ASM Metals HandBook Vol. 8 - Mechanical Testing and Evaluation

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66. W. Chen and G. Ravichandran, Dynamic Compressive Failure of a Glass Ceramic under Lateral

Confinement, J. Mech. Phys. Solids, Vol 45 (No. 8), 1997, p 1303–1328

67. D.R. Curran, L. Seaman, T. Cooper and D.A. Shockey, Micromechanical Model for Comminution and

Granular Flow of Brittle Material under High Strain Rate Application to Penetration of Ceramic

Targets, Int. J. Impact Eng., Vol 13, 1993, p 53–83

68. R.W. Klopp, D.A. Shockey, D.R. Curran, and T. Copper, “A Granular Flow Model for Developing

Smart Armor Ceramics,” Final Technical Report on Contract DAAH04-94-K-0001, Jan 1998, p 85

69. V.F. Nesterenko, M.A. Myers, and H.C. Chen, Shear Localization in High-Strain-Rate Deformation of

Granular Alumina, Acta Mater., Vol 44 (No. 5), 1996, p 2017–2026

70. C.J. Shih, V.F. Nesterenko, and M.A. Myers, High-Strain-Rate Deformation and Comminution of

Silicon Carbide, J. Appl. Phys., Vol 83 (No. 9), 1998, p 4660–4671

71. C.J. Shih, M.A. Meyers, and V.F. Nesterenko, High-Strain-Rate Deformation of Granular Silicon

Carbide, Acta Mater., Vol 46 (No. 11), 1998, p 4037–4065

High Strain Rate Shear Testing

Test Setup

As previously noted, the basic setup is configured for radial collapse of metallic specimens. The system uses

the controlled detonation of an explosive, which is placed coaxially along with the specimen. Denotation is

initiated at the top, propagating along the cylinder axis. The explosive parameters in the TWC method

(detonation velocity, density, and thickness) are carefully selected to provide a “smooth” pore collapse. The

geometry of the experiment should be such that implosion should be practically stopped due to the dissipation

processes inside the sample and driver tube. Wave reflection effects are minimized, and spalling of the internal

cylinder surface is nonexistent. A low-detonation explosive (detonation speed, D ≈ 4000 m/s, or 13,100 ft/s)

with low initial density (0.9 to 1 g/cm

) is typically used. A description of the method can be found in Ref 69,

70, 71, and 52, 53, 54, 55, 56, 57, 58. The velocity of the inner wall of the tube is measured by an

electromagnetic gage placed in the center of a cavity collapsing in external magnetic field (Ref 54, 55).

The collapse also can be driven by impulse magnetic field with detailed measurements of collapse geometry.

The great advantage of this type of driver is the minimization of axial movement and very well-controlled

boundary conditions represented by magnetic field (pressure) history on the outside surface of the driver tube.

The results obtained with this method for thin aluminum alloy shells (Ref 72) as well as for thick-walled

titanium and 304 stainless steel cylinders (Ref 73) are very encouraging. For example the dynamic of collapse

of the titanium and stainless steel cylinders was apparently different, despite the domination of total mass of the

assembly by the copper driver. That may be associated with damage induced by earlier shear localization in

titanium.

A simple model to calculate the kinematics of implosion driven by explosion can be used with the assumption

of instantaneous detonation. This assumption is applied because the velocity of the collapse is more than an

order of magnitude less than the detonation speed. In this model, the initial conditions in the detonation

products correspond to the uniform pressure, P

; density, ρ

; and sound speed, c

, as follows:

where ρ

is the initial density of explosive, D is the detonation speed, and k is the coefficient in the polytropic

law for detonation products. The dependence of sound speed, c, on current density, ρ, in expanding detonation

products has the following form:

Material of the cylinder is considered to be incompressible and elastic-plastic with strain, strain rate, and

temperature-dependent strength. This approach results in excellent agreement of calculations with measured

time of collapse 8 μs for standard conditions of testing where the outside diameter of explosive was 60 mm (2.4

in.), the inside diameter was 30 mm (1.2 in.), D = 4000 m/s (13,100 ft/s), ρ

= 1000 kg/m

, and k = 2.5.

References cited in this section

52. V.F. Nesterenko, A.N. Lazaridi, and S.A. Pershin, Localization of Deformation in Copper by Explosive

Compression of Hollow Cylinders, Fiz. Goren. Vzryva, Vol 25 (No. 4), 1989, p 154–155 (in Russian)

53. M.P. Bondar and V.F. Nesterenko, Strain Correlation at Different Structural Levels for Dynamically

Loaded Hollow Copper Cylinders, J. Phys. (France) IV Coll. C3, Vol 1, supplement to J. Phys.

(France) III, p 163–170

54. V.F. Nesterenko, M.P. Bondar, and I.V. Ershov, Instability of Plastic Flow at Dynamic Pore Collapse,

High-Pressure Science and Technology 1993 (Joint International Association for Research and

Advancement of High Pressure Science and Technology and American Physical Society Topical Group

on Shock Compression of Condensed Matter Conf.), (Colorado Springs), 28 June to 2 July 1993, AIP

Conf. Proceedings, Part 2 (No. 309), p 1173–1176

55. V.F. Nesterenko and M.P. Bondar, Localization of Deformation in Collapse of a Thick Walled Cylinder,

Fiz. Goren. Vzryva, Vol 30 (No. 4), 1994, p 99–111 (in Russian)

56. V.F. Nesterenko and M.P. Bondar, Investigation of Deformation Localization by the “Thick-Walled

Cylinder” Method, DYMAT J., Vol 1 (No. 3), 1994, p 245–251

57. V.F. Nesterenko, M.A. Myers, C.H. Chen, and J.C. LaSalvia, Controlled High-Rate Localized Shear in

Porous Reactive Media, Appl. Phys. Lett., Vol 65 (No. 24), 1994, p 3069–3071

58. V.F. Nesterenko, M.A. Myers, C.H. Chen, and J. LaSalvia, The Structure of Controlled Shear Bands in

Dynamically Deformed Reactive Mixtures, Metall. Mater. Trans. A, Vol 26, 1995, p 2511–2519

69. V.F. Nesterenko, M.A. Myers, and H.C. Chen, Shear Localization in High-Strain-Rate Deformation of

Granular Alumina, Acta Mater., Vol 44 (No. 5), 1996, p 2017–2026

70. C.J. Shih, V.F. Nesterenko, and M.A. Myers, High-Strain-Rate Deformation and Comminution of

Silicon Carbide, J. Appl. Phys., Vol 83 (No. 9), 1998, p 4660–4671

71. C.J. Shih, M.A. Meyers, and V.F. Nesterenko, High-Strain-Rate Deformation of Granular Silicon

Carbide, Acta Mater., Vol 46 (No. 11), 1998, p 4037–4065

72. J. Stokes, D. Oro, R.D. Fulton, D. Morgan, A. Obst, H. Oona, W. Anderson, E. Chandler, and P. Egan,

Material Failure and Pattern Growth in Shock-Driven Aluminium Cylinders at the Pegasus Facility,

Bull. Amer. Phys. Soc., Vol 44 (No. 2), 1999, p 33

73. J.L. Stokes, V.F. Nesterenko, J.S. Shlachter, and R.D. Fulton, Comparative Behavior of Ti and 304

Stainless Steel in Magnetically-Driven Implosion at the Pegasus-II Facility, personal communication

High Strain Rate Shear Testing

Test Factors

The implosion of relatively thin shell results in flow instability of geometrical nature (Ref 74, 75, 76, 77). In

conditions of thick-walled cylinder experiments, the number of waves due to the growth of small perturbations

on the inner wall caused by this geometric instability (any departures from circular form and uniform velocity

grow during collapse) can be evaluated and is less than 1. The geometry of the TWC method is selected in such

a way that material instability breaks the symmetry of the cylindrical collapse, resulting in shear localization.

The state of strain generated within the collapsing cylinder is one of pure shear before shear localization starts.

This is shown in Fig. 22, in which the distortion of an elemental cube at radius r

is followed as it moves toward

the axis of the cylinder. There is no rotation of the elemental cube. The strain in the axial direction of the

cylinder is zero, and the planes of maximum shear lie at 45° to a radius, as indicated at r

in Fig. 22. In pressure

insensitive materials, the planes are also inclined at 45° to the cylinder axis. The strains for material points

being collapsed to different final radii r

, are depicted on Fig. 23 for the initial inner and outer radii of a sample

7 and 10.5 mm (0.28 and 0.41 in.) respectively. Each line designates the strain of a material point as it

converges inward. The extremity of the line designates the radius and strain of the inside surface of a sample

with radius after collapse R

equal to 3.1 and 4.3 mm (0.12 and 0.17 in.). As the tube collapses inward, the inner

surfaces experience increasing strains as the final radius of the cavity (R

) approaches zero.

Fig. 22 Geometry of pure shear in incompressible thick-walled cylinder under uniform plastic

deformation. r

and r

are initial and final radii of element. Source: Ref 78

Fig. 23 Effective strain in titanium sample as a function of final radius of element r

in partially collapsed

geometry for two configurations with final inner sample radii R

equal to 3.1 and 4.3 mm (0.12 and 0.17

in.)

Strain Rates. The main step in establishing strain rates is connected with determination of velocity field in the

sample. The radial velocity of the cylindrical cavity ν(t) can be measured by an electromagnetic technique (Ref

54, 55). The insertion of samples from different materials inside a copper driver tube does not significantly

change the time of collapse in comparison with the uniform copper cylinder having the same geometrical

dimensions. This is because the replacement of the central part of copper by sample material does not

essentially change the overall mass, which is dominated by the mass of outside copper (and initial velocity) of

the assembly. That is why the velocity data obtained for a monolithic copper cylinder can be used as a first

approximation to calculate the strain rate. Figure 24 shows the shear strain rates during the collapse process for

two points corresponding to two values of final radii, which are the ends of shear bands in titanium. The strain

rate is seen to fluctuate around corresponding mean values, and the variation (±15%) is not significant. That is

why, to a first approximation, the strain rates for these material points can be considered as constant and equal

to 3.5 × 10

-1

and 6 × 10

-1

Fig. 24 Shear strain rate versus time in titanium sample as a function of final radii of element r

. Source:

Ref 78

Stresses in the TWC method are found experimentally or are based on the measured kinematics of implosion

and corresponding constitutive equations. The situation is the same as in the impact rod (Taylor) test. The

advantage of the TWC method is that it involves a very simple state of strain (pure shear during uniform flow

and plane strain after instability starts). Also in the TWC method, stress gages can be used more effectively for

the same reason. Although the main dynamic input to stresses is from inertial effects due to the gradient of

particle velocity in the cylinder, the effect of dynamic stresses on shear localization in solid materials can be

neglected to a first approximation. In general, the distribution of stresses during implosion depends on material

strength and cannot be controlled in this method. Therefore, numerical modeling should be used to extract the

stress field during the collapse process. Two-dimensional models reflecting plane-strain conditions can be

successfully applied, predicting the stress field and the relatively complex shape of the cavity (Ref 68, 79).

References cited in this section

54. V.F. Nesterenko, M.P. Bondar, and I.V. Ershov, Instability of Plastic Flow at Dynamic Pore Collapse,

High-Pressure Science and Technology 1993 (Joint International Association for Research and

Advancement of High Pressure Science and Technology and American Physical Society Topical Group

on Shock Compression of Condensed Matter Conf.), (Colorado Springs), 28 June to 2 July 1993, AIP

Conf. Proceedings, Part 2 (No. 309), p 1173–1176

55. V.F. Nesterenko and M.P. Bondar, Localization of Deformation in Collapse of a Thick Walled Cylinder,

Fiz. Goren. Vzryva, Vol 30 (No. 4), 1994, p 99–111 (in Russian)

68. R.W. Klopp, D.A. Shockey, D.R. Curran, and T. Copper, “A Granular Flow Model for Developing

Smart Armor Ceramics,” Final Technical Report on Contract DAAH04-94-K-0001, Jan 1998, p 85

74. S.V. Serikov, Stability of a Viscoplastic Ring, Zh. Prik. Mekh. Tekh. Fiz., Vol 25 (No. 1), 1984, p 157–

168 or Appl. Mech. Tech. Phys., July 1984, Vol 25, p 142–153

75. H.E. Lindberg, Buckling of a Very Think Cylindrical Shell due to an Impulsive Pressure, Trans. ASME

E, J. Appl. Mech., Vol 31, 1964, p 267–273

76. A.L. Florence and G. R. Abrahamson, Critical Velocity for Collapse of Viscoplastic Cylindrical Shells

without Buckling, J. Appl. Mech. (Trans. ASME), March 1977, p 89–94

77. A.G. Ivanov, V.A. Ogorodnikov, and E.S. Tyun'kin, The Behavior of Shells under Impulsive Loading:

Small Perturbations, Zh. Prikl. Mekh. Tekh. Fiz., Vol 33 (No. 6),1992, p. 112–115 or J. Appl. Mech.

Tech. Phys. (USSR), May 1993, p 871–874

78. V.F. Nesterenko, M.A. Meyers, and T.W. Wright, Self-Organization in the Initiation of Adiabatic Shear

Bands, Acta Mater., Vol 46 (No. 1), 1998, p 327–340

79. S. Nemat-Nasser, T. Okinaka, V.F. Nesterenko, and M. Liu, Dynamic Void Collapse in Crystals:

Computational Modeling and Experiments, Philos. Mag., Vol 78 (No. 5), 1998, p 1151–1174

High Strain Rate Shear Testing

Advantages and Limitations

The TWC method allows investigation of the material instability during the high-strain flow. The use of a TWC

(with a ratio of cylinder-wall thickness to hole diameter of approximately 1) is very important for reproducible

tests involving instability phenomena. A thick wall stabilizes the symmetry of implosion and creates

macroscopically uniform boundary conditions, even when instability breaks symmetry at the internal parts of

the setup.

Internal patterning induced by deformation in this method can also be clarified, depending on the precise

control of overall strain due to the developed, simple “soft” self-recovery setup. This allows comparative

analysis of the stages of shear localization in solids, chemical reactions, and deformation localization in

ceramics in high strain rate deformation. Also, it allows investigation of material instabilities from implosion

phenomena.

The TWC method could benefit from several improvements in the technique, such as these:

• In situ measurements of stresses during the collapse process, at least during the stage of uniform flow

before instability

• Development of driver systems (such as magnetically driven systems) with greater precision during

implosion to reduce the axial component of strain

• Development of continuous measurements of the kinematics of inner and outer surfaces of the assembly

• Methods to investigate the influence of boundary conditions on shear instability

• Methods to investigate nucleation sites using high-quality interfaces between the sample and the stopper

tube

High Strain Rate Shear Testing

Pressure-Shear Plate Impact Testing

Pressure-shear plate impact testing is a procedure that is used to obtain stress-strain curves at the highest strain

rates attainable under well-characterized loading conditions (Ref 80). Shear strain rates on the order of 10

-1

are obtained for specimens thinned to thicknesses of about 0.2 mm (0.008 in.). Even higher strain rates, up to

-1

, can be obtained for very thin specimens (with thicknesses of 2.5 μm) prepared by vapor deposition. In

addition to providing high strain rates, this type of testing provides simple shearing deformation and

controllable levels of nearly hydrostatic pressure.

Pressure-shear plate impact testing is limited to fine-grained materials, because the grain size must be small

compared to specimen thickness to ensure that a representative average polycrystalline response is measured.

The specimen material should also be soft relative to the plate materials used to impose the deformation so that

these plates will remain elastic under the impact loading conditions.

Pressure-shear experiments require a plate impact facility and instrumentation for measuring the shear waves

that are generated by pressure-shear impact. Such experiments are lengthy because of the time required for

specimen preparation. Although these limitations restrict the materials that can be studied and the facilities that

can perform the testing, the scientific and technical importance of understanding the plastic response of metals

at strain rates up to 10

-1

and above necessitates further development of the technique. The method has been

used successfully to obtain dynamic stress-strain curves of several face-centered cubic and body-centered cubic

metals at strain rates of 10

-1

and higher.

The method is based on concepts drawn from plate impact experiments and Kolsky bar experiments. From the

former is taken the concept that plane waves should be used in experiments designed to measure dynamic

material properties to simplify the interpretation of experimental results by making a one-dimensional wave

theory applicable. Plate impact experiments also provide the methodology and instrumentation required for

conducting such experiments. From Kolsky bar experiments is taken the concept of sandwiching a thin, soft

specimen between two hard, elastic materials in order to sustain high strain rates in the specimen and to allow

its response to be determined from measurements of wave profiles in the elastic materials. These measured

profiles are related to the stresses and nominal strain rates in the specimen by one-dimensional elastic wave

theory.

The pressure-shear loading configuration was introduced originally as a means of generating shear waves in

symmetric plate impact experiments involving the impact of two plates of the same material. Shear waves are

used, because shear wave profiles provide a more sensitive indication of the dynamic plastic response of

materials than do longitudinal waves generated in conventional normal impact of plates. Shear waves have been

generated by the impact of parallel plates inclined relative to their direction of approach (Ref 81, 82) and by the

normal impact of an anisotropic elastic plate of y-cut quartz (Ref 83, 84). Shear wave profiles have been

monitored by means of a transverse displacement interferometer (Ref 83), two normal-velocity interferometers

(Ref 85) with beams at nonnormal incidence (Ref 84), and, for nonmetallic targets, an embedded

electromagnetic gage (Ref 82).

Pressure-shear waves have been used to study the plastic response of 6061-T6 aluminum (Ref 84, 86, 87, 88)

and alpha-titanium (Ref 46). These studies are important because of the greater sensitivity of the shear wave

profiles to the plastic flow characteristics of materials and because of the information they provide on the

effects of nonproportional loading on the dynamic plastic response of materials. However, as with other wave

propagation experiments, limitations include the fact that the constitutive relation between stress, strain, and

strain rate is not obtained directly but must be inferred by comparison of the recorded wave profiles with those

predicted for various assumed constitutive models.

Another limitation of plastic wave propagation experiments is that, although high shear strain rates are

generated near the impact face, the recorded wave profiles are determined primarily by plastic response of the

material at remote positions where the wave profiles have spread and strain rates have decreased. Both of these

drawbacks to pressure-shear wave propagation experiments are overcome in the high strain rate pressure-shear

plate experiment.

The following is a brief description of the basic concepts of plate impact tests. More detailed discussions are in

the article “Low Velocity Impact Testing” in this Volume.

References cited in this section

46. L.W. Myer, Dynamic Behavior of Thermomechanically Treated Ultra High Strength Steel under Tensile

and Compressive Loading, I. Berman and J.W. Schroeder, Ed., High Energy Rate Fabrication, ASME,

1984, p 245–252

80. R.J. Clifton and R.W. Klopp, Pressure-Shear Plate Impact Testing, Mechanical Testing, Vol 8, Metals

Handbook, 9th ed., American Society for Metals, 1985, p 231–237

81. A.S. Abou-Sayed, R.J. Clifton, and L. Hermann, The Oblique Plate Impact Experiment, Exp. Mech.,

Vol 16, 1976, p 127–132

82. Y.M. Gupta, Shear Measurements in Shock Loaded Solids, Appl. Phys. Lett., Vol 29, 1976, p 694–697

83. K.S. Kim, R.J. Clifton, and P. Kumar, A Combined Normal and Transverse Displacement

Interferometer with an Application to Impact of Y-Cut Quartz, J. Appl. Phys., Vol 48, 1977, p 4132–

4139

84. L.C. Chhabildas, H.J. Sutherland, and J.R. Asay, Velocity Interferometer Technique to Determine

Shear-Wave Particle Velocity in Shock-Loaded Solids, J. Appl. Phys., Vol 50, 1979, p 5196–5201

85. L.M. Barker and R.E. Hollenbach, Laser Interferometer for Measuring High Velocities of Any

Reflecting Surface, J. Appl. Phys., Vol 43, 1972, p 4669–4675

86. K.S. Kim and R. J. Clifton, Pressure-Shear Impact of 6061-T6 Aluminum and Alpha-Titanium, J. Appl.

Mech., Vol 47, 1980, p 11–16

87. L.C. Chhabildas and J.W. Swegle, Dynamic Pressure-Shear Loading of Materials Using Anisotropic

Crystals, J. Appl. Phys., Vol 51, 1980, p 4799–4807

88. A. Gilat and R.J. Clifton, Pressure-Shear Waves in 6061-T6 Aluminum and Alpha-Titanium, J. Mech.

Phys. Solids, Vol 33, 1985, p 263–284

High Strain Rate Shear Testing

Basic Concepts

High strain rate pressure-shear impact testing is performed by impacting a thin, soft specimen plate with a hard,

elastic flyer plate inclined at an angle, θ, as shown in Fig. 25. The thin specimen plate is backed by a thick

elastic anvil plate, which creates a state of high pressure and high shear strain rate in the thin specimen.

Because the flyer and anvil plate remain elastic during the experiment, the stresses in the elastic plates can be

inferred by measuring the projectile velocity and the particle velocity at the rear surface of the anvil plate.

Fig. 25 Schematic representation of high strain rate pressure-shear impact configuration. u

is the initial

velocity of the flyer plate in the normal direction, and ν

is the initial velocity of the flyer plate in the

transverse direction. V

is the projectile velocity and θ is the flyer plate angle.

During early stages of the test, before release waves reach the center of the plate from the periphery, one-

dimensional wave theory applies. All states at the impact face of an elastic flyer plate must satisfy the following

characteristic relations (Ref 89):

-σ + ρc

u = ρc

(Eq 11)

τ + ρc

ν = ρc

(Eq 12)

All states at the impact face of an elastic anvil plate must satisfy:

-σ -ρc

u = 0

(Eq 13)

τ - ρc

ν = 0

(Eq 14)

where σ is the normal stress in the x-direction, τ is the shear traction in the transverse direction, ρc

is the

longitudinal acoustic impedance, ρc

is the shear impedance, u is the particle velocity in the normal direction,

and ν is the particle velocity in the transverse direction. The initial normal and transverse components of the

velocity of the flyer plate are u

and ν

, respectively.

The loci of stress and particle velocity states for the flyer and anvil plates are shown in Fig. 26 for the normal

components and in Fig. 27for the transverse components. At impact, shear waves and longitudinal waves are

sent forward into the anvil and backward into the specimen, as shown in the x-t diagram in Fig. 28. The flyer

and anvil plates are selected to have impedances that are greater than or equal to that of the specimen so that

unloading does not occur as the waves reflect back and forth.

Fig. 26 Loci of normal stress-particle velocity states for flyer, anvil, and specimen, ρc

is the longitudinal

acoustic impedance; u

is the particle velocity state for the flyer and anvil plates in the normal direction.

Fig. 27 Loci of shear stress/transverse particle velocity states for flyer, anvil, and specimen. ν

is the

anvil particle velocity and ν

is the transverse particle velocity at the free surface of the anvil. V

is the

particle velocity; ν

is the particle velocity state for the flyer and anvil plates in the transverse direction.

Fig. 28 x-t diagram illustrating the shear waves and longitudinal waves at impact

Because the waves in the specimen are plastic, they are quickly attenuated, and the stress state in the specimen

becomes nominally homogeneous. This has been verified computationally for aluminum specimens sandwiched

between steel plates (Ref 89). It was also verified that after a few reflections, the hydrostatic pressure in the

specimen becomes nearly equal to the value of the homogeneous normal stress (Ref 89). The homogeneous

normal stress attained is the value at the point of intersection of the two lines given by Eq 3 and 5.

(Eq 15)

This point, which is the apex in Fig. 26, is attained because the elastic resistance of the specimen to volume

change will not allow a finite normal velocity difference to be maintained across the thickness of the specimen,

while flow in the radial direction is prohibited.

It is possible, however, to maintain a finite transverse velocity difference across the thin specimen for the

duration of the experiment. If the specimen behaves viscoplastically and if the stress state is homogeneous, then

the shear stress will equilibrate at a value τ in Fig. 27, and the flyer and anvil particle velocities will be ν

and

, respectively. The nominal shear strain rate is:

(Eq 16)

where h is the specimen thickness. Note that:

- ν

= ν

- ν

(Eq 17)

where ν

is the transverse particle velocity at the free surface of the anvil. The shear strain rate can be

integrated over time to give the shear strain. From Eq 14, the shear stress is: