Limit analysis-based approach to determine the material plastic properties with conical indentation
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Norimasa Chiba Department of Mechanical Engineering, National Defense Academy, Hashirimizu, Yokosuka 239-8686, Japan
Xi Chena) Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, New York 10027-6699 (Received 16 September 2005; accepted 9 January 2006)
Representative strain plays an important role in indentation analysis; by using the representative strain and stress, the normalized indentation load becomes a function of one variable, which facilitates the reverse analysis of obtaining the material plastic properties. The accuracy of such function is critical to indentation analysis. Traditionally, polynomial functions are used to fit the function, which does not incorporate correct elastic/plastic limits and has no physical basis. In this paper, we have proposed a new limit analysis-based functional formulation based on the theoretical solutions of conical/wedge indentation on elastic and rigid plastic solids. It is found that both limits agree well with numerical results, and the new approach involves no—or at most one—fitting parameter, which can be obtained with much less effort compare with the traditional polynomial approach. Reverse analyses on five different materials have shown that the new and simple limit analysis-based formulation works better than the traditional polynomial fit. The new technique may be used to quickly and effectively measure material plastic properties for any conical indenter if the elastic modulus is known a priori.
I. INTRODUCTION
Instrumented indentation is widely used to probe the elastic and plastic properties of engineering materials. During the experiment, a sharp rigid indenter (with a half apex angle ␣) is penetrating normally into a homogeneous solid, where the indentation load P and displacement ␦ are continuously recorded during loading and unloading (Fig. 1). With the specimen Young’s modulus denoted by E and yield stress by y, without losing generality, the uniaxial stress–strain ( – ) curve of a stressfree solid can be expressed in a power-law form: = E⑀ for
⑀ 艋 y Ⲑ E ,
= R⑀n
⑀ 艌 y Ⲑ E
and for
,
(1)
a)
Address all correspondence to this author. e-mail: [email protected] DOI: 10.1557/JMR.2006.0108 J. Mater. Res., Vol. 21, No. 4, Apr 2006
where n is the work-hardening exponent and R ≡ y(E/ y)n is the work-hardening rate. When n is zero, Eq. (1) reduces to elastic-perfectly plastic material. For most metals and alloys n is between 0.1 and 0.5. From dimensional analysis1,2 P Ⲑ 共E␦2兲 = ⌿共y Ⲑ E,,n,␣兲 . (2) To simplify the analysis, the indenter is usually modeled as a rigid cone with ␣ ⳱ 70.3°, so that the ratio of cross-sectional area to depth is the same as for a Berkovich or Vickers indenter.3 In addition, the Poisson’s ratio is not an important factor in the indentation experiment,1,2,4 and for most engineering material ≈ 0.3. The dimensionless function ⌿ in Eq. (2), which now primarily depends on variable set (y/E,n), can be calculated by using extensive finite element analysis. To obtain the c
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