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Anisotropic microcrack nucleation in brittle materials
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0022-5096 90 %3.00+0.00 1990 Perpmon Presr plc
ANISOTROPIC MICROCRACK NUCLEATION IN BRITTLE MATERIALS PETER GUDMLJNDSON~ Departmentof Strengthof Materialsand Solid Mechanics. S-100 44 Stockholm,
(Received 9 Februaq,
The Royal Institute Sweden
of Technology,
1989)
ABSTRACT A CONSTITUTIVE model for anisotropic
microcracking in brittle materials is developed. The model is based on a stress controlled microcrack nucleation criterion, which can vary in a random way between different microcracks. The effects of microcrack closure and a random distribution of residual stresses are included
in the analysis. The resultant inelastic strains are determined using a standard homogenization technique. Numerical results are presented for three simple loading cases : pure tension. biaxial tension and triaxial tension. Crack tip shielding resulting from microcrack nucleation is also analysed. of K,,,/K, are presented for two different microcrack nucleation criteria.
1.
and numerical
results
INTRODUCTION
THE PRESENCEof microcracks has a large influence on the mechanical properties of a material. In this investigation microcracks in brittle materials will be analysed. The main application is perhaps structural ceramics but also the behaviour of rock and ice can be described in similar ways. The material is assumed to be polycrystalline and composed of one or several phases. Furthermore, it is assumed that the single crystals may exhibit an anisotropic behaviour. Microcracks have been observed in fracture mechanics specimens of zirconiatoughened alumina by RCJHLE et al. (1987) using transmission electron microscopy. The microcracks were formed around zirconia particles and extended on grain boundary facets in the alumina and on the interface between the zirconia particles and the alumina. Indirect evidences of microcracks have also been observed for various other ceramic materials by WV et al. (1978) and in rocks by HOAGLAND et al. (1973). The nucleation and later coalescence of microcracks is often the strength limiting mechanism in brittle materials. The microcracks can however also have a beneficial effect on toughness. Due to the formation of microcracks, the stress field close to the crack tip of a macrocrack is shielded. The nucleation of microcracks can in this case even enhance the fracture toughness, This toughening mechanism has been investigated in some recent papers (HOAGLAND and EMBURY, 1980; EVANS and Fu, 1985; HUTCHINSON, 1987; ORTIZ, 1987; CHARALAMBIDES and MCMEEKING, 1988 ; t Present address:
SICOMP,
Swedish
Institute
of Composites. 5.11
Box 271. S-941 26 Pitea, Sweden.
532
P. GUDMIXDSOE:
GIANNAKOPOULOS, 1988). A competing process is the degradation of the material due to microcracking (ORTIZ, 1988). It is however believed that in many cases the shielding mechanism outweighs the strength degradation. There are other fracture toughening mechanisms for ceramic materials which have been experimentally and theoretically investigated. A promising method is transformation toughening where the material contains a phase which undergoes a stressinduced phase transformation (EVANS and CANNON, 1986). This mechanism has been analysed by BUDIANSKY et al. (1983). Another strengthening mechanism is the introduction of tough particles (particulate toughening) in the ceramic matrix material. An analysis of this phenomenon was given by BUIXANSKY et al. (1988). The combined effect of transformation and particulate toughening was analysed by AMAZIGO and BUDIANSKY (1988). Previous work on the modelling of microcracked materials can be divided into two groups. a discrete or a continuous description of the microcrack density. In the discrete approach the microcracked material is simulated by a large but finite number of microcracks. The problem is then solved in an approximative way for each microcrack distribution, see HOAGLAND and EMBURY (1980) and KACHANOV (1986). In the continuous approach the effect of microcracks on the overall behaviour of the material is modelled by homogenization methods as developed by HILL (I 965) and BUDIANSKY (1965). The presence of microcracks is then reflected through changes of the elastic moduli and eventual release of residual stresses causing additional deformation. The continuous models can be divided into those applying isotropic (EVANS and Fu. 1985; HUTCHINSON, 1987; CHARALAMBIDESand MCMEEKING, 1988) and anisotropic (ORTIZ. 1987 ; GIANNAKOPOULOS, 1988 ; HOAGLAND et al., 1975 : Fu and EVANS, 1985) microcrack distributions respectively. Some of the models have been based on a phenomenological relationship between stress and strain for a microcracked material (HUTCHINSON. 1987 ; CHARALAMBIDESand MCMEEKING. 1988 ; ORTIZ, 1987 ; GIANNAKOPOULOS. 1988). In other work a microcrack nucleation criterion has been applied to individual potential microcracks (EVANS and Fu, 1985 ; HOAGLAND pt d.. 1975: Fu and EVANS. 1985). HOAGLAND et al. (1975) applied a critical normal stress criterion to determine the number of microcracks in front of a macrocrack. Fu and EVANS (1985) used the concept of a critical stress vector magnitude and included in an approximate way the influence of residual stresses to determine the microcrack density. which will generally have an anisotropic distribution. In the determination of the elastic moduli of the microcracked material, the microcrack distribution was however assumed to be isotropic (EVANS and Fu. 1985). In the present work, a micromechanical model for microcrack nucleation is developed. It will be assumed that the microcrack nucleation criterion can vary in a random way. Furthermore, residual stresses which may result from fabrication due to thermal mismatches between individual grains and/or second phase particles will be included in the analysis. It will be assumed that the residual stresses are described by random distributions. An approximate model for these random distributions in a single phase material was presented by ORTIZ and MOLINARI (1988). In the determination of the stress-strain relationship for a given microcrack distribution resulting from the model. anisotropic microcrack distributions and eventual microcrack closure effects will be properly modelled. The model is formulated in a general way and then
Anisotropic microcrack nucleation in brittle materials
533
FIG. 1. Definition of the spherical coordinate system.
specialized to certain cases. It is shown that for a specific nucleation criterion and monotonic, proportional loading a complementary strain energy density can be defined. The stress-strain relationship for a material with a critical normal stress as a nucleation criterion and rectangular random distributions of critical stress and residual stresses is also investigated in greater detail. Stress-strain curves and microcrack density distributions are graphically presented for simple loading cases. The crack tip shielding mechanism is analysed as well using the present model.
2.
MATERIAL MODEL
A virgin material without damage is considered first. It is assumed that under loading of the material, damage can develop due to the formation of microcracks. Concerning microcrack nucleation it will be assumed that there is an even distribution over all directions of potential microcracks. For simplicity only penny-shaped microcracks of a constant radius a will be considered. The microcrack direction is described by the normal vector of the crack surface. An arbitrary normal vector will here be defined as the normal of a unit sphere at the polar angles (fl, cp), see Fig. 1. It will later prove convenient to introduce a spherical coordinate system with the l-axis coinciding with the normal vector of the microcrack under consideration. The physical components of a tensor in this spherical coordinate system will be denoted by a bar, so that for example the components of the stress vector acting on a crack surface with directions defined by (0, cp) read
(2.1) where Ti, Fk denote the stress vector components in the global Cartesian and the spherical coordinate system respectively and nf the i-component of the kth unit direction in the spherical coordinate system. It is observed that the components T; can be identified with the normal stress o and the shear stress (5’ = TS’+ 5:) according to
and that the normal direction of a crack II, is given by 11:. If the number of potential microcracks per unit volume is p. then the density of microcracks with directions in the interval [H+dO], [q+dq] is p sin (0) dH dq, 2~. It has here been assumed that the n,-component of the microcrack direction always is positive. since there is no difference between a crack with normal vector jr; and (-II,) respectively. Due to fabrication of the material. residual stresses can exist in the material. Thus. on the prospective crack surfaces. residual stresses are acting which upon opening will release. This will result in a contribution to the overall strain which must be accounted for in the analysis. In addition, the residual stresses affect the microcrack nucleation criterion expressed in terms of the applied stresses. This effect should also be included in the model. It will be assumed that a potential microcrack will open according to a stress criterion. Due to microstructural variations it can be expected that the nucleation criterion will vary between microcracks. To model this effect. it is necessary to allow for a random distribution of fracture criteria. A microcracked material will be stiffer in compression in comparison to tension due to crack closure effects. This property will also be accounted for in the model. In the following sections the discussed phenomena will be analysed and combined to form a constitutive law taking all effects into account.
Residual stresses will appear in a material composed of anisotropic grains of random orientations. Temperature changes will result in mismatch of thermal strains between the individual grains due to differences in thermal expansion coefficients. The residual stresses will be of the order of EArAT, where E is a typical elastic modulus. AZ is the difference in thermal expansion between different directions or different phases and AT is the temperature change. It is observed that even small anisotropy can cause appreciable residual stresses if the temperature change is large. The residual stresses are self-equilibrating and hence the average stress components are vanishing. If the matrix is composed of grains of one phase of approximately the same size, the average stress in an individual grain will be zero. This is however not the case if the matrix is composed of several phases. In this case. the average stress for each phase will in general be nonvanishing. Since the distribution of the phases is assumed to be completely random, the only possibility for a nonvanishing average stress is a pure hydrostatic stress state in each phase. Due to the random character of the grain shapes and orientations. the residual stresses will vary in a random way. Concerning the microcrack nucleation criterion and the strain contribution from release of residual stresses. the residual stresses on potential crack planes will be of particular importance. The average residual stress vector T’ acting on a potential microcrack with normal vector n can be described by
Anisotropic
microcrack
nucleation
in brittle materials
535
a probability density function j;(c). Since the statistical properties of the normal stress generally are different from those of the shear stresses, it is convenient to employ the components of the stress vector referred to the spherical coordinate system as independent variables. see (2.2). The function .f, satisfies of course the normalization condition fr(?)dV,
= 1,
(2.3)
s ‘, where Vr denotes the whole r-space. Since the considered material is initially isotropic. the function ,f; must show certain symmetries. The probability density will only depend on the magnitude of the shear stress. Thus. the functional dependence on 7: and T’; (T; and r;) is restricted to (rr)’ = (7)’ + ( pJ) ‘. This implies that for each 7, (7, = or).
s
cA(r:,
dp? dT’; = 0.
(2.4)
c;
fork = 2. 3. If the residual stresses are the result of temperature changes the standard of fi will be of the order EAcrAT, see ORTIZ and MOLINARI(1988).
3 3 Microcrack _._.
nucleation
deviation
criterion
It will be assumed that the nucleation criterion for a single potential microcrack will depend on the normal and shear stress acting on the potential crack surface. Both the applied stresses and the residual stresses will contribute to the stress vector on the crack surface. It is further assumed that the fracture is controlled by a scalar function g which depends on the normal and shear stress. The condition for fracture of a microcrack can then be stated as g(T,+T)
> 8.
(2.5)
where T, denotes the applied stress vector and r the residual stress vector. The critical value of the function g is denoted by CJ’. Due to microstructural variations the critical value CJ’can vary between different microcracks and is most correctly modelled by a probability density function ./;(a’). It will be assumed that the probability functions ,f;.(a’) and .f,(c) are statistically independent. The introduction of ,f;(rY) means that the probability of fracture at a given loading defined by the function g is given by F. (9) where \ _ ~ j; (a”) do’ .
F,.(x) = s
(2.6)
2.3. Applied loads The applied loads are described by the stresses or,. The average stress over all grains must equal c,,. If there is more than one phase in the matrix, the average value of
P. GUDMUNDSON
536
stress in phase (X-). 6::). can differ from g,,. Due to the assumed of phases, the average stress 6::’ must be of the form @I _ AWa + B’Ua 6 r, 1, PP 11%
isotropic
distribution
(2.7)
where A’“’ and B’“’ denote dimensionless coefficients depending on volume fractions and material parameters. If the volume fraction of phase (k) is denoted by pl;. the parameters must fulfil
1 pl,B”’ = 0. As in the case of residual stresses, an applied stress o), will cause a random variation of stress from grain to grain. The average stress in each phase will be given by (3.7). In the case of a single phase material, the average stress must equal the applied stress and the standard deviation will scale with a,,AEjE. where AE denotes the difference in stiffness between directions. In a first-order approximation. it can be assumed that the standard deviation is small. This assumption will be made in the forthcoming analysis.
2.4. Crack closure Crack closure will have an influence on the stress-strain relationship for a microcracked material. The matrix will be stiffer if crack closure occurs. If a single microcrack is analysed the crack closure will depend on the applied stress and the residual stresses. Assuming that the stress field close to a microcrack is not influenced by neighbouring microcracks, the condition for crack closure can be expressed as a condition for the normal stress
T, + 7, < 0.
(3.9)
If a once opened crack is closed, there can still be sliding between the crack faces. This sliding will depend on the frictional forces. In the present analysis it will be assumed that the frictional forces are vanishing.
3.5. Comtitutiw
k7~.
The different phenomena which were defined in the previous sections will now be combined to establish a constitutive law. The derivation is based on the assumption of a small microcrack density so that the additional strain due to microcracking can be superposed on the strain which would have resulted without microcracking. Thus. E,, =
where C,,L, summarizes
the isotropic
C,k,O,,+ 4;,
compliances
determined
(2. IO)
by the elastic modulus
Anisotropic microcrack nucleation in brittle materials
E and the Poisson
ratio V. The additional
537
strain caused by microcracking
by c;. The microcracking will depend on the previous stress history, t denotes the current time. The fraction of opened microcracks defined by (8, cp) is determined from the history of the stress density functions A(c), A;.(&) and the microcrack nucleation
is denoted
a,-(s), for T < t,where with normal direction vector, the probability criterion according to
(2.5) (2.11)
g( r,(e, cp, t) + T) > 6(‘.
Equation (2.11) defines for each 0” and time T a region A’(a’, T, 8, cp) in the c-space for which the criterion (2.11) is fulfilled. The corresponding region in the c-space for which (2.11) has been fulfilled at least once up to time I is denoted by A(a’, r, 8, cp). The region A can be expressed as the union of all A’ for T < t.The fraction of opened microcracks (no) in direction (0, cp) at time t is therefore (2.12) Generally not all of the once-opened microcracks will be open at time t. The closed cracks of the once-opened microcracks will be denoted as sliding cracks. In the cspace, the region of sliding microcracks of direction (8, cp) at time t is denoted by B(r, 0,~) and is determined from (see 2.9) n : T, (t, 0, cp)+ c
< 0
and (2.13)
z E A(a’, t, 8, cp). The fraction
of sliding microcracks
(n,) in direction
(8, cp) at time f is therefore (2.14)
The additional strain caused by microcracking can be determined from the released complementary strain energy (IV) of a single penny-shaped crack, see BUDIANSKY and O’CONNELL (I 976). This energy is for an open crack W=
(2.15)
~aa3[(Ti+f:)(Ti+~)-~(T,+ii;)2],
and for a sliding crack (F, + F, < 0) W=
(2.16)
taa3[(~++)(~.++)-(~,+~,)‘],
where (2.17) and
T,;, c
denote
the components
of the applied
and residual
stress vector
in the
538
P. GLJIIMUWSON
spherical coordinate system. The constant u denotes the radius of the penny-shaped crack. The additional strain ~1,due to a single microcrack is determined from the derivative of W with respect to CJ,,
where ?W --= = ?T,
ro”[(~,+t;)-~s,,(~,+T;)].
(2.20)
for an open crack and (2.21)
for a sliding crack. The expectation value of the strain (0, cp) at time t will then be
contribution
The total strain contribution 6:;’from microcracking of microcracks per unit volume p and an integration P
E:; = j;
ss n2
”
~51,for a microcrack
in direction
is determined from the number over all microcrack directions
2n
”
E:, sin (0) d0 dq.
(2.23)
The additional strain caused by a given isotropic microcrack density was determined by BUDIANSKY and O’CONNELL (1976). HORII and NEMAT-NASSER (1983) took also crack closure into account in their analysis. They assumed however also an isotropic microcrack density. For some applications it is of interest to have the relationship between stress and strain rate. The additional strain rate ii’,’is determined directly from (2.23) as
(2.24)
where
Anisotropic
FIG. 2. Example
of a microcrack
microcrack
nucleation
nucleation
criterion
539
in brittle materials
which is defined by the energy release of a microcrack.
and
(2.26)
The surface T’, in the F-space is the part microcrack nucleation criterion is active, g(F;+
of the boundary
of A for which
the
(2.27)
E) = fJL’,
and (2.28) The rate relationship
between
stress and strain can be summarized
in the form (2.29)
9:: = Drl,nn~nm.
where D,,,, depends on the stress history according to (2.25). It is observed that generally Dij,, is not diagonally symmetric. The only case for which a diagonal symmetry can exist is when the microcrack nucleation function g is proportional to the energy release of a crack W, see Fig. 2. The condition for failure of a microcrack can then be formulated as W(i’,+C) and the probability
function
> w”,
f, will have w’ as an independent
(2.30) variable.
P. GUUMUNIXON
540
If in addition a proportional. monotonic strain rate C$’will only depend on the current strain energy density Cpcan be defined,
loading is considered, the strain c:y and stress state. In this case a complementary
(2.3 1) The potential
4(cri,) can be determined
from (2.23)
where 4,, denotes the linear elastic part of $J (chr = lC,,n-pl,~,,). It is observed that even in the case of proportional loading and g = W, the stressstrain relationship is of considerable complexity. The existence of a potential can however simplify the solution to certain problems. An application to the macrocrack shielding problem will be presented in Section 5.
3.
SATURATEDMICROCRACK DENSITY
If the loading history has been such that all potential microcracks have been developed, then the stress-strain relationship is considerably simplified. The region A in the expression for e; in (2.22-2.23) will then be the whole c-space. V,. The symmetry property, (2.4), can be utilized and (2.23) can be written as
where expressions for the differentiation of Ware given by (2.20-2.21). In a practical evaluation of (3.1), the T-space should be divided into two regions, one for which Fi;,+ P, > 0 and the remaining complementary region. An important case is when the applied load is vanishing (?;, = 0). Then there will be a permanent strain E:; according to (3.2) where (3.3) It is observed that a permanent strain can exist even though the average of the residual stresses vanishes. The reason is of course the crack closure effect. The micro-
Anisotropic
microcrack
nucleation
in brittle materials
541
cracks with negative residual normal stress will remain closed at vanishing applied load. Two other cases are of interest. Firstly, the case when all microcracks are open is considered. This is valid if the minimum principal stress (a?) is larger than --o~.~~“, where gR,m&n denotes the minimum residual normal stress. The stress-strain relation will then be s:; = pa3
16(1-v2) 3E
1 15(2 _ ,,) K10-2~V)&6j~ - VQ%,l~l,,.
(3.4)
Another well-defined situation is when all microcracks are sliding. Thus, the maximum principal stress (a,) is less than -oR,,,aX, where dR.mandenotes the maximum residual normal stress. The strain E; then reads E;
=
pa3
(3.5)
4.
CRITICALNORMAL STRESS
The complexity of the general equation (2.22-2.23) for the determination of the additional strain E: depends to a large extent on the microcrack nucleation criterion g. A simple but still for certain applications realistic nucleation criterion is based on the assumption that the microcrack fracture is solely dependent on the normal stress. In this case (2.5) will simplify to u+u’
> UC.
(4.1)
This specific failure criterion has been investigated in greater detail. An evaluation of (2.22) using (2.4, 4.1) and some algebraic manipulations results in P
E~=~cxa30
1 sCL 62
2n
o
&r,+n,T,)-
;ninjcT
-
1
h,(o,)
n,nj[ah,(a, a,,,) -h3(a, o,,,)]
(4.2)
where n,(8, rp) denotes the normal vector at (0, cp), T, = a,,n,, CJ= ~,~rz~n~, and grn the maximum experienced normal stress on the plane (0, cp). The functions h,, hz and h, are defined as m
ht (em)=
I
F,(d+u,)F,(a’)
da’,
(4.3)
Fc(a’ + a#,(~‘) s -Co
dar,
(4.4)
--(c -0
hz(a,g,)
=
542
s -1
/13(a,a,,,) =
F,(ar+a,)a’F,(a’)
do’,
(4.5)
--(r
where F,.(X) is defined in (2.6) and
(4.6) If the proportional but not necessarily monotonic loading cases are considered. then the maximum experienced normal stress (T,,,will be proportional to the actual loading u (a = qo,,,) for every direction (0, cp). Different loading cases of this type have been investigated in greater detail in Sections 4.14.3. To be able to derive quantitative results, the statistical functions F, and ,fl must be known. In the calculations, rectangular distributions have been considered. L 2aT
F,(a’) =
for
1~7~1 < aT
for
la’/ > aT.
ka
for
la’ -sot
< Aa
0
for
Ia’ -a01
> Aa.
i 0 J; (ar’) =
(4.7)
(4.8)
where aT, a,, and Aa are parameters
defining
the statistical
distributions.
4.1. Pure tension The only nonvanishing stress component is the stress a: in z-direction. The maximum experienced stress is denoted by a=,,). Equation (4.2) can in this case be expressed as
(1 -_.y2)1z1(g2a_,s2a,,,,)
d.u
The integrals in (4.9-4.10) can be expressed in closed form but the expressions too cumbersome to be of interest for an explicit presentation.
are
Anisotropic
microcrack
nucleation
in brittle materials
543
(b)
2
1
/ v
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
FIG. 3. (a) Fraction of opened microcracks (n) as a function of applied tensile load oI (ar/ao = 0.25, Ao:a,, = 0. 0.25, 0.5). (b) Fraction of opened microcracks in direction 0 (n(0)) at different tensile loads u_;o,, = I. 1.5.2. 2.5. 3 (uT/ao = 0.25. Au/u, = 0.25). (c) Stress-additional strain curves for tensile loading and unloading (-). Saturated microcrack density (- - -) (uT/uo = 0.5. Au/u, = 0.25).
In Fig. 3(aac) some numerical results are presented. Figure 3(a) shows the development of damage in the material as a function of applied tensile load CT._. The damage is here defined as the fraction of opened microcracks in the material. The damage as a function of microcrack orientation is presented in Fig. 3(b). It is observed that the
544
P. C&JDM~JNDS~N
damage is relatively slowly developing due to the strong variation in the normal stress with crack orientation. Figure 3(c) shows an example of a loading-unloading curve in comparison to a fully saturated case. One observes the permanent strain at zero loading and the additional flexibility at negative reversed stress.
4.2. Biaxial tension Biaxial tension has also been investigated, thus CJ,, = o,,. = crh and c=._= 0. The maximum experienced stress is denoted by oh,,,. Similarly the pure tension case. (4.2) can be expressed as
I-x’-
E
-
(1 -x’)‘hz(oh(l
-.~‘).a~,~(1
(1 -~‘)h~(cr~(l
+
&(ah(l
;(I-.~‘)*
-x2).
1
-s’))
ds
-.?).~&l
-s’))
~h,,,(l -x?))
1
x2h,(a,(l-s’),a,,,(l-.\-‘))ds
ds
(4. I I )
ds
(4.12)
As in the case of pure tension, the integrals (4.114.12) can be explicitly determined. The lengths of the expressions are however prohibitive. The analyses corresponding to the pure tension case have been carried out also for biaxial tension, see Fig. 4(aac). It is observed that the damage is more uniformly developed in the biaxial case. The transition from undamaged material to a certain level of damage is faster. The reason is of course the more uniform normal stress distribution for biaxial loading. The general behaviour of the curves is however similar.
4.3. Triaxial tension Triaxial tension is the simplest loading condition to be analysed. The stress state is described by CT,,= a,6,, and the maximum experienced stress by a,,,,. The additional strain is easily calculated from (4.2)
Anisotropic 1.0
microcrack
nucleation
545
in brittle materials
1
(a)
i
i
3 ‘JbPo
1.0
n(B)
0.5
(b)
i
1
I
__ 1s
’
ao-
Z-LL 45’
60’
75’
90’
0
FIG. 4. (a) Fraction of opened microcracks (n) as a function of applied biaxial load o,, (o,/a, = 0.25. Au/a,, = 0, 0.25. 0.5). (b) Fraction of opened microcracks in direction f? (n(O)) at different biaxial loads uh/uO = 1. 1.5. 2, 2.5. 3 (uTiu,, = 0.25, Au/u, = 0.25). (c) Stress-additional strain curves for biaxial loading and unloading (-). Saturated microcrack density (- - -) (ur/uo = 0.5, Au/o0 = 0.25).
546 1
.o-
n
0.5-
(a)
Fz. 5. (a) Fraction of opened microcracks (n) as a function of applied triaxial load CT,(a,,~,, Au/u, = 0. 0.25. 0.5). (b) Stress-additional strain curves for triaxial loading and unloading (c+;cT,, = 0.5. AC/U,, = 0.25).
~1;’= paa”
b,Vl*(~ )-/1?((T,,(1,,,)+/1~(~,,~,,,,)1~,,. ,,1,
In Fig. 5(a, b) the triaxial results are presented. The transition state to a fully damaged state is in this case the fastest possible.
5. CRACK
= 0.25. (-_)
(4.13)
from an undamaged
SHIELDING
It is known that the nucleation of microcracks can produce some shielding of a macrocrack, see, for example, HUTCHINSON (1987). This effect has been investigated with the aid of the present material model.
Anisotropic
FIG. 6. Definition
microcrack
nucleation
in brittle materials
of the path I- and the area A around
547
a crack tip.
A macroscopic crack of a length much larger than the microcracks under plane strain mode I loading is considered. It is further assumed that the damage zone around the macrocrack tip is much smaller than other relevant geometric measures. Thus. a so-called small-scale damage problem is considered. According to the assumptions, there will be a distance. say R, from the crack tip where the stress and strain are determined from the applied mode I stress intensity field (K,). Furthermore. asymptotically close to the crack tip, the material will be fully damaged. This material, which is under tension, can be described as an isotropic material with modified material parameters (,!?, C), see (3.4). The stress and strain field will then be described by a stress intensity factor K,,P. The problem of interest is to determine the relationship between K, and K,,p. For the solution of this problem it proves to be useful to apply the J-integral according to RICE(1968)
J =
s I-
[Un,
-a,jn,ui,,]dT,
(5.1)
where I’ denotes a path around the crack tip, nj the normal vector and II, the displacement vector. The quantity U is the stress work density and is determined from
(5.2)
where t denotes the time. If J is evaluated on a circle of radius R from the crack tip it Kf(l -v’)/E and if it is evaluated asymptotically close to the crack value K&( I - C’)/E. A closed contour r is now considered, see Fig. gence theorem is applied, a relation between K, and K,,, can be stated
K;(l
-v’) E
where
K&(1-+) ---IF-=
takes the value tip it takes the 6. If the diverin the form
I ’
(5.3)
548
P. GUDMUNDSON
bu.,k,-6,~
I=
r,.,
(5.4)
1dt dA> 1
and (A) denotes the area enclosed by F. To be able to evaluate the integral I, the stress and strain fields must be known in the area (A). For small inelastic strains, it is however possible to determine I in the form of a first-order perturbation to the linear elastic problem. According to the results in Section 2.5, the strains can be written as E,, = Cljh,Ok,+ For pa-’ + 0 the inelastic strains state will therefore be described Thus.
E:y.
(5.5)
are much smaller than the elastic ones. The stress by the elastic stress plus a small contribution o:,.
(r,,
=
a; + a;,.
(5.6)
Up to first order in pa3, the strains will hence be (5.7)
e,, = C,.,,,(~~,+o;,)+E~~(~~,),
where E:,‘(&) means that ~7 should be solely determined from &‘I according to (2.23). If only first-order terms are considered in (5.4) the integral I can be expressed as ’ [a~.,~::l(a~,)-~:;E:::,((~~,)]
I= A S[S
The strain
E:~(f$)
is the result of a proportional. m
EiJ.l =
.,>I E 1, =
according to (2.29). Equation (5.8) can therefore I=
dr
0
monotonic
D,jk/aZ/.
1
dA. loading.
a:;.,(D,,k,-
In this case
IT
Di,k~.
be rewritten
(5.8)
(5.9)
as DA,,,)& dt
I
dA.
(5.10)
It is observed that the integral I vanishes if the tensor D,,k, is diagonally symmetric. This is the case if there exists a complementary strain energy density 4, see (2.32). The only case for which 4 exists is when the microcrack nucleation criterion is determined from W, the energy release of a microcrack as was discussed in Section 2.5. In this particular case the I&, is determined from
For a Poisson value
ratio of 11= 0.3 the stress intensity
factor
at the crack tip takes the
Anisotropicmicrocrack nucleationin brittle materials Ktlp = &(I-
549
1.018pa3).
(5.12)
The integral I has also been evaluated for a case with a nonsymmetric tensor Dilk,, namely the model discussed in Section 4, where a critical normal stress was applied as a microcrack nucleation criterion. After an application of (4.224.8) in (5.10) the introduction or the operators
(.)=i$ $=I and a representation can be stated as I=
cos (p) g - i sin (p) $
of the stress field OF,in the form G$ = K,Ci,(/Q/j27rr
4pa3
-
3x2(2-v)
I’
K:(l-q E
i
(f2C,,
-BABY)
the integral I
(5.13)
sin (/J) sin (6) d/l dtI dp,
(5.14)
where ? and C denote the dimensionless shear and normal stress at an angle ,G from the crack plane on the microcrack plane defined by (0, cp). The stress intensity factor at the crack tip will then be 4
16(10-3v+8v2-3v3) +v(z$
45(2-v) The integral f has been numerically value of v = 0.3 that
evaluated
to be f=
-0.05695.
. 11
_
This gives for a
(5.16)
Ktip = Ki(1-1.016p~~). It is observed that the Z-integral has a negligible particular microcrack nucleation criterion.
6.
(5.15)
influence
on the K,,,-value
for this
DISCUSSION
A central assumption in the development of the present model was that the microcrack density in terms of pa3 should be small. The expressions for the additional strain caused by microcracking are only valid under this assumption of dilute microcracking. For larger microcrack densities self-consistent estimates of the additional strain could be computed in an analogous way as were done by BUDIANSKY and O’CONNELL (1976). HORII and NEMAT-NASSER (1983). Alternatively, the so-called differential scheme for the determination of modified elastic moduli could be applied, see MCLAUGHLIN (1977) and HASHIN (1988). The differential scheme is based on a model for incremental changes in moduli resulting from incremental changes in microcrack density. It seems that this procedure would be ideally suited to the present model, since the microcrack density is naturally formulated in incremental form. The problem of formulating a constitutive model allowing large microcrack densities is
550
P. GUDMUNDSON
however not only a problem of determining additional strains from a known microcrack distribution. The microcrack nucleation criterion and the microcrack closure criterion will also be influenced by a large microcrack density as a result of microcrack interactions. The present model is nevertheless applicable to many practical problems. If the number of potential microcracks is large, for example if all grain boundary facets in a single phase material are treated as possible microcrack sites. then the present model is only valid up to a certain critical, small microcrack density. The particular material will however fail long before the microcrack saturation density is reached, due to coalescence of neighbouring microcracks. The failure criterion can be expressed as another critical microcrack density. It is not unreasonable that the failure density is smaller than the limit density for the present model. In this case it makes no sense to include a much more complicated theory for large microcrack densities. Another situation where the present model can be directly applied is when microcracks are developing only in the vicinity of second phase particles. the density of which is small. Such microcrack patterns have been observed by R~JHLE et al. (1987) in zirconia-toughened alumina. In this case a saturation level may be well defined by the density of the second phase particles. One should observe however that for the present model to be valid only one potential microcrack direction is allowed for each particle. If there is more than one potential microcrack at each particle, where only one will develop, then the present statistical model has to be modified. It should be stressed that the present model predicts a permanent strain at vanishing stress, even though the average residual stresses are vanishing. The reason is that crack closure effects are taken into account and that a random variation of residual stresses exists. Another important feature of the model is that the degree of anisotropy is a direct consequence of the applied loading history. No phenomenological assumption whether an isotropic or anisotropic distribution results is necessary. The simple examples which were investigated showed that triaxiality of the loading is of importance concerning the resulting anisotropy. When the saturation level is reached, the microcrack density of the present model is always isotropic independently of the loading history. A maximum degree of anisotropy is therefore achieved at some intermediate load level. The crack tip shielding analysis showed that the present model predicts &,-values in accordance with alternative material models analysed by EVANS and Fu (1985) HUTCHINSON (1987), ORTIZ (1987), CHARALAMBIDESand MCMEEKING (1988), GIANNAKOPOULOS(1988). The critical normal stress nucleation criterion which was analysed showed a very small deviation from the simpler analysis with a complementary strain energy density. The present model is rather complicated to apply in a general structural mechanics problem. In each point of the solid the microcrack density in all directions must be known. This can be formulated as the fraction of opened cracks as a function of the spherical angles 0 and cp. Thus for each point in the solid a function of two variables is required. It would be simpler if the damage could be described by some lower order tensor quantities. ONAT (1984) has shown that a function of (8, cp) can be developed into tensors of different order. Approximately a function of (0, cp) can be modelled by a symmetric second order tensor. For the purpose of practical application it would therefore be of interest to investigate such an approximation.
Anisotropic microcrack nucleation in brittle materials
551
REFERENCES AMAZIGO, J. C. and BUDIANSKY, B. BUDIANSKY, B. BLJDIANSKY,B. and O’CONNELL, R. J. BUDIANSKY. B., HUTCHINSON,J. W. and LAMBROPOULOS,J. C. BUDIANSKY, B., AMAZIGO, J. C. and EVANS, A. G. CHARALAMBIDES,P. G. and MCMEEKING, R. M. EVANS, A. G. and Fu, Y. EVANS, A. G. and CANNON, R. M. Fu, Y. and EVANS, A. G. GIANNAKOPOULOS,A. E. HASHIN, Z. HILL, R. HOAGLAND, R. G., HAHN, G. T. and ROSENFIELD,A. R. HOAGLAND, R. G. and EMBURY, J. D. HOAGLAND, R. G., EMBURY, J. D. and GREEN, D. J. HORII, H. and NEMAT-NASSER, S. HUTCHINSON, J. W. KACHANOV, M. MCLAUGHLIN, R. ONAT, E. T. ORTIZ, M. ORTIZ, M. ORTIZ, M. and MOLINARI, A. RICE, J. R.
1988
J. Mech. P/I_w. Solids 36, 58 1.
1965 1976
J. Mech. Phys. Solids 13,223. I~I. J. Solids Struct. 12. 81.
1983
ht. J. Solids Struct. 19, 337.
1988
J. Mech. Phys. Solids 36, 167.
1988
J. Am. Ceram. Sot. 71,465.
1985 1986 1985 1988
Acta Acta Acta PhD
1988 1965 1973
Metall. 33, 1525. Metall. 34, 761. MetaN. 33, 1515. Thesis, Division of Engineering, versity. J. Mech. Phys. Solids 36, 719. J. Mech. Phys. Solids 13, 213. Rock Mech. 5, 77.
1980
J. Am. Ceram. Sot. 63,404.
1975
Scripta MetaN. 9, 907.
1983 1987 1986 1977 1984 1987 1988 1988 1968
J. Mech. Phys. Solids 31, 155. Acta Metali. 35, 1605. Int. J. Fracture 30, R65. Int. J. Engng Sci. 15, 237. Int. J. Engng Sci. 22, 1013. J. appl. Mech. 54,54. Jnt. J. Solids Struct. 24, 23 1. J. Mech. Phys. Solids 36, 385. In Fracture 4 (edited by H. LIEBOWITZ), p. 191,
1987
Acta Metall. 11. 2701.
1978
J. Mater. Sci. 13, 2659.
Academic R~HLE, M., EVANS, A. G., MCMEEKING, R. M., CHARALAMBIDES.P. G. and HUTCHINSON, J. W. WV, C. CM, FREIMAN, S. W., RICE, R. W. and MECHOLSKY. J. J.
Press, New York.
Brown Uni-
0022-5096 90 %3.00+0.00 1990 Perpmon Presr plc
ANISOTROPIC MICROCRACK NUCLEATION IN BRITTLE MATERIALS PETER GUDMLJNDSON~ Departmentof Strengthof Materialsand Solid Mechanics. S-100 44 Stockholm,
(Received 9 Februaq,
The Royal Institute Sweden
of Technology,
1989)
ABSTRACT A CONSTITUTIVE model for anisotropic
microcracking in brittle materials is developed. The model is based on a stress controlled microcrack nucleation criterion, which can vary in a random way between different microcracks. The effects of microcrack closure and a random distribution of residual stresses are included
in the analysis. The resultant inelastic strains are determined using a standard homogenization technique. Numerical results are presented for three simple loading cases : pure tension. biaxial tension and triaxial tension. Crack tip shielding resulting from microcrack nucleation is also analysed. of K,,,/K, are presented for two different microcrack nucleation criteria.
1.
and numerical
results
INTRODUCTION
THE PRESENCEof microcracks has a large influence on the mechanical properties of a material. In this investigation microcracks in brittle materials will be analysed. The main application is perhaps structural ceramics but also the behaviour of rock and ice can be described in similar ways. The material is assumed to be polycrystalline and composed of one or several phases. Furthermore, it is assumed that the single crystals may exhibit an anisotropic behaviour. Microcracks have been observed in fracture mechanics specimens of zirconiatoughened alumina by RCJHLE et al. (1987) using transmission electron microscopy. The microcracks were formed around zirconia particles and extended on grain boundary facets in the alumina and on the interface between the zirconia particles and the alumina. Indirect evidences of microcracks have also been observed for various other ceramic materials by WV et al. (1978) and in rocks by HOAGLAND et al. (1973). The nucleation and later coalescence of microcracks is often the strength limiting mechanism in brittle materials. The microcracks can however also have a beneficial effect on toughness. Due to the formation of microcracks, the stress field close to the crack tip of a macrocrack is shielded. The nucleation of microcracks can in this case even enhance the fracture toughness, This toughening mechanism has been investigated in some recent papers (HOAGLAND and EMBURY, 1980; EVANS and Fu, 1985; HUTCHINSON, 1987; ORTIZ, 1987; CHARALAMBIDES and MCMEEKING, 1988 ; t Present address:
SICOMP,
Swedish
Institute
of Composites. 5.11
Box 271. S-941 26 Pitea, Sweden.
532
P. GUDMIXDSOE:
GIANNAKOPOULOS, 1988). A competing process is the degradation of the material due to microcracking (ORTIZ, 1988). It is however believed that in many cases the shielding mechanism outweighs the strength degradation. There are other fracture toughening mechanisms for ceramic materials which have been experimentally and theoretically investigated. A promising method is transformation toughening where the material contains a phase which undergoes a stressinduced phase transformation (EVANS and CANNON, 1986). This mechanism has been analysed by BUDIANSKY et al. (1983). Another strengthening mechanism is the introduction of tough particles (particulate toughening) in the ceramic matrix material. An analysis of this phenomenon was given by BUIXANSKY et al. (1988). The combined effect of transformation and particulate toughening was analysed by AMAZIGO and BUDIANSKY (1988). Previous work on the modelling of microcracked materials can be divided into two groups. a discrete or a continuous description of the microcrack density. In the discrete approach the microcracked material is simulated by a large but finite number of microcracks. The problem is then solved in an approximative way for each microcrack distribution, see HOAGLAND and EMBURY (1980) and KACHANOV (1986). In the continuous approach the effect of microcracks on the overall behaviour of the material is modelled by homogenization methods as developed by HILL (I 965) and BUDIANSKY (1965). The presence of microcracks is then reflected through changes of the elastic moduli and eventual release of residual stresses causing additional deformation. The continuous models can be divided into those applying isotropic (EVANS and Fu. 1985; HUTCHINSON, 1987; CHARALAMBIDESand MCMEEKING, 1988) and anisotropic (ORTIZ. 1987 ; GIANNAKOPOULOS, 1988 ; HOAGLAND et al., 1975 : Fu and EVANS, 1985) microcrack distributions respectively. Some of the models have been based on a phenomenological relationship between stress and strain for a microcracked material (HUTCHINSON. 1987 ; CHARALAMBIDESand MCMEEKING. 1988 ; ORTIZ, 1987 ; GIANNAKOPOULOS. 1988). In other work a microcrack nucleation criterion has been applied to individual potential microcracks (EVANS and Fu, 1985 ; HOAGLAND pt d.. 1975: Fu and EVANS. 1985). HOAGLAND et al. (1975) applied a critical normal stress criterion to determine the number of microcracks in front of a macrocrack. Fu and EVANS (1985) used the concept of a critical stress vector magnitude and included in an approximate way the influence of residual stresses to determine the microcrack density. which will generally have an anisotropic distribution. In the determination of the elastic moduli of the microcracked material, the microcrack distribution was however assumed to be isotropic (EVANS and Fu. 1985). In the present work, a micromechanical model for microcrack nucleation is developed. It will be assumed that the microcrack nucleation criterion can vary in a random way. Furthermore, residual stresses which may result from fabrication due to thermal mismatches between individual grains and/or second phase particles will be included in the analysis. It will be assumed that the residual stresses are described by random distributions. An approximate model for these random distributions in a single phase material was presented by ORTIZ and MOLINARI (1988). In the determination of the stress-strain relationship for a given microcrack distribution resulting from the model. anisotropic microcrack distributions and eventual microcrack closure effects will be properly modelled. The model is formulated in a general way and then
Anisotropic microcrack nucleation in brittle materials
533
FIG. 1. Definition of the spherical coordinate system.
specialized to certain cases. It is shown that for a specific nucleation criterion and monotonic, proportional loading a complementary strain energy density can be defined. The stress-strain relationship for a material with a critical normal stress as a nucleation criterion and rectangular random distributions of critical stress and residual stresses is also investigated in greater detail. Stress-strain curves and microcrack density distributions are graphically presented for simple loading cases. The crack tip shielding mechanism is analysed as well using the present model.
2.
MATERIAL MODEL
A virgin material without damage is considered first. It is assumed that under loading of the material, damage can develop due to the formation of microcracks. Concerning microcrack nucleation it will be assumed that there is an even distribution over all directions of potential microcracks. For simplicity only penny-shaped microcracks of a constant radius a will be considered. The microcrack direction is described by the normal vector of the crack surface. An arbitrary normal vector will here be defined as the normal of a unit sphere at the polar angles (fl, cp), see Fig. 1. It will later prove convenient to introduce a spherical coordinate system with the l-axis coinciding with the normal vector of the microcrack under consideration. The physical components of a tensor in this spherical coordinate system will be denoted by a bar, so that for example the components of the stress vector acting on a crack surface with directions defined by (0, cp) read
(2.1) where Ti, Fk denote the stress vector components in the global Cartesian and the spherical coordinate system respectively and nf the i-component of the kth unit direction in the spherical coordinate system. It is observed that the components T; can be identified with the normal stress o and the shear stress (5’ = TS’+ 5:) according to
and that the normal direction of a crack II, is given by 11:. If the number of potential microcracks per unit volume is p. then the density of microcracks with directions in the interval [H+dO], [q+dq] is p sin (0) dH dq, 2~. It has here been assumed that the n,-component of the microcrack direction always is positive. since there is no difference between a crack with normal vector jr; and (-II,) respectively. Due to fabrication of the material. residual stresses can exist in the material. Thus. on the prospective crack surfaces. residual stresses are acting which upon opening will release. This will result in a contribution to the overall strain which must be accounted for in the analysis. In addition, the residual stresses affect the microcrack nucleation criterion expressed in terms of the applied stresses. This effect should also be included in the model. It will be assumed that a potential microcrack will open according to a stress criterion. Due to microstructural variations it can be expected that the nucleation criterion will vary between microcracks. To model this effect. it is necessary to allow for a random distribution of fracture criteria. A microcracked material will be stiffer in compression in comparison to tension due to crack closure effects. This property will also be accounted for in the model. In the following sections the discussed phenomena will be analysed and combined to form a constitutive law taking all effects into account.
Residual stresses will appear in a material composed of anisotropic grains of random orientations. Temperature changes will result in mismatch of thermal strains between the individual grains due to differences in thermal expansion coefficients. The residual stresses will be of the order of EArAT, where E is a typical elastic modulus. AZ is the difference in thermal expansion between different directions or different phases and AT is the temperature change. It is observed that even small anisotropy can cause appreciable residual stresses if the temperature change is large. The residual stresses are self-equilibrating and hence the average stress components are vanishing. If the matrix is composed of grains of one phase of approximately the same size, the average stress in an individual grain will be zero. This is however not the case if the matrix is composed of several phases. In this case. the average stress for each phase will in general be nonvanishing. Since the distribution of the phases is assumed to be completely random, the only possibility for a nonvanishing average stress is a pure hydrostatic stress state in each phase. Due to the random character of the grain shapes and orientations. the residual stresses will vary in a random way. Concerning the microcrack nucleation criterion and the strain contribution from release of residual stresses. the residual stresses on potential crack planes will be of particular importance. The average residual stress vector T’ acting on a potential microcrack with normal vector n can be described by
Anisotropic
microcrack
nucleation
in brittle materials
535
a probability density function j;(c). Since the statistical properties of the normal stress generally are different from those of the shear stresses, it is convenient to employ the components of the stress vector referred to the spherical coordinate system as independent variables. see (2.2). The function .f, satisfies of course the normalization condition fr(?)dV,
= 1,
(2.3)
s ‘, where Vr denotes the whole r-space. Since the considered material is initially isotropic. the function ,f; must show certain symmetries. The probability density will only depend on the magnitude of the shear stress. Thus. the functional dependence on 7: and T’; (T; and r;) is restricted to (rr)’ = (7)’ + ( pJ) ‘. This implies that for each 7, (7, = or).
s
cA(r:,
dp? dT’; = 0.
(2.4)
c;
fork = 2. 3. If the residual stresses are the result of temperature changes the standard of fi will be of the order EAcrAT, see ORTIZ and MOLINARI(1988).
3 3 Microcrack _._.
nucleation
deviation
criterion
It will be assumed that the nucleation criterion for a single potential microcrack will depend on the normal and shear stress acting on the potential crack surface. Both the applied stresses and the residual stresses will contribute to the stress vector on the crack surface. It is further assumed that the fracture is controlled by a scalar function g which depends on the normal and shear stress. The condition for fracture of a microcrack can then be stated as g(T,+T)
> 8.
(2.5)
where T, denotes the applied stress vector and r the residual stress vector. The critical value of the function g is denoted by CJ’. Due to microstructural variations the critical value CJ’can vary between different microcracks and is most correctly modelled by a probability density function ./;(a’). It will be assumed that the probability functions ,f;.(a’) and .f,(c) are statistically independent. The introduction of ,f;(rY) means that the probability of fracture at a given loading defined by the function g is given by F. (9) where \ _ ~ j; (a”) do’ .
F,.(x) = s
(2.6)
2.3. Applied loads The applied loads are described by the stresses or,. The average stress over all grains must equal c,,. If there is more than one phase in the matrix, the average value of
P. GUDMUNDSON
536
stress in phase (X-). 6::). can differ from g,,. Due to the assumed of phases, the average stress 6::’ must be of the form @I _ AWa + B’Ua 6 r, 1, PP 11%
isotropic
distribution
(2.7)
where A’“’ and B’“’ denote dimensionless coefficients depending on volume fractions and material parameters. If the volume fraction of phase (k) is denoted by pl;. the parameters must fulfil
1 pl,B”’ = 0. As in the case of residual stresses, an applied stress o), will cause a random variation of stress from grain to grain. The average stress in each phase will be given by (3.7). In the case of a single phase material, the average stress must equal the applied stress and the standard deviation will scale with a,,AEjE. where AE denotes the difference in stiffness between directions. In a first-order approximation. it can be assumed that the standard deviation is small. This assumption will be made in the forthcoming analysis.
2.4. Crack closure Crack closure will have an influence on the stress-strain relationship for a microcracked material. The matrix will be stiffer if crack closure occurs. If a single microcrack is analysed the crack closure will depend on the applied stress and the residual stresses. Assuming that the stress field close to a microcrack is not influenced by neighbouring microcracks, the condition for crack closure can be expressed as a condition for the normal stress
T, + 7, < 0.
(3.9)
If a once opened crack is closed, there can still be sliding between the crack faces. This sliding will depend on the frictional forces. In the present analysis it will be assumed that the frictional forces are vanishing.
3.5. Comtitutiw
k7~.
The different phenomena which were defined in the previous sections will now be combined to establish a constitutive law. The derivation is based on the assumption of a small microcrack density so that the additional strain due to microcracking can be superposed on the strain which would have resulted without microcracking. Thus. E,, =
where C,,L, summarizes
the isotropic
C,k,O,,+ 4;,
compliances
determined
(2. IO)
by the elastic modulus
Anisotropic microcrack nucleation in brittle materials
E and the Poisson
ratio V. The additional
537
strain caused by microcracking
by c;. The microcracking will depend on the previous stress history, t denotes the current time. The fraction of opened microcracks defined by (8, cp) is determined from the history of the stress density functions A(c), A;.(&) and the microcrack nucleation
is denoted
a,-(s), for T < t,where with normal direction vector, the probability criterion according to
(2.5) (2.11)
g( r,(e, cp, t) + T) > 6(‘.
Equation (2.11) defines for each 0” and time T a region A’(a’, T, 8, cp) in the c-space for which the criterion (2.11) is fulfilled. The corresponding region in the c-space for which (2.11) has been fulfilled at least once up to time I is denoted by A(a’, r, 8, cp). The region A can be expressed as the union of all A’ for T < t.The fraction of opened microcracks (no) in direction (0, cp) at time t is therefore (2.12) Generally not all of the once-opened microcracks will be open at time t. The closed cracks of the once-opened microcracks will be denoted as sliding cracks. In the cspace, the region of sliding microcracks of direction (8, cp) at time t is denoted by B(r, 0,~) and is determined from (see 2.9) n : T, (t, 0, cp)+ c
< 0
and (2.13)
z E A(a’, t, 8, cp). The fraction
of sliding microcracks
(n,) in direction
(8, cp) at time f is therefore (2.14)
The additional strain caused by microcracking can be determined from the released complementary strain energy (IV) of a single penny-shaped crack, see BUDIANSKY and O’CONNELL (I 976). This energy is for an open crack W=
(2.15)
~aa3[(Ti+f:)(Ti+~)-~(T,+ii;)2],
and for a sliding crack (F, + F, < 0) W=
(2.16)
taa3[(~++)(~.++)-(~,+~,)‘],
where (2.17) and
T,;, c
denote
the components
of the applied
and residual
stress vector
in the
538
P. GLJIIMUWSON
spherical coordinate system. The constant u denotes the radius of the penny-shaped crack. The additional strain ~1,due to a single microcrack is determined from the derivative of W with respect to CJ,,
where ?W --= = ?T,
ro”[(~,+t;)-~s,,(~,+T;)].
(2.20)
for an open crack and (2.21)
for a sliding crack. The expectation value of the strain (0, cp) at time t will then be
contribution
The total strain contribution 6:;’from microcracking of microcracks per unit volume p and an integration P
E:; = j;
ss n2
”
~51,for a microcrack
in direction
is determined from the number over all microcrack directions
2n
”
E:, sin (0) d0 dq.
(2.23)
The additional strain caused by a given isotropic microcrack density was determined by BUDIANSKY and O’CONNELL (1976). HORII and NEMAT-NASSER (1983) took also crack closure into account in their analysis. They assumed however also an isotropic microcrack density. For some applications it is of interest to have the relationship between stress and strain rate. The additional strain rate ii’,’is determined directly from (2.23) as
(2.24)
where
Anisotropic
FIG. 2. Example
of a microcrack
microcrack
nucleation
nucleation
criterion
539
in brittle materials
which is defined by the energy release of a microcrack.
and
(2.26)
The surface T’, in the F-space is the part microcrack nucleation criterion is active, g(F;+
of the boundary
of A for which
the
(2.27)
E) = fJL’,
and (2.28) The rate relationship
between
stress and strain can be summarized
in the form (2.29)
9:: = Drl,nn~nm.
where D,,,, depends on the stress history according to (2.25). It is observed that generally Dij,, is not diagonally symmetric. The only case for which a diagonal symmetry can exist is when the microcrack nucleation function g is proportional to the energy release of a crack W, see Fig. 2. The condition for failure of a microcrack can then be formulated as W(i’,+C) and the probability
function
> w”,
f, will have w’ as an independent
(2.30) variable.
P. GUUMUNIXON
540
If in addition a proportional. monotonic strain rate C$’will only depend on the current strain energy density Cpcan be defined,
loading is considered, the strain c:y and stress state. In this case a complementary
(2.3 1) The potential
4(cri,) can be determined
from (2.23)
where 4,, denotes the linear elastic part of $J (chr = lC,,n-pl,~,,). It is observed that even in the case of proportional loading and g = W, the stressstrain relationship is of considerable complexity. The existence of a potential can however simplify the solution to certain problems. An application to the macrocrack shielding problem will be presented in Section 5.
3.
SATURATEDMICROCRACK DENSITY
If the loading history has been such that all potential microcracks have been developed, then the stress-strain relationship is considerably simplified. The region A in the expression for e; in (2.22-2.23) will then be the whole c-space. V,. The symmetry property, (2.4), can be utilized and (2.23) can be written as
where expressions for the differentiation of Ware given by (2.20-2.21). In a practical evaluation of (3.1), the T-space should be divided into two regions, one for which Fi;,+ P, > 0 and the remaining complementary region. An important case is when the applied load is vanishing (?;, = 0). Then there will be a permanent strain E:; according to (3.2) where (3.3) It is observed that a permanent strain can exist even though the average of the residual stresses vanishes. The reason is of course the crack closure effect. The micro-
Anisotropic
microcrack
nucleation
in brittle materials
541
cracks with negative residual normal stress will remain closed at vanishing applied load. Two other cases are of interest. Firstly, the case when all microcracks are open is considered. This is valid if the minimum principal stress (a?) is larger than --o~.~~“, where gR,m&n denotes the minimum residual normal stress. The stress-strain relation will then be s:; = pa3
16(1-v2) 3E
1 15(2 _ ,,) K10-2~V)&6j~ - VQ%,l~l,,.
(3.4)
Another well-defined situation is when all microcracks are sliding. Thus, the maximum principal stress (a,) is less than -oR,,,aX, where dR.mandenotes the maximum residual normal stress. The strain E; then reads E;
=
pa3
(3.5)
4.
CRITICALNORMAL STRESS
The complexity of the general equation (2.22-2.23) for the determination of the additional strain E: depends to a large extent on the microcrack nucleation criterion g. A simple but still for certain applications realistic nucleation criterion is based on the assumption that the microcrack fracture is solely dependent on the normal stress. In this case (2.5) will simplify to u+u’
> UC.
(4.1)
This specific failure criterion has been investigated in greater detail. An evaluation of (2.22) using (2.4, 4.1) and some algebraic manipulations results in P
E~=~cxa30
1 sCL 62
2n
o
&r,+n,T,)-
;ninjcT
-
1
h,(o,)
n,nj[ah,(a, a,,,) -h3(a, o,,,)]
(4.2)
where n,(8, rp) denotes the normal vector at (0, cp), T, = a,,n,, CJ= ~,~rz~n~, and grn the maximum experienced normal stress on the plane (0, cp). The functions h,, hz and h, are defined as m
ht (em)=
I
F,(d+u,)F,(a’)
da’,
(4.3)
Fc(a’ + a#,(~‘) s -Co
dar,
(4.4)
--(c -0
hz(a,g,)
=
542
s -1
/13(a,a,,,) =
F,(ar+a,)a’F,(a’)
do’,
(4.5)
--(r
where F,.(X) is defined in (2.6) and
(4.6) If the proportional but not necessarily monotonic loading cases are considered. then the maximum experienced normal stress (T,,,will be proportional to the actual loading u (a = qo,,,) for every direction (0, cp). Different loading cases of this type have been investigated in greater detail in Sections 4.14.3. To be able to derive quantitative results, the statistical functions F, and ,fl must be known. In the calculations, rectangular distributions have been considered. L 2aT
F,(a’) =
for
1~7~1 < aT
for
la’/ > aT.
ka
for
la’ -sot
< Aa
0
for
Ia’ -a01
> Aa.
i 0 J; (ar’) =
(4.7)
(4.8)
where aT, a,, and Aa are parameters
defining
the statistical
distributions.
4.1. Pure tension The only nonvanishing stress component is the stress a: in z-direction. The maximum experienced stress is denoted by a=,,). Equation (4.2) can in this case be expressed as
(1 -_.y2)1z1(g2a_,s2a,,,,)
d.u
The integrals in (4.9-4.10) can be expressed in closed form but the expressions too cumbersome to be of interest for an explicit presentation.
are
Anisotropic
microcrack
nucleation
in brittle materials
543
(b)
2
1
/ v
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
/
FIG. 3. (a) Fraction of opened microcracks (n) as a function of applied tensile load oI (ar/ao = 0.25, Ao:a,, = 0. 0.25, 0.5). (b) Fraction of opened microcracks in direction 0 (n(0)) at different tensile loads u_;o,, = I. 1.5.2. 2.5. 3 (uT/ao = 0.25. Au/u, = 0.25). (c) Stress-additional strain curves for tensile loading and unloading (-). Saturated microcrack density (- - -) (uT/uo = 0.5. Au/u, = 0.25).
In Fig. 3(aac) some numerical results are presented. Figure 3(a) shows the development of damage in the material as a function of applied tensile load CT._. The damage is here defined as the fraction of opened microcracks in the material. The damage as a function of microcrack orientation is presented in Fig. 3(b). It is observed that the
544
P. C&JDM~JNDS~N
damage is relatively slowly developing due to the strong variation in the normal stress with crack orientation. Figure 3(c) shows an example of a loading-unloading curve in comparison to a fully saturated case. One observes the permanent strain at zero loading and the additional flexibility at negative reversed stress.
4.2. Biaxial tension Biaxial tension has also been investigated, thus CJ,, = o,,. = crh and c=._= 0. The maximum experienced stress is denoted by oh,,,. Similarly the pure tension case. (4.2) can be expressed as
I-x’-
E
-
(1 -x’)‘hz(oh(l
-.~‘).a~,~(1
(1 -~‘)h~(cr~(l
+
&(ah(l
;(I-.~‘)*
-x2).
1
-s’))
ds
-.?).~&l
-s’))
~h,,,(l -x?))
1
x2h,(a,(l-s’),a,,,(l-.\-‘))ds
ds
(4. I I )
ds
(4.12)
As in the case of pure tension, the integrals (4.114.12) can be explicitly determined. The lengths of the expressions are however prohibitive. The analyses corresponding to the pure tension case have been carried out also for biaxial tension, see Fig. 4(aac). It is observed that the damage is more uniformly developed in the biaxial case. The transition from undamaged material to a certain level of damage is faster. The reason is of course the more uniform normal stress distribution for biaxial loading. The general behaviour of the curves is however similar.
4.3. Triaxial tension Triaxial tension is the simplest loading condition to be analysed. The stress state is described by CT,,= a,6,, and the maximum experienced stress by a,,,,. The additional strain is easily calculated from (4.2)
Anisotropic 1.0
microcrack
nucleation
545
in brittle materials
1
(a)
i
i
3 ‘JbPo
1.0
n(B)
0.5
(b)
i
1
I
__ 1s
’
ao-
Z-LL 45’
60’
75’
90’
0
FIG. 4. (a) Fraction of opened microcracks (n) as a function of applied biaxial load o,, (o,/a, = 0.25. Au/a,, = 0, 0.25. 0.5). (b) Fraction of opened microcracks in direction f? (n(O)) at different biaxial loads uh/uO = 1. 1.5. 2, 2.5. 3 (uTiu,, = 0.25, Au/u, = 0.25). (c) Stress-additional strain curves for biaxial loading and unloading (-). Saturated microcrack density (- - -) (ur/uo = 0.5, Au/o0 = 0.25).
546 1
.o-
n
0.5-
(a)
Fz. 5. (a) Fraction of opened microcracks (n) as a function of applied triaxial load CT,(a,,~,, Au/u, = 0. 0.25. 0.5). (b) Stress-additional strain curves for triaxial loading and unloading (c+;cT,, = 0.5. AC/U,, = 0.25).
~1;’= paa”
b,Vl*(~ )-/1?((T,,(1,,,)+/1~(~,,~,,,,)1~,,. ,,1,
In Fig. 5(a, b) the triaxial results are presented. The transition state to a fully damaged state is in this case the fastest possible.
5. CRACK
= 0.25. (-_)
(4.13)
from an undamaged
SHIELDING
It is known that the nucleation of microcracks can produce some shielding of a macrocrack, see, for example, HUTCHINSON (1987). This effect has been investigated with the aid of the present material model.
Anisotropic
FIG. 6. Definition
microcrack
nucleation
in brittle materials
of the path I- and the area A around
547
a crack tip.
A macroscopic crack of a length much larger than the microcracks under plane strain mode I loading is considered. It is further assumed that the damage zone around the macrocrack tip is much smaller than other relevant geometric measures. Thus. a so-called small-scale damage problem is considered. According to the assumptions, there will be a distance. say R, from the crack tip where the stress and strain are determined from the applied mode I stress intensity field (K,). Furthermore. asymptotically close to the crack tip, the material will be fully damaged. This material, which is under tension, can be described as an isotropic material with modified material parameters (,!?, C), see (3.4). The stress and strain field will then be described by a stress intensity factor K,,P. The problem of interest is to determine the relationship between K, and K,,p. For the solution of this problem it proves to be useful to apply the J-integral according to RICE(1968)
J =
s I-
[Un,
-a,jn,ui,,]dT,
(5.1)
where I’ denotes a path around the crack tip, nj the normal vector and II, the displacement vector. The quantity U is the stress work density and is determined from
(5.2)
where t denotes the time. If J is evaluated on a circle of radius R from the crack tip it Kf(l -v’)/E and if it is evaluated asymptotically close to the crack value K&( I - C’)/E. A closed contour r is now considered, see Fig. gence theorem is applied, a relation between K, and K,,, can be stated
K;(l
-v’) E
where
K&(1-+) ---IF-=
takes the value tip it takes the 6. If the diverin the form
I ’
(5.3)
548
P. GUDMUNDSON
bu.,k,-6,~
I=
r,.,
(5.4)
1dt dA> 1
and (A) denotes the area enclosed by F. To be able to evaluate the integral I, the stress and strain fields must be known in the area (A). For small inelastic strains, it is however possible to determine I in the form of a first-order perturbation to the linear elastic problem. According to the results in Section 2.5, the strains can be written as E,, = Cljh,Ok,+ For pa-’ + 0 the inelastic strains state will therefore be described Thus.
E:y.
(5.5)
are much smaller than the elastic ones. The stress by the elastic stress plus a small contribution o:,.
(r,,
=
a; + a;,.
(5.6)
Up to first order in pa3, the strains will hence be (5.7)
e,, = C,.,,,(~~,+o;,)+E~~(~~,),
where E:,‘(&) means that ~7 should be solely determined from &‘I according to (2.23). If only first-order terms are considered in (5.4) the integral I can be expressed as ’ [a~.,~::l(a~,)-~:;E:::,((~~,)]
I= A S[S
The strain
E:~(f$)
is the result of a proportional. m
EiJ.l =
.,>I E 1, =
according to (2.29). Equation (5.8) can therefore I=
dr
0
monotonic
D,jk/aZ/.
1
dA. loading.
a:;.,(D,,k,-
In this case
IT
Di,k~.
be rewritten
(5.8)
(5.9)
as DA,,,)& dt
I
dA.
(5.10)
It is observed that the integral I vanishes if the tensor D,,k, is diagonally symmetric. This is the case if there exists a complementary strain energy density 4, see (2.32). The only case for which 4 exists is when the microcrack nucleation criterion is determined from W, the energy release of a microcrack as was discussed in Section 2.5. In this particular case the I&, is determined from
For a Poisson value
ratio of 11= 0.3 the stress intensity
factor
at the crack tip takes the
Anisotropicmicrocrack nucleationin brittle materials Ktlp = &(I-
549
1.018pa3).
(5.12)
The integral I has also been evaluated for a case with a nonsymmetric tensor Dilk,, namely the model discussed in Section 4, where a critical normal stress was applied as a microcrack nucleation criterion. After an application of (4.224.8) in (5.10) the introduction or the operators
(.)=i$ $=I and a representation can be stated as I=
cos (p) g - i sin (p) $
of the stress field OF,in the form G$ = K,Ci,(/Q/j27rr
4pa3
-
3x2(2-v)
I’
K:(l-q E
i
(f2C,,
-BABY)
the integral I
(5.13)
sin (/J) sin (6) d/l dtI dp,
(5.14)
where ? and C denote the dimensionless shear and normal stress at an angle ,G from the crack plane on the microcrack plane defined by (0, cp). The stress intensity factor at the crack tip will then be 4
16(10-3v+8v2-3v3) +v(z$
45(2-v) The integral f has been numerically value of v = 0.3 that
evaluated
to be f=
-0.05695.
. 11
_
This gives for a
(5.16)
Ktip = Ki(1-1.016p~~). It is observed that the Z-integral has a negligible particular microcrack nucleation criterion.
6.
(5.15)
influence
on the K,,,-value
for this
DISCUSSION
A central assumption in the development of the present model was that the microcrack density in terms of pa3 should be small. The expressions for the additional strain caused by microcracking are only valid under this assumption of dilute microcracking. For larger microcrack densities self-consistent estimates of the additional strain could be computed in an analogous way as were done by BUDIANSKY and O’CONNELL (1976). HORII and NEMAT-NASSER (1983). Alternatively, the so-called differential scheme for the determination of modified elastic moduli could be applied, see MCLAUGHLIN (1977) and HASHIN (1988). The differential scheme is based on a model for incremental changes in moduli resulting from incremental changes in microcrack density. It seems that this procedure would be ideally suited to the present model, since the microcrack density is naturally formulated in incremental form. The problem of formulating a constitutive model allowing large microcrack densities is
550
P. GUDMUNDSON
however not only a problem of determining additional strains from a known microcrack distribution. The microcrack nucleation criterion and the microcrack closure criterion will also be influenced by a large microcrack density as a result of microcrack interactions. The present model is nevertheless applicable to many practical problems. If the number of potential microcracks is large, for example if all grain boundary facets in a single phase material are treated as possible microcrack sites. then the present model is only valid up to a certain critical, small microcrack density. The particular material will however fail long before the microcrack saturation density is reached, due to coalescence of neighbouring microcracks. The failure criterion can be expressed as another critical microcrack density. It is not unreasonable that the failure density is smaller than the limit density for the present model. In this case it makes no sense to include a much more complicated theory for large microcrack densities. Another situation where the present model can be directly applied is when microcracks are developing only in the vicinity of second phase particles. the density of which is small. Such microcrack patterns have been observed by R~JHLE et al. (1987) in zirconia-toughened alumina. In this case a saturation level may be well defined by the density of the second phase particles. One should observe however that for the present model to be valid only one potential microcrack direction is allowed for each particle. If there is more than one potential microcrack at each particle, where only one will develop, then the present statistical model has to be modified. It should be stressed that the present model predicts a permanent strain at vanishing stress, even though the average residual stresses are vanishing. The reason is that crack closure effects are taken into account and that a random variation of residual stresses exists. Another important feature of the model is that the degree of anisotropy is a direct consequence of the applied loading history. No phenomenological assumption whether an isotropic or anisotropic distribution results is necessary. The simple examples which were investigated showed that triaxiality of the loading is of importance concerning the resulting anisotropy. When the saturation level is reached, the microcrack density of the present model is always isotropic independently of the loading history. A maximum degree of anisotropy is therefore achieved at some intermediate load level. The crack tip shielding analysis showed that the present model predicts &,-values in accordance with alternative material models analysed by EVANS and Fu (1985) HUTCHINSON (1987), ORTIZ (1987), CHARALAMBIDESand MCMEEKING (1988), GIANNAKOPOULOS(1988). The critical normal stress nucleation criterion which was analysed showed a very small deviation from the simpler analysis with a complementary strain energy density. The present model is rather complicated to apply in a general structural mechanics problem. In each point of the solid the microcrack density in all directions must be known. This can be formulated as the fraction of opened cracks as a function of the spherical angles 0 and cp. Thus for each point in the solid a function of two variables is required. It would be simpler if the damage could be described by some lower order tensor quantities. ONAT (1984) has shown that a function of (8, cp) can be developed into tensors of different order. Approximately a function of (0, cp) can be modelled by a symmetric second order tensor. For the purpose of practical application it would therefore be of interest to investigate such an approximation.
Anisotropic microcrack nucleation in brittle materials
551
REFERENCES AMAZIGO, J. C. and BUDIANSKY, B. BUDIANSKY, B. BLJDIANSKY,B. and O’CONNELL, R. J. BUDIANSKY. B., HUTCHINSON,J. W. and LAMBROPOULOS,J. C. BUDIANSKY, B., AMAZIGO, J. C. and EVANS, A. G. CHARALAMBIDES,P. G. and MCMEEKING, R. M. EVANS, A. G. and Fu, Y. EVANS, A. G. and CANNON, R. M. Fu, Y. and EVANS, A. G. GIANNAKOPOULOS,A. E. HASHIN, Z. HILL, R. HOAGLAND, R. G., HAHN, G. T. and ROSENFIELD,A. R. HOAGLAND, R. G. and EMBURY, J. D. HOAGLAND, R. G., EMBURY, J. D. and GREEN, D. J. HORII, H. and NEMAT-NASSER, S. HUTCHINSON, J. W. KACHANOV, M. MCLAUGHLIN, R. ONAT, E. T. ORTIZ, M. ORTIZ, M. ORTIZ, M. and MOLINARI, A. RICE, J. R.
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Acta Metall. 11. 2701.
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Brown Uni-