STRAIN-HARDENING EXPONENT OF HIGH STRENGTH STEEL SHEETS UNDER TENSILE DEFORMATION. Ziegelheimova, J., Janovec, J., Cejp, J.

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1 STRAIN-HARDENING EXPONENT OF HIGH STRENGTH STEEL SHEETS UNDER TENSILE DEFORMATION Ziegelheimova, J., Janovec, J., Cejp, J. Czech Technical University in Prague Faculty of Mechanical Engineering Department of Materials Engineering Karlovo namesti 13, Praha 2 ziegelhe@ .cz Abstract It has been experimentally proved that strain hardening exponent (n) has very strong sensitivity to structure. In deep-drawing materials is possible to state strain-hardening exponent from 10 to 20 % of deformation. Using high strength steel sheets the deformation does not reach such values, so it is essential to establish the strain-hardening exponent using different method. Strain-hardening exponent is related to strain as well as strain-rate. Considering the structure sensitivity in plasticity this paper deduces the analytical and experimental expressions for n values of high strength MSW, Docol, BTR, Boron, CP steel sheets used in automotive industry. 1 INTRODUCTION The concept of strain-hardening exponent commonly uses Hollomon s exponential equation of n from experience in metal tensile deformation. [1] There are some other formulae for measuring n, but they fail to express clearly the mechanical essence of n [2]. The advance in plastic forming makes it possible for further research the mechanical meaning and exact measuring method of n. [2] The standardized measuring formulae of n are given as follows: The strain-hardening exponent at constant strain-rate[3]: n Equation 1 = ( / log) The strain-hardening exponent at constant velocity[3]: n ( / log) v = Equation 2 v Using the above formulae and exact measuring methods, a set of experimental curves can be obtained for each material. The purpose of this paper is to establish a functional relationship of n v and n vs. the change of from the basic theory and further reveal the mechanical essence of n. 1.1 Mechanical analysis of n value Constrain equation of n and m To express mathematically the mechanical parameters of deformation, suppose that: I. The volume of sample is uncompressed. Plastic deformation is homogeneous, and the stress σ distributes uniformly on cross-section A.

2 II. True stress σ, true strain and true strain-rate are mechanical state parameters of tensile deformation. III. There is no effect of temperature, i.e. all tests are carried out by the same room temperature Then the state equations of tensile deformation can be expressed as: σ= σ(, ) Equation 3 Differentiate the eequation 3 and divide it by σ on both sides, then [4]: d log σ log log = Equation 4 Equation 4 can be transformed into[4]: log log = Equation 5 = log d d log log log Equation 6 In eq. 5 and 6, and are defined as general strain-hardening exponent n and general strain-rate sensitivity exponent m respectively, and are defined log log as strain-hardening exponent n at constant strain-rate and strain-rate sensitivity exponent m under constant strain respectively, and and are determined by forming paths. [4] Using the above-mentioned equations we can re-write: n n m = Equation 7 m = n m Equation 8 Finally we get [4]:. m n. m m. n = Equation 9 n Thus equation 9 is just the constrain equation of n and m on arbitrary forming-path Analysis expression of n value along typical forming path According to precondition (I) the volume of sample is uncompressed. Plastic deformation is homogeneous, and the stress σ distributes uniformly on cross-section A. Then we have d log A = d logl Equation 10 The relation among stress σ, load F and sample length is: d log = d log F d log l σ Equation 11

3 l 0 With the definition of true strain = ln (where l0 is the initial length of sample), we get: l d logl d log = Equation 12 With the relation of v at strain-rate (=v/l), we have[5]: = d logv d logl Equation 13 Substituting this into the general definition formulae of strain-hardening exponent [5]: d log σ d log F d logl = d logl Equation 14 Load F, sample gauge length l and crosshead speed v are experimental parameters, and stress σ, strain and strain-rate are mechanical parameters in tension. Constant strain means fixed length in substance. The n value is insignificant when strain is fixed. In our case one forming path is possible: v= const. v=const: n = n Equation 15 m. 2 Experimental part 2.1 Materials being tested Basically 3 kinds of multiphase steel sheets were explored. Their chemical composition can be seen from table 1. Their basic description is given below. BTR, BORON, DOCOL and MSW represent ultra high strength steels with martensitic (MSW) or dual phase structure particularly suitable for cars crumple zones, because of these steels high energy absorbing properties CPW Complex Phase steel consist of a very fine microstructure of ferrite and a higher volume fraction of hard phases. Table 1: Chemical composition of investigated materials Element Material C Si Mn P S Nb Cr Mo Ti B Al BTR Max Max Max Max MS-W Max Max BORON Max0.025 Max0.010 DOCOL Max0.015 Max CP W Max Max Max

4 2.2 Tensile tests Shape and size of specimens was made with respect to the standard ČSN EN Using tensile-test machines (Instron in CTU, Faculty of Mechanical Engineering, Department of Materials Engineering) it was possible to change crosshead speed from 5 to 500 mm/min. Suppose, that crosshead speed=const., the calculated strain-rates are shown in table 2. Table 2: Strain-rates during tensile testing Cross- Head For each material two logσ log curves were evaluated Speed of [s -1 1 = ] s -1 and 2 = s -1 see example for [mm/min] material Docol in Figure The dependence of log σ-log, material Docol =0,0012s -1 =0,0595s -1 3,3 3,28 3,26 3,24 3,22 3,2 3,18 3,16 3,14 3,12 3,1-1,8-1,6-1,4-1,2-1 Figure 1: Example of logσ-log curves, Docol material Then simulating the curves by a computer, we get the following polynomials: Table 3: The polynomials of log σ log curves Material [s -1 ] Polynomials BTR log σ = (log) (log) (log) log σ = (log) (log) (log) MS-W log σ = (log) (log) (log) log σ = (log) (log) (log) BORON log σ = (log) (log) (log) log σ = (log) (log) (log) DOCOL log σ = (log) (log) (log) log σ = (log) (log) (log) CP W log σ = (log) (log) (log)

5 log σ = (log) (log) (log) log is identical to log(0.0595)-log(0.0012), and from above mentioned polynomials logσ correspondig to every are obtained. Substituting log and logσ into: = log logσ log m Equation 16 We can get m for any. Curves m are interpreted see figure 2 example for material Docol. m 0,06 0,05 0,04 0,03 0,02 0,01 0 m Docol steel sheet 0,08 0,09 0,1 0,11 0,12 0,13 0,14 0,15 0,16 Figure 2: m curve of Docol material Table 4: Equation for m calculation Material Equation of m BTR m = 8, ,1281 0,0281 MSW m = 25,66 2-2,3098 0,0312 BORON m = 15, ,7327 0,127 DOCOL m = 9, ,83 0,1093 CPW m = -1, ,4305-0,0166 With above mentioned polynomials and corresponding to m are substituted into equation 15. Then we have the theoretical curves n-. Typical n- curves can be seen from figure 3. 0,15 0,1 0,05 n [-] 0 =0,0012 s-1 n- curves of Docol material =0,0595s -1-0,05-0,1-0,15 0,08 0,12 [ ] 0,16 Figure 3: Typical n- curves for Docol material

6 3 Disscussion in the theoretical equation 15 is the sublimate even strain, whereas what is gained by measuring is not homogeneous, even within the gauge length. There exist deviation between the theoretical curves and the experimental curves n-, even if the strain is small and n is affected little by unevenness. m is generally very small and the product m. is also small. 4 Conclusions The basic change law of n value is revealed by the change of n vs.. n decreases with the increasing. When necking occurs, n tends to zero and can even decrease to negative value before necking occurs. The change of n vs. and m meets equation 15 n=n -m. The values of strain-hardening exponent and strain-rate hardening exponent in dependence on strain for high strength steel sheet materials were succesfully established Even though that the values of m are rather small, its affection to n values (and consequently to the deformation behaviour of materials) is in the case of HSS materials strong. Using menthod described in this paper we can evaluate the values of strain-hardening exponent n and strain-rate hardening exponent m for any strain. 5 References [1] Pan, Chi-Ling and Yu, Wei Wen, Design of Automotive Structural Components Using High Strength Sheet Steels: Influence of Strain Rate on the Mechanical Properties of Sheet Steels and Structural Performance of Cold-Formed Steel Members, Eighteenth Progress Report, Civil Engineering Study 92-3, University of Missouri-Rolla, December, [2] Ghosh, A. K.,: The Influence of Strain Hardening and Strain Rate Sensitivity on Sheet Metal Forming, Research Laboratories, GM Corp., Warren, Mich., Transaction of the ASME, July 1977, pp [3] Marciniak, Z., Kurzynski, K., Limit Strain in Process of Stretch Forming Sheet Metal, Int. J. Mech. Sci.,15, 1973, p.789 [4] Song, Y., Hai, J., Mechanical Analysis of Strain-hardening Exponent under Tensile Deformation, Science in China, December 2001, Vol.44, No.6, p [5] N. W. Dowling, "Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue", 2nd Edition, Prentice Hall, 1998