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Cover
Title Page
Preface
Introduction
Part 1: Stress Waves in Solids
1. 1 Elastic Waves
  1. 1.1 Elastic Wave in a Uniform Circular Bar
  2. 1.2 Types of Elastic Wave
  3. 1.3 Reflection and Interaction of Waves
  4. Questions 1
  5. Problems 1
2. 2 Elastic‐Plastic Waves
  1. 2.1 One‐Dimensional Elastic‐Plastic Stress Wave in Bars
  2. 2.2 High‐Speed Impact of a Bar of Finite Length on a Rigid Anvil (Mushrooming)
  3. Questions 2
  4. Problems 2
Part 2: Dynamic Behavior of Materials under High Strain Rate
1. 3 Rate‐Dependent Behavior of Materials
  1. 3.1 Materials’ Behavior under High Strain Rates
  2. 3.2 High‐Strain‐Rate Mechanical Properties of Materials
  3. 3.3 High‐Strain‐Rate Mechanical Testing
  4. 3.4 Explosively Driven Devices
  5. 3.5 Gun Systems
  6. Problems 3
2. 4 Constitutive Equations at High Strain Rates
  1. 4.1 Introduction to Constitutive Relations
  2. 4.2 Empirical Constitutive Equations
  3. 4.3 Relationship between Dislocation Velocity and Applied Stress
  4. 4.4 Physically Based Constitutive Relations
  5. 4.5 Experimental Validation of Constitutive Equations
  6. Problems 4
Part 3: Dynamic Response of Structures to Impact and Pulse Loading
1. 5 Inertia Effects and Plastic Hinges
  1. 5.1 Relationship between Wave Propagation and Global Structural Response
  2. 5.2 Inertia Forces in Slender Bars
  3. 5.3 Plastic Hinges in a Rigid‐Plastic Free–Free Beam under Pulse Loading
  4. 5.4 A Free Ring Subjected to a Radial Load
  5. Questions 5
  6. Problems 5
2. 6 Dynamic Response of Cantilevers
  1. 6.1 Response to Step Loading
  2. 6.2 Response to Pulse Loading
  3. 6.3 Impact on a Cantilever
  4. 6.4 General Features of Traveling Hinges
  5. Problems 6
3. 7 Effects of Tensile and Shear Forces
  1. 7.1 Simply Supported Beams with no Axial Constraint at Supports
  2. 7.2 Simply Supported Beams with Axial Constraint at Supports
  3. 7.3 Membrane Factor Method in Analyzing the Axial Force Effect
  4. 7.4 Effect of Shear Deformation
  5. 7.5 Failure Modes and Criteria of Beams under Intense Dynamic Loadings
  6. Questions 7
  7. Problems 7
4. 8 Mode Technique, Bound Theorems, and Applicability of the Rigid‐Perfectly Plastic Model
  1. 8.1 Dynamic Modes of Deformation
  2. 8.2 Properties of Modal Solutions
  3. 8.3 Initial Velocity of the Modal Solutions
  4. 8.4 Mode Technique Applications
  5. 8.5 Bound Theorems for RPP Structures
  6. 8.6 Applicability of an RPP Model
  7. Problems 8
5. 9 Response of Rigid‐Plastic Plates
  1. 9.1 Static Load‐Carrying Capacity of Rigid‐Plastic Plates
  2. 9.2 Dynamic Deformation of Pulse‐Loaded Plates
  3. 9.3 Effect of Large Deflection
  4. Problems 9
6. 10 Case Studies
  1. 10.1 Theoretical Analysis of Tensor Skin
  2. 10.2 Static and Dynamic Behavior of Cellular Structures
  3. 10.3 Dynamic Response of a Clamped Circular Sandwich Plate Subject to Shock Loading
  4. 10.4 Collision and Rebound of Circular Rings and Thin‐Walled Spheres on Rigid Target
References
Index
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List of Tables

Chapter 01
1. Table 1.1 Typical longitudinal wave speed in solid materials
Chapter 02
1. Table 2.1 A numerical example of Taylor’s approach (Johnson, 1972)
Chapter 06
1. Table 6.1 Discontinuities at plastic hinges
Chapter 07
1. Table 7.1 Summary of the dynamic response of a simply supported beam with no axial constraint
Chapter 10
1. Table 10.1 The equivalent material properties of some planar lattices

List of Illustrations

Chapter 01
1. Figure 1.1 The propagation of a compressive elastic wave in a uniform circular bar.
2. Figure 1.2 The equilibrium of a representative element of the bar.
3. Figure 1.3 Particle displacements produced by a forward wave.
4. Figure 1.4 The transverse wave in a circular bar produced by suddenly releasing the clamp.
5. Figure 1.5 Diagram of a Rayleigh wave.
6. Figure 1.6 A straight beam under bending. (a) Beam under bending; (b) free body diagram of an element.
7. Figure 1.7 Reflection and refraction when a longitudinal wave encounters an interface.
8. Figure 1.8 The reflection and transmission of a longitudinal wave in a one‐dimensional cylinder.
9. Figure 1.9 One‐dimensional wave reflection at a free end.
10. Figure 1.10 One‐dimensional wave reflection at a fixed end.
11. Figure 1.11 A rectangular pulse in a semi‐infinite long bar.
12. Figure 1.12 The interaction of a compressive pulse and a tensile pulse.
13. Figure 1.13 The interaction of two tensile pulses.
14. Figure 1.14 The normal collinear collision of two identical bars. (a) Stress distribution; (b) space–time diagram.
Chapter 02
1. Figure 2.1 Stress wave in a semi‐infinite long bar.
2. Figure 2.2 The stress–strain curve of an elastic plastic material.
3. Figure 2.3 The wave speed related to the stress–strain curve.
4. Figure 2.4 Stress–strain curve and the wave speed of the linear strain‐hardening material.
5. Figure 2.5 A semi‐infinite bar made of linear strain‐hardening material subjected to a step load at its free end. (a) Schematic of wave propagation; (b) stress–strain curve; (c) space–time diagram.
6. Figure 2.6 A semi‐infinite bar made of decreasingly strain‐hardening material subjected to a monotonically increasing load at its free end. (a) Stress–strain curve; (b) space–time diagram.
7. Figure 2.7 A semi‐infinite bar made of increasingly strain‐hardening material subjected to a monotonically increasing load at the free end. (a) Stress–strain curve; (b) space–time diagram.
8. Figure 2.8 Schematic of the formation of a shock wave.
9. Figure 2.9 Unloading wave in a semi‐infinite bar made of linear strain‐hardening material. (a) Stress–strain curve; (b) space–time diagram.
10. Figure 2.10 Distribution of different wave regions in a semi‐infinite bar subjected to a rectangular pulse.
11. Figure 2.11 Tensile stress–strain curve of a material with ultimate strength.
12. Figure 2.12 A uniform bar impinges on a rigid flat anvil.
13. Figure 2.13 A bar made of elastic linear strain‐hardening material impacts a rigid flat anvil. (a) Space–time diagram; (b) distribution of wave regions.
14. Figure 2.14 A bar made of elastic linear strain‐hardening material impacts a rigid flat anvil. (a) Stress–strain curve; (b) stress–particle velocity diagram.
15. Figure 2.15 A short cylindrical bar impinges a rigid anvil: Taylor’s model.
16. Figure 2.16 Velocity of the residual rigid region v versus the plastic strain at interface e.
17. Figure 2.17 Final profiles given by Taylor’s approach.
18. Figure 2.18 Hawkyard’s energy approach to the Taylor bar.
19. Figure 2.19 Final profiles of the Taylor bar using Hawkyard’s approach.
Chapter 03
1. Figure 3.1 Examples of strain rate calculations. (a) A specimen under dynamic tension; (b) a projectile hits a rigid wall.
2. Figure 3.2 Schematic classification of testing techniques according to strain rate.
3. Figure 3.3 Effect of strain rate on the stress–stain response under compression.
4. Figure 3.4 Stress–strain curves of steel under uniaxial tension: (a) engineering stress–strain; (b) true stress–strain.
5. Figure 3.5 Variation of dynamic yield stress with strain rate for mild steel under uniaxial tension.
6. Figure 3.6 Dependence of flow stress (at different plastic strains) on the shear strain rate for 1100‐0 aluminum.
7. Figure 3.7 Schematic of intermediate‐strain‐rate compressed‐gas machines.
8. Figure 3.8 CEAST 9350 Drop Tower Impact System.
9. Figure 3.9 High‐speed tension machine using rotary flywheel.
10. Figure 3.10 INSTRON VHS8800 High Strain Rate System.
11. Figure 3.11 Intermediate‐strain‐rate materials testing apparatus. Top: photograph of the apparatus; middle and bottom: schematic of the apparatus. 1, front travel switch; 2, rear travel switch; 3, front retainer; 4, hydraulic cylinder; 5, piston rod; 6, rear retainer; 7, connector; 8, impact hammer; 9, load cell; 10, rear grip; 11, specimen; 12, front grip.
12. Figure 3.12 Bertram Hopkinson experimental device: (a) schematic diagram of the original Hopkinson pressure bar; (b) stress wave with finite length.
13. Figure 3.13 Split Hopkinson pressure bar. (a) Schematic of experimental device; (b) stress waves.
14. Figure 3.14 Setups for testing specimens in a Hopkinson bar in tension mode: (a) hat‐shaped specimen; (b) collared specimen; (c) inertial‐bar configuration; (d) collar impacting bolt head.
15. Figure 3.15 (a) Torsion Hopkinson bar for pure shear configuration; (b) hat‐shaped specimen for shear stresses in compressive Hopkinson bar.
16. Figure 3.16 Expanding‐ring technique: (a) steel block with explosive in core; (b) sections of cylinder just after detonation and during flight of ring; (c) section of ring.
17. Figure 3.17 Triangular line‐wave generator.
18. Figure 3.18 Conical explosive lens.
19. Figure 3.19 Mousetrap‐type plane‐wave generator.
20. Figure 3.20 Yoshida system using shock superposition.
21. Figure 3.21 Schematic diagram of the ‘business end’ of a plate impact shock loading gun.
22. Figure 3.22 Schematic of the inclined impact of a flat plate.
23. Figure 3.23 Schematic representation of a one‐stage light gas gun.
24. Figure 3.24 Two‐stage gas gun: (a) phase 1; (b) phase 2; (c) phase 3; (d) phase 4.
25. Figure 3.25 Basic operation of a direct‐current electric rail gun.
Chapter 04
1. Figure 4.1 Lower yield stress versus strain rate of mild steel.
2. Figure 4.2 Effects of temperature and strain rate on the yield stress of iron.
3. Figure 4.3 Dislocations in metal crystal. (a) Movement of one dislocation causing displacement b; (b) movement of an array of dislocations causing shear strain .
4. Figure 4.4 Shear stress changing with dislocation velocity.
5. Figure 4.5 Compilation of results in the literature relating dislocation velocities and stress.
6. Figure 4.6 Schematic representation of the stress–velocity behavior of nickel.
7. Figure 4.7 Different defects in crystalline materials.
8. Figure 4.8 Peierls–Nabarro force. (a) Movement of dislocation from one equilibrium position to next; (b) applied stress versus distance.
9. Figure 4.9 Dislocation cutting through a dislocation forest.
10. Figure 4.10 Barriers encountered by a dislocation on its course.
11. Figure 4.11 (a) Overcoming of barriers by thermal energy; (b) stress required to overcome an obstacle as a function of temperature.
12. Figure 4.12 Flow stress as a function of temperature for different strain rates; thermal and athermal components of stress indicated.
13. Figure 4.13 Effect of velocity on self‐energy of a dislocation.
14. Figure 4.14 Three regimes of dislocation response.
15. Figure 4.15 Strain rate/homologous temperature deformation map for titanium. BCC, body‐centered cubic. H.T., high temperature; L.T., low temperature.
Chapter 05
1. Figure 5.1 A beam subjected to transverse dynamic loading.
2. Figure 5.2 Pin‐jointed slender link under impact.
3. Figure 5.3 The definition of internal force components.
4. Figure 5.4 The general motion of a beam.
5. Figure 5.5 Inertia force on a small segment adjacent to point P.
6. Figure 5.6 Inertia forces and moments during the collapse of a chimney.
7. Figure 5.7 Forces on an element of the chimney and the plot of the equivalent transversely distributed load.
8. Figure 5.8 Force diagram of a pendulum subjected a concentrated impact force.
9. Figure 5.9 Distribution of the transversely distributed load in the impact pendulum.
10. Figure 5.10 Distribution of shear force in the impact pendulum.
11. Figure 5.11 (a) Stress–strain relationship of rigid‐perfectly plastic materials; (b) bending moment–curvature relationship of rigid‐perfectly plastic beams.
12. Figure 5.12 A free–free beam subjected to a concentrated step force at its midpoint.
13. Figure 5.13 Deformation mechanism and acceleration of a free–free beam subjected to a concentrated force () at its midpoint.
14. Figure 5.14 Distribution of bending moment of a free–free beam subjected to a concentrated load () at its midpoint.
15. Figure 5.15 Deformation mechanism of a free–free beam subjected to a concentrated load () at its midpoint.
16. Figure 5.16 A triangular‐shaped pulse load.
17. Figure 5.17 Deformation mechanism of a ring subjected to a radial dynamic load.
18. Figure 5.18 Force analysis of a circular ring subjected to a radial impact.
19. Figure 5.19 Deformation mechanism of a circular ring subjected to opposite radial forces.
Chapter 06
1. Figure 6.1 A cantilever beam under step loading.
2. Figure 6.2 The rigid body motion and the velocity distribution diagram of a cantilever under a moderate step force at its tip ().
3. Figure 6.3 The shear force and bending moment of a cantilever under a moderate step force F ()applied at its tip.
4. Figure 6.4 The rigid body motion and the velocity distribution diagram of a cantilever under an intense step force () at its tip.
5. Figure 6.5 The shear force and bending moment of a cantilever under an intense step force ().
6. Figure 6.6 The acceleration at the tip of a cantilever varies with the magnitude of the applied force.
7. Figure 6.7 Rectangular pulse.
8. Figure 6.8 The plastic hinge position and tip velocity for a cantilever under a rectangular pulse.
9. Figure 6.9 The normalized final deformation profiles of a cantilever under rectangular pulse.
10. Figure 6.10 Schematic of a general pulse.
11. Figure 6.11 A cantilever impacted by a concentrated mass at the free end.
12. Figure 6.12 The hinge location and tip velocity versus time for a cantilever impacted by a mass at its free end.
13. Figure 6.13 The velocity, acceleration, shear force, and bending moment distribution along the cantilever.
14. Figure 6.14 The final profiles of a cantilever under impacts from different mass ratios.
15. Figure 6.15 Partitioning of the energy dissipation in two phases versus mass ratio.
16. Figure 6.16 Schematic of a function Φ(x).
Chapter 07
1. Figure 7.1 A simply supported beam with no axial constraint under uniform impulsive loading.
2. Figure 7.2 Response phase I of a simply supported beams with no axial constraint.
3. Figure 7.3 Response phase II of a simply supported beam with no axial constraint.
4. Figure 7.4 The stress distribution on a cross‐section of a rectangular beam under bending moment M.
5. Figure 7.5 The stress distribution on a cross‐section of a rectangular beam under tensile force N.
6. Figure 7.6 The stress distribution on a cross‐section of a rectangular beam subjected to both bending moment M and tensile force N.
7. Figure 7.7 The limit locus on the M–N plane.
8. Figure 7.8 Response of a simply supported beam with axial constraint.
9. Figure 7.9 Schematic on the compatibility relation of a beam with axial constraint.
10. Figure 7.10 Equilibrium of a plastic string.
11. Figure 7.11 Several limit loci of a rectangular beam made of rigid‐perfectly plastic material.
12. Figure 7.12 Final deflections predicted by different limit loci.
13. Figure 7.13 The initial position of the traveling hinge.
14. Figure 7.14 The velocity distribution of a fully clamped beam subjected to collision of a mass at the midpoint.
15. Figure 7.15 Non‐dimensional final deflection varies with non‐dimensional input energy .
16. Figure 7.16 Beam of infinite length impinged by a projectile.
17. Figure 7.17 Response of a beam of infinite length impinged by a projectile, analyzed by bending‐only theory.
18. Figure 7.18 Response of a beam of infinite length impinged by a projectile, analyzed by bending‐shear theory.
19. Figure 7.19 History of the transverse velocity of beam tip v(t) and that of the projectile ṽ(t).
20. Figure 7.20 A clamped beam subjected to a uniformly distributed velocity v₀.
21. Figure 7.21 Failure modes: (a) mode I; (b) mode II; (c) mode III.
22. Figure 7.22 A schematic map of failure mode occurrence.
23. Figure 7.23 Velocity field for an impulsively loaded, fully clamped beam with shear sliding at supports.
24. Figure 7.24 The effective length of the plastic hinge (a) bending only (b) with the influence of tensile force.
25. Figure 7.25 (a) Schematic of shear sliding over a finite width W; (b) yield locus for the sliding section.
Chapter 08
1. Figure 8.1 History of velocity given by the complete solution and the modal solution.
2. Figure 8.2 The modal solution of the Parkes problem. (a) A cantilever under impact at its tip; (b) distribution of initial velocity.
3. Figure 8.3 Distribution of bending moment by the modal solution.
4. Figure 8.4 Comparison between the complete solution and the modal solution of the Parkes problem. (a) History of the beam tip velocity; (b) final tip deflections versus the mass ratio.
5. Figure 8.5 Modal solution of a fully clamped beam under local impulsive loading.
6. Figure 8.6 Comparison between the complete solution and the modal solution of a fully clamped beam under local impulsive loading. (a) History of beam’s central velocity; (b) final central deflections versus the size of the loading region.
7. Figure 8.7 Mechanical models of material idealizations. (a) An elastic‐perfectly plastic model; (b) a rigid‐perfectly plastic model.
8. Figure 8.8 Force–displacement relationship for perfectly plastic models. (a) Elastic‐plastic model; (b) rigid‐plastic model.
Chapter 09
1. Figure 9.1 Rigid segments connected by a plastic hinge line.
2. Figure 9.2 Simply supported square plate loaded by a concentrated force F at its center.
3. Figure 9.3 The deformation mechanism of a square plate. (a) Angular velocities of rigid segments; (b) hodograph of angular velocity.
4. Figure 9.4 A rectangular plate under transverse load. (a) Deformation mechanism; (b) hodograph of angular velocity.
5. Figure 9.5 The deformation mechanism of a regular polygonal plate with n sides. (a) Angular velocities of rigid segments; (b) hodograph of angular velocity.
6. Figure 9.6 An annual plate subjected to distributed load on its inner boundary. (a) Deformation mechanism; (b) vectors of angular velocity of rigid segments; (c, d) hodograph of angular velocity.
7. Figure 9.7 The equivalent of an arbitrary pulse.
8. Figure 9.8 Square plate under uniformly distributed pressure of a rectangular pulse shape. (a) Rectangular pulse; (b) deformation mechanism.
9. Figure 9.9 Triangular segment rotating with respect to its bottom edge.
10. Figure 9.10 Central velocity of a square plate versus time.
11. Figure 9.11 Annular circular plate subjected to a rectangular pulse on its inner boundary.
12. Figure 9.12 Cross‐section along a diameter of a simply supported circular plate.
13. Figure 9.13 Stress distribution at the cross‐section along a diameter of a simply supported circular plate.
14. Figure 9.14 Determination of the position of the I–I line. (a) ; (b) .
15. Figure 9.15 Load‐carrying capacity of circular plates versus the central deflection.
16. Figure 9.16 Initial kinetic energy versus the normalized maximum central deflection.
Chapter 10
1. Figure 10.1 The schematic of a tensor skin.
2. Figure 10.2 Three response stages of a tensor skin. (a) Elastic deformation stage; (b) unfolding stage of tensor ply; (c) stretching stage of tensor ply.
3. Figure 10.3 Deformation mechanisms of unfolding process of tensor ply, (a) , (b) , (c) .
4. Figure 10.4 Tensor ply under uniform distributed pressure.
5. Figure 10.5 The normalized critical static pressure versus angle (a) and normalized central deflection (b).
6. Figure 10.6 Various pulses that approximate to a rectangular pulse , . Rec., rectangular pulse; Lin. Dec., linearly decreased pulse; Cos., cosine pulse; Exp., explosive pulse; Lin. Inc., linearly increased pulse; Tri., triangular pulse.
7. Figure 10.7 The dynamic response of the tensor skin (k = 0.5) under pulses approximated by the reference rectangular pulse Ī = 1.3q̄⁰_s, t̄_de = 1. (a) Angle; (b) normalized central deflection; (c) normalized energy dissipation; (d) normalized kinetic energy. Rec., rectangular pulse; Lin. Dec., linearly decreased pulse; Cos., cosine pulse; Exp., explosive pulse; Lin. Inc., linearly increased pulse; Tri., triangular pulse.
8. Figure 10.8 A piece of hexagonal honeycomb and its representative cell.
9. Figure 10.9 A typical stress–strain curve for a hexagonal honeycomb under in‐plane loading.
10. Figure 10.10 Schemes of some generalized honeycomb structures. (a) Hexagonal; (b) rhombus; (c) square; (d) triangular; (e) Kagome.
11. Figure 10.11 Collapse mechanisms of lattices under the X₂ uniaxial compression. (a) Hexagonal; (b) rhombus; (c) square; (d) triangular and Kagome.
12. Figure 10.12 Equivalent compression stress versus the equivalent strain.
13. Figure 10.13 Deformation profiles of hexagonal under compression. (a) X₂ direction: equivalent strains (from left to right) , 1/3, 2/3; (b) X₁ direction: strains (from left to right) , .
14. Figure 10.14 Two types of structure. (a) Response curves; (b) type I: a circular ring; (c) type II: pre‐bent plates.
15. Figure 10.15 The equivalent stress–strain curves of some lattices under compression in small strain stage.
16. Figure 10.16 Rigid‐perfectly plastic shock theory. (a) A typical stress–strain curve for cellular material with rigid‐perfectly plastic idealization. (b) Sketch of a mass G impacting an initially stationary cylinder.
17. Figure 10.17 A hexagonal lattice under dynamic compression. (a) Velocity 3.5 m/s; (b) velocity 28 m/s. ① ε = 0.05, ② ε = 0.11, ③ ε = 0.30, ④ ε = 0.46, ⑤ ε = 0.57, ⑥ ε = 0.80.
18. Figure 10.18 Average stress of different lattices with same relative density.
19. Figure 10.19 The representative cell of the generalized honeycombs, (a) Hexagonal; (b) rhombus; (c) square; (d) triangular; (e) Kagome.
20. Figure 10.20 Geometry of the clamped sandwich plate.
21. Figure 10.21 The exact, inscribing, and circumscribing yield loci of the sandwich plate.
22. Figure 10.22 Design chart for a clamped sandwich plate with , and under .
23. Figure 10.23 The structural response time and core compression time as a function of applied impulse Ī ( and ). FE, finite element analysis.
24. Figure 10.24 The maximum central deflections for clamped sandwich plates and beams, as a function of applied impulse Ī. ( and ). FE, finite element analysis.
25. Figure 10.25 Design chart for a clamped sandwich plate with a fixed .
26. Figure 10.26 Schematic drawing of the collision of a circular ring on a rigid target.
27. Figure 10.27 Impact force versus the displacement of mass center of the ring (a) and versus time (b). η = 1/30, Y = 150 MPa, and v =1.5.
28. Figure 10.28 Velocity histories at different locations of the ring. η = 1/30, Y = 150 MPa, v = 1.5.
29. Figure 10.29 Energy partitioning: η = 1/30, Y = 150 MPa, and v = 0.1.
30. Figure 10.30 Energy dissipation: η = 1/30, Y = 150 MPa, and v = 0.3.
31. Figure 10.31 Schematic drawing of the four‐hinge mode (a) and the five‐hinge mode (b).
32. Figure 10.32 Deformed configurations with bending moment per unit width of the ring wall. η = 1/30, Y = 150 MPa, and v = 3.0.
33. Figure 10.33 Rebounding velocity (a) and the coefficient of restitution (COR) (b) as functions of the initial velocity.
34. Figure 10.34 Experimental setup for ring collision.
35. Figure 10.35 Final shapes of the ring specimens after collision with various velocities.
36. Figure 10.36 Rebounding velocity (a) and the coefficient of restitution (COR) (b) as functions of the initial velocity. COR, coefficient of restitution.
37. Figure 10.37 Dynamic collision of a ping pong ball onto a solid target.
38. Figure 10.38 Variation of the contact diameter.
39. Figure 10.39 (a) Impact force versus time; (b) displacements versus time; (c) impact force versus compression ratio of the ball, when V₀ = 0.5 m/s.
40. Figure 10.40 Scheme of the ball flattened by a rigid target.
41. Figure 10.41 Distribution of the elastic strain energy density and configurations of a deformed ball at different time instants when v = 1.2, side view.
42. Figure 10.42 (a) Coefficient of restitution (COR) versus v = V₀/V_Y; (b) impact duration versus v = V₀/V_Y.