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Micromechanical modelling of porous materials under dynamic loading
Journal of the Mechanics and Physics of Solids 49 (2001) 1497 – 1516
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Micromechanical modelling of porous materials under dynamic loading A. Molinari ∗ , S. Mercier Laboratoire de Physique et Mecanique des Materiaux, I.S.G.M.P., Universite de Metz, Ecole Nationale d’Ingenieurs, Ile du Saulcy, 57045 Metz, France Received 16 May 2000; received in revised form 9 November 2000; accepted 2 January 2001
Abstract The behaviour of porous material under dynamic conditions is assessed by a micromechanical approach. By averaging, a general form for the dynamic macrostress is proposed which recovers the static de0nition when inertia e1ects are neglected. In this work, a representative volume element for the porous material is de0ned as a hollow sphere. Using an approximation of the velocity 0eld and the principle of virtual work, an explicit relationship is found between the macroscopic stress and strain rate. The macrostress tensor is proved to be symmetric, in the present formulation proposed for porous materials. Illustrations are shown for hydrostatic tension or compression and also for axisymmetric loading. In the latter case, the e1ect of stress triaxiality c 2001 Elsevier Science Ltd. All rights reserved. is captured. Keywords: A. Dynamics; B. Constitutive behaviour; B. Porous material; B. Viscoplastic material; Micromechanical approach
1. Introduction The dynamic response of a porous ductile material is analysed in the present work. During straining, voids are nucleated in metals, mainly by decohesion at the particle– matrix interfaces. Later in the deformation process, these voids evolve in the matrix until coalescence; this phenomenon triggers ductile fracture of metals. In the opposite direction, in the consolidation of metallic powders, granular material is compacted to reduce the porosity. For high strain rate processes (such as shock compaction), the microinertia e1ects inherited from the rapid growth or collapse of voids has an in
Corresponding author. Tel.: +33-03-87-31-53-69; fax: +33-03-87-31-53-66. E-mail address: [email protected] (A. Molinari).
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porosity evolution and the dynamic mechanical response is crucial for the modelling of impacted structures and of rapid material processing. McClintock (1968) was the 0rst to consider the evolution of a single cylindrical void in an in0nite matrix subjected to axisymmetric loading at the remote boundary. Rice and Tracey (1969) used a variational approach to investigate the response of an isolated spherical void in an in0nite medium. The material is loaded by a uniform tensile extension 0eld with a superposed hydrostatic pressure. For both authors, the matrix is rigid perfectly plastic and a static analysis is performed. Carroll and Holt (1972) proposed a dynamic solution for the growth of spherical voids in a 0nite medium. The material is still assumed perfectly plastic but inertia e1ects are accounted for. Numerous authors have proposed improvement of those three pioneering works. Johnson (1981) considered a linear viscous matrix and concluded that the stabilizing e1ect of inertia is negligible in comparison with the viscous stabilization. KlFocker and Montheillet (1988) performed numerical simulations for the dynamic growth of an isolated void in an in0nite viscoplastic medium. They observed that inertia is signi0cant for large strain rate. Ortiz and Molinari (1992) proposed a description of the dynamic expansion of a spherical void in an unbounded solid under hydrostatic tension. At the early stage of the deformation, the void growth is controlled by strain hardening and strain rate sensitivity. Inertia dominates the long term response of the medium. Cortes (1992) extended the analysis of Carroll and Holt (1972) by taking into account strain rate sensitivity, strain hardening and thermal softening. It is shown that the e1ects of strain rate sensitivity and of strain hardening are important whereas the e1ect of thermal softening is really weak, in the conditions analysed by Cortes (1992). Tong and Ravichandran (1993, 1995) have incorporated thermal softening in the matrix behaviour. The temperature evolution of the material is governed by adiabatic heating, since rapid deformation of porous medium was considered. Their results showed that the thermal softening accelerates the deformation process and illustrated the stabilizing aspect of strain hardening, strain rate sensitivity and inertia. These approaches, based on the hollow sphere model, are able to describe the response of porous media for particular con0guration and loading conditions. They do not provide a general constitutive relationship for porous solids. Gurson (1977) has proposed an approximate yield criterion for ductile porous media, using a micromechanical approach. The matrix material is rigid perfectly plastic and virtual velocity 0elds are considered to model the deformation in the matrix. This model and its derivatives, the GTN (Tvergaard, 1982; Needleman and Tvergaard, 1984) model for example, have been widely used to describe the growth and the coalescence of voids in solids under static conditions. To de0ne the range of validity of micromechanical models, Ponte-Casta˜neda (1991) and Michel and Suquet (1992) derived bounds for voided materials. Recently, Lee and Mear (1994) and KlFocker and Montheillet (1992) proposed virtual velocity 0elds for ellipso˙Ldal voids in an in0nite viscoplastic matrix. Following a Gurson’s like approach, Gologanu et al. (1993, 1994) considered a spheroidal cavity embedded in a 0nite volume. The e1ects of shape and of triaxiality are analysed. In the previous micromechanical models, inertia is neglected. Since inertia has a signi0cant role for high strain rates processes, a Gurson’s type approach for dynamic applications has to be de0ned. In an attempt by Wang (1997), Wang and Jiang (1997), admissible virtual velocity 0elds are adopted for the problem of the hollow sphere
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subjected to high strain rates at the remote boundary. Using the principle of virtual work, a relationship is obtained between the macroscopic stress, the porosity and the applied kinematic boundary condition. Two contributions (static and dynamic) appear in the de0nition of the macroscopic stress. The author demonstrates the e1ect of inertia on the yield criterion. In his approach, Wang adopted the static de0nition for the macroscopic stress, in the sense that the macroscopic stress is still de0ned as the average value of the microscopic stress. In our opinion, this static de0nition is not appropriate in dynamic conditions. The purpose of the present work is to formulate a dynamic constitutive formulation for porous materials. In Sections 2 and 3, a general form for the dynamic macrostress is de0ned by averaging. It is also built up of two components: a static viscoplastic part and a dynamic part. In the limiting case where inertia is neglected, the static de0nition for the macrostress is retrieved. In Section 4, we focus our attention on the dynamic behaviour of a porous material. We assume that a representative volume element can be represented by a hollow sphere. As in the Gurson’s approach, an approximate velocity 0eld, which is kinematically admissible, is considered. Assuming a rigid viscoplastic behaviour for the matrix, an explicit macroscopic stress strain-rate relationship is derived from the principle of virtual work and provides the dynamic constitutive behaviour for the porous material. In Section 5, illustrations of the proposed approach are presented for a porous medium subjected to hydrostatic tension or compression and to axisymmetric loading. The present approach is shown to agree with previous works (Carroll and Holt, 1972; Ortiz and Molinari, 1992; Tong and Ravichandran, 1993, 1995), when spherical loadings are of interest. In addition, the e1ect of triaxiality is clearly captured.
2. Denition of the macroscopic elds A representative volume element (RVE) of a porous material is considered. A cartesian coordinate system is adopted in the present analysis. Mechanical 0elds (Cauchy stress and strain rate) acting at a material point of the RVE are ij and dij . The equation of motion is the following: ij; j =
dvi dt
with ij = ji ;
(1)
where (:; j ) represents the partial derivative with respect to the coordinate xj , is the density of the material, vi the velocity components. The strain rate tensor is de0ned by dij = 12 (vi; j + vj; i ):
(2)
The following kinematic condition is applied to the boundary @ of the RVE: C=D·x
on @ ;
(3)
where “·” represents the simple contracted product. D is supposed to be uniform and represents the macroscopic strain rate tensor. It is straightforward that the macroscopic
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strain rate D is the volume average of the corresponding microscopic quantity: D = d;
(4)
where · = (1=| |) · d and | | represents the volume of the RVE. The macroscopic stress is de0ned by = + ⊗ x;
(5)
where ⊗ denotes the tensorial product and = dC=dt is the acceleration of a particle. This relationship is proposed so that and D are work conjugated quantities through the principle of virtual work (see Appendix A): 1 d|C|2 ; (6) : D = : d + 2 dt where (:) is the twice contracted product. Note that, due to dynamic conditions, the static de0nition of the macrostress = is not valid; an inertial term is added. When boundary conditions consist in prescribing surface tractions as ·n=·n
on @ ;
uniform;
(7)
then it is easily proved (Appendix A) that the macrostress is related to corresponding microscopic stress by Eq. (5). In that case, the macroscopic strain rate D is de0ned by relationship (4). As for kinematic boundary conditions, macroscopic quantities and D are work conjugated, since relation (6) is preserved (see Appendix A). 3. Macroscopic constitutive law To obtain the constitutive behaviour, a link between the two macroscopic quantities and D has to be developed. For that purpose, the kinematic boundary conditions (3) are prescribed at the remote boundary of the RVE. The matrix material is supposed to be incompressible. The local velocity 0eld and strain rate tensor are considered to be related to the macroscopic strain rate tensor as follows: C = C(x; D);
d = d(x; D):
(8)
In addition, the local material behaviour is speci0ed by the introduction of a stress potential (x; d). Thus, the stress tensor is expressed in terms of the strain-rate tensor d via the following relationship: =
@ (x; d): @d
(9)
A relation between the macrostress and the macroscopic strain rate D is sought using a variational approach. The volume average of the increment of rate of work, evaluated in terms of the stress potential, has the form @ : d = (x; d) : d ; (10) @d
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where d represents the increment of strain rate. Using expression (8), the stress potential can be expressed in terms of the macroscopic strain rate D: ˆ D): (x; d(x; D)) = (x;
(11)
By means of this relation, the average of the increment of rate of work is written as @ @ˆ @ @d : D = : d ; (12) : D = : @D @d @D @d where D represents the increment of macroscopic strain rate. Therefore, the macroscopic potential de0ned by 1 ˆ = ˆ d (13) | | satis0es the following relationship: ˆ @ : D: (14) @D The average increment of rate of work can also be evaluated by means of the principle of virtual work when C, d and D are replaced by the corresponding increments C, d and D. Thus it follows (same calculations as in Appendix A) that 1 · C d : (15) : d = : D − | | : d =
The combination of Eqs. (14) and (15) provides ˆ @ 1 : D = · C d : : D + @D | |
(16)
From Eq. (8), the increment of the velocity 0eld appears to be a linear function of the strain rate: @C C = : D = K(x; D) : D: (17) @D Thus, Eq. (16) leads to the following expression: ˆ @ 1 − · K d : D = 0: (18) − @D | | In Eq. (18), D is an arbitrary symmetric tensor. Therefore, the symmetric part of the macrostress is given by s = static + (∗ )s ;
(19)
where
static
ˆ @ = @D
and
1 = | | ∗
· K(x; D) d :
(·)s represents the symmetric part of the second order tensor (·).
(20)
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The viscoplastic behaviour of the RVE under dynamic conditions is de0ned by expression (19), in which the macrostress is related to the macroscopic strain rate. The macrostress is composed of a viscoplastic stress calculated from the quasistatic ˆ and an inertia-dependent term. In the application to porous materials, it potential will be demonstrated that the macrostress is indeed a symmetric tensor. 4. Applications to porous materials The purpose of the present work is to propose a modelling of the constitutive behaviour for porous materials with incompressible matrix, subjected to dynamic loading. To go further in the analysis of the previous section, we have to evaluate in Eq. (19) the two contributions to the macrostress static and ∗ . Numerous static stress potentials could have been used in the present analysis to evaluate static (see for example Gurson, 1977; Olevsky and Skorohod, 1988 or Leblond et al., 1994). The ˆ adopted in the following analysis is the one proposed macroscopic stress potential by Olevsky and Skorohod (1988) since it has been used successfully to model powder compaction (see Olevsky and Molinari, 2000 for a review). Olevsky and Skorohod have developed a phenomenological formulation for the potential of porous materials with incompressible matrix: ˆ = where
A (1 − f)W m+1 ; m+1
W= =
2 + e2 ; 1−f
D : D
and
(21)
= (1 − f)2 ;
=
2 (1 − f)3 ; 3 f
e = tr(D):
(22)
D represents the deviatoric part of the macroscopic strain rate, f is the porosity (volume fraction of voids) and m is the strain rate sensitivity of the matrix. The ˆ has been derived assuming that the skeleton (the matrix) of the porous potential material has a nonlinear viscoplastic behaviour de0ned by the following stress potential: A m+1 (23) ; m+1 s √ where s = d : d ; d being the strain rate in the skeleton; A is a scaling factor for the yield stress. The evolution law for the porosity is given by =
f˙ = 3(1 − f)Dm ;
(24) ·
where Dm stands for tr D=3. ( ) represents the material time derivative operator. The static macrostress tensor is linked to the macroscopic strain rate D by ijstatic =
ˆ @ = AW m−1 (Dij + @Dij
eij ):
(25)
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Fig. 1. Schematic representation of the porous material. A hollow sphere (internal radius a and external radius b) is taken as the representative volume element.
3 From Eq. (25), it can be seen that 2 A corresponds to the yield stress in tension when the matrix has a rigid perfectly plastic behaviour. Relationship (19) de0nes the symmetric part of the macrostress providing that the acceleration 0elds and the concentration tensor K can be calculated. Porous materials have complex microstructures, with voids of di1erent sizes and di1erent shapes. However, in the present approach, the material is represented as in Gurson’s model (1977), by a composite hollow sphere with internal and external radii a and b, with porosity f = a3 =b3 , see Fig. 1. In this representative volume element, a single void is embedded in the matrix. It has to be noticed that interactions between voids are not taken into account; therefore, only material with low porosity can be modelled. The velocity 0eld, solution of the proposed boundary value problem, cannot usually be obtained in a closed analytical form. In this analysis as in the Gurson’s model (1977), an approximate representation for the velocity 0eld is assumed: vi =
Dij xj
3 b + Dm xi ; r
(26)
where r is the norm of the position vector. Inside the void, the velocity 0eld is extended from the matrix velocity (26) in an arbitrary way, but preserving the continuity of the velocity at the void surface. The approximate velocity 0eld is split in two parts: shape change at constant volume and volume change at constant shape. It is easily shown that the proposed velocity 0eld satis0es both the incompressibility and the kinematic boundary conditions (3). Note that the shape of voids is spherical in the proposed formulation. Therefore for consistency, the dilatation part in Eq. (26) (scaled by the mean strain rate) has to overwhelm the shape change part (represented by the deviator of the strain rate), see Rice and Tracey (1969). This restricts the range of applications of the model to cases with large stress triaxiality. To evaluate , we can use relationship (16) and calculate the inertia term. Since the velocity 0eld is given by Eq. (26), the term (1=| |) · C d can be determined explicitly. In that case, only the symmetric part of can be evaluated. A di1erent way is adopted so that both the symmetry and the expression of are obtained. The analysis of Sections 2 and 3 with the more general boundary condition C = L · x is performed, where L represents the macroscopic velocity gradient. Some details of the
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analysis are provided in Appendix B. Finally, the macroscopic stress is obtained as
1 static 2 1 −2=3 ˙ (f = + a − f) D + C − tr(C)I + (f−2=3 − 1)Dm D 3 5 1 + (f−2=3 − 1)(D : D )I − (f−2=3 − f−1 )D˙ m I 6
5 1 + 3f−1 − f−2=3 − f−2 Dm2 I ; 2 2
(27)
where C = D · D ; a is the current void radius and I is the second order identity tensor. The term scaled by a2 de0nes the dynamic contribution to the macrostress, noted (∗ )s in (19), and is related to the current porosity f, to the macroscopic strain rate and its time derivative. Note that is symmetric. Using the principle of virtual work, it can be easily shown that satis0es the macroscopic equation of motion, and that the relationship between traction force t and stress tensor is preserved (t = · n). Relationship (27), providing the behaviour of a porous material subjected to dynamic loading, has been obtained with the kinematic boundary conditions (3) prescribed at remote boundaries. Nevertheless by inversion of relationships (27), cases where tractions of form (7) are applied at the boundary, can be easily treated. Therefore the present model predicts the behaviour of porous material under stress or kinematic control. Compared to other models developed in the literature, there is no limitation to pure hydrostatic paths; complex loadings can be considered. 5. Results In the following, we consider a material representative of some aluminium. This material is subjected to a linearly increasing stress loading. Di1erent remote 0elds are applied: hydrostatic expansion, hydrostatic compression and axisymmetric loadings. The static analysis can be obtained as a limiting case of the dynamic analysis when the mass density vanishes → 0. This way is not followed because of the important computational time needed to reach the limit. A direct solution of the quasistatic problem is obtained by solving Eqs. (24) and (25). 5.1. Hydrostatic path In this section, time-dependent hydrostatic loadings are considered. On the boundary @ , the traction force t = · n is · n = pn:
(28)
The pressure is considered to be a linear function of time, with initial pressure p = 0. In the following, p˙ will designate the pressure rate. To validate the proposed approach, the dynamic compression of a rate-independent material is simulated. The con0guration adopted is consistent with the case analysed by Carroll and Holt (1972). Voids are spherical with initial radius ai = 20 m, initial
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Fig. 2. Dynamic compression of a porous aluminium. Results of the dynamic analysis of Carroll and Holt (1972) are retrieved. Initial radius ai =20 m, initial porosity fi =0:23. The pressure rate is p=−10 ˙ MPa=ns. Material constants are: A = 245 Int. Syst., m = 0:005 and = 2700 kg=m3 .
porosity fi = 0:23. The tensile yield stress of the aluminium considered here is 300 MPa. Therefore, the material parameters used in Eq. (25) are set to A = 245 × 106 Int. Syst., m = 0:005. The low value of m has been chosen as in Tong and Ravichandran (1993), in order to be close to the rate-independent assumption. The mass density is = 2700 kg=m3 . The hydrostatic compression loading rate is assumed constant p˙ = −10 MPa=ns. Fig. 2 represents the evolution of the distention factor 1=(1 − f) versus time. A close agreement is observed between the present approach and the Carroll and Holt (1972) dynamic analysis. The e1ects of strain rate sensitivity and of loading rate on pore collapse are investigated now. Dynamic and static analyses are compared for hydrostatic compression tests and for the aluminium material previously considered. The initial radius is ai = 20 m, the porosity fi = 0:1. For the relatively low pressure rate p˙ = −10 MPa=ns, the pore collapse is controlled by viscosity (cf. Fig. 3(a)). Results are identical for static and dynamic approaches (when m = 0:05 and 0.15). In addition, an increase of the strain rate sensitivity, which stabilizes the
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Fig. 3. Dynamic compression of a porous aluminium (A = 245 Int. Syst., = 2700 kg=m3 , initial porosity de0ned by ai = 20 m and fi = 0:1). Comparisons between dynamic and static analyses for two loading rates and two strain rate sensitivities (m=0:05 and 0:15) are presented: (a) p=−10 ˙ MPa=ns, (b) p=−250 ˙ MPa=ns.
values below 0.3 because it is known that the coalescence of voids occurs when the porosity f reaches 0.1– 0.3. It is observed that an increase of the loading rate (from 10 to 250 MPa=ns) triggers the process of rapid expansion earlier. For a given loading rate, an increase of m delays the void growth. It is interesting to note that, contrary to hydrostatic compression, inertia acts in a signi0cant way. At early stage of the deformation, static and dynamic analysis are identical. The void growth is controlled by the viscosity of the material. Later, the static solution grows faster than the dynamic one. Inertia dominates the late stage of the deformation (Ortiz and Molinari, 1992).
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Fig. 4. Dynamic hydrostatic extension of a porous aluminium (A = 245 Int. Syst., = 2700 kg=m3 , initial porosity de0ned by ai = 20 m and fi = 0:1). A comparison between static and dynamic analyses is carried out for di1erent loading rates (p˙ = 10, 250 MPa=ns) and strain rate sensitivities (m = 0:05 and 0:15).
For m = 0:05, and p˙ = 10 MPA=ns, the porosity f = 0:3 is reached for t = 376 ns in the static analysis, compared to t = 455 ns in the dynamic one. To illustrate the e1ect of di1erent parameters on the porosity evolution, a parametric study is carried out. The hydrostatic expansion (stress rate p˙ = 10 MPa=ns) of porous aluminium (voids de0ned by ai = 20 m and fi = 0:01) is chosen as the reference test. The e1ect of the strain rate sensitivity has already been clari0ed in Fig. 4 and is shown to delay the rapid evolution of porosity. In Fig. 5 results are obtained by varying the mass density, the
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Fig. 5. E1ects of the mass density, of the initial radius, of the initial porosity and of the
with q˙ a constant term. The triaxiality factor T is related to by Sm 1+2 ; (30) T= e = S 3|1 − | where Sm = 13 tr is the hydrostatic stress and Se = 32 Sij Sij the e1ective stress. When = 1, hydrostatic loading is prescribed, while = 0 represents simple tension or compression. During axisymmetric loading, the shape of voids will evolve toward spheroids. Numerous authors have studied the shape evolution (Budiansky et al., 1982; Koplik and Needleman, 1988; Lee and Mear, 1994; Gologanu et al., 1994; Briottet et al., 1996). The evolution is shown to be a complicated function of the triaxiality T . Budiansky et al. (1982) and Lee and Mear (1994) have performed a parametric study on T and have proved that a spherical void evolves towards an asymptotic shape among the following: needle, cylinder, prolate and oblate spheroids, sphere, crack and point. Through a comparison between predictions based upon models including void evolution and models assuming that voids remain spherical, Lee and Mear (1994) have concluded that little di1erence is observed when 0:6 ≤ ≤ 1. Budiansky et al. (1982) have observed that as soon as the triaxiality factor is larger than unity T ≥ 1 ( ≥ 0:4), voids can be considered as spherical during the deformation. Since in the proposed approach, no shape evolution is considered, only results for ≥ 0:5 are presented. For low triaxiality, voids evolve towards a nonspherical shape. In dynamic loading, inertia e1ects act di1erently in the direction of loading and in the transverse direction. Therefore, results concerning the evolution of the shape can be modi0ed by inertia. An extension of the present analysis to spheroidal voids is necessary to clarify the e1ects of inertia on
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Fig. 6. In
ovalization. Applications of the present model for low triaxiality loading (for example, simple tension or compression = 0) should be avoided. The same aluminium as in the previous section is considered (A = 245 Int. Syst., m = 0:05 and = 2700 kg=m3 ). Attention is focused on triaxiality e1ects (through the parameter ). Two cases are investigated; 0rst, a compression test is performed with lateral con0nement (q˙ = −10 MPa=ns and varying from 0.5 to 1). In Fig. 6, it is observed that the closure of pores is strongly in
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Fig. 7. In
Fig. 8. In
6. Conclusion A model has been proposed to describe the behaviour of a porous material loaded under dynamic conditions. Using averaging methods and assuming homogeneous kine-
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matic boundary conditions, a relationship between macrostress, macroscopic strain rate and porosity has been obtained. The macrostress is shown to be composed of a static and a dynamic part. For the static contribution, the phenomenological potential developed by Olevsky and Skorohod (1988) has been adopted. Matrix incompressibility is an essential hypothesis in the theory presented. Predictions of the present model to various stress loading paths are compared to previous results. In hydrostatic loading, the e1ects of material parameters (
(A.1)
(A.2)
Two cases are investigated; kinematic boundary conditions are 0rst considered: C=D·x
on @
(vi = Dik xk );
(A.3)
where D represents the macroscopic strain rate. Eqs. (A.2) – (A.3), with the use of the Gauss’ theorem provide 1 dvi 1 : d = ij; j xk d + Dik ik − vi Dik d : (A.4) | | | | dt
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Using the equation of motion to modify the 0rst term on the right side, relationship (A.4) is transformed into
1 1 d|C|2 : (A.5) Dik ik + i xk d = : d + 2 dt | | The de0nition of the macrostress (relationship (5)) is recognized and relationship (6) is obtained. When stress boundary conditions are applied ik nk = ik nk
on @ ;
(A.6)
relationship (5) can be proved easily. The mean value ij is expressed in the form 1 1 1 ij = ij d = (ik xj ); k d − ik; k xj d : (A.7) | | | | | | With the use of equation of motion (1) and Gauss’ theorem, it follows that 1 ij = xj ik nk ds − i xj : | | @
(A.8)
Since tractions (A.6) are prescribed at the boundary, relationship (A.8) becomes ij = ij − i xj
(A.9)
which proves (5). The combination of Eq. (A.2) with the boundary relation (A.6) leads to 1 dvi 1 : d = ij vi nj ds − vi d : | | @ | | dt
(A.10)
Gauss’ theorem is applied to the 0rst term on the right side of the previous relation. Using the de0nition of the macroscopic strain rate (4), relationship (6) is retrieved: 1 d|C|2 : (A.11) : d = : D − 2 dt Therefore, macroscopic quantities and D, related to microscopic values and d by relationships (4) and (5), are work conjugated for each boundary condition considered. Appendix B. Symmetry and formulation of the macroscopic stress tensor To prove the symmetry of the macroscopic tensor and to derive the constitutive law, the analysis proposed in Sections 2 and 3 is developed on the basis of a more general kinematic boundary condition C = L · x;
(B.1)
where L is the macroscopic velocity gradient. The macroscopic strain rate D = 12 (L + t L), the spin tensor , = 12 (L − t L) and the velocity gradient L are related to the local tensors d; and l by D = d;
, = ;
L = l:
(B.2)
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t
( ) denotes the transpose operator. The macroscopic stress is still de0ned by Eq. (5) so as and L are work conjugated: 1 d|C|2 t : (B.3) : L = : d + 2 dt With the boundary condition (B.1), the local velocity 0eld and strain rate tensor are related to the macroscopic velocity gradient and strain rate tensor C = C(x; L);
d = d(x; D):
(B.4)
The local material behaviour is still de0ned by (9). As a consequence, with use of the principle of virtual work (B.3), relationship (16) becomes ˆ 1 @ : D + : L = | | @D t
· C d
(B.5)
with C =
@C : L = K(x; L) : L: @L
(B.6)
From Eqs. (B.5) and (B.6), it follows that ˆ 1 @ 1 · K d : D − + · K d : , = 0: − − @D | | | | (B.7) In Eq. (B.7), D and , are arbitrary. Therefore, the macrostress which is not a symmetric tensor in general, is given by s = static + (∗ )s
a = −(∗ )a ;
(B.8)
where static =
ˆ @ @D
and
∗ =
1 | |
· K(x; L) d :
(B.9)
(·)a represents the antisymmetric parts of the second order tensor (·). To evaluate the dynamic contribution ∗ to the macrostress, the porous material is still represented by a composite hollow sphere. The solution for the exact velocity 0eld cannot be reached in a closed form. Therefore, a simpli0ed velocity 0eld is considered: C = L · x + Bx
(B.10)
with L being the deviatoric velocity gradient, B = Dm (b=r)3 and Dm = 13 tr D. The kinematic boundary condition (B.1) is satis0ed together with the incompressibility condition in the matrix.
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From (B.10), the acceleration of a material particle is
˙ · x + L · C + Bx ˙ + BC: (x) = L
(B.11)
The variation of the velocity due to the change L in the boundary condition is C = L · x + Bx:
(B.12)
Thus, the integral term I = · C d of (B.5) can be written as ˙ · x + C · t L · L · x ˙ · L · x + Bx · L · C d = (x · t L I=
˙ · L · x + BBx ˙ + Bx · L · C + Bx · x + BC · L · x + BBC · x) d : (B.13) The RVE, , is represented by a spherical shell (inner radius a and outer radius b), see Fig. 1. The calculations, which are not presented here in detail, are conducted in spherical coordinates; the following result is obtained: i=3 k=3 1 1 ˙ ii + Cii − (f−2=3 − f) D a2 Ckk + (f−2=3 − 1)Dm Dii I= 5 3 i=1
k=1
1 + (f−2=3 − 1)(L : L ) − (f−2=3 − f−1 )D˙ m 6
5 −2=3 1 −2 −1 Dm2 Dii − f + 3f − f 2 2 1 + a2 (f−2=3 − f)(L˙ij + Cij ) + (f−2=3 − 1)Dm Lij Lij ; 5
(B.14)
1≤i≤3
1≤j≤3 i=j
where C=L ·L . By identi0cation (the components Lij being considered as independent terms) the macroscopic stress tensor is obtained from (B.7) as
1 s s static 2 1 −2=3 ˙ = + a − f) D + C − tr(C)I + (f−2=3 − 1)Dm D (f 5 3 1 + (f−2=3 − 1)(L : L )I − (f−2=3 − f−1 )D˙ m I 6
5 1 + 3f−1 − f−2=3 − f−2 Dm2 I ; 2 2 a
= a
2
1 −2=3 a −2=3 ˙ − f)(, + C ) + (f − 1)Dm , : (f 5
(B.15a)
(B.15b)
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When spin e1ects vanish, a =0. Then, the macroscopic stress is shown to be symmetric and relationship (27) is retrieved:
1 static 2 1 −2=3 ˙ = + a − f) D + C − tr(C)I + (f−2=3 − 1)Dm D (f 5 3 1 + (f−2=3 − 1)(D : D )I − (f−2=3 − f−1 )D˙ m I 6
5 1 + 3f−1 − f−2=3 − f−2 Dm2 I ; 2 2
(B.16)
where C = D · D . References Briottet, L., KlFocker, H., Montheillet, F., 1996. Damage in a viscoplastic material Part I: Cavity growth. Int. J. Plasticity 12, 481–505. Budiansky, B., Hutchinson, J.W., Slutsky, S., 1982. Void growth and collapse in viscous solids. In: Hopkins, H.G., Sewell, M.J. (Eds.), Mechanics of Solids. Pergamon Press, Oxford. Carroll, M.M., Holt, A.C., 1972. Static and dynamic pore-collapse relations for ductile porous materials. J. Appl. Phys. 43, 1626–1636. Cortes, R., 1992. The growth of microvoids under intense dynamic loading. Int. J. Solids Struct. 29, 1339– 1350. Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing nonspherical voids – case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754. Gologanu, M., Leblond, J.B., Devaux, J., 1994. Approximate models for ductile metals containing nonspherical voids – case of axisymmetric oblate ellipsoidal cavities. J. Engng. Mater. Technol. 116, 290–297. Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth: Part I – Yield criteria and
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Ortiz, M., Molinari, A., 1992. E1ect of strain hardening and rate sensitivity on the dynamic growth of a void in a plastic material. J. Appl. Mech. 114, 48–53. Ponte-Casta˜neda, P., 1991. The e1ective mechanical properties of nonlinear isotropic materials. J. Mech. Phys. Solids 39, 45–71. Rice, J.R., Tracey, D.M., 1969. On the ductile enlargement of voids in triaxial stress 0elds. J. Mech. Phys. Solids, 17, 201–217. Tong, W., Ravichandran, G., 1993. Dynamic pore collapse in viscoplastic materials. J. Appl. Phys. 74, 2425–2435. Tong, W., Ravichandran, G., 1995. Inertial e1ects on void growth in porous viscoplastic materials. J. Appl. Mech. 62, 633–639. Tvergaard, V., 1982. On localization in ductile materials containing spherical voids. Int. J. Fract. 18, 237–252. Wang, Z.P., 1997. Void-containing nonlinear materials subject to high-rate loading. J. Appl. Phys. 81, 7213–7227. Wang, Z.P., Jiang, Q., 1997. A yield criterion for porous ductile media at high strain rate. J. Appl. Mech. 64, 503–509.
www.elsevier.com/locate/jmps
Micromechanical modelling of porous materials under dynamic loading A. Molinari ∗ , S. Mercier Laboratoire de Physique et Mecanique des Materiaux, I.S.G.M.P., Universite de Metz, Ecole Nationale d’Ingenieurs, Ile du Saulcy, 57045 Metz, France Received 16 May 2000; received in revised form 9 November 2000; accepted 2 January 2001
Abstract The behaviour of porous material under dynamic conditions is assessed by a micromechanical approach. By averaging, a general form for the dynamic macrostress is proposed which recovers the static de0nition when inertia e1ects are neglected. In this work, a representative volume element for the porous material is de0ned as a hollow sphere. Using an approximation of the velocity 0eld and the principle of virtual work, an explicit relationship is found between the macroscopic stress and strain rate. The macrostress tensor is proved to be symmetric, in the present formulation proposed for porous materials. Illustrations are shown for hydrostatic tension or compression and also for axisymmetric loading. In the latter case, the e1ect of stress triaxiality c 2001 Elsevier Science Ltd. All rights reserved. is captured. Keywords: A. Dynamics; B. Constitutive behaviour; B. Porous material; B. Viscoplastic material; Micromechanical approach
1. Introduction The dynamic response of a porous ductile material is analysed in the present work. During straining, voids are nucleated in metals, mainly by decohesion at the particle– matrix interfaces. Later in the deformation process, these voids evolve in the matrix until coalescence; this phenomenon triggers ductile fracture of metals. In the opposite direction, in the consolidation of metallic powders, granular material is compacted to reduce the porosity. For high strain rate processes (such as shock compaction), the microinertia e1ects inherited from the rapid growth or collapse of voids has an in
Corresponding author. Tel.: +33-03-87-31-53-69; fax: +33-03-87-31-53-66. E-mail address: [email protected] (A. Molinari).
c 2001 Elsevier Science Ltd. All rights reserved. 0022-5096/01/$ - see front matter PII: S 0 0 2 2 - 5 0 9 6 ( 0 1 ) 0 0 0 0 3 - 5
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porosity evolution and the dynamic mechanical response is crucial for the modelling of impacted structures and of rapid material processing. McClintock (1968) was the 0rst to consider the evolution of a single cylindrical void in an in0nite matrix subjected to axisymmetric loading at the remote boundary. Rice and Tracey (1969) used a variational approach to investigate the response of an isolated spherical void in an in0nite medium. The material is loaded by a uniform tensile extension 0eld with a superposed hydrostatic pressure. For both authors, the matrix is rigid perfectly plastic and a static analysis is performed. Carroll and Holt (1972) proposed a dynamic solution for the growth of spherical voids in a 0nite medium. The material is still assumed perfectly plastic but inertia e1ects are accounted for. Numerous authors have proposed improvement of those three pioneering works. Johnson (1981) considered a linear viscous matrix and concluded that the stabilizing e1ect of inertia is negligible in comparison with the viscous stabilization. KlFocker and Montheillet (1988) performed numerical simulations for the dynamic growth of an isolated void in an in0nite viscoplastic medium. They observed that inertia is signi0cant for large strain rate. Ortiz and Molinari (1992) proposed a description of the dynamic expansion of a spherical void in an unbounded solid under hydrostatic tension. At the early stage of the deformation, the void growth is controlled by strain hardening and strain rate sensitivity. Inertia dominates the long term response of the medium. Cortes (1992) extended the analysis of Carroll and Holt (1972) by taking into account strain rate sensitivity, strain hardening and thermal softening. It is shown that the e1ects of strain rate sensitivity and of strain hardening are important whereas the e1ect of thermal softening is really weak, in the conditions analysed by Cortes (1992). Tong and Ravichandran (1993, 1995) have incorporated thermal softening in the matrix behaviour. The temperature evolution of the material is governed by adiabatic heating, since rapid deformation of porous medium was considered. Their results showed that the thermal softening accelerates the deformation process and illustrated the stabilizing aspect of strain hardening, strain rate sensitivity and inertia. These approaches, based on the hollow sphere model, are able to describe the response of porous media for particular con0guration and loading conditions. They do not provide a general constitutive relationship for porous solids. Gurson (1977) has proposed an approximate yield criterion for ductile porous media, using a micromechanical approach. The matrix material is rigid perfectly plastic and virtual velocity 0elds are considered to model the deformation in the matrix. This model and its derivatives, the GTN (Tvergaard, 1982; Needleman and Tvergaard, 1984) model for example, have been widely used to describe the growth and the coalescence of voids in solids under static conditions. To de0ne the range of validity of micromechanical models, Ponte-Casta˜neda (1991) and Michel and Suquet (1992) derived bounds for voided materials. Recently, Lee and Mear (1994) and KlFocker and Montheillet (1992) proposed virtual velocity 0elds for ellipso˙Ldal voids in an in0nite viscoplastic matrix. Following a Gurson’s like approach, Gologanu et al. (1993, 1994) considered a spheroidal cavity embedded in a 0nite volume. The e1ects of shape and of triaxiality are analysed. In the previous micromechanical models, inertia is neglected. Since inertia has a signi0cant role for high strain rates processes, a Gurson’s type approach for dynamic applications has to be de0ned. In an attempt by Wang (1997), Wang and Jiang (1997), admissible virtual velocity 0elds are adopted for the problem of the hollow sphere
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subjected to high strain rates at the remote boundary. Using the principle of virtual work, a relationship is obtained between the macroscopic stress, the porosity and the applied kinematic boundary condition. Two contributions (static and dynamic) appear in the de0nition of the macroscopic stress. The author demonstrates the e1ect of inertia on the yield criterion. In his approach, Wang adopted the static de0nition for the macroscopic stress, in the sense that the macroscopic stress is still de0ned as the average value of the microscopic stress. In our opinion, this static de0nition is not appropriate in dynamic conditions. The purpose of the present work is to formulate a dynamic constitutive formulation for porous materials. In Sections 2 and 3, a general form for the dynamic macrostress is de0ned by averaging. It is also built up of two components: a static viscoplastic part and a dynamic part. In the limiting case where inertia is neglected, the static de0nition for the macrostress is retrieved. In Section 4, we focus our attention on the dynamic behaviour of a porous material. We assume that a representative volume element can be represented by a hollow sphere. As in the Gurson’s approach, an approximate velocity 0eld, which is kinematically admissible, is considered. Assuming a rigid viscoplastic behaviour for the matrix, an explicit macroscopic stress strain-rate relationship is derived from the principle of virtual work and provides the dynamic constitutive behaviour for the porous material. In Section 5, illustrations of the proposed approach are presented for a porous medium subjected to hydrostatic tension or compression and to axisymmetric loading. The present approach is shown to agree with previous works (Carroll and Holt, 1972; Ortiz and Molinari, 1992; Tong and Ravichandran, 1993, 1995), when spherical loadings are of interest. In addition, the e1ect of triaxiality is clearly captured.
2. Denition of the macroscopic elds A representative volume element (RVE) of a porous material is considered. A cartesian coordinate system is adopted in the present analysis. Mechanical 0elds (Cauchy stress and strain rate) acting at a material point of the RVE are ij and dij . The equation of motion is the following: ij; j =
dvi dt
with ij = ji ;
(1)
where (:; j ) represents the partial derivative with respect to the coordinate xj , is the density of the material, vi the velocity components. The strain rate tensor is de0ned by dij = 12 (vi; j + vj; i ):
(2)
The following kinematic condition is applied to the boundary @ of the RVE: C=D·x
on @ ;
(3)
where “·” represents the simple contracted product. D is supposed to be uniform and represents the macroscopic strain rate tensor. It is straightforward that the macroscopic
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A. Molinari, S. Mercier / J. Mech. Phys. Solids 49 (2001) 1497 – 1516
strain rate D is the volume average of the corresponding microscopic quantity: D = d;
(4)
where · = (1=| |) · d and | | represents the volume of the RVE. The macroscopic stress is de0ned by = + ⊗ x;
(5)
where ⊗ denotes the tensorial product and = dC=dt is the acceleration of a particle. This relationship is proposed so that and D are work conjugated quantities through the principle of virtual work (see Appendix A): 1 d|C|2 ; (6) : D = : d + 2 dt where (:) is the twice contracted product. Note that, due to dynamic conditions, the static de0nition of the macrostress = is not valid; an inertial term is added. When boundary conditions consist in prescribing surface tractions as ·n=·n
on @ ;
uniform;
(7)
then it is easily proved (Appendix A) that the macrostress is related to corresponding microscopic stress by Eq. (5). In that case, the macroscopic strain rate D is de0ned by relationship (4). As for kinematic boundary conditions, macroscopic quantities and D are work conjugated, since relation (6) is preserved (see Appendix A). 3. Macroscopic constitutive law To obtain the constitutive behaviour, a link between the two macroscopic quantities and D has to be developed. For that purpose, the kinematic boundary conditions (3) are prescribed at the remote boundary of the RVE. The matrix material is supposed to be incompressible. The local velocity 0eld and strain rate tensor are considered to be related to the macroscopic strain rate tensor as follows: C = C(x; D);
d = d(x; D):
(8)
In addition, the local material behaviour is speci0ed by the introduction of a stress potential (x; d). Thus, the stress tensor is expressed in terms of the strain-rate tensor d via the following relationship: =
@ (x; d): @d
(9)
A relation between the macrostress and the macroscopic strain rate D is sought using a variational approach. The volume average of the increment of rate of work, evaluated in terms of the stress potential, has the form @ : d = (x; d) : d ; (10) @d
A. Molinari, S. Mercier / J. Mech. Phys. Solids 49 (2001) 1497 – 1516
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where d represents the increment of strain rate. Using expression (8), the stress potential can be expressed in terms of the macroscopic strain rate D: ˆ D): (x; d(x; D)) = (x;
(11)
By means of this relation, the average of the increment of rate of work is written as @ @ˆ @ @d : D = : d ; (12) : D = : @D @d @D @d where D represents the increment of macroscopic strain rate. Therefore, the macroscopic potential de0ned by 1 ˆ = ˆ d (13) | | satis0es the following relationship: ˆ @ : D: (14) @D The average increment of rate of work can also be evaluated by means of the principle of virtual work when C, d and D are replaced by the corresponding increments C, d and D. Thus it follows (same calculations as in Appendix A) that 1 · C d : (15) : d = : D − | | : d =
The combination of Eqs. (14) and (15) provides ˆ @ 1 : D = · C d : : D + @D | |
(16)
From Eq. (8), the increment of the velocity 0eld appears to be a linear function of the strain rate: @C C = : D = K(x; D) : D: (17) @D Thus, Eq. (16) leads to the following expression: ˆ @ 1 − · K d : D = 0: (18) − @D | | In Eq. (18), D is an arbitrary symmetric tensor. Therefore, the symmetric part of the macrostress is given by s = static + (∗ )s ;
(19)
where
static
ˆ @ = @D
and
1 = | | ∗
· K(x; D) d :
(·)s represents the symmetric part of the second order tensor (·).
(20)
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The viscoplastic behaviour of the RVE under dynamic conditions is de0ned by expression (19), in which the macrostress is related to the macroscopic strain rate. The macrostress is composed of a viscoplastic stress calculated from the quasistatic ˆ and an inertia-dependent term. In the application to porous materials, it potential will be demonstrated that the macrostress is indeed a symmetric tensor. 4. Applications to porous materials The purpose of the present work is to propose a modelling of the constitutive behaviour for porous materials with incompressible matrix, subjected to dynamic loading. To go further in the analysis of the previous section, we have to evaluate in Eq. (19) the two contributions to the macrostress static and ∗ . Numerous static stress potentials could have been used in the present analysis to evaluate static (see for example Gurson, 1977; Olevsky and Skorohod, 1988 or Leblond et al., 1994). The ˆ adopted in the following analysis is the one proposed macroscopic stress potential by Olevsky and Skorohod (1988) since it has been used successfully to model powder compaction (see Olevsky and Molinari, 2000 for a review). Olevsky and Skorohod have developed a phenomenological formulation for the potential of porous materials with incompressible matrix: ˆ = where
A (1 − f)W m+1 ; m+1
W= =
2 + e2 ; 1−f
D : D
and
(21)
= (1 − f)2 ;
=
2 (1 − f)3 ; 3 f
e = tr(D):
(22)
D represents the deviatoric part of the macroscopic strain rate, f is the porosity (volume fraction of voids) and m is the strain rate sensitivity of the matrix. The ˆ has been derived assuming that the skeleton (the matrix) of the porous potential material has a nonlinear viscoplastic behaviour de0ned by the following stress potential: A m+1 (23) ; m+1 s √ where s = d : d ; d being the strain rate in the skeleton; A is a scaling factor for the yield stress. The evolution law for the porosity is given by =
f˙ = 3(1 − f)Dm ;
(24) ·
where Dm stands for tr D=3. ( ) represents the material time derivative operator. The static macrostress tensor is linked to the macroscopic strain rate D by ijstatic =
ˆ @ = AW m−1 (Dij + @Dij
eij ):
(25)
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Fig. 1. Schematic representation of the porous material. A hollow sphere (internal radius a and external radius b) is taken as the representative volume element.
3 From Eq. (25), it can be seen that 2 A corresponds to the yield stress in tension when the matrix has a rigid perfectly plastic behaviour. Relationship (19) de0nes the symmetric part of the macrostress providing that the acceleration 0elds and the concentration tensor K can be calculated. Porous materials have complex microstructures, with voids of di1erent sizes and di1erent shapes. However, in the present approach, the material is represented as in Gurson’s model (1977), by a composite hollow sphere with internal and external radii a and b, with porosity f = a3 =b3 , see Fig. 1. In this representative volume element, a single void is embedded in the matrix. It has to be noticed that interactions between voids are not taken into account; therefore, only material with low porosity can be modelled. The velocity 0eld, solution of the proposed boundary value problem, cannot usually be obtained in a closed analytical form. In this analysis as in the Gurson’s model (1977), an approximate representation for the velocity 0eld is assumed: vi =
Dij xj
3 b + Dm xi ; r
(26)
where r is the norm of the position vector. Inside the void, the velocity 0eld is extended from the matrix velocity (26) in an arbitrary way, but preserving the continuity of the velocity at the void surface. The approximate velocity 0eld is split in two parts: shape change at constant volume and volume change at constant shape. It is easily shown that the proposed velocity 0eld satis0es both the incompressibility and the kinematic boundary conditions (3). Note that the shape of voids is spherical in the proposed formulation. Therefore for consistency, the dilatation part in Eq. (26) (scaled by the mean strain rate) has to overwhelm the shape change part (represented by the deviator of the strain rate), see Rice and Tracey (1969). This restricts the range of applications of the model to cases with large stress triaxiality. To evaluate , we can use relationship (16) and calculate the inertia term. Since the velocity 0eld is given by Eq. (26), the term (1=| |) · C d can be determined explicitly. In that case, only the symmetric part of can be evaluated. A di1erent way is adopted so that both the symmetry and the expression of are obtained. The analysis of Sections 2 and 3 with the more general boundary condition C = L · x is performed, where L represents the macroscopic velocity gradient. Some details of the
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analysis are provided in Appendix B. Finally, the macroscopic stress is obtained as
1 static 2 1 −2=3 ˙ (f = + a − f) D + C − tr(C)I + (f−2=3 − 1)Dm D 3 5 1 + (f−2=3 − 1)(D : D )I − (f−2=3 − f−1 )D˙ m I 6
5 1 + 3f−1 − f−2=3 − f−2 Dm2 I ; 2 2
(27)
where C = D · D ; a is the current void radius and I is the second order identity tensor. The term scaled by a2 de0nes the dynamic contribution to the macrostress, noted (∗ )s in (19), and is related to the current porosity f, to the macroscopic strain rate and its time derivative. Note that is symmetric. Using the principle of virtual work, it can be easily shown that satis0es the macroscopic equation of motion, and that the relationship between traction force t and stress tensor is preserved (t = · n). Relationship (27), providing the behaviour of a porous material subjected to dynamic loading, has been obtained with the kinematic boundary conditions (3) prescribed at remote boundaries. Nevertheless by inversion of relationships (27), cases where tractions of form (7) are applied at the boundary, can be easily treated. Therefore the present model predicts the behaviour of porous material under stress or kinematic control. Compared to other models developed in the literature, there is no limitation to pure hydrostatic paths; complex loadings can be considered. 5. Results In the following, we consider a material representative of some aluminium. This material is subjected to a linearly increasing stress loading. Di1erent remote 0elds are applied: hydrostatic expansion, hydrostatic compression and axisymmetric loadings. The static analysis can be obtained as a limiting case of the dynamic analysis when the mass density vanishes → 0. This way is not followed because of the important computational time needed to reach the limit. A direct solution of the quasistatic problem is obtained by solving Eqs. (24) and (25). 5.1. Hydrostatic path In this section, time-dependent hydrostatic loadings are considered. On the boundary @ , the traction force t = · n is · n = pn:
(28)
The pressure is considered to be a linear function of time, with initial pressure p = 0. In the following, p˙ will designate the pressure rate. To validate the proposed approach, the dynamic compression of a rate-independent material is simulated. The con0guration adopted is consistent with the case analysed by Carroll and Holt (1972). Voids are spherical with initial radius ai = 20 m, initial
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Fig. 2. Dynamic compression of a porous aluminium. Results of the dynamic analysis of Carroll and Holt (1972) are retrieved. Initial radius ai =20 m, initial porosity fi =0:23. The pressure rate is p=−10 ˙ MPa=ns. Material constants are: A = 245 Int. Syst., m = 0:005 and = 2700 kg=m3 .
porosity fi = 0:23. The tensile yield stress of the aluminium considered here is 300 MPa. Therefore, the material parameters used in Eq. (25) are set to A = 245 × 106 Int. Syst., m = 0:005. The low value of m has been chosen as in Tong and Ravichandran (1993), in order to be close to the rate-independent assumption. The mass density is = 2700 kg=m3 . The hydrostatic compression loading rate is assumed constant p˙ = −10 MPa=ns. Fig. 2 represents the evolution of the distention factor 1=(1 − f) versus time. A close agreement is observed between the present approach and the Carroll and Holt (1972) dynamic analysis. The e1ects of strain rate sensitivity and of loading rate on pore collapse are investigated now. Dynamic and static analyses are compared for hydrostatic compression tests and for the aluminium material previously considered. The initial radius is ai = 20 m, the porosity fi = 0:1. For the relatively low pressure rate p˙ = −10 MPa=ns, the pore collapse is controlled by viscosity (cf. Fig. 3(a)). Results are identical for static and dynamic approaches (when m = 0:05 and 0.15). In addition, an increase of the strain rate sensitivity, which stabilizes the
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Fig. 3. Dynamic compression of a porous aluminium (A = 245 Int. Syst., = 2700 kg=m3 , initial porosity de0ned by ai = 20 m and fi = 0:1). Comparisons between dynamic and static analyses for two loading rates and two strain rate sensitivities (m=0:05 and 0:15) are presented: (a) p=−10 ˙ MPa=ns, (b) p=−250 ˙ MPa=ns.
values below 0.3 because it is known that the coalescence of voids occurs when the porosity f reaches 0.1– 0.3. It is observed that an increase of the loading rate (from 10 to 250 MPa=ns) triggers the process of rapid expansion earlier. For a given loading rate, an increase of m delays the void growth. It is interesting to note that, contrary to hydrostatic compression, inertia acts in a signi0cant way. At early stage of the deformation, static and dynamic analysis are identical. The void growth is controlled by the viscosity of the material. Later, the static solution grows faster than the dynamic one. Inertia dominates the late stage of the deformation (Ortiz and Molinari, 1992).
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Fig. 4. Dynamic hydrostatic extension of a porous aluminium (A = 245 Int. Syst., = 2700 kg=m3 , initial porosity de0ned by ai = 20 m and fi = 0:1). A comparison between static and dynamic analyses is carried out for di1erent loading rates (p˙ = 10, 250 MPa=ns) and strain rate sensitivities (m = 0:05 and 0:15).
For m = 0:05, and p˙ = 10 MPA=ns, the porosity f = 0:3 is reached for t = 376 ns in the static analysis, compared to t = 455 ns in the dynamic one. To illustrate the e1ect of di1erent parameters on the porosity evolution, a parametric study is carried out. The hydrostatic expansion (stress rate p˙ = 10 MPa=ns) of porous aluminium (voids de0ned by ai = 20 m and fi = 0:01) is chosen as the reference test. The e1ect of the strain rate sensitivity has already been clari0ed in Fig. 4 and is shown to delay the rapid evolution of porosity. In Fig. 5 results are obtained by varying the mass density, the
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Fig. 5. E1ects of the mass density, of the initial radius, of the initial porosity and of the
with q˙ a constant term. The triaxiality factor T is related to by Sm 1+2 ; (30) T= e = S 3|1 − | where Sm = 13 tr is the hydrostatic stress and Se = 32 Sij Sij the e1ective stress. When = 1, hydrostatic loading is prescribed, while = 0 represents simple tension or compression. During axisymmetric loading, the shape of voids will evolve toward spheroids. Numerous authors have studied the shape evolution (Budiansky et al., 1982; Koplik and Needleman, 1988; Lee and Mear, 1994; Gologanu et al., 1994; Briottet et al., 1996). The evolution is shown to be a complicated function of the triaxiality T . Budiansky et al. (1982) and Lee and Mear (1994) have performed a parametric study on T and have proved that a spherical void evolves towards an asymptotic shape among the following: needle, cylinder, prolate and oblate spheroids, sphere, crack and point. Through a comparison between predictions based upon models including void evolution and models assuming that voids remain spherical, Lee and Mear (1994) have concluded that little di1erence is observed when 0:6 ≤ ≤ 1. Budiansky et al. (1982) have observed that as soon as the triaxiality factor is larger than unity T ≥ 1 ( ≥ 0:4), voids can be considered as spherical during the deformation. Since in the proposed approach, no shape evolution is considered, only results for ≥ 0:5 are presented. For low triaxiality, voids evolve towards a nonspherical shape. In dynamic loading, inertia e1ects act di1erently in the direction of loading and in the transverse direction. Therefore, results concerning the evolution of the shape can be modi0ed by inertia. An extension of the present analysis to spheroidal voids is necessary to clarify the e1ects of inertia on
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Fig. 6. In
ovalization. Applications of the present model for low triaxiality loading (for example, simple tension or compression = 0) should be avoided. The same aluminium as in the previous section is considered (A = 245 Int. Syst., m = 0:05 and = 2700 kg=m3 ). Attention is focused on triaxiality e1ects (through the parameter ). Two cases are investigated; 0rst, a compression test is performed with lateral con0nement (q˙ = −10 MPa=ns and varying from 0.5 to 1). In Fig. 6, it is observed that the closure of pores is strongly in
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Fig. 7. In
Fig. 8. In
6. Conclusion A model has been proposed to describe the behaviour of a porous material loaded under dynamic conditions. Using averaging methods and assuming homogeneous kine-
A. Molinari, S. Mercier / J. Mech. Phys. Solids 49 (2001) 1497 – 1516
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matic boundary conditions, a relationship between macrostress, macroscopic strain rate and porosity has been obtained. The macrostress is shown to be composed of a static and a dynamic part. For the static contribution, the phenomenological potential developed by Olevsky and Skorohod (1988) has been adopted. Matrix incompressibility is an essential hypothesis in the theory presented. Predictions of the present model to various stress loading paths are compared to previous results. In hydrostatic loading, the e1ects of material parameters (
(A.1)
(A.2)
Two cases are investigated; kinematic boundary conditions are 0rst considered: C=D·x
on @
(vi = Dik xk );
(A.3)
where D represents the macroscopic strain rate. Eqs. (A.2) – (A.3), with the use of the Gauss’ theorem provide 1 dvi 1 : d = ij; j xk d + Dik ik − vi Dik d : (A.4) | | | | dt
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Using the equation of motion to modify the 0rst term on the right side, relationship (A.4) is transformed into
1 1 d|C|2 : (A.5) Dik ik + i xk d = : d + 2 dt | | The de0nition of the macrostress (relationship (5)) is recognized and relationship (6) is obtained. When stress boundary conditions are applied ik nk = ik nk
on @ ;
(A.6)
relationship (5) can be proved easily. The mean value ij is expressed in the form 1 1 1 ij = ij d = (ik xj ); k d − ik; k xj d : (A.7) | | | | | | With the use of equation of motion (1) and Gauss’ theorem, it follows that 1 ij = xj ik nk ds − i xj : | | @
(A.8)
Since tractions (A.6) are prescribed at the boundary, relationship (A.8) becomes ij = ij − i xj
(A.9)
which proves (5). The combination of Eq. (A.2) with the boundary relation (A.6) leads to 1 dvi 1 : d = ij vi nj ds − vi d : | | @ | | dt
(A.10)
Gauss’ theorem is applied to the 0rst term on the right side of the previous relation. Using the de0nition of the macroscopic strain rate (4), relationship (6) is retrieved: 1 d|C|2 : (A.11) : d = : D − 2 dt Therefore, macroscopic quantities and D, related to microscopic values and d by relationships (4) and (5), are work conjugated for each boundary condition considered. Appendix B. Symmetry and formulation of the macroscopic stress tensor To prove the symmetry of the macroscopic tensor and to derive the constitutive law, the analysis proposed in Sections 2 and 3 is developed on the basis of a more general kinematic boundary condition C = L · x;
(B.1)
where L is the macroscopic velocity gradient. The macroscopic strain rate D = 12 (L + t L), the spin tensor , = 12 (L − t L) and the velocity gradient L are related to the local tensors d; and l by D = d;
, = ;
L = l:
(B.2)
A. Molinari, S. Mercier / J. Mech. Phys. Solids 49 (2001) 1497 – 1516
1513
t
( ) denotes the transpose operator. The macroscopic stress is still de0ned by Eq. (5) so as and L are work conjugated: 1 d|C|2 t : (B.3) : L = : d + 2 dt With the boundary condition (B.1), the local velocity 0eld and strain rate tensor are related to the macroscopic velocity gradient and strain rate tensor C = C(x; L);
d = d(x; D):
(B.4)
The local material behaviour is still de0ned by (9). As a consequence, with use of the principle of virtual work (B.3), relationship (16) becomes ˆ 1 @ : D + : L = | | @D t
· C d
(B.5)
with C =
@C : L = K(x; L) : L: @L
(B.6)
From Eqs. (B.5) and (B.6), it follows that ˆ 1 @ 1 · K d : D − + · K d : , = 0: − − @D | | | | (B.7) In Eq. (B.7), D and , are arbitrary. Therefore, the macrostress which is not a symmetric tensor in general, is given by s = static + (∗ )s
a = −(∗ )a ;
(B.8)
where static =
ˆ @ @D
and
∗ =
1 | |
· K(x; L) d :
(B.9)
(·)a represents the antisymmetric parts of the second order tensor (·). To evaluate the dynamic contribution ∗ to the macrostress, the porous material is still represented by a composite hollow sphere. The solution for the exact velocity 0eld cannot be reached in a closed form. Therefore, a simpli0ed velocity 0eld is considered: C = L · x + Bx
(B.10)
with L being the deviatoric velocity gradient, B = Dm (b=r)3 and Dm = 13 tr D. The kinematic boundary condition (B.1) is satis0ed together with the incompressibility condition in the matrix.
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A. Molinari, S. Mercier / J. Mech. Phys. Solids 49 (2001) 1497 – 1516
From (B.10), the acceleration of a material particle is
˙ · x + L · C + Bx ˙ + BC: (x) = L
(B.11)
The variation of the velocity due to the change L in the boundary condition is C = L · x + Bx:
(B.12)
Thus, the integral term I = · C d of (B.5) can be written as ˙ · x + C · t L · L · x ˙ · L · x + Bx · L · C d = (x · t L I=
˙ · L · x + BBx ˙ + Bx · L · C + Bx · x + BC · L · x + BBC · x) d : (B.13) The RVE, , is represented by a spherical shell (inner radius a and outer radius b), see Fig. 1. The calculations, which are not presented here in detail, are conducted in spherical coordinates; the following result is obtained: i=3 k=3 1 1 ˙ ii + Cii − (f−2=3 − f) D a2 Ckk + (f−2=3 − 1)Dm Dii I= 5 3 i=1
k=1
1 + (f−2=3 − 1)(L : L ) − (f−2=3 − f−1 )D˙ m 6
5 −2=3 1 −2 −1 Dm2 Dii − f + 3f − f 2 2 1 + a2 (f−2=3 − f)(L˙ij + Cij ) + (f−2=3 − 1)Dm Lij Lij ; 5
(B.14)
1≤i≤3
1≤j≤3 i=j
where C=L ·L . By identi0cation (the components Lij being considered as independent terms) the macroscopic stress tensor is obtained from (B.7) as
1 s s static 2 1 −2=3 ˙ = + a − f) D + C − tr(C)I + (f−2=3 − 1)Dm D (f 5 3 1 + (f−2=3 − 1)(L : L )I − (f−2=3 − f−1 )D˙ m I 6
5 1 + 3f−1 − f−2=3 − f−2 Dm2 I ; 2 2 a
= a
2
1 −2=3 a −2=3 ˙ − f)(, + C ) + (f − 1)Dm , : (f 5
(B.15a)
(B.15b)
A. Molinari, S. Mercier / J. Mech. Phys. Solids 49 (2001) 1497 – 1516
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When spin e1ects vanish, a =0. Then, the macroscopic stress is shown to be symmetric and relationship (27) is retrieved:
1 static 2 1 −2=3 ˙ = + a − f) D + C − tr(C)I + (f−2=3 − 1)Dm D (f 5 3 1 + (f−2=3 − 1)(D : D )I − (f−2=3 − f−1 )D˙ m I 6
5 1 + 3f−1 − f−2=3 − f−2 Dm2 I ; 2 2
(B.16)
where C = D · D . References Briottet, L., KlFocker, H., Montheillet, F., 1996. Damage in a viscoplastic material Part I: Cavity growth. Int. J. Plasticity 12, 481–505. Budiansky, B., Hutchinson, J.W., Slutsky, S., 1982. Void growth and collapse in viscous solids. In: Hopkins, H.G., Sewell, M.J. (Eds.), Mechanics of Solids. Pergamon Press, Oxford. Carroll, M.M., Holt, A.C., 1972. Static and dynamic pore-collapse relations for ductile porous materials. J. Appl. Phys. 43, 1626–1636. Cortes, R., 1992. The growth of microvoids under intense dynamic loading. Int. J. Solids Struct. 29, 1339– 1350. Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing nonspherical voids – case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754. Gologanu, M., Leblond, J.B., Devaux, J., 1994. Approximate models for ductile metals containing nonspherical voids – case of axisymmetric oblate ellipsoidal cavities. J. Engng. Mater. Technol. 116, 290–297. Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth: Part I – Yield criteria and
1516
A. Molinari, S. Mercier / J. Mech. Phys. Solids 49 (2001) 1497 – 1516
Ortiz, M., Molinari, A., 1992. E1ect of strain hardening and rate sensitivity on the dynamic growth of a void in a plastic material. J. Appl. Mech. 114, 48–53. Ponte-Casta˜neda, P., 1991. The e1ective mechanical properties of nonlinear isotropic materials. J. Mech. Phys. Solids 39, 45–71. Rice, J.R., Tracey, D.M., 1969. On the ductile enlargement of voids in triaxial stress 0elds. J. Mech. Phys. Solids, 17, 201–217. Tong, W., Ravichandran, G., 1993. Dynamic pore collapse in viscoplastic materials. J. Appl. Phys. 74, 2425–2435. Tong, W., Ravichandran, G., 1995. Inertial e1ects on void growth in porous viscoplastic materials. J. Appl. Mech. 62, 633–639. Tvergaard, V., 1982. On localization in ductile materials containing spherical voids. Int. J. Fract. 18, 237–252. Wang, Z.P., 1997. Void-containing nonlinear materials subject to high-rate loading. J. Appl. Phys. 81, 7213–7227. Wang, Z.P., Jiang, Q., 1997. A yield criterion for porous ductile media at high strain rate. J. Appl. Mech. 64, 503–509.