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Structural intensity study of plates under low-velocity impact
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International Journal of Impact Engineering 31 (2005) 957–975 www.elsevier.com/locate/ijimpeng
Structural intensity study of plates under low-velocity impact Z.S. Liua, H.P. Leeb,, C. Lua a
Computational Solid Mechanics Division, Institute of High Performance Computing, 1 Science Park Road, #01-01 The Capricorn, Singapore Science Park II, 117528, Singapore b Institute of High Performance Computing, 1 Science Park Road, #01-01 The Capricorn, Singapore Science Park II, 117528, Singapore Received 18 December 2003; received in revised form 10 May 2004; accepted 16 June 2004 Available online 10 August 2004
Abstract The transmissions of transient energy flow and dynamic transient response of plate structures under lowvelocity impact are presented. The structural intensity approach is used to study the transient dynamic characteristics of plate structures under low-velocity impact. In the dynamic impact response analysis, ninenode degenerated shell elements with assumed shear and membrane strain fields are adopted to model the target and impactor. The dynamic contact-impact algorithm and the governing equations for both the target and impactor are derived based on the updated Lagrangian approach. Explicit integration algorithm has been adopted in the time integration process. The novel structural intensity streamline representation is introduced to interpret energy flow paths for transient dynamic response of plates under low-velocity impact. The effects of plates with and without structural damping on the energy flow and energy path are discussed. Numerical results, including contact force, deflection histories and transient energy flow vectors as well as structural intensity streamlines, show that the present method and representation are an efficient approach for exploring dynamic response for plate structures subjected to low-velocity impact. r 2004 Elsevier Ltd. All rights reserved. Keywords: Dynamic response; Finite element method; Low-velocity impact; Plate; Structural intensity
Corresponding author. Tel.: +65-6419-1288; fax: +65-6778-0522.
E-mail addresses: [email protected] (Z.S. Liu), [email protected] (H.P. Lee), [email protected] (C. Lu). 0734-743X/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2004.06.010
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1. Introduction There are many practical situations where plate structures are subjected to transient impact loading of high intensity. Transient deformations are induced ranging from small-deflection linear elastic behaviour to large-deflection elasto-plastic behaviour with permanent deformation. Local damage and even failure of plates may take place. In such cases, it is essential to perform impact analysis for the purpose of plate design and/or verification. Structural intensity occurs as a result of dynamic impact loading. Despite their importance, however, impact effects are usually ignored or oversimplified in the dynamic analysis of plate structures. One reason is the complexity involved as contact-impact problems involve unknown boundary conditions. Specifically, the contact surface or points and the associated stresses and displacements are all unknown prior to the solution of the problem. Consequently, mathematical models of plate impact problems typically involve systems of inequalities and non-linear equations. Understandably, most contactimpact analyses concerning plates have to be carried out numerically such as by means of the finite element method (FEM). Traditionally, the impact analyses normally provide transient deformation and transient stress level. As the high-intensity transient phenomenon involved in impact precesses, the information of stress and deformation may be insufficient for further plate structure failure prediction and analysis. In order to provide more information for plate impact problems, the structural intensity approach can be adopted in the impact dynamic response. The concept of structural intensity was introduced to extend the vector acoustics approach to energy flow in structure-born sound fields [1]. The development of structural intensity formulations by Pavic [2] led to a growing interest in this field during the last two decades. Structural intensity is the power flow per unit cross-sectional area in elastic medium and it is analogous to acoustic intensity in a fluid medium due to structural vibration or dynamic response. Since the structural intensity field indicates the magnitude and direction of vibrational and transient energy flow at any point of a structure, it is very interesting to investigate structural intensity from a practical point of view. In addition, the structural intensity distribution can offer full information of energy transmission paths and positions of sources and sinks of mechanical energy. Dissipative elements, mechanical modification and active vibration control can be used for an alteration of energy flow paths within the structure and the amount of mechanical energy injected into the structure. Therefore, it is imperative to investigate the energy flow paths for the damage detection in plates subjected to transient dynamic loading. The structural intensity approach has been successfully used in the field of plate vibration to determine major vibration energy transmission paths in complex and noisy structures. As best as we know, very few reports have been published in the study transient impact phenomenon by using structural intensity approach [3]. The impact response of plate structures is often analysed by segmenting the problem into different response phenomena that occur. This generally involves considering the ‘‘global’’ and ‘‘local’’ responses of the structures by the impactor. The global response refers to the dynamic structural response of the plate configuration, whereas the local response refers to the indentation caused by impact. Because both phenomena occur simultaneously during the impact event, it makes the problems more complex. Consideration of both phenomena is important for accurate prediction of dynamic response of plate structures under impact loading. Prediction of the impact
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response of structures requires the knowledge of the following key parameters, namely, the impact force history, the pressure distribution between the impactor and the structure and the contact area. In previous studies [4–6], it is generally assumed that impact duration is large compared to the stress wave transient time in the target. That is, when the impact velocity is very low, dynamic response in the local contact area and response of wave problems can be ignored. However, the stress wave phenomenon should be considered in the normal low-velocity impact analysis of plate. By adopting the structural intensity approach, more information including transient energy transmission and path can be determined and it provides more transient phenomena in the plate impact analysis. In this paper, the methodology for predicting the dynamic response and structural intensity of a plate under low-velocity impact is presented. The FEM is adapted to perform dynamic impact response. In the impact process, the degenerated shell element is used to model plates and impactor. The dynamic contact-impact algorithm and the governing equations for both the target and impactor are derived based on the updated Lagrangian approach. Explicit integration algorithm has been adopted in the time integration process impact. The computational method for structural intensity is illustrated and formulas of structural intensity for plates are given in the time domains. Several low-velocity impact problems of plate structures with/without damping impacted by steel spherical projectiles are analysed and the results are interpreted by the structural intensity approach. The energy transmission and flow path in plates which are subjected to transient impact loading are presented. The novel structural intensity streamline representation is introduced to interpret energy flow path. Using the structural intensity technique and structural intensity streamline representation, the detailed transmission pattern of transient energy flows and the flow paths from the impact source of excitation to the sink in plates are demonstrated. The changes of energy flow in plates with and without dampers are discussed.
2. Formulations of contact-impact of plates and shells To treat the impact problem, normally the contact force method and energy balance method are used. Among various contact force methods, Hertzian method is the simplest in terms of formulation by using Hertzian’s contact law or its modification. Several researchers, including Sun and Yang [7], Koller and Busenhart [8], Aggour and Sun [9], Nosier and Reddy [10], Lee and Kwak [6], and Liu and Swaddiwudhipong [11], have adopted the Hertzian’s contact law in their studies. While the Hertzian method provides the global impact response of plates satisfactorily at a relatively small computing cost, its extension to tackle more complex problems becomes difficult locally, i.e. in the contact area. In order to deal with the local effect of impact, other methods, such as the Lagrange multiplier method, penalty method and augmented Lagrange multiplier method, are proposed. In the Lagrange multiplier method, the contact force is modelled by means of Lagrange parameter and contact conditions are satisfied exactly [12]. The penalty method assumes from the outset that the impenetrability constraints will be violated [13,14]. The augmented Lagrange multiplier method combines the penalty and Lagrange multiplier methods [15]. As the penalty method has many advantages that it can be extended to treat tangential force due to friction and the number of unknowns does not increase due to the enforcement of the
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contact constraints and that the system of equations remains positive definite, the penalty method fits very well into an explicit time integration of impact analysis and its implementation is straightforward [16,17]. In this study, the penalty method will be adopted to perform impact response analysis. In the modelling for treating impact problem of plates, different types of elements can be used. Liu and Swaddiwudhipong [11] derived a new formulation for the analysis of contact in degenerate shell elements. This formulation accounts for the transverse stress and strain through the shell thickness. Large deformations and rotations can be accounted for by invoking appropriate stress and strain measures. El-Abbasi and Meguid [18] presented another formulation for the analysis of contact in degenerate shell elements, allowing doublesided shell contact. Based on recent studies, it appears that degenerated shell elements offer an effective means to treat impact problems of plates and are adopted in the present study. As plate structures often undergo large deflection, nonlinear transient analysis of such structures under a low-velocity impact load is considered, adopting a layered approach [19]. Deformation components, strains and stresses are defined in local coordinate system. Updated Lagrangian approach is adopted in the derivation of formulation of the incremental form of governing equations. Finite element formulation incorporating the penalty algorithm is used to evaluate the contact force history. Explicit time integration algorithms may be adopted in time integration.
2.1. Degenerated shell element model On the basis of Mindlin plate theory, degenerated shell element can be formulated. By adopting isoparametric geometric description, the element can be used to represent thin and thick shell components with arbitrary shapes, circumventing the complexities of classical shell theory and differential geometry. The two main assumptions adopted in the degeneration process are: (i) normal to mid-surface before deformation remains straight after deformation and (ii) normal stress component is negligible in the constitutive equation. Transverse shear deformations are considered in the analysis. The Cartesian coordinates at any point of the shell can be uniquely defined in terms of nodal and thickness coordinates, as follows [20]: 2 3 2 x3 2 3 tk xk x n n X X z6 y7 6 7 6 7 ð1Þ N k 4 yk 5 þ N k hk 4 tk 5; x ¼ 4y5 ¼ 2 z k¼1 k¼1 tk zk z where k denotes the node number and n is the number of nodes in the element, N k the element shape function defined in surface z ¼ constant, z the natural coordinate in the direction perpendicular to the middle surface, and hk is the shell thickness. The thickness vector on the right-hand side of Eq. (1) is constructed from the nodal coordinates of the top and bottom surfaces. In the present study, five degrees of freedom are specified at each nodal point. They are the three displacement components (uk ; vk ; wk ) and two rotations (b1k ; b2k ) at node k as shown in
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Fig. 1. Nine-node quadratic degenerated shell element.
Fig. 1. The element displacement field can be expressed as 3 2 3 2 x 2 3 rk sxk uk u n n X X z6 y 6 7 6 7 y 7 b1k N k 4 vk 5 þ N k hk 4 rk sk 5 ; u¼4v5¼ 2 z b2k k¼1 k¼1 z rk sk w wk
ð2Þ
where rk and sk are the unit vectors of local Cartesian coordinates. For moderately thick-shell problems, degenerated shell element performs well generally. For thin-shell problems, however, over-stiff solutions arise due to shear and membrane-locking phenomena when full integration is carried out to evaluate the stiffness matrix. Zienkiewicz et al. [21] and Hughes et al. [22] proposed the reduced and selective integration techniques to alleviate the over-constraining effects of shear and membrane stiffness. An alternative approach is to adopt assumed strain fields as introduced by Bathe and Dvorkin [23] for four-node shell element and by Huang and Hinton [19] for higher order shell elements. Swaddiwudhipong and Liu [24] also adopted this approach to study the dynamic response of large-strain elasto-plastic plate and shell structures. In the present study, a new contact-impact formulation is developed based on degenerated shell element model. In order to consider contact-impact, the top and bottom surfaces of element are defined as exterior and interior surfaces, respectively. It is assumed that only the exterior surface of the shell elements has possible contact with another shell structure. The displacements and geometry of the shell element at contact surface can be defined as follows: 3 2 c3 2 3 2 x rk sxk uk u n n 16 y 6 7 X 6 7 X y 7 b1k r s uc ¼ 4 vc 5 ¼ ; ð3Þ N k 4 vk 5 þ N k hk 4 k k5 2 z b2k k¼1 k¼1 z c rk sk w wk
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2
xc
3
2
xk
3
2
txk
3
n n 16 y 7 6 7 X 6 7 X x c ¼ 4 yc 5 ¼ N k 4 yk 5 þ N k hk 4 tk 5: 2 z k¼1 k¼1 tk zc zk
ð4Þ
The contact is thus governed by the mid-surface nodes of the element, and no additional degrees of freedom are introduced in the system equation. 2.2. Finite element discretization for contact-impact problem According to the Hu-Washizu variational principle, the minimization problem can be stated as determination of nodal displacement or velocity, such that the potential energy is minimized while not violating the principles of impenetrability of equal and opposite reaction [25]. Introducing a suitable stress tensor in updated Lagrangian formulation and considering non-linear contact equations, the incremental linearized form of the virtual work, which includes the contact contribution, can be expressed as [16,26] Z Z Z n n tij dn Zij dV þ tij dn eij dV n Dijkl ekl dn eij dV þ t t t O O O Z Z ðdui ðrtþDt bi ru€ i Þ dV ðttþDt Þ du ds Ot Gtt Z þ ½ðn n dðduc Þ þ gtþDt dnÞ þ tN dðduc Þ dG ¼ 0: (5) N Gc
In the above equation, the quadratic terms of the contact virtual work are ignored. The detailed deriving of Eq. (5) can be referred to Liu et al’s paper [16]. Considering a displacement-based finite element formulation and applying the semidiscrete FEM to the variational Eq. (5) for a contact system, the incremental form of the dynamic FEM equations can be written as ð6Þ M tþDt d€ þ C tþDt d_ þ ðt K L þ t K t þ t K c ÞtþDt Dd ¼ tþDt R t F t F c ; t
t
t
t
t
where M and C are mass and damping matrices, ðtt K L þ tt K t þ tt K c Þ the global consistent tangent stiffness matrices. tt K L is the linear stiffness matrix; tt K t is the initial stress stiffness matrix; tt K c the global contact stiffness matrix. tþDt R the consistent force vector. tt F the equivalent nodal force vector resulting from the presence of the initial stresses in the element, and tt F c and tt F ec the equivalent nodal force vector from the contribution of global and element contact in the contact surface, respectively. The detailed expression for the above matrices can be referred to Liu and Swaddiwudhipong [11,16]. 2.3. Time-stepping algorithm for contact-impact problem Either implicit or explicit time integration can be used for solving non-linear contact system [16]. According to Wriggers et al. [27], for contact-impact problems without inherent shock waves, implicit integration can be used with considerable larger time steps than those adopted in the explicit scheme. Therefore, the Newmark’s method can be used for low-velocity impact problem.
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The effective tangent stiffness and effective residuals will be used in the algorithm as updated expression of this scheme. Within the solution algorithm, Newton–Raphson method will be used to achieve equilibrium within a time step. In the implicit time integration, a common technique used is the trapezoidal rule, which is Newmark’s method with parameters d ¼ 0:5 and a ¼ 0:25 [26]. On the other hand, explicit methods are often applied for short-duration impact contact problem. Explicit time integration is also suitable for contact-impact problems as the small time steps imposed by numerical stability can treat the discontinuities in contact impact. Another advantage of explicit algorithms is that the shells can first be integrated completely independently, as if they were not in contact. This uncoupled solution correctly indicates which parts of the shell are in contact. The detailed expression of time-stepping algorithm can be referred from the Liu et al’s paper [16,17].
3. Structural intensity in plates The instantaneous intensity component is a time-dependent vectorial quantity equal to the change of energy density in a given infinitesimal volume. Its ith component in time domain can be defined as [28,29] I i ðtÞ ¼ sij ðtÞ vj ðtÞ;
i; j ¼ 1; 2; 3;
ð7Þ
where sij ðtÞ is the component of the stress tensor at a point where i represents the normal direction of the area on which stress acts, j represents the direction of stress (i; j ¼ 1; 2; 3 corresponding to x, y, and z directions in a Cartesian co-ordinate system), and vj ðtÞ is velocity in the j-direction at time t. Based on pure bending plate theory, Pavic [2] derived the structural intensity formulas for a thin plate. In the impact dynamic response, large deformation may take place, and the more general formulas for structural intensity should be considered. The general 3-D structural intensity formulations for plate were developed by Ramano et al. [30] on the basis of the definition of structure intensity and 3-D elasticity. In the specific case of the plate mid-surface lying in the x-, yplane, the structure intensity can be expressed as [3] _ _ x þ v_N xy ; y_ y M x y_ x M xy þ uN I x ¼ ½wQ x
_ y y_ y M yx y_ x M y þ uN _ yx þ v_N y ; I y ¼ ½wQ
(8)
where M x , M y , M xy , M yx , N yx , N xy , Qx , Qy , N x , N y represent bending moments, twisting _ v_, w_ and y_ x , y_ y moments, shear forces, membrane forces, and in-plane shear force, respectively. u, are translational and rotational velocities, respectively. Compared to Pavic structural intensity _ yx þ v_N y , which corresponding to _ x þ v_N xy and uN formulas, Ramano’s has two more terms uN longitudinal and in-plane shear waves. The energy associated with these terms is not coupled with the bending wave energy in plates. Thus, the two terms represent a significant difference between Romano’s and Pavic’s formulation only when the in-plane motion of the structure is significant, which can occur with in-plane force. The difference and suitability of these two types of formulas were discussed in detail by Zhang and Adin [31].
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It should be noted that, in the present study, as the degenerated shell element is selected in the analysis, the five translational and rotational displacements or velocities are independent. As no other displacement relation assumption is made, the process of directly using the above equations should be more accurate than that adopted in the previous studies [30,31]. The reason the shear effects should be considered is that the classical theory, which is based on the pure bending assumption, is only applicable for the waves that are long in comparison with the thickness of the plates. As the wavelength diminishes, the velocity in the 3-D theory has as its upper limit the velocity of Rayleigh surface waves. Hence, the classical plate theory cannot be expressed to give good results for sharp dynamic transient response, i.e. in the impact case. The above formulas will be adopted in the interpreting energy flow in the plates.
4. Structural intensity streamline concept Vector fields may be either steady or changing with time. A single data cube represents a field at an instant of time and is necessarily equivalent to a steady field. To represent a changing or dynamic field, it is necessary to have a stack sampling the field at a set of time steps. The streamline technique displays the flow as lines everywhere parallel to the velocity field. The relative spacing of the lines indicates the speed of the flow. For steady flows, path lines and streamlines are identical. As the structural intensity is a vector field in the space, it is similar to the velocity field in the flow. Inspired by flow line representation in fluid mechanics, the structural intensity streamline representation is adopted to interpret the structural intensity transmission path. It is convenient to study the structural intensity or energy flow using this technique. Similar to fluid mechanics definition, the structural intensity streamline can be expressed as dr Iðr; tÞ ¼ 0;
ð9Þ
where r is energy flow particle position. The structural intensity of an energy flow element on such a streamline is perpendicular to r and parallel to dr. The cross product can be written as i j k ð10Þ I x I y I z ¼ 0: dx dy dz Thus, the differential equation describing structural intensity streamline is dx dy dz þ þ ¼0 Ix Iy Iz
ð11Þ
for 2-D plate, the structural intensity streamline reduces to dx dy þ ¼ 0: Ix Iy
ð12Þ
Every particle that passes any particular point will follow the streamline that goes through that point. A streamline is a line which is everywhere tangential to an energy flow particle’s structural intensity. By default, this means a streamline is the path an energy flow particle travels.
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5. Numerical examples and discussion 5.1. A steel plate hit by a steel sphere As shown in Fig. 2, consider a clamped 100 100 mm2 steel plate hit by a steel sphere at its centre. Initial conditions of zero displacement and zero velocity are assumed for the plate. The initial thickness of the plate, h, is 0.2 mm, while the properties of materials considered for this example are given in Table 1. The radius of the sphere is R ¼ 5:0 mm, and its velocity just before impact is V 0 ¼ 10:0 m=s. Two types of plate conditions are studied. For case 1, the plate has no any damping. For case 2, four dampers with damping ratio of 1200 N s/m are connected to plate, as shown in Fig. 2. The sphere has been modelled using shell elements which have same mass with sphere. The transient deflection at the centre of the plate is presented in Fig. 3. The maximum displacement at the centre of the plate takes place at about 0.25 ms. From Fig. 3, it can be seen that the time histories of maximum displacements are nearly same at before 0.35 ms for a plate with and without damping. After 0.35 ms, substantial difference can be seen for cases with and without dampers. In damping case, the central deformation of the plate rebounds to initial position and further to positive direction. The computed structural intensity vectors for the cases without damping and with four dampers are shown in Figs. 4a–h and 5a–h at different times, which correspond to 0.05–0.40 ms after impact, with an incremental time step of 0.05 ms. From Figs. 4 and 5, it can be seen that, with the time increase, the energy flow or structural intensity increases and moves toward to the plate boundary. Comparing Figs. 4 and 5, it can be demonstrated that, in the case of having dampers,
Fig. 2. Plate model which was subjected to impact by a sphere at the plate centre.
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Table 1 Material properties used for numerical analysis Properties
Steel (for plate target)
Steel (for sphere impactor)
Young’s modulus (GPa) Shear modulus (GPa) Density (kg/m3) Poisson’s ratio
206.0 79.85 7833.0 0.29
206.0 79.85 7833.0 0.29
0.4
Displacement (mm)
0.2 0.0 -0.20.0 -0.4 -0.6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
No damper Damping
-0.8 -1.0 -1.2 -1.4
Time (ms)
Fig. 3. Displacement history of plate at the centre.
the structural intensity from the source position flows to damper locations. The energy flow source and sinks can be clearly located. For no damping case, the structural intensity or energy flow moves from impact source towards the plate boundary. When it reaches the plate boundary, the energy flow is reflected and becomes quite uniformly distributed as shown in Fig. 4h. On the other hand, for the damper case, the energy flow pours to sink locations. This means that the damper can redirect the energy flow distribution. In order to demonstrate the structural intensity flow path, the structural intensity streamlines and Von Mises stress contour are depicted in Figs. 6 and 7 for a plate having no damper and having dampers at different time moments. From Figs. 6 and 7, it can be clearly seen that the transient energy flow transmits from impact area toward the boundary. The structural intensity streamlines detailed the energy flow transmission paths in the plate. It can also be seen that the stress wave propagating trends are different from the energy flow path. It can be seen that the pattern of energy flow varies with the time. At a duration of 0.25 ms after impact, at which the maximum impact force has been reached, the calculated structural intensity vectors demonstrate that there exist the transient virtual energy flow sources or sinks. This phenomenon is due to stress wave reflection from the plate boundary. Although there are no real sources in these areas, the transient structural intensity and its streamlines provide the possible weak points for transient dynamic response. As time increases, the damper’s effects on the structural intensity become more significant. This phenomenon can also be observed in the displacement time–history curve, as shown in Fig. 3.
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Fig. 4. (a–h) Structural intensity vectors at different times (no damper).
5.2. Cutout steel plate impacted by steel sphere To study the transient structural intensity flow phenomena, an irregular steel plate is analysed. The cutout plate is that of a rectangular plate with the dimensions of 50 125 mm2, with two rectangular cutouts of dimensions 25 12.5 mm2. The irregular plate is subjected to low-velocity impact by a steel sphere. The impact arrangement layout and analysis model are shown in Fig 8. In this case, the plate material properties are the same as the example in Section 5.1. The radius of
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Fig. 5. (a–h) Structural intensity vectors at different times (with dampers).
the sphere is R ¼ 5 mm, and its velocity just before impact is V 0 ¼ 10 m=s. In the model, the plate is clamped along all edges. The displacement history of the plate at impact centre is shown in Fig. 9. Fig. 9 shows that the maximum displacement takes place at about 0.23 ms. The computed structural intensity vectors, structural intensity streamlines and Von Mises stress contour are depicted in Figs. 10 and 11 for different time instances, which correspond to an after-impact time of 0.03–0.38 ms, with
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Fig. 6. (a–h) Structural intensity streamlines and Von Mises stress contours at different times (no damper).
increments of 0.05 ms, respectively. From Fig. 10, it can be clearly seen that the transient energy flow transmits from the impact area towards the boundary. From the structural intensity vectors plot, the energy flow source can be clearly located. When the energy flow reaches the boundary, the structural intensity is reflected and some virtual sources or sinks can be constructed. As the neck looks like a flow pipe, the energy passes through the neck to the other part of the plate with time increase. Finally, the constructed virtual sink or sources are destroyed and the structural
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Fig. 7. (a–h) Structural intensity streamlines and Von Mises stress contours at different times (with dampers).
intensity is redistributed. The structural intensity streamlines give the energy flow transmission paths in the plate in detail. It can also be seen that the stress wave propagating trends are different from the energy flow path. From Figs. 10 and 11, it can be seen that the pattern of energy flow varies with the time. After an impact time of 0.18 ms, the calculated structural intensity vectors demonstrate that there exist the virtual energy flow sources or sinks. This phenomenon is due to stress wave reflection from the boundary of the plate. Although there are no real sources in these
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Fig. 8. Irregular plate impacted by a steel sphere.
Displacement (mm)
0.4 0.2 0.0 -0.20.0 -0.4 -0.6
0.1
0.2
0.3
0.4
0.5
0.6
-0.8 -1.0 -1.2 -1.4
Time (ms)
Fig. 9. Displacement histories of plate at impact point centres.
areas, the transient structural intensity and structural intensity streamlines provide the possible weak points for transient dynamic response. In the transient dynamic response analysis, the structural intensity magnitude and direction should be considered for further studying the impact dynamic damage mechanism. In order to reduce the failure possibility in the impact case, these areas also should be paid more attention.
6. Concluding remarks In this paper, the governing equations of impact for plate structures are formulated. Degenerated nine-node shell elements with assumed membrane and transverse shear strains are
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Fig. 10. (a–h) Structural intensity vectors of the irregular plate at different moments.
employed to treat impact problems under low-velocity impact. A dynamic contact-impact algorithm, employing penalty constraint enforcement, is presented in the finite element context. The structural intensity method has been adopted to predict the information of energy flow for the plates subjected to impact dynamic loading. The structural intensity streamline is introduced to interpret the energy flow path. The present results demonstrate that the structural intensity fields and structural intensity streamlines can be used to clearly indicate the source, the sink and the
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Fig. 11. (a–h) Structural intensity streamlines and Von Mises stress contours at different times.
direction of energy flow from the source to the sink. It also can be an effective tool for transient dynamic response control for plates, provided the power flow pattern, energy density, and structural intensity path in plates can be modified and controlled by properly arranging dampers. The structural intensity method can be adopted in the design of plate structures and may act as a new criterion for plate structural design under dynamic loading. The study demonstrates that the structural intensity method and structural intensity streamline technique are potential means for studying dynamic failure and fracture of plates subjected to low-velocity impact. The proposed
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method provides a basis of further study in terms of impact contact damage and for composite plate and shell structures under low-velocity impact.
References [1] Noiseux DU. Measurement of power flow in uniform beams and plates. J Acoust Soc Am 1970;47:238–47. [2] Pavic G. Measurement of structure borne wave intensity, Part I: formulation of the methods. J Sound Vibration 1976;49(2):221–30. [3] Lee HP, Lu C, Liu ZS, Xu XD. The structural intensity analysis of plates under dynamic loading. In: Mathisen KM, Kvamsdal T, Okstad KM, editors. Proceedings of the 16th Nordic Seminar on Computational Mechanics. Trondheim, Norway: Trondheim; 2003. p. 145–8. [4] Sun CT, Liou WJ. Investigation of laminated composite plates under impact dynamic loading using a threedimensional hybrid stress finite element method. Comput Struct 1989;33:879–84. [5] Mittal RK. A simplified analysis of the effect of transverse shear on the response of elastic plates to impact loading. Int J Solids Struct 1987;23:1191–203. [6] Lee DI, Kwak BM. An analysis of low-velocity impact of spheres on elastic curved-shell structures. Int J Solids Struct 1993;30:2879–93. [7] Sun CT, Yang SH. Contact law and impact response of laminated composite. NASA CR-159884. Washington, DC: NASA; 1980. [8] Koller MG, Busenhart M. Elastic impact of spheres on thin shallow spherical shells. Int J Impact Eng 1986;4:11–21. [9] Aggour H, Sun CT. Finite element analysis of a laminated composite plate subjected to circularly distributed central impact loading. Comput Struct 1988;28:729–36. [10] Nosier A, Kapania RK, Reddy JN. Low-velocity impact of laminated composites using a layerwise theory. Comput Mech 1994;13:360–79. [11] Liu ZS, Swaddiwudhipong S. Response of plate and shell structures due to low velocity impact. J Eng Mech ASCE 1997;123(12):1230–7. [12] Hughes TJR, Taylor RL, Sackman JL, Curnier A, Kanoknukulchai W. A finite element method for a class of contact/impact problems. Comput Meth Appl Mech Eng 1976;8:249–76. [13] Belytschko T, Neal M. Contact-impact by the pinball method algorithm with penalty and Lagrangian. Int J Numer Meth Eng 1991;31:547–72. [14] Belytschko T, Yeh I. The splitting pinball method for contact impact problems. Comput Meth Appl Mech Eng 1993;105:375–93. [15] Wriggers P, Zavarise G. Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interfaces. Commun Numer Methods Eng 1993;9(10):815–24. [16] Liu ZS, Swaddiwudhipong S, Koh CG. Contact-impact dynamic response of shell structure under low velocity impact. In: Obrebski JB, editor. Proceedings of the International Symposium on Lightweight Structures in Civil Engineering. Warsaw, Poland: IASS; 2002. p. 377–84. [17] Swaddiwudhipong S, Koh CG, Liu ZS. Numerical study of shell structures under low velocity impact. J Int Assoc Shell Spatial Struct 2002;43(140):159–70. [18] El-Abbasi N, Meguid SA. Large deformation analysis of contact in degenerate shell elements. Int J Numer Meth Eng 1998;43:1127–41. [19] Huang HC, Hinton E. A nine nodes Lagrangian Mindlin plate element with enhanced shear interpolation. Eng Comput 1984;1:369–79. [20] Huang HC. Static and dynamic analyses of plates and shells. London, Berlin: Springer; 1989. [21] Zienkiewicz OC, Taylor RL, Too JM. Reduced integration technique in general analysis of plates and shells. Int J Numer Meth Eng 1971;3:275–90. [22] Hughes TJR, Cohen M, Haroun M. Reduced and selective integration techniques in finite element analysis of plate. Nucl Eng Design 1978;46:203–22.
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[23] Bathe KJ, Dvorkin EN. A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. Int J Numer Meth Eng 1985;21:367–83. [24] Swaddiwudhipong S, Liu ZS. Dynamic response of large strain elasto-plastic plate and shell structures. ThinWalled Struct 1996;26:223–39. [25] Belytschko T, Liu WK, Moran B. Nonlinear finite elements for continua and structure. Baffins Lane, Chichester, England: Wiley; 2000. [26] Bathe KJ. Finite element procedures. Englewood Cliffs, NJ: Prentice-Hall; 1996. [27] Wriggers P, Vu VT, Stein E. Finite element formulation of large deformation impact-contact problems with friction. Comput Struct 1990;37(3):319–31. [28] Pavic G. Vibroacoustical energy flow through straight pipes. J Sound Vibration 1992;154(3):411–29. [29] Gavric L, Pavic G. A finite element method for computation of structural intensity by the normal mode approach. J Sound Vibration 1993;164(1):29–43. [30] Romano AJ, Abraham PB, Williams EG. A Poynting vector formulation for thin shell and plate, and its application to structural intensity and source localization, Part 1: theory. J Acoust Soc Am 1990;1166–75. [31] Zhang Y, Adin Mann III J. Measuring the structural intensity and force distribution in plate. J Acoust Soc Am 1996;99(1):345–53.
International Journal of Impact Engineering 31 (2005) 957–975 www.elsevier.com/locate/ijimpeng
Structural intensity study of plates under low-velocity impact Z.S. Liua, H.P. Leeb,, C. Lua a
Computational Solid Mechanics Division, Institute of High Performance Computing, 1 Science Park Road, #01-01 The Capricorn, Singapore Science Park II, 117528, Singapore b Institute of High Performance Computing, 1 Science Park Road, #01-01 The Capricorn, Singapore Science Park II, 117528, Singapore Received 18 December 2003; received in revised form 10 May 2004; accepted 16 June 2004 Available online 10 August 2004
Abstract The transmissions of transient energy flow and dynamic transient response of plate structures under lowvelocity impact are presented. The structural intensity approach is used to study the transient dynamic characteristics of plate structures under low-velocity impact. In the dynamic impact response analysis, ninenode degenerated shell elements with assumed shear and membrane strain fields are adopted to model the target and impactor. The dynamic contact-impact algorithm and the governing equations for both the target and impactor are derived based on the updated Lagrangian approach. Explicit integration algorithm has been adopted in the time integration process. The novel structural intensity streamline representation is introduced to interpret energy flow paths for transient dynamic response of plates under low-velocity impact. The effects of plates with and without structural damping on the energy flow and energy path are discussed. Numerical results, including contact force, deflection histories and transient energy flow vectors as well as structural intensity streamlines, show that the present method and representation are an efficient approach for exploring dynamic response for plate structures subjected to low-velocity impact. r 2004 Elsevier Ltd. All rights reserved. Keywords: Dynamic response; Finite element method; Low-velocity impact; Plate; Structural intensity
Corresponding author. Tel.: +65-6419-1288; fax: +65-6778-0522.
E-mail addresses: [email protected] (Z.S. Liu), [email protected] (H.P. Lee), [email protected] (C. Lu). 0734-743X/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2004.06.010
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1. Introduction There are many practical situations where plate structures are subjected to transient impact loading of high intensity. Transient deformations are induced ranging from small-deflection linear elastic behaviour to large-deflection elasto-plastic behaviour with permanent deformation. Local damage and even failure of plates may take place. In such cases, it is essential to perform impact analysis for the purpose of plate design and/or verification. Structural intensity occurs as a result of dynamic impact loading. Despite their importance, however, impact effects are usually ignored or oversimplified in the dynamic analysis of plate structures. One reason is the complexity involved as contact-impact problems involve unknown boundary conditions. Specifically, the contact surface or points and the associated stresses and displacements are all unknown prior to the solution of the problem. Consequently, mathematical models of plate impact problems typically involve systems of inequalities and non-linear equations. Understandably, most contactimpact analyses concerning plates have to be carried out numerically such as by means of the finite element method (FEM). Traditionally, the impact analyses normally provide transient deformation and transient stress level. As the high-intensity transient phenomenon involved in impact precesses, the information of stress and deformation may be insufficient for further plate structure failure prediction and analysis. In order to provide more information for plate impact problems, the structural intensity approach can be adopted in the impact dynamic response. The concept of structural intensity was introduced to extend the vector acoustics approach to energy flow in structure-born sound fields [1]. The development of structural intensity formulations by Pavic [2] led to a growing interest in this field during the last two decades. Structural intensity is the power flow per unit cross-sectional area in elastic medium and it is analogous to acoustic intensity in a fluid medium due to structural vibration or dynamic response. Since the structural intensity field indicates the magnitude and direction of vibrational and transient energy flow at any point of a structure, it is very interesting to investigate structural intensity from a practical point of view. In addition, the structural intensity distribution can offer full information of energy transmission paths and positions of sources and sinks of mechanical energy. Dissipative elements, mechanical modification and active vibration control can be used for an alteration of energy flow paths within the structure and the amount of mechanical energy injected into the structure. Therefore, it is imperative to investigate the energy flow paths for the damage detection in plates subjected to transient dynamic loading. The structural intensity approach has been successfully used in the field of plate vibration to determine major vibration energy transmission paths in complex and noisy structures. As best as we know, very few reports have been published in the study transient impact phenomenon by using structural intensity approach [3]. The impact response of plate structures is often analysed by segmenting the problem into different response phenomena that occur. This generally involves considering the ‘‘global’’ and ‘‘local’’ responses of the structures by the impactor. The global response refers to the dynamic structural response of the plate configuration, whereas the local response refers to the indentation caused by impact. Because both phenomena occur simultaneously during the impact event, it makes the problems more complex. Consideration of both phenomena is important for accurate prediction of dynamic response of plate structures under impact loading. Prediction of the impact
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response of structures requires the knowledge of the following key parameters, namely, the impact force history, the pressure distribution between the impactor and the structure and the contact area. In previous studies [4–6], it is generally assumed that impact duration is large compared to the stress wave transient time in the target. That is, when the impact velocity is very low, dynamic response in the local contact area and response of wave problems can be ignored. However, the stress wave phenomenon should be considered in the normal low-velocity impact analysis of plate. By adopting the structural intensity approach, more information including transient energy transmission and path can be determined and it provides more transient phenomena in the plate impact analysis. In this paper, the methodology for predicting the dynamic response and structural intensity of a plate under low-velocity impact is presented. The FEM is adapted to perform dynamic impact response. In the impact process, the degenerated shell element is used to model plates and impactor. The dynamic contact-impact algorithm and the governing equations for both the target and impactor are derived based on the updated Lagrangian approach. Explicit integration algorithm has been adopted in the time integration process impact. The computational method for structural intensity is illustrated and formulas of structural intensity for plates are given in the time domains. Several low-velocity impact problems of plate structures with/without damping impacted by steel spherical projectiles are analysed and the results are interpreted by the structural intensity approach. The energy transmission and flow path in plates which are subjected to transient impact loading are presented. The novel structural intensity streamline representation is introduced to interpret energy flow path. Using the structural intensity technique and structural intensity streamline representation, the detailed transmission pattern of transient energy flows and the flow paths from the impact source of excitation to the sink in plates are demonstrated. The changes of energy flow in plates with and without dampers are discussed.
2. Formulations of contact-impact of plates and shells To treat the impact problem, normally the contact force method and energy balance method are used. Among various contact force methods, Hertzian method is the simplest in terms of formulation by using Hertzian’s contact law or its modification. Several researchers, including Sun and Yang [7], Koller and Busenhart [8], Aggour and Sun [9], Nosier and Reddy [10], Lee and Kwak [6], and Liu and Swaddiwudhipong [11], have adopted the Hertzian’s contact law in their studies. While the Hertzian method provides the global impact response of plates satisfactorily at a relatively small computing cost, its extension to tackle more complex problems becomes difficult locally, i.e. in the contact area. In order to deal with the local effect of impact, other methods, such as the Lagrange multiplier method, penalty method and augmented Lagrange multiplier method, are proposed. In the Lagrange multiplier method, the contact force is modelled by means of Lagrange parameter and contact conditions are satisfied exactly [12]. The penalty method assumes from the outset that the impenetrability constraints will be violated [13,14]. The augmented Lagrange multiplier method combines the penalty and Lagrange multiplier methods [15]. As the penalty method has many advantages that it can be extended to treat tangential force due to friction and the number of unknowns does not increase due to the enforcement of the
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contact constraints and that the system of equations remains positive definite, the penalty method fits very well into an explicit time integration of impact analysis and its implementation is straightforward [16,17]. In this study, the penalty method will be adopted to perform impact response analysis. In the modelling for treating impact problem of plates, different types of elements can be used. Liu and Swaddiwudhipong [11] derived a new formulation for the analysis of contact in degenerate shell elements. This formulation accounts for the transverse stress and strain through the shell thickness. Large deformations and rotations can be accounted for by invoking appropriate stress and strain measures. El-Abbasi and Meguid [18] presented another formulation for the analysis of contact in degenerate shell elements, allowing doublesided shell contact. Based on recent studies, it appears that degenerated shell elements offer an effective means to treat impact problems of plates and are adopted in the present study. As plate structures often undergo large deflection, nonlinear transient analysis of such structures under a low-velocity impact load is considered, adopting a layered approach [19]. Deformation components, strains and stresses are defined in local coordinate system. Updated Lagrangian approach is adopted in the derivation of formulation of the incremental form of governing equations. Finite element formulation incorporating the penalty algorithm is used to evaluate the contact force history. Explicit time integration algorithms may be adopted in time integration.
2.1. Degenerated shell element model On the basis of Mindlin plate theory, degenerated shell element can be formulated. By adopting isoparametric geometric description, the element can be used to represent thin and thick shell components with arbitrary shapes, circumventing the complexities of classical shell theory and differential geometry. The two main assumptions adopted in the degeneration process are: (i) normal to mid-surface before deformation remains straight after deformation and (ii) normal stress component is negligible in the constitutive equation. Transverse shear deformations are considered in the analysis. The Cartesian coordinates at any point of the shell can be uniquely defined in terms of nodal and thickness coordinates, as follows [20]: 2 3 2 x3 2 3 tk xk x n n X X z6 y7 6 7 6 7 ð1Þ N k 4 yk 5 þ N k hk 4 tk 5; x ¼ 4y5 ¼ 2 z k¼1 k¼1 tk zk z where k denotes the node number and n is the number of nodes in the element, N k the element shape function defined in surface z ¼ constant, z the natural coordinate in the direction perpendicular to the middle surface, and hk is the shell thickness. The thickness vector on the right-hand side of Eq. (1) is constructed from the nodal coordinates of the top and bottom surfaces. In the present study, five degrees of freedom are specified at each nodal point. They are the three displacement components (uk ; vk ; wk ) and two rotations (b1k ; b2k ) at node k as shown in
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Fig. 1. Nine-node quadratic degenerated shell element.
Fig. 1. The element displacement field can be expressed as 3 2 3 2 x 2 3 rk sxk uk u n n X X z6 y 6 7 6 7 y 7 b1k N k 4 vk 5 þ N k hk 4 rk sk 5 ; u¼4v5¼ 2 z b2k k¼1 k¼1 z rk sk w wk
ð2Þ
where rk and sk are the unit vectors of local Cartesian coordinates. For moderately thick-shell problems, degenerated shell element performs well generally. For thin-shell problems, however, over-stiff solutions arise due to shear and membrane-locking phenomena when full integration is carried out to evaluate the stiffness matrix. Zienkiewicz et al. [21] and Hughes et al. [22] proposed the reduced and selective integration techniques to alleviate the over-constraining effects of shear and membrane stiffness. An alternative approach is to adopt assumed strain fields as introduced by Bathe and Dvorkin [23] for four-node shell element and by Huang and Hinton [19] for higher order shell elements. Swaddiwudhipong and Liu [24] also adopted this approach to study the dynamic response of large-strain elasto-plastic plate and shell structures. In the present study, a new contact-impact formulation is developed based on degenerated shell element model. In order to consider contact-impact, the top and bottom surfaces of element are defined as exterior and interior surfaces, respectively. It is assumed that only the exterior surface of the shell elements has possible contact with another shell structure. The displacements and geometry of the shell element at contact surface can be defined as follows: 3 2 c3 2 3 2 x rk sxk uk u n n 16 y 6 7 X 6 7 X y 7 b1k r s uc ¼ 4 vc 5 ¼ ; ð3Þ N k 4 vk 5 þ N k hk 4 k k5 2 z b2k k¼1 k¼1 z c rk sk w wk
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2
xc
3
2
xk
3
2
txk
3
n n 16 y 7 6 7 X 6 7 X x c ¼ 4 yc 5 ¼ N k 4 yk 5 þ N k hk 4 tk 5: 2 z k¼1 k¼1 tk zc zk
ð4Þ
The contact is thus governed by the mid-surface nodes of the element, and no additional degrees of freedom are introduced in the system equation. 2.2. Finite element discretization for contact-impact problem According to the Hu-Washizu variational principle, the minimization problem can be stated as determination of nodal displacement or velocity, such that the potential energy is minimized while not violating the principles of impenetrability of equal and opposite reaction [25]. Introducing a suitable stress tensor in updated Lagrangian formulation and considering non-linear contact equations, the incremental linearized form of the virtual work, which includes the contact contribution, can be expressed as [16,26] Z Z Z n n tij dn Zij dV þ tij dn eij dV n Dijkl ekl dn eij dV þ t t t O O O Z Z ðdui ðrtþDt bi ru€ i Þ dV ðttþDt Þ du ds Ot Gtt Z þ ½ðn n dðduc Þ þ gtþDt dnÞ þ tN dðduc Þ dG ¼ 0: (5) N Gc
In the above equation, the quadratic terms of the contact virtual work are ignored. The detailed deriving of Eq. (5) can be referred to Liu et al’s paper [16]. Considering a displacement-based finite element formulation and applying the semidiscrete FEM to the variational Eq. (5) for a contact system, the incremental form of the dynamic FEM equations can be written as ð6Þ M tþDt d€ þ C tþDt d_ þ ðt K L þ t K t þ t K c ÞtþDt Dd ¼ tþDt R t F t F c ; t
t
t
t
t
where M and C are mass and damping matrices, ðtt K L þ tt K t þ tt K c Þ the global consistent tangent stiffness matrices. tt K L is the linear stiffness matrix; tt K t is the initial stress stiffness matrix; tt K c the global contact stiffness matrix. tþDt R the consistent force vector. tt F the equivalent nodal force vector resulting from the presence of the initial stresses in the element, and tt F c and tt F ec the equivalent nodal force vector from the contribution of global and element contact in the contact surface, respectively. The detailed expression for the above matrices can be referred to Liu and Swaddiwudhipong [11,16]. 2.3. Time-stepping algorithm for contact-impact problem Either implicit or explicit time integration can be used for solving non-linear contact system [16]. According to Wriggers et al. [27], for contact-impact problems without inherent shock waves, implicit integration can be used with considerable larger time steps than those adopted in the explicit scheme. Therefore, the Newmark’s method can be used for low-velocity impact problem.
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The effective tangent stiffness and effective residuals will be used in the algorithm as updated expression of this scheme. Within the solution algorithm, Newton–Raphson method will be used to achieve equilibrium within a time step. In the implicit time integration, a common technique used is the trapezoidal rule, which is Newmark’s method with parameters d ¼ 0:5 and a ¼ 0:25 [26]. On the other hand, explicit methods are often applied for short-duration impact contact problem. Explicit time integration is also suitable for contact-impact problems as the small time steps imposed by numerical stability can treat the discontinuities in contact impact. Another advantage of explicit algorithms is that the shells can first be integrated completely independently, as if they were not in contact. This uncoupled solution correctly indicates which parts of the shell are in contact. The detailed expression of time-stepping algorithm can be referred from the Liu et al’s paper [16,17].
3. Structural intensity in plates The instantaneous intensity component is a time-dependent vectorial quantity equal to the change of energy density in a given infinitesimal volume. Its ith component in time domain can be defined as [28,29] I i ðtÞ ¼ sij ðtÞ vj ðtÞ;
i; j ¼ 1; 2; 3;
ð7Þ
where sij ðtÞ is the component of the stress tensor at a point where i represents the normal direction of the area on which stress acts, j represents the direction of stress (i; j ¼ 1; 2; 3 corresponding to x, y, and z directions in a Cartesian co-ordinate system), and vj ðtÞ is velocity in the j-direction at time t. Based on pure bending plate theory, Pavic [2] derived the structural intensity formulas for a thin plate. In the impact dynamic response, large deformation may take place, and the more general formulas for structural intensity should be considered. The general 3-D structural intensity formulations for plate were developed by Ramano et al. [30] on the basis of the definition of structure intensity and 3-D elasticity. In the specific case of the plate mid-surface lying in the x-, yplane, the structure intensity can be expressed as [3] _ _ x þ v_N xy ; y_ y M x y_ x M xy þ uN I x ¼ ½wQ x
_ y y_ y M yx y_ x M y þ uN _ yx þ v_N y ; I y ¼ ½wQ
(8)
where M x , M y , M xy , M yx , N yx , N xy , Qx , Qy , N x , N y represent bending moments, twisting _ v_, w_ and y_ x , y_ y moments, shear forces, membrane forces, and in-plane shear force, respectively. u, are translational and rotational velocities, respectively. Compared to Pavic structural intensity _ yx þ v_N y , which corresponding to _ x þ v_N xy and uN formulas, Ramano’s has two more terms uN longitudinal and in-plane shear waves. The energy associated with these terms is not coupled with the bending wave energy in plates. Thus, the two terms represent a significant difference between Romano’s and Pavic’s formulation only when the in-plane motion of the structure is significant, which can occur with in-plane force. The difference and suitability of these two types of formulas were discussed in detail by Zhang and Adin [31].
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It should be noted that, in the present study, as the degenerated shell element is selected in the analysis, the five translational and rotational displacements or velocities are independent. As no other displacement relation assumption is made, the process of directly using the above equations should be more accurate than that adopted in the previous studies [30,31]. The reason the shear effects should be considered is that the classical theory, which is based on the pure bending assumption, is only applicable for the waves that are long in comparison with the thickness of the plates. As the wavelength diminishes, the velocity in the 3-D theory has as its upper limit the velocity of Rayleigh surface waves. Hence, the classical plate theory cannot be expressed to give good results for sharp dynamic transient response, i.e. in the impact case. The above formulas will be adopted in the interpreting energy flow in the plates.
4. Structural intensity streamline concept Vector fields may be either steady or changing with time. A single data cube represents a field at an instant of time and is necessarily equivalent to a steady field. To represent a changing or dynamic field, it is necessary to have a stack sampling the field at a set of time steps. The streamline technique displays the flow as lines everywhere parallel to the velocity field. The relative spacing of the lines indicates the speed of the flow. For steady flows, path lines and streamlines are identical. As the structural intensity is a vector field in the space, it is similar to the velocity field in the flow. Inspired by flow line representation in fluid mechanics, the structural intensity streamline representation is adopted to interpret the structural intensity transmission path. It is convenient to study the structural intensity or energy flow using this technique. Similar to fluid mechanics definition, the structural intensity streamline can be expressed as dr Iðr; tÞ ¼ 0;
ð9Þ
where r is energy flow particle position. The structural intensity of an energy flow element on such a streamline is perpendicular to r and parallel to dr. The cross product can be written as i j k ð10Þ I x I y I z ¼ 0: dx dy dz Thus, the differential equation describing structural intensity streamline is dx dy dz þ þ ¼0 Ix Iy Iz
ð11Þ
for 2-D plate, the structural intensity streamline reduces to dx dy þ ¼ 0: Ix Iy
ð12Þ
Every particle that passes any particular point will follow the streamline that goes through that point. A streamline is a line which is everywhere tangential to an energy flow particle’s structural intensity. By default, this means a streamline is the path an energy flow particle travels.
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5. Numerical examples and discussion 5.1. A steel plate hit by a steel sphere As shown in Fig. 2, consider a clamped 100 100 mm2 steel plate hit by a steel sphere at its centre. Initial conditions of zero displacement and zero velocity are assumed for the plate. The initial thickness of the plate, h, is 0.2 mm, while the properties of materials considered for this example are given in Table 1. The radius of the sphere is R ¼ 5:0 mm, and its velocity just before impact is V 0 ¼ 10:0 m=s. Two types of plate conditions are studied. For case 1, the plate has no any damping. For case 2, four dampers with damping ratio of 1200 N s/m are connected to plate, as shown in Fig. 2. The sphere has been modelled using shell elements which have same mass with sphere. The transient deflection at the centre of the plate is presented in Fig. 3. The maximum displacement at the centre of the plate takes place at about 0.25 ms. From Fig. 3, it can be seen that the time histories of maximum displacements are nearly same at before 0.35 ms for a plate with and without damping. After 0.35 ms, substantial difference can be seen for cases with and without dampers. In damping case, the central deformation of the plate rebounds to initial position and further to positive direction. The computed structural intensity vectors for the cases without damping and with four dampers are shown in Figs. 4a–h and 5a–h at different times, which correspond to 0.05–0.40 ms after impact, with an incremental time step of 0.05 ms. From Figs. 4 and 5, it can be seen that, with the time increase, the energy flow or structural intensity increases and moves toward to the plate boundary. Comparing Figs. 4 and 5, it can be demonstrated that, in the case of having dampers,
Fig. 2. Plate model which was subjected to impact by a sphere at the plate centre.
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Table 1 Material properties used for numerical analysis Properties
Steel (for plate target)
Steel (for sphere impactor)
Young’s modulus (GPa) Shear modulus (GPa) Density (kg/m3) Poisson’s ratio
206.0 79.85 7833.0 0.29
206.0 79.85 7833.0 0.29
0.4
Displacement (mm)
0.2 0.0 -0.20.0 -0.4 -0.6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
No damper Damping
-0.8 -1.0 -1.2 -1.4
Time (ms)
Fig. 3. Displacement history of plate at the centre.
the structural intensity from the source position flows to damper locations. The energy flow source and sinks can be clearly located. For no damping case, the structural intensity or energy flow moves from impact source towards the plate boundary. When it reaches the plate boundary, the energy flow is reflected and becomes quite uniformly distributed as shown in Fig. 4h. On the other hand, for the damper case, the energy flow pours to sink locations. This means that the damper can redirect the energy flow distribution. In order to demonstrate the structural intensity flow path, the structural intensity streamlines and Von Mises stress contour are depicted in Figs. 6 and 7 for a plate having no damper and having dampers at different time moments. From Figs. 6 and 7, it can be clearly seen that the transient energy flow transmits from impact area toward the boundary. The structural intensity streamlines detailed the energy flow transmission paths in the plate. It can also be seen that the stress wave propagating trends are different from the energy flow path. It can be seen that the pattern of energy flow varies with the time. At a duration of 0.25 ms after impact, at which the maximum impact force has been reached, the calculated structural intensity vectors demonstrate that there exist the transient virtual energy flow sources or sinks. This phenomenon is due to stress wave reflection from the plate boundary. Although there are no real sources in these areas, the transient structural intensity and its streamlines provide the possible weak points for transient dynamic response. As time increases, the damper’s effects on the structural intensity become more significant. This phenomenon can also be observed in the displacement time–history curve, as shown in Fig. 3.
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Fig. 4. (a–h) Structural intensity vectors at different times (no damper).
5.2. Cutout steel plate impacted by steel sphere To study the transient structural intensity flow phenomena, an irregular steel plate is analysed. The cutout plate is that of a rectangular plate with the dimensions of 50 125 mm2, with two rectangular cutouts of dimensions 25 12.5 mm2. The irregular plate is subjected to low-velocity impact by a steel sphere. The impact arrangement layout and analysis model are shown in Fig 8. In this case, the plate material properties are the same as the example in Section 5.1. The radius of
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Fig. 5. (a–h) Structural intensity vectors at different times (with dampers).
the sphere is R ¼ 5 mm, and its velocity just before impact is V 0 ¼ 10 m=s. In the model, the plate is clamped along all edges. The displacement history of the plate at impact centre is shown in Fig. 9. Fig. 9 shows that the maximum displacement takes place at about 0.23 ms. The computed structural intensity vectors, structural intensity streamlines and Von Mises stress contour are depicted in Figs. 10 and 11 for different time instances, which correspond to an after-impact time of 0.03–0.38 ms, with
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Fig. 6. (a–h) Structural intensity streamlines and Von Mises stress contours at different times (no damper).
increments of 0.05 ms, respectively. From Fig. 10, it can be clearly seen that the transient energy flow transmits from the impact area towards the boundary. From the structural intensity vectors plot, the energy flow source can be clearly located. When the energy flow reaches the boundary, the structural intensity is reflected and some virtual sources or sinks can be constructed. As the neck looks like a flow pipe, the energy passes through the neck to the other part of the plate with time increase. Finally, the constructed virtual sink or sources are destroyed and the structural
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Fig. 7. (a–h) Structural intensity streamlines and Von Mises stress contours at different times (with dampers).
intensity is redistributed. The structural intensity streamlines give the energy flow transmission paths in the plate in detail. It can also be seen that the stress wave propagating trends are different from the energy flow path. From Figs. 10 and 11, it can be seen that the pattern of energy flow varies with the time. After an impact time of 0.18 ms, the calculated structural intensity vectors demonstrate that there exist the virtual energy flow sources or sinks. This phenomenon is due to stress wave reflection from the boundary of the plate. Although there are no real sources in these
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Fig. 8. Irregular plate impacted by a steel sphere.
Displacement (mm)
0.4 0.2 0.0 -0.20.0 -0.4 -0.6
0.1
0.2
0.3
0.4
0.5
0.6
-0.8 -1.0 -1.2 -1.4
Time (ms)
Fig. 9. Displacement histories of plate at impact point centres.
areas, the transient structural intensity and structural intensity streamlines provide the possible weak points for transient dynamic response. In the transient dynamic response analysis, the structural intensity magnitude and direction should be considered for further studying the impact dynamic damage mechanism. In order to reduce the failure possibility in the impact case, these areas also should be paid more attention.
6. Concluding remarks In this paper, the governing equations of impact for plate structures are formulated. Degenerated nine-node shell elements with assumed membrane and transverse shear strains are
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Fig. 10. (a–h) Structural intensity vectors of the irregular plate at different moments.
employed to treat impact problems under low-velocity impact. A dynamic contact-impact algorithm, employing penalty constraint enforcement, is presented in the finite element context. The structural intensity method has been adopted to predict the information of energy flow for the plates subjected to impact dynamic loading. The structural intensity streamline is introduced to interpret the energy flow path. The present results demonstrate that the structural intensity fields and structural intensity streamlines can be used to clearly indicate the source, the sink and the
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Fig. 11. (a–h) Structural intensity streamlines and Von Mises stress contours at different times.
direction of energy flow from the source to the sink. It also can be an effective tool for transient dynamic response control for plates, provided the power flow pattern, energy density, and structural intensity path in plates can be modified and controlled by properly arranging dampers. The structural intensity method can be adopted in the design of plate structures and may act as a new criterion for plate structural design under dynamic loading. The study demonstrates that the structural intensity method and structural intensity streamline technique are potential means for studying dynamic failure and fracture of plates subjected to low-velocity impact. The proposed
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method provides a basis of further study in terms of impact contact damage and for composite plate and shell structures under low-velocity impact.
References [1] Noiseux DU. Measurement of power flow in uniform beams and plates. J Acoust Soc Am 1970;47:238–47. [2] Pavic G. Measurement of structure borne wave intensity, Part I: formulation of the methods. J Sound Vibration 1976;49(2):221–30. [3] Lee HP, Lu C, Liu ZS, Xu XD. The structural intensity analysis of plates under dynamic loading. In: Mathisen KM, Kvamsdal T, Okstad KM, editors. Proceedings of the 16th Nordic Seminar on Computational Mechanics. Trondheim, Norway: Trondheim; 2003. p. 145–8. [4] Sun CT, Liou WJ. Investigation of laminated composite plates under impact dynamic loading using a threedimensional hybrid stress finite element method. Comput Struct 1989;33:879–84. [5] Mittal RK. A simplified analysis of the effect of transverse shear on the response of elastic plates to impact loading. Int J Solids Struct 1987;23:1191–203. [6] Lee DI, Kwak BM. An analysis of low-velocity impact of spheres on elastic curved-shell structures. Int J Solids Struct 1993;30:2879–93. [7] Sun CT, Yang SH. Contact law and impact response of laminated composite. NASA CR-159884. Washington, DC: NASA; 1980. [8] Koller MG, Busenhart M. Elastic impact of spheres on thin shallow spherical shells. Int J Impact Eng 1986;4:11–21. [9] Aggour H, Sun CT. Finite element analysis of a laminated composite plate subjected to circularly distributed central impact loading. Comput Struct 1988;28:729–36. [10] Nosier A, Kapania RK, Reddy JN. Low-velocity impact of laminated composites using a layerwise theory. Comput Mech 1994;13:360–79. [11] Liu ZS, Swaddiwudhipong S. Response of plate and shell structures due to low velocity impact. J Eng Mech ASCE 1997;123(12):1230–7. [12] Hughes TJR, Taylor RL, Sackman JL, Curnier A, Kanoknukulchai W. A finite element method for a class of contact/impact problems. Comput Meth Appl Mech Eng 1976;8:249–76. [13] Belytschko T, Neal M. Contact-impact by the pinball method algorithm with penalty and Lagrangian. Int J Numer Meth Eng 1991;31:547–72. [14] Belytschko T, Yeh I. The splitting pinball method for contact impact problems. Comput Meth Appl Mech Eng 1993;105:375–93. [15] Wriggers P, Zavarise G. Application of augmented Lagrangian techniques for non-linear constitutive laws in contact interfaces. Commun Numer Methods Eng 1993;9(10):815–24. [16] Liu ZS, Swaddiwudhipong S, Koh CG. Contact-impact dynamic response of shell structure under low velocity impact. In: Obrebski JB, editor. Proceedings of the International Symposium on Lightweight Structures in Civil Engineering. Warsaw, Poland: IASS; 2002. p. 377–84. [17] Swaddiwudhipong S, Koh CG, Liu ZS. Numerical study of shell structures under low velocity impact. J Int Assoc Shell Spatial Struct 2002;43(140):159–70. [18] El-Abbasi N, Meguid SA. Large deformation analysis of contact in degenerate shell elements. Int J Numer Meth Eng 1998;43:1127–41. [19] Huang HC, Hinton E. A nine nodes Lagrangian Mindlin plate element with enhanced shear interpolation. Eng Comput 1984;1:369–79. [20] Huang HC. Static and dynamic analyses of plates and shells. London, Berlin: Springer; 1989. [21] Zienkiewicz OC, Taylor RL, Too JM. Reduced integration technique in general analysis of plates and shells. Int J Numer Meth Eng 1971;3:275–90. [22] Hughes TJR, Cohen M, Haroun M. Reduced and selective integration techniques in finite element analysis of plate. Nucl Eng Design 1978;46:203–22.
ARTICLE IN PRESS Z.S. Liu et al. / International Journal of Impact Engineering 31 (2005) 957–975
975
[23] Bathe KJ, Dvorkin EN. A four-node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation. Int J Numer Meth Eng 1985;21:367–83. [24] Swaddiwudhipong S, Liu ZS. Dynamic response of large strain elasto-plastic plate and shell structures. ThinWalled Struct 1996;26:223–39. [25] Belytschko T, Liu WK, Moran B. Nonlinear finite elements for continua and structure. Baffins Lane, Chichester, England: Wiley; 2000. [26] Bathe KJ. Finite element procedures. Englewood Cliffs, NJ: Prentice-Hall; 1996. [27] Wriggers P, Vu VT, Stein E. Finite element formulation of large deformation impact-contact problems with friction. Comput Struct 1990;37(3):319–31. [28] Pavic G. Vibroacoustical energy flow through straight pipes. J Sound Vibration 1992;154(3):411–29. [29] Gavric L, Pavic G. A finite element method for computation of structural intensity by the normal mode approach. J Sound Vibration 1993;164(1):29–43. [30] Romano AJ, Abraham PB, Williams EG. A Poynting vector formulation for thin shell and plate, and its application to structural intensity and source localization, Part 1: theory. J Acoust Soc Am 1990;1166–75. [31] Zhang Y, Adin Mann III J. Measuring the structural intensity and force distribution in plate. J Acoust Soc Am 1996;99(1):345–53.