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On the Determination of Residual Stress and Mechanical Properties by Indentation Xi Chen Columbia University Cleveland State University EngagedScholarship@CSU Mechanical Engineering Faculty Publications Mechanical Engineering Department 1-25-2006 On the Determination of Residual Stress and Mechanical Properties by Indentation Xi Chen Columbia University Jin Yan Yan University of Delaware Anette M. Karlsson Cleveland State University, [email protected] Follow this and additional works at: https://engagedscholarship.csuohio.edu/enme_facpub Part of the Mechanical Engineering Commons How does access to this work benefit oy u? Let us know! Publisher's Statement NOTICE: this is the author’s version of a work that was accepted for publication in Materials Science and Engineering A. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Materials Science and Engineering A, 416, 1-2, (01-25-2006); 10.1016/j.msea.2005.10.034 Original Citation Chen, X., Yan, J., and Karlsson, A. M., 2006, "On the Determination of Residual Stress and Mechanical Properties by Indentation," Materials Science & Engineering A, 416(1-2) pp. 139-149. This Article is brought to you for free and open access by the Mechanical Engineering Department at EngagedScholarship@CSU. It has been accepted for inclusion in Mechanical Engineering Faculty Publications by an authorized administrator of EngagedScholarship@CSU. For more information, please contact [email protected]. On the determination of residual stress and mechanical properties by indentation Xi Chen a,*, Jin Van b, Anette M. Karlsson b • D"/Hlr/lllenl ofCMI Engitweri,rg lillli Engi/leering Meciumil:s. en/Wl1bill UtJi\'ersil)~ New York. NY JOO27·6699. USA b Depllrlmelll ofMt'ciJalliw/ Ellgineering. Ulli>'ersily ofIJd(IWW7. Ne""o rk. DE 19716·)140. USA I. lntroduction the specimen must be detennined, frequently by measurements, since associated material properties are many times unknown. The mechanical reliability ofbulk materials and coatings (e.g. There are many challenges associated with measuring prop- fatigue, fracture, corrosion and wear) is strongly affected by their erties of small scale structures, but instrumented indentation is residual stress, commonly introduced by thermal mismatch, or widely used to probe mechanical properties such as Young's mechanical and thenna! processing. The most critical residual modulus, E, and yield strength, ay, for most engineering mate- stress component in coatings is the in-plane cqui-biaxial residual rials. Therefore, we will extend the usefulness of thi s method to stress Iypically caused by laHice spacing mismatch and thermal measure the residual stress, arcs. expansion mismatch between the film and substrate. Moreover, the mechanical properties and residual stress of all materials 1.1. A brief review of fhe illdentmioll Techlliqu e are temperature dependent. For thin comings and substrates, the dependency becomes exacerbated with a change in tempera- A variety of techniques have been developed for measuring ture. This difference in behavior due to temperature change can mechanical properties from indentation load-displacement data be critical in service, since mismatch between coating and sub- of an elastic-perfectly plastic, slresslree bulk material lS- 8], strate leads to high residual stress and ultimately coating failure which will be review briefly in this subsection. Instrumented [11. For example, coatings used for thennal protection in gas tur- indentation is characterized by a sharp rigid indenter (with a bine engines, experience local compressive stresses as high as half apex angle a) penetrating normally into a homogeneous several GPa at ambient [2-4J. Thus, in order to achieve a reliable solid where the indentation load, P, and displacement, 8, are engineering design, the actual level and sign ofresidual stress in continuously recorded during one complete cyele ofloading and unloading (Fig. l a and b). To simplify the analysis, the inden- • Corresponding aUlhor. Te l. : +1 212854 3787; fa~ : +1 2128546267. ter is usually modeled as a ri gid cone with 0' = 70. 3°, so that E-mail add",ss: xiehen@;eil·il.columbia.cdu(X. Chen). the ratio of cross-sectional area 10 depth is the same as for a Fig. 1. Schematic of instrumented indentation with a sharp indentation: (a) indentation on a homogeneous, isotropic semi-infinite substrate; (b) typical load- displacement curves obtained from an indentation experiment; (c) conical indentation on a specimen with equi-biaxial in-plane residual stress. Berkovich or Vickers indenter [9]. By neglecting friction and As the indenter penetrates the specimen, the materials either the finite compliance of the measuring system and the indenter produce plastic pile-up at the crater rim (when the yield strain, tip, the equations used to extract the hardness H and indentation σy/E, is small), or exhibit the elastic sink-in effect (when σy/E modulus M are is large) [11]. The amount of pile-up/sink-in is denoted as δp (Fig. 1a). For conical indenters, the projected contact area A is H = P/A = c σ , b y (1) given by and = 2 = 2 2= 2 A πa π(tan α) δc 24.5δc , (3) 2 √ S = γβ √ M A. (2) where the contact depth: π = + Here, the hardness H is defined as the ratio between indenta- δc δ δp. (4) P A tion load , projected contact area and yield stress σ. The Eq. (4) contains contributions of both plastic pile-up around the M indentation modulus is given by the plane-strain modulus, indenter and elastic sink-in, which is counted negative. It is obvi- ¯ ≡ − 2 E E/1 ν , for isotropic materials and by a more compli- ous from Eqs. (3) and (4), that the projected contact area depends cated weighted average of the elastic constants for anisotropic on δp. Oliver and Pharr [5] proposed an elastic model for deter- E v materials [10], where and are the Young’s modulus and mining the contact area in which plastic pile-up is neglected: Poisson’s ratio of the bulk material, respectively. The contact � � stiffness S =dP/dδ is obtained from the slope of initial por- � � = εPmax δp (5) tion of the elastic unloading curve (Fig. 1b). The constant cb Smax in Eq. (1) is a constraint factor that depends on indenter shape where ε = 0.75 for a conical indenter. Alternatively, the pile-up and material properties: c increases with E/σ and approaches a b and contact area can be measured experimentally or determined constant (≈3) when E¯ tan α/σ > 20 [9,11]. β is a shape fac- y from the numerical analysis (such as the finite element method tor, with β = 1 for axisymmetric indenters and β = 1.03–1.05 or molecular dynamics). Once A is determined properly, the for indenters with square or rectangular cross-sections [12]. π/4+0.155 cot α(1−2ν/4(1−ν)) hardness is then obtained from Eq. (1) and the stiffness can be γ = π is a correction factor for the (π/2−0.831 cot α(1−2ν/4(1−ν)))2 derived from Eq. (2), which allows one to measure E¯ and σy by conical indenter [13]. Both hardness and stiffness are indepen- using the indentation technique. dent of the indentation depth if the strain gradient effect is 1 ignored [14]. 1.2. The effect of in-plane residual stress Several attempts have been made to incorporate residual 1 For metals it is observed that the hardness increases with decreasing inden- tation depth, when the penetration is in the sub-micron regime. This is known as stress measurements from instrumented indentation. Tsui et al. strain gradient plasticity. We ignore such effect by assuming that the indentation 1996 [15] used standard nanoindentation techniques to inves- depth is sufficiently deep. tigate the influence of uniaxial and biaxial in-plane stresses (applied to the specimen by bending) on the hardness and elas- 2. Model definition tic modulus measurements of aluminum. The projected contact areas were measured by optical techniques. It was found that the 2.1. Dimensional analysis hardness and elastic modulus thus calculated were essentially independent of the residual stress level, which was verified by The axisymmetric model for indentation on a bulk material a parallel finite element simulation [16]. Based on the assump- or thick coating with in-plane residual stress is shown in Fig. 1a tion that the hardness is unaffected by tensile or compressive and c. The substrate is taken to be infinitely deep. The half ◦ elastic residual stress, several general methods was proposed apex angle of the rigid conical indenter is α = 70.3 . The coat- for determining the in-plane equi-biaxial residual stress by con- ing is elastic-perfectly plastic, applicable to most high-strength ical or spherical indentation. Most approaches compare the alloys and ceramic coatings (the effect of work hardening will contact depth (or contact area, or load-displacement curve) of be explored elsewhere). Recent work by Mesarovic and Fleck the stressed and unstressed specimens, from which σres can be [23] has shown that the Poisson’s ratio is a minor factor for estimated [17–20]. In some other studies [21,22], the residual static indentation. Dimensional analysis dictates the following compression was seen to enhance the pile-up and residual ten- relationship for hardness and stiffness: δ [] sion reduced p. The ratio between the projected and nominal P σ σ contact areas, i.e. (δ /δ)2, was calculated as a function of σ by = f res ,,y c res 2 (6) πa σy σy E using finite element analysis. Therefore, the authors suggests, [] the in-plane residual stress can be obtained if δ can be mea- p S = σres σy sured properly for a given material [21,22].
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