Analysis of the Deformation and Damage of Clamped Square Plates under Near-Field Explosion Loads

*e deformation and damage mechanism of shell structures under near-field explosion loads has been of great significance in the theoretical study of impact dynamics andmay serve as a dependable theoretic basis for the antiexplosion design of shell structures. In this paper, the plastic zone and crevasse size of clamped square plates under near-field explosion loads were discussed based on the plastic hinge law and energy theory. *e crevasse size of a plate moving at motion modes Ι and ΙΙ under medium load was obtained according to the ultimate plastic strain criterion. Furthermore, the plastic zone under a high load was determined in terms of the movement law of a plastic hinge line.When the applied load ended, the crevasse sizes of the plates at motionmodes III and IV were deduced on the basis of the principle of energy conservation. Finally, numerical simulation was used to analyse the deformation and damage mechanism of the shell structures under near-field explosion loads. *e theory and method proposed in this paper are verified using ANSYS software and compared with the experimental results. *is study verifies the validity of the proposed approach for analysis of the deformation and damage of a clamped square plate under near-field explosion loads.


Introduction
Shell structures are widely used in naval vessels, aircraft, and tanks due to their special properties, such as simple construction, being light weight, and good bearing capacity [1][2][3][4].However, shell structures are often subjected to explosion loads or shock waves in practical applications [5][6][7].It is necessary to investigate the plastic dynamic response, residual deformation, and damage of the thin plate subjected to an explosion load.
Generally, the dynamic response of thin plates under explosion loads depends on the applied load, material properties, and boundary conditions [8][9][10].Considerable efforts have been made to investigate the dynamic response of materials under explosion loads.Houlston and Desrochers [11] calculated the dynamic plastic response of a clamped square plate subjected to blast loading by applying the ADINA procedure.Spranghers et al. [12] proposed an inverse method that uses full-field optical measurements taken during the first milliseconds of a free air explosion to identify the plastic response of aluminum plates subjected to sudden blast loads, and the results verify that the proposed methods can be successfully used to analyse the plastic behaviour of metals subjected to blast waves.Breslavsky et al. [13] experimentally and numerically studied the deformation and fracture of impact-loaded thin steel square plates and discussed the numerical-experimental approach for fracture prediction in plates under repetitive impact loading.Zheng et al. [14] investigated the large deflection behaviour of clamped stiffened plates subjected to confined blast loads through experiments, theoretical analysis, and numerical simulations; it was proven that the proposed calculation model is of high precision and practicability by comparing the numerical results.Li et al. [15,16] studied the blast response of plates with preformed holes under blast loading; furthermore, the author discussed the damage level at places where cracks might form, the dynamic stress concentration coefficient, and stress status change and analysed the influence of preformed holes on the failure mode.Kazanci and Mecitoglu [17] used the virtual work principle to analyse and discuss the nonlinear dynamic response of a laminated composite plate subjected to blast load, and finite element results in the literature show good agreement with their approximate numerical analysis results.Yao et al. [18] developed a new dimensionless number that takes into account the influence of blast load, strength of material, and structural dimensions for the dynamic response of box-shaped structures subjected to internal blast loading based on the governing equations of a fully clamped plate.However, a theoretical analysis on the deformation and damage of clamped square plates under near-field explosion load is still limited and requires further study.
In this paper, the plastic zone and crevasse size of a clamped square plate with different types of motion modes are discussed under medium and high loads.After theoretical analysis, the deformation and damage of the plate under the action of near-field explosion load are simulated.
e numerical simulation results are in good agreement with the theoretical analysis results, and the parameter correctness of various material models and state equations has been verified.ese studies have laid a foundation for the numerical simulation of explosion problems and provide theoretical guidance for practical applications.

Explosion Load Simplification.
e explosion load can be simplified to a triangle load with an ascent stage, as shown in Figure 1, and can be defined as follows: where p m is the peak pressure of the explosion load, t m is the time required for the load to reach the peak value, and t s is the duration of the applied load.

Yield Criterion.
To simplify the calculations, the yield curve in a square shape plate (Figure 2) is used.It is the expansion of the Tresca hexagonal yield curve, which can be defined as follows: (2) In (2), h is the half thickness of the plate.e length and thickness of the clamped square plate are 2l and 2h, respectively.e near-field explosion loads are applied to the square plate uniformly with a value of p. e coordinate system selected is shown in Figure 3, and a new coordinate is proposed defined as follows: e ultimate pressure of the plate subjected to a horizontal, static, uniformly distributed load is p 0 � 12M s /l 2 .Under a moderate load of p 0 < p m < p b � 2p 0 , the moving hinge line will not appear in the plate, and the corresponding velocity field is in the form of a tetrahedral pyramid.Under a high load of p m > p b � 2p 0 , the moving hinge line will occur in the plate with a hexahedron velocity field.
Under moderate load, the square plates move in the form of a tetrahedral pyramid. is movement means that when p 0 < p m ≤ p mb1 , at the time of t � t 0 , the movement of the deformable mechanism with a pyramid begins, the velocity of the plate increases first and then decreases, but it stops before the end of the explosive load.While p mb1 < p m ≤ 2p 0 , at the end of the load, the plate will continue to move because the speed of the plate is not zero; under the action of inertia, the velocity gradually drops to zero, and the plate is static.
Under high load, at first, the velocity field is in the form of a tetrahedral pyramid, the moving hinge line appears in the plate when the load suddenly disappears, and the hinge line moves to the centre of the plate; thus, the top platform will gradually shrink until it vanishes.

2
Shock and Vibration

Plastic Deformation and Crevasse Size of the Square Plate under a Medium Load
Under the load of p 0 < p m < p b , an immobile plastic hinge line appears along the four edges and diagonals of the plate, whereas the remaining area of the plate is rigid.e deformation of the plate is shown in Figure 4. e square plate cracks first in the middle section, and then, the plastic hinge lines located at the diagonals of the plate continue to crack, finally leaving a cracked hole on the plate.When the plate stops moving, the deflection does not increase, and the cracks do not propagate.Hence, the plastic strain is not greater than the ultimate plastic strain of a point on the plastic hinge line.e point located at the outer edge of the crevasse can further determine the size of the crevasse.
On the plastic hinge line OQ, the stretching effect along the y direction can be ignored because the crevasse occurs in the middle of the plate.As a result, the plastic strain of each point at the plastic hinge line consists of the tensile strain caused by transverse displacement and the flexural strain caused by bending.
Considering a point on OQ with a y distance to origin, the tensile strain ε xt as a result of the transverse displacement can be calculated.
e calculated deformation diagram is shown in Figure 5.
As shown in Figure 5, the distance of the selected point with the y distance to the origin point becomes y 1 after the load is applied.us, e transverse tensile strain in the x direction is calculated as follows: e calculated diagram of bending strain is shown in Figure 6. e flexural strain of a point at the plastic hinge line is calculated as follows: where κ x is the bending curvature of OQ and can be defined by the following formula: where ϕ 1 is the relative angle of the plate on both sides of the plastic hinge and C is the width of the bending deformation zone.e value of 4h was assigned to C in this paper [19].
Using the Law of Cosines, in ΔOCD,

Shock and Vibration
Hence, the bending strain can be expressed as follows: where OO ′ is the residual deflection of the plate centre.
According to [11], the deflection of the plate centre point can be expressed as follows: en, the total plastic strain at this point is the superposition of tensile strain and bending strain: When the plastic strain of one point in the plastic hinge line reaches its limiting value, the plate stops cracking and the crevasse size reaches a maximum: 3.1.Crevasse Size of the Plate at Motion Mode Ι.Under the applied load of p 0 < p m < p mb1 , p mb1 is the load when the velocity of the plate is zero, p mb1 � 2p 0 − (p 0 t 0 /t s ) [20], the plate is at motion mode Ι. us, its motion is restricted to only one phase, that is, the impact load and inertial force are responsible for the plate motion.e plate will stop moving when these loads become zero.
According to [20], the final residual deflection of the plate centre can be calculated as follows: where t d is the time at the end of the plate movement, t s is the explosion load duration, t m is the time when the load reaches the peak pressure, and t 0 refers to the time when plate moves as a pentahedron, which can be defined by t 0 � p 0 t m /p m .By substituting ( 13) into ( 12), the crevasse size at motion mode Ι can be obtained as (14).

Crevasse Size of the Plate at Motion Mode ΙΙ.
Under the explosion load of p mb1 < p m ≤ 2p 0 , the plate motion at motion mode ΙΙ has two phases.e first phase is when the plate moves under the impact load and inertial force from time t 0 to t s .e second phase is when the plate moves under the inertia force from the time t s to t d and stops moving at t d .
e deflection at the end of the first phase satisfies the idea of the plate at motion mode Ι and can be obtained according to (10) as (15).
en, the crevasse size of the plate can be calculated as (16).
When t � t s , the impact load value becomes zero.After that, the plate continues to move under an inertial force and crevasse continues to expand until the motion ends at (17).
e final residual deflection of the plate at motion mode ΙΙ can be calculated as (18).
Finally, the maximum crevasse size can be determined as ( 19) 4 Shock and Vibration

Plastic Deformation and Crevasse Size of a Square Plate under a High Load
Let p(t b ) � p b and p(t m ) � p m , then t m > t b when p m > p b .When t 0 < t ≤ t b , the explosion load is the medium load, and the plastic zone and crevasse size can be obtained using the relevant functions presented in the previous chapter.When t > t b , a plastic platform that changes with the applied load appears in the plate, which results in two different plastic zones, as shown in Figure 7. Zone _ is the plastic platform expressed for 0 ≤ z ≤ φ, where φ is a straight line paralleled to PQ to define the limits of zone _.Zone __ is a rigid deformation zone divided by the diagonal plastic hinges and can be expressed for φ ≤ z ≤ 1.

Plastic Zone of the Plate under a High
Load. e plastic zone of the plate is determined when the moving hinge line reaches its limited state.It is assumed that the position of the moving hinge line is φ c at the time t c .e plastic state of zone _ is defined as follows: e plastic state of zone __ is defined as follows: where f(z, t) ≤ 0. Apparently, the boundary condition and speed continuity condition are both satisfied by the velocity field in the above functions.In addition, the plastic hinge line is plastic since zv/zα is discontinuous at z � φ.
Substituting t � t b � 2t 0 into (10) and its first derivative, the initial velocity and initial displacement conditions of the plate movement under a high load can be obtained: Considering the initial force, the equations of motion of the plate can be calculated as follows: Substituting (20) and the initial conditions in ( 22) and (23) into the above equations of motion, the velocity and displacement of zone _ can be calculated as follows: Similarly, substituting ( 21) into (24), the differential equation of f(z, t) can be obtained as (27).
Hence, the general solution of f(z, t) is expressed as (28): Shock and Vibration e integration constants C 3 and C 4 can be calculated according to the initial force continuity of the plastic hinge line z � θ(t): Considering the clamped boundary condition Substituting the analytic expression of f(z) into the above equations, Here, substituting (25) into the expression of A, Consequently, combining (1) and ( 25) into (32), the expression for φ � φ(t) can be obtained to determine the size of the plastic zone.
e expression of φ(t) should be calculated based on different stages since the explosion load is simplified into the triangular load with an ascent stage.
First, considering the ascent stage at 0 ≤ t ≤ t m , Substituting ( 33) into (32), we get the following: e initial condition of the above equation is φ(t)| t�2t 0 � 0, and the solutions of (33) that satisfy 0 ≤ φ(t) ≤ 1 can be calculated with the assistance of the Maple software package.
e moving law of the plastic hinge line at the ascent stage of the simplified explosion load is obtained.e firstorder derivative is equal to the first function in (35), which represents the moving velocity of the moving hinge.Furthermore, the plastic zone can be determined if we assign a value of zero to the velocity of the moving hinge line.However, the velocity of the moving hinge has not reached zero at t � t m .us, it is necessary to discuss the following:  Shock and Vibration e condition of t ≥ t m since the plastic deformation at this time has not reached its maximum value.
e moving law of the plastic hinge line during the entire applied explosion load progress has been determined.To investigate the maximum plastic zone, the time required for the plastic hinge line velocity to become zero should be determined.en, substituting the time value into (31), we get the following: e time when the moving hinge line velocity becomes zero can be calculated by substituting the second expression of the simplified explosion load and (38) into (40): e plastic zone can be determined as (42) by substituting (41) into (38):

Crevasse Size of the Square Plate under a High Load.
Under a high load of p m > p b , there are two different motion zones in the plate.e plate is at a plastic state in zone Ι and the central plate is damaged when the vertical displacement reaches a certain value.As the vertical displacement increases, the cracks continue to propagate along the diagonal line of the plate forming four rigid moving pieces.e motion energy of each piece transforms into bending energy to rotate around the plastic hinge line and fracture energy for the crack expansion.e crevasse size reaches its maximum value when the plastic hinge line expands to its maximum.e four fractured pieces [21] are shown in Figure 8. e angle of AOB is π/2, and OA represents the crack length.AB Shock and Vibration refers to the plastic hinge line at the root of the fractured pieces, which moves towards the boundary when the crack OA expands along the y-axis.e moving velocity of AB can be expressed as _ L, and L shows the triangle height of AB.At this moment, the triangular piece OAB warps, as shown in Figure 9. e curve OR represents the deformation of piece OAB and can be regarded as an arc with a circular centre at P and radius R. e moving velocity of the plate in zone Ι can be calculated with (27).erefore, the energy of motion of the piece OAB can be expressed as follows: where l AB is the length of AB and ϕ 2 represents the rotating angle of AB. e variation ratio of the bending energy with time can be calculated as follows: Here, R refers to the radius of the piece and can be expressed as R � 0.8L 0.6 h 0.4 and ψ is the angle of the triangular piece with ψ � π/4.e variation ratio of the fracture energy of the plate can be calculated as [8]: where δ h has the value of 1 and Y refers to the length of the boundary to the side of the crevasse, which is in a direct line with L. e plate in zone Ι is in a plastic state under a high load.e energy of motion of each fracture piece will transform into both bending energy and fracture energy: where t s refers to the moment when the moving hinge line at the root of the fracture piece begins to move (t s � 2t 0 under a high load) and t f is the moment when the hinge line stops moving (t f � t c ).
Consequently, the crevasse size can be obtained as follows:

Crevasse Size of the Plate at Motion Mode ΙΙΙ.
Under the explosion load of 2p 0 < p m < p mb2 , where p mb2 � 4p 0 − 4p 0 t 0 /t s , the plate is at motion mode ΙΙΙ.When the applied explosion load becomes zero at t � t s , the velocity of the plate in zone Ι can be expressed as follows: Geometrically, When the applied load is over, the motion energy of each fracture piece can be obtained: Substituting (50) into (45) and combing the values of ψ and R, the equation of the variation ratio of the bending energy can be simplified as follows: Similarly, the equation of the variation ratio of the fracture energy can be simplified as follows:  When the explosion load becomes zero at t � t s , the velocity of the plate in zone I can be calculated as follows [11]: Combing ( 43), (49), and (55) and t � t s , the motion energy of each fracture piece at the end of the explosion load can be calculated as follows: Substituting ( 52), (53), and (56) with t s � 2t 0 and t f � t c into (48), the crevasse size of the plate at motion mode IV can be obtained:

Comparison Results between Experiment and eoretical
Analysis.To verify the theoretical analysis of the plastic zone of a clamped square plate, the experimental results from the literature are considered [22].e comparison results are listed in Table 1.
In the experiment, the slit scanning method of optical measurement technology was used to photograph the deformation process of whole test plate.
e size of the experimental plates is shown in Table 1 and experimental installation is shown in Figure 10.A symmetrical set of coordinates is drawn on the symmetry line of the test plate to make an angle between the plate and the horizontal plane.
When the plate is deformed, the trajectories of the coordinate points varying with the deformation of the test plate are captured by a high-speed camera, by which the displacement and velocity fields of the test plate can be obtained.
As shown in Table 1, when the plastic hinge line moves to the plate centre, the plastic platform size from the theoretical results is relatively larger than that of the experimental results; however, the errors are quite small when the plastic hinge is located away from the plate centre.Overall, the theoretical results in this paper and the experimental results are nearly identical.

Material Model.
LS-DYNA software is a general purpose nonlinear dynamic analysis finite element programme, which is mainly explicit and supplemented by implicit; it is especially suitable for coping with nonlinear dynamic impact problems such as high-speed collisions, explosion, and metal forming of various 2D and 3D nonlinear structures.Meanwhile, heat transfer, fluid, and fluid-solid coupling problems also exist.erefore, 8 groups of numerical simulation analyses were carried out by ANSYS/LS-DYNA to compare the results.
In the simulation, the explosive was a spherical TNT charge, and the plate was made of low-carbon steel with four clamped sides.e air was assumed to be an ideal gas with the size of 60 cm × 60 cm × 0.3 cm.Considering the symmetry of the question, a quarter of the plate and air model was created.e explosive, air, and steel plates were all simulated with unit SOLID164.e model HIGH_EXPLOSIVE_BURN and the JWL equation of state were employed for the explosive: where p is the explosion pressure; A, B, R 1 , R 2 , and ω are the material constants relating to the type of explosive; V is the relative volume; and E 0 is the internal energy per unit volume.e LINEAR_POLYNOMIAL model was employed for the air: For ideal air, c is the ideal gas isentropic adiabatic exponent, ρ is the density of the air, and E 0 is the internal energy per unit volume.

Shock and Vibration
Steel plates are typically made of ordinary low-carbon steel; because the explosive impact load reaches great pressure in a very short period of time, the influence of strain rate and reinforcement of structural materials must be considered.
e nonlinear plasticity model PLASTIC KINEMATIC model material is adopted for consideration of the strain rate effect.e material model is based on the relation of Cowper-Symonds: where σ 0 is the initial yield strength, C and P are the material constants, and β is the enhancement parameter, which varies from 0 to 1.When β � 0, the model is the plastic follow-up strengthening model; when β � 1, the model is the isotropic reinforcement model, and E p is the plastic reinforcement modulus; E is the modulus of elasticity and E t is the tangent modulus.

Meshing of the Model.
After the establishment of the geometric model, the next step is to mesh the geometric model.e problems studied in this paper have the characteristics of high speed and large deformation, so highquality grids are required.erefore, when meshing, the body is first cut to ensure the satisfaction of the mapping grid conditions.e mapping grid is then used to divide the cells into hexahedrons.e finite element model after meshing is shown in Figure 11.
e size of the air grid around the explosives is smaller during the meshing process because of the drastic changes in the explosives and air.e steel plate also uses a fine meshing size due to the effect of the explosion shock wave.To acquire the appropriate grid size, the sensitivity of the mesh size was studied for a plate with a 2.0 cm charge radius and 0.054 kg dynamite, and the influence of grid size on the size precision of the fracture is shown in Table 2.
According to the relative error of the four experiments, when the mesh size is 1.5 cm and 2.0 cm, the simulation value is closer to the theoretical value.In addition, considering that too fine a mesh will result in long computing time, the mesh size of 2.0 cm is finally used and the air grid away from explosives is relatively large.Explosives and air are modelled by common joints, while steel and air are modelled by noncommon joints.
e fluid-solid coupling is then defined to apply explosive loads to the steel plate.In addition, unreflected boundary conditions are applied at the air boundary to avoid the propagation law of the air shock wave affected by the reflection of the shock wave at the air boundary.
e way to apply the no reflection boundary condition method is to choose the outer surface of the air model and select the nodes on all nodes which is defined as a group; the no reflection boundary condition on the node group is defined.

Comparison between the Numerical Simulation and
eoretical Analysis.To analyse the characteristics of the spherical charge near the field explosion load and the peak overpressure of blast wave, a numerical simulation of two different loading radii were carried out.e radius of the load on working condition 1 is 3.5 cm and the loading amount is 0.29 kg.
e radius of the load on working condition 2 is 7.0 cm and the loading quantity is 2.30 kg.
e propagation process of shock waves generated by explosive explosion in the air is shown in Figure 12.A blast wave caused by a spherical charge exploding in the air is distributed uniformly in space, and it propagates in all directions in the form of spherical waves.It is similar to the propagation in the free air field before it hits the steel plate.
is outcome is in line with the basic theory of explosions, which verifies the accuracy of the numerical simulation of explosions in this paper.
According to the theoretical analysis shown in Sections 3 and 4, the explosion load is divided into high load and medium load, and each load has four motion modes.
e fracture size of the square plate under each motion mode has been studied theoretically to verify the correctness of the theoretical analysis, and this section presents eight groups of numerical simulation analyses for every motion mode.e comparison results of the theoretical analysis and numerical simulations are listed in Table 3.As shown in Table 3, the theoretical results are generally larger than the numerical simulation results, mainly because of the ignored strengthening effects of the materials and the effects of the surface force in the theoretical analysis.On the contrary, the results from the numerical study, which took these effects into account, were more accurate than those from the experiment.However, the theoretical analysis in this paper would always be more conservative, so the present approach could provide theoretical guidance for practical engineering.

Conclusions
In summary, the plastic zones and crevasse sizes of the plate under medium and high loads were discussed based on the ultimate plastic strain criteria.e following conclusions can be drawn from the analysis.Under medium load, the plastic zone consisted of the four sides and two diagonals of the square plate.Based on the plate motion when the explosive load becomes zero, the motion was divided into two modes.With reference to the plastic response of the clamped square plate and ultimate plastic strain criteria, the crevasse size formulas of the plate at two different motion models were derived.
Under high load, the plastic hinge lines and the corresponding movement law were discussed, and the plastic zone of the plate was determined.e crevasse sizes of the plate at motion modes III and IV after the explosion load becomes zero were calculated according to the principle of the conservation of energy.
e present theoretical analysis agrees with the experimental and simulation results; this outcome verifies the validity of our approach.is approach is also expected to be extendable to applications in practical engineering.

Figure 2 :
Figure 2: Approximate yield curve of the plate.

Figure 7 :
Figure 7: Two different plastic format areas of the plate.

Figure 12 :
Figure 12: Waveform of a spherical charge explosion.Explosion waveform of the (a) working condition 1 and (b) working condition 2.
4ρ 0 t 2 0   t s t m − 2t 0 t s + t 2 0  2 φ 2 t s  − 3.77M s  l 0.6 h 0.4    Crevasse Size of the Plate at Motion Mode IV.Under the explosion load of p m > p mb2 , the plate is at motion mode IV.e difference in calculating the crevasse size for modes III and IV is the different moving velocity when the applied load becomes zero, which results in different motion energy.

Table 1 :
Comparisons of theoretical and experimental results.

Table 2 :
Sensitivity of mesh size on the fracture of plate.

Table 3 :
Comparisons of theoretical and numerical simulation results.