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A model for dynamic ductile behavior applicable for arbitrary triaxialities
C. R. Acad. Sci. Paris, t. 328, Série II b, p. 381–386, 2000 Endommagement, fatigue, rupture/Damage, fatigue, rupture
A model for dynamic ductile behavior applicable for arbitrary triaxialities Jean-Baptiste LEBLOND a , Gilles ROY b a
Laboratoire de modélisation en mécanique, Université P. et M. Curie, 4, place Jussieu, 75252 Paris cedex 05, France b Laboratoire de mécanique et physique des matériaux, ENSMA, site du Futuroscope, BP 109, 86960 Futuroscope cedex, France E-mail: [email protected]; [email protected] (Reçu le 2 mars 2000, accepté le 6 mars 2000)
Abstract.
Previous models for ductile behavior incorporating dynamic effects were derived assuming the triaxiality to be very high. Such is indeed the case in experiments of impact of plates, but not in those of expansion of rings or shells. Here we propose a model for dynamic ductile behavior applicable for arbitrary triaxialities. The material is schematized as a porous, viscoplastic Norton medium. The essential approximation made consists in accounting for this acceleration arising from growth of the voids, but not for that arising from their change of shape. Within the framework of this approximation, the special case of a hollow cylinder loaded axisymmetrically in generalized plane strain is treated exactly. This special, analytic solution is used as a guide to propose a model for the practically more significant case of spherical cavities. This model is finally extended in a heuristic way to incorporate elasticity effects. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS ductile behavior / dynamic effects / viscoplastic material / spherical voids / void expansion
Un modèle de comportement ductile dynamique applicable pour des triaxialités arbitraires Résumé.
Les modèles précédents de comportement ductile incorporant les effets dynamiques ont été obtenus en supposant la triaxialité très élevée. C’est bien le cas dans les expériences d’impact de plaques, mais pas dans celles d’expansion d’anneaux ou de coques. Nous proposons ici un modèle de comportement ductile dynamique applicable pour des triaxialités arbitraires. Le matériau est schématisé comme un milieu viscoplastique de Norton poreux. L’approximation essentielle faite consiste à prendre en compte l’accélération due à l’expansion des vides, mais pas celle due à leur changement de forme. Dans le cadre de cette approximation, le cas particulier d’un cylindre creux chargé axisymétriquement en conditions de déformation plane généralisée est traité exactement. Cette solution analytique particulière est utilisée comme un guide pour proposer un modèle pour le cas, plus intéressant en pratique, de cavités sphériques. Ce modèle est finalement étendu de manière heuristique de façon à incorporer les effets élastiques. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS comportement ductile / effets dynamiques / matériau viscoplastique / cavités sphériques / expansion des vides
Note présentée par Jean-Baptiste L EBLOND. S1620-7742(00)00048-9/FLA 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
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J.-B. Leblond, G. Roy
Version française abrégée Les modèles de rupture ductile dynamique proposés jusqu’à présent, et notamment celui, classique, de Johnson [5] ne sont très généralement applicables que pour des états de contraintes purement hydrostatiques (triaxialité infinie). Pourtant, on rencontre souvent en pratique des triaxialités modérées, notamment dans les expériences d’expansion d’anneaux ou de coques. Le seul modèle connu des auteurs visant à couvrir ces cas est celui de Cortes [11]. Mais ce modèle souffre de deux défauts : premièrement, l’hypothèse faite d’un comportement visqueux linéaire est souvent irréaliste en pratique ; deuxièmement, il ne fournit que l’expression de la partie moyenne des contraintes et non celle de leur partie déviatorique. On propose dans cette Note un modèle applicable lui aussi à des triaxialités arbitraires mais ne présentant pas ces défauts. Une brève revue de la littérature conduit à la conclusion qu’il est suffisant de schématiser la matrice métallique comme un milieu visqueux de Norton sans seuil : équation (1). L’approximation fondamentale faite consiste à prendre en compte la seule accélération due à la croissance des vides, celle due à leur changement de forme étant négligée. Deux arguments peuvent être avancés en faveur de cette hypothèse : premièrement, les observations post mortem de spécimens fracturés font apparaître un grossissement des cavités beaucoup plus important que leur changement de forme ; deuxièmement, il ne serait pas logique de tenir compte de l’accélération due au changement de forme dans la mesure où l’on n’incorpore pas l’effet de la forme des cavités elle-même dans le modèle (développé uniquement pour des vides sphériques). On envisage d’abord le problème-modèle d’un cylindre creux chargé axisymétriquement et dynamiquement en déformation plane généralisée. Dans le cadre de l’approximation fondamentale consentie (et pour une porosité faible), ce problème admet une solution analytique complète : équations (4) à (7). On constate sur cette solution que pour introduire les effets d’inertie dans la solution statique, il suffit d’ajouter à l’expression de la seule contrainte (Σxx ) responsable de la croissance de la cavité le « terme dynamique » correspondant au cas d’un chargement purement hydrostatique. On considère ensuite le cas de cavités sphériques. On donne d’abord la solution exacte du problème de la sphère creuse sous chargement hydrostatique et dynamique en (8) et (9). On donne ensuite une solution approchée du problème de la sphère creuse sous chargement arbitraire mais statique, de (10) à (12). Enfin, on étend ces deux solutions au cas d’un chargement arbitraire et dynamique en utilisant la remarque, supposée s’appliquer aussi au cas sphérique, faite à propos du cas cylindrique ((13), (9) et (11)). Enfin, on incorpore heuristiquement, suivant la démarche proposée en [5], l’élasticité au modèle, de manière à pouvoir rendre compte des effets de propagation d’ondes (par exemple dans les expériences d’impact de plaques). La partie moyenne de la loi d’élasticité suit la loi d’état non-linéaire de Mie– Grüneisen (18), rendue nécessaire par les très hautes pressions possibles, cependant que sa partie déviatorique est donnée par une loi d’hypoélasticité linéaire classique (19).
1. Introduction Gurson’s model [1] for the quasistatic behavior of porous ductile plastic solids has now gained wide acceptance. Several proposals, among which [2], have also been made for the case of viscoplastic solids. Such models or similar ones have been employed for instance by [3] and [4] to analyze dynamic rupture. However they were not meant for such use insofar as they disregard the influence of dynamic effects upon void growth. Several models for dynamic ductile behavior have been proposed for the case of purely hydrostatic stresses applied (infinite triaxiality), for instance the classical one of Johnson [5], based on the previous work of Carroll and Holt [6]. However, there are some cases, for instance in experiments of dynamic expansion of circular rings or spherical shells, where the triaxiality is far from infinite. One model, due to [7], has been proposed for dynamic ductile behavior under arbitrary stresses. However, this model suffers from
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at least two drawbacks: first, the sound (void-free) matrix is assumed to be linearly viscous, which is not quite a realistic assumption in many cases; second, it is incomplete since only the mean stress is given an analytical expression, no similar formula for the deviatoric one being proposed. The aim of the present paper is to propose a new model for dynamic ductile behavior applicable for all triaxialities but free of these defects. The question naturally arises of what features should be included in this model. Several authors, among whom Klöcker and Montheillet [8], Ortiz and Molinari [9], Tong and Ravichandran [10], have shown that the influence of inertia upon void growth can be important. Strain hardening has been concluded by [10] to be relatively unimportant; also, it should be noted that for the high strain rates we have in mind, the behavior of the sound matrix does not generally exhibit strong strain hardening. Thermal effects due to adiabatic heating arising from plastic dissipation are generally thought not to play an important role; see, e.g. [11]. These elements lead to the conclusion that the sound matrix may safely be schematized as a simple viscoplastic Norton material without threshold. This last simplifying assumption arises from the consideration that introduction of some threshold would be necessary only if low strain rates (leading to plastic, not viscoplastic, behavior) were of interest, which is not the case here. However, influence of dynamic effects upon void growth should be incorporated. The one fundamental approximation made in the derivation of our model is that whereas we account for these accelerations arising from growth of the cavities, we disregard those arising from their change of shape. Two arguments of different nature can be put forward to justify such an assumption. First, from a physical point of view, the enlargement of voids in fractured specimens observed post mortem is generally much more important than their change of shape. Second, from a purely logical point of view, it would make no sense to account for accelerations arising from that change of shape whereas no influence of the void shape itself is to be incorporated into the model (which will be developed only for spherical cavities). The paper is organized as follows. First, we treat the special case of a hollow cylinder loaded axisymmetrically in generalized plane strain; the treatment is exact within the framework of the fundamental approximation just mentioned (and provided that the porosity is small, see below). Then we consider the case of spherical cavities. A model is proposed for such voids, based on some exact or approximate solutions pertaining to two special cases involving a hollow sphere, and on the previous solution for a cylindrical void. Finally elasticity is introduced in a heuristic way in order to account for wave propagation phenomena (as encountered, e.g. in plate impact experiments). 2. The case of a hollow cylinder We consider a hollow cylinder of internal radius a, external radius b, porosity f ≡ a2 /b2 ; f will be assumed to be small (this hypothesis is made only for the sake of simplicity and could easily be removed if necessary, but is generally quite safe in practice). This cylinder is loaded in generalized plane strain conditions by some axisymmetric macroscopic stresses Σxx = Σyy , Σzz . The behavior of the matrix obeys Norton’s law: 3 σeq n−1 σ 0 d = (1) ε˙0 2 σ0 σ0 where d denotes the strain rate tensor, σ 0 the deviator of the stress tensor σ, σeq the von Mises equivalent stress and ε˙0 , σ0 , n material constants. Because of rotational symmetry, axial invariance and incompressibility of the matrix, the velocity field is of the form, in cylindrical coordinates: vr = A/r − Br/2,
vz = Bz,
where A = 32 b2 Dm ,
B = Dzz
(2)
In equations (2)1 and (2)2 , A and B are constants depending only on time and related to the mean (Dm ) and axial (Dzz ) parts of the macroscopic strain rate through equations (2)3 and (2)4 . The term proportional to A represents the expansion of the void whereas that proportional to B represents its change of shape.
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Thus our fundamental approximation will consist in neglecting those accelerations arising from the terms proportional to B. The acceleration field is then given by: ˙ − Avr /r2 ≈ A/r ˙ − A2 /r3 , γr ≈ A/r
γz ≈ 0
(3)
The approximation γz ≈ 0, which results from our fundamental hypothesis, is compatible with the assumed axial invariance insofar as the axial equation of dynamics, ∂σzr /∂r + σzr /r + ∂σzz /∂z = ργz (where ρ is the mass per unit volume), with σzr = 0 and σzz independent of z, precisely implies γz = 0. Integration of the radial equation of dynamics, ∂σrr /∂r + (σrr − σθθ )/r = ργr , using the flow rule (1), then yields upon a somewhat lengthy calculation: 1/n |Dzz | n 1 Σxx = √ (4) I(X) + XI 0 (X) + τ 2 Q f, f˙, f¨ σ0 ε˙0 2 3 n+1 √ Z 2X/f n+1 dx 3 Dm ; I(X) ≡ 2(1 + x2 ) 2n 2 (5) X≡ 2 |Dzz | x 2X ρa20 1 ¨ f˙2 2 ˙ ¨ (6) ; Q f, f , f ≡ ln f− τ ≡ 4σ0 f0 f 2f where f0 and a0 denote the initial porosity and void radius, respectively. Also, calculating Σzz − Σxx as 0 0 0 − (σrr + σθθ )/2, and using equation (1), one gets: the mean value of σzz − (σxx + σyy )/2 = σzz 1/n |Dzz | 1 Σzz − Σxx sgn(Dzz ) = X I(X) − nXI 0 (X) σ0 n+1 ε˙0
(7)
where sgn(Dzz ) denotes the sign of Dzz . Inertia effects are entirely confined within the term τ 2 Q(f, f˙, f¨) of the expression (4) of Σxx /σ0 ; note that if one sets it to zero, one recovers the exact static solution derived by Leblond et al. [2]. Note also that this term is independent of the loading so that it takes the same value as for a purely hydrostatic one. One thus sees that in order to introduce inertia effects into the static solution, one just needs to add, in the sole expression of that stress (Σxx ) governing void growth, the ‘dynamic term’ corresponding to some purely hydrostatic loading. 3. The case of spherical voids 3.1. Exact dynamic solution for a hollow sphere loaded hydrostatically We begin by deriving the exact dynamic solution for a hollow sphere of inner radius a, outer radius b, porosity f ≡ a3 /b3 1, subjected to some hydrostatic (positive) stress Σm and obeying Norton’s constitutive law (1). It is analogous to that provided by Johnson [5] except that here the viscous behavior is nonlinear. The velocity is purely radial and given by vr = A/r2 where A is a constant depending only on time and related to the mean macroscopic strain rate Dm by the relation A = b3 Dm . The ˙ 2 − 2A2 /r4 . Integrating the radial equation of dynamics, (radial) acceleration is given by γr = A/r ∂σrr /∂r + 2(σrr − σθθ )/r = ργr , one then gets: 2Dm 1/n Σm 2n −1/n f = −1 + τ 2 Q f, f˙, f¨ (8) σ0 3 ε˙0 2/3 (9) 3σ0 f0 ; Q f, f˙, f¨ ≡ f −1/3 − 1 f¨ − 16 f −4/3 f˙2 τ 2 ≡ ρa20 Note that the terms −1 in the expressions f −1/n − 1 (8) and f −1/3 − 1 (9) have been retained in spite of the hypothesis f 1, because f −1/n and f −1/3 may be close to unity in view of the smallness of the
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A model for dynamic ductile behavior
exponents −1/n and −1/3. Note also that in the static case (obtained by discarding the term τ 2 Q(f, f˙, f¨)), one recovers the (exact) solution provided by Wilkinson and Ashby [12]. 3.2. Approximate static solution for a hollow sphere under arbitrary loading Such a solution has been derived by Leblond et al. [2] for an axisymmetric loading. It is easy to extend it to an arbitrary loading by evaluating the approximate stress potential corresponding to the classical trial velocity field given by v = Dm b3 er /r2 + D 0 · x, where D0 denotes the deviator of the macroscopic strain rate. The results read as follows: 1/n Deq Σm 1 n ' (10) I(X) + XI 0 (X) σ0 3 n + 1 ε˙0 1/n 0 Deq D Σ0 2 1 ' X I(X) − nXI 0 (X) (11) σ0 3 n + 1 ε˙0 Deq Z 2X/f n+1 dx Dm ; I(X) ≡ 2 1 + x2 2n 2 (12) X≡ Deq x 2X where Σ 0 denotes the deviator of the macroscopic stress tensor and Deq the equivalent macroscopic strain rate. Note the formal analogy here with the solution (equations (4), (5) and (7)) for a hollow cylinder in the static case (τ 2 Q(f, f˙, f¨) ≡ 0). Also, it can be verified that equation (10) for Σm /σ0 does reduce to (8) (with τ 2 Q(f, f˙, f¨) ≡ 0) for a purely hydrostatic loading (Deq → 0). 3.3. Approximate dynamic solution for spherical voids under arbitrary loading In view of the difficulty of solving such a problem, even approximately, we resort to a simpler approach, based on the remark made at the end of Section 2 about the cylindrical case, which consists in assuming that as in that case, inertia effects can be introduced simply by adding to the static expression of the sole stress (Σm ) governing void growth, the dynamic term arising from a purely hydrostatic loading. The expression (10) of Σm /σ0 then becomes: 1/n Deq Σm 1 n ' I(X) + XI 0 (X) + τ 2 Q f, f˙, f¨ σ0 3 n + 1 ε˙0
(13)
where τ 2 and Q(f, f˙, f¨) are given by (9), whereas that (11) of Σ 0 /σ0 remains unchanged. 4. A dynamic model for spherical voids incorporating elasticity In the static case, elasticity generally plays a minor role. In the dynamic one, in contrast, it becomes essential to introduce it since it governs wave propagation, for instance in plate impact experiments. The relation connecting the mean stress and the mean elastic strain must be nonlinear in view of the extremely high pressures which can be encountered in practice. On the other hand, the stress deviator is much smaller so that a linear relation is sufficient to describe the deviatoric part of the elasticity law. We essentially follow here the same heuristic approach as in [5]. Thus, let v (≡ 1/ρ) denote the current specific volume and v0 its initial value in the reference, stress-free state. The variation v − v0 is assumed to consist of a reversible (elastic) part and an irreversible (plastic) part, and similarly for the strain rate D: v − v0 ≡ v e + v p ;
D ≡ De + Dp ,
(14)
the rates of v e and v p being connected to D e and Dp by the relations: v˙ e /v = tr De ;
v˙ p /v = tr Dp .
(15)
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Note that equations (14) and (15) are compatible with the classical relation v/v ˙ = tr D. The variation of porosity arising from elasticity being as usual neglected, f˙ is given by the classical formula: f˙ = (1 − f ) tr Dp
(16) p
The flow rule is supposed to be the same as before, D being replaced by D ; thus equations (13) and (11) become: p 1/n Deq Σm 1 n = I(X) + XI 0 (X) + τ 2 Q f, f˙, f¨ p Dm σ0 3 n + 1 ε˙0 (17) ; X ≡ 0 p 1/n p Deq Deq Dp Σ0 2 1 0 = p X I(X) − nXI (X) σ0 3 n + 1 ε˙0 Deq where I(X), τ 2 and Q(f, f˙, f¨) are still given by equations (12)2 and (9). With regard to the elasticity law, the relation between the mean stress (opposite of pressure) and the elastic part of the variation of specific volume is supposed to be given by the Mie–Grüneisen equation of state: Σm = K0
2 Γ0 ve Γ0 v e ve − E 1+ 1+s v0 2 v0 v0 v0
(18)
where K0 denotes the compressibility coefficient at zero pressure, Γ0 the Grüneisen coefficient in the same conditions, s a dimensionless constant and E the specific internal energy. Also, the deviatoric part of the (hypo)elasticity law is expressed as: ˇ 0 = 2G0 De0 Σ
(19)
where ˇ denotes the Jaumann derivative and G0 the shear modulus in the reference state. Finally, the evolution of E in equation (18) is given by the first principle of thermodynamics (in the adiabatic case): E˙ = v Σ : D
(20)
References [1] Gurson A.L., Continuum theory of ductile rupture by void nucleation and growth: Part I – Yield criteria and flow rules for porous ductile media, ASME J. Eng. Materials Technol. 99 (1977) 2–15. [2] Leblond J.-B., Perrin G., Suquet P., Exact results and approximate models for porous viscoplastic solids, Int. J. Plasticity 10 (1994) 213–235. [3] Needleman A., Tvergaard V., An analysis of dynamic, ductile crack growth in a double edge cracked specimen, Int. J. Fracture 49 (1991) 41–67. [4] Nemes J.A., Eftis J., Pressure-shear waves and spall fracture described by a viscoplastic-damage constitutive model, Int. J. Plasticity 8 (1992) 185–207. [5] Johnson J.N., Dynamic fracture and spallation in ductile solids, J. Appl. Phys. 52 (1981) 2812–2825. [6] Carroll M.M., Holt A.C., Static and dynamic pore collapse relations for ductile porous materials, J. Appl. Phys. 43 (1972) 1626–1636. [7] Cortes R., Dynamic growth of microvoids under combined hydrostatic and deviatoric stresses, Int. J. Solids Structures 29 (1992) 1637–1645. [8] Klöcker H., Montheillet F., Influence of flow rule and inertia on void growth in a rate sensitive material, J. Phys. IV C3 (1991) 733–738. [9] Ortiz M., Molinari A., Effect of strain hardening and rate sensitivity on the dynamic growth of a void in a plastic material, ASME J. Appl. Mech. 59 (1992) 48–53. [10] Tong W., Ravichandran G., Inertial effects on void growth in porous viscoplastic materials, ASME J. Appl. Mech. 62 (1995) 633–639. [11] Cortes R., The growth of microvoids under intense dynamic loading, Int. J. Solids Structures 29 (1992) 1339–1350. [12] Wilkinson D.S., Ashby M.F., Pressure sintering by power law creep, Acta Metall. 23 (1975) 1277–1285.
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A model for dynamic ductile behavior applicable for arbitrary triaxialities Jean-Baptiste LEBLOND a , Gilles ROY b a
Laboratoire de modélisation en mécanique, Université P. et M. Curie, 4, place Jussieu, 75252 Paris cedex 05, France b Laboratoire de mécanique et physique des matériaux, ENSMA, site du Futuroscope, BP 109, 86960 Futuroscope cedex, France E-mail: [email protected]; [email protected] (Reçu le 2 mars 2000, accepté le 6 mars 2000)
Abstract.
Previous models for ductile behavior incorporating dynamic effects were derived assuming the triaxiality to be very high. Such is indeed the case in experiments of impact of plates, but not in those of expansion of rings or shells. Here we propose a model for dynamic ductile behavior applicable for arbitrary triaxialities. The material is schematized as a porous, viscoplastic Norton medium. The essential approximation made consists in accounting for this acceleration arising from growth of the voids, but not for that arising from their change of shape. Within the framework of this approximation, the special case of a hollow cylinder loaded axisymmetrically in generalized plane strain is treated exactly. This special, analytic solution is used as a guide to propose a model for the practically more significant case of spherical cavities. This model is finally extended in a heuristic way to incorporate elasticity effects. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS ductile behavior / dynamic effects / viscoplastic material / spherical voids / void expansion
Un modèle de comportement ductile dynamique applicable pour des triaxialités arbitraires Résumé.
Les modèles précédents de comportement ductile incorporant les effets dynamiques ont été obtenus en supposant la triaxialité très élevée. C’est bien le cas dans les expériences d’impact de plaques, mais pas dans celles d’expansion d’anneaux ou de coques. Nous proposons ici un modèle de comportement ductile dynamique applicable pour des triaxialités arbitraires. Le matériau est schématisé comme un milieu viscoplastique de Norton poreux. L’approximation essentielle faite consiste à prendre en compte l’accélération due à l’expansion des vides, mais pas celle due à leur changement de forme. Dans le cadre de cette approximation, le cas particulier d’un cylindre creux chargé axisymétriquement en conditions de déformation plane généralisée est traité exactement. Cette solution analytique particulière est utilisée comme un guide pour proposer un modèle pour le cas, plus intéressant en pratique, de cavités sphériques. Ce modèle est finalement étendu de manière heuristique de façon à incorporer les effets élastiques. 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS comportement ductile / effets dynamiques / matériau viscoplastique / cavités sphériques / expansion des vides
Note présentée par Jean-Baptiste L EBLOND. S1620-7742(00)00048-9/FLA 2000 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
381
J.-B. Leblond, G. Roy
Version française abrégée Les modèles de rupture ductile dynamique proposés jusqu’à présent, et notamment celui, classique, de Johnson [5] ne sont très généralement applicables que pour des états de contraintes purement hydrostatiques (triaxialité infinie). Pourtant, on rencontre souvent en pratique des triaxialités modérées, notamment dans les expériences d’expansion d’anneaux ou de coques. Le seul modèle connu des auteurs visant à couvrir ces cas est celui de Cortes [11]. Mais ce modèle souffre de deux défauts : premièrement, l’hypothèse faite d’un comportement visqueux linéaire est souvent irréaliste en pratique ; deuxièmement, il ne fournit que l’expression de la partie moyenne des contraintes et non celle de leur partie déviatorique. On propose dans cette Note un modèle applicable lui aussi à des triaxialités arbitraires mais ne présentant pas ces défauts. Une brève revue de la littérature conduit à la conclusion qu’il est suffisant de schématiser la matrice métallique comme un milieu visqueux de Norton sans seuil : équation (1). L’approximation fondamentale faite consiste à prendre en compte la seule accélération due à la croissance des vides, celle due à leur changement de forme étant négligée. Deux arguments peuvent être avancés en faveur de cette hypothèse : premièrement, les observations post mortem de spécimens fracturés font apparaître un grossissement des cavités beaucoup plus important que leur changement de forme ; deuxièmement, il ne serait pas logique de tenir compte de l’accélération due au changement de forme dans la mesure où l’on n’incorpore pas l’effet de la forme des cavités elle-même dans le modèle (développé uniquement pour des vides sphériques). On envisage d’abord le problème-modèle d’un cylindre creux chargé axisymétriquement et dynamiquement en déformation plane généralisée. Dans le cadre de l’approximation fondamentale consentie (et pour une porosité faible), ce problème admet une solution analytique complète : équations (4) à (7). On constate sur cette solution que pour introduire les effets d’inertie dans la solution statique, il suffit d’ajouter à l’expression de la seule contrainte (Σxx ) responsable de la croissance de la cavité le « terme dynamique » correspondant au cas d’un chargement purement hydrostatique. On considère ensuite le cas de cavités sphériques. On donne d’abord la solution exacte du problème de la sphère creuse sous chargement hydrostatique et dynamique en (8) et (9). On donne ensuite une solution approchée du problème de la sphère creuse sous chargement arbitraire mais statique, de (10) à (12). Enfin, on étend ces deux solutions au cas d’un chargement arbitraire et dynamique en utilisant la remarque, supposée s’appliquer aussi au cas sphérique, faite à propos du cas cylindrique ((13), (9) et (11)). Enfin, on incorpore heuristiquement, suivant la démarche proposée en [5], l’élasticité au modèle, de manière à pouvoir rendre compte des effets de propagation d’ondes (par exemple dans les expériences d’impact de plaques). La partie moyenne de la loi d’élasticité suit la loi d’état non-linéaire de Mie– Grüneisen (18), rendue nécessaire par les très hautes pressions possibles, cependant que sa partie déviatorique est donnée par une loi d’hypoélasticité linéaire classique (19).
1. Introduction Gurson’s model [1] for the quasistatic behavior of porous ductile plastic solids has now gained wide acceptance. Several proposals, among which [2], have also been made for the case of viscoplastic solids. Such models or similar ones have been employed for instance by [3] and [4] to analyze dynamic rupture. However they were not meant for such use insofar as they disregard the influence of dynamic effects upon void growth. Several models for dynamic ductile behavior have been proposed for the case of purely hydrostatic stresses applied (infinite triaxiality), for instance the classical one of Johnson [5], based on the previous work of Carroll and Holt [6]. However, there are some cases, for instance in experiments of dynamic expansion of circular rings or spherical shells, where the triaxiality is far from infinite. One model, due to [7], has been proposed for dynamic ductile behavior under arbitrary stresses. However, this model suffers from
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A model for dynamic ductile behavior
at least two drawbacks: first, the sound (void-free) matrix is assumed to be linearly viscous, which is not quite a realistic assumption in many cases; second, it is incomplete since only the mean stress is given an analytical expression, no similar formula for the deviatoric one being proposed. The aim of the present paper is to propose a new model for dynamic ductile behavior applicable for all triaxialities but free of these defects. The question naturally arises of what features should be included in this model. Several authors, among whom Klöcker and Montheillet [8], Ortiz and Molinari [9], Tong and Ravichandran [10], have shown that the influence of inertia upon void growth can be important. Strain hardening has been concluded by [10] to be relatively unimportant; also, it should be noted that for the high strain rates we have in mind, the behavior of the sound matrix does not generally exhibit strong strain hardening. Thermal effects due to adiabatic heating arising from plastic dissipation are generally thought not to play an important role; see, e.g. [11]. These elements lead to the conclusion that the sound matrix may safely be schematized as a simple viscoplastic Norton material without threshold. This last simplifying assumption arises from the consideration that introduction of some threshold would be necessary only if low strain rates (leading to plastic, not viscoplastic, behavior) were of interest, which is not the case here. However, influence of dynamic effects upon void growth should be incorporated. The one fundamental approximation made in the derivation of our model is that whereas we account for these accelerations arising from growth of the cavities, we disregard those arising from their change of shape. Two arguments of different nature can be put forward to justify such an assumption. First, from a physical point of view, the enlargement of voids in fractured specimens observed post mortem is generally much more important than their change of shape. Second, from a purely logical point of view, it would make no sense to account for accelerations arising from that change of shape whereas no influence of the void shape itself is to be incorporated into the model (which will be developed only for spherical cavities). The paper is organized as follows. First, we treat the special case of a hollow cylinder loaded axisymmetrically in generalized plane strain; the treatment is exact within the framework of the fundamental approximation just mentioned (and provided that the porosity is small, see below). Then we consider the case of spherical cavities. A model is proposed for such voids, based on some exact or approximate solutions pertaining to two special cases involving a hollow sphere, and on the previous solution for a cylindrical void. Finally elasticity is introduced in a heuristic way in order to account for wave propagation phenomena (as encountered, e.g. in plate impact experiments). 2. The case of a hollow cylinder We consider a hollow cylinder of internal radius a, external radius b, porosity f ≡ a2 /b2 ; f will be assumed to be small (this hypothesis is made only for the sake of simplicity and could easily be removed if necessary, but is generally quite safe in practice). This cylinder is loaded in generalized plane strain conditions by some axisymmetric macroscopic stresses Σxx = Σyy , Σzz . The behavior of the matrix obeys Norton’s law: 3 σeq n−1 σ 0 d = (1) ε˙0 2 σ0 σ0 where d denotes the strain rate tensor, σ 0 the deviator of the stress tensor σ, σeq the von Mises equivalent stress and ε˙0 , σ0 , n material constants. Because of rotational symmetry, axial invariance and incompressibility of the matrix, the velocity field is of the form, in cylindrical coordinates: vr = A/r − Br/2,
vz = Bz,
where A = 32 b2 Dm ,
B = Dzz
(2)
In equations (2)1 and (2)2 , A and B are constants depending only on time and related to the mean (Dm ) and axial (Dzz ) parts of the macroscopic strain rate through equations (2)3 and (2)4 . The term proportional to A represents the expansion of the void whereas that proportional to B represents its change of shape.
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Thus our fundamental approximation will consist in neglecting those accelerations arising from the terms proportional to B. The acceleration field is then given by: ˙ − Avr /r2 ≈ A/r ˙ − A2 /r3 , γr ≈ A/r
γz ≈ 0
(3)
The approximation γz ≈ 0, which results from our fundamental hypothesis, is compatible with the assumed axial invariance insofar as the axial equation of dynamics, ∂σzr /∂r + σzr /r + ∂σzz /∂z = ργz (where ρ is the mass per unit volume), with σzr = 0 and σzz independent of z, precisely implies γz = 0. Integration of the radial equation of dynamics, ∂σrr /∂r + (σrr − σθθ )/r = ργr , using the flow rule (1), then yields upon a somewhat lengthy calculation: 1/n |Dzz | n 1 Σxx = √ (4) I(X) + XI 0 (X) + τ 2 Q f, f˙, f¨ σ0 ε˙0 2 3 n+1 √ Z 2X/f n+1 dx 3 Dm ; I(X) ≡ 2(1 + x2 ) 2n 2 (5) X≡ 2 |Dzz | x 2X ρa20 1 ¨ f˙2 2 ˙ ¨ (6) ; Q f, f , f ≡ ln f− τ ≡ 4σ0 f0 f 2f where f0 and a0 denote the initial porosity and void radius, respectively. Also, calculating Σzz − Σxx as 0 0 0 − (σrr + σθθ )/2, and using equation (1), one gets: the mean value of σzz − (σxx + σyy )/2 = σzz 1/n |Dzz | 1 Σzz − Σxx sgn(Dzz ) = X I(X) − nXI 0 (X) σ0 n+1 ε˙0
(7)
where sgn(Dzz ) denotes the sign of Dzz . Inertia effects are entirely confined within the term τ 2 Q(f, f˙, f¨) of the expression (4) of Σxx /σ0 ; note that if one sets it to zero, one recovers the exact static solution derived by Leblond et al. [2]. Note also that this term is independent of the loading so that it takes the same value as for a purely hydrostatic one. One thus sees that in order to introduce inertia effects into the static solution, one just needs to add, in the sole expression of that stress (Σxx ) governing void growth, the ‘dynamic term’ corresponding to some purely hydrostatic loading. 3. The case of spherical voids 3.1. Exact dynamic solution for a hollow sphere loaded hydrostatically We begin by deriving the exact dynamic solution for a hollow sphere of inner radius a, outer radius b, porosity f ≡ a3 /b3 1, subjected to some hydrostatic (positive) stress Σm and obeying Norton’s constitutive law (1). It is analogous to that provided by Johnson [5] except that here the viscous behavior is nonlinear. The velocity is purely radial and given by vr = A/r2 where A is a constant depending only on time and related to the mean macroscopic strain rate Dm by the relation A = b3 Dm . The ˙ 2 − 2A2 /r4 . Integrating the radial equation of dynamics, (radial) acceleration is given by γr = A/r ∂σrr /∂r + 2(σrr − σθθ )/r = ργr , one then gets: 2Dm 1/n Σm 2n −1/n f = −1 + τ 2 Q f, f˙, f¨ (8) σ0 3 ε˙0 2/3 (9) 3σ0 f0 ; Q f, f˙, f¨ ≡ f −1/3 − 1 f¨ − 16 f −4/3 f˙2 τ 2 ≡ ρa20 Note that the terms −1 in the expressions f −1/n − 1 (8) and f −1/3 − 1 (9) have been retained in spite of the hypothesis f 1, because f −1/n and f −1/3 may be close to unity in view of the smallness of the
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exponents −1/n and −1/3. Note also that in the static case (obtained by discarding the term τ 2 Q(f, f˙, f¨)), one recovers the (exact) solution provided by Wilkinson and Ashby [12]. 3.2. Approximate static solution for a hollow sphere under arbitrary loading Such a solution has been derived by Leblond et al. [2] for an axisymmetric loading. It is easy to extend it to an arbitrary loading by evaluating the approximate stress potential corresponding to the classical trial velocity field given by v = Dm b3 er /r2 + D 0 · x, where D0 denotes the deviator of the macroscopic strain rate. The results read as follows: 1/n Deq Σm 1 n ' (10) I(X) + XI 0 (X) σ0 3 n + 1 ε˙0 1/n 0 Deq D Σ0 2 1 ' X I(X) − nXI 0 (X) (11) σ0 3 n + 1 ε˙0 Deq Z 2X/f n+1 dx Dm ; I(X) ≡ 2 1 + x2 2n 2 (12) X≡ Deq x 2X where Σ 0 denotes the deviator of the macroscopic stress tensor and Deq the equivalent macroscopic strain rate. Note the formal analogy here with the solution (equations (4), (5) and (7)) for a hollow cylinder in the static case (τ 2 Q(f, f˙, f¨) ≡ 0). Also, it can be verified that equation (10) for Σm /σ0 does reduce to (8) (with τ 2 Q(f, f˙, f¨) ≡ 0) for a purely hydrostatic loading (Deq → 0). 3.3. Approximate dynamic solution for spherical voids under arbitrary loading In view of the difficulty of solving such a problem, even approximately, we resort to a simpler approach, based on the remark made at the end of Section 2 about the cylindrical case, which consists in assuming that as in that case, inertia effects can be introduced simply by adding to the static expression of the sole stress (Σm ) governing void growth, the dynamic term arising from a purely hydrostatic loading. The expression (10) of Σm /σ0 then becomes: 1/n Deq Σm 1 n ' I(X) + XI 0 (X) + τ 2 Q f, f˙, f¨ σ0 3 n + 1 ε˙0
(13)
where τ 2 and Q(f, f˙, f¨) are given by (9), whereas that (11) of Σ 0 /σ0 remains unchanged. 4. A dynamic model for spherical voids incorporating elasticity In the static case, elasticity generally plays a minor role. In the dynamic one, in contrast, it becomes essential to introduce it since it governs wave propagation, for instance in plate impact experiments. The relation connecting the mean stress and the mean elastic strain must be nonlinear in view of the extremely high pressures which can be encountered in practice. On the other hand, the stress deviator is much smaller so that a linear relation is sufficient to describe the deviatoric part of the elasticity law. We essentially follow here the same heuristic approach as in [5]. Thus, let v (≡ 1/ρ) denote the current specific volume and v0 its initial value in the reference, stress-free state. The variation v − v0 is assumed to consist of a reversible (elastic) part and an irreversible (plastic) part, and similarly for the strain rate D: v − v0 ≡ v e + v p ;
D ≡ De + Dp ,
(14)
the rates of v e and v p being connected to D e and Dp by the relations: v˙ e /v = tr De ;
v˙ p /v = tr Dp .
(15)
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Note that equations (14) and (15) are compatible with the classical relation v/v ˙ = tr D. The variation of porosity arising from elasticity being as usual neglected, f˙ is given by the classical formula: f˙ = (1 − f ) tr Dp
(16) p
The flow rule is supposed to be the same as before, D being replaced by D ; thus equations (13) and (11) become: p 1/n Deq Σm 1 n = I(X) + XI 0 (X) + τ 2 Q f, f˙, f¨ p Dm σ0 3 n + 1 ε˙0 (17) ; X ≡ 0 p 1/n p Deq Deq Dp Σ0 2 1 0 = p X I(X) − nXI (X) σ0 3 n + 1 ε˙0 Deq where I(X), τ 2 and Q(f, f˙, f¨) are still given by equations (12)2 and (9). With regard to the elasticity law, the relation between the mean stress (opposite of pressure) and the elastic part of the variation of specific volume is supposed to be given by the Mie–Grüneisen equation of state: Σm = K0
2 Γ0 ve Γ0 v e ve − E 1+ 1+s v0 2 v0 v0 v0
(18)
where K0 denotes the compressibility coefficient at zero pressure, Γ0 the Grüneisen coefficient in the same conditions, s a dimensionless constant and E the specific internal energy. Also, the deviatoric part of the (hypo)elasticity law is expressed as: ˇ 0 = 2G0 De0 Σ
(19)
where ˇ denotes the Jaumann derivative and G0 the shear modulus in the reference state. Finally, the evolution of E in equation (18) is given by the first principle of thermodynamics (in the adiabatic case): E˙ = v Σ : D
(20)
References [1] Gurson A.L., Continuum theory of ductile rupture by void nucleation and growth: Part I – Yield criteria and flow rules for porous ductile media, ASME J. Eng. Materials Technol. 99 (1977) 2–15. [2] Leblond J.-B., Perrin G., Suquet P., Exact results and approximate models for porous viscoplastic solids, Int. J. Plasticity 10 (1994) 213–235. [3] Needleman A., Tvergaard V., An analysis of dynamic, ductile crack growth in a double edge cracked specimen, Int. J. Fracture 49 (1991) 41–67. [4] Nemes J.A., Eftis J., Pressure-shear waves and spall fracture described by a viscoplastic-damage constitutive model, Int. J. Plasticity 8 (1992) 185–207. [5] Johnson J.N., Dynamic fracture and spallation in ductile solids, J. Appl. Phys. 52 (1981) 2812–2825. [6] Carroll M.M., Holt A.C., Static and dynamic pore collapse relations for ductile porous materials, J. Appl. Phys. 43 (1972) 1626–1636. [7] Cortes R., Dynamic growth of microvoids under combined hydrostatic and deviatoric stresses, Int. J. Solids Structures 29 (1992) 1637–1645. [8] Klöcker H., Montheillet F., Influence of flow rule and inertia on void growth in a rate sensitive material, J. Phys. IV C3 (1991) 733–738. [9] Ortiz M., Molinari A., Effect of strain hardening and rate sensitivity on the dynamic growth of a void in a plastic material, ASME J. Appl. Mech. 59 (1992) 48–53. [10] Tong W., Ravichandran G., Inertial effects on void growth in porous viscoplastic materials, ASME J. Appl. Mech. 62 (1995) 633–639. [11] Cortes R., The growth of microvoids under intense dynamic loading, Int. J. Solids Structures 29 (1992) 1339–1350. [12] Wilkinson D.S., Ashby M.F., Pressure sintering by power law creep, Acta Metall. 23 (1975) 1277–1285.
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