf{X}={\left[\begin{array}{ccccccc}{a}_0& a_1& a_2& a_3& a_4& a_5& a_6\end{array}\right]}^{\text{T}}$$

29

$$\mathbf{Y}={\left[\begin{array}{ccccc}ln\left(\frac{W_1}{H_1}\right)& ln\left(\frac{W_2}{H_2}\right)& ln\left(\frac{W_3}{H_3}\right)& \cdots & ln\left(\frac{W_m}{H_m}\right)\end{array}\right]}^{\text{T}}$$

Equation (26) can be solved by the singular value decomposition method [43]. Further information regarding this method can be found in Ref. [32, 34].

Two existing methods for comparison

Section 4 compares the proposed structural response models with the models in the reference [43]. Therefore, the empirical models in the reference [43] are also briefly introduced here. They are stated as follows:

30

$$\frac{W}{H}=a_0{\lambda }^{a_1}{\theta }_1^{a_2}{\overline{R}}^{a_3}{\overline{S}}^{a_4}$$where $\lambda$,${\theta }_1$,$\overline{R}$, and $\overline{S}$ are expressed as follows:

31

$$\left\{\begin{array}{c}\lambda =\frac{4n_1\rho V_0^2}{{\sigma }_0}{\left(\frac{R}{H}\right)}^2\hfill \\ {\theta }_1={\left(\frac{I}{3\sqrt{2}\pi \rho R^{4}D}\right)}^{\frac{1}{q}}\hfill \\ \overline{R}=n_1\frac{R_0}{H_t}\hfill \\ \overline{S}=n_1\frac{S_{}^{}}{H_t}\hfill \end{array}\right)$$

Comparing Eq. (33) with Eqs. (18)–(21), there are three different points. The first is the impulsive loading replaced by the charge mass. The second is the denominator of $\overline{R}$ and $\overline{S}$. They are the plate's radius or length, while in the reference [43], they are the plate thickness. In our proposed general form, the plate's radius or length is used because $\overline{R}$ and $\overline{S}$ are used to describe the localized property of the blast loading like the reference [16]. The third is the strain effect, which can be considered by both the Cowper–Symonds and Johnson–Cook constitutive equations in our general form, while in the reference, the strain effect is considered mainly by the Cowper–Symonds constitutive equation.

In subSect. 4.1, the theoretical method in the reference [45] is also used for comparison. This method is briefly introduced here.

32

$${\left(\frac{W}{H}\right)}_i=2i\xi m{\left(\frac{W}{H}\right)}_{i-1}+\frac{{(1+2{\lambda }_{sy,i}/3)}^{\frac{1}{2}}-1}{2}$$where the subscript $i$ represents the $i$th blast. ${\left(\frac{W}{H}\right)}_i$ is the ratio of the displacement to thickness for the $i$th blast. m is the charge mass. $\xi$ is the modified coefficient.${\lambda }_{sy,i}$ is the dimensionless number for the $i$th blast. Much more details can be found in the reference [48].

Results and discussion

In this section, the unknown coefficients of the empirical models in Table 1 are solved by the experimental results summarized in Sect. 2. Then, the calculated results by the proposed empirical models are compared with the experiment results and other existing methods. The unknown coefficients' values in Eq. (26) are solved by the method shown in Eq. (27). The calculated results are given in Table 2.

Table 2. The calculated results for unknown coefficients in Eq. (26)

Plate configuration		Output $a_0$	Input $a_1$	Input $a_2$	Input $a_3$	Input $a_4$	Input $a_5$	Input $a_6$
Circular plate	Single-layered	4.5730	0.7501	− 1.7494	–	− 2.0976	1.2124	–
		2.1430	0.5744	0.1111	− 1.3363	− 0.6042	1.0554	–
	Double-layered	3472.3	0.2540	− 0.0345	–	1.1836	− 1.8915	–
	Triple-layered	0.1663	0.2278	− 0.0974	–	1.1381	− 2.1121	–
Rectangular plate	Welded	1.1505	0.7779	0.0124	− 0.6766	− 0.2686	0.3541	− 0.1080

Single-layered circular plate

The experimental results for the single-layered circular plates under repeated blast loading in Table 3 of Appendix are used. The material parameters for the Cowper–Symonds constitutive equation are listed below: ${\sigma }_0=750\text{MPa}$, $q=5$, $D=40.4{\text{s}}^{-1}$, $\rho =7800{\text{kg/m}}^3$. The material parameters for the Johnson–Cook constitutive equation are listed below: ${\sigma }_0=750\text{MPa}$, $B=270.6\text{Mpa}$,$n=0.263$, ${\dot{\epsilon }}_0=0.001{\text{s}}^{-1}$,$C=0.014$. In Table 1, the first line for a single-layered circular plate is the empirical model based on the Cowper–Symonds constitutive equation, while the second line for a single-layered circular plate is the empirical model based on the Johnson–Cook constitutive equation. The former is referred to as the C–S model, and the latter is referred to as the J–C model. The other empirical method in the reference [43] is also adopted for comparison. It is referred to as Ref-43.

Here, a mean relative error (MRE) is defined to compare the accuracy of each method. MRE is given by

33

$$\text{MRE}=\sum _{i=1}^m\frac{\left|{\left(W\left./\phantom{WH}\right)H\right)}_{p,i}-{\left(W\left./\phantom{WH}\right)H\right)}_{e,i}\right|}{{\left(W\left./\phantom{WH}\right)H\right)}_{e,i}}$$

The values of the unknown coefficients are given in Table 2. The calculated results for the proposed method and the method in the reference [45] are presented in Fig. 6. It is referred to as Ref-45.

[See PDF for image]

Fig. 6

The calculated results for single-layered circular plate

Figure 7 gives the calculated results for 3-mm-thick circular plates by our proposed C–S model, Ref-43 and Ref. 45. Figure 8 presents the calculated results by the Johnson–Cook constitutive equation, and they are compared with the calculated results by the Johnson–Cook constitutive equation. Figure 9 compares the calculated results by these two constitutive equations with and without considering the strain effect. In Figs. 6, 8, and 9, the 1:1 line with a slope of 1 is where the calculated results are the same as the experimental results. Also, two black dotted lines describing the $\pm 10\%$ error are plotted to assess the accuracy of the empirical model.

[See PDF for image]

Fig. 7

Result comparison for 3-mm-thick single-layered circular plate

[See PDF for image]

Fig. 8

The calculated results for the J–C model and result comparison between the C–S model and the J–C model

[See PDF for image]

Fig. 9

Result comparison for the case where strain effect is considered or not

According to the results in Fig. 6, MREs for these two methods are, respectively, 9.24% (our C–S model) and 8.7% [43]. The lines in Fig. 7 show that the increasing trend for the results from the derived empirical model, the two existing methods, and the experiment is consistent. MRE for the results in Fig. 7 is, respectively, 2.33% [45], 6.35% (Our C–S model), and 5.23% [43]. Although the MRE values for the other two methods are smaller than ours, our method does not require an impulsive load. For the other two methods, special equipment is required to measure the impulsive load, as shown in Figs. 1, 2, and 3. Therefore, it is hard to predict the plastic deformation before the explosion for the other two methods. However, our method directly builds the relationship between the charge mass and the structural response of the target plates. This gives it the advantage of predicting the plastic deformation of the target plates without any special equipment to measure the impulsive load. Besides, the theoretical method in the reference [45] (e.g., Ref-45) is especially suitable for single-layered circular plates and not for other plates. The proposed modeling framework in Sect. 3.1 can produce a couple of empirical models for different plate configurations. The proposed modeling framework is more universally adaptive.

Figure 8b shows that the calculated results by both the C–S and the J–C models are very similar to each other. According to Fig. 9, it is evident that considering the strain effect can obtain better calculated results. Besides, most of the calculated results are within the error line range in their figures. Based on the analysis conducted, it can be inferred that the proposed empirical models considering the strain effect, the charge mass, and two constitutive equations can be expected to acquire good calculated results for the structural response of single-layered plates under repeated blast loading.

Double-layered circular plate

This subsection uses 31 series of experimental results for double-layered plates in Table 4 of Appendix [43]. The plate configurations for double-layered circular plates are shown in Fig. 10. In Fig. 10, D represents the double-layered configuration, S and A, respectively, denote the steel material and aluminum material, and the number behind the material is the plate thickness (33 means the thickness of each plate is 3 mm), 200 and 300 are the stand-off distances. Much more details can be found in the reference [43]. The material parameters for steel material are listed below: ${\sigma }_0=350\text{MPa}$, $q=5$, $D=40.4{\text{s}}^{-1}$, $\rho =7850{\text{kg/m}}^3$. The material parameters for aluminum material are listed below: ${\sigma }_0=110\text{MPa}$, $q=4$, $D=6500{\text{s}}^{-1}$, $\rho =2700{\text{kg/m}}^3$.

[See PDF for image]

Fig. 10

The plate configurations for double-layered circular plates

The values of the unknown coefficients are given in Table 2.

The calculated results by our C–S model and Ref-44 are given in Fig. 11. Results comparison for 3-mm-thick double-layered plates are presented in Fig. 12. Similarly, to conduct a comprehensive comparison of the model's accuracy, MRE values for these two methods are 1.85% (our C–S model) and 2.02% [43] through the results in Fig. 11. Also, all calculated results are located in the range of $\pm 10\%$ error lines for our C–S model. For a given repeated blast case, the increasing trend for the results from the derived empirical model, the method in the reference [43], and the experiment is also consistent. Hence, the proposed C–S model can reasonably calculate the structural response of double-layered circular plates.

[See PDF for image]

Fig. 11

The calculated results for the double-layered plate

[See PDF for image]

Fig. 12

Result comparison for 3-mm-thick double-layered circular plate

[See PDF for image]

Fig. 13

The plate configurations for triple-layered circular plates

Triple-layered circular plate

In this subsection, 42 series of experimental results for triple-layered plates in Table 5 of Appendix are used. The plate configurations for triple-layered circular plates are shown in Fig. 13. In Fig. 13, T represents the triple-layered configuration, TM represents the triple-layered mixed configuration, S and A, respectively, denote the steel material and aluminum material, the number behind the material is the plate thickness (333 means the thickness of each plate is 3 mm), 200 and 300 are the stand-off distances. Much more details can be found in the reference [43]. The material parameters are the same as those in subSect. 4.2.

The calculated results for the two models are shown in Fig. 14. The comparison for four plate configurations is given in Fig. 15. For different plate configurations, the increasing trend for the results from the derived empirical model, the method in the reference [43], and the experiment is identical. MRE values for these two methods are 5.91% (our C–S model) and 8.11% [43]. Most of the calculated results are located in the range of $\pm 10\%$ error lines for our C–S model. The MRE value of our C–S model is smaller than that of Ref-43. Therefore, the proposed C–S model can well calculate the structural response of triple-layered circular plates.

[See PDF for image]

Fig. 14

The calculated results for the triple-layered circular plate

[See PDF for image]

Fig. 15

Result comparison for different plate configurations

Welded rectangular plate

In this subsection, 34 series of experimental results for welded rectangular plates in Table 6 of Appendix are used. The plate configurations for triple-layered circular plates are shown in Fig. 16. In Fig. 16, three different categories of 0 W (a plate without any weld seam), 1 W (a plate with a single-weld seam), and 2 W (a plate with double-weld seams) are presented. The material parameters for the Johnson–Cook constitutive equation are listed below: ${\sigma }_0=350\text{MPa}$, $B=789\text{Mpa}$, $n=0.830$, ${\dot{\epsilon }}_0=1{\text{s}}^{-1}$,$C=0.022$. Much more details can be found in the reference [44].

[See PDF for image]

Fig. 16

The plate configuration

The calculated results by the proposed J–C empirical model are shown in Fig. 17. Results comparison for four plate configurations are given in Fig. 18. Clearly, all calculated results are near the 1:1 line and are located in the range of $\pm 10\%$ error lines. The MRE value of our J–C model is 1.30%. For different welded numbers, the increasing trend for the results from the derived empirical model, the method in the reference [43], and the experiment is identical. It can be inferred that the proposed J–C model can calculate the structural response of welded rectangular plates very well.

[See PDF for image]

Fig. 17

The calculated results for the welded rectangular plate by our J–C model

[See PDF for image]

Fig. 18

Result comparison for different charge masses and welded numbers

Conclusion

This study proposes a general non-dimensionless analysis framework on the plastic deformation of typical plates under repeated blast loading. A couple of empirical structural response analysis models are derived from this general framework. The calculated accuracy of these empirical models is validated by comparison with the experimental results and other existing methods. Some conclusions can be drawn as follows.

1. Comparison with the other two existing methods and the experimental results shows that the derived empirical models can calculate the structural response of single-layered, double-layered, triple-layered circular, and welded rectangular plates well without measuring the impulsive load.

2. For a given repeated blast, the calculated results based on the derived empirical models increase with the blast number. This observation is consistent with the experimental case.

3. The proposed framework has the advantage of predicting the plastic deformation of the target plates without any special equipment to measure the impulsive load. For the other two existing methods, the impulsive load must be acquired by experimental equipment so that the structural response can be calculated by their models.

4. The proposed modeling framework can produce a couple of empirical models for different plate configurations, while the method in the reference [45] is especially suitable for single-layered circular plates and not for other plates. The proposed modeling framework is more universally adaptive.

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 12472394).

Declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

João Marciano Laredo dos Reis

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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