Laurent Aguerreche
email: laurent.aguerreche@irisa.fr-thierry.duval@irisa.fr-anatole.lecuyer@inria.fr

Thierry Duval

Anatole Lécuyer

Evaluation of a Reconfigurable Tangible Device for Collaborative Manipulation of Objects in Virtual Reality

Keywords: (according to ACM CCS): H.5.1 [Information Interfaces and Presentation (e.g. HCI)]: Multimedia Information Systems-Artificial, augmented, and virtual realities I.3.6 [Computer Graphics]: Methodology and Techniques-Interaction techniques

In this paper we introduce an evaluation of a Reconfigurable Tangible Device (RTD) for collaborative manipulation of objects in virtual environments. The considered RTD, called RTD-3, has a triangular shape that naturally provides three points of manipulation. The shape of the tangible triangle can be reconfigured at any time as its branches can be shrunk or stretched by users at will. Thanks to this simple shape the RTD-3 can be easily attached to any 3D virtual object and fully defines its virtual motion in 6 Degrees of Freedom. We have conducted an experiment to assess the potential of the RTD-3 and compare it with classical techniques used for collaborative virtual manipulation. Participants were asked to manipulate and assemble, in a collaborative manner, virtual parts. Our results suggest that the physical manipulation proposed by the tangible device is significantly preferred by participants in terms of immersion, realism of interaction and preparation to the real task. Although our approach is slightly slower than the other tested methods, it produces the fewest collisions.

Introduction

Object manipulation is one of the most fundamental tasks of 3D interaction in Virtual Reality (VR) [BKLP04]. Collaborative manipulation of virtual objects by multiple users is a very promising area for Collaborative Virtual Environments (CVE) [BGRP01]. Collaborative manipulation seems indeed necessary in many different applications of VR such as virtual prototyping, training simulations or assembly and maintenance simulations [START_REF] Ruddle R | Symmetric and Asymmetric Action Integration during Cooperative Object Manipulation in Virtual Environments[END_REF]. In such virtual collaborative tasks, all the users are expected to participate naturally and efficiently in the manipulation of virtual objects.

Although most collaborative systems support simultaneous manipulation of different objects by different users, generally only one user at a time can manipulate a virtual object. Interaction metaphors that are usually used for single-user 3D interaction, such as virtual hands, virtual rays or virtual 3D cursors, must be adapted for collaborative 3D virtual manipulation. This is why we have proposed a new physical device named Reconfigurable Tangible Device (RTD) [START_REF] Aguerreche | Reconfigurable Tangible Devices for 3D Virtual Object Manipulation by Single or Multiple Users[END_REF] to enable collaboration on a shared virtual object through a tangible device and precise positioning of users' real hands. The RTD maintains the distance between users hands. Besides users can modify the shape of the RTD to better fit the virtual object they intend to manipulate.

We compared the RTD-3 (a tirangular RTD) with two classical techniques for manipulating collaboratively (with two users) virtual objects: the Mean technique and the Separation of Degrees of Freedom (DoF). The first technique simply averages translations and rotations applied by the two users. The second technique allows two users to make substantially different actions: one is controlling the orientation of a virtual object while the other applies translations. This paper is structured as follows: first, we give an overview of related work in the field of collaborative interaction in virtual environments and the use of tangible devices in virtual reality. Second, we briefly describe the concept r a f t d r a f t and implementation of our RTD-3. Third, we describe the evaluation of the RTD-3 and its comparison with the Mean technique and the Separation of DoF. The paper ends with a general discussion and a conclusion.

Related Work

Here we present a state of the art about metaphors and tangible devices for collaborative 3D interactions.

Two-Handed Object Manipulation

Several 3D interaction techniques have been proposed to manipulate virtual objects with the two hands of a single user [HPPK98]. But only a few of them, such as "grab-and-carry", "grab-and-twirl" and "trackball" techniques [START_REF] Cutler | Two-Handed Direct Manipulation on the Responsive Workbench[END_REF], enable users to position and rotate virtual objects. The "grab-and-carry" technique [START_REF] Cutler | Two-Handed Direct Manipulation on the Responsive Workbench[END_REF] is a 5 Degrees-of-Freedom (DoF) bimanual symmetric tool that enables users to carry and turn an object around with both hands. Object roll is not supported since it is not possible to determine rotation around the axis defined by the positions of the two hands. The "grab-and-twirl" technique extends the "grab-and-carry" technique, adding the sixth DoF using either the left hand's roll, the right hand's roll, or a combination of both. The "trackball" technique is a bimanual asymmetric tool that enables users to use the non-dominant hand to position a virtual object while the dominant hand rotates this object around its center.

In our opinion, these techniques are not very representative of real world interactions considering users' movements. In addition, they could probably not be used to simulate interactions with large or cumbersome objects that a user cannot manipulate alone.

Multi-User Object Manipulation

Several approaches are suitable to combine two users' movements to obtain the final movement of a virtual object [START_REF] Ruddle R | Symmetric and Asymmetric Action Integration during Cooperative Object Manipulation in Virtual Environments[END_REF]. A first approach consists in adding the two motions (asymmetric integration of movements). A second approach is to average the two motions. A third approach aims at keeping only the common part (intersection) of the two motions (symmetric integration of movements). But none of these combinations seems ideal. Indeed, the intersection technique is the more relevant when the two users have to perform a very similar action, whilst the average technique is preferred when users have to perform different tasks [START_REF] Ruddle R | Symmetric and Asymmetric Action Integration during Cooperative Object Manipulation in Virtual Environments[END_REF].

The Bent-Pick-Ray [RHWF06] metaphor enables several users to simultaneously co-manipulate a virtual object. This technique merges users' inputs according to the amount of hand movement a user does with one input device. Rotations are computed with a spherical linear interpolation (Slerp)

[Sho85], while the translations are interpolated using only offset transformations.

The SkeweR technique enables multiple users to simultaneously grab any part of a virtual object through special points called "crushing points" [DLT06]. To determine the translation and the rotation of a grabbed object, SkeweR considers positions of those points. A problem remains for determining the rotation along the axis linking the two crushing points. A similar technique seems to be used to construct a virtual gazebo [START_REF] Roberts | Constructing a Gazebo: Supporting Teamwork in a Tightly Coupled, Distributed Task in Virtual Reality[END_REF]. Two users manipulate a beam by grabbing its extremities. But no solution is proposed for the sixth DoF. This beam manipulation has been reproduced by using two virtual hands but simply using their average position in order to provide a position for the manipulated virtual beam [START_REF] García | Enhancing Collaborative Manipulation through the Use of Feedback and Awareness in CVEs[END_REF]. In [SJF09], Salzmann et al. use two optical markers to let two users manipulate a windshield. Authors also use the simple averaging of translation and rotation to move the virtual windshield.

Another kind of collaborative manipulation consists in splitting the task among users [PBF02]. In this case, the number of DoF that each user can access and control is limited: one user controls rotation of the object while the other one is limited to translation. This can be compared to the Two-Hand "trackball" technique [START_REF] Cutler | Two-Handed Direct Manipulation on the Responsive Workbench[END_REF].

The concept of a 3-Hand Manipulation Technique has been introduced in [START_REF] Aguerreche | 3-Hand Manipulation of Virtual Objects[END_REF]. It is a 3D interaction technique for 6 DoF multi-user collaborative manipulation of 3D objects. It enables the determination of virtual object position and orientation through only positions of three non-aligned manipulation points placed on the surface of this object. These manipulation points can be used by three different "hands" of two or three users.

Tangible Devices for Object Manipulation

A tangible device (or prop) is a real object that users can hold to move a virtual object and feel a passive tactile feedback. Such tangible interfaces are often preferred by people over non-physical interfaces [SJF09]. However, several studies show that they do not always lead to better performance [HTP * 97] [WR99]. Nevertheless, passive tactile feedback can be used to increase presence and improve training effectiveness in virtual environments [IMWB01].

Many adhoc tangible interfaces have been proposed to mimic real objects in a virtual world. In this case, users may have to hold a scale model for interaction with the virtual environment as in [START_REF] Hinckley | Passive real-world interface props for neurosurgical visualization[END_REF]. If some tangible interfaces let users use both hands, to our knowledge usual tangible interfaces are generally limited to single-user interactions.

Tangibles interfaces can also be designed for helping people to coordinate their movements during a collaborative manipulation. In same distance. As such, the prop acts as haptic link between them. As position and orientation are given by only one optical marker on the top of the prop, this technique is limited to one point of manipulation. The shape of the prop also restricts its use to cases where users use only one of their hands or keep their two hands very close. Some tangible interfaces can be re-designed and modeled, such as MERL bricks [AFM * 99] or the Active-Cubes [WIA * 04] where users can assemble several tangible blocks that the system would later match with 3D virtual models. Other approaches consist in deforming a malleable TUI [START_REF] Smith R | Digital Foam Interaction Techniques for 3D Modeling[END_REF] or balance between malleability and rigidity such as the Senspectra TUI [LPI07] or the Glume TUI [PLI06]. However, none of these reconfigurable tangible devices are rigid enough to be shared by several users.

Conclusion

Separate motions of several users' inputs (from several hands or users) can be used to define the final motion of a virtual object. However, due to the complexity of current VR interfaces, no universal software solution has been proposed to naturally apply a motion to a co-manipulated object.

While tangible interfaces may be interesting for better collaboration between users, a lack of a universal tangible device for collaborative manipulation is apparent: to the authors' best knowledge no previous work has been done in the area of reconfigurable tangible user interfaces to find a good balance between easy reconfigurability and rigidity for the purpose of collaborative object manipulation in VR.

We assume that coupling collaborative interaction techniques with a rigid reconfigurable tangible device can improve the overall collaboration and provide a haptic passive link for collaborative manipulations between users. This will be addressed in the following sections with the evaluation of the Reconfigurable Tangible Device.

The Reconfigurable Tangible Device

The Reconfigurable Tangible Device (RTD) has been introduced in [START_REF] Aguerreche | Reconfigurable Tangible Devices for 3D Virtual Object Manipulation by Single or Multiple Users[END_REF]. Its aim is to propose a universal physical interface that can match the shape of any virtual object. Besides RTD has been designed to let at least two users interact together with the same virtual object, and to let them precisely place their hands on the virtual object to grab it. Toward this goal, the RTD provides stretchable physical links between physical handles. Each of these handles matches one virtual handle in order to obtain a virtual object that is somehow "embodied" in the RTD.

A Reconfigurable Triangle for Three-Point Manipulation

In this paper, we use the instance of the RTD that is based on only three points: the RTD-3. This choice has been inspired by the 3-Hand Manipulation technique which has been introduced in [START_REF] Aguerreche | 3-Hand Manipulation of Virtual Objects[END_REF]. This technique determines the position and orientation of a virtual object through only positions of three non-aligned manipulation points placed on it. More generally, the RTD-3 could enable to grab any part of an object to manipulate even if it is in an inner or outer part of the virtual object. As a result, two or three users can move, resize or reshape a virtual object thanks to this device.

The RTD-3 is made up of three branches connected together by a pivot. Each branch can be shrunk or stretched (see Figure 1) by users by pulling a button to unblock/block a branch. When users want to attach the device to an object, they start to set the relevant branch lengths. Then they move the three virtual points associated with the device in order to touch the virtual object to manipulate. After selecting the object, users can begin to manipulate it. Varying branch lengths and using many angles let users obtain small or large triangles with arms length varying from 38 cm up to 95 cm and with angles varying from 20 degrees up to 130 degrees as a result. Users are able to grasp not only flat objects but also long, round or cubic objects. Objects can be grasped horizontally or vertically.

Implementation

The Reconfigurable Tangible Device has been evaluated within a virtual reality centre involving an ART optical tracking system. The infrared cameras were placed around a large screen facing users to track positions and orientations.

Optical markers are placed at each corner of the RTD-3. Three virtual pointers are associated with the positions of these markers, they are used to point and touch the manipulated object to begin the manipulation. Our implementation uses ray-casting coming from one pointer to add a manipulation point where the ray hits the virtual object.

To manipulate the RTD-3, one user can put one hand on one corner while the other user puts their hands on the two remaining corners. They can move the triangle seamlessly together by applying movements to the device. The beginning of the manipulation is triggered by an external button click. Buttons could also be incorporated directly on the RTD-3 to activate/release each virtual pointer individually. r a f t d r a f t

Evaluation

The objective of our evaluation was to compare the RTD-3 with two classical techniques for collaborative virtual manipulation: the Mean and Separation of DoF techniques. The proposed task is a "pick-and-place" task involving two users in the manipulation of a virtual car hood described in [START_REF] Aguerreche | Comparison of Three Interactive Techniques for Collaborative Manipulation of Objects in Virtual Reality[END_REF]. We collected task completion time, number of collisions, distance covered by the virtual hood, distance covered by the users' hands, and a questionnaire on users' subjective preferences. Preliminary results of this evaluation about subjective preferences have been briefly presented in [START_REF] Aguerreche | Reconfigurable Tangible Devices for 3D Virtual Object Manipulation by Single or Multiple Users[END_REF]; here, we now evaluate the results in full detail.

Three Interaction Techniques to Compare

We now analyse three different interaction techniques, with each technique covered in a separate section.

Technique 1: the Reconfigurable Tangible Device

(RTD-3)

The RTD-3 was implemented as previously described. One user supports two corners of the RTD-3 with his hands while the other user supports the remaining corner (Figure 2). Before the selection, users can adjust the shape of the RTD-3 to fit the shape of the virtual object. When the manipulation has begun, users are not allowed any longer to modify the shape of the RTD-3. The Mean technique [RSJ02] combines user's actions by averaging positions and orientations that they provide. In our implementation [START_REF] Aguerreche | Comparison of Three Interactive Techniques for Collaborative Manipulation of Objects in Virtual Reality[END_REF], this technique is only concerned with users' movements and not absolute positions. So users are free to place themselves anywhere in the tracked area and (for instance) to stand far from their counterpart. Each user holds only one marker (Figure 3).

The result of interaction will be as follows: If users do opposite movements then the virtual object will remain almost stationary. If one user stays inactive then the other user must move excessively to apply motion. If users do almost the same movements at the same time then the car hood will move similarly. Users do not have to point to the hood before selecting it, instead they have to ask an operator to initiate the manipulation. The Separation of DoF splits the control of the degrees of freedom of the motion among users [START_REF] Ruddle R | Symmetric and Asymmetric Action Integration during Cooperative Object Manipulation in Virtual Environments[END_REF]. In our implementation, this technique shares common aspects with the Mean technique for placement and movement of users since people act through their movements, rather than absolute positions in space. Actions of users are separated: one is restricted to manipulating translations whilst the other one is bounded to rotations (Figure 4).

As seen with the Mean technique, each user holds only one optical marker and users do not need to select the hood before manipulating it. One marker is dedicated to translations and the other is dedicated to rotations.

Method

We now describe the method behind the experiment in terms of apparatus required, procedure carried out and the experimental plan.

Apparatus

Users were staying in front of a large screen with stereoscopic images (3 m large and 2 m height). The tracked area for the ART infrared cameras was 4 x 4 m. Users wore shutter glasses and they shared the same point of view (heads were not tracked).

The simulation used Bullet for physics and Ogre3D for 3D graphics. We provided a virtual moving camera that was following the car hood such that users did not have to move their body out of the tracked area. Thus, users had to walk a maximum of two steps in front of them during the simulation to achieve the task.

No sound was provided but yellow particles were emitted visually during virtual collisions. They were triggered when two objects collided. Shadows were provided to help users to perceive the scene depth.

Procedure

24 volunteers to participated in our study (20 male, 4 female). Their average age was 26.7 years old. Few users had experience with 3D interaction. Most of the users were computer science students, software engineers, computer science researchers or teachers.

The task to complete is described in Figure 5. When manipulation starts, users have to move the virtual car hood outside a Z-shape. For users, this shape forces them to frequently rotate the hood to pass the Z-shape. Therefore, users have to coordinate their movements to translate and rotate the object.

Once the virtual car hood is out of the Z-shape, users have to walk one or two steps in front of them to place the hood on the final virtual support. This support is also an abstract shape but made up of two stems that have to be aligned with the holes of the virtual hood. Furthermore, a T-shape is placed on one side of this support to force users to 1) orientate the virtual hood almost vertically, 2) align the hole of the virtual hood with the stem at the same side than the T-shape, 3) move the virtual hood towards the ground, 4) at the same time continuing to translate the virtual hood and orientate it horizontally.

For each technique, users received explanations about how it works. Then users had few minutes to practice the technique before doing the measured task. They were free to ask any question of the instructor during the practice.

An experimental task in the real world was also proposed to help users in understanding the usefulness of practicing in the virtual world before trying to make a similar task in the real world. This "real-world task" is illustrated Figure 6. Steps are as follows: 1) initial position, 2) passing the "elbow" of the "Z-shape", 3) passing between the "T-shape" and a stem, 4) reaching the final position.

Experimental Plan

Experiments were conducted with 12 pairs of users. Each pair of users tested the three techniques. Six groups of user corresponding to the 6 orders of presentation of the 3 techniques participated. Two virtual scenes were available: one being the "mirror" of the other. The type of the scene was selected randomly. Each pair of users had to pass 4 times each technique, which gives a total of 12 trials. The total duration of the experiment was 40 minutes.

For each technique and each trial, we measured the time needed to complete the task and the number of collisions between the manipulated virtual object and the other objects of the world. We also measured the distance covered by the virtual car hood and by the users' hands. For each pair of users and each technique, we recorded the 2 best attempts of the users in terms of time spent to complete the task. We also collected the answers of participants to the subjective questionnaire.

For each criterion, we have performed a global single factor ANalysis Of VAriance (ANOVA) to compare the three r a f t d r a f t

Results

Table 1 shows the time spent (in seconds) to complete each task with each technique, and the number of collisions between the car hood and the other objects of the virtual world. Table 2 shows the distance covered by the car hood and the users' hands during the task with each technique.

Task Completion Time

A global single factor ANOVA revealed a significant main effect of the technique used on the completion time of the task (F(2, 69) = 4.9, p = 0.0102). The difference was highly significant between RTD-3 and Mean (F(1, 46) = 9.63, p = 0.0033), but not between RTD-3 and Separation (F(1, 46) = 2, p = 0.1639) nor between Mean and Separation (F(1, 46) = 3.1, p = 0.0848). The preliminary conclusion about task completion time is that RTD-3 seems significantly slower than Mean.

Collisions

A global single factor ANOVA revealed a significant main effect of the technique used on the number of collisions during the task (F(2, 69) = 14.57, p < 0.0001). The difference was not significant between RTD-3 and Mean (F(1, 46) = 1.16, p = 0.28), but it was highly significant between RTD-3 and Separation (F(1, 46) = 26.48, p < 0.0001) and between Mean and Separation (F(1, 46) = 13.99, p = 0.0005). The preliminary conclusions about collisions is that RTD-3 and Mean seem significantly more precise than Separation. Table 2: Average distance (in meters) covered by the car hood and the users' hands.

Distance Covered by the Users' Hands

For the RTD-3, we considered the three hands involved in the manipulation, for the Mean we considered the two hands involved in the manipulation, and for the Separation we considered only the hand of the user who was translating the car hood. Here again, a global single factor ANOVA revealed a significant main effect of the technique used on the distance covered by the hands during the task (F(2, 141) = 8.52, p = 0.0003). The difference was significant between RTD-3 and Mean (F(1, 118) = 12.2, p = 0.0027) and between RTD-3 and Separation (F(1, 94) = 9.09, p = 0.0033), but not between Mean and Separation (F(1, 70) = 0, p = 0.97).

Subjective Ratings

We recall here the subjective results that have already been presented in [START_REF] Aguerreche | Reconfigurable Tangible Devices for 3D Virtual Object Manipulation by Single or Multiple Users[END_REF]. At the end of the experiment, each user was asked to fill a questionnaire with subjective ratings (using a 7-point Likert scale) for the 3 techniques according to the following criteria: training for the same task in the real world (Training), realism of the manipulation (Realism), feeling of presence in the virtual environment (Presence), fatigue during the manipulation (Fatigue), and how much they liked the technique (Like). Table 3: Absolute rating of the techniques using a 7-point Likert scale.

An Analysis of Variance (global ANOVA) revealed that the technique used had a significant effect on the rating of the Realism rating (F(2, 69) = 22.45, p < 0.00001). For this criterion, the RTD was found significantly better rated than the Mean technique (F(1, 46) = 29.02, p < 0.00001), and than the Separation (F(1, 46) = 35.71, p < 0.00001). The ANOVA also revealed that the technique used had a significant effect on the Presence criterion (F(2, 69) = 9.72, p = 0.0002). For this criterion, the RTD was again found significantly better rated than the Mean technique (F(1, 46) = 8.41, p = 0.0057), and Separation (F(1, 46) = 16.73, p = 0.0002). Last, the ANOVA also revealed that the technique used had a significant effect on the Training criterion (F(2, 69) = 9.71, p = 0.0002). For this criterion, the RTD was again found significantly better rated than the Mean technique (F(1, 46) = 11.06, p = 0.0017), and Separation (F(1, 46) = 16.66, p = 0.0002). No significant effects were found for the other criteria: Fatigue (F(2, 69) = 0.03, p = 0.85) and Like (F(2, 69) = 0.04, p = 0.9584).

Discussion

Our evaluation aimed at assessing the potential of the RTD-3 and comparing it with other classical interaction techniques used for collaborative virtual manipulations. From our results we can first stress that the subjective questionnaire of participants turns strongly in favor of the RTD-3. These results complement findings of Salzmann et al. [SJF09] wherein users tend to prefer prop-based interactions over pure virtual interactions. On top of that, the RTD-3 is a good candidate for training people to work on two-user manipulation tasks as our users tend to feel themselves more prepared for a real task when using it. In [SJF09], participants were manipulating a windshield.

The RTD-3 seems to be more realistic mainly for the user who can use two hands. In his point of view, the two hands move in a very realistic way to move an object: raising one hand while lowering the other one rotates a virtual object. We think that this gesture is also responsible for the feeling of a better preparation for the real task. All the same, the user using only one hand still feels him involved in the interaction because of the haptic link of the prop.

Althought the distance covered by the hood is nearly the same with the three techniques, the distance covered by the users' hands is significantly more important with the RTD-3. It is not a surprise because users have to move their hands in opposite ways to make the hood rotate. This leads to more body movements, so it can explain that the RTD-3 can be slower than the two other techniques and that it can also be more tiring. Nevertheless, these large movements also provide a better precision during the manipulation.

We observed that users were very concerned about achieving the correct movements with the RTD-3. Usually, users were helping each other with verbal exchanges. Unrealistic movements with the Mean technique favored it. By contrast, the RTD-3 provides many precise ways to orientate a virtual object while the Mean technique can act seamlessly only when both users are coordinated. So the RTD-3 aims at letting each user interact with the same capabilities, and the Mean technique leads to a compromise solution. In fact, some users described the Mean technique as a kind of lowpass filter. Salzmann et al. [SJF09] found that a pure virtual technique takes longer to complete a manipulation task than a prop-based technique. This may be explained by the chosen implementations of the Mean technique. The implementation of [SJF09] averages users' positions while our implementation averages users' movements. Perhaps our implementation lets users adopt a more confortable posture since they were not constrained to remain close each other. We note however that Salzmann et al. [SJF09] observed that some pairs of users were able to reach times comparable to the prop manipulation with their implementation of the Mean technique.

Finally, results suggests that the RTD-3 provides users with a better feeling of training, realism and presence than the Separation technique. Indeed, this last technique carried the most mixed users' opinions in the reporting questionnaire and during informal discussions with them. Some pairs of users were acting in a very mechanical fashion: one user was translating the hood whilst the other was turning it exactly at the right time. For other pairs, one user was faster than the other and thus provided help with vocal orders to the second. With such pairs of users and no help to coordinate their gestures, the Separation technique could lead to difficulties for someone who is unable to anticipate his partner's gestures.

Conclusion

The Reconfigurable Tangible Device (RTD) is a universal physical interface that can match many shapes of virtual objects in collaborative virtual manipulations. The RTD-3 is an instance of the RTD device that uses the translation motions of three manipulation points to fully determine the resulting 6 DoF motion of the manipulated object.

We conducted an experiment to compare the RTD-3 with classical collaborative interaction techniques: the Mean of interactions and the DoF Separation. Objective results show that the RTD-3 ends with slower task completion time than the Mean technique probably due to the complex movements of the two-handed user, and that the RTD-3 is as precise as the Mean technique. Subjective results show that our technique provides a better sense of immersion and better realism. According to users, the RTD-3 provides a better knowledge transfer to the real world.

The evaluation suggests that the RTD-3 could be used in many collaborative applications such as for training people in virtual environments to do assembly or maintenance tasks in a collaborative manner.

Future Work

We have constructed an instance of RTD with four coplanar points to enable each user to use two hands in a symmetric way, we must now conduct new evaluations to compare it to the RTD-3. We have also constructed another instance of RTD with four non-coplanar points that draw a tetrahedron which could better match complex 3D shapes. We could also construct other instances of RTD with more points, for example to allow more than two users to interact with the same virtual object. We also plan to use the RTD for dynamically resizing or reshaping virtual objects.

  [SJF09], Salzmann et al. propose a prop for two-user manipulation that maintains the users' hands at the , T. Duval & A. Lécuyer / Evaluation of the RTD-3 for Collaborative Manipulation of Objects in Virtual Reality

Figure 1 :

 1 Figure 1: Minimal and maximal configurations of the Triangular Reconfigurable Tangible Device (RTD-3).

Figure 2 :

 2 Figure 2: Use of the RTD-3 technique.

Figure 3 :

 3 Figure 3: Use of the Mean technique. Users are standing in a similar way since they have to synchronize their gestures to optimize the task.

Figure 4 :

 4 Figure 4: Use of the DoF separation technique. The user standing on the left is orientating the hood. The other participant is translating it.

Figure 5 :

 5 Figure 5: Experimental task. Column on the left: two users are achieving the task with the RTD-3. Column on the right: movements of the virtual car hood.Steps are as follows: 1) initial position, 2) passing the "elbow" of the "Z-shape", 3) passing between the "T-shape" and a stem, 4) reaching the final position.

Figure 6 :

 6 Figure 6: Real-world task: manipulation of a "real" hood made of cardboard.

4. 3

 3 .3. Distance Covered by the Car Hood A global single factor ANOVA revealed a significant main effect of the technique used on the distance covered by the hood during the task (F(2, 69) = 5.5, p = 0.0061). The difference was significant between RTD-3 and Mean (F(1, 46) = 5.13, p = 0.0283) and between Mean and Separation (F(1, 46) = 0.71, p = 0.0032), but it was not significant between RTD-3 and Separation (F(1, 46) = 1.34, p = 0.2525).

  , T. Duval & A. Lécuyer / Evaluation of the RTD-3 for Collaborative Manipulation of Objects in Virtual Reality

Table 1 :

 1 Average time spent (in seconds) and number of collisions to complete the task in the virtual environment.

		Time (in seconds)	Number of collisions
		Mean	SD	Mean	SD
	RTD	26.22	σ = 9.7	151.88	σ = 38.59
	Mean	18.34	σ = 7.38	166.38	σ = 51.63
	Separation	22.44	σ = 8.39	227.54	σ = 59.03

Table 3

 3

	shows the results of the
	questionnaire.

c The Eurographics Association 2011.


J Dean

C S Dunleavy

P M Brown

T W Clyne

Energy Absorption during Projectile Perforation of Thin Steel Plates and the Kinetic Energy of Ejected Fragments

Keywords: Ballistic, Impact, Perforation, ABAQUS, Johnson & Cook, 304 stainless steel

This paper concerns energy absorption in thin (0.4 mm) steel plates during perforation by spherical projectiles of hardened steel, at impact velocities between 200 and 600 m s -1 . Absorbed energies have been obtained from measured incident and emergent projectile velocities. These tests were simulated using ABAQUS/Explicit, using the Johnson and Cook plasticity model. Astrain rate-dependent, critical plastic strain fracture criterion was employed to model fracture.

Good agreement is obtained between simulations and experiment and the model successfully captures the transitions in failure mode as projectile velocity increases. At velocities close to the ballistic limit, the plates fail by dishing and discing. As the incident velocity is increased, there are two transitions in failure mode, firstly to shear plugging and secondly to fragmentation and petalling. The simulations also show that, during the latter mode of failure, the kinetic energy of ejected debris is significant, and failure to include this contribution in the energy balance leads to a substantial over-estimate of the energy absorbed within the sheet. Information is also presented relating to the strain rates at which plastic deformation occurs within the sample under different conditions. These range up to about 10 5 s -1 , with the corresponding strain rate hardening effect being quite substantial (factor of 2-3 increase in stress).

Introduction

Perforation of plates during projectile impact is a complex process, commonly involving elastic and plastic deformation, strain and strain rate hardening effects, thermal softening, crack formation, adiabatic shearing, plugging, petalling and even shattering. These effects depend on the properties and geometries of projectile and target and on the incident velocity. There have been many theoretical and experimental studies in this area, which are covered in the comprehensive reviews of Corran et al [START_REF] Corran | Impact Loading of Plates -An Experimental Investigation[END_REF], Anderson and Bodner [START_REF] Anderson | Ballistic Impact: The Status of Analytical and Numerical Modelling[END_REF] and Corbett et al [START_REF] Corbett | Impact Loading of Plates and Shells by Free Flying Projectiles[END_REF].

Specific studies have been undertaken on petalling phenomena [START_REF] Wierzbicki | Petalling of Plates under Explosive and Impact Loading[END_REF][START_REF] Landkof | Petalling of Thin Metallic Plates During Penetration by Cylindro-Conical Projectiles[END_REF], shear plugging failure [START_REF] Atkins | Necking and Radial Cracking Around Perforations in Thin Sheets at Normal Incidence[END_REF][START_REF] Borvik | Numerical Simulation of Plugging Failure in Ballistic Penetration[END_REF] and dishing [START_REF] Lee | Fracture Prediction of Thin Plates under Localised Impulsive Loading. Part 1: Dishing[END_REF] in thin metallic plates, while the effects of target thickness, impact obliquity [START_REF] Goldsmith | Non-Ideal Projectile Impact on Targets[END_REF] and projectile nose shape [START_REF] Gupta | Experimental and Numerical Studies on the Behaviour of Thin Aluminium Plates Subjected to Impact by Blunt-and Hemispherical-Nosed Projectiles[END_REF][START_REF] Borvik | Perforation of 12mm Thick Steel Plates by 20mm Diameter Projectiles with Flat, Hemispherical and Conical Noses, Part 1: Experimental Study[END_REF] have also been studied.

The main failure modes are ductile hole enlargement (lateral displacement of material), petal formation, plugging failure via through-thickness shear (often accompanied or preceded by adiabatic heating), dishing, which is characterised by large plate deflection, membrane stretching and tensile tearing (discing) and, finally, fragmentation. It is not uncommon for a combination of these failure modes to operate simultaneously. Furthermore, the failure mode often changes with increasing projectile velocity [START_REF] Teng | Transition of Failure Modes in Round-Nosed Mass-to-Beam Impact[END_REF].

The Finite Element Method (FEM) has become an increasingly useful tool for the analysis of impact events, and current commercial codes are well suited to dealing with problems involving large deformations and elevated strain rates. The necessary computational resources are also routinely available. However, as highlighted by Zukas [START_REF] Zukas | Some Common Problems in the Numerical Modeling of Impact Phenomena[END_REF], erroneous results can be obtained.

Common errors include the use of models that are ill-suited to the problem, failure to recognise numerical instabilities (which may be attributed to physical phenomena), poor mesh specification, and use of inappropriate property data and constitutive relations. Of these, the latter are amongst the most common and, potentially, most significant.

The constitutive behaviour of most metals is, in general, fairly well understood and standard expressions are routinely implemented into FEM codes. These normally incorporate the effects of strain, strain rate and temperature on the effective stress. Examples include the phenomenological model of Johnson & Cook [START_REF] Johnson | Fracture Characteristics of Three Metals Subjected to Various Strains, Strain Rates, Temperatures and Pressures[END_REF] and the more theoretical, dislocation dynamics-based models of [START_REF] Zerilli | Dislocation-Mechanics-Based Constitutive Relations for Material Dynamics Calculations[END_REF]. The Johnson & Cook plasticity model is well suited to high rate deformation, and in general is reported to capture the main features of penetration and perforation -an example being the studies of Borvik et al [START_REF] Borvik | Numerical Simulation of Plugging Failure in Ballistic Penetration[END_REF][START_REF] Borvik | Experimental and Numerical Study on the Perforation of AA6005-T6 Panels[END_REF][START_REF] Borvik | Ballistic Penetration of Steel Plates[END_REF] on projectile penetration of steel and aluminium sheets. Their simulations agree reasonably well with experimental data, particularly at high velocities, with high mesh densities and when using an adaptive re-meshing algorithm.

Zerilli and Armstrong

Gupta et al [START_REF] Gupta | Experimental and Numerical Studies on the Behaviour of Thin Aluminium Plates Subjected to Impact by Blunt-and Hemispherical-Nosed Projectiles[END_REF] also used a Johnson & Cook elastic-viscoplastic model, to simulate penetration experiments on 1 mm thick Al sheets. Their model included the effects of linear thermoelasticity, yielding, plastic flow, isotropic strain hardening, strain rate hardening and softening due to adiabatic heating. They successfully predicted the residual velocities of penetrating projectiles, as well as ballistic limit velocities.

Simulations of this type commonly employ fracture criteria. These can be either instantaneous, time-dependent or micromechanical [START_REF] Zukas | Some Common Problems in the Numerical Modeling of Impact Phenomena[END_REF]. Micromechanical models are often preferred, despite their complexity, since they are usually more reliable [START_REF] Zukas | Some Common Problems in the Numerical Modeling of Impact Phenomena[END_REF]. In a comparative study of six fracture conditions, including a simple critical fracture strain, Teng and Wierzbicki [START_REF] Teng | Evaluation of Six Fracture Models in Hgh Velocity Perforation[END_REF] found that those based on accumulated equivalent plastic strain were most reliable. One such fracture criterion is that of Johnson & Cook, which has been widely employed [START_REF] Gupta | Experimental and Numerical Studies on the Behaviour of Thin Aluminium Plates Subjected to Impact by Blunt-and Hemispherical-Nosed Projectiles[END_REF][START_REF] Borvik | Experimental and Numerical Study on the Perforation of AA6005-T6 Panels[END_REF][START_REF] Borvik | Ballistic Penetration of Steel Plates[END_REF][START_REF] Borvik | Perforation Of 12mm Thick Steel Plates By 20mm Diameter Projectiles With Flat, Hemispherical And Conical Noses Part 2: Numerical Simulations[END_REF] for simulating the perforation of metallic plates. It can be expressed in the form

ε f , pl = D 1 + D 2 exp D 3 σ * ( ) ⎡ ⎣ ⎤ ⎦ 1 + D 4 ln dε / dt ( ) pl dε / dt ( ) 0 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1 + D 5 θ ( ) (1)
(A nomenclature listing is included in this paper as an Appendix.) However, whilst the Johnson & Cook plasticity parameters are relatively easy to determine from simple mechanical tests (at least for low strain rates), the fracture constants (D 1 …..D 5 ) are more difficult to obtain. Multiple tests must be conducted over a wide range of strain rate and stress triaxiality. In the current study, a strain rate-dependent, critical plastic strain criterion is employed, which requires relatively few parameters to be evaluated. This fracture criterion was coupled with the Johnson and Cook plasticity model ( § 3.2.2).

A further potential complication at high strain rates is the possible formation of adiabatic shear bands. These are bands of localised plastic shear strain, within which large temperature rises can occur. If the associated thermal softening exceeds the strain hardening, then continued loading is likely to result in failure at stress levels well below the static strength of the material at ambient temperature [START_REF] Wright | The Physics and Mathematics of Adiabatic Shear Bands[END_REF][START_REF] Bai | Adiabatic Shear Localization[END_REF]. Adiabatic shear band formation has been modelled numerically [START_REF] Borvik | Numerical Simulation of Plugging Failure in Ballistic Penetration[END_REF][START_REF] Roessig | Adiabatic Shear Localisation in the Dynamic Punch Test, Part 2: Numerical Simulations[END_REF][START_REF] Chou | Experimentation and Finite Element Simulation of Adiabatic Shear Bands in Controlled Penetration Impact[END_REF].

For example, Chou et al [START_REF] Chou | Experimentation and Finite Element Simulation of Adiabatic Shear Bands in Controlled Penetration Impact[END_REF] showed, using Johnson & Cook constitutive relations, that shear bands are commonly expected if thermal softening is taken into account. They noted, however, that a relatively fine mesh size is often required in order to capture their formation. In steels, for example, they are typically 10-100 μm in width, so local mesh dimensions need to be correspondingly fine.

The experimental work in the present study concerns 304 stainless steel. To the authors' best knowledge, the only previous study on this material, over the impact velocity range covered here, is that of Yungwirth et al [24], who examined the ballistic performance of sandwich panels with pyramidal lattice cores and compared it with that of monolithic plate of equal areal density. As impact velocities were increased beyond the ballistic limit, they observed a small decrease in the specific absorbed energy by the monolithic plates, followed by a monotonic increase.

Experimental Procedures

Tensile test specimens were cut using Electric Discharge Machining (EDM), from 0.4 mm thick sheets of grade-304 stainless steel that had been annealed in inert atmosphere at a temperature of 1195°C for 1.5 hours. Tensile tests were conducted using a 10 kN ESH servo-hydraulic testing machine at a strain rate of 0.01 s -1 . Strain was measured using a linear variable displacement transducer (LVDT). Stress-strain responses were recorded and material property data were inferred.

Circular disk specimens were cut, using EDM, from the stainless steel sheets. Specimens, of 76.5 mm diameter, were rigidly clamped around the periphery, such that a circle of 60 mm diameter was exposed. Specimens were impacted at normal incidence, with a range of incident velocities, by spherical projectiles of hardened steel, 8 mm in diameter and of 2 g mass. At low projectile velocities (170-250 m s -1 ), the incident velocity was measured using a series of light gates, comprising three light emitting diodes and three light receiving photodiodes. The residual times were in the ranges 10-36 µs and 6-9 µs. The projectile velocity was calculated from the inter-frame time and the distance travelled, using the Photo-sonics Phantom software. At high projectile velocities (∼350-600 m s -1 ), a high speed framing camera was employed (since a shorter exposure time was required). The high speed photographic images were taken with an Ultranac FS501 image converter camera, capturing 12 sequential images on Polaroid film. The projectiles were back-illuminated, using a Bowens flash (with a light duration of several ms). The exposure time for all of the images was 1 µs and the inter-frame times ranged from 10-70 µs. Over the critical period (∼170 µs), 12 polaroid images were captured. After penetration, impacted specimens were sectioned using EDM and photographed. Specimens were weighed before and after testing, and mass losses (due to the ejection of plugs or multiple fragments) were calculated.

3 Numerical Modelling

Mesh Formulation and Boundary Conditions

The steel sheets were modelled as elastic-plastic shells and meshed with reduced integration, linear quadrilateral shell elements -type S4R. The shell element formulation is well suited to dynamic analyses involving large plastic bending strains [START_REF] Abaqus | ABAQUS/Explicit User's Manuals[END_REF]. The mesh density was refined in the region directly beneath the impact site. The sensitivity of the predicted results to mesh density was a key concern during the simulations (particularly since large strain gradients were expected), and there exists a threshold mesh density value for solution convergence. Table III shows the predicted absorbed energy as a function of mesh density and minimum element size for an impact velocity of 600 m/s, which is insensitive to further refinements beyond 2457 elements.

An encastré boundary condition was specified to simulate the clamping conditions. The projectile was modelled as an analytical rigid body and simply assigned mass. This assumes that projectile deformation is negligible and observations of projectiles after impact confirmed that there was no noticeable plastic deformation. Penalty contact was defined between the projectile and the plate. Friction between the projectile and the plate was taken as negligible, which is consistent with the results of Krafft [START_REF] Krafft | Surface Friction in Ballistic Penetration[END_REF], who showed experimentally that sliding friction typically absorbs less than 3% of the total incident energy.

Material Response Characterisation

Johnson & Cook Plasticity

The Johnson & Cook plasticity formulation, which defines the flow stress as a function of equivalent plastic strain, strain rate and temperature, was employed in all simulations The dynamic flow stress is expressed by the following relation [START_REF] Johnson | Fracture Characteristics of Three Metals Subjected to Various Strains, Strain Rates, Temperatures and Pressures[END_REF]

σ d = A + B(ε pl ) n ⎡ ⎣ ⎤ ⎦ 1 + C ln dε / dt ( ) pl dε / dt ( ) 0 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ (1 -θ) (3)
where σ d is the dynamic flow stress, ε pl is the equivalent plastic strain, dε / dt ( ) pl is the equivalent plastic strain rate, dε / dt ( ) 0 is a reference strain rate, A, B, n, m and C are material parameters and θ is the non-dimensional temperature given by: ( ) ( )

transition transition 0 < ˆ/ 1 transition melt transition melt mel θ θ θ θ θ θ θ θ θ θ θ θ ≡ - - ≤ ≤ > t ( 4
)
where θ is the current temperature, melt θ is the melting temperature and transition θ is the transition temperature defined as the one at or below which there is no temperature dependence on the expression of the yield stress. The constant A is the yield stress under quasi-static conditions, B

and n are strain hardening parameters, m controls the temperature dependence and C the strain rate dependence. The values of the Johnson & Cook parameters used in the present work, which were obtained from Lee et al [START_REF] Lee | Dynamic Failure of Metallic Pyramidal Truss Core Materials -Experiments and Modelling[END_REF], are listed in Table I, and the data are plotted in Fig. 1. This does, of course, assume that the material used in this study is similar to that of Lee et al [START_REF] Lee | Dynamic Failure of Metallic Pyramidal Truss Core Materials -Experiments and Modelling[END_REF]. This isn't entirely satisfactory, since microstructural differences could affect the mechanical properties. However, since these data [START_REF] Lee | Dynamic Failure of Metallic Pyramidal Truss Core Materials -Experiments and Modelling[END_REF] are reportedly [START_REF] Maloy | The Influence of Explosive-Driven "Taylor Wave" Shock Pre-straining on the Structure/Property Behaviour of 304 Stainless Steel[END_REF] representative of annealed 304 stainless steel, it seems likely that they are at least broadly appropriate for the present work.

Fracture

A strain rate-dependent, critical plastic strain-based fracture criterion was employed. The equivalent plastic strain at the onset of fracture ε D. pl is assumed to be a function of the equivalent plastic strain rate and stress triaxiality, ie ε D, pl dε / dt ( ) pl , Σ ( ) . The stress triaxiality is the ratio of the mean stress to the equivalent stress, and thus characterises the degree to which plasticity is being promoted, which is known to affect the likelihood of fracture -as the triaxiality increases, the strain to fracture decreases [START_REF] Borvik | On the Influence of Stress Triaxiality and Strain Rate on the Behaviour of a Structural Steel. Part 2. Numerical Study[END_REF]. The base level (maximum) triaxiality can be calculated using Bridgman's analysis [START_REF] Hill | The Mathematical Theory of Plasticity[END_REF], from the initial notch diameter, although to determine this parameter over a range of strain rate and temperature requires a significant number of tests [START_REF] Dey | On the Influence of Fracture Criterion in Projectile Impact of Steel Plates[END_REF].

In any case, it has previously been reported that [START_REF] Borvik | On the Influence of Stress Triaxiality and Strain Rate on the Behaviour of a Structural Steel. Part 2. Numerical Study[END_REF] Bridgman's theory can lead to inaccurate estimates of the stress triaxiality, and no such tests were conducted during the current work.

Instead, a triaxiality of 1/3 was assumed, based on the work of Zhou [START_REF] Zhou | Impact Response of Lightweight Sandwich Panels[END_REF]. This is a plausible approximation, assuming that the plate material can be treated as a shell, with plane stress conditions satisfied. A stress triaxiality of 1/3 does, however, imply that the stress state of the plate is dominated by uniaxial tension, which is not strictly true. It is also worth noting that, in the parametric study conducted by Borvik et al [START_REF] Borvik | On the Influence of Stress Triaxiality and Strain Rate on the Behaviour of a Structural Steel. Part 2. Numerical Study[END_REF], the stress triaxiality was nearly insensitive to strain rate and inertia. The stress triaxiality was also relatively insensitive to temperature.

The damage, once initiated, is assumed to evolve linearly until the material is fully degraded and can no longer sustain load. The damage, D , accumulates in an element according to the following relation:

D = Δε pl ε f . pl ∑ (5)
where Δε pl is the increment of accumulated equivalent plastic strain and ε f , pl is the plastic strain to failure. Failure occurs when D ≥ 1 ; once this condition is satisfied in an element, it is removed from the mesh. The fracture strain is established as a tabular function of strain rate. However, since high strain rate fracture data were unavailable, and following the work of Lichtenfeld et al [33] on type-304 stainless steel, the fracture strain was assumed to decrease by 20% (compared to the quasi-static value -measured experimentally) for a strain rate of 10 4 s -1 . A progressive reduction in fracture strain was assumed over the strain rate range, as shown in Fig. 1.

Extraction of Global Information about Strain Rate Distributions

For each volume element, during each time interval, the average strain rate at which deformation occurred was recorded, as was the strain induced and the plastic work done. These increments of energy were accumulated into a series of bins covering specific strain rate ranges.

For cases in which the strain rate varied during the time interval beyond the range covered by a single bin, the energy was partitioned between the bins concerned using a proportional algorithm.

This algorithm is relatively crude, but it was ensured that the time interval of the computations was short enough to ensure that the associated error was small. (As with other aspects of the computation, it was confirmed that the meshing and time interval selection were sufficiently refined to ensure that they had no significant effect on the predictions.)

A "characteristic strain rate", representing the mean strain rate throughout the specimen for a given incident velocity, was calculated by considering all volume elements, across the complete time domain, weighted by the fraction of energy absorbed in a given element during a given time step. It was obtained by simply multiplying each strain rate bin upper boundary by the corresponding fraction of energy absorbed within that strain rate bin. A summation was then made for all bins.

Model Predictions and Comparisons with Experimental Data

Effects of Projectile Incident Velocity

A summary is shown in Table II of the experimental data obtained from each run. The table includes incident and residual projectile velocities, corresponding loss of projectile kinetic energy (ΔU), loss of specimen mass after impact, associated fragment kinetic energy (obtained assuming that all fragments acquire the residual projectile velocity) and the energy absorbed within the specimen (difference between energy loss of projectile and kinetic energy of fragments). A comparison is shown in Fig. 2(a) between experimental data and predictions, obtained using the The main focus of the present study is on the energy absorbed by the plate, and the associated deformation and fracture mechanisms. The energy absorbed in the plate is expected to be (predominantly) dissipated via plastic deformation and fracture, i.e. bulging, dishing, discing, petalling, plugging or fragmentation. For an individual element, the energy dissipated is given by:

U element = σ ε, dε / dt ( ) ,T ( ) dε 0 ε ≤ε f ∫ (6)
so that the total work done is equal to

U plate = U element 1 n ∑ (7)
Fig. 2(b) shows the experimental data of Fig. 2(a), re-plotted as absorbed energies (difference between incident and emergent projectile kinetic energies). It can be seen that agreement with experiment is good over the complete range of (sub-sonic) incident velocities being investigated here. Also plotted in Fig. 2(b) is the predicted energy absorbed within the specimen by plastic deformation and fracture -i.e. excluding the kinetic energy associated with expelled fragments (Eq.6). It can be seen that, as expected, this contribution to the absorbed energy becomes significant at higher incident velocities (>~300 m s -1 ), such that fragment expulsion starts to become significant. At an incident velocity of the order of 600 m s -1 , it constitutes over 50% of the total energy absorbed, although admittedly this energy now represents just a small fraction of the kinetic energy of the incident projectile.

The failure modes observed during these experiments are: (i) dishing and discing, (ii) shear plugging, and (iii) fragmentation. Fig. 3 shows the predicted deformation histories for impact speeds of 200, 316 and 580 m s -1 . Fig. 4 shows the experimental and predicted specimen shapes after projectile penetration. It can be seen that there is excellent agreement in terms of specimen appearance, and the transitions in failure mode with incident velocity are accurately predicted.

Dishing and discing-type failures occur at low velocities, just above the ballistic limit (~160 m s -1 ), with associated bulging, deflection and membrane stretching (tensile tearing) of the sheet. Plugging failure occurs at higher impact velocities (>~250 m s -1 ), with little or no dishing (but with some petalling), characterised by the removal of a plug of material with a diameter similar to that of the projectile (often slightly smaller for thin plates). A sequence of high speed video images (Fig. 5) show plug ejection, for an impact velocity of 316 m s -1 . At higher impact velocities (>~400 m s -1 ), failure tends to occur by a combination of extensive petal formation and fragmentation.

Kinetic Energy of Ejected Plugs or Fragments

It is clear from the predictions shown in Fig. 3(b) that, for incident velocities sufficient to create plugging or fragmentation failure, the kinetic energy associated with these fragments can constitute a significant proportion of the total energy absorbed by the specimen, and indeed represents over half of the total for very high incident velocities (~600 m s -1 ). The high speed photograph shown in Fig. 6 confirms that fragments can be seen emerging after penetration of the specimen and the mass loss data shown in Table II confirms that this is occurring. It's fairly clear that plug and fragment velocities can be in a similar range to the residual projectile velocity. Fig. 7 shows predicted velocity histories for the projectile and for the plug which is ejected by it, for a case in which plugging occurs.

A comparison is presented in Fig. 8 between model predictions and experimental data for the fragment kinetic energy, as a function of the incident projectile velocity. These experimental data were obtained from measured mass loss data (see Table II), so the agreement exhibited here constitutes evidence of the reliability of the model which is independent of the comparison shown in Fig. 3(b). Of course, the assumption that all fragments acquire a velocity equal to the residual velocity of the projectile is a rather crude one, which is probably responsible for some of the scatter in the experimental data in Fig. 8, but in general it is expected to be a fair approximation and this is consistent with the level of agreement between predicted and experimental data in Fig. 8.

Deformation Mechanisms and Local Strain Rates

The predictions shown in Fig. 3 increased. This is not unexpected, since the specimen volume within which plastic deformation occurs does not change much with increasing projectile velocity (indeed, it tends to fall, compared with the behaviour at low impact velocities -see below), and there is a limit to the levels of plastic strain which can be generated within this volume.

It is of interest to consider the strain rates experienced by different regions while deformation takes place. Fig. 9 shows, for three increasing projectile velocities, the proportion of the plastic work absorbed within the specimen, as a function of the strain rate at which this plastic deformation occurred. Naturally, the strain rates at which most of the deformation takes place tend to rise as the projectile velocity is increased.

However, some interesting effects can be identified on examining these data more closely. For example, Fig. 10(a) shows the predicted dependence on projectile velocity of the characteristic (mean) strain rate within a specimen (see §3.2.5). It can be seen that, while there is a trend for this strain rate to increase as the incident velocity goes up, there is actually a dip over the range of 300-400 m s -1 . This is the approximate range over which plugging occurs -see Fig. 8. It's certainly plausible that the onset of this failure mechanism should lead to a reduction in the average strain rate, since deformation rates will tend to be relatively low within a plug being pushed by a projectile, while straining in neighbouring material will also tend to be curtailed at a relatively early stage as the plug becomes detached. These changes in mechanism are also reflected in the curve shown in Fig. 10 (b), which shows how the volume of material subjected to plastic deformation changes as the projectile velocity increases. It can be seen that this has a peak value at the ballistic limit, falls off as the velocity increases within the regime of dishing and discing (and the dishing becomes more localised), remains low while plugging failure occurs and rises slightly as fragmentation replaces plugging.

It's also of interest to note the absolute values of typical strain rates created during this kind of deformation. It can be seen in Fig. 9 that these range up to about 3 10 4 s -1 , for a projectile velocity of 200 m s -1 , and up to about 3 10 5 s -1 , for a velocity of 600 m s -1 . Naturally, these strain rate values will tend to vary with projectile size and sheet thickness (and also with sheet material), but these calculations certainly give an indication of the nature of the deformation taking place within the specimen under this type of loading. It's of interest to note (Fig. 1) that, according to the Johnson-Cook formulation for this material, strain rates of ~10 5 s -1 generate stress level enhancements (compared to the quasi-static case) by factors of at least about 2 -i.e. substantial strain rate hardening occurs under these experimental conditions.

Of course, it must be borne in mind that these predictions would themselves be affected if a different strain rate sensitivity were to be incorporated into the constitutive equations employed.

There's no doubt that it is preferable to use experimental stress-strain data obtained over a wide range of imposed strain rate, although obtaining these data can be time-consuming and difficult.

In fact, due to the lack of literature data relating to strain rates above 10 4 s -1 , the constitutive relation for that strain rate (see Fig. 1) was employed in the present work for all higher strain rates as well, which is clearly not really appropriate or satisfactory. Nevertheless, the FEM predictions presented here do appear to capture most of the main features of the response of this particular material to this type of ballistic loading. In order to investigate further and more deeply, it would be advisable to bring the micro-mechanisms of deformation clearly into the picture, rather than simply trying to identify the most reliable constitutive relations without reference to the effects responsible for them.

Conclusions

The following conclusions can be drawn from this work.

(a) Thin (0.4 mm) plates of 304 stainless steel were perforated by spherical steel projectiles, with a range of (sub-sonic) incident velocities (up to 600 m s -1 ). Absorbed energies were obtained experimentally from incident and emergent projectile velocities. The tests were simulated using the Explicit FEM code in ABAQUS/CAE. The Johnson and Cook plasticity algorithm was employed. This was coupled with a strain rate-dependent, critical plastic strain fracture criterion.

(b) The predicted amount of energy absorbed by the specimen was found to agree well with the experimental data over the complete range of projectile velocity. The model also captured the transitions in failure mode, from dishing at low velocities, to plugging failure at around 300-400 m s -1 and a fragmentation process at higher velocities. It is shown that, towards the upper end of the velocity range investigated, the contribution to the total energy absorbed represented by the kinetic energy of escaping small fragments of the specimen becomes highly significant (>50% of the total). (c) A study has been made of the strain rates at which the deformation takes place within specimens of this type, during ballistic impact. While, in general, there is a tendency for strain rates to increase at higher impact velocities, it has been shown that the failure mode is important. For example, the sequence of events during plugging failure is such that the mean strain rate tends to fall somewhat as the velocity becomes high enough for it to become dominant. The strain rates that are operative while most of the plastic deformation occurs during ballistic impact fall mostly in the range of 10 2 -10 4 s -1 for an incident velocity of 200 m s -1 and 10 3 -10 5 s -1 for an incident velocity of 600 m s -1 . At strain rates of this magnitude, this steel exhibits substantial strain rate hardening and it's clearly important to have reliable input data concerning this effect if these impact processes are to be realistically modelled. There is currently a shortage of such data for most materials.

T m (K) T t (K) C (-) dε / dt ( ) 0 (-) α (-) c p (J kg -1 K -1 )

Figure Captions

Fig. 1 Stress-strain relationships for 304 steel, based on the Johnson-Cook plasticity relation, using parametric values obtained by Radford et al [START_REF] Radford | The Response of Clamped Sandwich Plates with Metallic Foam Cores to Simulated Blast Loading[END_REF]and fracture strain information given by Lichtenfeld [START_REF] Lichtenfeld | Effect of Strain Rate on Stress-Strain Behaviour of Alloy 309 and 304L Austenitic Stainless Steel[END_REF].

Fig.6

High speed photograph taken during projectile emergence, after penetrating the specimen with an incident velocity of 592 m s -1 , with several ejected fragments, as well as the projectile, visible in flight.

Fig.7

Predicted histories of projectile and plug velocity, for an impact velocity of 316 m s -1 .

Fig.8

Comparison between the model predictions for the aggregate kinetic energy of ejected (single or multiple) fragments and experimental data, obtained by measurement of the lost mass and based on the assumption that all fragments emerge with the same velocity as the projectile.

  Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al p-3-

  Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al p-4-

  Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al p-6-

  M A N U S C R I P T A C C E P T E D ARTICLE IN PRESS Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al p-7-

  Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al p-8-

9 -

 9 Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al p-Johnson and Cook plasticity formulation, for the residual velocity as a function of the incident velocity. It can be seen that the level of agreement is very good.

  Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al p-10-

  (b) indicate that the energy absorbed within the specimen (by plastic deformation, fracture etc), shows a tendency to reach a plateau as the projectile velocity is M A N U S C R I P T A C C E P T E D ARTICLE IN PRESS Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al p-11-

  Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al p-12-

  Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al p-13-

  Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al p-16-Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al p-17-Tablesρ (kg m -3 ) E (GPa) ε f (-) A (MPa) B (MPa)

Fig. 2

 2 Fig.2 Comparison between measured and predicted dependence on incident velocity of (a) residual velocity (and (b) absorbed energy, based on Johnson-Cook input data (with and without the kinetic energy of specimen fragments being included in the predicted plots).

Fig. 3

 3 Fig.3Model predictions, for the impact velocities shown, of the progression of fracture and the surface stress field, viewed from the rear of the specimen, for the times shown (after initial projectile contact).

Fig. 4

 4 Fig.4 Comparison between the modelled (left) and experimentally-observed (right) appearance of specimens after projectile penetration, as a function of incident velocity, showing (a) dishing, (b) shear plugging and (c) fragmentation modes of failure.

Fig. 5 A

 5 Fig.5A sequence of high speed video images, showing the ejection of a plug following plate perforation at an incident velocity of 316 m s -1 .

Fig. 9

 9 Fig.9Predicted work of deformation distributions over the range of strain rate, showing how the proportion of the total work done within a specimen (subjected to impact by a projectile with a given velocity) occurs within a series of bins covering the strain rate range.

  Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al

				ARTICLE IN PRESS ARTICLE IN PRESS ARTICLE IN PRESS
	Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al Energy Absorption during Projectile Penetration of Thin Steel Plates …..Dean et al
	No. Of Elements	Minimum Element Size (μm)			Absorbed Energy (J)
	220			815				49.36
	Run 585 1452	V in (m s -1 )	V res (m s -1 )	ΔU 433 (J) 132	m frag (g)	U frag (J)	U plate (J)	Failure 45.89 Mode 43.43
	1673 Table I Johnson & Cook plasticity and thermal property parameters for 304 stainless steel 293 0.07 0.01 0.95 440 [27] M A N U S C R I P T 2 176.3 109.9 19.0 0 0 19.0 Hinged Cap 3 179.2 121.8 17.3 0 0 17.3 4 Petals 4 181.2 112.9 20.1 0 0 20.1 Hinged Cap 5 184.5 121.3 19.3 0 0 19.3 Hinged Cap 6 190.3 127.5 19.9 0 0 19.9 Hinged Cap 7 214.73 178.5 14.2 0 0 14.2 6 Petals 8 217.9 172.4 17.7 0 0 17.7 7 Petals 9 220.7 172.9 18.8 0 0 18.8 4 Petals 10 222.3 187.5 14.3 0.098 1.71 12.5 Plugging 11 228 188.6 16.4 0 0 16.4 Hinged Cap 12 310 286.09 14.3 0.156 6.38 7.9 Plugging/Petalling 13 316 290.21 15.6 0.250 10.54 5.1 Plugging/Petalling 14 318 292.24 15.7 0.199 8.50 7.2 Plugging/Petalling 15 576.5 541.53 39.1 0.222 32.58 6.5 Petalling/Fragments 16 580 544.43 40.0 0.202 29.91 10.1 Petalling/Fragments 17 583 545.98 41.8 0.237 35.34 6.4 Petalling/Fragments M A N U S C R I P T 2457 120 42.075 4516 83 41.913 5818 41 41.863 6602 32 41.822 Table III: Mesh sensitivity analysis showing the predicted absorbed energys for an impact speed of 600 m s -1 M A N U S C R I P T
	A C C E P T E D 18 592 560.47 36.3 0.114 Table II Measured projectile velocities and mass losses, inferred energy changes and observed 17.94 18.4 Petalling/Fragments failure modes for individual runs. A C C E P T E D A C C E P T E D
					p-18-p-19-

p-20-

Acknowledgements

Financial support has come from DSTL (for JD), and from EPSRC (for CSD). The DSTL contract number under which part of the work was done is RD033-3169. The authors are grateful for useful discussions with a number of researchers with an interest in this area, including Richard Jones (DSTL) and Andrew Wallwork, Giles Aldrich-Smith and Neil Bourne (AWE).


The nature of sexual desire, by James Giles, University Press of America, Inc. 2008, 218 pp. In writing this book, the author has attempted, with success, to deconstruct the complexities of the nature of human sexual desire. The method and style of writing is from an inquiring perspective, the philosophical roots of which are based on the premise that reality is composed of objects and events that are within human consciousness and Giles attempts to search for the underlying meanings within sexual desire. Overall, the intricacies of what constitutes sexual desire are evident within the chapters, which cover areas such as the sexual process, sexual desire, gender and being in love. There is clear and thoughtful discussion and review of sexual desire and the continuums present in each of these areas from which it is formed. Of particular interest is the focus on deconstructing ideas from differing theoretical, philosophical, psychological, cultural and social constructionist perspectives, which creates a lively, if at times slightly complicated, narrative throughout the book. These differences in professional opinion, however, leave the reader with useful insights and points to consider whilst reading.

Initially, the reader is led through an analysis of why contemplating and discussing sexual desire is, in itself, a complex issue. Giles presents the components of sexual desire, whilst raising the issue of what it is that we desire is a central, yet existential problem to define. This chapter gives us a multi-faceted understanding, with a number of conflicting theories presented.

The following two chapters, focus on the sexual process and the object of sexual desire. Giles discusses the physiological, behavioural and experiential events involved. This chapter may be of particular interest to those working within a sexual health context and certainly provokes thought into the wider components of sex itself and human desires. In deconstructing, in detail, the object of sexual desire, Giles skilfully presents theories and their challenges. There is also an interesting commentary on the ideas of vulnerability and sex, which again I feel is an interesting area to explore clinically in working within a sexual health setting. The penultimate chapter is centred around the structure of being in love, again approached from a detailed analytical perspective. The theories of love, from two-component models to passion and commitment, are described and critiqued.

In describing the non-exclusivity of love, Giles raises interesting cultural constructions related to the ideas around singular and multiple partnerships.

Overall, this chapter raises the reader's awareness of the consideration of being in love in detail, when it is generally a concept that is discussed and written about as if it is a 'known' idea.

The final chapter of this book turns our attention to the social construction of sexual desire -proposing that it only exists in the culture which has constructed it.

In concluding the chapter, the author posits that perhaps it is an existential human need and one that is the "core human distinction".

There are two main issues I found within this book that are worthy of note. Firstly, due to the very nature of its detailed, deconstructive and discursive approach, at times it can get extremely specific and this might leave the reader finding it hard to keep the thread of the narrative. Secondly, although the book does not purport to be a therapeutic aid, as a clinician, I was left feeling that its usefulness in a clinical setting may be limited and it would have been valuable to have additional discussions on working within sexual settings and applying the ideas discussed. If you are looking for a detailed analysis, from a multi-theoretical and philosophical perspective, on human sexual desire, this text stands alone for its innovative and comprehensive approach.

Dr Nick King, September 2009