mathematical modeling of large-amplitude dynamic ...

MATHEMATICAL MODELING OF LARGE-AMPLITUDE DYNAMIC-PLASTIC BEHAVIOR OF CIRCULAR PLATES SUBJECTED TO IMPULSIVE LOADS A. Zajkani *

H.R. Sefidi Shirkoohi *

A. Darvizeh ***

M. Darvizeh **

H. Gharababaei *

Department of Mechanical Engineering Engineering Faculty, University of Guilan, Rasht, Iran ABSTRACT In this paper, an analysis of the large-amplitude dynamic-plastic behavior of the circular plates with a rigid perfectly plastic material is presented. The plate is subjected to a short-time high-intensity impulsive load uniformly distributed over the surface. Modeling is complemented by using specific convex yield criteria. Corresponding to boundary conditions of the plate, it can be deformed through more than one mechanism, so, the mathematical formulation is based on the principle of calculus of variations in which the transverse displacement fields are assumed as a combination of appropriate paths. Based on the upper bound approach, the different terms of kinetic and consumed plastic energies likewise the applied impulse energy derived to produce an energy functional with unknown coefficients which is minimized through the displacement path. Finally, calculating the constants maximum residual deflection and strain distribution are obtained. Results of present model show satisfactory correlation with the empirical data for the different levels of the pulsed loads. Keywords : Dynamic plastic, Explosive forming, Circular plate, Calculus of variations.

1.

INTRODUCTION

Beams, plates, shells are basic structural elements for the numerous protective parts in framework of ships, aeronautic equipments, large scale machine-building, pressure vessels and other construction industries. In recent decades there have been widespread theoretical and experimental investigations with different target dimensions on the dynamic plastic behavior of deformable solids, in particular of plates, subjected to diversity of dynamic loadings. Through emphasis on the efficiency of strength analysis of structural members contributes to the development of solutions on direct and inverse problems to identify internal and external operative parameters such as geometries, materials, boundary conditions and nature of dynamic loading sources. Under high-intensity dynamic loading, short duration pressure pulse, typically explosive blast loading, the formability and tearing of plates, determine possibility of subsequent usages in sheet metal forming as well as in the evaluation of unexpected situations and risks. With overview in the theoretical branch of high rate deflection modeling of plates, will be obvious that circular plate has been regarded as prototype in which various concepts could be conveniently used. Analytical, Numerical and experimental studies of deformations of plates subjected to impulses imposed by explosive charge and other sources of blast loading have been extensively reported in the some past litera*

Ph.D. student

**

Professor

***

ture reviews such as Nurick et al. [1-5], Jones [6-7], Menkes and Opat [8], Wierzbicki [9] and Torn [10]. Because of simplification of analysis, energy approach has been exerted of these case studies of dynamic rigidplastic behavior of material, frequently. Several deformed shape functions taken by researchers mentioned in the literature by Wierzbicki [9]. Substantial work performed by Symonds and Wierzbicki [11] brought about the extension of approximate methods. The transverse shear force effects on the dynamic behavior of the circular plates were considered by many researchers for example; Jones and Oliveira [12], Jones [13] (see report [10]). In order to establish the dynamic-plastic failure criterion for the elements, three modes of failure are proposed for circular plates in these studies. Gupta [14] and Wierzbicki [9] reported these studies perfectly. Several constitutive equations of materials have been used for the dynamic responses of metallic circular plates subjected to impulsive loading. Wen [15] studied the response of circular plates with work-hardening material. However the power-law stressstrain correlation is exerted in the some works to predict the deformations and tearing of plates, Zaera et al. [16] applied the several non-linear approximating models of the viscoplastic materials for yield condition. The theory of unified strength is used in the considerations of Guowei Ma et al. [17] and Yan-bin Wang et al. [18]. Shen and Jones [7] and Zaera [16] used the Cow-

Professor, corresponding author

Journal of Mechanics, Vol. 26, No. 4, December 2010

431

perSymonds constitutive equation to regard strain rate effects. Weichert and Stoffel [19] presented numerical simulation and experimental results considering transverse shear deformation, rotary inertia and geometrical nonlinear effects with Chaboche and Bodner Partom viscoplastic constitutive rules. As an alternative method in consideration of the transient response of membrane, wave form solution introduced by Mihailescu-Suliciu and Wierzbicki [20] to calculate the deflections versus the intensity of the applied impulses and radius of area where central loading is performed. Their approach eliminated the discontinuity due to an initial-boundary value problem in the wave equation. Bodner and Symonds [21], Nurick and Martin [22] and Teeling-Smith and Nurick [1] carried out experimental tests on clamped circular plates under uniformly distributed impulses in order to obtain the comparable tools for validating the analytical and numerical investigations by other researchers. Balden and Nurick [5] using finite element code ABAQUS characterized the experimental results taken by [1] (uniform blast loading) and [2] (localized blast loading) for deformation and post-failure response. Lee and Wierzbicki [9,23] introduced studies on the dynamic response and fracture of thin plates under a localized pulse loading in three stages of deformation, i.e. dishing, discing and petalling. In Part I [9], they investigated the dishing stage as predicting start of fracture and presented various types of shape function to the mode solution [11] and also with numerical simulations verified analytical predictions [11,20] considering the effect of various spatial and temporal distribution of dynamic pressure with focus on the local strain fields. In Part II [23], a ductile fracture criterion was introduced using equivalent plastic strain and discing and petalling stages of thin circular plates were predicted. Also, the beginning of circumferential cracking and propagation of radial cracks were formulated, truly. Gupta [14] used the finite element analysis (ANSYS) to calculate the temporary dynamic behavior of circular plates under the uniform pulse pressure with variable supports and then compared results with testing data. The mathematical approach presented in this paper tracks the new contributions. In order to achieve more compatible results with experimental data, various types of shape profiles for each case of the boundary condition of the plate is recommended. In this model, the changes of the plastic hinges movement are produced to describe its time history. The generality and complexity of proposed displacement function combined by hinges procedure’s descriptions although beget the copious calculations, but create the comprehensive lookout for the transient responses of thin plates under impulsive loading verified by experimental data in the literature being mentioned. Using this method, the effects of bending moments, membrane forces, rotational inertia and strain rate on the dynamic-plastic response of the plate is considered. The purpose of the present consideration is to derive a set of integrals and proposes the functional of energy employing technique of modified Ritz method as a variational method to obtain the unknown quantities. Furthermore, these equations are solved numerically and the displacement func432

tion is obtained. The effect of two spatial distribution of dynamic pressure on the behavior of plates is presented with particular focus on the deflection and local strain fields.

2.

FORMULATION OF PROCESS

A circular plate with radius R, thickness H0 and density  has been subjected to the high- intensity loading over the surface of the plate. Assuming of being symmetric loading, pressure profile caused by explosion and also edge constraints are independent of angular variable . Hence, displacement function is independent of this parameter. On the other hand according to [11-16], the ratio of deflection in the radial direction to transverse direction is negligible. So, displacement is function of radius, r, and time, t. Besides, the magnitude of total impulse is obtained as;

I 

 0



R 0

p( r , t ) 2r dr dt

(1)

The imposed pressure profile can be presented as; p  p (r , t )  p0 p1 (r ) p2 (t )

(2)

where p0 is overpressure peak, p1(r) and p2(t) are as the spatial and temporal distribution, respectively. Function p1(r) is depends on style that explosive material is positioned. According to the previous studies, function of time distribution of pulse can be represented as exponential form [6]; t

p2 (t )  e 

(3)

The time constant  is a function of the weight of explosive charge and refers to the time when the pressure is reduced to 1/3 of the maximum pressure. This function is true up to the time t   and can be called the time of effective loading or the time of duration of blast wave [24,25]. Dynamic response of circular plate can be changed by the variation of boundary conditions, significantly [24,25]. 2.1

Deflection of Clamped Plate

By instant application of the dynamic load on the circular plate, a peripheral plastic hinge is formed quickly and in the cited area the plate becomes flat while traversing across as is evident in Fig. 1 [6,24,26]. This takes place while the position of the plastic hinge varies quickly during loading. It should be mentioned that the position of the above hinge comes closer to the support as the amount of dynamic load increases. By terminating the loading stage, , the first phase of the motion comes to an end. In this case, the distance of the formed hinge from the centre of the plate is stated by 1(t)R, while 1(t) is a dimensionless parameter indicating the position of the formed hinge in the distance 0  1(t)  1 and 0  0. At the moment t  0, hinge is in the position 1(t) = 1 on the edge of the plate and Journal of Mechanics, Vol. 26, No. 4, December 2010

written as follows: w (r , t ) I  21 (t ) R 2



1 (t ) R 1  12 (t )



 n 1



2 1 cos n1 (t )  (1)n 2 n





(7)

n cos r R

The Newton’s second law is used for the flat central area:



1 ( t ) R

0

c (t ) I p (r , t ) 2 rdr  H 0 R 2 12 (t ) w

(8)

c (t ) I is the acceleration of the centre of the in which w disk which is derived from Eq. (7) for r = 0: (a) Overall schema

 1  0  1 (t )  c (t ) I  R  (1  e0t )  w  0   1  e   2 0 e0t  1 (t )   02 e0t 1 (t )





(9)

Putting the pressure pulse profile in the Eq. (8) and also putting Eqs. (5) and (9) in the right term of same relation, an inhomogeneous second order differential equation with nonlinear coefficient based on the function and derivations of 1(t) can be resulted as the initial value provided; 1(0)  0. Moreover the differential equations are presented and solved in terms of 1(t) for two states of the local uniform distribution (rectangular) of pressure and triangular distribution whereas time profile for pressure pulse p2(t) is regarded as similar to Eq. (3). For the rectangular distribution of pressure following equation can be deduced:   p0  1  e 0  H 0 R  1  0

(b) Details

Fig. 1

Deformation shape in the second phase (Parabolic displacement)

reaches to 0 at the moment t  , quickly. In this case marginal area, | r |  | R(1  1(t)) | moves as generating of frustum and the central area, | r |  | R1(t) |, remain nearly flat. The important thing to note is that based on the previous empirical studies [11] the velocity of the hinge motion becomes nearly constant at the end of this phase: 1 (0)  1 ,

1 ()  0

(4)

Therefore, based on the above resultant hinge condition, an exponential function can be proposed for the motion of the hinge:

1  0 0  e0   0 t 1 (t )  e  1  e0  1  e0 

(5)

Slope of angle between generating of a frustum and horizontal axis is shown as: 1 (t )  tan 1 (t )

(6)

In the first phase, utilizing an extension of Fourier series for frustum of cones, the function of deformation is Journal of Mechanics, Vol. 26, No. 4, December 2010

 (1  e

 0 t

 t  e  

1 (t )  2 0 e 0t  1 (t )   02 e 0t 1 (t ) )



(10)

The solution is as follows:

1 (t ) 

p0 2  1  e 0   H 0 R  1   0

 t t    e   1     0 t   1 e    

(11)

For the triangular pressure distribution, following equation is achieved: t p0  1  e 0   0 t      2(1 e ) e   3H 0 R  1  0   t   0 t  1 (t )   02 e0t 1 (t ) (1  e 0 )  1 (t )  2 0 e





(12)

The solution is as follows: 1 (t ) 

 t p0    1  e0   2  ( e   t  )   3H 0 R   1  0  t      e  2 1        t 0 t 2     1  e 0 t  (13)    (1 ) e 0     

433

After the end of loading phase and the beginning of the second phase, the resultant hinge moves towards the centre at the velocity of  2 (t ) R . Regarding experimental results mentioned in the literature [6], deformation profile for the central region can be approximated to various parabolic curves that the parabola parameter will be always a descending function. (See Figs. 1(a) and 1(b)). The deflection functions are determined for each part as follow;  1  1  2 (t ) 2 w(r , t ) II   2 (t ) R 1  2 (t )   r 2   2  2 (t ) R

(14)

and: w( r , t ) II   2 (t )( R  r )

w(t , r ) II 



 2 (t ) R 3   (t ) 6

r 2   w(r , t )  Z 0   rc2 2

w r

2

 

2

 1  1  n sin n 2 (t )  (1) n  cos r  2  R   n 1 n  n 2 (t ) (16)





  2 (t )

Z 0   2 (t ) R 1   2 (t )  

(18)

2 (t ) R  2 (t )

rc   2 (t ) R 1 

1  22 (t )

R 22 (t ) 1  22 (t )  22 (t )

(20)

Using Eqs. (17) to (20) the displacement function in the unified form of Fourier series can be presented in the interval [R, R] for second phase;

 22 (t )  1   2 (t )  2 (t )   2   sin 1 2 2   2 (t ) 2 2 (t )   2 (t )  1 ( t ) 1   2    22 (t )  1   22 (t )  1  R  1  22 (t )              2 ( t ) 1 ( t ) 2 ( t )sin n ( t ) ( t ) J n ( t )    2 2 2 2 2 1 2   n 1 n   2 (t )  22 (t )  22 (t )       2 (t )  1    n  22 (t )  1   2 2 (t )  2 (t )     2   22 (t )  2 2    2sin 1   J 0  n2 (t ) 2 2   2  2 (t )     2 (t )    2 (t )  1   t ( ) 1  2     2         (t )  1  1  2 (t )  2 (t )  1     J 2 k  n2 (t ) 2 2 sin  2(k  1) sin 1 sin  2(k  1)sin 1   2 (t )   k  1  k 1  22 (t )  1  k  1   22 (t )  1       2 (t )     n 2    cos r  sin  2k sin 1 2     k R  2 (t )  1       w(t , r ) II 

(19)

and:



As an alternative, the deformation profile of the central region between the resultant hinges and the centre of plate can be regarded as Spherical dome-shape function

hinge

Combing the Eqs. (17) and (18):

   2 (t ) R   

(17)

where rc and Z0 indicate radius and centre of the spheres consecutively that both vary with respect to the time. On the other hand, from equality of the derivative of spherical segment cap with the generating of cone slope in the hinge point can be deduced:

(15)

Since point of separating two above curves is indeterminate now, in order to utilize displacement function as a unified form to compute energy integrals introduced further (section 5), the Fourier series can be employed in the interval [R, R] given as; 2 2

functions that vary proportionate to time for conical shape (see Fig. 2). Consequently the deformation profile is assumed as arcs from spheres which have the largest radius at the beginning of the phase and as time passes and the hinge advances the passing spheres radii become smaller and come close to zero at the end of the phase. The equation of the resultant spheres is written in the following form:

 R22 (t )



(21)



In the above equation Jv is the Bessel function of the first kind of order . hinge motion in the second phase satisfies the following conditions:  2 (t )  0  2 ()  1 ()  0

The function 2(t) which represents the

(22)

The final deflection of the hinge equals f. Assuming that the rate of the hinge motion becomes zero at the moment which deformation is terminated. Whatever position of hinge moves closer to the centre (f → 0), the more perfect conical shape is expected. Utilizing normalized ratio of several relative functions i2 (t ) under the hinge conditions, a more exact hinge function can be proposed under the conditions of Eq. (22) as; 434

Journal of Mechanics, Vol. 26, No. 4, December 2010

Although it will be expected that for achievement more accuracy and higher the degree of freedom for proposed explanatory hinge function, the more unknown coefficients can be added in relation of 2(t), but for abbreviation of computations, ms = 2 is considered. Therefore Eq. (23) is resulted as;  e  a1t  a1 te  a1 t f   e  a2t  a2 te  a2 t f   2 (t )  0   a     a2   a1 t f  a t  1  a2  e 2 f   e  a1  e  e

(26)

In the second phase of the motion, the following relations are written for the generating slope of a frustum:

 2 (t )  0 ,

 2 (t )  0 

(27)

In this case, an appropriate function of generating slope of frustum 2(t) is chosen which satisfies the above conditions describing the behavior of deformation;

 2 (t )  C1 (t f  t ) 2  C2

(a) Overall schema

(28)

If the condition of continuity 2(t) is utilized in the final point of first phase (t  );  2 ()  1 ()

(29)

The final time of the process can be calculated versus other parameters as an explicit relation for any state of pressure pulse. For example; for uniform (rectangular) distribution p1(r) = 1 and triangular distribution p1(r) = 1 r/R, can be obtained respectively: tf 

1  p0 2  1  C2   C1  e H 0 R  1  0

    

(30)

and:  p0 2   1  e0   1  t f    C2   2  (2e  1)    3e H 0 R   1  0   C1 

(b) Details

Fig. 2

 2  2  (1   0 )

Deformation shape in the second phase (Spherical dome-shape displacement)

 2 (t )  0 1  

ms



 i 1

i2

 (t )  

(23)

with following conditions:   i2 (t f )  i2 ()  0   i2 (t )  0 , 0  i2 (t )  1

(24)

where tf is the final time of the process. Establishing the above conditions, synonymous normalizing func tions i2 (t ), can be proposed as; a t  e ai   ai (  t ) e i f  e ai t i2 (t )  a t e ai   ai  e i f

Journal of Mechanics, Vol. 26, No. 4, December 2010

(25)

2.2

  2e 0 1  1    1      0   0   e  1 e

1 2

        (31)   

Deflection of Simply Supported Plate

Regarding the empirical studies on the large intensity dynamic-plastic responses of simply supported circular plate such as [6] a standpoint can be introduced that the deflection profile is composed of parabolas and ellipsoids shown on the Fig. 3. This configuration supports an opinion that hinge formation is in the junction of two sets of curves. Therefore, central area profiles between hinge and apex of the plate change in the form of various parabolic shapes. So, the central area profile is receded from horizontal axis upon progressing hinge toward the apex of parabolas on which parameter of these parabolic curves will be descended function. In this case, the central area profile moves and falls with ellipsoid curve. Because of hinge’s formation, deformation procedure varies in its position and 435

and depend on the smooth deformation’s procedure. Relevant to the amplitude of initial pulse and the other parameters such as geometric, material and boundary condition, hinge may be reach to centre. With Controlling of process to prevent any failure, it is expected that final shape can nearly be a parabola. In this state difference of phases are in the type of suggested function for hinge’s movement. During actuation of impulse, situation of plastic hinge varies rapidly in the first phase. It is notable that whatever intensity of dynamic load is increased, position of hinge is closer to the support. If C1, C0 and Z0 indicate the radiuses and centre of ellipse, consecutively: Z 0  OC , C1  CD, C0  CD 

(32)

Then equation of ellipsoid is as follows: (a) Overall schema

r 2  w(r , t )  Z 0   1 C12 C02 2

(33)

Equation of parabola with order 2m,(m  0.5) in the interval [R, R] can be introduced:



w( r , t )  C2 R 2 m  | r |2 m



(34)

By taking into account the continuity condition of hinge and using Eqs. (33), (34), the following relation is obtained:





Z 0  C2 R 2 m 1  2 m (t ) 

C0  2mC1 C2  (t ) R 

Fig. 3 Deformation of circular simply supported plate (combined parabolas and ellipsoids) decreases corresponding to the rate of hinge falling and passes aboard ellipsoid instead of the parabola. Slopes of two curves are equal in the place of hinge’s formation



 

2m2

C12  2 (t ) R 2

(36)

Substituting Z0 and C0 from Eqs. (35), (36) into the Eq. (33) and regarding Eq. (34), the general equation for displacement can be written in unified form in terms of Fourier series in the overall domain as;



w(r , t )  C2 (t ) R 2 m 1  (t ) 2 m  2m(t )2 m 1 R 2 m  2 C12   2 (t ) R 2  m(t ) 2 m  2 R 2 m 3C12

(35)

The curves have same slope angle at the hinge point, Thus identity of derivatives of Eqs. (33) and (34) at this point leads to obtain C0:

(b) Details



C0 C12  2 (t ) R 2 C1



 (t ) R (t ) R  (t ) 2 m 1 R 2 m 2 m 1 C12   2 (t ) R 2  2 C12   2 (t ) R 2  sin 1  R   (t ) R 2 m  C 2 m  1 C 1   1



 1  2C 2m 2    2 R 2 m 1  (t ) 2 m  2m  (t ) R  C12   2 (t ) R 2 sin  n(t )   n  n 1 2 (t ) R   n  C n   n   2(t ) R C1 J1  C1   1  J 0  C1   C12   2 (t ) R 2    2sin 1  2 R 2 R R C1  C    1  







     (t ) R  (t ) R  2 1  n   1 1 (t ) R     J 2 k  C1   sin  2( k  1) sin 1 sin  2(k  1) sin 1    sin  2k sin    C1  k  1  C1  k C1     R   k 1   k 1 2 m  k k  2m 2 R 2 m C2 k   (1) n R 2 m n   k    2m   R  (t ) R  sin  n(t )   2nC2   k !   sin  n(t )  sin    cos r   k  1 k 1  k 1  k   2 2  R ( ) ( )   n n      

436

(37)

Journal of Mechanics, Vol. 26, No. 4, December 2010

where value of m can be selected arbitrary in the simply supported plate (for example; m  2) and hinge function (t) should be evaluated by Eq. (5) in the interval 0  t   and Eq. (26) in the interval   t  tf. Considering the velocity of the centre of the plate equal to zero, the final time of the process will be obtained: w (0, t f )  w c (t f )  0

3.

(38)

STRESS ANALSIS

The resultant stresses in the circular plate under symmetric loading: radial force Nr, tangential force N, shear force Qr, radial moments Mr, and tangent moments M. But if R / H 0  2 will be true, the work resulting from the Qr can be ignored [6,16]. When the displacement be small along the radial direction, the work resulting from the N can be neglected [15,16]. Figure 4 displays the deformed elements of a circular plate in the coordinate plane r.

4.

Fig. 4

The deformed elements of a circular plate in two plates r

STRAIN ANALSIS

The radial strain can be defined based on the Fig. 5 as follows: dl  dr  w   1   1 dr  r  2

r 

(39)

The ratio of longitudinal change in the circumferential direction can be considered and the circumferential strain defined:  

 d   rd  1 d   1 rd  r  d 

(40)

where  is circumferential curvature with radii  (see Fig. 4): w 1 r    2 1/ 2 r  w   1       r  

(41)

The details in distribution of thickness strains

Corresponding to the principle of constancy of volume for plastic deformation [27], the thickness strain can be written:  w  2 1  w   r t  1   2    r   w     r 

(44)

As alternative the thickness strain is formulated by logarithmic feature:

The change of angle of the deformed element can be considered in the middle surface of thin plate be equal to; d  1 d

Fig. 5

(42)

t  Ln

h( r , t ) H0

(45)

During plastic deformation the vertical distance H(t) can be expressed in the whole deformed element at any moment; (see Fig. 5)

Thus circumferential strain is deduced;   w   1     r      w r

2

  

1

h(r , t )  H (t ) cos  (r , t ) 

2

1

Journal of Mechanics, Vol. 26, No. 4, December 2010

(43)

H (t )  w(r , t )  1    r 

2

(46)

It is quite clear that H(t) is a descending function with 437

its initial value, H0, and final value, H . Finally, the ratio of the final plate thickness to the initial thickness is obtained using Eqs. (44) to (46) with respect to final angle of generating of a frustum, f ; h 

   H  1  1  (cos  f ) e  H0 

   2 sin   f   tan  f  2  4  

   

(47)

r 



  w  2   1       r  

S

CALCULATING OF ENERGIES

According to the principle of conservation of energy rate [2,16], the following equation is always true while deforming the plate: We (t )  E c (t )  W p (t )

(48)

We is the rate of the external work done by the pressure p at the moment t, E c is the rate of kinetic energy and W p is the rate of work dissipated during plastic deformation. Each component of energy can be expressed in terms of the position function w(r, t).The external work imposed on the plate by applied external pressure is equal to:

We (t ) 

2

  0

R

p (r , t ) w (r , t ) r dr d 

0

(49)

The kinetic energy in the plate is resulted from the both translational and rotational motions. The kinetic energy resulting from the translational movement is; 1 Ect (t )  H 0 2

2

  0

R 0

2

 w    r drd   t 

(50)

6.

2

Ecr (t )  

1 H 30 24



 0

(51)

The plastic work Wp(t) dissipated during deformation is summation of plastic works resulting from different components of resultant stresses. For the radial force Nr can be written: W pN r 





R

0 0

N r  r r dr d 

(52)

The work due to the bending moment Mr in domain S, is: W pM r 

 M

r

S





 0

R

0

d r rd  



R

0

0

 

Mr

dl r d r

2

 w  M r r 1    r dr d   r 

where r is the radial curvature with radii r as; 438

(53)

 0



R 0

M  (  1)  r dr

(55)

CONSTITUTIVE EQUATIONS OF MATERIAL

Take into account of the strain rate influence; it will be convenient to substitute the dynamic-yield stress of the material d instead of the static-yield stress y. For this purpose, the Coper-Symound correlation can be employed [6]:    1/ q   d   y 1   m      0  

(56)

 m is average strain rate at the overall time and radial coordinate r and can be written in terms of radial strain rate only:

 m 

1 Rtf

  tf 0

R

0

 r dr dt

(57)

q and  0 are the constants of the material. The values of plastic force and moment in fully yielded section of plate are equal to: N p  d H 0 ,

Also, the kinetic energy resulting from the rotational motion in the whole plate is [16]:  2 w    R  r t  r dr d  2 0   w    1       r  

(54)

3/ 2

In the similar way, using Eq. (40) the work due to M in domain S, can be calculated: W pM    M  d  dr  

5.

2 w r 2

Mp 

d H 02 4

(58)

In order to analyze the forces and moments imposed on a plate under the explosive loading, the model of rigidperfectly plastic material can be utilized [2,6,7,11,16]. Consequently, the loaded plate deforms only when the applied stresses on its parts is beyond the surface of yield criterion. In this study, relation produced by [16] has been utilized as a designator of yield surface which includes all three resultant stresses with due consideration of the convexity condition (the Drucker’s postulate [27]). Now, using the flow rule and principle of normality of the strain rate vectors on the yield surface, the values of resultant stress applied on the plate can be rewritten according to the strain rate vectors similar to study done by [16] as follow; Nr  3

N p2  m 1/ 2

3N p2  2m  4 M p2 (  r2   2   r   )  M p2 (2 r    ) 2 3 Mr  3 3N 2  2  4 M 2 (  2   2    ) 1/ 2  p r r    p m 2 M p (  r  2  ) 2 3 M  3 3 N 2  2  4 M 2 (  2   2    ) 1/ 2  p r r    p m

(59)

Journal of Mechanics, Vol. 26, No. 4, December 2010

7.

Table 1

CALCULAS OF VARIATION

Specifications of the plates Material

On the base of approximating variational technique, Hamilton’s principle applied to derivations of fundamental equations in the large class of problems in mechanics covered in many textbooks [28]. Integrating and arranging of the Eqs. (49) to (59), the Lagrangian energy functional can be written as follows;   Ec (t )  W p (t )  We (t )

(60)

(61)

So, the total expression of the Lagrangian energy functional needs to be minimized in search of stationary values. These search require taking its derivatives with respect to the unknown coefficients and setting equal to zero (modified Ritz method) as follow; g  0, 0 g  0, C2

g g  0, 0  0 C1 g g  0, 0 a1 a2

(62)

To solve above nonlinear equations systems, the numerical calculus techniques specially; iterative Jacobian method should be utilized. It is notable that approximating Chebyshev’s and Legendre’s polynomials used to evaluate some terms of integrands in the calculating of energy integrals [29]. All of these calculations are carried out by MAPLE programming procedure. Whatever the number of minimization procedure be carried out by the more times, the more exact answers can be expected. For simply supported plate, Eq. (38) should be coupled with the Eqs. (62) and be solved the sets of equations until the final time be obtained simultaneously.

8.

RESULTS AND DISCUSSION

The purpose of this study is to investigate the large deflection on the dynamic-plastic behavior of circular plates and present its variation with different coefficient. The specifications of the utilized plates are presented in the Table 1. A rectangular (uniform) loading impulse (I 12N.s) caused by explosive charge with loading duration  10s applied over the steel (ASTM.A415) plate has been used in overall results in present analysis unless otherwise stated. In the schedule of solution, choosing Journal of Mechanics, Vol. 26, No. 4, December 2010

Aluminum 1100 [6]

Steel Titanium (ASTM-A415) (Ti 50A) [21] [21] 31.8 31.8

R (mm)

40

H0 (mm)

3

2.34

1.93

 (kg/m )

2260

4520

7850

Y (MPa)

120

251

223

 0 (s )

6500

120

40

q

4

9

5

3

1

To determine the displacement functions by Eqs. (7), (16) or (21) and also (37), passing from specific boundary points, they need to be directly substituted in the above Lagrangian energy functional. Corresponding to the applied displacement fields, the strains values substituted into Eq. (60). Then choosing input parameters of problem such as geometrical and material properties of plate, the energy functional (60) will be reduced in the following function:   g (0 ,  0 , C1 , C2 , a1 , a2 )

Specifications

infinitesimal time increment, t, the unknown coefficients are obtained at current times; ti = ti1  t where t0 = 0. This procedure is resumed with increasing i and determining coefficients until the velocity of centre of plate becomes zero. It is notable that determining exact criteria to calculate the final time of process is rather difficult, because; considering fluctuation of central region of plate is complicated phenomenon at the process. According to the Fig. 6, the membrane model [11] with including strain rate effects and the present approach produce different final time for different values of impulses. The present analysis with Spherical dome-shape approximation shows longer final time in the interval of I  1.9N.s as compared to parabolic approximation. In the Fig. 7, the time history of (t) and (t) are plotted. The functions of Spherical domes and parabolic deformations are used. Thus, at the initial stages of the process, (t) has a rapid growth (almost equal to 50) so that, the second phase begins; the motion takes a slower rate. It is clear that the final dimensionless values of (t) are equals to f  0.11 and f  0.08 with a parabolic and Spherical dome-shape approximation, respectively. It means that the resultant conical shapes aren’t 100 perfect and there is a small distance to form a perfect cone. In the Fig. 8, the variations in the final quantities in hinge’s place and final angles of generating of a cone are drawn based on different impulses over the clamped circular aluminum plate. In the Fig. 9, the variation in the amounts of centre deflection of clamped circular plate by approximating parabolic and Spherical dome-shape displacement functions are shown. On the Figs. 9(a) and 9(b), results for a steel (ASTM.A415) and a titanium plate as compared to the empirical data [21] and other models [7,16] respectively, indicating good agreement with the results. As evident in graphs, in lower impulses parabolic shape functions provide better results than Spherical domeshape functions and in higher impulses the Spherical dome-shape functions produce fewer errors. Results showed that for constant values of impulse, the effect of time duration of impulse on the magnitude of deflection of plate is negligible. Indeed with regarding to relations (1-3) at any state, the intensity of blast load is just efficacious. 439

Fig. 6

(a) Steel (ASTM.A415)

The final time in terms of impulse

(b) Ttitanium

Fig. 7

Time history of hinge and angle of generating of cone

Fig. 9

Centre displacement of the clamped edge plate vs. impulse

where subscripts “M” and “E” represent the modeling and experimental data at instance i respectively and N is the number of available experiments. The normalized root mean squared deviation or error (NRMSD or NRMSE) is the RMSD divided by the range of observed values, or: NRMSE or NRMSD 

Fig. 8

Final hinge’s position and angle of generating of cone vs. impulse

The root mean square deviation (RMSD) or root mean square error (RMSE) is a frequently-used measure of the differences between values predicted by a model or an estimator and the values actually observed from the experiments [30]. RMSD is a good measure of accuracy. These individual differences are also called residuals, and the RMSD serves to aggregate them into a single measure of predictive power and it can be applied for dimensionless centre deflection of plate as follow; RMSE or RMSD 

440

1 N

2

  wcf   wcf        (63)   H 0 i , E  i 1   H 0 i , M N



RMSE or RMSD  wcf   wcf       H 0  N , E  H 0 1, E

(64)

Also, in order to compare obtained results in detail for dimensionless centre deflection of plate between experimental and analytical modeling, the coefficient of determination (r2) index was calculated by using statistical analysis ToolPak of EXCEL version 2007. Table. 2 summarize these results for centre deflection of plate between experimental and analytical trajectories and show good correlation between these trajectories for both materials. The values of NRMSE for spherical dome shape model are relatively smaller than parabolic shape model. But the values of coefficient of determination (r2) for parabolic shape model are more close to unity. Also both models produce better results for the titanium plate compared on the steel plate. This matter has justified by other theories [7,16] in the Fig. 9. Vertical velocities of aluminum clamped plate (summarized in Table 1) are compared for two spatial distributions loading in the Fig. 10. It is presupposed that displacement fields is approximated by parabolic forms. Journal of Mechanics, Vol. 26, No. 4, December 2010

Table 2

NRMSE and coefficient of determination (r2) Parabolic shape NRMSE ()

r

2

Spherical dome shape NRMSE ()

r2

Steel

24.6900

0.9563

21.6801

0.9379

Titanium

15.2869

0.9475

10.3471

0.9436

(a)

(a) Rectangular distributed loading

(b)

(b) Triangular distributed loading

Fig. 10 Vertical velocity of deformation vs. time and dimensionless distance from centre Immediately when pressure pulse is applied, the velocity of plate is almost constant but afterwards, the change of slope and jump in the values of velocities are observed in the initial stages of deformation, significantly. In this domain driven plastic hinge has progressed after the beginning of loading. The plate’s velocity reaches to maximum quantity about 50s. Repercussions of velocities are sensible in the ultimate subsequent stages of motion. In the triangular distribution, velocity profile accedes to maximum value in lower time (more plate’s acceleration) and value of maximum velocity is greater than uniform distribution. Another output of this model is the possibility of determining resultant stress during time of the forming. Also, considering Fig. 11, the effects of membrane and bending in deformation of plate are obvious in different areas of plate. In this state, impulse is distributed over the clamped aluminum plate uniformly. In the initial stages of deformation (first phase) radial forces are low and with blow over the time become higher and exceed the yield. The maximum values of radial forces are observed in the region between centre and edge. In domain between edge and centre, the portion of bending moments is rather high at the beginning of process then reduces Journal of Mechanics, Vol. 26, No. 4, December 2010

(c)

Fig. 11

Resultant stress

duration the time particularly at ending of process. However it has no much change in the neighbor of centre at the overall times. The negative bending moments is observed especially adjacent edge (Figs. 11 (b) and 11(c)). This situation can be explained by principle of normality. The principle resultant stress of the plate makes a 3D surface in which inside of this surface plate is not yielded whereas out of surface yield is occurred and principle strain rate vector is always normal to yield surface. Regarding to Eqs. (57) and (59), the values of resultant stress depend on sign and values of rates of radial strain, radial curvature and circumferential curvature. In the Figs. 12 ~ 15, the distribution of thickness, circumferential, and radial strains for clamped and simply 441

(a)

(b)

(b)

(c)

Fig. 13 Strains in the rectangular distribution loading (simply supported plate)

(c)

Fig. 12 Strains in the rectangular distribution loading (clamped edge plate) (a)

(a)

442

(b)

Journal of Mechanics, Vol. 26, No. 4, December 2010

(c)

Fig. 14 Strains in the triangular distribution loading (clamped edge plate)

supported aluminum plate are illustrated for two type of spatial distribution pulse, respectively. In these graphs the parabolic displacement field is regarded as mechanism of deformation for clamped plate. What is shown by these curves is the formation of a negative circumferential strain near the edges of clamped plate because of hinge formation near the edges. The influence of hinge on distribution of circumferential strain is more evident in the triangular loading. In the case of radial strain distribution, there is a sudden jump in the sections near the edge and the formed hinge. This reflects an enormous variation in the radial length in these parts of plate. Although, the effect of hinge on the thickness strains is low, but changes of thickness strains are rather considerable. The strain is greater in the centre of plate as compared to the other places. Variations of radial and thickness strain have faster rate in the central area whereas; the amounts of circumferential strains are negligible and nearly are uniform in this area. For simply supported plate; near edge of plate radial strains increase rapidly and reach maximum value in edge. By genesis of negative strains in three directions, annular deformed elements of plate are constringed and shrunk in some radial distance and moments.

9.

(a)

(b)

CONCLUSION

Utilization of present mathematical modeling based on the calculus of variation seems to be having advantages such as simplicity and generality. Because, the effects of bending moments, membrane forces, rotational inertia and the influence of high strain rate on the behavior of circular plates can be regarded, simultaneously. Results have been verified successfully with the available empirical and other analytical data. Also, results have been shown that variations of values of final shape and strains for two type spatial distribution of pressure pulse are rather significant. The types of in-plane boundary condition such as immovable (clamped) and simply supported (movable) affect in the strains values particularly; radial strains. This upper bound solution accompanied by external geometrical approaches into the mechanism of deformation of plates help to take into account time historical response of circular plates which actually provide the essential accuracy and capacity of precision in the analyzing explosive forming as compared to lower bound solutions with intrinsic complexities. Besides, it can be confirmed that the calculus of variation may be very useful to develop analysis of other aspects of dynamic-plastic response of structures by selecting appropriate shape functions.

REFERENCES (c)

Fig. 15 Strains in the triangular distribution loading (simply supported plate) Journal of Mechanics, Vol. 26, No. 4, December 2010

1. Teeling-Smith, R. G. and Nurick, G. N., “The Deformation and Tearing of Thin Circular Plates Subjected to

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(Manuscript received July 7, 2009, accepted for publication December 14, 2009.)

Journal of Mechanics, Vol. 26, No. 4, December 2010