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Plastic instabilities in porous cylinders under remote triaxial loading
European Journal of Mechanics A/Solids 37 (2013) 193e199
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European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol
Plastic instabilities in porous cylinders under remote triaxial loadingq Tal Cohen*, David Durban Faculty of Aerospace Engineering, Technion, Haifa 32000, Israel
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 March 2012 Accepted 13 June 2012 Available online 11 July 2012
Plastic instabilities under remote triaxial loading, including cylindrical cavity expansion, are studied in context of large strain plasticity and constrained plane-strain. Material hardening is taken into consideration and porosity is incorporated using the Gurson (1977) plasticity model. Spontaneous growth of central cavity due to constant applied triaxial remote load (cavitation instability) is examined in an infinite cylindrical medium. It is found that the remote field exhibits instability at a definite value of applied radial tension which may occur before onset of cavitation. That maximum is proposed as an approximation for field stability limit while, as shown, cavitation in the porous solid can occur only for relatively low values of initial porosity and within a limited range of remote loading triaxiality. Analysis is facilitated upon choice of effective stress as independent time-like parameter. Ó 2012 Elsevier Masson SAS. All rights reserved.
Keywords: Cavitation Finite strain Porous plasticity
1. Introduction Void growth and nucleation are commonly accepted as the main mechanisms in ductile fracture, as established experimentally in the pioneering study by Rogers (1960), revealing also that high triaxiality conditions accelerate void growth. Since then, there has been extensive experimental and analytical research on the relation between porosity and fracture. One of the earlier studies, by McClintock (1968), pursued a criterion for ductile fracture by growth of holes. Analyzing the plane-strain behavior of a representative cylindrical cell, McClintock (1968) suggested that fracture will occur if the cylindrical cavity grows beyond cell boundaries. Rice and Tracey (1969) examined a similar problem with a spherical void in an infinite medium. Those studies confirmed Rogers (1960) observation that stress triaxiality has a profound effect on void growth rates. An important study, also motivated by ductile fracture phenomena, given by Gurson (1977), reviews experimental data that supports the role of void growth in ductile fracture. Gurson (1977) developed yield criteria with associated flow rule for porous ductile materials, via upper bound approximation of deformation rate of a representative unit cube with void volume fraction identical to that of the aggregate. In the present paper we deal with the border line case of cavity expansion where remote constant load promotes spontaneous growth of an embedded void. Nucleation and coalescence of voids
q This work is based on part of a PhD Thesis to be submitted to the Technion. * Corresponding author. Tel.: þ972 4 829 3819; fax: þ972 4 829 2030. E-mail address: [email protected] (T. Cohen). 0997-7538/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.euromechsol.2012.06.005
are not incorporated in the material response. It is conceivable that the load required to induce such cavitation serves as an upper bound for ductile fracture. Furthermore, under high triaxiality conditions, spontaneous growth of a single void may be a primary failure mechanism, as experimentally demonstrated by Ashby et al. (1989) in constrained metal wires. An interesting numerical reconstruction of those experiments was given by Tvergaard (2004). To study the effect of triaxiality in quasistatic cavitation we consider an axisymmetric infinite medium subjected to both radial and axial tension in the remote field with a centrally embedded cylindrical cavity. The main focus of the present study is to assess the role of initial porosity on cavitation, employing the Gurson (1977) model. A comprehensive review on cavitation in elastic solids was given by Horgan and Polignone (1995). Though most available literature is limited to spherical geometries, cylindrical cavitation in hyperelastic solids under triaxial conditions was studied by Cohen and Durban (2010), a short review of some experimental observations is also given therein. Recently, Lopez-Pamies et al. (2011) presented a new approach to incorporate pre-existing defects, such as voids, in cavitation of elastomeric solids allowing general 3D loading. Masri and Durban (2007) studied plane-strain cylindrical cavitation in compressible Mises and Tresca solids due to internal pressure at the cavity wall and included an extensive review of previous work. Cylindrical cavitation in pressure sensitive geomaterials subjected to internal pressure was studied by Durban and Papanastasiou (1997). Another review of cavitation instabilities in ductile metals was given by Tvergaard (2007). Cylindrical cavitation under remote triaxiality in incompressible J2 solids was examined by Tvergaard et al. (1992). While deformation theory
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T. Cohen, D. Durban / European Journal of Mechanics A/Solids 37 (2013) 193e199
predicted cavitation instability even when far field is in plastic range, no such cavitation has been found with flow theory. Tvergaard and Vadillo (2007) examined cavitation of a spherical cavity under high triaxiality conditions in porous solids revealing maximum stress values, followed by numerical instability in the cavity expansion process, as opposed to expected asymptotic behavior in quasistatic cavitation. Influence of interaction between neighboring voids on cavitation instabilities is studied in a recent paper by Tvergaard (2012). The present study deals with cavitation of a cylindrical cavity under triaxial conditions using an analytical approach to better understand the failure mechanism which develops. The paper begins with brief formulation of field equations and constitutive response followed by investigation of remote field boundary conditions. Limitations on the stability of the remote field and thus on the entire cylindrical medium are found. Cavitation fields are then studied though restricted to remote elastic conditions. It is established that the remote field stability limit serves as an accurate measure of entire field stability.
1981; Nahson and Hutchinson, 2008) are not considered here, however analysis can easily be generalized to include them as well. Proceeding with the standard plasticity theory principles, the associated elastoplastic constitutive law can be cast in the form
D ¼
g2 ¼ 1 2f cosh q þ f 2
(1)
with
q¼
s 3sh 1 3 ; g ¼ ; sh ¼ I,,s; s2 ¼ S,,S; S ¼ s sh I se 2se 3 2
(2)
following the notation in Cohen and Durban (in press) where f is the void volume fraction (porosity), se e the effective stress, sh e the macroscopic hydrostatic stress, I e the second order unit tensor, s e the Mises stress, S e the stress deviator and s is the stress tensor. Notice that g reflects the deviation of effective stress from the Mises stress. Available modifications of the Gurson (1977) model (Tvergaard,
(3)
where D denotes the Eulerian strain rate, (E, G, n) are the usual 7 elastic constants, s denotes the Jaumann stress rate and the superposed dot represents differentiation with respect to a monotonous time-like parameter. The effective plastic strain (of matrix) 3 p is a known function of se determined experimentally. Specifying flow theory constitutive equation (3), for constrained plane-strain axially-symmetric cylindrical fields, reduces the tensorial equation to just three scalar relations 3_ r
_r n S _ z þ zr h3_ p _q þS ¼ S
(4)
3_ q
_ z þ zq h3_ p _r þS ¼ S_ q n S
(5)
3_ z
_z n S _ q þ zz h3_ p ¼ 0 _r þS ¼ S
(6)
2. Problem formulation and constitutive response Consider spontaneous axially-symmetric expansion of a cylindrical cavity in a radially unbounded porous solid due to applied remote radial tension with completely constrained axial motion. The active Cauchy stress components are (sr, sq, sz) using conventional notation. The deformation path assumed is that of an unbounded cylindrical medium loaded simultaneously by remote radial stress sr ¼ t and axial stress sz ¼ s. These stresses increase in a fixed proportional ratio (t/s ¼ r as r / N). However, while the remote radial stress remains uniform, the axial stress component admits a radial profile that assures a generalized plane-strain response. No distortion of cross-sections perpendicular to the z axis is permissible, so deformation field is with axial homogeneity. We search for critical pairs of remote stress (tc, sc) which induce plane-strain cavitation characterized by spontaneous expansion of the cavity while remote stresses and strains remain constant. Once that cavitation instability sets in, a separate analysis is required for each zone. The inner, near cavity, field deforms in a self-similar steady-state pattern with radially decaying effect. In that framework we bypass the precavitation stress history with all field variables depending only on the effective stress. The remote field is obtained from the solution of the uniformly strained circular cylinder due to proportional loading by radial and axial stress. We expect that this uniform field describes the conditions far away from the expanding cavity. Loss of stability in the remote field provides a failure mechanism which competes with cavitation instability. That global instability, induced by material porosity, appears as the dominant failure mode apart from a definite range of load triaxiality where cavitation sets in first. The porous solid is modeled by the Gurson (1977) macroscopic yield surface
1 7 3n 1 3S=s þ f sinh qI s s_ h I þ ð1 f Þ_3 p 2 e E 2G 2 g þ f q sinh q
where logarithmic strain components are denoted by (3 r, 3 q, 3 z), the nondimensional stresses are defined by (Sr, Sq, Sz, Sh, S) ¼ (sr, sq, sz, sh, se)/E and
h¼
1f g2 þ f q sinh q
zi ¼
3 Si 1 þ f sinh q 2 S 3
i ¼ r; q; z
(7)
where Si are the nondimensional (with respect to the elastic modulus E) normal components of the stress deviator. The condition 3_ z ¼ 0 in equation (6) is required to maintain plane-strain response during cavitation. Adding equations (4)e(6) and inserting void growth rate
3 f_ ¼ ð1 f Þf ðsinh qÞh3_ p 2
(8)
as obtained from the first invariant of the plastic branch of equation (3), yields the useful connection for rate of dilation 3_ r
þ 3_ q þ 3_ z ¼ 3bS_ h þ
f_ ; 1f
b ¼ 1 2n
(9)
which is integrated to obtain the closed form holonomic relation
f ¼ 1 ð1 fo Þe3bSh ð3 r þ3 q þ3 z Þ
(10)
where fo is the initial stress free porosity. We assume that steady-state cavitation occurs at critical combinations of remote stressing (sr ¼ tc, sz ¼ sc) along a proportional pre-cavitation loading path tc ¼ rsc, with r reflecting level of external loading triaxiality. Assuming self similarity of cavitation field, transformation to effective stress S as single independent variable is permissible. Accordingly, we proceed with equations (4)e(6) and (8) under the understanding that the effective stress (S) by itself can serve as the time like parameter, namely
_ ¼ dðÞ ðÞ dS
(11)
Cavity expansion velocity is assumed sufficiently small to omit inertia terms from the radial equilibrium equation, thus
r
dSr þ ðSr Sq Þe3 r 3 q ¼ 0 dr
(12)
T. Cohen, D. Durban / European Journal of Mechanics A/Solids 37 (2013) 193e199
where r is the Lagrangian radial coordinate. Recalling however that the logarithmic strain components are compatible if
r
d3 q e3 r 3 q þ 1 ¼ 0 dr
(13)
we combine equations (12) and (13) to eliminate the radial coordinate, resulting in the differential relation
e3 q 3 r 1 dSr þ ðSq Sr Þd3 q ¼ 0
(14)
While differentials in equation (14) are spatial, they transform in self similar fields to derivatives with respect to S, thus we rewrite equation (14) as
_ r þ 2m3_ q ¼ 0 S
(15)
where the apparent nondimensional shear modulus m is defined by
m ¼
1 Sq Sr 2 e3 q 3 r 1
(16)
To sum up, upon eliminating porosity ratio f from formulation via equation (10), the governing system consists of four differential equations, (4)e(6) and (15) and one algebraic relation (1), with five unknowns (Sr, Sq, Sz, 3 r, 3 q) and one independent variable S. The constant longitudinal strain (3 z) is dictated by remote boundary data which will be discussed at the end of this section. To simplify the numerical solution it is possible to differentiate equation (1) and combine with relation (8) to obtain a fifth differential equation
1f zr S_ r þ zq S_ q þ zz S_ z ¼ 2 3h2 ðcosh q f Þf ðsinh qÞS_3 p 2h (17) Notice that in this formulation 3_ p ¼ d3 p =dSð ¼ E=ET 1Þ is a known function of S with ET denoting the tangent modulus. Though the formulation accounts for arbitrary hardening response, in the present study we use the modified Ludwik law
3p
8 <0 ¼ S 1=n : Sy S Sy
for
S < Sy
for
S Sy
(18)
n being the strain hardening index and Sy the nondimensional yield stress. A different yet analogous formulation of the quasistatic cylindrical plane-strain cavitation field for Mises solid was given by Masri and Durban (2007) with earlier reference to Durban and Fleck (1997). Therein, strain rate components were replaced with kinematic relations which depend on the nondimensional radial coordinate (x) and the nondimensional radial velocity (V). In steady-state cavity expansion x can serve as independent variable, hence the transformation $
ðÞ ¼
_ A dðÞ ðV xÞ A dx
(19)
where A is the deformed cavity radius and A_ the constant cavity wall velocity, can be implemented to formulate rate relations like (4)e(6) resulting in a system of equations equivalent to the present system with strain rate components replaced as 3_ r
¼
dR_ ¼ dR
_ A dV ; A dx
3_ q
¼
_ A V R_ ¼ A x R
(20)
where R is the deformed radial coordinate. On balance, the present formulation is preferable in facilitating the numerical solution,
195
particularly in handling the strong gradient zone near the cavity. Also, choice of effective stress as independent variable, instead of x, enables investigation of remote zone (x / N) with finite values of S. Turning to boundary data, assuming that remote field is unaffected by cavity induced disturbance, conditions are as in a proportionally loaded uniform cylinder, with no internal cavity,
S ¼ SN :
Sr ¼ Sq ¼ rSc ;
Sz ¼ Sc ;
3r
¼ 3q ¼
3N
(21)
where Sc ¼ sc/E, Tc ¼ rSc ¼ tc/E, the subscript ()N refers to the corresponding value in the remote field and the stress ratio r is kept constant. Clearly, imposing remote field conditions (21) requires a separate solution for the proportional loading path, in generalized plane-strain, which will provide the dependence of Sc and 3 N on SN for given triaxiality r. That solution, determined in the next section assumes that cavity presence does not influence remote uniform field. It is hoped that critical pairs of constant applied remote load (Tc, Sc) that induce spontaneous cavity expansion can be exposed, with unbound values of effective stress (S) at the cavity wall where the radial stress vanishes
S/N :
Sr ¼ 0
(22)
In the remote field, expression for the apparent shear modulus (16) is determined by a simple limit procedure with S / SN and L’Hopital’s rule, resulting in the linear elastic relation
mN ¼
1 2ð1 þ nÞ
(23)
as expected from flow theory. It is worth mentioning in this context that with a deformation type theory mN would depend on SN. Inserting remote field conditions (21), together with (23), in equations (4)e(6), (15) and (17) reveals a singularity (vanishing of determinant of first order derivative coefficients) of the nonhomogenous system at the remote boundary. It follows that the assumption that remote field is unaffected by internal cavity is inconsistent. This observation can be explained by considering the quasistatic process as a border line dynamic process. In the dynamic process application of remote load induces a wave which propagates up to the cavity wall and is then reflected back to the remote field. In the steady-state quasistatic process the reflected wave will unite with the remote boundary implying that remote field is influenced by internal cavity. However, if remote field is elastic, singularity vanishes and it is possible to obtain a consistent solution with remote field conditions (21). Tvergaard et al. (1992) studied the incompressible cylindrical cavity expansion process in elastoplastic solids with J2 flow theory and reported difficulties in approaching cavitation state when remote conditions are plastic, thus supporting the present argument. Though it is possible to solve field equations with plastic remote conditions, this vastly complicates solution procedure. Accordingly, in this study we limit cavitation analysis to fields with remote elastic conditions and therefore to high triaxiality levels. Indeed, earlier solutions of cylindrical cavitation under remote tension are limited as well to high triaxiality conditions; Tvergaard and Vadillo (2007) limited their results to the range of 0.85 r 1 and the solution by Durban (1979) was limited to the incompressible plane-strain response in a Mises solid, implying that effective stress vanishes in the remote field, thus restricting analysis to elastic remote conditions. The limit value of applied remote load is deduced directly from yield condition (1) with f ¼ fo and SN ¼ Sy, given value of r,
ð1 rÞ2
S Sy
1 þ 2r S þ fo2 2 Sy
2 ¼ 1 2fo cosh
(24)
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T. Cohen, D. Durban / European Journal of Mechanics A/Solids 37 (2013) 193e199
Fig. 1. Applied radial tension vs effective stress in porous cylinder with no cavity (Sy ¼ 0.003, n ¼ 0.1, n ¼ 0.3).
Explicit solutions of this equation for S exist when r ¼ 1 and for the nonporous Mises solid with fo ¼ 0
S 2 ¼ lnðfo Þ Sy 3 2 S 1 fo ¼ 0/ ¼ Sy ð1 rÞ2
r ¼ 1/
(25)
3. The remote field In this section we analyze the pre-cavitation, proportionally stressed, homogenous axially-symmetric field which develops when subjecting a porous circular cylinder, with no cavity, to uniform radial and axial stresses (Sr ¼ Sq ¼ T, Sz ¼ S) increasing with a prescribed ratio r ¼ T/S. We shall see also that as a result of material porosity this uniform field can become unstable before cavitation sets in, thus providing the dominant failure mode. While not influenced by presence of expanding cavity, the remote field generates the deriving energy needed to sustain steady-state cavitation. Applied radial stress (T) is kept positive as discussion is limited to cavitation fields under remote radial tension. Since conditions in the remote field of a cylinder with an embedded cavity are expected to be identical, this uniform field provides data for remote field behavior in the cavitation problem. Here we distinguish between the preexisting cylindrical macroscopic cavity
in the cavitation problem and the microscopic voids which are incorporated in the constitutive response. The uniform stress field relations are therefore
Sh ¼
1 þ 2r 1 þ 2r S d; g2 ¼ ð1 rÞ2 d2 ; d ¼ S; q ¼ S 3 2
(26)
While constitutive equations (4)e(6) remain unchanged, all field variables vary monotonously with applied load and differentiation with respect to effective stress (S) is equivalent to differentiation with respect to a time like parameter along loading path. Specifying equations (4)e(6) under these remote field conditions we find that the first two are identical, thus verifying the equality 3 r ¼ 3 q ¼ 3 N. We write the remaining independent equations as 3_ N 3_ z
¼ ðr nð1 þ rÞÞS_ þ zr h3_ p ¼ ð1 2nrÞS_ þ zz h3_ p
(27) (28)
here 3_ z s0 since 3 z varies along loading path. The two differential equations (27) and (28) together with yield surface relation (1) and holonomic hydrostat (10) provide the required system of equations needed to determine the dependence of field variables (3 N, 3 z, S, f) on the applied load (S). Note that equilibrium equation (15) is fulfilled identically by uniformity of axially symmetric field.
Fig. 2. Maximum applied stress for varying values of stress ratio r and initial porosity fo (Sy ¼ 0.003, n ¼ 0.1, n ¼ 0.3). Spherical field maximum is marked by at r ¼ 1.
T. Cohen, D. Durban / European Journal of Mechanics A/Solids 37 (2013) 193e199
197
a
b
Fig. 3. Maximum applied stress for varying values of stress ratio r and (a) hardening index n, (b) yield stress Sy. Parent material fo ¼ 0.1, Sy ¼ 0.003, n ¼ 0.1, n ¼ 0.3. Spherical field maximum is marked by at r ¼ 1.
Integration begins with zero applied load (T ¼ S ¼ 0) where (3 N, S ¼ 0, f ¼ fo) and progresses with increasing effective stress. For S < Sy the field is elastic (h ¼ 1, f ¼ fo) and equations (27) and (28) are integrated analytically to obtain the linear relations
3 z,
3N
3z
¼ ðr nð1 þ rÞÞS ¼ ð1 2nrÞS
(29) (30)
For S Sy plasticity sets in and integration continues numerically. Dependence of applied radial tension on effective stress for various values of initial porosity ratio fo and stress ratio r is shown on Fig. 1. While the Mises solid with fo ¼ 0 does not exhibit unstable behavior, even the slightest initial porosity has a considerable effect on stability and maximum stress values are obtained. Notice that compression (r < 0) reduces levels of stress at instability. Though cavitation field solution is limited to remote elastic conditions, analysis here is beyond elastic limit to determine stability limit of entire field. A similar instability was encountered in the spherical hydrostatic tension field studied by Cohen and Durban (in press). That work revealed that a non-porous spherical Mises cell with central cavity exhibits cavitation instability at remote tension identical to the maximum load in hydrostatic expansion of the full porous sphere, implying that the instability which occurs in the porous media can be characterized as local cavitation of internal voids. It is emphasized that data displayed on Fig. 1 is for the uniform cylindrical field absence of central cavity. Maximum applied stress values, for varying stress ratio r, with investigation of sensitivity to initial porosity (fo), hardening index (n) and yield stress (Sy) are shown on Figs. 2 and 3. In both figures the corresponding spherical field maximum is labeled by () markers (at r ¼ 1) for comparison. It is observed that in all cases the spherical value is only slightly above the cylindrical field maximum at r ¼ 1. We also notice that near r ¼ 1 the positive branch of r (axial tension) exhibits a maximum value in the triaxial cylindrical field. As initial porosity decreases, that maximum increases and for the Mises solid with fo ¼ 0 no such maximum exists. At r ¼ 0 remote radial tension vanishes and the field reduces to the uniaxial stress field. In the negative branch of r no maximum values are obtained for 0.5 r < 0. To explain this result we return to relations (26) and find that the hydrostatic stress vanishes at r ¼ 0.5 (Sh ¼ q ¼ 0) and therefore porosity remains constant (f ¼ fo) along
the loading path, as observed from void growth relation (8). Now, returning to yield condition (1) we notice that in this case the ratio between axial pressure and effective stress is constant
2 2 S 2 ¼ 1 þ fo2 S 3
(31)
so no maximum value exists. Further examination for 0.5 < r < 0 where Sh ¼ q < 0 shows that porosity ratio decreases monotonically to zero as loading progresses, hence at high levels of tension a constant value of S/S is attained, as in the Mises field, and no instability is observed. Tvergaard and Vadillo (2007) calculated curves similar to those shown on Fig. 2 for the maximum obtained in the spherical cavity expansion process under high triaxiality conditions (figure 6 therein). Comparison (Fig. 4) shows that present results for maximum applied remote tension are slightly above the Tvergaard and Vadillo (2007) data. Apparently Tvergaard and Vadillo (2007) approached a similar material instability which explains the reported numerical difficulties near the maximum. In the absence of a maximum value in the non-porous solid (fo ¼ 0), the dashed line
Fig. 4. Maximum applied tension for varying values of stress ratio r and initial porosity fo. Dashed line represents cavitation stress in non-porous solid (Sy ¼ 0.003, n ¼ 0.1, n ¼ 0.3).
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T. Cohen, D. Durban / European Journal of Mechanics A/Solids 37 (2013) 193e199
Fig. 5. Maximum radial stress values vs stress ratio r for solid with fo ¼ 0.0003, 0.001 (Sy ¼ 0.003, n ¼ 0.1, n ¼ 0.3).
represents the corresponding cavitation stress. Cavitation will occur in the porous solid only if cavitation stress is below the maximum applied tension in the remote field. Notice the inversion between the dashed line (fo ¼ 0) and the curve for initial porosity (fo ¼ 0.001). While it is unlikely that the porous solid is more stable than the non-porous material, this inversion implies that a different instability will set in at stress values lower than the maximum remote stress for r > 0.92, fo ¼ 0.001. That instability is discussed in the next section. 4. Cavitation results The objective of cavity expansion field solution is to determine the critical level of load pairs (Tc, Sc) which induce quasi-static cavitation. Numerical solution is performed using a simple shooting method. Applied load in the remote field is assumed and once remote field values are obtained integration of field equations (4)e(6), (15) and (17) begins at the remote field with boundary conditions (21). Integration is continued until Sr ¼ 0 where effective stress should become infinite, as dictated by condition (22). This procedure is executed for gradually increasing values of applied remote load until all cavity boundary conditions are fulfilled. If iterative solution continues up to the maximum remote field value, yet with finite circumferential strain at cavity wall, then cavitation will not occur and remote field instability sets in. Notice that since central cavity grows in steady-state self similar pattern there is no specific inner radius (of cavity) that needs to be considered. Cavity expansion fields were analyzed for 0.0003 fo 0.3 but cavitation instability was found only for very low porosity levels and only within a definite range of stress ratio (r) due to remote field limitation. As an example, results for fo ¼ 0.001 and for fo ¼ 0.0003 are shown on Fig. 5 together with curves for: maximum applied tension in remote field; remote field elastic limit as deduced from equation (24); and cavitation stress in the nonporous medium. For 0.92 < r < 1.2 cavitation instability sets in while beyond this range the remote field maximum serves as
a limit of field stability. As expected, instability of the porous solid is manifested before that in the non-porous Mises field, progressively with increasing fo. Upshot is that remote field maximum serves as an accurate limit for field stability. 5. Concluding remarks Analysis of plastic instabilities under triaxial loading conditions in an infinite porous medium shows appreciable sensitivity to initial porosity, as studied in the context of large strain plasticity. Material hardening is taken into consideration and porosity is incorporated using the Gurson (1977) plasticity model. While cavitation instability can occur in absence of initial porosity, presence of even small material porosity may change the mode of instability, depending on triaxiality conditions with global failure as dominant mode. That instability is apparently due to local cavitation of internal voids in the remote field. Furthermore, it is found that cavitation instability can occur only at relatively low values of initial porosity and under high triaxiality loading conditions. Implying that maximum value of applied load in the homogeneous field can serve as a practical bound for field stability. This observation is in agreement with available FEM calculations. Thus, results obtained by extensive numerical study of entire field can be assessed with the aid of simple analysis of limit loads in the remote field. References Ashby, M., Blunt, F., Bannister, M., 1989. Flow characteristics of highly constrained metal wires. Acta Metall. 37, 1847e1857. Cohen, T., Durban, D., 2010. Cavitation in elastic and hyperelastic sheets. Int. J. Eng. Sci. 48, 52e66. Cohen, T., Durban, D. Hypervelocity cavity expansion in porous elastoplastic solids. J. Appl. Mech., in press. Durban, D., 1979. Large strain solution for pressurized elasto/plastic tubes. J. Appl. Mech. 46, 228e230. Durban, D., Fleck, N.A., 1997. Spherical cavity expansion in a drucker-prager solid. J. Appl. Mech. 64, 743e750. Durban, D., Papanastasiou, P., 1997. Cylindrical cavity expansion and contraction in pressure sensitive geomaterials. Acta Mech. 122, 99e122.
T. Cohen, D. Durban / European Journal of Mechanics A/Solids 37 (2013) 193e199 Gurson, A.,1977. Continuum theory of ductile rapture by void nucleation and growth: part I e yield criteria and flow rules for porous ductile media. J. Eng. Mater. Tech. 99, 2e15. Horgan, C., Polignone, D., 1995. Cavitation in nonlinearly elastic solids: a review. Appl. Mech. Rev. 48, 471e485. Lopez-Pamies, O., Idiart, M., Nakamura, T., 2011. Cavitation in elastomeric solids: IA defect-growth theory. J. Mech. Phys. Solids 59, 1464e1487. Masri, R., Durban, D., 2007. Cylindrical cavity expansion in compressible mises and tresca solids. Eur. J. Mech. Solids 26, 712e727. McClintock, F., 1968. A criterion of ductile fracture by the growth of holes. J. Appl. Mech. 35, 363e371. Nahson, K., Hutchinson, J., 2008. Modification of the Gurson model for shear failure. Eur. J. Mech. Solids 27, 1e17. Rice, J., Tracey, D., 1969. On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids 17, 201e217.
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Rogers, H., 1960. The tensile fracture of ductile metals. Trans. Met. Soc. AIME 218, 498e506. Tvergaard, V., 1981. Influence of voids on shear band instabilities under plane strain conditions. Int. J. Fract. 17, 389e407. Tvergaard, V., 2004. Effect of residual stress on cavitation instabilities inconstrained metal wires. J. Appl. Mech. 71, 560e566. Tvergaard, V., 2007. Analysis of cavitation instabilities in ductile metals. Key Eng. Mater. 340e341, 49e57. Tvergaard, V., 2012. On cavitation instabilities with interacting voids. Eur. J. Mech. Solids 32, 52e58. Tvergaard, V., Huang, Y., Hutchinson, J., 1992. Cavitation instabilities in power hardening elasticeplastic solids. Eur. J. Mech. Solids 11, 215e231. Tvergaard, V., Vadillo, G., 2007. Influence of porosity on cavitation instability predictions for elasticeplastic solids. Int. J. Mech. Sci. 49, 210e216.
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European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol
Plastic instabilities in porous cylinders under remote triaxial loadingq Tal Cohen*, David Durban Faculty of Aerospace Engineering, Technion, Haifa 32000, Israel
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 March 2012 Accepted 13 June 2012 Available online 11 July 2012
Plastic instabilities under remote triaxial loading, including cylindrical cavity expansion, are studied in context of large strain plasticity and constrained plane-strain. Material hardening is taken into consideration and porosity is incorporated using the Gurson (1977) plasticity model. Spontaneous growth of central cavity due to constant applied triaxial remote load (cavitation instability) is examined in an infinite cylindrical medium. It is found that the remote field exhibits instability at a definite value of applied radial tension which may occur before onset of cavitation. That maximum is proposed as an approximation for field stability limit while, as shown, cavitation in the porous solid can occur only for relatively low values of initial porosity and within a limited range of remote loading triaxiality. Analysis is facilitated upon choice of effective stress as independent time-like parameter. Ó 2012 Elsevier Masson SAS. All rights reserved.
Keywords: Cavitation Finite strain Porous plasticity
1. Introduction Void growth and nucleation are commonly accepted as the main mechanisms in ductile fracture, as established experimentally in the pioneering study by Rogers (1960), revealing also that high triaxiality conditions accelerate void growth. Since then, there has been extensive experimental and analytical research on the relation between porosity and fracture. One of the earlier studies, by McClintock (1968), pursued a criterion for ductile fracture by growth of holes. Analyzing the plane-strain behavior of a representative cylindrical cell, McClintock (1968) suggested that fracture will occur if the cylindrical cavity grows beyond cell boundaries. Rice and Tracey (1969) examined a similar problem with a spherical void in an infinite medium. Those studies confirmed Rogers (1960) observation that stress triaxiality has a profound effect on void growth rates. An important study, also motivated by ductile fracture phenomena, given by Gurson (1977), reviews experimental data that supports the role of void growth in ductile fracture. Gurson (1977) developed yield criteria with associated flow rule for porous ductile materials, via upper bound approximation of deformation rate of a representative unit cube with void volume fraction identical to that of the aggregate. In the present paper we deal with the border line case of cavity expansion where remote constant load promotes spontaneous growth of an embedded void. Nucleation and coalescence of voids
q This work is based on part of a PhD Thesis to be submitted to the Technion. * Corresponding author. Tel.: þ972 4 829 3819; fax: þ972 4 829 2030. E-mail address: [email protected] (T. Cohen). 0997-7538/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.euromechsol.2012.06.005
are not incorporated in the material response. It is conceivable that the load required to induce such cavitation serves as an upper bound for ductile fracture. Furthermore, under high triaxiality conditions, spontaneous growth of a single void may be a primary failure mechanism, as experimentally demonstrated by Ashby et al. (1989) in constrained metal wires. An interesting numerical reconstruction of those experiments was given by Tvergaard (2004). To study the effect of triaxiality in quasistatic cavitation we consider an axisymmetric infinite medium subjected to both radial and axial tension in the remote field with a centrally embedded cylindrical cavity. The main focus of the present study is to assess the role of initial porosity on cavitation, employing the Gurson (1977) model. A comprehensive review on cavitation in elastic solids was given by Horgan and Polignone (1995). Though most available literature is limited to spherical geometries, cylindrical cavitation in hyperelastic solids under triaxial conditions was studied by Cohen and Durban (2010), a short review of some experimental observations is also given therein. Recently, Lopez-Pamies et al. (2011) presented a new approach to incorporate pre-existing defects, such as voids, in cavitation of elastomeric solids allowing general 3D loading. Masri and Durban (2007) studied plane-strain cylindrical cavitation in compressible Mises and Tresca solids due to internal pressure at the cavity wall and included an extensive review of previous work. Cylindrical cavitation in pressure sensitive geomaterials subjected to internal pressure was studied by Durban and Papanastasiou (1997). Another review of cavitation instabilities in ductile metals was given by Tvergaard (2007). Cylindrical cavitation under remote triaxiality in incompressible J2 solids was examined by Tvergaard et al. (1992). While deformation theory
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predicted cavitation instability even when far field is in plastic range, no such cavitation has been found with flow theory. Tvergaard and Vadillo (2007) examined cavitation of a spherical cavity under high triaxiality conditions in porous solids revealing maximum stress values, followed by numerical instability in the cavity expansion process, as opposed to expected asymptotic behavior in quasistatic cavitation. Influence of interaction between neighboring voids on cavitation instabilities is studied in a recent paper by Tvergaard (2012). The present study deals with cavitation of a cylindrical cavity under triaxial conditions using an analytical approach to better understand the failure mechanism which develops. The paper begins with brief formulation of field equations and constitutive response followed by investigation of remote field boundary conditions. Limitations on the stability of the remote field and thus on the entire cylindrical medium are found. Cavitation fields are then studied though restricted to remote elastic conditions. It is established that the remote field stability limit serves as an accurate measure of entire field stability.
1981; Nahson and Hutchinson, 2008) are not considered here, however analysis can easily be generalized to include them as well. Proceeding with the standard plasticity theory principles, the associated elastoplastic constitutive law can be cast in the form
D ¼
g2 ¼ 1 2f cosh q þ f 2
(1)
with
q¼
s 3sh 1 3 ; g ¼ ; sh ¼ I,,s; s2 ¼ S,,S; S ¼ s sh I se 2se 3 2
(2)
following the notation in Cohen and Durban (in press) where f is the void volume fraction (porosity), se e the effective stress, sh e the macroscopic hydrostatic stress, I e the second order unit tensor, s e the Mises stress, S e the stress deviator and s is the stress tensor. Notice that g reflects the deviation of effective stress from the Mises stress. Available modifications of the Gurson (1977) model (Tvergaard,
(3)
where D denotes the Eulerian strain rate, (E, G, n) are the usual 7 elastic constants, s denotes the Jaumann stress rate and the superposed dot represents differentiation with respect to a monotonous time-like parameter. The effective plastic strain (of matrix) 3 p is a known function of se determined experimentally. Specifying flow theory constitutive equation (3), for constrained plane-strain axially-symmetric cylindrical fields, reduces the tensorial equation to just three scalar relations 3_ r
_r n S _ z þ zr h3_ p _q þS ¼ S
(4)
3_ q
_ z þ zq h3_ p _r þS ¼ S_ q n S
(5)
3_ z
_z n S _ q þ zz h3_ p ¼ 0 _r þS ¼ S
(6)
2. Problem formulation and constitutive response Consider spontaneous axially-symmetric expansion of a cylindrical cavity in a radially unbounded porous solid due to applied remote radial tension with completely constrained axial motion. The active Cauchy stress components are (sr, sq, sz) using conventional notation. The deformation path assumed is that of an unbounded cylindrical medium loaded simultaneously by remote radial stress sr ¼ t and axial stress sz ¼ s. These stresses increase in a fixed proportional ratio (t/s ¼ r as r / N). However, while the remote radial stress remains uniform, the axial stress component admits a radial profile that assures a generalized plane-strain response. No distortion of cross-sections perpendicular to the z axis is permissible, so deformation field is with axial homogeneity. We search for critical pairs of remote stress (tc, sc) which induce plane-strain cavitation characterized by spontaneous expansion of the cavity while remote stresses and strains remain constant. Once that cavitation instability sets in, a separate analysis is required for each zone. The inner, near cavity, field deforms in a self-similar steady-state pattern with radially decaying effect. In that framework we bypass the precavitation stress history with all field variables depending only on the effective stress. The remote field is obtained from the solution of the uniformly strained circular cylinder due to proportional loading by radial and axial stress. We expect that this uniform field describes the conditions far away from the expanding cavity. Loss of stability in the remote field provides a failure mechanism which competes with cavitation instability. That global instability, induced by material porosity, appears as the dominant failure mode apart from a definite range of load triaxiality where cavitation sets in first. The porous solid is modeled by the Gurson (1977) macroscopic yield surface
1 7 3n 1 3S=s þ f sinh qI s s_ h I þ ð1 f Þ_3 p 2 e E 2G 2 g þ f q sinh q
where logarithmic strain components are denoted by (3 r, 3 q, 3 z), the nondimensional stresses are defined by (Sr, Sq, Sz, Sh, S) ¼ (sr, sq, sz, sh, se)/E and
h¼
1f g2 þ f q sinh q
zi ¼
3 Si 1 þ f sinh q 2 S 3
i ¼ r; q; z
(7)
where Si are the nondimensional (with respect to the elastic modulus E) normal components of the stress deviator. The condition 3_ z ¼ 0 in equation (6) is required to maintain plane-strain response during cavitation. Adding equations (4)e(6) and inserting void growth rate
3 f_ ¼ ð1 f Þf ðsinh qÞh3_ p 2
(8)
as obtained from the first invariant of the plastic branch of equation (3), yields the useful connection for rate of dilation 3_ r
þ 3_ q þ 3_ z ¼ 3bS_ h þ
f_ ; 1f
b ¼ 1 2n
(9)
which is integrated to obtain the closed form holonomic relation
f ¼ 1 ð1 fo Þe3bSh ð3 r þ3 q þ3 z Þ
(10)
where fo is the initial stress free porosity. We assume that steady-state cavitation occurs at critical combinations of remote stressing (sr ¼ tc, sz ¼ sc) along a proportional pre-cavitation loading path tc ¼ rsc, with r reflecting level of external loading triaxiality. Assuming self similarity of cavitation field, transformation to effective stress S as single independent variable is permissible. Accordingly, we proceed with equations (4)e(6) and (8) under the understanding that the effective stress (S) by itself can serve as the time like parameter, namely
_ ¼ dðÞ ðÞ dS
(11)
Cavity expansion velocity is assumed sufficiently small to omit inertia terms from the radial equilibrium equation, thus
r
dSr þ ðSr Sq Þe3 r 3 q ¼ 0 dr
(12)
T. Cohen, D. Durban / European Journal of Mechanics A/Solids 37 (2013) 193e199
where r is the Lagrangian radial coordinate. Recalling however that the logarithmic strain components are compatible if
r
d3 q e3 r 3 q þ 1 ¼ 0 dr
(13)
we combine equations (12) and (13) to eliminate the radial coordinate, resulting in the differential relation
e3 q 3 r 1 dSr þ ðSq Sr Þd3 q ¼ 0
(14)
While differentials in equation (14) are spatial, they transform in self similar fields to derivatives with respect to S, thus we rewrite equation (14) as
_ r þ 2m3_ q ¼ 0 S
(15)
where the apparent nondimensional shear modulus m is defined by
m ¼
1 Sq Sr 2 e3 q 3 r 1
(16)
To sum up, upon eliminating porosity ratio f from formulation via equation (10), the governing system consists of four differential equations, (4)e(6) and (15) and one algebraic relation (1), with five unknowns (Sr, Sq, Sz, 3 r, 3 q) and one independent variable S. The constant longitudinal strain (3 z) is dictated by remote boundary data which will be discussed at the end of this section. To simplify the numerical solution it is possible to differentiate equation (1) and combine with relation (8) to obtain a fifth differential equation
1f zr S_ r þ zq S_ q þ zz S_ z ¼ 2 3h2 ðcosh q f Þf ðsinh qÞS_3 p 2h (17) Notice that in this formulation 3_ p ¼ d3 p =dSð ¼ E=ET 1Þ is a known function of S with ET denoting the tangent modulus. Though the formulation accounts for arbitrary hardening response, in the present study we use the modified Ludwik law
3p
8 <0 ¼ S 1=n : Sy S Sy
for
S < Sy
for
S Sy
(18)
n being the strain hardening index and Sy the nondimensional yield stress. A different yet analogous formulation of the quasistatic cylindrical plane-strain cavitation field for Mises solid was given by Masri and Durban (2007) with earlier reference to Durban and Fleck (1997). Therein, strain rate components were replaced with kinematic relations which depend on the nondimensional radial coordinate (x) and the nondimensional radial velocity (V). In steady-state cavity expansion x can serve as independent variable, hence the transformation $
ðÞ ¼
_ A dðÞ ðV xÞ A dx
(19)
where A is the deformed cavity radius and A_ the constant cavity wall velocity, can be implemented to formulate rate relations like (4)e(6) resulting in a system of equations equivalent to the present system with strain rate components replaced as 3_ r
¼
dR_ ¼ dR
_ A dV ; A dx
3_ q
¼
_ A V R_ ¼ A x R
(20)
where R is the deformed radial coordinate. On balance, the present formulation is preferable in facilitating the numerical solution,
195
particularly in handling the strong gradient zone near the cavity. Also, choice of effective stress as independent variable, instead of x, enables investigation of remote zone (x / N) with finite values of S. Turning to boundary data, assuming that remote field is unaffected by cavity induced disturbance, conditions are as in a proportionally loaded uniform cylinder, with no internal cavity,
S ¼ SN :
Sr ¼ Sq ¼ rSc ;
Sz ¼ Sc ;
3r
¼ 3q ¼
3N
(21)
where Sc ¼ sc/E, Tc ¼ rSc ¼ tc/E, the subscript ()N refers to the corresponding value in the remote field and the stress ratio r is kept constant. Clearly, imposing remote field conditions (21) requires a separate solution for the proportional loading path, in generalized plane-strain, which will provide the dependence of Sc and 3 N on SN for given triaxiality r. That solution, determined in the next section assumes that cavity presence does not influence remote uniform field. It is hoped that critical pairs of constant applied remote load (Tc, Sc) that induce spontaneous cavity expansion can be exposed, with unbound values of effective stress (S) at the cavity wall where the radial stress vanishes
S/N :
Sr ¼ 0
(22)
In the remote field, expression for the apparent shear modulus (16) is determined by a simple limit procedure with S / SN and L’Hopital’s rule, resulting in the linear elastic relation
mN ¼
1 2ð1 þ nÞ
(23)
as expected from flow theory. It is worth mentioning in this context that with a deformation type theory mN would depend on SN. Inserting remote field conditions (21), together with (23), in equations (4)e(6), (15) and (17) reveals a singularity (vanishing of determinant of first order derivative coefficients) of the nonhomogenous system at the remote boundary. It follows that the assumption that remote field is unaffected by internal cavity is inconsistent. This observation can be explained by considering the quasistatic process as a border line dynamic process. In the dynamic process application of remote load induces a wave which propagates up to the cavity wall and is then reflected back to the remote field. In the steady-state quasistatic process the reflected wave will unite with the remote boundary implying that remote field is influenced by internal cavity. However, if remote field is elastic, singularity vanishes and it is possible to obtain a consistent solution with remote field conditions (21). Tvergaard et al. (1992) studied the incompressible cylindrical cavity expansion process in elastoplastic solids with J2 flow theory and reported difficulties in approaching cavitation state when remote conditions are plastic, thus supporting the present argument. Though it is possible to solve field equations with plastic remote conditions, this vastly complicates solution procedure. Accordingly, in this study we limit cavitation analysis to fields with remote elastic conditions and therefore to high triaxiality levels. Indeed, earlier solutions of cylindrical cavitation under remote tension are limited as well to high triaxiality conditions; Tvergaard and Vadillo (2007) limited their results to the range of 0.85 r 1 and the solution by Durban (1979) was limited to the incompressible plane-strain response in a Mises solid, implying that effective stress vanishes in the remote field, thus restricting analysis to elastic remote conditions. The limit value of applied remote load is deduced directly from yield condition (1) with f ¼ fo and SN ¼ Sy, given value of r,
ð1 rÞ2
S Sy
1 þ 2r S þ fo2 2 Sy
2 ¼ 1 2fo cosh
(24)
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Fig. 1. Applied radial tension vs effective stress in porous cylinder with no cavity (Sy ¼ 0.003, n ¼ 0.1, n ¼ 0.3).
Explicit solutions of this equation for S exist when r ¼ 1 and for the nonporous Mises solid with fo ¼ 0
S 2 ¼ lnðfo Þ Sy 3 2 S 1 fo ¼ 0/ ¼ Sy ð1 rÞ2
r ¼ 1/
(25)
3. The remote field In this section we analyze the pre-cavitation, proportionally stressed, homogenous axially-symmetric field which develops when subjecting a porous circular cylinder, with no cavity, to uniform radial and axial stresses (Sr ¼ Sq ¼ T, Sz ¼ S) increasing with a prescribed ratio r ¼ T/S. We shall see also that as a result of material porosity this uniform field can become unstable before cavitation sets in, thus providing the dominant failure mode. While not influenced by presence of expanding cavity, the remote field generates the deriving energy needed to sustain steady-state cavitation. Applied radial stress (T) is kept positive as discussion is limited to cavitation fields under remote radial tension. Since conditions in the remote field of a cylinder with an embedded cavity are expected to be identical, this uniform field provides data for remote field behavior in the cavitation problem. Here we distinguish between the preexisting cylindrical macroscopic cavity
in the cavitation problem and the microscopic voids which are incorporated in the constitutive response. The uniform stress field relations are therefore
Sh ¼
1 þ 2r 1 þ 2r S d; g2 ¼ ð1 rÞ2 d2 ; d ¼ S; q ¼ S 3 2
(26)
While constitutive equations (4)e(6) remain unchanged, all field variables vary monotonously with applied load and differentiation with respect to effective stress (S) is equivalent to differentiation with respect to a time like parameter along loading path. Specifying equations (4)e(6) under these remote field conditions we find that the first two are identical, thus verifying the equality 3 r ¼ 3 q ¼ 3 N. We write the remaining independent equations as 3_ N 3_ z
¼ ðr nð1 þ rÞÞS_ þ zr h3_ p ¼ ð1 2nrÞS_ þ zz h3_ p
(27) (28)
here 3_ z s0 since 3 z varies along loading path. The two differential equations (27) and (28) together with yield surface relation (1) and holonomic hydrostat (10) provide the required system of equations needed to determine the dependence of field variables (3 N, 3 z, S, f) on the applied load (S). Note that equilibrium equation (15) is fulfilled identically by uniformity of axially symmetric field.
Fig. 2. Maximum applied stress for varying values of stress ratio r and initial porosity fo (Sy ¼ 0.003, n ¼ 0.1, n ¼ 0.3). Spherical field maximum is marked by at r ¼ 1.
T. Cohen, D. Durban / European Journal of Mechanics A/Solids 37 (2013) 193e199
197
a
b
Fig. 3. Maximum applied stress for varying values of stress ratio r and (a) hardening index n, (b) yield stress Sy. Parent material fo ¼ 0.1, Sy ¼ 0.003, n ¼ 0.1, n ¼ 0.3. Spherical field maximum is marked by at r ¼ 1.
Integration begins with zero applied load (T ¼ S ¼ 0) where (3 N, S ¼ 0, f ¼ fo) and progresses with increasing effective stress. For S < Sy the field is elastic (h ¼ 1, f ¼ fo) and equations (27) and (28) are integrated analytically to obtain the linear relations
3 z,
3N
3z
¼ ðr nð1 þ rÞÞS ¼ ð1 2nrÞS
(29) (30)
For S Sy plasticity sets in and integration continues numerically. Dependence of applied radial tension on effective stress for various values of initial porosity ratio fo and stress ratio r is shown on Fig. 1. While the Mises solid with fo ¼ 0 does not exhibit unstable behavior, even the slightest initial porosity has a considerable effect on stability and maximum stress values are obtained. Notice that compression (r < 0) reduces levels of stress at instability. Though cavitation field solution is limited to remote elastic conditions, analysis here is beyond elastic limit to determine stability limit of entire field. A similar instability was encountered in the spherical hydrostatic tension field studied by Cohen and Durban (in press). That work revealed that a non-porous spherical Mises cell with central cavity exhibits cavitation instability at remote tension identical to the maximum load in hydrostatic expansion of the full porous sphere, implying that the instability which occurs in the porous media can be characterized as local cavitation of internal voids. It is emphasized that data displayed on Fig. 1 is for the uniform cylindrical field absence of central cavity. Maximum applied stress values, for varying stress ratio r, with investigation of sensitivity to initial porosity (fo), hardening index (n) and yield stress (Sy) are shown on Figs. 2 and 3. In both figures the corresponding spherical field maximum is labeled by () markers (at r ¼ 1) for comparison. It is observed that in all cases the spherical value is only slightly above the cylindrical field maximum at r ¼ 1. We also notice that near r ¼ 1 the positive branch of r (axial tension) exhibits a maximum value in the triaxial cylindrical field. As initial porosity decreases, that maximum increases and for the Mises solid with fo ¼ 0 no such maximum exists. At r ¼ 0 remote radial tension vanishes and the field reduces to the uniaxial stress field. In the negative branch of r no maximum values are obtained for 0.5 r < 0. To explain this result we return to relations (26) and find that the hydrostatic stress vanishes at r ¼ 0.5 (Sh ¼ q ¼ 0) and therefore porosity remains constant (f ¼ fo) along
the loading path, as observed from void growth relation (8). Now, returning to yield condition (1) we notice that in this case the ratio between axial pressure and effective stress is constant
2 2 S 2 ¼ 1 þ fo2 S 3
(31)
so no maximum value exists. Further examination for 0.5 < r < 0 where Sh ¼ q < 0 shows that porosity ratio decreases monotonically to zero as loading progresses, hence at high levels of tension a constant value of S/S is attained, as in the Mises field, and no instability is observed. Tvergaard and Vadillo (2007) calculated curves similar to those shown on Fig. 2 for the maximum obtained in the spherical cavity expansion process under high triaxiality conditions (figure 6 therein). Comparison (Fig. 4) shows that present results for maximum applied remote tension are slightly above the Tvergaard and Vadillo (2007) data. Apparently Tvergaard and Vadillo (2007) approached a similar material instability which explains the reported numerical difficulties near the maximum. In the absence of a maximum value in the non-porous solid (fo ¼ 0), the dashed line
Fig. 4. Maximum applied tension for varying values of stress ratio r and initial porosity fo. Dashed line represents cavitation stress in non-porous solid (Sy ¼ 0.003, n ¼ 0.1, n ¼ 0.3).
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T. Cohen, D. Durban / European Journal of Mechanics A/Solids 37 (2013) 193e199
Fig. 5. Maximum radial stress values vs stress ratio r for solid with fo ¼ 0.0003, 0.001 (Sy ¼ 0.003, n ¼ 0.1, n ¼ 0.3).
represents the corresponding cavitation stress. Cavitation will occur in the porous solid only if cavitation stress is below the maximum applied tension in the remote field. Notice the inversion between the dashed line (fo ¼ 0) and the curve for initial porosity (fo ¼ 0.001). While it is unlikely that the porous solid is more stable than the non-porous material, this inversion implies that a different instability will set in at stress values lower than the maximum remote stress for r > 0.92, fo ¼ 0.001. That instability is discussed in the next section. 4. Cavitation results The objective of cavity expansion field solution is to determine the critical level of load pairs (Tc, Sc) which induce quasi-static cavitation. Numerical solution is performed using a simple shooting method. Applied load in the remote field is assumed and once remote field values are obtained integration of field equations (4)e(6), (15) and (17) begins at the remote field with boundary conditions (21). Integration is continued until Sr ¼ 0 where effective stress should become infinite, as dictated by condition (22). This procedure is executed for gradually increasing values of applied remote load until all cavity boundary conditions are fulfilled. If iterative solution continues up to the maximum remote field value, yet with finite circumferential strain at cavity wall, then cavitation will not occur and remote field instability sets in. Notice that since central cavity grows in steady-state self similar pattern there is no specific inner radius (of cavity) that needs to be considered. Cavity expansion fields were analyzed for 0.0003 fo 0.3 but cavitation instability was found only for very low porosity levels and only within a definite range of stress ratio (r) due to remote field limitation. As an example, results for fo ¼ 0.001 and for fo ¼ 0.0003 are shown on Fig. 5 together with curves for: maximum applied tension in remote field; remote field elastic limit as deduced from equation (24); and cavitation stress in the nonporous medium. For 0.92 < r < 1.2 cavitation instability sets in while beyond this range the remote field maximum serves as
a limit of field stability. As expected, instability of the porous solid is manifested before that in the non-porous Mises field, progressively with increasing fo. Upshot is that remote field maximum serves as an accurate limit for field stability. 5. Concluding remarks Analysis of plastic instabilities under triaxial loading conditions in an infinite porous medium shows appreciable sensitivity to initial porosity, as studied in the context of large strain plasticity. Material hardening is taken into consideration and porosity is incorporated using the Gurson (1977) plasticity model. While cavitation instability can occur in absence of initial porosity, presence of even small material porosity may change the mode of instability, depending on triaxiality conditions with global failure as dominant mode. That instability is apparently due to local cavitation of internal voids in the remote field. Furthermore, it is found that cavitation instability can occur only at relatively low values of initial porosity and under high triaxiality loading conditions. Implying that maximum value of applied load in the homogeneous field can serve as a practical bound for field stability. This observation is in agreement with available FEM calculations. Thus, results obtained by extensive numerical study of entire field can be assessed with the aid of simple analysis of limit loads in the remote field. References Ashby, M., Blunt, F., Bannister, M., 1989. Flow characteristics of highly constrained metal wires. Acta Metall. 37, 1847e1857. Cohen, T., Durban, D., 2010. Cavitation in elastic and hyperelastic sheets. Int. J. Eng. Sci. 48, 52e66. Cohen, T., Durban, D. Hypervelocity cavity expansion in porous elastoplastic solids. J. Appl. Mech., in press. Durban, D., 1979. Large strain solution for pressurized elasto/plastic tubes. J. Appl. Mech. 46, 228e230. Durban, D., Fleck, N.A., 1997. Spherical cavity expansion in a drucker-prager solid. J. Appl. Mech. 64, 743e750. Durban, D., Papanastasiou, P., 1997. Cylindrical cavity expansion and contraction in pressure sensitive geomaterials. Acta Mech. 122, 99e122.
T. Cohen, D. Durban / European Journal of Mechanics A/Solids 37 (2013) 193e199 Gurson, A.,1977. Continuum theory of ductile rapture by void nucleation and growth: part I e yield criteria and flow rules for porous ductile media. J. Eng. Mater. Tech. 99, 2e15. Horgan, C., Polignone, D., 1995. Cavitation in nonlinearly elastic solids: a review. Appl. Mech. Rev. 48, 471e485. Lopez-Pamies, O., Idiart, M., Nakamura, T., 2011. Cavitation in elastomeric solids: IA defect-growth theory. J. Mech. Phys. Solids 59, 1464e1487. Masri, R., Durban, D., 2007. Cylindrical cavity expansion in compressible mises and tresca solids. Eur. J. Mech. Solids 26, 712e727. McClintock, F., 1968. A criterion of ductile fracture by the growth of holes. J. Appl. Mech. 35, 363e371. Nahson, K., Hutchinson, J., 2008. Modification of the Gurson model for shear failure. Eur. J. Mech. Solids 27, 1e17. Rice, J., Tracey, D., 1969. On the ductile enlargement of voids in triaxial stress fields. J. Mech. Phys. Solids 17, 201e217.
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Rogers, H., 1960. The tensile fracture of ductile metals. Trans. Met. Soc. AIME 218, 498e506. Tvergaard, V., 1981. Influence of voids on shear band instabilities under plane strain conditions. Int. J. Fract. 17, 389e407. Tvergaard, V., 2004. Effect of residual stress on cavitation instabilities inconstrained metal wires. J. Appl. Mech. 71, 560e566. Tvergaard, V., 2007. Analysis of cavitation instabilities in ductile metals. Key Eng. Mater. 340e341, 49e57. Tvergaard, V., 2012. On cavitation instabilities with interacting voids. Eur. J. Mech. Solids 32, 52e58. Tvergaard, V., Huang, Y., Hutchinson, J., 1992. Cavitation instabilities in power hardening elasticeplastic solids. Eur. J. Mech. Solids 11, 215e231. Tvergaard, V., Vadillo, G., 2007. Influence of porosity on cavitation instability predictions for elasticeplastic solids. Int. J. Mech. Sci. 49, 210e216.