Conical indentation of strain-hardening solids

European Journal of Mechanics A/Solids 27 (2008) 210–221

Conical indentation of strain-hardening solids David Durban , Rami Masri ∗,1 Faculty of Aerospace Engineering, Technion, Haifa 32000, Israel Received 20 February 2007; accepted 25 May 2007 Available online 5 June 2007

Abstract Conical indentation of strain hardening solids is examined within the spherical cavity expansion simulation pattern in finite strain plasticity. Analysis accounts for elastic compressibility and arbitrary strain hardening. Unlike existing studies of indentation processes that assume a definite yield point, the present formulation applies also to smooth elastoplastic transition. Approximate hardness formulae are derived, at different levels of accuracy, and compared with available finite element calculations. Effects of pile-up, or sink-in, and external friction have been ignored. It is suggested that test data over a range of cone angles can be used to reconstruct the axial stress–strain curve. The relation between cavitation and conical indentation is discussed. It is shown that the cylindrical Tresca cavitation yield stress serves as a natural scaling stress in estimating hardness of strain hardening solids. © 2007 Elsevier Masson SAS. All rights reserved. Keywords: Cavity expansion; Hardness; Cavitation

1. Introduction A central result in conical indentation (Fig. 1) has been derived by Johnson (1985) within the framework of smallstrain elastic/perfectly-plastic response. That analysis is based on the spherical cavity expansion model (Bishop et al., 1945) and leads to the hardness relation  E  cot α + 4β 2 H = 2 + ln Y , Y 3 3(1 + β)

(1)

where H is the hardness, Y – yield stress, E – elastic modulus, β = 1 − 2ν with ν denoting the Poisson ratio, α – half included angle. Relation (1) is expected to be valid in the elastoplastic transition zone before full plasticity sets in at about H = 2.8Y . In recent years, there have been a few attempts to generalize (1) into the strain hardening range. Jayaraman et al. (1998) investigated the hardness of hard materials with relatively high levels of Y/E ratio. Based on FE calculations (α = 70.3◦ , ν = 0.3) for power hardening solids the authors suggest a best fit formula, in elastoplastic transition zone, * Corresponding author. Fax: +972 4 829 2030.

E-mail address: [email protected] (R. Masri). 1 This work is based on a part of a PhD thesis to be submitted to the Technion.

0997-7538/$ – see front matter © 2007 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2007.05.007

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Fig. 1. Conical indentation. Included angle is 2α, imprint radius – a, hypothetical radial displacement – w. Inner core is bounded by r = a.

  σ0.07 0.84 H = 1.34 , Y E

(2)

where σ0.07 is the stress value at a strain of 0.07. A detailed FE study on conical indentation, over a range of material properties, has been made by Casals and Alcalá (2005), enhancing earlier work by Mata et al. (2002), Mata and Alcalá (2003) and Mata et al. (2006). For the standard indenter (α = 70.3◦ ) and power law response the authors present a best fit relation within the elastic-plastic zone (ν = 0.3)   E H , (3) = 1.440 + (0.264) ln σ0.1 σ0.1 where σ0.1 is the stress value at strain level of 0.1. A further generalization of (3), in powers of ln(E/σ0.1 ), can cover the fully plastic regime as well (Mata and Alcalá, 2003). Relations (2)–(3) make use of predefined representative strains (0.07 in (2) and 0.1 in (3)). It is worth mentioning that Tabor (1951) has suggested to calculate deep plastic hardness of a strain-hardening material from the fully plastic relation H = 2.8Y , but with the yield stress replaced with the flow stress at a representative strain of  = 0.2 cot α which is about  = 0.0716 for α = 70.3◦ . Gao et al. (2006) have derived analytical results for the hardness of elastoplastic solids with particular hardening characteristics (linear-hardening and power-hardening). The analysis is based on a variant of Johnson’s model and neglects elastic compressibility. For power law hardening they obtain the hardness relation     2 3 1 cot α n 1 H = + (4) − +1 , Y 3 4 n 3Y/E n where n is the hardening index. No analytically derived formula appears to be available in the literature for estimating hardness of compressible elastoplastic solids with arbitrary strain hardening. In this study we attempt to generalize existing work by combining Johnson’s (1985) cavity expansion pattern with accurate continuum plasticity solutions of pressurized cavities. This procedure generates new relations for H with strain-hardening response, valid for the entire family of J2 plasticity. Governing equations are introduced in next section where we show that hardness estimation centers on two nonlinear equations. The spherical cavity expansion pattern beneath the conical indenter is formulated as an eigen-cell phenomena with averaged field quantities reflecting plasticity level and hardness. Then, in Section 3, we derive a few useful small strain approximations and compare with relations available in the literature. A new incremental energy connection is used to analyze the reverse problem of reconstructing the uniaxial stress strain curve from indentation test data.

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A detailed comparison with finite element calculations, given in Section 4, reveals a nice agreement with our results, over a wide range of hardening parameters. Finally, in Section 5, we examine the relation between hardness and cavitation pressure. It is suggested that the cavitation yield stress, of a pressurized cylindrical cavity embedded in a Tresca plastic medium, serves as an appropriate scaling stress for hardness estimation. 2. Eigen-cell simulation With the notation of Fig. 1 we examine a generic state of the elastoplastic field induced by a conical indenter of included angle 2α. Denoting the radius of the imprint by a, we follow Johnson’s (1985) analysis and assume that the hemispherical surface with deformed radius r = a has been displaced uniformly from a (hypothetical) undeformed reference configuration by amount w. The self-similar eigen-cell induced by the indenter is represented by an internally pressurized spherical cavity at r = a, while beneath the indenter, for r < a, only average stress components are considered. Neglecting elastic compressibility inside the inner core (r < a) we find from volume conservation that  1/3 w 1 1 − = 1 − cot α , (5) a 2 which is limited to large cone angles above 90◦ . In the external zone (r > a) we assume large-strain spherical symmetric conditions of a pressurized spherical cavity. That problem has been solved by Durban and Baruch (1976, 1977) within the framework of finite strain J2 theory. For internal wall pressure p, wall displacement w and deformed radius a (note that in Durban and Baruch (1976, 1977) a denotes the initial, undeformed, radius of the cavity), we have the relation    w 1 1 1 − = exp − a + β P − Σa , (6) a 2 2 where P = p/E, Σa is the effective Mises (Tresca) stress at the cavity (nondimensionalized with respect to E), and a is the total strain at the cavity and a known function of Σa . This solution is valid for arbitrary hardening characteristics with the internal pressure related to cavity wall plastification by the integral Σa P=

(d/dΣ + β)Σ dΣ 3

0

β

e 2 − 2 Σ − 1 + 2βΣ

,

(7)

where the independent variable Σ is the Mises (Tresca) effective stress divided by E and the total strain  depends on Σ. In integrating (7) one can shift to  as the independent variable and, in fact, account for possible strain softening. Identifying P with the interfacial radial stress at r = a in the eigen-cell simulation of the indentation test (Fig. 1), we equate (5) with (6) to obtain   1 2 a + βΣa = − ln 1 − cot α + 2βP . (8) 3 2 The pair (7)–(8) can be solved, for given strain hardening and elastic compressibility, to determine the dependence of P and Σa on cone angle. As in Johnson’s (1985) original cavity model we seek now a relation between hardness (average pressure applied by indenter) and interfacial pressure P . To this end we assume that inside the inner core (r < a) the average effective Mises (Tresca) stress retains it’s interface value Σa while the average hydrostatic stress Σ¯ h remains constant and is equal to interface radial pressure (−P ). Thus, with average spherical stress components inside the core denoted by (Σ¯ r , Σ¯ θ = Σ¯ φ ) we have Σ¯ θ − Σ¯ r = Σa ,

(9)

and 1 Σ¯ h = (Σ¯ r + 2Σ¯ θ ) = −P . 3

(10)

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Identifying the nondimensional hardness H /E with (−Σ¯ r ) we find from (9)–(10) that 2 H = P + Σa . (11) E 3 An identical expression can be obtained by Johnson’s suggestion (Johnson, 1985) to examine the inner core stress state in cylindrical coordinates. The Mises yield criterion in cylindrical coordinates, with average effective stress Σa and average hydrostatic stress (−P ), can be written as √ 3 Σa = (Σ¯ θ − Σ¯ r )2 + 3(Σ¯ z + P )2 . (12) 2 Here (Σ¯ r , Σ¯ θ , Σ¯ z ) are the average cylindrical (principal) stress components inside the core. Identifying nondimensional hardness H /E with (−Σ¯ z ) and neglecting in-plane shear we arrive again at relation (11). Representing the hydrostatic core field by average stress components is useful in bypassing deviations from spherical symmetry beneath the indenter. The present exact finite strains formulation, (7)–(8) and (11), accounts for elastic compressibility and for arbitrary strain hardening, including absence of hardening as a special case. For elastic/perfectly-plastic response Σa = Y/E and relation (11) agrees with Johnson’s (1985) modification of the connection between hardness and interfacial pressure. The present formulation should be applicable even when the ratio Y/E is not negligible, say in range of 0.05–0.1. The elastic and plastic parts of (7) are then evaluated separately, resulting in  Σy  3 exp( 3−β 2Σy (1 + β)Σ dΣ 2 Σy ) − (1 − 2βΣy ) exp[− 2 (a − Σy )] + ln P= , 3−β 3−β 3(1 − 2βΣy ) exp( Σy ) − 1 + 2βΣy exp( Σ) − 1 + 2βΣ 2

0

(13)

2

when Σy = Y/E is the nondimensional yield stress, in the elastic range  = Σ , and in the plastic range Σ ≡ Σy with  serving as the integration variable. Now, a straightforward expansion of (13) in powers of Σy , for incompressible solids (β = 0), leads to the asymptotic approximation, via (8),     3 H 2 cot α + Σy , (14) = 2 + ln Y 3 3Σy 8 which practically (Σy  1) coincides with (1) at the incompressibility limit, but differs from (4), at the limit of n = 0, by an extra term. Actually, for an incompressible solid a simple exact solution of (7)–(8) is available for any strain hardening (softening) behavior, including perfectly-plastic response as a special case, H = E

a 0

2 + Σa 3  e2 − 1 3 Σ d

  1 2 with a = − ln 1 − cot α . 3 2

(15)

It is clear that hardness increases with decreasing of included angle due to an increase in cavity strain level. The present model simulates the indentation process by assuming a self-similar eigen-cell in the spirit of Johnson’s (1985) original work for elastic/perfectly-plastic materials. That constraint is here relaxed by considering strain hardening (softening). Finite strains are allowed and no definite yield point needs to be defined. Hardness is identified with average radial stress (spherical coordinates) or average axial stress (cylindrical coordinates) within the inner (incompressible) core, yet elastic compressibility is maintained in the external cavity field. This formulation leads to a natural connection between hardness and cavitation pressure, as suggested originally by Bishop et al. (1945) and recently discussed by Mata et al. (2006). While effects of pile-up, or sink-in, and external friction have been ignored, this study still provides useful relations for estimating the hardness of compressible elastoplastic solids with arbitrary strain hardening response. 3. Small strain approximations For large cone angles ( 12 cot α  1) we may assume that strains are sufficiently small (  1) and replace (7)–(8) by the approximations

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2 P= 3

Σa 0

d ( dΣ + β)Σ dΣ ,  + βΣ

a + βΣa =

(16)

1 cot α + 2βP . 3

(17)

With some specific hardening characteristics it is possible to derive analytical solutions of (16)–(17). The simplest case – though not within the scope of this study – is the linear elastic solid where  = Σ . We find that 2 cot α P = Σa = , 3 3(1 + ν)

(18)

implying the hardness 2 cot α H = , E 3(1 + ν)

(19)

which differs from the classical Sneddon solution by a mere factor of 4(1 − ν)/3; Sneddon’s result is recovered exactly when ν = 1/4. A common hardening function is described by the elastic/linear-hardening model =Σ

for Σ  Σy

and  = Σy + η(Σ − Σy )

for Σ > Σy ,

(20)

where η = E/ET , with ET denoting the constant tangent modulus. The solution of (16)–(17) is now obtained from the transcendental equations    (η + β)Σa − (η − 1)Σy (η − 1)Σy 2 , (21) Σa + ln P= 3 η+β (1 + β)Σy (η + β)Σa − (η − 1)Σy =

1 cot α + 2βP , 3

(22)

with the associated hardness evaluated from (11). A simplified relation can be extracted by assuming incompressibility (β = 0), namely       H 2 cot α 2 η−1 cot α + . (23) = 2 + ln Y 3 η 3Σy η 3Σy This relation is similar to the result obtained by Gao et al. (2006) but predicts higher values with the factor 2 (inside the brackets) instead of 7/4 in their work, and for elastic/perfectly-plastic response (η → ∞) it coincides with (1) at the incompressibility limit. Expanding (16) in powers of the compressibility parameter β we find the leading terms 2 P= 3

Σa



Σ 1 1−β −  d/dΣ



 d Σ dΣ + · · · dΣ , 

(24)

0

with an analogous approximation for Σa and a . As an example, by applying (24) to the power law solid,  1/n Σ  = Σ for Σ  Σy and  = Σy for Σ > Σy , Σy where n is the hardening index, we obtain       Σy 1/n−2 2 1−n 1 P= . Σy + (Σa − Σy ) − βΣy 1− 3 n 1 − 2n Σa

(25)

(26)

Combining (26) with a similar approximation of (17) we arrive at the first order approximation for hardness of compressible power-law materials

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Fig. 2. Comparison of hardness predictions for power law solids. α = 70.3◦ , ν = 0.3. 1 – best fit FE data (3), 2 – best fit FE data (2), 3 – present model approximation (27), 4 – formula (1) with representative strain 0.0716, 5 – incompressible approximation (4). (a) n = 0.1. (b) n = 0.4.

   1 cot α n 1 +1 − +1 n 3Σy n       2 1−n 4(1 − n2 ) cot α n−1 4 − 4n − 8n2 + 6n3 cot α 2n−1 . − β + − 3 1 − 2n 3n 3Σy 3n(1 − 2n) 3Σy

2 H = Y 3



(27)

At the limit of elastic incompressibility (β = 0) we arrive with a higher hardness estimation than (4). At the limit of vanishing hardening (n = 0) we have          H cot α cot α 4 3Σy 2 −β 1− +1 . (28) = 2 + ln ln Y 3 3Σy 3 cot α 3Σy The different predictions of hardness for power law solids (25) are compared in Fig. 2 over a range of Y/E values and for two different hardening exponents. It is apparent from the curves in Fig. 2 that while all five hardness relations are in close agreement at low levels of Y/E, considerable differences emerge for hard solids. For n = 0.4 the present analysis (27) remains close to the finite element relation (2), and (4) gives nearly the same results as (1) when Y is replaced, according to Tabor’s recipe, by the flow stress at representative strain of 0.0716. It is noted that numerical data behind the best fit curves, (2)–(3), is available for about Y/E < 0.04, hence curves 1–2 in Fig. 2 are extrapolated for higher values of Y/E, according to (3)–(2) respectively. While all curves in Fig. 2 are for a specific hardening characteristic, it is noted that the present analysis is not restricted to any particular stress–strain relation. A rather simple result is obtained from (16)–(17) for incompressible materials (β = 0) with arbitrary hardening where a Σ d 1 2 and a = cot α. (29) P= 3  3 0

It follows that hardness is directly connected to plastic moduli by 

a  a H ES + ET 2 Σ 2 = d + Σa = d, E 3  3 E 0

(30)

0

where (ES , ET ) are the secant and tangent moduli, respectively. It is a matter of ease to recover from (30) the hardness of incompressible linear-hardening and power-hardening solids. Two particular cases of (29) were integrated analytically in Gao et al. (2006), but as a general result, Eqs. (29)–(30) appear here for the first time. The hardness of strain-softening materials can be obtained from (30) with any plastic response characteristic. Consider for example the relation  n  e−k(−y ) for  > y = Σy , (31) Σ = y y

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while linear elastic response is maintained for   y . Softening is exhibited for strains larger than  = n/k and material hardness is given by

 n−1  n  a   H 2 −k(−y ) (1 + n) e − ky d , (32) = 2y + E 3 y y y

where a is given in (29). The integral in (32) can be evaluated numerically for given material parameters (y , n, k). We can cast (30) in the form a H 2 d(Σ) 1 = with a = cot α, (33) E 3  3 0

where Σ is a known function of  (or vice versa). Just to give an example, with the hardening law Σ =  + k n , we find      2 2 cot α 1 cot α n H . = +k +1 E 3 3 n 3

(34)

(35)

Finally, we note that the reverse problem, of deducing the stress–strain curve from hardness test data, can be investigated with the theory provided here. For brevity, we shall limit the discussion to the simplified model (33) which implies, at r = a, the energy differential relation   H 3 . (36) d(Σ) = d 2 E A series of tests should provide the curve H /E as a function of (≡ a ), or after integrating    H 3 Σ = d + A, 2 E

(37)

0

where 0 is the lowest indentation test strain. The integration constant A is fitted through the compatibility requirement at Σ =  = Σy or, if elastic phase is included, at Σ =  = 0. This procedure is illustrated with the conjectured test result   4a H = h tanh , (38) E 3h where h is the test hardness data asymptote. Inserting (38) in (37) and integrating over the entire elastic-plastic path results in the stress–strain relation           4 3h 4 8 2 3 − ln cosh ≈ 1− 2 for  2  h2 . Σ = h tanh (39) 2 3h 4 3h 9h The linear elastic branch is recovered at once from (39) at low strains, while plastic strain hardening is apparent as  increases. Implementing the proposed procedure of recovering the stress–strain curve from hardness measurement depends on availability of data over a sufficiently wide range of cone angles to enable integration of (37). 4. Comparison with numerical results Both accuracy and validity of present study can be assessed by comparison with available numerical data. Detailed finite elements calculations have been performed recently by Casals and Alcalá (2005) and by Jayaraman et al. (1998) for cones with α = 70.3◦ . Calculations were made for the plastic power law (25) over a range of material parameters, and Poisson ratio of 0.3. Fig. 3 compares hardness values obtained from (11), through the solution of (7)–(8), with FE data for different values of n. Also shown in Fig. 3 are predictions obtained from the small strain approximation (27). Present analysis

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Fig. 3. Comparison with FE calculations, for power law solids, from Jayaraman et al. (1998) (black symbols) and Casals and Alcalá (2005) (empty symbols), normalized by the yield stress. Markers denote: triangle-up n = 0.4, six-pointed star n = 0.3, circle n = 0.2, square n = 0.1, diamond n = 0. Solid line – present model (7)–(8) and (11), dashed line – approximation (27). For all data α = 70.3◦ , ν = 0.3.

appears to agree well with numerical results in the elastic-plastic transition zone (before full plasticity sets in). The eigen-cell pattern does not account for slip line fields as is indeed evident from Fig. 3: as E/Y increases theoretical hardness consistently overestimates calculated levels with lower values of n. Thus, validity limit is about E/Y < 300 for n = 0.1 and E/Y < 1000 for n = 0.2. Correlation between present theory and numerics for n = 0.4 is very good over the entire range up to E/Y = 4000. 5. Cavitation and indentation For sufficiently large values of Σa the spherical cavity expansion pressure P of (7) approaches its asymptotic value (Σa → ∞) known as the spherical cavitation pressure (Durban and Baruch, 1976). The connection between hardness and cavitation has been examined in the pioneering study of Bishop et al. (1945). The authors conjectured that hardness is bounded from below by cylindrical cavitation pressure and from above by spherical cavitation pressure. Calculations were made with an approximated Tresca plastic model and linear strain hardening. Comparison with experimental data for two metals has indeed confirmed the suggested bounds. Since highly accurate formulae are now available for both spherical cavitation pressure (Durban and Baruch, 1976) and cylindrical cavitation pressure (Masri and Durban, 2007), it is possible to examine the validity of the cavitation bounds over a wide range of material properties. The spherical cavitation pressure, obtained from (7), is ∞ d ( dΣ + β)Σ dΣ SP , (40) Pc = β 3 e 2 − 2 Σ − 1 + 2βΣ 0

and while this formula is valid for both Mises and Tresca models, there is a difference between the two models in cylindrical cavitation. With the Tresca solid (Masri and Durban, 2007) we have ∞ d 2 [ dΣ − ( 1−β 2 ) ]Σ dΣ T . (41) Pc = 3−β 2− 1+β 2 Σ −1+ e βΣ 2 0 For high strength aerospace solids (Masri and Durban, 2007) the ratio PcSP /PcT is slightly above 1.3, and about 1.4 for more ductile solids. Cavitation pressure values reflect, for given pattern, only material properties and can be viewed as measures of extreme strength of elastoplastic solids. Fig. 4(a) displays calculated values of hardness normalized with respect to PcSP and PcT , respectively. We have used accurate numerical evaluations of (40)–(41) and numerical data, for power-law solids (n = 0.1, 0.2, 0.3, 0.4), was

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Fig. 4. Hardness values, Jayaraman et al. (1998) (black symbols) and Casals and Alcalá (2005) (empty symbols), normalized by cavitation pressure, for power law solids with n = 0.1, 0.2, 0.3, 0.4. For all data α = 70.3◦ , ν = 0.3. Triangles indicate normalization with respect to cylindrical Tresca cavitation pressure. Circles indicate normalization with respect to spherical cavitation pressure. Squares indicate normalization with respect to cylindrical Mises cavitation pressure.

taken from Casals and Alcalá (2005) and from Jayaraman et al. (1998). We may conclude from Fig. 4(a) that spherical cavitation pressure provides an upper bound on hardness (circle markers), while cylindrical Tresca cavitation pressure appears to predict a clear lower bound, for materials with hardening index between 0.1 and 0.4, only within a limited range of 100 < E/Y < 300. Also, most of the hardness results are scattered in a band width of ±10% around the cylindrical Tresca cavitation pressure (triangle markers). Similar calculations with the cylindrical cavitation pressure for a Mises material (Masri and Durban, 2007) ∞ PcM

=

√ 0

e

d ( dΣ + 3− √β Σ 3

1−2κ 3 β)Σ

−1+

dΣ

2(1−κ) √ βΣ 3

with κ = −0.4725,

(42)

reveal a tighter upper bound (square markers), which is about 15% below the spherical cavitation (upper) bound as shown in Fig. 4(b). Yet, scatter of data in Fig. 4 is similar with all three normalizations. A more detailed analysis of Fig. 4 shows the influence of hardening exponent n (not indicated on figures), but is omitted here for brevity. The relation between cavitation pressure and hardness can be pursued further to obtain a useful observation. To this end we define the cavitation yield stress (Masri, in preparation) of a comparison elastic/perfectly-plastic solid, so that cavitation pressure is identical with that of the hardening solid. For elastic/perfectly-plastic response the cavitation yield stress is the actual yield stress Y , and the cylindrical Tresca cavitation pressure (41) can be expressed as (Masri and Durban, 2007)    Σy 2 T for Σy  1. 1 + ln Pc = (43) 2 (3 − β)(1 + β)Σy If Y is not much smaller than E, one has to evaluate numerically integral (41) after separating the elastic and perfectly plastic branches. The cylindrical Tresca cavitation yield stress (YcT ) is obtained by equating the cavitation pressure (41) for the hardening solid with the cavitation pressure of the comparison perfectly-plastic solid, given by (43) when Σy  1. Masri and Durban (2005) have used the definition of the spherical cavitation yield stress to include strainhardening effects in a dynamic spherical cavitation model. In Fig. 5 we have plotted numerical data of hardness values normalized with respect to the cavitation yield stress of a cylindrical cavity in a Tresca solid. Similar figures with hardness data normalized with respect to cylindrical Mises cavitation yield stress, or the spherical cavitation yield stress, display considerable scatter. The band wise appearance of Fig. 5 is most coherent with the cylindrical Tresca cavitation yield stress. Calculated values in Fig. 5, by comparison to Fig. 3, appear to concentrate within a narrow band up to about E/YcT = 100. Beyond that limit effects of full plasticity become apparent (more pronounced for low values of n) and the cavity expansion model is less applicable. Also shown in Fig. 5 are results obtained from the perfect plasticity (n = 0) finite strain theory ((8) and (13)), with hardness evaluated from (11) at Σa = Σy , and from small strain

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Fig. 5. Hardness values, Jayaraman et al. (1998) (black symbols) and Casals and Alcalá (2005) (empty symbols), normalized by cylindrical Tresca cavitation yield stress. Markers denote: triangle-up n = 0.4, six-pointed star n = 0.3, circle n = 0.2, square n = 0.1, diamond n = 0. Solid line – present model for n = 0 ((8), (11) and (13)), dashed line – approximation (28) for n = 0, dash-dot line – approximation (46). For all data α = 70.3◦ , ν = 0.3.

plasticity theory (28) approximation. Those solutions appear to provide a clear upper bound on normalized hardness values (H /YcT ). For elastic/perfectly-plastic solids within the transition zone (E/Y < 100) we can examine the relation between hardness and cavitation pressure in a simple way. Starting with the cylindrical Tresca cavitation pressure (43) we calculate the ratio (H /E)/PcT where hardness is given by (28). With (α = 70.3◦ , β = 0.4) we find that this ratio for E/Y = (100, 75, 50) is nearly constant (1.128, 1.125, 1.124). For smaller values of E/Y it will be better to use, instead of (43), the exact elastic/perfectly-plastic solution. A similar observation emerges for the spherical cavitation pressure, obtained from (40),    2Σy 2 SP for Σy  1, 1 + ln (44) Pc = 3 3(1 + β)Σy with the respective results (0.871, 0.870, 0.872). With the cylindrical Mises cavitation pressure (Masri and Durban, 2007), for Σy  1, √    Σy 3 M for κ = −0.4725, (45) Pc = √ 1 + ln (3 + (1 − 2κ)β)Σy 3 we obtain practically the same values as predicted by (28), namely (1.013, 1.013, 1.016). Those relative proportions will change with α but not significantly in the common testing range. Turning to strain hardening solids, we show in Fig. 6 hardness data, for three different values of n, in a range where sufficient FE calculations are available for deep plastic zone. Comparison of numerical results for H with cylindrical Tresca cavitation pressure reveals a remarkable agreement as n increases. In fact, cylindrical Tresca cavitation pressure appears to be more accurate than the cavity expansion model predictions. The cylindrical Mises cavitation pressure provides reasonable upper bounds for n = 0.1 and 0.2, but overestimates when n = 0.4 (not shown). Finally, we have shown on Fig. 5 a restricted version of (28), obtained by neglecting coupling between elastic compressibility (β) and strain ratio cot α/3Σy     2 cot α E H = 2 + ln −β , (46) Y 3 3 Y where Y is identified with YcT . This hardness relation falls below the more accurate results (solid and dashed lines in Fig. 5) but appears to provide a practical estimation for hardness of strain hardening solids before full plasticity sets in. At the limit of the elastoplastic transition zone (E/YcT ≈ 100) we recover from (46), for α = 70.3◦ and ν = 0.3, that H ≈ 2.7YcT which falls between Tabor’s (1951) figure of 2.8Y and that of Casals and Alcalá (2005) of 2.6Y .

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Fig. 6. Comparison with FE calculations of hardness, for power law solids, from Jayaraman et al. (1998) (black symbols) and Casals and Alcalá (2005) (empty symbols), normalized by the yield stress (H /Y ). Markers denote: triangle-up n = 0.4, circle n = 0.2, square n = 0.1, asterisk – limit of elastoplastic transition zone (E/YcT = 100). Solid line (H /Y ) – present model (7)–(8) and (11), dash-dot line – PcT /Y , dashed line – PcM /Y . For all data α = 70.3◦ , ν = 0.3.

6. Concluding remarks A blend of the pragmatic spherical cavity expansion model and accurate continuum plasticity theory has been used to study the hardness of strain-hardening solids, in conical indentation. The eigen-cell field beneath the indenter is modeled by average stress values leading to a simple relation for average pressure that reflects plastic hardening. Small strain approximations have been derived and compared with existing studies. A useful incremental energy connection can be used, in that context, to determine the uniaxial stress–strain curve from indentation tests with different cone angles. Comparison with finite element calculations of hardness, for α = 70.3◦ and ν = 0.3, over a wide range of plastic parameters, reveals good agreement (before full plasticity sets in) for low hardening and very good agreement for high hardening. Since the range of material parameters examined spans a broad family of solids, we may conclude that both spherical and cylindrical Mises pressures provide upper bounds on hardness, with the cylindrical value being the lower upper bound. Cylindrical Tresca cavitation pressure gives a lower bound on hardness within a limited range of E/Y , but for most cases predicts H values within ±10% around FE results. In fact, cylindrical Tresca cavitation pressure appears to be more accurate than the cavity expansion model predictions. A specific reference stress has been defined as the cylindrical Tresca cavitation yield stress. This representative stress serves as a useful scaling value in estimating hardness of strain hardening solids in elastoplastic transition range. It appears that the limit of the elastoplastic transition zone is reached at E/YcT ≈ 100, regardless of hardening characteristics. In that range, with absence of strain hardening, hardness is nearly proportional to cavitation pressure levels. In the deep plastic zone hardness is predicted by cylindrical Tresca cavitation pressure, at high levels of strain hardening, with remarkable accuracy. Acknowledgements It is our pleasure to thank Prof. Jorge Alcalá for useful discussions and for providing us with the finite element calculations. Part of this study was supported by the Asher Peled Memorial Research Fund. References Bishop, R.F., Hill, R., Mott, N.F., 1945. The theory of indentation and hardness. Proc. Phys. Soc. 57, 147–159. Casals, O., Alcalá, J., 2005. The duality in mechanical property extractions from Vickers and Berkovich instrumented indentation experiments. Acta Mater. 53, 3545–3561.

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