Mechanical and fracture behaviour of porous materials

Engineering Fmcrun Mechanics Vol. 2X. No. S/6, pp. 627442. Printed in Great Britain.

MECHANICAL

1987

0013-7944187 $3.00 + .ocl @ 1987 Pergamon Journals Ltd.

AND FRACTURE BEHAVIOUR POROUS MATERIALS

OF

Ph. BOMPARD, DAN WEI, T. GUENNOUNI and D. FRANCOIS Ecole Centrale des Arts et Manufactures, Grande Voie des Vignes, F 92290 Chatenay Malabry, France CNRS-GRECO

Grandes deformations

et endommagement.

Abstract-The elastic moduli of porous materials represented as a combination of spherical, cylindrical or disk shaped holes or solid elements was calculated using a self consistent method. A yield criterion could be found by stating that the elastic distortion energy evaluated with these moduli was equal to a critical value. Another yield criterion could be approximated from the results of finite elements computations using the homogenization technique. These criteria possess the same homogeneity as the yield criterion of the dense material. In the presence of a secondary population of smaller cavities the yield criterion was found to be different from the one obtained by simply adding the two volume fractions of holes. The hole growth rate deduced from these calculations was proportional to the equivalent strain rate and to the stress triaxiality ratio, inkeeping with the evolution of damage proposed by Iemaitre. This hole growth rate was enhanced by a population of small secondary cavities. In general the strain hardening and the damage are coupled. However for sintered porous nickel the introduction of an initial damage parameter proportional to the porosity suihced to describe the strain hardening behaviour. The condition for hole coalescence could not be connected to either a critical value of the porosity, or of the damage parameter, or a necking condition. The yield criterion modified by replacing the yield stress by the fracture stress could describe experimental results. Yet a critical local strain criterion also gave a good fit. The fracture toughness of syntactic foams could be explained by the local stress distribution in the glass spheres. In the case of ductile porous nickel the COD at initiation decreased as the fracture strain. This material exhibited a large tearing modulus whose decrease when the porosity was increased could be taken into account by the damage parameter. Fatigue crack propagation rates could also be rationalized with the use of the damage parameter and by reducing the surface of the material to be fractured.

INTRODUCTION AS FOR ANY other materials the fracture properties of a porous material can be measured using the usual normalized methods. What we would like to understand is how they are related to the porosity, and to the mechanical behaviour of the matrix. T. Yokobori was one of the first to study the relations between fracture toughness or fatigue crack growth resistance and the microscopic mechanical behaviour of the material and the physical fracture processes. The method he used remains an efficient one and we would like to follow it in the case of porous materials. It is then needed to study constitutive equations of the porous material and then to try to describe as precisely as possible the mechanisms which take place at crack tips in monotonic or in cyclic loadings. ELASTIC BEHAVIOUR

OF A POROUS MATERIAL

The elastic behaviour of porous material was studied by many authors[l-41. They can also be deduced from the elastic properties of two phases materials by decreasing indefinitely the moduli of one phase[5-131. This problem was studied again by Frappier[l4] who used self consistent method to calculate the elastic moduli of a material containing holes of various shapes: spherical, cylindrical or disk shaped, or made of aggregates of spherical, cylindrical or disk shaped solid elements considered as N different phases. Using Eshelby’s theory of the inclusion[l5] he obtained the following equations for the shear and bulk moduli of a porous material containing holes of given shapes: G=

Go

1-a

K = K.

1 - “c: ; GO EPW 28:5/6-J

627

1-e

(1)

Ph. BOMPARD

628

ef crl.

Table I. Shape factors Disks

Cylinders

Spheres

1

1

K,+;G,

K,+;C,+G

11%

1

1

t

G,+cYK+SG ’

j

6 K+2G

2

1

s

3K+G

G,+G------

;j(,,:-G.j

2 +SC,

4 3

’

2 1 +-5G,+G

3K+7G

where GO and k; are the shear and bulk moduli of the matrix, CYis the porosity, cri the volume fraction of type i porosity and nj and ki are shape parameters given in Table 1. For a porous material made of solid elements of various shapes (opened porosity):

(2)

where 8: is the volume fraction of type i element. YIELD CRITERION Stress concentrations appear at the border of the holes or in the neck of ligaments and these are the locations where microplasticity begins. However macroplasticity is exhibited only when complete yielding takes place. Various methods can be used to predict the yield criterion of the material. Frappier[l4] extended Mises criterion to porous materials by simply stating that yielding should take place when the elastic distortion energy in the solid matrix reached a critical value. He obtained the elastic distortion energy as: (3) where (Yis the porosity, Y Poisson’s ratio, Y the Young’s Modulus of the porous material, Xeceq the Mises equivalent stress, &, the mean stress and K and KC,the bulk moduli of the porous material and of the matrix material respectively. Equating this expression with the critical value in uniaxial tension he thus obtained the yield criterion as a function of porosity. zj+pCf,=a;(l+p)

(4)

with

cry being the yield stress of the dense material. The morphology of the porous material is introduced in the /3 factor through its influence on the elastic moduli, v and K. Frappier’s yield criterion is similar to Khun and Downey’s[l6] and to Rosenberg’s[l7] who used the plastic Poisson’s ratio of the porous material and the total elastic energy rather than the distortion energy, an hypothesis which is unusual. A slightly modified yield criterion is also proposed by Frappier in the case where the elastic

Fracture behaviour of porous materials

629

limits are not the same in tension and in compression. Gurson[ 181 sought kinematical admissible velocity fields and reached the following yield criterion:

(5) a@ being the yield stress of the matrix. C1 and C2 depend upon the stress state: plane strain or axisymmetric conditions. However Guennouni[l9] demonstrated that for a viscoplastic Norton-Hoff material with a stress exponent q, the yield criterion must be homogeneous of degree q. For a perfectly plastic material the yield criterion must be homogeneous of degree I, a condition which is not met by Gurson’s criterion. Guennouni and Frangois[20] used numerical computations and homogeneization technique in order to obtain the yield criterion of a porous medium with a periodic arrangement of cylindrical cavities. The calculations were carried out in plane strain for various stress states characterized by the ratio of the principal stresses. The primitive cell was made of a perfectly plastic medium. The porosity was varied according to the size of the hole relative to that of the cell. Figure 1 shows the yield criterion obtained as compared with Gurson’s. A large difference was found for intermediate stress triaxiaiity ratios. It comes from the velocity field chosen by Gurson which corresponds to a general flow of the whole medium differing from the gliding of rigid blocks as suggested by the computed behaviour. The following yield criterion approximated well the computed result.

where A and B are functions of the hole volume fraction (Y: A(Q) = - L (1 - 1.92 + 5.5’7~~~- 6.03~~) In a, J3 B(a) = 1 - 1.33~uO.~ if B(a)=

a s 0.1256

l.l89[exp(-4.9(y)-4.4~

lW2]

if

~y>O.l256.

Guennouni computed also the yield criterion of a porous material containing in addition to the large holes a second population of small cavities, increasing the compressibility of the matrix. It was found that owing to the interaction between the two populations the effect is not simply additive (Fig. 2).

0

1

2

3

G

5

6

{3X,/ Cry0 Fig. 1. The yield criterion obtained by Guennouni and FranGois[ZO] (formula 6) as compared with Gurson[lS].

Ph. BOMPARD

630

et al.

Rice and Tracey

0.28

I

Fig. 2. Experimental

10-Z

10-j

10-4

W

I 10-l

preexponential factor of cavity growth models as a function of the volume fraction of void forming inclusions [from 25 and 261.

In this case the B parameter in the yield criterion remains the same if expressed as a function of the total hole volume fraction (Yewhereas the A parameter becomes: A(cr, oo) = -L(l &

- 1.92a,+5.57a:-6.03a:)ln[a+ao-

1.25((u(uo)0~7+0.227(aao)0~33] (7)

where a0 is the volume (a,=cu+a!g-(YQO).

fraction

FRACTURE

of small holes in the matrix excluding

BEHAVIOUR

GROWTH

the large holes

OF HOLES

In the case of a porous plastic material the deformation in a tensile test is accompanied by the growth of the holes. The conservation of the mass shows that the rate of hole growth d is a function of the mean plastic strain rate &m (in plane strain &,, = &.:,,+ &,,,) &=(1-(Y)&

(8)

(neglecting the elastic strain). &, can be deduced from the yield criterion using the associated flow rule. When using Gurson’s criterion, the cavity growth rate is found to be very similar to Rice and Tracey’s[21] who analysed the growth of an isolated spherical cavity of radius R in an infinite rigid plastic medium: dR F=0.283&,exp

(9)

& being the equivalent plastic strain rate. Various experiments [22-251 showed that this law could represent the results except that the Rice and Tracey preexponential factor 0.283 was usually exceeded (Fig. 2). A possible interpretation of this trend relies on the interaction between cavities which is not incorporated in Rice and Tracey’s model. However Bompard[26] testing porous sintered nickel found still a different behaviour (Fig. 3). He measured a cavity growth rate which decreased when the porosity was increased.

Fracture behaviour of porous materials

.

631

density

t\

Fig. 3. Experimental preexponential factor of the hole growth rate measured by various techniques as a function of the porosity of sintered nickel[26].

From Guennouni’s computations the associated flow rule yields a cavity growth rate which indeed checks this latter result. It is given by the formula: dR f=3x3--_

l-a CY

(10)

Figure 4 shows a comparison between this formula and Rice and Tracey’s model or Budiansky, Hutchinson and Slusky’s mode1[27]. A linear variation of the cavity growth rate with the stress triaxiality ratio is found, as it comes out of the necessary homogeneity of the yield criterion. This contrasts with the predictions of models which consider a single cavity growing in a medium deforming homogeneously. On the contrary the finite element computations leading to formula (10) show that the interaction between cavities induces a flow localization as evidenced on Fig. 5. As seen on Fig. 4 the difference between a linear and an exponential influence of the stress triaxiality ratio is difficult to prove experimentally. It is worth mentioning that these results are in good agreement with Lemaitre’s model of damage[28]. According to this model in a tensile test the damage parameter D would be given by the simple expression: D E-E, -_=Dc ER-Eo

(11)

where ,Dc is the critical value of the damage parameter when the elongation E reaches the fracture strain E,, E. being a threshold. This simple result is the consequence of the conservation of the mass (formula 8). For porous nickel Eo-is considered to be zero whereas the material contains an initial damage Do which is a function of the porosity. If the damage parameter D is related to the change in Young’s modulus Y by D=l-$

(12)

this relation 11 predicts a linear decrease of Young’s modulus with the deformation. The experimental results are in good agreement with this as shown on Fig. 6 which discloses also the equivalence between the initial porosity and the growing holes during deformation. Finally, Guennouni and Fransois’s[20] computations show that, in connection with the modification of the yield criterion, the growth rate of large holes is enhanced by a population of smaller cavities.

Ph. BOMPARD ef ul.

632

1

2

stress triaxidty

0

0

1

ratio

3 &j/X,

2

3

stress tr~xia~ity ratio &$F.Q Fig, 4. Comparison between the cavity growth rate computed by Guennouni and Francois[20] compared with Rice and Tracey[ZI] or Budiansky, Hutchinson and Slusky[27].

Fig. 5. One instance of Row localization obtained by the computations of Guennouni and Fran$ois[ZO] in one cell of a periodic porous medium.

Fracture behaviour of porous materials

633

\ ‘“i\

0

0.2

0.4

I I I ‘\ 0.6 0.8 1

. S * 1.2 I4 1.5

PLASTIC DEFORMATION

Fig. 6. Linear decrease of Young’s modulus of porous sintered nickel as a function of the deformation in a tensile test[26]. Translation of the data points to take into account the initial porosity discloses the equivalence between this parameter and the plastic growth of holes.

PLASTIC BEHAVIOUR The stress-strain curve of a material results from the combination of the strain hardening due to dislocations multiplication and to the strain softening from void initiation and growth, In general these two phenomena are strongly coupled, the free energy of the medium being of the form proposed by Guennouni: ,=;L(D)&e+g(a,

D)

(13)

where L(D) is the Hookeian tensor modified by the damage parameter D and a the strain hardening parameter (the plastic deformation in an uniaxial stress state). In the case of nickel of full density it can be shown that the coupling component ag/aD of the thermodynamic damage force represents 97% of the total[26]. On the contrary it is negligible in the case of porous

theoretical curve

u-

0

~

0.1

02

0.3

0.4

0.5

PLASTIC DEFOfj+lATlON

Fig. 7. Coincidence between the experimental stress strain curves of sintered nickel of various porosities with the stress strain curve deduced from the hehaviour of full density nickel using an initial damage parameter D,, (eq. 14)[26].

et al.

Ph. BOMPARD

634

nickel. In that case the plastic behaviour

follows rather well the simple law:

I: = (1 - &)(ao+

kEZ)

(14)

u0 + kE; representing the stress strain law of the nickel of full density, and Do the initial damage due to the porosity (26) (Fig. 7). HOLE COALESCENCE The condition for coalescence is yet poorly known. A simple idea is that it should take place when the porosity reaches a critical value. Bompard having found that Young’s modulus decreased proportionally to the porosity for sintered nickel, a quick glance at Fig. 6 clearly shows that the large differences between Young’s modulus at fracture ruin this hypothesis[29]. The damage parameter being itself a function of Young’s modulus, reaching a critical value of this parameter is not the condition for fracture[29]. Hole coalescence can be the result of plastic instability of the remaining ligaments. Using Considere’s approach on the load carried by these ligaments: F = uS( 1 - D),

(15)

S being the total section of the specimen, and m the local stress in the ligaments, and the volume conservation of the solid material

v, = SL( 1 -

a)

(16)

where L is the length of the specimen, the following necking condition is obtained: (17) It was experimentally found that D = 2.3~~ for porous nickel. Using the cavity growth law found by Guennouni (eq. 10) this last condition gives a strain at instability ER which decreases very slowly when the porosity is increased and which is much larger than the experimental values. It thus clearly appears that the fracture occurs by another mechanism. Guennouni tried to simply replace in the yield criterion (eq. 7) the yield stress by the fracture stress of the dense material and he obtained a rather good fit with various experimental results (Fig. 8).

*UR 02 iron n sintered

1

$ 0.6 vl $

0.4

5 +

0.2

&ntered +cermet

0 0

01

0.2

0.3

0.4

porosity

Fig. 8. Plot of the experimental

0.5

0.6

0.7

o<

fracture stress of different materials vs porosity compared Guennouni’s criterion (eq. 10).

with

Fracture behaviour of porous materials

635

ER’

Fig. 9. Plot of the fracture strain of porous nickel vs the porosity (eq. 20).

Lastly another simple fracture condition consists in stating that the local strain in the ligaments must reach a critical value en as suggested by Hwang[30]. The local strain E must be related to the surface fraction 1 - (Y’of solid matter along the path of maximum porosity. Putting (Y”” = pc~“~ and assuming that e = E/(1 - a’) a good fit with the experimental results of Bompard on porous nickel is obtained with p = 1.75 for low closed porosity and p = 1.3 for high opened porosity (Fig. 9). INITIATION

OF CRACK PROPAGATION

The fracture toughness K rC of a brittle porous material should be related to its fracture stress &. If the porosity is considered as being an initial damage D the effective stress in the porous material is ,:

(18)

fleff = 1_D

and the fracture stress would be reached when this effective stress becomes equal to rrff the microscopic local fracture stress. As the damage is related to the Young’s modulus Y i-+-

(1% m

Y, being the Young’s modulus of the dense material, the fracture stress would simply be given by

ZR=

Okg

(20)

m

Using formulae 1 a relation is then obtained between the fracture stress of the brittle porous material and the porosity. The stress intensity factor Kr is equal to K,, if the fracture stress & is reached over a distance d related to the micros~ucture: for instance the size of the holes, or their mean distance. Dan Wei[3 l] studied the brittle fracture of syntactic foams, a material made of hollow glass spheres imbedded in an epoxy resin, Table 2 gives the results he obtained for various foams.

Ph. BOMPARD

636

Table 2. Characteristics

Largest diameter of the glass spheres d,,, (pm)

of syntactic foams

A

B

2400 18 0.52 133

3040 37 0.84 82

2650 28.5 0.82 93

2740 33.5 0.78 82

133

82

132

86

150

90

130

90

Foam type Young’s modulus Y (MPa) (theoretical and experimental) Fracture stress X, (MPa) K,, (MPaJm) Y/Z,

et al.

Ct[l]

D

tC foam also contained glass fibers.

Table 3. Critical CTOD of porous nickel compared with the fracture strains Porosity Q ai

E, WE, (pm)

0.09 14 0.23 61

0.2 5.7 0.12 47

0.3 3.6 0.06 60

0.4 1.16 0.03 37

Taking into account the dispersion which is an inherent property of brittle materials, the critical distance d is found to be of the order of the largest glass spheres diameter, while the ratio Y/I& which, according to relation (20) should be proportional to Y,,,/v~, matches this diameter. This means that CR is material dependent. Microscopic observations showed indeed that fracture began by the cracking of the largest glass spheres and calculations revealed that the local stress applied to these spheres increased as their diameter. For ductile materials it is expected that the critical crack tip opening displacement at initiation Si should be related to the fracture strain ER. Bompard measured the fracture toughness JIc of porous nickel and deduced values of 6i given in Table 3. Although the trend is right, the ratio &/ER is not constant. This ratio must be equal to some critical distance over which the strain ER must be reached. The main conclusion is that the CTOD at initiation is very small especially when the porosity is opened. However as will be seen next, the tearing modulus is high, so that these ductile porous materials are tougher than what would be expected from their low resistance to the initiation of crack propagation. STABLE

CRACK

GROWTH

The investigations of Bompard[26] on porous nickel revealed that the slope of the fracture energy with respect to the crack growth, in other words the tearing modulus, decreased strongly as the porosity was increased as shown in Table 4. A major part of this evolution could be accounted for by using an effective J

(21)

Table 4. Evolution of the tearing modulus T,,, = (Y/u:) (d&/da) as a function of the porosity of sintered nickel Porosity (I T,

0.09 267

0.2 59.6

0.3 28.4

0.4 11.4

Fracture behaviour of porous materials

Fig. 12. Micrograph of a growing crack in nickel of 0.4 porosity[26].

637

Fracture behaviour of porous materials

639

JR : 1/(1-D)

4

5 6 Ba (mm)

Fig. 10. Jeff defined by eq. (24) vs the crack extension Aa[26].

where J is the measured value of Rice’s strain energy release rate and D the damage parameter (Fig. 10). According to Bui and Ehrlacher[32] the size h of the damaged zone, where microcracks develop, is given by the following equation: 2

1

(22)

h=(2E,/E4)

where &, is the flow stress Z. = (cry + u&i? and E0 the corresponding value of the strain. Figure 11 shows the size of the damaged zone thus evaluated which corresponds well with micrographic

l;~Lrnaxi

j,o-l

1o-2i (y ; 10-j 0

0.1

0.3

0.2

0.4

0.5

OC Fig. 11. Evolution of the size h of the damaged zone (eq. 25) vs porosity a [26].

Ph. BOMPARD

640

et al.

E 6.5 z’ 5

6’p

l

’

.

.

.

0.3

0.4

2: 5.5 5

0

0.1

0.2

0.5

o<

Fig. 13. Evolution (da/dN=

of the exponent m and of the constant CAKF) andof Cocorrected by thedamage

C of the da/dN Co=

vs AK1 curve,

C(l-D)%+‘[31].

10-2k syntactic

foam

A

. syntactic

foam

6

l

A epoxy

: p=O.66 : p=O28

resin

n l * l

10-71

I

10-I

I

.

& i

b1411

1 Ah (MPaJMM)

Fig. 14. Fatigue crack propagation rate da/dN vs AK, for syntactic foams containing spheres (A), small glass spheres (B) and for resin.

large glass

Fracture behaviour of porous materials

641

observations (Fig. 12). A large increase of this size is observed when increasing the porosity from 9% to 20%, that is at the transition between closed and opened porosity. This difference in the microcracking behaviour might well explain why on Fig. 10, all Jea vs Au curves merge together with the exception of the one corresponding to 9% porosity. FATIGUE

CRACK

PROPAGATION

As for stable crack growth the use of the effective stress related to the porosity allows to obtain good correlations of the fatigue crack growth versus AK, curves. As shown by Bompard and Francois[32] plotting (1 - D) da/dN as a function of A&/( 1 - D) brings all the curves together Fig. 13). This is justified by the fact that the fatigue crack needs to cross only a reduced surface S(l - D) while the stress (34) which opens it is the effective stress Z/(1 - D). Figure 12 shows that there still remains a variation of C,, with the porosity. It is explained by the stress concentrating influence of the holes leading to accelerated growth at low porosities, and by the blunting of the cracking tip and reduction of the plastic constraint when the cavities are interconnected at high porosities. A somewhat similar situation was observed for syntactic foams although the dispersion was larger and the conclusion depended upon the way by which the crack propagated. In foams containing large glass spheres these broke and behaved as holes. On the contrary the crack went around small spheres which acted as hard inclusions, and the crack velocity was reduced (Fig. 14). Acknowledgements-The

authors are grateful to Professor Hwang Keh Chih for very useful discussions.

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[ 151 J. D. Eshelby, The determination of the elastic fields of an ellipsoidal inclusion and related problems. Pmt. R. Sot. 241, 376 (1957). [ 161 H. A. Khun and C. L. Downey, Deformation characteristics and plasticity theory of sintered powder materials. Znr. /. Powder Merall. 7, 15 (1971). [ 171 J. M. Rosenberg, Ductile Fracture and Plasticity of Porous Solids. QfI Naval Research (197 1). [ 181 A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth yield criteria and flow rules for porous ductile media. J. Engng Mater. Technol. 99, 2 (1977). [ 191 T. Guennouni, Front&e d’tcoulement des matdriaux h&rogbnes a constituants rigides parfaitement plastiques. Cas des materiaux poreux ou lissur6s. J. M&z. ‘Z’hkor.A&., submitted. [20] T. Guennouni and D. Franeois, Constitutive equations and cavity growth rate for a porous plastic medium. ZCM 5, Beijing (1987). [21] J. R. Rice and D. M. Tracey, On the ductile enlargement of voids in triaxial stress fields. J. Mech. Plays.Solids 17, 201-217 (1969). [22] F. M. Beremin, Study of fracture criteria for ductile rupture of A 508 steel. Progress in Fracture Research. Fracture 81 (Edited by D. Francois), p. 809. Pergamon Press, Oxford (1981). [23] Sun Yao Qing and D. Francois, Influence du taux de triaxialit6 des contraintes sur la coalescence des cavites et la morphologic des cupules de la fonte a graphite spheroIdal. Revue de M6tallurgie CIT. p. 809 (Qctobre 1984). [24] Sun Yao Qing, J. M. Detraux, G. Touzot and D. FranGois, Mem. Scient. Reu. Met., p. 183 (April 1983). [25] B. Marini, Croissance des cavit& en plasticid. Rupture sous chargement non radiaux et en mode mixte. These de docteur ingenieur. Ecole Nationale Sup&ieure des Mines de Paris (1984). [26] P. Bompard, Effets endommageants de la porosit6 sur la propagation des fissures dans le nickel fritte. These de doctorat d’btat. Universite de Technologie de Compibgne (1986).

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Ph. BOMPARD

ef al.

[27] B. Budiansky, J. W. Hutchinson and S. Slusky, Void growth and collapse in viscous solids. In Mechanics of Solids (Edited by H. G. Hopkins and M. J. Swell), p. 13. Pergamon Press, Oxford (1982). [28] J. Lemaitre, A continuous damage mechanics model for ductile fracture. J. Engng Mater. Technol. 107,83-89 (1985). [29] P. Bompard and D. Francois, In Advances in Fracture Research (Edited by S. R. Valluri et al.), p. 1279. Pergamon Press, Oxford (1984). [30] Hwang Keh Chih, Private communication. [31] Dan Wei, Thesis. Ecole Centrale des Arts et Manufactures. To be. published. [32] P. Bompard and D. Francois, In Advances in Fracrure Research (Edited by S. R. Valluri et al.), p. 2049. Pergamon Press, Oxford (1984). [33] H. D. Bui and A. Ehrlacher, Advances in Fracture Research. ZCF5 (Edited by D. Francois et al.), p. 535. Pergamon Press, Oxford (1981). [34] N. A. Fleck and R. A. Smith, Powder Metal 3, 126-130 (1981).