Cylindrical cavity expansion in compressible Mises and Tresca solids

European Journal of Mechanics A/Solids 26 (2007) 712–727

Cylindrical cavity expansion in compressible Mises and Tresca solids Rami Masri ∗,1 , David Durban Faculty of Aerospace Engineering, Technion, Haifa 32000, Israel Received 4 April 2006; accepted 1 December 2006 Available online 17 January 2007

Abstract The elastoplastic field induced by quasi-static expansion in steady-state plane-strain conditions of a pressurized cylindrical cavity (cylindrical cavitation) is investigated. Material behavior is modeled by Mises and Tresca large strain flow theories formulated as hypoelastic. Both models account for elastic-compressibility and allow for arbitrary strain-hardening (or softening). For the Mises solid analysis centers on the axially-hydrostatic assumption (axial stress coincides with hydrostatic stress) in conjunction with a controlled error method. Introducing an error control parameter we arrive at a single-parameter-dependent quadrature expression for cavitation pressure. Available results are recovered with particular values of that parameter, and an optimal value is defined such that the cavitation pressure is predicted with high accuracy. For the Tresca solid we obtain an elegant solution with the standard model when no corner develops in the yield surface. Under certain conditions however a corner zone exists near the cavity and the solution is accordingly modified revealing a slight difference in cavitation pressure. Comparison with numerical solutions suggests that the present study establishes cylindrical cavitation analysis on equal footing with existing studies for spherical cavitation. © 2007 Elsevier Masson SAS. All rights reserved. Keywords: Cavity expansion; Cavitation; Plasticity

1. Introduction The pioneering study by Bishop et al. (1945) on cavitation phenomena in elastoplastic solids, suggests that indentation pressure is bounded between cylindrical and spherical cavitation pressures. In recent years there has been increasing interest in elastoplastic cavitation as a basic physical model underlying penetration behavior. However, most of existing literature on elastoplastic cavitation deals with spherical patterns and quite a few issues concerned with cylindrical cavitation have remained opened. Available studies on spontaneous (self-similar) growth of pressurized cylindrical cavities are usually limited to incompressible Mises solids (Durban, 1979), or elastic/perfectly-plastic Tresca materials (Hill, 1950). Attempts to include compressibility effects in Mises media and strain-hardening behavior for the Tresca model were made by Durban (1988) and Durban and Kubi (1992), respectively.

* 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.2006.12.003

R. Masri, D. Durban / European Journal of Mechanics A/Solids 26 (2007) 712–727

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In Bishop et al. (1945) the constitutive model for Mises media was approximated by neglecting elastic volumetric changes in the plastic zone. Following this assumption, approximate solutions for spherical and cylindrical cavitation pressure in Mises solids were derived including strain-hardening. Hill (1950) presented approximate solutions for spherical and cylindrical cavitation pressure in compressible elastic/perfectly-plastic Mises materials without neglecting elastic volumetric changes in the plastic zone. These solutions provide good approximations under the practical assumption of very small yield stress compared to elastic modulus. Durban and Baruch (1976) solved the exact governing system for spherical cavity expansion in a compressible Mises solid with arbitrary strain-hardening. Durban (1979) derived an exact solution for cylindrical cavity expansion in an incompressible Mises solid with any hardening characteristic. Elastic-compressibility has been considered in Durban (1988) where the J2 deformation theory was employed. Recently, Masri and Durban (2006) introduced several approximate solutions, from a variety of points of view, for cylindrical cavitation pressure in compressible strain-hardening Mises media, with comparison to accurate numerical calculations. In the present paper we investigate self-similar elastoplastic fields induced by quasi-static expansion of a pressurized cylindrical cavity under the assumption of plane-strain. Material behavior is modeled by hypoelastic flow theories for Mises and Tresca solids. The theory accounts for elastic-compressibility and allows for arbitrary hardening (or softening) in the plastic range. We begin, in the next section, by recapitulating basic relations that describe steady-state expansion of cylindrical cavities in continuous media. Governing equations for Mises solid are given in Section 3 with a discussion of the axially-hydrostatic assumption. The controllable error method is introduced in Section 4 and used to derive an approximate solution in terms of quadratures for the Mises model. It turns out that the axially-hydrostatic assumption combined with the error controlled approximation, lead to a single-parameter-dependent expression for the cavitation pressure. The latter covers earlier results in the literature, derived from a variety of simplifications. Finally, we suggest an optimal value of the error control parameter such that the Mises cavitation pressure can be predicted with high accuracy from a simple formula. In Section 5 exact quadrature solutions for cylindrical cavitation in standard (no corner) Tresca solid are presented, and compared with the Mises results. Possible emergence of a corner zone near the cavity is discussed and the solution is accordingly modified in Section 6. For common metals, there is little influence of the corner zone on the cavitation pressure. 2. Cylindrical cavity expansion in steady-state conditions A circular cylindrical cavity, of instantaneous radius A, is expanding quasi-statically (with constant expansion velocity A˙ sufficiently small to neglect inertia) under constant internal cavitation pressure pc , in an infinite, remotely unstressed, elastoplastic compressible medium. We assume that an axially-symmetric field is induced in the medium by the expanding cavity, and that plane-strain deformation is maintained by appropriate axial stresses. Locating the origin (denoted by O in Fig. 1) of a cylindrical system (R, θ, Z), where R is the Eulerian radial coordinate, at the center of the cavity, the only equilibrium equation to be considered is in the radial direction. In steady-state expansion (cavitation) we assume a self-similar field with the nondimensional coordinate ξ = R/A as the single independent variable (Fig. 1) so the radial equilibrium equation is 1 Σr + (Σr − Σθ ) = 0, ξ

(1)

where (Σr , Σθ , Σz )=(σr , σθ , σz )/E are the nondimensional stress components (with respect to elastic modulus E) and differentiation with respect to ξ is denoted by a superposed prime. The radial stress increases monotonously with ξ , implying by (1) that Σθ > Σr within the entire deformation zone. The strain rate components are simply  ˙  ˙ dR˙ R˙ A A V  V , ˙θ = = ˙r = (2) = , ˙z = 0, dR A R A ξ ˙ A˙ is the nondimensional where the axial strain rate ˙z vanishes on account of the plane-strain constraint, V = R/ velocity and a superposed dot denotes differentiation with respect to a time-like parameter. Finally, we exploit a time derivative transformation by the similarity relation (Durban and Papanastasiou, 1997; Masri and Durban, 2006)  ˙  R A˙ d( ) A˙ d( ) d( ) ˙ ˙ = −ξ = (V − ξ ) . (3) ()=ξ dξ A A dξ A dξ

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This transformation will be used in the next sections to facilitate, together with constitutive relations for Mises and Tresca media, analytical solutions for the cavitation pressure. 3. Cylindrical cavitation in compressible Mises media The elastoplastic J2 flow theory can be written in the hypoelastic form     3˙p ν 1+ν σˇ − (I · ·σˇ )I + S. D= E E 2σe

(4)

In (4) tensor D is the Eulerian strain rate, σ – the Cauchy stress tensor, σˇ – the Jaumann stress rate, S – the stress deviator, I – the second order unit  tensor, ν – Poisson’s ratio and p is the effective plastic strain and a known function of the Mises effective stress σe = 32 S · ·S, which for an axially-symmetric field takes the nondimensional form √ 3 σe = (Σθ − Σr )2 + 3(Σz − Σh )2 , (5) Σ= E 2 where Σh is the nondimensional hydrostatic stress

1 Σh = (Σr + Σθ + Σz ). (6) 3 Consequently, in the absence of material spin, the tensorial constitutive equation (4) separates into just three scalar relations, with the aid of (2)–(3),     3 Σr − Σh   (7) p , V = (V − ξ ) Σr − ν(Σθ + Σz ) + 2 Σ    3 Σθ − Σh  V (8) = (V − ξ ) Σθ − ν(Σz + Σr ) + p , ξ 2 Σ    3 Σz − Σh  p . 0 = (V − ξ ) Σz − ν(Σr + Σθ ) + (9) 2 Σ Conservation of matter implies the universal density ratio ρ = e−Θ with Θ = 3βΣh and β = 1 − 2ν, ρ0

(10)

where ρ is the deformed state density, ρ0 denotes the reference density of the undeformed state and parameter β is a measure of compressibility. Here we use ρ = ρ0 and Σh = 0 as stress free conditions at infinity. Subtracting (8) from (7) we write the in-plane shear strain rate 

  V 3 Σθ − Σr  = − (1 + ν)(Σθ − Σr ) + p , (11) ln 1 − ξ 2 Σ with which we prefer to replace the radial relation (7). At this stage, we have five governing equations in (1), (8)–(9), (10) and (11) with five unknowns (Σr , Σθ , Σz , V , ρ) whose dependence on ξ should be determined using definitions (5) and (6). Integration of that system is carried from the cavity’s wall (ξ = 1) where V = 1 and p → ∞, to infinity (ξ → ∞) where ρ = ρ0 and both velocity and radial stress should vanish. When no remote external loads are applied, we expect all stress components to vanish at infinity. The density can be determined separately from (10) after solving for the stresses, hence in the subsequent analysis we consider only Eqs. (1), (8)–(9) and (11). Our main interest in this study lies in the cavitation pressure Pc = pc /E = −Σr (ξ = 1) which drives the cavitation process (Fig. 1). For strain-hardening with a definite yield point, like the modified Ludwik power-hardening law, plastic response is activated at the elastic/plastic interface ξ = ξi (Fig. 1), where p vanishes and Σ reaches the value of the nondimensional yield stress Σy (Σy = Y/E with Y denoting the yield stress). However, for strain-hardening response without a definite yield point, like the Ramberg–Osgood power-hardening law, the plastic branch is active within the entire deformation zone but becomes negligible compared with the elastic branch when approaching infinity. In the special case of elastic/perfectly-plastic response p is not known a-priori and an extra algebraic equation (Σ ≡ Σy ) is obtained from (5), in the post yield range.

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Fig. 1. Scheme of self-similar field in quasi-static expansion of a circular cylindrical cavity in compressible elastoplastic media. Cavitation pressure is pc . The radial coordinate ξ is nondimensionalized with respect to the current radius of the cavity. Plastic yielding occurs at the elastic/plastic interface ξ = ξi . The remote boundary at infinity is stress-free.

Most of existing literature on pressurized tubes of Mises solids centers on the assumption of elastic incompressibility (ν = 12 ⇒ β = 0) in conjunction with elastic/perfectly-plastic response. It is indeed evident from (9) that with ν = 12 the axial stress coincides with the hydrostatic stress, implying from (6) that 1 Σz = Σh = (Σr + Σθ ), (12) 2 hence, from (5), √ 3 (Σθ − Σr ). (13) Σ= 2 Consequently, the governing equations are simplified and (11) admits a direct integration with a quadrature solution for the cavitation pressure for any strain-hardening (or softening) characteristic ∞ PcM (inc) = 0

Σ d

e

√

3

−1

(14)

.

Here we have introduced the total elastoplastic strain  = Σ + p , which is a known function of the effective stress Σ . An identical expression has been obtained by Durban (1979) as the asymptotic limit of the cavity expansion process under monotonously increasing internal pressure. Now, for compressible Mises media a comparative review of several analytical approximations for cylindrical cavitation pressure has been given recently by Masri (2007) and Masri and Durban (2006) – henceforth designated as [MD 2006] – with practical lower and upper bounds, based on ad-hoc compressibility assumptions Pc↓

∞ = 0

(d/dΣ + (2/3)β)Σ dΣ √ √ , √ e 3−(β/ 3)Σ − 1 + 3 βΣ

Pc↑

∞ = 0

e

√

(d/dΣ + (2/3)β)Σ dΣ . √ − 1 + (4/ 3)βΣ

√ 3−(2/ 3)βΣ

(15)

The accuracy of (15) has been examined in [MD 2006] over a wide range of material parameters, revealing that ↓ ↑ (15) provides lower and upper bounds (hence the notations Pc and Pc respectively) on the exact results, obtained by numerical solutions, with deviations of less than 0.5%. By comparison, the incompressible result (14) predicts values of up to 4.5% above the exact results [MD 2006]. Yet, in this work we attempt at relaxing the incompressibility assumption, facilitating an elegant derivation of accurate relations for cylindrical cavitation in compressible Mises media. The various approximations for the numerically evaluated Mises cavitation pressure PcM are based on material orthotropic models (both elastic and plastic) discussed in Masri (2007) and on ad-hoc compressibility assumptions

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Fig. 2. Numerical quasi-static cylindrical cavitation solution for steel 5CrMoV using Mises plasticity model. Metal characteristics are presented in Table 1. Strain-hardening behavior is modeled by the Ramberg–Osgood power-hardening law ( = Σ + KΣ n ).

(like (15)) and the axially-hydrostatic assumption (12), both discussed in [MD 2006]. Surely, the plane-strain constraint (9) is violated by (12) when compressibility is accounted for, yet theoretical analysis supported by detailed numerical studies [MD 2006] shows that the axial stress is very close to (12). The idea of axial stress being equal to the arithmetic mean of radial and circumferential stress components dates back to Nadai (1931). In Fig. 2 we present an example of the numerical solution for the stress profiles in steel 5CrMoV, taken from [MD 2006], which clearly demonstrates the validity of the axially-hydrostatic assumption. Also, calculated differences between the Mises effective stress Σ , defined in (5), and the approximate expression (13) are too small to be observed. Further illustrations that support those observations are given in Masri (2007). In this study we proceed with an entirely different approach that centers on controlling the error induced by using (12) (and hence also (13)) for a compressible material. It turns out that existing approximations are just particular cases of a more general result. Thus, a unified treatment of cylindrical cavitation pressure formula leads to a singleparameter-dependent expression for the cavitation pressure that covers earlier results in the literature including the ↓ ad-hoc lower bound (Pc ). Practically, the controllable error method suggested here would give an excellent and more solid-based approximation than the ad-hoc bounds (15). 4. Controllable error approximation We begin with replacing the circumferential constitutive relation (8) by a relation obtained from multiplying (9) with a constant parameter κ and subtracting the result from (8) 

3 Σθ − κΣz − (1 − κ)Σh  V    (16) = (V − ξ ) (1 + νκ)Σθ − (1 − κ)νΣr − (κ + ν)Σz + p , ξ 2 Σ while Eqs. (1), (9) and (11) remain unchanged. Of course, the exact solution of this new governing system, with (16) replacing (8), is independent of parameter κ. Now, taking (13) as an approximation for the effective stress in compressible media, we can integrate (11) with the stress free conditions at infinity to obtain the approximate radial velocity √ √ V = 1 − e−( 3−(β/ 3)Σ) . ξ

(17)

Here we have used the total elastoplastic strain  = Σ + p and the compressibility measure β = 1 − 2ν. Notice that the approximate solution (17) satisfies the wall boundary condition V (ξ = 1) = 1 because (ξ = 1) → ∞. Also, the equilibrium equation (1), when combined with (13), takes the approximated differential form

R. Masri, D. Durban / European Journal of Mechanics A/Solids 26 (2007) 712–727

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√   dξ 3 dΣr dΣ = , ξ 2 dΣ Σ

(18)

Inserting (17) in (16), and using assumptions (12)–(13), we arrive at the simpler approximate equation √ √ √  3  1 (1 − 2κ)β    = 1 − e 3−(β/ 3)Σ . (19) (1 − κ)βΣr + Σ + √ 2 ξ 2 3 The couple (18)–(19), which forms a system with two unknowns (Σr , Σ), admits a quadrature type solution, with the effective stress Σ as the independent variable. This solution satisfies the stress free conditions at infinity together with the total strain asymptotic tendency to infinity at the cavity’s wall, namely ∞

 √ ln ξ 2 = 3

Σ κ

J (s) ds

and Σr = −

Σ

J κ (s)s ds,

(20)

0

where d/dΣ + ((1 − 2κ)/3)β √ . √ e 3−(β/ 3)Σ − 1 + (2(1 − κ)/ 3)βΣ

J κ (Σ) =

√

(21)

However, Eqs. (18)–(19) are not accurate due to use of approximations (12)–(13) and hence solution (20)–(21) depends on parameter κ. This solution is the accurate solution of the approximate equations (18)–(19) but it is an approximation for the exact system described in Section 3. Yet, the error induced in the solution can be controlled by parameter κ, now-on labeled as the error control parameter. In deriving that solution we have not necessarily assumed the existence of a definite yield point. This simplifies mathematical steps and leads to a universal solution, gradually covering the entire elastoplastic behavior. The solutions for V , Σθ , Σz and ρ can be obtained immediately from (17), (13), (12) and (10), respectively. The asymptotic value (Σ → ∞) of −Σr (the second of (20)) is the cavitation pressure for arbitrary strain-hardening ∞ Pc (κ) =

∞ J Σ dΣ = κ

0

0

e

√

(d/dΣ + ((1 − 2κ)/3)β)Σ dΣ , √ − 1 + (2(1 − κ)/ 3)βΣ

√ 3−(β/ 3)Σ

(22)

while from the first of (20) the location of the elastic/plastic interface ξi , for solids with a definite yield point, is

 √ ln ξi2 = 3

∞ J κ dΣ.

(23)

Σy

The error involved in calculating Pc from (22) is controlled by parameter κ and one may hope, in view of existing approximations for the cavitation pressure, to minimize the induced error. Exceptionally, when β = 0 relation (22) reduces to the exact incompressible result (14), independent of κ. A similar (exact) expression for quasi-static spherical cavitation pressure has been obtained by Durban and Baruch (1976), as the asymptotic limit of the cavity expansion process under monotonously increasing internal pressure. For the spherical cavitation pattern the effective stresses for Mises and Tresca solids are identical hence the cavitation pressure is the same. The spherical cavitation pressure is brought here for the sake of comparison, namely sp Pc

∞ =

(d/dΣ + β)Σ dΣ . − 1 + 2βΣ

(24)

e(3/2)−(β/2)Σ 0

Notice that for incompressible media (β = 0) the spherical result sp Pc (inc) =

∞

Σ d , −1

(25)

e(3/2) 0

becomes identical to the (exact) cylindrical analogue (14) with (Σ, ) in the latter replaced by ( √2 Σ, 3

√

3 2 ).

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Expressions (22) and (23) can be used to generate simple results for elastic/perfectly-plastic materials by considering separately elastic ( ≡ Σ) and elastic/perfectly-plastic (Σ ≡ Σy ) deformation zones. This procedure gives exact relations which can be approximated, under the practical assumption Σy  1, by √

2  Σy  3 2 Pc (κ) = √ 1 + ln ξi . (26) with ξi = [3 + (1 − 2κ)(1 − 2ν)]Σy 3 Relations (26) remain invariant with respect to the interchange ν ↔ κ. Turning to specific examples, we begin with the observation that when κ = 1 one can deduce from (26) the cavitation pressure result derived by Bishop et al. (1945) √  

Σy 3 . (27) Pc (κ = 1) = √ 1 + ln 2(1 + ν)Σy 3 A later and more accurate result is Hill’s (Hill, 1950) ad-hoc classical formula which is here obtained from (26) with κ = − 12 , √ 

   Σy 1 3 . (28) = √ 1 + ln Pc κ = − 2 (5 − 4ν)Σy 3 In fact, the ad-hoc lower bound in (15) is recovered from the general expression (22) when κ = − 12 , namely ↓ Pc = Pc (κ = − 12 ). Moreover, with vanishing of the error control parameter, κ = 0, we arrive at the benchmark approximation which deviates by up to 2% above the exact value PcM [MD 2006]. Also, Masri (2007) has shown that when the isotropic elastic branch in (7)–(9) is replaced by an elastic orthotropic branch, with transverse Poisson ratios of νrz = νθz = 12 , an exact quadrature type solution exists. The corresponding cavitation pressure is then identical with (22) when κ = 12 . This elastic-orthotropic solution gives cavitation pressure values above the exact results, slightly higher than the incompressible relation (14). Similarly, if the isotropic plastic branch in (7)–(9) is replaced by an orthotropic plastic branch, based on an associated model with effective stress (13) (but not employing assumption (12)), an exact quadrature type solution for Pc is obtained again (Masri, 2007). That cavitation pressure appears as an upper bound on the exact result PcM , with a deviation of up to 1%, and can be derived here as a particular case of (22) with κ = −ν ∞ Pc (ort) = Pc (κ = −ν) = 0

e

√

[d/dΣ + (2 − β)β/3]Σ dΣ , √ − 1 + ((3 − β)β)/ 3Σ

√ 3−(β/ 3)Σ

(29)

and by neglecting β 2 terms we obtain the lower bound in (15). We ask now if it is possible to locate an optimal value of the error control parameter κ, such that (22) will predict the cavitation pressure with high accuracy. To this end we have performed a numerical search, calculating values of Pc (κ), over a wide range of material properties, with different values of parameter κ. The main finding is that for all cases examined here the exact value of the cavitation pressure, determined numerically by the solution of the original system (1), (8)–(9) and (11), is predicted extremely well by (22) with values of κ falling in a narrow range. This observation is clearly supported by Fig. 3 where we show, for several representative metals, how the cavitation pressure (22) changes with κ. It is apparent from Fig. 3 that an optimal value of κ can be recommended, with very little sensitivity of (22) to small variations near that value. Based on our numerical study, which includes materials beyond those shown in Fig. 3, we suggest the value of κ = −0.4725. Thus, expression (22) can be used with that optimal choice to predict, with excellent accuracy of up to an error of less than 0.1%, the quasi-static cylindrical cavitation pressure in Mises media, ∞ PcM

Pc (κ = −0.4725) = 0

(d/dΣ + 0.648β)Σ dΣ

√ √ e 3−(β/ 3)Σ

− 1 + 1.7βΣ

.

(30)

For elastic/perfectly-plastic response, substituting κ = −0.4725 in (26) gives an approximation which is very close to Hill’s ad-hoc solution (28) suggested more than fifty years ago!

R. Masri, D. Durban / European Journal of Mechanics A/Solids 26 (2007) 712–727

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Fig. 3. Cavitation pressure (22), normalized by the exact solution PcM , versus error control parameter κ is demonstrated here for several metals. The highest relative errors for the choice of κ = −0.4725 are shown by up and down arrows. The numbers are for: 1. Stainless steel, 2. AL 2014-T6, 3. Titanium B120VCA, 4. AL 7075 T6, 5. ST X-200 and 6. ST 5CrMoV. Material parameters are shown in Tables 1–3. Table 1 Cylindrical and spherical cavitation data for strain-hardening solids Metal

ST 5CrMoV

AL 7075 T6

AL killed steel

Reference

Durban (1979)

Durban (1979)

Durban and Birman (1982)

E [GPa]

194

72.4

207

K

5.23 × 1031

3.94 × 1021

6.43 × 1011

n

16.67

10.9

4.505

ν

0.26

0.32

0.27

sp Pc (inc) (25) sp Pc (24) PcM (inc) (14) PM c PcT (inc) (56) PT c (43) PcV sp Pc /PcM PcM /PcT

3.660 × 10−2

2.936 × 10−2

0.564 × 10−2

3.440 × 10−2

2.813 × 10−2

0.552 × 10−2

3.069 × 10−2

2.459 × 10−2

0.470 × 10−2

2.943(2) × 10−2

2.389 × 10−2

0.463 × 10−2

2.706 × 10−2

2.153 × 10−2

0.398 × 10−2

2.619 × 10−2

2.102 × 10−2

0.394 × 10−2

2.618 × 10−2

2.102 × 10−2

0.394 × 10−2

1.169

1.177

1.192

1.124

1.137

1.175

Pv (58)

1.683 × 10−2

1.825 × 10−2

0.203 × 10−2

ξv

2.221

1.312

2.933

Pv /PcV

0.643

0.868

0.515

Figures denoted by ‘( )’ represent the fourth significant digit, in the highly accurate approximation (30), only if it differs from the fourth significant digit of the exact numerical solution PcM .

In Tables 1–3 we present, for ten metals, the exact cavitation pressure PcM , calculated from numerical solutions, together with results obtained from the incompressible solution (14), which are above the exact solution. Metal characteristics for Ramberg–Osgood power hardening law ( = Σ + KΣ n ) are detailed in the tables. The values in ‘( )’ for PcM represent the fourth significant digit in the highly accurate approximation (30), only if it differs from the fourth significant digit in the exact solution PcM . It appears that (30) is more accurate than the ad-hoc upper and lower bounds (15) with a deviation of less than 0.1%. Also shown in the tables is the corresponding spherical cavisp tation pressure Pc , given by (24), which is in the range of 1.17 to 1.21 times above the Mises cylindrical cavitation sp pressure PcM . The incompressible spherical cavitation pressure Pc (inc) attains the highest values in all cases.

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Table 2 Cylindrical and spherical cavitation data for strain-hardening solids Metal

ST AISI 4340

ST X-200

ST D6AC

AL 2014-T6

Reference

Durban (1979)

Durban (1979)

Durban (1979)

Durban (1979)

E [GPa]

201

208

213

69

K

7.61 × 1054

8.85 × 1066

2.52 × 1055

6.08 × 1031

n

27.6

34.1

28

15.62

0.28

0.27

0.27

0.33

3.273 × 10−2

3.486 × 10−2

3.339 × 10−2

2.815 × 10−2

3.084 × 10−2

3.271 × 10−2

3.138 × 10−2

2.695 × 10−2

2.751 × 10−2

2.930 × 10−2

2.806 × 10−2

2.363 × 10−2

2.642 × 10−2

2.806 × 10−2

2.690(89) × 10−2

2.294(5) × 10−2

2.434 × 10−2

2.596 × 10−2

2.483 × 10−2

2.079 × 10−2

2.356 × 10−2

2.509 × 10−2

2.401 × 10−2

2.028 × 10−2

2.356 × 10−2

2.508 × 10−2

2.400 × 10−2

2.028 × 10−2

1.167

1.166

1.167

1.175

1.121

1.118

1.120

1.131

Pv (58)

1.546 × 10−2

1.611 × 10−2

1.530 × 10−2

1.687 × 10−2

ξv

2.276

2.350

2.377

1.463

Pv /PcV

0.656

0.642

0.637

0.832

ν sp

Pc (inc) (25) sp Pc (24) PcM (inc) (14) PM c PcT (inc) (56) PT c (43) PcV sp Pc /PcM PcM /PcT

Figures denoted by ‘( )’ represent the fourth significant digit, in the highly accurate approximation (30), only if it differs from the fourth significant digit of the exact numerical solution PcM . Table 3 Cylindrical and spherical cavitation data for strain-hardening solids Metal

Titanium B120VCA

Soft aluminum

Stainless steel

Reference

Masri and Durban (2005)

Durban and Birman (1982)

Masri and Durban (2005)

E [GPa]

106

69

206

K

2.4 × 1029

1.27 × 1010

5.78 × 104

n

16.5

3.718

3

ν

1 3

0.33

0.30

4.606 × 10−2

0.402 × 10−2

4.029 × 10−2

4.397 × 10−2

0.398 × 10−2

3.938 × 10−2

3.856 × 10−2

0.334 × 10−2

3.295 × 10−2

3.737(8) × 10−2

0.331 × 10−2

3.247 × 10−2

3.406 × 10−2

0.280 × 10−2

2.769 × 10−2

3.317 × 10−2

0.278 × 10−2

2.739 × 10−2

1.127

1.202

1.213

1.177

1.191

1.185

sp

Pc (inc) (25) sp Pc (24) PcM (inc) (14) PM c PcT (inc) (56) PT c (43) sp Pc /PcM PcM /PcT

Figures denoted by ‘( )’ represent the forth significant digit, in the highly accurate approximation (30), only if it differs from the forth significant digit of the exact numerical solution PcM . No corner zone develops for the metals of this table.

5. Cylindrical cavitation in compressible Tresca media The standard Tresca analysis of internally pressurized cylindrical tubes (Durban and Kubi, 1992) is based on the intermediate axial stress assumption Σr < Σz < Σθ , which implies the effective stress (identical with plastic potential)

(31)

R. Masri, D. Durban / European Journal of Mechanics A/Solids 26 (2007) 712–727

Σ = Σθ − Σr .

721

(32)

Plastic incompressibility along with plastic power equivalence and the plane-strain condition, detailed in Durban and Kubi (1992), generate the constitutive relations. The elastic branch of the constitutive relations remains here the same as in (7)–(9), but no plastic response is activated in the axial direction. Thus, in steady state expansion, with the aid of (2)–(3), the constitutive connections for the Tresca solid are    (33) V  = (V − ξ ) Σr − ν(Σθ + Σz ) − p ,    V = (V − ξ ) Σθ − ν(Σz + Σr ) + p , (34) ξ   (35) 0 = (V − ξ ) Σz − ν(Σr + Σθ ) . These relations are the self-similar (steady-state) analogue of the relations used by Durban and Kubi (1992) for the problem of a thick-walled cylindrical tube subjected to internal pressure. Now, under the stress free conditions at infinity, the plane-strain constraint (35) dictates the axial stress Σz = ν(Σr + Σθ ),

(36)

which should comply with the initial assumption (31), namely Σr < Σz ⇒ (1 − ν)Σr < νΣθ ,

Σz < Σθ ⇒ νΣr < (1 − ν)Σθ .

(37)

Subtracting (34) from (33), and using (32), we obtain an integrable equation with the solution (accounting for vanishing stresses and velocity at infinity) V = 1 − e−(2−((1+β)/2)Σ) , ξ

(38)

that satisfies wall boundary condition V (ξ = 1) = 1 since (ξ = 1) → ∞. Substituting (38) back in (34) results in, with the aid of (32) and (36),    3−β 1−β 2  1 βΣr − Σ +   = 1 − e2−((1+β)/2)Σ . (39) 2 2 ξ Observing that the equilibrium equation (1) takes here the differential form   dΣr dΣ dξ = , ξ dΣ Σ

(40)

on account of (32), we arrive at the quadrature type solution of (39)–(40)

 ln ξ 2 = 2

∞

Σ and Σr = −

T

J (s) ds Σ

J T (s)s ds,

(41)

0

where J T (Σ) =

d/dΣ − ((1 − β)/2)2 . − 1 + ((3 − β)/2)βΣ

e2−((1+β)/2)Σ

(42)

Stress free conditions at infinity, and total strain asymptotic tendency to infinity at the cavity’s wall, are satisfied by this solution. The solutions for V , Σθ , Σz and ρ can be obtained immediately from (38), (32), (36) and (10), respectively. The corresponding cavitation pressure in a Tresca solid follows, from the second of (41), as ∞ PcT

=

∞ J Σ dΣ = T

0

0

[d/dΣ − ((1 − β)/2)2 ]Σ dΣ , e2−((1+β)/2)Σ − 1 + ((3 − β)/2)βΣ

and for solids with a definite yield point the location of the elastic/plastic interface ξi is, from the first of (41),

(43)

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R. Masri, D. Durban / European Journal of Mechanics A/Solids 26 (2007) 712–727

 ln ξi2 = 2

∞ J T dΣ.

(44)

Σy

Specific values of the Tresca cavitation pressure are shown in Tables 1–3 and compared to the (higher) Mises results. Expressions (43) and (44) can be used to derive specific formula for elastic/perfectly-plastic materials by considering separately elastic ( ≡ Σ) and elastic/perfectly-plastic (Σ ≡ Σy ) deformation zones  −1



 1 3−β , ln ξi2 = ln 1 − 1 − βΣy e−((3−β)/2)Σy (1 − ((3 − β)/2)βΣy ) 2 PiT

1 = 2

Σy 0

((3 − β)/2)(1 + β)Σ dΣ , exp(((3 − β)/2)Σ) − 1 + ((3 − β)/2)βΣ

Σy 2  PcT = ln ξi + PiT . 2

(45)

Here, PiT = −Σr (Σ = Σy ) denotes the radial pressure at the elastic/plastic interface ξi for solids with a definite yield point (not necessarily perfectly-plastic), which is also the internal pressure required to cause initial yielding at the cavity’s wall for Tresca solids (Durban and Kubi, 1992). Solution (45) can be further simplified for common materials (Σy  1) 

 Σy 2 1 and ξi2 = with PiT = = . PcT = PiT 1 + ln ξi2 2 (5 − 4ν − β 2 )Σy 2(1 − ν 2 )Σy

(46)

Hill (1950) derived a similar, though less accurate (without the β 2 term), expression and suggested the transformation 2 Σy → √ Σy , 3

(47)

to obtain the Mises equivalent relation (28). However, without neglecting the β 2 term in (46) the transformation which converts the Mises error controlled solution (26) to the Tresca relation (46) is √ 3 Σy , κ = −ν. (48) Σy → 2 This analogy can be carried further to transform the strain-hardening Mises error controlled solution (22) to the solution for the Trseca material (43), using κ = −ν along with the following exchange √   3 T 2 1 T M M T Σ → Σ (49)  →√  − Σ , 2 4 3 where Σ M and  M denote Σ and  in (22), while Σ T and  T are for (43). Moreover, by replacing (Σ, ) in the (exact) spherical solution (24) by   3−β 4 β2 + β + 4 Σ,  − Σ 4 3 12 we recover, up to a scaling factor of 3−β 3 (≡ 1 for incompressible media), the (exact) Tresca solution (43). Applying solution (41)–(42) to elastic/perfectly-plastic response gives particular relations with the independent variable . For the elastic zone where  ≡ Σ  Σy , 1 Σr = − 2

Σ 0

((3 − β)/2)(1 + β)s ds Σ

− exp(((3 − β)/2)s) − 1 + ((3 − β)/2)βs 2

(for Σy  1),

 2 Σy   Σy ξ ((3 − β)/2)(1 + β) ds = ln

ln (for Σy  1), ξi exp(((3 − β)/2)s) − 1 + ((3 − β)/2)βs Σ Σ

(50)

R. Masri, D. Durban / European Journal of Mechanics A/Solids 26 (2007) 712–727

and for the elastic/perfectly-plastic zone where   Σ ≡ Σy ,   2

Σy ξi ln + PiT , Σr = − 2 ξ  −1



2 1 3−β −(2−((1+β)/2)Σy ) ln ξ = , ln 1 − 1 − βΣy e (1 − ((3 − β)/2)βΣy ) 2

723

(51)

where ξi and PiT are calculated from (45). It is instructive to examine the controllable error method when applied to the Tresca model. To this end, we replace (34) by a relation obtained from multiplying (35) with κ and subtracting the result from (34)   V = (V − ξ ) (1 + νκ)Σθ − (1 − κ)νΣr − (κ + ν)Σz + p . ξ

(52)

Inserting (38) in (52), and using (32) along with the axially-hydrostatic assumption (12) instead of solution (36), we arrive at the approximate equation  1 − (1 − 2κ)β  1 Σ +   = 1 − e2−((1+β)/2)Σ , 4 ξ

(1 − κ)βΣr −

(53)

which should be solved together with (40). Actually, for incompressible media (ν = 12 ) the axially-hydrostatic assumption is accurate for Mises (12) and Tresca (36) models and remains accurate, for both models, in the compressible linear elastic zone [MD 2006]. Yet, in the deep plastic zone assumption (12) is an excellent approximation only for the compressible Mises model [MD 2006]. Comparing (53) with the exact equation (39) we find that the choice κ = −ν leads to the exact solution for the Tresca model (41)–(42) and hence completely vanishes the error induced by using assumption (12). This interesting observation supports the use of the controllable error method for the Mises material, particularly in view of the accuracy of (29). We turn now to examine the validity of the intermediate axial stress assumption (31) via inequalities (37). The first of these (Σr < Σz ) can be put in the form, using the second of (41) and the effective stress definition (32), 1−β Σ +β 2

Σ J T (s)s ds > 0,

(54)

0

which is valid for common metals in the entire field. The other inequality (Σz < Σθ ) becomes   Σ 1 Σ > J T (s)s ds = −Σr (Σ), 1+ β 2

(55)

0

and is likely to fail for certain metals. This is illustrated in Figs. 4–6 for three different metals where solid lines represent the RHS of inequality (55) and dashed lines represent the LHS. Initially, for small values of Σ (elastic zone) inequality (55) is uniformly valid because in the elastic phase the RHS of (55) is given essentially by Σ/2, as shown in (50), while the factor (1 + 1/β) at the LHS is greater than one. Eventually, as plasticity progresses and the cavity’s wall is approached, the inequality can be violated. This is apparent in Figs. 4 and 5, while Fig. 6 demonstrates a case where (55) remains valid at any level of the effective stress. Much of that constitutive sensitivity depends on Poisson ratio, as demonstrated by the factor (1 + 1/β), but inequality (55) is uniformly valid when β = 0 (incompressible media) regardless of plastic response (Σ). The cylindrical cavitation pressure for incompressible Tresca solids is, from (43), ∞ PcT (inc) = 0

(d/dΣ − 1/4)Σ dΣ . e2−(1/2)Σ − 1

(56)

Comparing (56) with the analogous Mises result PcM (inc), from (14), it is obvious that (56) gives, for the same hardening characteristic, lower values. This is shown in Tables 1–3 where the ratio PcM (inc)/PcT (inc) falls in the

724

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Fig. 4. Inequality (55) illustrated for ST 5CrMoV (Table 1). Vertex values from (58) are denoted by (Σv , Pv ) and marked with a black circle.

Fig. 5. Inequality (55) illustrated for AL killed steel (Table 1). Vertex values from (58) are denoted by (Σv , Pv ) and marked with a black circle.

range of 1.13 to 1.19. For incompressible Tresca solids with a definite yield point we find from (44) the elastic/plastic interface location −1

, ξi2 = 1 − e−(3/2)Σy

(57)

which is independent of the hardening characteristics and reduces, for Σy  1, to Hill’s (Hill, 1950) elastic/perfectlyplastic result ξi2 = 2/(3Σy ). Actually, incompressible cavitation pressures for spherical and cylindrical (Mises and Tresca models) expansions depend on strain-hardening characteristics but the elastic/plastic interface locations are independent of hardening.

R. Masri, D. Durban / European Journal of Mechanics A/Solids 26 (2007) 712–727

725

Fig. 6. Inequality (55) illustrated for Titanium B120VCA (Table 3). For this material the standard solution is valid in the entire deformation zone with cavitation pressure PcT = 0.03317.

6. The corner zone solution Once the intermediate stress assumption fails, a Tresca corner zone (vertex solution) will develop from the vertex radius ξ = ξv to the cavity ξ = 1. The standard Tresca solution breaks down at the vertex interface, where Σ = Σv , and from (55) 1+β Σv = 2β

Σv J T Σ dΣ = Pv ,

(58)

0

where Pv = −Σr (ξ = ξv ) denotes the radial pressure at ξv , and is certainly higher than Σv . The couple (58) determines the values of Σv and Pv whenever a corner solution exist, as illustrated in Figs. 4–5. Now, evaluating (58) for the ten metals reveals that a corner zone exists for the seven metals of Tables 1–2, while for the other three metals, shown in Table 3, the standard solution is valid. Nevertheless, Tables 1–2 give the standard cavitation pressure values PcT for further comparison with the corner solution, which will be discussed shortly. Also, values of Pv , as obtained from (58), are shown in Tables 1–2. A simple estimation of Σv is derived by Masri (2007) for the Ramberg–Osgood power-hardening law

−1   1 − ν 2 1/(n−1) 1 . (59) Σv (est) = 1 − (n − 1)β K This simplified relation provides a sound working first guess which is close to the exact solution of (58). For example, applying (59) to steel 5CrMoV (Fig. 4) we get Σv (est) = 1.086 × 10−2 (exact result from (58) is 1.092 × 10−2 ), while for the aluminum killed steel (Fig. 5) we find Σv (est) = 0.110 × 10−2 (0.128 × 10−2 ), for steel AISI 4340 Σv (est) = 0.942 × 10−2 (0.945 × 10−2 ), for steel X-200 Σv (est) = 1.014 × 10−2 (1.015 × 10−2 ) and for aluminum 7075-T6 Σv (est) = 0.906 × 10−2 (0.966 × 10−2 ). While the standard solution for the radial stress (the second of (41)) is valid in the outer range (ξ  ξv ) when a corner zone exists, the (ξ, Σ) mapping (the first of (41)) is replaced by



 ln ξ 2 = ln ξv2 + 2

Σv J T (s) ds,

(60)

Σ

where, for now, the vertex location ξv is unknown. For solids with a definite yield point, where Σv > Σy and ξv < ξi , we find from (60)

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R. Masri, D. Durban / European Journal of Mechanics A/Solids 26 (2007) 712–727



ξi ln ξv

2

Σv = 2 J T dΣ.

(61)

Σy

Within the inner corner zone (1  ξ  ξv ) the plastic normality rule is replaced by the vertex relations Σθ = Σz

and Σ = Σθ − Σr = Σz − Σr .

(62)

Plastic incompressibility along with plastic power equivalence and plane-strain condition, detailed in Durban and Kubi (1992), combined with relations (2)–(3) for steady-state expansion, generate the two (in-plane) constitutive relations

 V  = (V − ξ ) βΣr + βΣ  −   , (63)

 V = (V − ξ ) 2βΣr + βΣ  +   . (64) ξ The corner zone equations, (63)–(64) and (1), need to be solved for the three unknowns Σθ , Σr and V , under continuity conditions at the vertex interface ξ = ξv and the velocity condition V = 1 at the cavity’s wall where  → ∞. The axial stress in the corner zone is identical, by (62), with the circumferential stress and the density can be calculated from (10) after solving for the stresses. Subtracting (64) from (63) and integrating over ξ , we find the corner zone velocity profile V = 1 − V0 e−(2+βΣr ) , ξ

(65)

where V0 is an integration constant. This solution, like previous velocity profile solutions (17) and (38), satisfies the wall boundary condition V (ξ = 1) = 1. However, (65) is limited to the corner zone and depends explicitly on the radial stress profile Σr . Continuity of radial velocity requires that (65) will equal (38) at the vertex interface ξ = ξv and hence, with the aid of (58), we obtain V0 = 1. A further substitution of (65) in (64) gives an equation for the stresses Σr and Σ  1 2βΣr + βΣ  +   = 1 − e2+βΣr , (66) ξ with a second equation supplied by (40). These equations can be combined to form a single nonlinear first order differential equation for the radial stress Σr , with the effective stress Σ as the independent variable, (d/dΣ + β)Σ dΣr = − 2+βΣ , r − 1 + 2βΣ dΣ e

(67)

in conjunction with the continuity condition at the vertex interface Σr (Σ = Σv ) = −Pv . The corner cavitation pressure PcV is the limit value of the corner zone solution PcV = −Σr (Σ → ∞). Now, integration of (40), accounting for the singular behavior at the cavity’s wall, gives the (ξ, Σ) mapping in the corner zone  ∞

2 dΣr (s) ds −2 , (68) ln ξ = ds s Σ

where Σr = Σr (Σ) is the solution of (67), thus completing the numeric procedure for the corner zone. The location of the vertex interface is  ∞

2 dΣr dΣ −2 , (69) ln ξv = dΣ Σ Σv

which can be used in (60) for the (ξ, Σ) mapping in the outer zone. Particularly, using (61), we obtain the elastic/plastic interface location for solids with a definite yield point when a corner zone exists. In Tables 1–2 we present the corner cavitation pressure PcV , obtained from numerical solution of (67), and compare it to the standard solution PcT (43). It is seen that the difference between the two solutions is negligible, yet the cavitation pressure with the corner zone solution PcV is slightly lower than the standard solution PcT . Commonly, this small

R. Masri, D. Durban / European Journal of Mechanics A/Solids 26 (2007) 712–727

727

difference increases with increasing corner zone size ξv up to a deviation of less than 0.1%. By neglecting that minor difference the corresponding ratio between quasi-static cylindrical Mises and Tresca cavitation pressures (PcM /PcT ) is shown in Tables √ 1–2, and in Table 3 with no corner zone, for ten metals. This ratio deviates from Hill’s (Hill, 1950) suggestion of 2/ 3 ( 1.155) for elastic/perfectly-plastic materials by up to 3%. Also, Tables 1–2 show that the pressure at the vertex interface Pv is about 0.51 to 0.87 times the corner cavitation pressure PcV . These conclusions on the influence of the corner zone are supported by a full analytical corner zone analysis for elastic/perfectly-plastic media in Masri (2007) and here we bring only the approximate (Σy  1) condition for the existence of a corner zone and the approximate solution for the cavitation pressure, respectively, (3 − β)(1 + β)e1/β <

2 , Σy

PcV = PcT −

(1 − 2ν)2 −1/(1−2ν) Σy , e 2 4(1 − ν 2 )

(70)

with PcT given by (46). 7. Concluding remarks In this paper a detailed analysis of quasi-static cylindrical cavitation in compressible Mises and Tresca solids is presented. While the Mises analysis is approximate, and based on the axially-hydrostatic assumption with an error control parameter, an exact standard solution is derived for Tresca materials along with a corner zone analysis when a vertex interface exists. For the Mises model it turns out that available approximations, derived from a variety of simplifications, are just particular cases of a more general result. Based on the error control formulation, a highly accurate cavitation pressure expression is suggested. For the Tresca model it has been demonstrated, by accurate numerical calculations, that the standard cavitation pressure solution is slightly reduced in the presence of a corner zone solution for common metals but for incompressible media no corner zone is exist regardless of plastic hardening response. Simple expressions for elastic/perfectly-plastic solids (70), derived in Masri (2007), reveal that, while for an incompressible solid the standard solution is uniformly valid, a corner zone solution exists for common metals, and the slight difference in cavitation pressures (PcT − PcV ) decreases with decreasing compressibility and yield stress. Unlike spherical cavitation fields, which are identical for the two plasticity models, in cylindrical cavitation the Mises solid has a higher cavitation pressure and the location of the elastic/plastic interface is closer to the cavity’s wall. Material compressibility lowers the cavitation pressure and is more pronounced in the Mises material. However, cylindrical cavitation pressure is less sensitive to elastic compressibility than the spherical analogue. This study establishes cylindrical cavitation analysis on equal footing with existing studies for spherical cavitation. In fact, we can summarize the main findings of this study by the inequalities sp

PcV PcT < PcM Pc (κ = −0.4725) < Pc .

(71)

Acknowledgements One of us (D.D.) wishes to acknowledge the support of the Sydney Goldstein Chair in Aerospace Engineering. Part of this study was supported by the fund for the promotion of research at the Technion. References Bishop, R.F., Hill, R., Mott, N.F., 1945. The theory of indentation and hardness. Proc. Phys. Soc. 57, 147–159. Durban, D., Baruch, M., 1976. On the problem of a spherical cavity in an infinite elasto-plastic medium. J. Appl. Mech. 43, 633–638. Durban, D., 1979. Large strain solution for pressurized elasto/plastic tubes. J. Appl. Mech. 46, 228–230. Durban, D., Birman, V., 1982. On the elasto-plastic stress concentration at a circular hole in an anisotropic sheet. Acta Mech. 43, 73–84. Durban, D., 1988. Finite straining of pressurized compressible elasto-plastic tubes. Int. J. Engrg. Sci. 26, 939–950. Durban, D., Kubi, M., 1992. A general solution for the pressurized elastoplastic tube. J. Appl. Mech. 59, 20–26. Durban, D., Papanastasiou, P., 1997. Cylindrical cavity expansion and contraction in pressure sensitive geomaterials. Acta Mech. 122, 99–122. Hill, R., 1950. The Mathematical Theory of Plasticity. Oxford University Press, London. Masri, R., PhD Thesis, Technion, Israel, in preparation. Masri, R., Durban, D., 2005. Dynamic spherical cavity expansion in an elastoplastic compressible Mises solid. J. Appl. Mech. 72, 887–898. Masri, R., Durban, D., 2006. Quasi-static cylindrical cavity expansion in an elastoplastic compressible Mises solid. Int. J. Solids Struct. 43, 7518– 7533. Nadai, N., 1931. Plasticity. McGraw-Hill, New York.