Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 49, 1, pp. 159-173, Warsaw 2011 IMPACT TEST ANALYSIS OF DYNAMIC MECHANICAL PARAMETERS OF A RIGID-PLASTIC MATERIAL WITH LINEAR STRAIN HARDENING Edward Włodarczyk Faculty of Mechatronics, Military University of Technology, Warsaw, Poland e-mail: edward.wlodarczyk@wat.edu.pl Asimple engineeringmethod to determine dynamicmechanical parame- ters of a rigid-plastic material with linear strain hardening is presented in this paper. For this purpose, Taylor’s impact test, i.e., perpendicular impact of a long rodonaflat rigid targethas beenused.The rod ismade of the testedmaterial. The transcendental equationswhich explicitly de- terminate the dynamic yield stress and the plastic strain of the rod have been derived. The general trend of the obtained results is observation of the appearance of higher strengths at higher impact velocities, which is in agreementwith expectations. The applied in literature approxima- tion of dynamic properties ofmetalswith strainhardeningbymeans of a perfectly plastic material is far-reaching simplification disagreeing with reality. Key words: impact loads, Taylor impact test, dynamic yield stress 1. Introduction Beginning from the second half of the last century, there has beenmuch inte- rest in dynamic initial boundary-valueproblemsof the theory plasticity. Those problems were thoroughly addressed in monographs by Kolsky (1953), Bro- berg (1956), Goldsmith (1960), Shewmon and Zackay (1961), Rakhmatulin and Demyanov (1961), Zukas (1962), Perzyna (1966), Cristescu (1967), Lin- dholm (1968), Kinslow (1970), Nowacki (1974), Kaliski et al. (1992), Meyers (1994). The investigations were necessary in order to obtain reliable operation of various machine elements and special objects which are exposed to impact loadings in extreme conditions.The importantplace in suchproblemsoccupies the Taylor test. 160 E. Włodarczyk Taylor published his theory in 1948 (Taylor, 1948). Originally, one- dimensional analysis of Taylor was used by Whiffin (1948) to estimate the dynamic yield stress of specimens. There has been much interest in impact testing and estimating dynamic yield stress since then. A selective review of the literature with respect to the Taylor impact test is in the papers by Jones et al. (1987, 1997), and is not necessary to be discussed in this paper. The present opinion seems to be that Taylor’s theory fails to provide re- liable yield stress estimates, especially for tests conducted at higher velocities (Jones et al., 1987, 1997). For this reason, many investigators correlate their results with sophisticated computer analyses which are capable of utilizing several complex forms of constitutive eqations. These programs can match geometry of the post-test specimen with very high accuracy and give very re- liable estimates for material properties. The drawback is that these programs are expensive and often require substantial amounts of time to execute. The Taylor impact test is a useful experiment for estimating material be- haviour at high strain rates (Meyers, 1994). The test is reproducible and is reasonably economical after the initial investment has beenmade. Jones et al. (1987) assert that simple engineering theories, such as that given by Taylor, still have considerable value. Such theories frequently give investigators insight into the interaction of physical parameters and their re- lationship with the outcome of the event. Most often, these interactions are difficult to ascertain from complex computer outputs. As a result, simple en- gineering theories often provide the basis for the design of experiments and are frequently used to refine the areas in which computing is to be done. Bearing inmind theabove-mentioned facts, theTaylor problem, for a rigid- plastic material with linear strain hardening, loaded by an impact has been solved in a closed form in this paper.The engineering transcendental equations which explicitly determine the dynamic yield stress and plastic strain have been derived. 2. Formulation of the problem The corrected Taylor theory represented by Jones et al. (1987, 1997) has been used in this paper. Consider a uniform rod of the initial dimensions: length L and cross- sectional area A0, which impacts against a rigid boundary. Let x denote a Lagrangian coordinate aligned with the axis of the rod and having its origin at the end of the rod opposite to the impacted end. The initial velocity of Impact test analysis of dynamic mechanical... 161 the rod is denoted by V . Assuming that V is large enough, a portion of the rod will deform plastically. Let X represent the time-dependent extent of the plastic zonemeasured relative to the original configuration of the rod, Fig.1a, S be the time-dependent displacement of the back end of the rod, as shown in Fig.1b, and h – the time-dependent extent of the plastic zone measured relative to the deformed configuration of the rod. Define l as L−X so that l+X =S+ l+h=L. Fig. 1. Schematic illustration of the rod impacting a rigid boundary: (a) original configuration of the rod; (b) deformed configuration of the rod Assume that the rodmaterial is rigid-plastic with linear strain hardening, Fig.2a, and is incompressible.The rigid-perfectly plasticmaterial, Fig.2b,was considered in papers by Taylor (1948) and Jones et al. (1987). The material model shown in Fig.2a sufficiently approximates dynamicmechanical proper- ties of somemetals, e.g. alloy steels, especially chromium-nickel steels (Ashby and Jones, 1993; Lee and Tupper, 1951). The influence of transverse strain and friction force between the target surface and the impacted end of the rod upon its longitudinal motion are neglected. Consider the motion of the undeformed section of the rod. At some ti- me t, during the deformation, suppose that the undeformed section length is L−X(t) and suppose that Ṡ(t) = υ(t) is its speed, see Fig.3a. As shown in Fig.3b, the undeformed section has lost an increment ∆X to plastic de- formation at some later time t+∆t. This increment has undergone plastic deformation and now has a new cross-sectional area A1. Themass of the pla- stic element, however, remains ρA0∆X, where ρ is thematerial densitywhich is assumed to be constant throughout the deformation process. The speed of this element is denoted by u, which will generally be different from υ+∆υ, i.e. the speed of the remaining rigid end. The forces that act on this new pla- 162 E. Włodarczyk Fig. 2. Stress-strain curves for ductile materials: (a) rigid-plasticmaterial with linear strain hardening; (b) rigid-perfectly plastic; (c) true curve for chromium-nickel steel Fig. 3. Schematic illustration of the rear portion of the rod. The undeformed section shown in (a) has transferred somemass to the deformed section after a time interval ∆t, as indicated in (b) stic element are in interaction with the underformed section, denoted by F in Fig.3b, and its interaction with the previously deformedmaterial denoted as P +∆P in the same figure. The change in linearmomentumof the system from configuration 3a to 3b must equal the net impulse. Thus ρA0∆Xu+ρA0(L−X−∆X)(υ+∆υ)−ρA0(L−X)υ= 2P +∆P 2 ∆t (2.1) Impact test analysis of dynamic mechanical... 163 where (2P+∆P)/2denotes themeanvalue of forces P and P+∆P .Dividing both sides of this equation by ∆t and neglecting small quantities of higher order, and taking limits (∆t→ 0), gives (L−X)υ̇− Ẋ(υ−u)= P ρ A0 (2.2) where the superposed dots denote derivatives with rispect to time. However P =σA1 =σ(εp) A0 1+εp (2.3) where σ and εp are, respectively, the engineering stress and strain at the deformed cross section. Combining Egs. (2.2) and (2.3), gives (L−X)υ̇− Ẋ(υ−u)= 1 ρ f(εp) (2.4) where f(εp)= σ(εp) 1+εp (2.5) In terms of the undeformed section length l, Eq. (2.4) becomes lυ̇+ l̇(υ−u)= 1 ρ f(εp) (2.6) where Ẋ =−l̇. This expression is the equation ofmotion of the undeformed (rigid) section of the rod. This equation has been solved in the following considerations. 3. General solution to simplified (u=0) equation of motion (2.6) Previously, it was assumed that the rod material is rigid-plastic with linear strain hardening. This rod impacts the rigid target with a large enough spe- ed V . At such a condition, the plastic wave is generated in the rod during the impact process. This wave propagates along the rod from the rigid target towards of its free end. The velocity of the wave is determined by the formula a1 = √ E1 ρ (3.1) where E1 is the modulus of strain hardening. 164 E. Włodarczyk It follows from experimental results and numerical calculations (Lee and Tupeer, 1951) aswell as fromanalytical solutions (Włodarczyk andJackowski, 2010) that themodulus E1 is poorly dependent on the impact speed and one can assume that the velocity a1 ≈ const. The plastically deformed portion of the rod is placed between the rigid target and plastic wave front. That portion of the rod is motionless. The elastic strain is equal to zero (εs =0) in the rigid plastic model, Fig.2a, and for that reason the velocity u = 0 in the deformed plastically portion of the rod contacting with rigid target. For u=0, fromEq. (2.6), we obtain d(lυ) dt = f(εp) ρ (3.2) In turn, the continuity of the Lagrangian component of displacement across the rigid-plastic interface gives U(X,t) =−S Differentiating this equation with respect to time, leads to −εpẊ+u= εpl̇+u= Ṡ = υ (3.3) where εp = ∂U ∂X u= ∂U ∂t For u=0, from (3.3), we have dl dt = υ εp (3.4) Next, according to Fig.1, there is S+ l+h=L and after differentiation, we obtain Ṡ+ l̇+ ḣ=0 (3.5) or l̇= dl dt =−(a1+υ) (3.6) where ḣ= a1 and Ṡ= υ, a1 > 0. Impact test analysis of dynamic mechanical... 165 From Eqs. (3.4) and (3.6), it follows that the plastic strain εp can be determined bymeans of the formula εp =− υ a1+υ (3.7) Combining Egs. (3.2), (3.4) and (3.7), gives d(lυ) dl =−f[−υ/(a1+υ)] ρ(a1+υ) (3.8) Equation (3.8) has separable variables and can be immediately integrated, leading to an explicit dependence of l upon υ, namely ln l L =− υ ∫ V a1+y y(a1+y)+ 1 ρ f[−y/(a1+y)] dy (3.9) or l= l(υ)=Lexp ( V ∫ υ a1+y y(a1+y)+ 1 ρ f[−y/(a1+y)] dy ) (3.10) 4. Dynamic mechanical parameters of the rigid-plastic material with linear strain hardening According to Fig.2a, we have σ=σs+E1εp =σs+ρa 2 1εp (4.1) and then the function f(εp) has the form f(εp)= σ(εp) 1+εp = σs+ρa 2 1 ( − υ a1+υ ) 1− υ a1+υ or f ( − υ a1+υ ) = σs(a1+υ) a1 −ρa1υ (4.2) where σs and εp denote, respectively, the yield stress and longitudinal plastic strain of the rod. 166 E. Włodarczyk In order to simplify the quantitative analysis of particular dynamicmecha- nical parameters of the deformed rod, the following dimensionless quantities have been introduced α= ρV σs α∗ = ρV 2 YW β∗ = a1 V γ= υ V ξ= l L ξf = lf L ξp = hf L ξL = Lf L (4.3) where the symbols YW and lf denote, respectively, the dynamic yield stress of the rigid-plastic material with linear strain hardening, and the final length of the undeformed portion of the rod; Lf is the final overall length of the specimen and hf is the final length of the deformed portion of the rod. Substituting function (4.2) into Eq. (3.9) and using dimensionless quanti- ties (4.3), we get lnξ= 1 ∫ γ β∗+γ1 γ21 + 1 αβ∗ γ1+ 1 α dγ1 (4.4) The right-hand side of Eq.(4.4) can be expressed by the following functions: — for ∆> 0 F1(α,γ) = 1√ ∆ ( β∗− 1 2αβ∗ ) ln ∣ ∣ ∣ ∣ ∣ ( 2+ 1 αβ∗ − √ ∆ 2+ 1 αβ∗ + √ ∆ )( 2γ+ 1 αβ∗ + √ ∆ 2γ+ 1 αβ∗ − √ ∆ ) ∣ ∣ ∣ ∣ ∣ + (4.5) + 1 2 ln ∣ ∣ ∣ ∣ ∣ 1+ 1 αβ∗ + 1 α γ2+ 1 αβ∗ γ+ 1 α ∣ ∣ ∣ ∣ ∣ — for ∆< 0 F2(α,γ) = 1√ −∆ ( 2β∗− 1 αβ∗ ) arctan 2γ+ 1 αβ∗√ −∆ + 1 2 ln ∣ ∣ ∣γ2+ 1 αβ∗ γ+ 1 α ∣ ∣ ∣ (4.6) — for ∆=0 F3(α,γ) = 1 2 ln ∣ ∣ ∣ ∣ ∣ 1+ 1 αβ∗ + 1 α γ2+ 1 αβ∗ γ+ 1 α ∣ ∣ ∣ ∣ ∣ − β∗ 1+2αβ∗γ + β∗ 1+αβ∗ (4.7) where ∆= ( σs ρV 2 )2(V a1 )2 − 4σs ρV 2 = ( 1 αβ∗ )2 − 4 α (4.8) Impact test analysis of dynamic mechanical... 167 In the investigated problem, the parameter α is negative (α < 0), and in accordance with expression (4.8) the quantity ∆ is positive (∆> 0). In this context, in the following considerations only the function F1(α,γ)will beused. Relationships (4.4) and (4.5) lead to an explicit dependence of l(ξ= l/L) upon υ(γ = υ/υ0) for a rigid-plastic material with linear strain hardening, namely lnξ= 1 2 ln ∣ ∣ ∣ ∣ ∣ 1+ 1 αβ∗ + 1 α γ2+ 1 αβ∗ γ+ 1 α ∣ ∣ ∣ ∣ ∣ + (4.9) + 1√ ∆ ( β∗− 1 2αβ∗ ) ln ∣ ∣ ∣ ∣ ∣ ( 2+ 1 αβ∗ − √ ∆ 2+ 1 αβ∗ + √ ∆ )( 2γ+ 1 αβ∗ + √ ∆ 2γ+ 1 αβ∗ − √ ∆ ) ∣ ∣ ∣ ∣ ∣ or ξ= ∣ ∣ ∣ ∣ ∣ 1+ 1 αβ∗ + 1 α γ2+ 1 αβ∗ γ+ 1 α ∣ ∣ ∣ ∣ ∣ 1 2 ∣ ∣ ∣ ∣ ∣ 2+ 1 αβ∗ − √ ∆ 2+ 1 αβ∗ + √ ∆ 2γ+ 1 αβ∗ + √ ∆ 2γ+ 1 αβ∗ − √ ∆ ∣ ∣ ∣ ∣ ∣ 1 √ ∆ ( β∗− 1 2αβ∗ ) (4.10) At the end of the impact process, l= lf(ξ= ξf) and υ=0 (γ =0). Then Eq. (4.10) reduces to ξf =Φ(α)= ∣ ∣ ∣1+α+ 1 β∗ ∣ ∣ ∣ 1 2 ∣ ∣ ∣ ∣ ∣ 2+ 1 αβ∗ − √ ∆ 2+ 1 αβ∗ + √ ∆ 1 αβ∗ + √ ∆ 1 αβ∗ − √ ∆ ∣ ∣ ∣ ∣ ∣ 1√ ∆ ( β∗− 1 2αβ∗ ) (4.11) Thereby,we obtain a transcendental equationwhichuniquelydeterminates the value of the parameter α= α∗, see Fig.4. In accordance with expression (4.3)2, we have YW = ρV 2 |α∗| (4.12) A similar equation for rigid perfectly plasticmaterials, Fig.2b,was derived in te paper by Jones et al. (1987), namely ξp = βJ(1− ξf)2 1+βJ(1− ξf) − [ βJξf(1− ξf) 1+βJ(1− ξf)2 ] ln βJξf 1+βJ (4.13) and αJ = 1− ξf βJ YJ = ρV 2 αJ (4.14) The value of the parameter βJ is found from Eq. (4.13). 168 E. Włodarczyk Fig. 4. Schematic illustration of determination of the parameter α∗ Taylor derived an approximate formula (Taylor, 1948) to calculate the dynamic yield stress for the rigid perfectly plastic material in the following form YT = 1−ξf 2(1− ξf) 1 ln(1/ξf) ρV 2 (4.15) Values of the parameters YW , YJ and YT calculated by means of Eqs. (4.11), (4.13) and (4.15) are presented in the next section. Let us now consider the plastic strain εp. This parameter is determined by the formula εp(ξ)=− υ(ξ) a1+υ(ξ) =− γ(ξ) β∗+γ(ξ) (4.16) where the quantity γ is the real root of the following transcendental equation ξ= ∣ ∣ ∣ ∣ ∣ 1+ 1 α∗β∗ + 1 α∗ γ2+ 1 α∗β∗ γ+ 1 α∗ ∣ ∣ ∣ ∣ ∣ 1 2 ∣ ∣ ∣ ∣ ∣ 2+ 1 α∗β∗ − √ ∆∗ 2+ 1 α∗β∗ + √ ∆∗ 2γ+ 1 α∗β∗ + √ ∆∗ 2γ+ 1 α∗β∗ − √ ∆∗ ∣ ∣ ∣ ∣ ∣ 1 √ ∆∗ ( β∗− 1 2α∗β∗ ) (4.17) where ∆∗ = ( 1 α∗β∗ )2 − 4 α∗ ξf ¬ ξ¬ 1 0¬ γ ¬ 1 (4.18) It seems that value of the expression ϕ(γ)= ∣ ∣ ∣ ∣ ∣ 1+ 1 α∗β∗ + 1 α∗ γ2+ 1 α∗β∗ γ+ 1 α∗ ∣ ∣ ∣ ∣ ∣ 1 2 0¬ γ ¬ 1 (4.19) is contained within the interval 1¬ϕ(γ)¬ δ∗ = ∣ ∣ ∣1+α∗+ 1 β∗ ∣ ∣ ∣ 1 2 Impact test analysis of dynamic mechanical... 169 In turn, δ∗ ≈ 1with the accuracy of several per cent (see Table 1). Therefore, with an accuracy sufficient for technical purposes, one may assume that ∣ ∣ ∣ ∣ ∣ 1+ 1 α∗β∗ + 1 α∗ γ2+ 1 α∗β∗ γ+ 1 α∗ ∣ ∣ ∣ ∣ ∣ 1 2 ≈ 1 Then Eq. (4.17) reduces to γ(ξ)= 1 2 a− bdξ 1 c dξ 1 c −1 (4.20) where a= 1 α∗β∗ + √ ∆∗ b= 1 α∗β∗ − √ ∆∗ c= 1√ ∆∗ ( β∗− 1 2α∗β∗ ) d= 2+ 1 α∗β∗ + √ ∆∗ 2+ 1 α∗β∗ − √ ∆∗ (4.21) Finally, the strain εp can be found bymeans of the formula εp(ξ)= a− bdξ 1 c d(2β∗− b)ξ 1 c +a−2β∗ (4.22) ξf ¬ ξ¬ 1 εp(ξf)= 0 |εp(1)|= |εpmax|= 1 β∗+1 Thereby, we obtain analytical expressions enabling analysis of the dynamic parameters YW and εp for the rigid-plastic material with linear strain harde- ning. 5. Example In the examinations, uniform steel rods of the initial dimensions: length L = 56mm and diameter D = 8mm were used. The following mechanical parameters of steel were assumed: density ρ=7800kg/m3, modulus of strain hardening E1 = 5GPa, engineering static yield stress Re = 1255MPa. The rodswere driven by a firing gun to initial speeds contained within the interval 120-210m/s. The rods impacted perpendicularly against a flat rigid target. Pictures of deformed rods after impact are shown in Fig.5. 170 E. Włodarczyk Fig. 5. Pictures of deformed rods Fig. 6. Variations of parameters YW , YJ and YT in function of the impact initial speed V ; YT – dynamic yield stress for rigid perfectly plastic material, YJ – dynamic yield stress according to the model by Jones et al. (1987), YW – dynamic yield stress for rigid plastic material with linear strain hardening Experimental data obtained from the impact tests of the above-mentioned steel and results of calculations are given in Fig.6 and listed in Table 1. The following conclusions result from the obtained data. • Strain hardening of a material influences to a considerable degree the dynamic yield stress (see Fig.6 – YW andTable 1). The general trend of the obtained results indicate higher strengths at higher impact velocities which is in agreement with the analytical expressions. The parameter YW increases almost linearly together with the increase of the impact velocity in the range 100-200m/s. Impact test analysis of dynamic mechanical... 171 Table 1. Comparison of the experimental and analytical results from Taylor tests for chromium-nickel steel V ξL ξf ξp α ∗ β∗ δ∗ [m/s] 120 0.973 0.60 0.373 −0.075 6.667 1.037 140 0.961 0.58 0.381 −0.093 5.714 1.040 172 0.945 0.56 0.385 −0.120 4.651 1.046 209 0.922 0.53 0.392 −0.158 3.828 1.050 |εpmax| YW YJ YT YW Re YJ Re YT Re[MPa] [MPa] [MPa] 0.085 1500 2176 1824 1.195 1.733 1.453 0.094 1649 2059 1702 1.314 1.641 1.356 0.109 1920 2127 1800 1.530 1.695 1.434 0.124 2152 2141 1804 1.715 1.706 1.437 • In this range of the impact velocity, the parameters YJ and YT nearly do not change. Due to error in the Taylor theory (Jones et al., 1987) the inequality YJ >YT is fulfilled. • The approximation of dynamic properties of metals with strain harde- ning bymeans of themodel of a perfectly plasticmaterial is far-reaching simplification disagreeing with the reality. References 1. Ashby M.F., Jones D.R.H., 1993, Engineering Materials, Cambridge Uni- versity England 2. BrobergK.B., 1956,ShockWaves in Elastic and Elastic-PlasticMedia, Stoc- kholm 3. Cristescu N., 1967,Dynamic Plasticity, North-Holland, Amsterdam 4. GoldsmithW., 1960, Impact. The Theory and Physical Behavior of Colliding Solids, E. Arnold, London 5. Jones S.E., Gillis P.P., Foster J.C. Jr., 1987,On the equation ofmotion of the undeformed section of a Taylor impact specimen, J. Appl. Phys., 61, 2 6. Jones S.E., Maudlin P.J., Foster J.C. Jr., 1997, An engineering analysis of plastic wave propagation in the Taylor test, Int. J. Impact Engng., 19, 2 172 E. Włodarczyk 7. Kaliski S., Rymarz C., Sobczyk K., Wlodarczyk E., 1992, Waves, El- servier, Amsterdam-Oxford-NewYork-Tokyo 8. Kinslow R., Edit., 1970,High-Velocity Impact Phenomena, Academic Press, NewYork 9. Kolsky H., 1953, Stress Waves in Solids, Oxford 10. Lee E., Tupper S., 1951, Analysis of inelastic deformation in a steel cylinder striking a rigid target, J. of Appl. Mech., 21, 1 11. Lindholm U.S., 1968,Mechanical Behavior of Materials under Dynamic Lo- ads, Springer, NewYork 12. Meyers M.A., 1994, Dynamic Behavior of Materials, John Wiley and Sons, Inc. NewYork-Chichester-Brisbane-Toronto-Singapore 13. Nowacki W., 1974,Wave Problems in Plasticity, PWN,Warsaw 14. Perzyna P., 1966,Viscoplasticity, PWN,Warsaw [in Polish] 15. Rakhmatulin Kh.A., Demyanow Yu.A., 1961, Strength under Intense Short-Term Loadings, Gostizdat, Moskva [in Russian] 16. Shewmon P.G., Zackay V.F., Eds., 1961,Response of Metals to High Ve- locity Deformation, Interscience Publishers, NewYork-London 17. Taylor G., 1948, The use of flat-ended projectiles for determining dynamic yield stress. I. Theoretical considerations,Proc. Roy. Soc., Series A, London 18. WhiffinA.C., 1948,Theuse of flat-endedprojectiles for determiningdynamic yiel stress, 2. Tests on various metallic materials, Proc. Roy. Soc., Serias A, London 19. Wlodarczyk E., Jackowski A., 2010, On wavemethod of determining dy- namic yield stress of the elastic-plasticmaterial with linear strain hardening by means of Taylor test,Acad. Bul.,LIX, 2 20. Zukas J.A., 1962, Impact Dynamics, Wiley-Interscience NewYork Inżynierska analiza dynamicznych parametrów mechanicznych sztywno-plastycznego materiału z liniowym wzmocnieniem dla testu udarowego Streszczenie Wpracyprzedstawionoprostą inżynierskąmetodę określaniadynamicznychpara- metrówmechanicznych sztywno-plastycznychmateriałów z liniowymwzmocnieniem. Zastosowano do tego celu uderzeniowy test Taylora, tj. prostopadłe uderzenie prę- ta, wykonanego z testowanego materiału, w nieodkształcalną płytę. Wyprowadzono Impact test analysis of dynamic mechanical... 173 przestępne równania, z którychmożna określić explicite dynamiczną granicę plastycz- ności YW i odkształcenie plastyczne εp materiału pręta. Uzyskano wzrost parame- tru YW , wraz ze wzrostem prędkości uderzenia pręta, co jest zgodne z oczekiwa- niem. Stosowana w literaturze aproksymacja dynamicznych właściwości metali mo- delem idealnie-plastycznym jest daleko idącym uproszczeniem, niezgodnym z rzeczy- wistością. Manuscript received February 18, 2010; accepted for print July 12, 2010