Contents

1. One-Dimensional Theory

2. Dispersion Correction and SHB calibration

3. Fundamentals of (Split) Hopkinson Bar

4.  Constitutive models

1. One Dimensional Theory 

(1) Evolution of Specimen Strain Rate in Split Hopkinson Bar Test

 Proc. Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233(13), 4667–4687, 2019. 

Open-Access URL: https://doi.org/10.1177/0954406218813386

This paper reveals the physical origin of the varying nature of the specimen strain rate in the SHB test, which is fundamental for exploiting the SHB as a tool to investigate the high-strain-rate behavior (properties) of materials. It is highly worthwhile for researchers to read at least the introduction of the paper, which will reward them knowledge on what were unknown (unsolved) thus far in the SHB technology and what are solved in the current paper for (1) understanding the physical origin of the varying nature of the specimen strain rate, (2) verifying the SHB test result, (3) predicting the specimen strain rate before the SHB test, (4) predicting the maximum specimen strain, (5) achieving a pseudo-constant strain rate, and (6) predicting the  operatability of the SHB instrument (which is: Eq (10) >0).

(2) One-Dimensional Analyses of Striker Impact on Bar with Different General Impedance

Proc. Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 234 (2), 589–608, 2020.

Publisher’s URL: https://doi.org/10.1177/0954406219877210

This paper analyzes the physics/mechanics of bar impact using the formulated one dimensional equations here based on (1) momentum conservation, (2) energy conservation, and (3) Newton’s second law. It explains the nature of the wave profile in the striker and bar depending on the relative general impedance. The formulated one-dimensional equations and predicted wave profiles are verified using numerical analysis (explicit finite element analysis).

(3) Theory of Schroeder and Webster 

 Sliding surface

To describe behavior of sliding surface, SW employed the law of Coulomb friction:

They used Eq. (1) in the consideration of the stress equilibrium of an axisymmetric stress element being compressed and established a differential equation,

By integrating Eq. (2) from  to  (the left side) and from  to  (the right side), the profile of the normal pressure, , was obtained as a function of the radial position ( ):

Eq. (3) is called the ‘SW-sliding profile of the normal pressure’ in this study. An example of this profile is illustrated in Fig. 1(a) as curve a. 

The average value of  is noted as . In experiment, the value of  is measured.  can be calculated from the normal pressure profile ( ) shown in Fig. 1(a) (curve a) using the equation,

By plugging Eq. (3) into Eq. (4), the friction compensation model is obtained as: 

where . In this study, Eq. (5) is called the ‘SW-sliding compensation model’. When the friction-free property, , is obtained from the measured value of  under the frictional condition, equations like Eq. (5) are called the friction-compensation models. If the load required for upsetting ( ) is predicted from knowledge of , equations like Eq. (5) are called the prediction models for the upsetting load.

Sticking surface

The term sticking used in the study of SW did not necessarily mean adhesion at the interface; it indicates the state that the contact surface of the material does not move relative to the surface of the platen.  (the shear stress on the contact surface due to the normal pressure, ) will not increase to a higher value than the value necessary to produce yielding or plastic flow. When the value of  increases to a level for invoking plastic flow, relative sliding between the platen and specimen surface stops (sticking takes place). Then, spreading action of the specimen occurs because of the shear strain of the surface layer that makes the new contact surface with the platen at the radial end region (rollover). In such a shear stress regime where the value of  is high, because the platen restrains the free sliding of the surface layer of the specimen, a condition approximating a hydrostatic pressure is produced; the magnitude of the hydrostatic pressure equals to , upon which value a shear stress  is superimposed. Fig. 1(b) schematically illustrates such a stress state.

The von Mises yield criterion in such a stress state (Fig. 1(b)) is 

where  is the maximum value of , which is the yield stress in pure shear. From Eq. (6), the relationship between  and ,

is established ( ). Eq. (7) is the constant friction law that is applied when shear yielding takes place. The relationship between the Coulomb friction law and the constant friction law are illustrated in Fig. 1(c).

The condition under which the sticking occurs can be found by considering the critical moment when  ( ) reaches  ( ):

where  is the critical normal pressure above which the law of constant friction is applied (Fig. 1(c)) and the shear stress reaches the level for invoking plastic flow (sticking occurs). Because, , if  reaches , ; the law of constant friction starts to be applied (sticking occurs) when the normal pressure ( ) reaches the flow stress of the specimen ( ). In this regard, SW considered that if  sticking occurs everywhere in the specimen.

SW used Eq. (7), the law of constant friction, in the consideration of the stress equilibrium of an axisymmetric stress element being compressed and established a differential equation,

The profile of the normal pressure ( ) for the sticking surface can be found by integrating Eq. (8) from  to  (the left side) and from  to  (the right side):

Eq. (10) is called the ‘SW-sticking profile of the normal pressure’ in this study. An example of this profile is illustrated in Fig. 1(a) as curve b. By plugging Eq. (10) into Eq. (4), the compensation model is:

Eq. (11) is called the ‘SW-sticking compensation model’ in this study.

Mixed surface

SW considered that, even when , sticking occurs on the surface portion when the normal pressure  is higher than its critical value ; the equality  (constant friction law) is applied in any region where  . Because  is higher in the central region than the radial end, the view that sticking occurs when  naturally results in the concept of the critical radius ( ) below which sticking occurs and above which sliding occurs. The concept of  and associated  is illustrated in Fig. 2(a)). When ,  and thus sticking occurs. When ,   and thus sliding takes place. We call this surface as the mixed contact surface in that sticking and sliding occurs simultaneously on a given contact surface. The value of  can be obtained if we substituting  for  in the normal pressure profile (Eq. (3); the blue curve in Fig. 2(a)):

The change in  with  is illustrated in Fig. 2(b) for the range of  values.

The contact condition, i.e., whether  or  when , is graphically illustrated in Fig. 3(a). This contact map was established from the viewpoint of the magnitude of the normal pressure; we took a look at the phenomenon of the mixed contact surface (the curves in Fig. 2(a)) from the ordinate ( ).

If we take a look at the phenomenon of the mixed contact surface (the curves in Fig. 2(a)) from th e abscissa ( ), a different version of the contact map can be established as follows.  For the mixed surface,  should be positive. When , Eq. (12) transforms to 

When ,  and thus mixed surface is obtained. On the other hand, When  ,  and thus the whole contact surface is sliding. The contact map established using Eq. (13) is graphically illustrated in Fig. 3(b).  The two versions of the contact map (Fig. 3(a) and Fig. 3(b)) are the same; it depends on whether we take a look at the phenomenon of the mixed contact surface (the curves in Fig. 2(a)) from the abscissa ( ) or from the ordinate ( ). 

The normal pressure profile ( ) when  can be found by integrating Eq. (9) over the range, , i.e., integration from  to  (the left side) and from  to  (the right side):

The pressure profile over the range  is given by Eq. (3). The ‘SW-mixed profile of the normal pressure’ called in this study is composed of two parts: Eq. (14) over the range,  (e.g., the red curve in Fig. 2(a)) and Eq. (3) over the range,  (e.g., the blue curve in Fig. 2(a)).

The compensation model for the mixed surface is obtained if we plug Eq. (14) into Eq. (4) over the range,  and plug Eq. (3) into Eq. (4) over the range, :

where  is the same as before and . Eq. (15) is called the ‘SW-mixed compensation model’ in this study.

(4)  Compensation models for sliding surfaces

While SW derived the compensation models for each of the three types of contact surfaces, especially the compensation model for the sliding surface has received much interest of other researchers as well. In this subsection, the compensation models for the sliding surface are reviewed in chronological order for convenience’s purpose.

In as early as 1923, Siebel [14] derived the average additional stress ( ) necessary to deform a 2D (plane strain) solid when uniform friction exists at the contact surfaces: 

where  is the current width and  is the current height of the work piece. Because , Eq. (16) transforms to

The compensation model of Siebel (Eq. (17)) is introduced here because, in the literature [10, 16-18, 31], the model of Hill (Eq. (18)) or even the SW-sliding model (Eq. (5)) is often referred as the Siebel’s model [14-15]. However, reference [14] reported Eq. (17) only; reference [15] dealt with the axial compression of the specimen with a conically grooved surface.

In 1949, as already reviewed, Schroeder and Webster derived Eq. (5) as the compensation model for the sliding surface. However, books [18, 19] written later did not cite the study of SW explicitly or introduced Eq. (5) as the Siebel’s model. Thus, some recent studies using Eq. (5) [16, 20] could not cite the study of SW. To our knowledge, Eqs. (3)-(5) were derived first in the study of Schroeder and Webster. Han [16], Christiansen et al. [20], and Smith and Kassner [21] used Eq. (5) to compensate their quasi-static stress-strain curves. Recently, Altinbalik et al. [22] used the SW theory to predict press loads in closed-die upsetting. 

Hill [23] provided an approximate solution for Eqs. (3) and (4) in his book published in 1950:

Richardson [18] pointed out that the error of Eq. (18) is less than 1% compared with Eq. (5) if . Kamler et al. [31] used Eq. (18) to compensate their stress-strain curves measured using the Kolsky-Hopkinson pressure bar.

Rand [24] also presented a solution for Eqs. (3) and (4):

where  and  are the initial diameter and initial height, respectively, and  is the engineering strain. Bertholf and Karnes [32] used Eq. (19) to compensate their stress-strain curves obtained using the Kolsky-Hopkinson pressure bar.

Cha et al. [33] considered the energy equilibrium of the cylinder specimen being compressed under sliding contact condition, from which a compensation model was derived:

They used Eq. (20) in correcting the stress-strain curves determined from Kolsky-Hopkinson pressure bar signals.

(5) References

[1] Voce, E., 1948, “The Relationship between Stress and Strain for Homogeneous Deformation,” J. Inst. Metals. 74, pp. 537–562.

[2] Johnson G. R., and Cook W. H., 1983, “A Constitutive Model and Data for Metals Subjected to Large Strains, High Strain Rates and High Temperatures,” Proc. of the 7th International Symposium on Ballistics. Hague: Organizing committee of the 7th ISB; pp. 541–547.

[3] Shin, H., and Kim, J.-B., 2010, “A Phenomenological Constitutive Equation to Describe Various Flow Stress Behaviors of Materials in Wide Strain Rate and Temperature Regimes,” J. Eng. Mater. Technol., 132, pp. 021009.

[4] Nakamura, T., Tanaka, S., Hayakawa, K., Fukai, Y., 2000, “A Study of the Lubrication Behavior of Solid Lubricants in the Upsetting Process,” J. Tribol., 122(4), pp. 803–808.

[5] Azushima, A., 2000, “FEM Analysis of Hydrostatic Pressure Generated Within Lubricant Entrapped into Pocket on Workpiece Surface in Upsetting Process,” J. Tribol., 122(4), pp.822–827.

[6] Azushima A., Yanagida A., Tani, S., 2010, “Permeation of Lubricant Trapped Within Pocket into Real Contact Area on the End Surface of Cylinder,” J. Tribol., 133(1), 011501.

[7] Mizuno T., Hasegawa K, 1982, “Effects of Die Surface Roughness on Lubricating Conditions in the Sheet Metal Compression-Friction Test,” J. of Lubrication Tech. 104(1), pp. 23–28.

[8] Ramaraj, T. C., Shaw, M. C., 1985, “A New Method of Evaluating Metal-Working Lubricants,” J. Tribol., 107(2), pp. 216–219.

[9] Krishna, C. H., Davidson, M. J., Nagaraju, C, and Kumar, P. R., 2015, “Effect of Lubrication in Cold Upsetting Using Experimental and Finite Element Modeling,” J. Test. Eval., 43(1), pp. 53–61.

[10] Misirili, C., 2014, “On Materials Flow Using Different Lubricants in Upsetting Process,” Ind. Lub. Tribol., 66(5), pp. 623–631.

[11] Banerjee, J. K., 1985, “Barreling of Solid Cylinders Under Axial Compression,” J. Eng. Mater. Technol., 107(2), pp. 138–144.

[12] Banerjee, J. K., and Cárdenas, Gerardo., 1985, “Numerical Analysis on the Barreling of Solid Cylinders Under Axisymmetric Compression,” J. Eng. Mater. Technol., 107(2), pp. 145–147.

[13] Lee C. H., and Altan T., 1972, “Influence of Flow Stress and Friction Upon Metal Flow in Upset Forging of Rings and Cylinders,” J. Eng. Ind., 94(3), pp. 775–782. 

[14] Schroeder, W., and Webster, D. A., 1949, “Press-Forging Thin Sections: Effect of Friction, Area, and Thickness on Pressure Required,” J. Appl. Mech., 16, pp. 289–294.

[15] Siebel, E., 1923, “Grundlagen zur Berechnung des Kraft- und Arbeitsbedarfs beim Schmieden und Walzen (Basics for Calculating the Force and Work Requirements of Forging and Rolling),” Stahl. Und. Eisen., 43(41), pp. 1295–1298.

[16] Siebel E., and Pomp A., 1927, “Die Ermittlung der Formänderungsfestigkeit von Metallen durch den Stauchversuch (Determination of the Deformation Strength of Metals by the Compression Test),” Mitt. Kaiser. Wilhelm. Inst. Eisenforsch., 9(8), pp. 157–171.

[17] Han, H., 2002, “The Validity of Mechanical Models Evaluated by Two-Specimen Method under the Known Coefficient of Friction and Flow Stress,” J. Mater. Process. Technol., 122, pp. 386–396.

[18] Loizou, N., and Sims, R. B., 1953, “The Yield Stress of Pure Lead in Compression,” J. Mech. Phys. Solids., 1(4), pp. 234–243.

[19] Richardson, G. R., Hawkins, D. N., and Sellars, C. M., 1985, “Worked Examples in Metal Working,” Inst. Metals, London.

[20] Thompsen, E. G., Yang, C. T., and Kobayashi, S., 1965, “Mechanics of Plastic Deformation in Metal Processing,” Macmillan, New York.

[21] Christiansen, P., Martins, P. A. F., and Bay, N., 2016, “Friction Compensation in the Upsetting of Cylindrical Test Specimens,” Exp. Mech., 56(7), pp. 1271–1279.

[22] Smith, K. K., and Kassner, M. E., 2016, “Through-Thickness Compression Testing of Commercially Pure (Grade II) Titanium Thin Sheet to Large Strains,” J. Metall., 2016, 6178790.

[23] Altinbalik, T., Akata, H., and Can, Y., 2007, “An Approach for Calculation of Press Loads in Closed-Die Upsetting of Gear Blanks of Gear Pumps,” Mater. Des., 28(2), pp. 730–734.

[24] Hill, R., 1950, “The Mathematical Theory of Plasticity,” Oxford University Press, London., pp. 262–281. 

[25] Rand, J. L., 1967, “An Analysis of the Split Hopkinson Pressure Bar,” Technical Report (NOLTR 67-156), US Naval Ordnance Laboratory, Silver Spring, Maryland.

[26] Cook, M., and Larke, E. C., 1945, “Resistance of Copper and Copper Alloys to Homogeneous Deformation in Compression,” J. Inst. Metals., ​71(12), pp. ​371–390.

[27] Avitzur, B., 1968, “Metal forming; processes and analysis,” McGraw-Hill, New York.

[28] Schey, J. A., Venner, T. R., and Takomana, S. L., 1982, “The Effect of Friction on Pressure in Upsetting at Low Diameter-to-Height Ratios,” J. Mech. Work. Technol., 6(1), pp. 23–33.

[29] Hartley, P., Sturgess, C. E. N., and Rowe, G. W., 1980, “Influence of Friction on the Prediction of Forces, Pressure Distributions and Properties in Upset Forging,” Int. J. Mech. Sci., 22(12), pp. 743–753.

[30] Bugini, A., Maccarini, G., Giardini, C., Pacagnella, R., and Levi, R., 1993, “The Evaluation of Flow Stress and Friction in Upsetting of Rings and Cylinders,” CIRP Annals., 42(1), pp. 335–338.

[31] Tan, X., Zhang, W., and Bay, N., 1999, “A New Friction Test Using Simple Upsetting and Flow Analysis,” Adv. Technol. Plast., 1(6), pp. 365–370.

[32] Kamler, F., Niessen, P., and Pick, R. J, 1995, “Measurement of the Behaviour of High Purity Copper at Very High Rates of Straining,” Canad. J. Phys., 73(5-6), pp. 295–303.

[33] Bertholf, L. D., and Karnes, C. H., 1975, "Two-Dimensional Analysis of the Split Hopkinson Pressure Bar System," J. Mech. Phys. Solids., 23(1), pp. 1–19.

[34] Cha, S. H., Shin, H., and Kim, J. B., 2010, “Numerical Investigation of Frictional Effects and Compensation of Frictional Effects in Split Hopkinson Pressure Bar (SHPB) Test (in Korean),” Trans. Korean Soc. Mech. Eng., A., 34(5), pp. 511–518.

[35] Gorham, D. A., Pope, P. H., and Cox, O., 1984, “Sources of Error in Very High Strain Rate Compression Tests,” Inst. Phys. Conf. Ser., 70, pp. 151–158.

[36] Hall, I. W., and Guden, M., 2003, “Split Hopkinson Pressure Bar Compression Testing of an Aluminum Alloy: Effect of Lubricant Type,” J. Mater. Sci., 22(21), pp. 1533–1535.

[37] Mori, L. F., Krishnan, N., Cao, J., and Espinosa, H. D., 2007, “Study of the Size Effects and Friction Conditions in Microextrusion—Part II: Size Effect in Dynamic Friction for Brass-Steel Pairs,” J. Manuf. Sci. Eng., 129(4), pp. 677–689.

[38] Wang, Z. J., and Cheng, L. D., 2009, “Experimental Research and Numerical Simulation of Dynamic Cylinder Upsetting,” Mater. Sci. Eng., 499(1-2), pp. 138–141.

[39] Jankowiak, T., Rusinek, A., Lodygowski, T., 2011, “Validation of the Klepaczko–Malinowski Model for Friction Correction and Recommendations on Split Hopkinson Pressure Bar,” Finite Elem. Analysis Design., 47(10), pp. 1191–1208.

[40] Iwamoto, T., and Yokoyama, T., 2012, “Effects of Radial Inertia and End Friction in Specimen Geometry in Split Hopkinson Pressure Bar Tests: A Computational Study,” Mech. Mater., 51, pp. 97–109.

[41] Lu, Y., and Zhang, S., 2013, “Study on Interface Friction Model for Engineering Materials Testing in Split Hopkinson Pressure Bar Tests,” Mod. Mech. Eng., 3(1), pp. 27–33.

[42] Siviour, C. R., and Walley, S. M., 2018, “Inertial and Frictional Effects in Dynamic Compression Testing,” in: The Kosky-Hopkinson Bar Machine, pp. 205–247. 

2. Measurement of a Nearly Friction-Free Stress–Strain Curve of Silicone Rubber up to a Large Strain in Compression Testing 

This section provides an executive summary of the study carried out by Kim et al. (Experimental Mechanics 2018; 58(9): 1479–1484).

(1) Introduction

A reliable constitutive model and its precise calibration are two prerequisites to an accurate computer simulation of the mechanical deformation of materials and structures [1-2]. Rubber is often subjected to compressive loading up to a large strain (e.g., up to the nominal strain of even 90%). For the simulation of such compressive deformation of rubber, it is desirable to calibrate a constitutive model using a compressive stress–strain curve constructed up to a large strain [3-9]. 

A stress–strain curve measured under a friction-free condition is required to calibrate a constitutive model. Otherwise, the obtained stress–strain curve is not the true material property but merely a load-displacement indicator of a compression event in the testing machine that takes place under the influence of friction; the specimen deforms non-uniformly in shape and is no longer in the uniaxial and homogeneous stress state. Thus, determining the friction-free stress–strain curve of materials under uniaxial compression is not a simple task. For a plastically deforming metallic material (the Poisson’s ratio is approximately 0.5), barreling and rollover (non-uniform deformation) of the specimen usually takes place because of friction, indicating an inhomogeneous and non-uniaxial stress state. The phenomena of barreling and rollover lead to overestimation of the stress–strain curve [19–26]. 

The stress–strain curves of rubber are different notably depending on the L/D ratio. Such a result imposes a difficulty in selecting a curve to be used for the calibration of a constitutive model as the friction-free curve. Although there has been considerable interest in measuring the compressive stress–strain curve of rubber [3-9], it is not easy to find studies that aimed to obtain the friction-free property, probably because an appropriate methodology was unavailable. In this regard, the necessity of developing a method for measuring the friction-free stress–strain curve is high. 

This study sets up a procedure for measuring the nearly friction-free stress–strain curve of rubber as follows. If one can measure the current contact area of rubber with the platens simultaneously with the measurement of the compressive stress–strain curve, the curve of the current contact area of the real specimen can be compared with that of the ideal specimen that deforms in the friction-free state; the current contact area of the ideal (friction-free) specimen can be calculated suitably by considering the volume constancy of an incompressible solid (a Poisson’s ratio of approximately 0.5): Ac Lc=Ao Lo, where A and L are cross-sectional area and length, respectively, and the subscripts c and o denote the current and initial values, respectively. If the current contact area of a series of specimens having a range of L/D ratios are compared with that (Ac) of the ideal specimen, it is postulated that the desirable (optimal) L/D ratio that results in the closest current contact area to that of the ideal specimen can be determined. The stress–strain curve measured at the optimal L/D ratio is closest to the friction-free stress–strain curve; it can be regarded as a nearly friction-free stress–strain curve.

The current contact area of rubber can be measured during the compression test using an appropriate apparatus based on various techniques, such as digital image correlation and 3D position measurement. This study introduces an alternative method for measuring the current contact area as follows. If we apply stamp ink onto the sidewall of the specimen at the beginning of the compression test, it is hypothesized that traces of ink will be left on the platen after the tests to allow measurement of the contact area and identification of the rollover-induced area. This method of measuring the current contact area is easy to carry out (handy) and economical. Its efficiency is to be tested in this study.

Once the above procedure and method for measuring the contact area are experimentally demonstrated, they may be useful for measuring nearly friction-free compressive stress–strain curves of various types of rubber material. The purpose of this study is to demonstrate that a nearly friction-free stress–strain curve of silicon rubber can be obtained successfully up to a fairly large compressive strain (a nominal strain of 0.9) using the above procedure and method. 

(2) σn-εn curves dependent on the L/D ratios of silicone rubber specimen 

The nominal stress–strain curves were highly dependent on the L/D ratio, as shown in Fig. 1. This L/D-ratio dependency results from the influence of friction between the specimen and platen; the influence of friction increases with the decrease of the L/D ratio [19–26] 

The specimen that deforms uniformly without volume change in the friction-free state is named the ideal specimen (or the friction-free specimen). The stress–strain curve of the ideal specimen is called the friction-free stress–strain curve. From the findings in Fig. 1 (described in the above paragraph), it is difficult to select the stress–strain curve that is closest to the friction-free stress–strain curve of silicone rubber. 

(3) Experimental methods

The nominal stress of the specimen (σn) was determined based on the equation σn=F/A0, where  F is the current load measured using the load cell, and  A0 is the initial cross-sectional area of the specimen. The nominal strain of the specimen (εn) was determined based on the relationship εn=(lc-ln )/l0, where  lc  and l0  are the current and initial lengths of the specimen, respectively. The true stress of the specimen (σt) was determined using the relationship σt=F/Ac, where  Ac is the current cross-sectional area of the specimen. The true strain (εt) of the specimen was determined using the equation εt=ln⁡(lc/l0 ). Tests for the series of specimens with a range of L/D ratios were carried out at a nominal strain rate (ε ̇n ) of 10-3 s-1. Once the desirable (optimal) L/D ratio (1.0 for silicone rubber, as will be shown later in this study) was determined based on the measured current contact area, additional tests at nominal strain rates of 10-2 and 10-4 s-1 were carried out for specimens with the optimum L/D value (=1.0). The speed of the cross head (dL/dt ) was constant, and its value at a given nominal strain rate (ε ̇n) was determined using the relationship dL/dt=L0 ε ̇n . All tests were carried out at ambient temperature. 

At the beginning of the compression test, stamp ink (purchased at a local stationary store) was applied to the sidewall of the specimen. After the test, the image of the ring-type ink mark that appeared on the steel platen was photographed (Examples of the ring-type mark will be presented later in Fig. 2.). The average value of the outer diameter of the ring mark was quantified using image-processing software (Image J). For this purpose, first, information on the number of pixels per centimeter (Nc) was obtained from the image of the ruler included in the same photograph. Second, three diagonal lines were artificially drawn on the image of the ring mark, resulting in six contact points (60 degree apart) between the diagonal lines and outer periphery of the ring mark. Third, the number of pixels (N) between the contact points on a given diagonal line was counted using Image J. Finally, the diameter of the ring mark (d) was determined using the relationship d=N/Nc. For each ring mark, the diameter was determined in this way three times using the three diagonal lines; the areas of the ring mark calculated from the three diameter values were averaged.

(4) Measurement of current contact areas according to the L/D ratios 

Fig. 2. Examples of the ink mark remaining on the surface of the stainless-steel platen after the compression test up to a nominal strain of 0.9. (a) L/D =0.2, (b) L/D =0.4, (c) L/D =0.8, and (d) L/D =1.0. M: outer periphery of the ink mark. G: grooved ring on the steel platen. Cross lines at 60o interval were drawn to find contact points (marked as circled numbers) for measurement of the diameter in image processing software. 

The ink mark remaining on the steel platen after the compression test was measured to examine the observed stress–strain behavior. Examples of the ink mark remaining on the surface of the stainless-steel platen are illustrated in Fig. 2. The stamp ink was only applied to the sidewall of the specimen, so any trace of ink on the platen would be direct evidence of rollover of the sidewall caused by high friction barreling. 

From the above findings, this study quantified only the total contact area. The averaged total contact area quantified by the method described in section 3 is presented in Fig. 3 as a function of the nominal strain. Included in Fig. 3 is the ideal current contact area of the friction-free specimen (Ac), that was calculated from the initial area of the specimen (A0) based on the assumption of the volume constancy condition in the friction-free state: AcLc=A0L0.

Fig. 3. Comparison of the measured total contact area (dashed curves with data points) with the ideal contact area (solid curves) for the case when the L/D ratio is (a) 0.2, (b) 0.4, (c) 0.8, and (d) 1.0. The dashed curves were constructed using measured data points based on the Levenberg-Marquardt algorithm implemented in commercial software (OriginLab).

The total contact area, including the rollover-induced contact area, was less than the ideal contact area for L/D ratios less than 1.0, as shown in Fig. 3. The total contact area of the specimen was reasonably consistent with the contact area of the ideal (friction-free) specimen for an L/D ratio of 1. Therefore, the stress–strain curve of this specimen (L/D=1.0) is closest to the friction-free stress–strain curve among the curves presented in Fig. 1.

After the optimal L/D ratio of 1.0 was determined at a nominal strain rate of 10-3 s-1, and additional tests were carried out at 10-2 and 10-4 s-1 to check whether the measured contact area is independent of the strain rate. The results are included in Fig. 3. Figure 3 shows that the measured contact areas at the nominal strain rates of 10-2 and 10-4 s-1 are reasonably consistent with that at 10-3 s-1. This finding indicates that the speed of rollover of the side wall (where ink was applied) does not influence the size of the ring mark that forms during loading.  Therefore, the method for measuring the total contact area using stamp ink can be used in quasi-static tests in a wide range of strain rates (at least up to a strain rate of 10-2 s-1). No reason can be found why formation of the outer periphery should be influenced by a slower strain rate than 10-4 s-1; there seems to be no lower limit of the strain rate for use of stamp ink. As for the maximum limit of the strain rate up to which the current method can be applied, further study is needed. 

(5) Conclusion

The nominal stress–strain curve of the specimen with an L/D ratio of 1.0 (Fig. 1) was converted to the curve of true stress vs. true strain by assuming that the specimen is ideal (friction free), as shown by the solid red curve in Fig. 4. The dashed curve (red) of Fig. 4 was constructed based on the measured contact area based on the dashed curve in Fig. 3(d). The two red curves for the specimen with L/D=1.0 are reasonably consistent up to the measured strain, which indicates that the nominal stress–strain curve of this specimen (Fig. 1) and the true stress–strain curve (shown as a dashed curve in red in Fig. 4) are close to the friction-free curve. 

This study presented a procedure for measuring the friction-free compressive stress–strain curve by comparing the measured current contact area with the ideal (friction-free) contact area calculated from the condition of volume constancy of the friction-free specimen. It also presented a handy method to quantify the total contact area. The application of stamp ink to the sidewall of a rubber specimen appears to be efficient and informative for checking the reliability of an experimentally measured stress–train curve of rubber from the viewpoint of the proximity of the two area curves. It was demonstrated that a nearly friction-free stress–strain curve of silicon rubber can be obtained successfully up to a fairly large compressive strain (up to the nominal strain of 0.9) using the presented methodology. 

The results in this study may be useful for future studies aiming at measuring friction-free compressive stress–strain curves of various types of rubber material; the following are tips (recommendations) for such future studies. 

Fig. 4. True stress–strain curves (solid curves) converted from the nominal curves in Fig. 1 by assuming that the specimen is ideal.

Dashed curves were constructed based on the measured total contact area (dashed curves in Fig. 3). 

(6) References

[1] Ju ML, Jmal H, Dupuis R, Aubry E (2015) Visco-hyperelastic constitutive model for modeling the quasi-static behavior of polyurethane foam in large deformation. Polym Eng Sci 55:1795–1804

[2] Kim W-D, Kim W-S, Woo C-S, Lee H-J (2004) Some considerations on mechanical testing methods of rubbery materials using nonlinear finite element analysis. Polym Int 53:850–856 

[3]  Gent AN, Discenzo FM, Suh JB (2009) Compression of rubber disks between frictional surfaces. Rubber Chem Technol 82:1–17

[4]  Bradley GL, Chang PC, Mckenna GB (2001) Rubber modeling using uniaxial test data. J Appl Polym Sci 81:837–848

[5]  Williams JG, Gamonpilas C (2008) Using the simple compression test to determine Young’s modulus, Poisson’s ratio and the Coulomb friction coefficient. Int J Solids Struct 45:4448–4459

[6]  Doman DA, Cronin DS, Salisbury CP (2006) Characterization of polyurethane rubber at high deformation rates. Exp Mech 46:367–376

[7]  Dar UA, Zhang W, Xu Y, Wang J (2014) Thermal and strain rate sensitive compressive behavior of polycarbonate polymer-experimental and constitutive analysis. J Polym Res 21:519

[8]  Mansouri MR, Darijani H, Baghani M (2017) On the correlation of FEM and experiments for hyperelastic elastomers. Exp Mech 57:195–206

[9]  Mesa-Múnera E, Ramírez -Salazar JF, Boulanger P, Branch JW (2012) Inverse-FEM characterization of a brain tissue phantom to simulate compression and indentation. Ing y Cienc 8:11–36

[10] Johnson GR (1983) A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. In: Proceedings of the 7th International Symposium on Ballistics, The Hague, Netherlands, 1983

[11] Shin H, Kim J-B (2010) A phenomenological constitutive equation to describe various flow stress behaviors of materials in wide strain rate and temperature regimes. J Eng Mater Technol 132:21009

[12] Mooney M (1940) A theory of large elastic deformation. J Appl Phys 11:582–592

[13] Rivlin RS, Saunders DW (1951) Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Phil Trans R Soc Lond A 243:251–288

[14] Treloar LRG (1943) The elasticity of a network of long-chain molecules I. Trans Faraday Soc 39:36–41

[15] Ogden RW (1972) Large deformation isotropic elasticity - on the correlation of theory and experiment for incompressible rubberlike solids. Proc R Soc London 326:565–584

[16] Gent AN, Thomas AG (1958) Forms for the stored (strain) energy function for vulcanized rubber. J Polym Sci 28:625–628

[17] Yeoh OH (1990) Characterization of elastic properties of carbon-black-filled rubber vulcanizates. Rubber Chem Technol 63:792–805

[18] Arruda EM, Boyce MC (1993) A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J Mech Phys Solids 41:389–412

[19] Schroeder W, Webster DA (1949) Press-forging thin sections-effect of friction, area, and thickness on pressures required. J Appl Mech ASME 16:289–294

[20] Hill R (1998) The mathematical theory of plasticity. Oxford University Press Inc., New York

[21] Rand JL (1967) An analysis of the split Hopkinson pressure bar. NOLTR 67-156, White Oak, MD

[22] Cha SH, Shin H, Kim JB (2010) Numerical investigation of frictional effects and compensation of frictional effects in split hopkinson pressure bar (SHPB) test. Trans Korean Soc Mech Eng A 34:511–518

[23] Banerjee JK (1985) Barreling of solid cylinders under axial compression. J Eng Mater Technol 107:138–144

[24] Christiansen P, Martins PAF, Bay N (2016) Friction compensation in the upsetting of cylindrical test specimens. Exp Mech 56:1271–1279

[25] Khan AS, Balzer JE, Wilgeroth JM, Proud WG (2014) Aspect ratio compression effects on metals and polymers. J Phys Conf Ser 500

[26] Espinosa HD, Patanella A, Fischer M (2000) A novel dynamic friction experiment using a modified Kolsky bar apparatus. Exp Mech 40:138–153

3. Determination of the Flow Stress–Strain Curve of Metals using Compressive Load–Displacement Curves of a Hat-Type Specimen

This section provides an executive summary of the study carried out by Lee et al. (Journal of Applied Mechanics, in press).

(1)   INTRODUCTION

The precise calibration of the constitutive model [1–6] is a prerequisite for an accurate computer simulation of the deformation behavior of solids and structures. To calibrate a constitutive model, stress–strain curves of materials measured in a friction-free stress state (the uniform and uniaxial stress state) are needed. 

Compression testing using a cylindrical specimen is probably the simplest and most handy methodology for measuring the stress–strain curve. In the compression test, however, the uniform and uniaxial stress state is manifested only when there is no friction between the specimen and platen. When there is friction, barreling and rollover of the side wall of the specimen usually takes place [7–14]. Then, the stress state of the specimen ceases to be uniaxial, becoming multi-axial and non-uniform. As a result, the stress–strain curve measured in the compression test is overestimated; the degree of overestimation increases as the height-to-diameter ( ) ratio decreases [15–18]. 

The ratio of the measured flow stress ( ) to the friction-free flow stress ( ) is described by an equation called the friction-compensation equation, : , where  is the Coulomb friction coefficient. An example of the friction-compensation equation developed by Schroeder and Webster (SW) [15] for the sliding contact surface is 

where . Once  is known,  can be obtained from the measured flow stress under a frictional condition ( ). Thus far, a number of other friction compensation equations have been developed [16–18]. To use these friction-compensation equations to obtain the friction-free stress–strain curve, the Coulomb friction coefficient needs to be measured separately using appropriate methodologies such as the ring compression test [19–20] and barreling compression test [21–22]; however, the reliabilities of these test methodologies have not been completely verified [20–21].

If a reliable friction coefficient is obtained, its application to the cylindrical specimen must be done carefully. According to Schroeder and Webster (SW) [15], there are three types of contact surfaces between a cylindrical specimen and the platen: (i) the entire contact surface is sliding, (ii) there is no relative sliding on the inner zone while the outer annular zone is sliding, and (iii) no relative sliding occurs on the entire contact surface. SW provided compensation equations for each contact surface. As for the first case when the entire contact surface is sliding, besides SW, a number of researchers have provided compensation equations [16–18]. To accurately compensate for the effect of friction in the measured stress–strain curve of the cylindrical specimen, information on not only the value of the friction coefficient but also the contact mechanism (the types of the contact surface) is necessary. Knowledge of the most reliable friction-compensation equation among the available equations is also necessary [15–18].

Despite the above limitations, most of the studies in the literature used cylindrical specimens to measure the compressive stress–strain curve and compensated for the influence of friction using the compensation models developed for a sliding-contact surface. For instance, Han [23], Christiansen et al. [24], and Smith and Kassner [25] used Eq. (1), i.e., the compensation equation of Schroeder and Webster [15], to compensate for the influence of friction in their quasi-static stress-strain curves. In the study of Altinbalik et al. [26], Eq. (1) was used to predict press loads in closed-die upsetting. To compensate the dynamic stress–strain curves measured using the cylindrical specimen in the split Hopkinson pressure bar test [27–28], Kamler et al. [29], Bertholf and Karnes [30], and Cha et al. [18] used the compensation models of Hill [16], Rand [17], and Cha et al. [18], respectively. Except for the study by Christiansen et al. [24], the friction coefficient was arbitrarily assumed.

Under the above circumstances, eliminating the influence of friction itself in the experimental stage of specimen deformation may be one of the desirable approaches for obtaining a friction-free stress–strain curve in the compression test. In this regard, this study considers a hat-type specimen in compression testing. To the best of our knowledge, no study has been carried out to determine a friction-free stress–strain curve using the hat-type specimen. As will be presented later in this paper, the load–displacement curve of the hat-type specimen is manifested mainly from the plastic deformation of the localized shear zone that is far from the contact surfaces; friction at the contact surfaces hardly influences local deformation in the shear zone. In this regard, this study aims to determine the friction-free stress–strain curves of an aluminum alloy and tantalum by comparing the experimental and simulation curves of the load–displacement of a hat-type specimen subjected to a compression test. 

(2) Optimization of the constitutive parameters

The shape and axi-symmetric finite element model of the hat-type specimen considered in this study are presented in Fig. 1. The numerical experiment as well as the real experiment was carried out using the specimen.

The flow chart for the optimization process used to determine the friction-free stress–strain curve is illustrated in Fig. 2. The load–displacement curve of the hat-type specimen of either aluminum alloy (AA) or tantalum (Ta) was used as the target function. The stress–strain curves of the specimen materials to be determined were modeled using either the Ludwik or Voce constitutive law.

A governing program written in Python script selects the constitutive parameters using an optimization algorithm (the function “fminsearch” built in MATLAB) while initially guessed parameters are set by the user of the program. The program then runs the finite element (FE) package (ABAQUS) using the selected parameters. After the simulation is finished, the program compares the simulated load–displacement curve with the target function to calculate the value of the error.

If the value of the error is larger than the pre-defined level ( ), the governing program reselects the constitutive parameters based on the optimization algorithm and reruns the FE package. If the error value is less than the predefined value, it outputs the constitutive parameters that were used in the final simulation. The stress–strain curve of the specimen material is constructed (uncovered) using the optimized constitutive parameters that were used in the final simulation. 

(3) Numerical verification of the methodology

To numerically verify the presented methodology, the flow stress–strain curves of AA and Ta available in the literature [31, 32] were used as the input properties of the specimen materials. The deformed shapes of the hat-type specimens (for AA and Ta) obtained via numerical simulation (implicit finite element analysis) of the compression test are presented in Fig. 3 with contours of the equivalent plastic strain ( = 0). As can be observed in Fig. 3, deformation is concentrated in the shear zone, which is far from the contact surfaces. When the friction coefficient was set as unity, no appreciable differences were observed in the deformed shape and contour (not shown).

The load–displacement curves obtained in the numerical simulation are presented in Fig. 4(a) for AA and Ta. Although the friction coefficient was changed from zero to unity (extreme case), the load–displacement curves are almost the same regardless of the value of the friction coefficient at the contact surfaces. This is because the deformation of the shear zone is mainly responsible for the manifestation of the load–displacement curve, while friction at the contact surfaces does not influence the shear deformation that occurs in local areas far from the contact surfaces. If the stress–strain curve is extracted from such a friction-independent load–displacement curve, the obtained stress–strain curve should be a friction-free property of the material.

To determine the flow stress–strain curves that resulted in the respective load–displacement curves in Fig. 4, the flow stress–strain curves of AA and Ta were modelled using the Ludwik and Voce constitutive laws, respectively. Then, a numerical simulation of the compression test of the hat-type specimen was carried out by assuming appropriate constitutive parameters. The constitutive parameters were optimized via the process described in Fig. 2. The stress–strain curves constructed using the determined parameters are presented in Fig. 5 as closed squares. Fig. 5 shows the solid curves that indicate the input properties of the respective material used to obtain the load–displacement curves of Fig. 4 via the simulation (the target function).

As can be observed in Fig. 5, the stress–strain curves determined via the parameter optimization process (closed squares) are consistent with the input properties used for obtaining the target function via the simulation (the load–displacement curves in Fig. 4). This finding verifies that the determined stress–strain curves of AA and Ta from the load–displacement curves of the hat-type specimen via the parameter optimization process are reliable. The stress–strain curve obtained in this way is a friction-free property of the specimen material.

 Based on the numerical verification of the method as above, experiment for AA and TA specimens were carried out. The result is available in the finally published article.

(4) Shape design of the hat-type specimen

For the geometry considered in this study (Fig. 1), the load–displacement curve was independent of the friction coefficient between the hat-type specimen and the top/bottom platen (Fig. 4). Such a friction-independent load–displacement curve can be obtained using other geometries of the hat-type specimen as well. Based on a separate simulation (not shown), the hat-type geometry with a  value of 1 mm and  value of 2 mm also yielded the friction-independent load–displacement curve (the definition of  and  are shown in Fig. 1). If the value of  is insufficient at a given value of , the load–displacement curve tends to show friction dependency. Using this information, researchers may suitably design their own geometries of the hat-type specimen to achieve a friction-independent load–displacement curve and verify the geometries via FE analysis (Fig. 4).

(5) Conclusion

The numerical simulation of the compression test for the hat-type specimen employed in this study indicates that the load–displacement curve is independent of friction between the top/bottom platens and the specimen. This finding means that, if a stress–strain curve is extracted from the measured load–displacement curve, the obtained stress–strain curve is a friction-free property of the material. 

This study numerically verified the following process for obtaining a friction-free stress–strain curve of a specimen material from the load–displacement curve of the hat-type specimen. The flow stress–strain curve of the specimen material to be determined is modelled using either the Ludwik or Voce constitutive law. Then, a numerical simulation for the experimental process is carried out by assuming appropriate constitutive parameters of the specimen material. The simulation is repeated until the simulated load–displacement curve is reasonably coincident to the experimentally obtained load–displacement curve (the target function). The constitutive parameters are extracted when the simulated load–displacement curve is consistent with the target function. The optimized constitutive parameters in this way are used to construct the friction-free stress–strain curve.

Based on the above process, friction-free stress–strain curves of aluminum alloy (Al 6061-T6) and tantalum were experimentally determined. For the case of the aluminum alloy, the experimentally determined flow stress–strain curve in this study was consistent with the one reported in the literature, verifying the reliability of the proposed process using the hat-type specimen. 

As for the design guideline for the hat-type specimen, if the value of  is insufficient at a given value of  (  and  are defined in Fig. 1), the load–displacement curve tends to show friction-dependency. With this knowledge, researchers may suitably design their own geometries of the hat-type specimen to achieve a friction-independent load–displacement curve and verify the geometries via FE analysis. 

 In case the strain rate of the determined stress–strain curve is needed, it is desirable to carry out the optimization process using an implicit analysis, such as in this study, and carry out the explicit analysis later (separately) using the parameters determined in the implicit analysis. After the explicit analysis, the average value of the equivalent plastic strain in the shear zone can be extracted as a function of time, and the slope of the curve in the plastic regime can be averaged to obtain the strain rate.

(6) REFERENCES

[1] Ludwik, P., 1909, Elemente der Technologisehen Mechanik (Elements of Technological Mechanics), Julius Springer, Berlin, p. 32. ISBN: 978-3-662-39265-2.

[2] Voce, E., 1948, “The Relationship between Stress and Strain for Homogeneous Deformation,” J. Inst. Metals. 74, pp. 537–562.

[3] Johnson G. R., and Cook W. H., 1983, “A Constitutive Model and Data for Metals Subjected to Large Strains, High Strain Rates and High Temperatures,” Proc. of the 7th International Symposium on Ballistics. Hague: Organizing committee of the 7th ISB; pp. 541–547.

[4] Shin, H., and Kim, J.-B., 2010, “A Phenomenological Constitutive Equation to Describe Various Flow Stress Behaviors of Materials in Wide Strain Rate and Temperature Regimes,” J. Eng. Mater. Technol., 132, pp. 021009.

[5] Gross, A. J., and Ravi-Chandar K., 2015, “On the Extraction of Elastic–Plastic Constitutive Properties from Three-Dimensional Deformation Measurements,” J. Appl. Mech., 82(7) pp. 071013.

[6] Yafu F., Wang Q.-d., Ning J.-s., Chen J., and Wei, J., 2010, “Experimental Measure of Parameters: The Johnson–Cook Material Model of Extruded Mg–Gd–Y Series Alloy,” J. Appl. Mech., 77(5) pp. 051902.

[7] Felling, A. J., and Doman, D. A., 2018, “A New Video Extensometer System for Testing Materials Undergoing Severe Plastic Deformation,” J. Eng. Mater. Technol., 140(3), pp. 031005.

[8] Krishna, C. H., Davidson, M. J., Nagaraju, C, and Kumar, P. R., 2015, “Effect of Lubrication in Cold Upsetting Using Experimental and Finite Element Modeling,” J. Test. Eval., 43(1), pp. 53–61.

[9] Misirili, C., 2014, “On Materials Flow Using Different Lubricants in Upsetting Process,” Ind. Lub. Tribol., 66(5), pp. 623–631.

[10] Banerjee, J. K., 1985, “Barreling of Solid Cylinders Under Axial Compression,” J. Eng. Mater. Technol., 107(2), pp. 138–144.

[11] Banerjee, J. K., and Cárdenas, G., 1985, “Numerical Analysis on the Barreling of Solid Cylinders Under Axisymmetric Compression,” J. Eng. Mater. Technol., 107(2), pp. 145–147.

[12] Lee C. H., and Altan T., 1972, “Influence of Flow Stress and Friction Upon Metal Flow in Upset Forging of Rings and Cylinders,” J. Eng. Ind., 94(3), pp. 775–782. 

[13] Kim, S., Kim, M., Shin, H., and Rhee, K. Y., 2018, “Measurement of a Nearly Friction-Free Stress–Strain Curve of Silicone Rubber up to a Large Strain in Compression Testing,” Exp. Mech., 58(9), pp. 1479–1484.

[14] Lee, J.-H., Shin, H., Kim, J.-B., Seo, S.-J., Lee, J.-G., Lee, J.-E., Yoon T.-S., and Jung, C.-S., “Numerical Verification of Contact Conditions and Friction-Compensation Models for the Axisymmetric Compression Test,” J Tribology, Submitted.

[15] Schroeder, W., and Webster, D. A., 1949, “Press-Forging Thin Sections: Effect of Friction, Area, and Thickness on Pressure Required,” J. Appl. Mech., 16, pp. 289–294.

[16] Hill, R., 1950, “The Mathematical Theory of Plasticity,” Oxford University Press, London, pp. 262–281. 

[17] Rand, J. L., 1967, “An Analysis of the Split Hopkinson Pressure Bar,” Technical Report (NOLTR 67-156), US Naval Ordnance Laboratory, Silver Spring, Maryland.

[18] Cha, S.-H., Shin, H., and Kim, J.-B., 2010, “Numerical Investigation of Frictional Effects and Compensation of Frictional Effects in Split Hopkinson Pressure Bar (SHPB) Test (in Korean),” Trans. Korean Soc. Mech. Eng., A., 34(5), pp. 511–518.

[19] Hu, C., Ou, H., Zhao, Z., 2015, “An Alternative Evaluation Method for Friction Condition in Cold Forging by Ring with Boss Compression Test,” J. Mater. Process. Technol., 224, pp. 18–25.

[20] Sofuoglu, H., Gedikli, H., Rasty, J., 2001 “Determination of Friction Coefficient by Employing the Ring Compression Test,” J. Eng. Mater. Technol., 123, pp. 338–348.

[21] Solhjoo, S., 2010, “A note on ‘‘Barrel Compression Test”: A Method for Evaluation of Friction,” Comput. Mater. Sci., 49, pp. 435–438.

[22] Yao, Z., Mei, D., Shen, H., Chen, Z., 2013, “A Friction Evaluation Method Based on Barrel Compression Test,” Tribol. Lett., 51, pp. 525–535.

[23] Han, H., 2002, “The Validity of Mechanical Models Evaluated by Two-Specimen Method under the Known Coefficient of Friction and Flow Stress,” J. Mater. Process. Technol., 122, pp. 386–396.

[24] Christiansen, P., Martins, P. A. F., and Bay, N., 2016, “Friction Compensation in the Upsetting of Cylindrical Test Specimens,” Exp. Mech., 56(7), pp. 1271–1279.

[25] Smith, K. K., and Kassner, M. E., 2016, “Through-Thickness Compression Testing of Commercially Pure (Grade II) Titanium Thin Sheet to Large Strains,” J. Metall., 2016, 6178790.

[26] Altinbalik, T., Akata, H., and Can, Y., 2007, “An Approach for Calculation of Press Loads in Closed-Die Upsetting of Gear Blanks of Gear Pumps,” Mater. Des., 28(2), pp. 730–734.

[27] Al-Mousawi, M. M., Reid, S. R., and Deans, W. F., 1997, “The Use of the Split Hopkinson Pressure Bar Techniques in High Strain Rate Materials Testing,” Proc. IMechE, Part C: J. Mech. Eng. Sci., 211(4), pp. 273–292.

[28] Shin, H., and Kim, J-B., 2018, “Evolution of Specimen Strain Rate in Split Hopkinson Bar Test,” Proc. IMechE, Part C: J. Mech. Eng. Sci., In press, 2018.

[29] Kamler, F., Niessen, P., and Pick, R. J., 1995, “Measurement of the Behaviour of High Purity Copper at Very High Rates of Straining,” Canad. J. Phys., 73(5-6), pp. 295–303.

[30] Bertholf, L. D., and Karnes, C. H., 1975, "Two-Dimensional Analysis of the Split Hopkinson Pressure Bar System," J. Mech. Phys. Solids., 23(1), pp. 1–19.

[31] Ambriz, R. R., Mesmacque, G., Ruiz, A., Amrouche. A., Lopez. V. H., 2010, “Effect of welding profile generated by the modified indirect electric arc technique on the fatigue behavior of 6061-T6 aluminum alloy,” Mat. Sci. Eng. A., 527, pp. 2057–2064.

[32] Chen, S. R., Gray Ill, G. T., 1996, “Constitutive Behavior of Tantalum and Tantalum-Tungsten Alloys,” Metall. Mater. Trans., A, 27, pp. 2994–3006.

[33] Lemanski, S. L., Petrinic, N., Nurick,  G. N., 2013, “Experimental Characterisation of Aluminium 6082 at Varying Temperature and Strain Rate,” Strain, 49, pp. 147–157. 

[34] Hoge, K. G., 1966, “Influence of Strain Rate on Mechanical Properties of 6061-T6 Aluminum under Uniaxial and Biaxial States of Stress,” Exp. Mech., 6, pp. 204–211.