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Micromechanical damage model taking loading-induced anisotropy into account
Aerospace Science and Technology, 1997, no 1, 65-76
Micromechanical Damage Model Taking Loading-Induced Anisotropy into Account C. Vinet (l), P. Priou 12) (I) Bordeaux-I
University, Airospatiale @) Akrospatiale
(Space Division), (Space Division),
Manuscript
Vinet C., Priou P., Aerospace Abstract
Science and Technology,
12, 1995; accepted February
France. France. 22, 1996.
1997, no 1, 65-76.
Micromechanical
model - Ceramics - Anisotrop
damage constitutive
law - Dynamics.
ModGe d’endommagement micromkcanique prenant en compte I’anisotropie induite par le chargement. Cet article presente un modirle d’endommagement construit a partir d’une thtorie destinee aux materiaux fragiles comportant des microfissures. Le modele est implante dans le code 3D de dynamique rapide DYNA3D. L’etude phenomenologique montre l’importance de prendre en compte les effets des micromecanismes pour modeliser la rupture macroscopique du materiau. Le modele developpe relie l’endommagement a des parambtres microscopiques (rayon de microfissures, densite de fissures...) et prend en compte le caractitre anisotrope de l’endommagement induit par le chargement, en reliant a des orientations privilegiees l’evolution des microfissures. Le caractbre unilateral (difference de comportement en traction et en compression) est consider& lors du critere de croissance des microfissures. La degradation progressive des proprietes du materiau, induite par la presence des microfissures, est obtenue a partir de la theorie des modules effectifs de Margolin et le materiau est pulverise lorsque la densite de microfissures depasse une valeur critique. Un calcul de l’energie dissipee par endommagement est propose. Le modele, applique a une ceramique SiC/SiAlyon est valid6 en comparant les resultats des essais aux barres d’Hopkinson et les simulations numeriques. Une comparaison entre un modble fragile macroscopique et le modele d’endommagement present& montre la capacite de ce dernier a predire les effets des microfissures sur le comportement a la rupture des ceramiques. Mots-elks
Aerospace
received December
78130 Les Mureaux, 78130 Les Mureaux,
This paper deals with a micromechanical damage model, based on a constitutive theory for brittle materials weakened by microcracks. The model is implemented in the DYNA3D three-dimensional explicit finite element code. The phenomenological study shows the importance of taking micromechanical effects into account to model macroscopic failure of the material. The constitutive model relates damage to microscopic parameters (size of microcracks, cracks density etc.) and takes loading-induced anisotropy damage into account by correlating microcrack growth to preferential orientations. The unilateral character (behaviour difference between tension and compression) is treated by the microcrack growth criterion. The progressive reduction in material stiffness due to the presence of microcracks is modelled using Margolin’s effective modulus expressions, and the material is pulverised if the microcrack density exceeds a critical value. Determination of the energy dissipated by damage is proposed. The constitutive model applied to SiC/SiAlYON ceramics is validated by a comparaison of the results between a Hopkinson’s Bar Test and numerical simulation. Comparing the macroscopic brittle model results with the damage model results shows the ability of the second to predict microcrack effects on the dynamic failure behaviour of ceramics. Keywords:
RikumC
66, route de Vemeuil, 66, route de Vwneuil,
Science and Technology,
: Modele micromecanique
0034-1223,
97/01/$
7.00/O
- Gramique
Gauthier-Villars
- Loi endommagement
anisotrope - Dynamiques.
66
C. Vinet, P. Priou
NOTATIONS macroscopic strain tensor mean strain tensor local strain tensor effective compliance tensor of the microcracked material Cauchy’s macroscopic stress tensor mean stress tensor local stress tensor mean radius of the cracks with normal n’i (i = 1,3) maximum radius of the cracks with normal ?zi (i = 1,3) number of cracks with normal n’i per unit volume damage parameter in direction i normal to the crack in direction i statistical distribution of the radii of the cracks with normal ?Li;in the element crack radius Poisson’s ratio of the undamaged material Young’s modulus of the undamaged material microcrack growth rate of the crack with radius a Rayleigh wave speed critical energy release rate toughness
I - INTRODUCTION Ceramics can exhibit inelastic strain under compressive, tensile and shear loading of the grains. Under mechanical loading, they rapidly incur diffuse degrading, called damage. This results in a gradual loss of their physical properties. The presence of microcracks and the type of loading determine the growth of the damage. The shape, distribution and spacing of the microcracks all affect growth of these microcracks and consequently the physical quantities involved. The state of damage is described by one or more appropriate internal variables. Several types of models can be distinguished. The phenomenological approach of continuum damage mechanics [l-lo] relies on the original Kachanov model of damage variables [1 l] and on the concept of the effective stress. It’s a macroscopic view of the damage process which depends on the microstructural kinematics. The theory of continuum damage mechanics is developed based on the general concepts of the thermodynamics of irreversible processes with internal variables. The thermodynamic potential defines the material damage state and the potential of dissipation describes the evolution of the irreversible damage. This approach requires the perfect knowing of the damage topology in order to determine the internal variables (damage tensor) and the kinematics laws governing the
damage evolution. The microscopic models [ 12-241 establish a relationship between the damage parameter (crack density) to the microstructure (geometry, grain and pore distribution...). These models tend to protect physical phenomena but for programming efficiency, it’s necessary to assume several simplifying hypotheses on crack form, orientation and distribution in the material. The damage state (chronology and distribution) not known accurately, a microphysical approach of the problem, based on microscopic descriptions and measurements of the material, was preferred over an empirical or phenomenological approach to analyse the microcracking process and simulate the material stress state associated with microcrack initiation. The properties and behaviour of a volume element of the material were obtained by integrating the microscopic results on the macroscopic volume element using mathematical homogenisation techniques. Rajendran [23, 241 propose a 3D constitutive model based on microcrack growth and pore collapse. Inelastic strains are introduced to account for microcrack opening and sliding. The pore compaction is modelled but the crack closing under compression is not considered. The formulation is based on Margolin’s effective modulus theory [25] and Budiansky-OConnell’s theory [26]. Margolin gives the expression for the effective isotropic stiffness tensor for randomly oriented, penny-shaped non-interacting microcracks of different sizes. This model appears to be well design for porous ceramics subjected to multiaxial loading. However, the fact that the damage is isotropic remains a problem. The damage model developed and implemented in DYNA3D is based on Rajendran model but the description of the microstructure is modified to introduce anisotropic damage depending on the loading and preferential microcrack directions. The emphasis was placed on determining the statistical crack distribution to allow loading-induced damage anisotropy to be taken into account. This paper describes this distribution as well as adaptation of the growth law and calculation of the damaged elastic moduli to this distribution. Calculation of the energy dissipated by the damage is proposed. The model is then used to predict macroscopic fracture of an SiC/SialYON test specimen subjected to a Hopkinson’s bar test. The numerical results are compared with the experimental data. The predefinition of three privileged orientations of microcracks proves conclusive in the case of the test presented but it does not allow to model the anisotropy induced by complex loading. The damage material symmetries are predefined by the choice of the global axes and consequently the failure planes must be known in order to have a good model. Aerospace
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Micromechanical Damage Model Taking Loading-Induced Anisotropy into Account/ Mod2le d’endommagement microme’canique prenant en compte l’anisotropie induite par le chargement
II - DESCRIPTION
where N (ai; ?Li)) is the statistical distribution of the radii of cracks with normal n’i in the element. The total microcrack density is given by:
OF THE MODEL
11.1 - General Since the ceramic investigated has an initial porosity of only 2% and plastification does not occur during the tests, closing of the pores and plastic strains were neglected. The strain process applied is therefore a brittle process for which the elastic strains are assumed infinitesimal and the anelastic strains are due only to the presence of microcracks. The macroscopic strains are defined by:
where E$ is the elastic strain tensor of the undamaged matrix, E;~ is the inelastic strain tensor accounting for the microcracks. The elastic strains of the matrix and the microcrack strains are proportional to the stress fields Ckl :
The elements of the effective compliance matrix C’ijkl are obtained from the effective modulus theory based on Margolin’s theory [25]. II.2 - Statistical
crack distribution
The model is constructed from microscopic observations of the material. Damage is described based on statistical distribution of the microcracks. The cracks are assumed penny-shaped. The interaction between cracks is neglected. This assumption remains verified for [25]: Na3 << I where N is the number of cracks per unit volume and a is the crack radius. In Rajendran’s model [23, 241 as well as in most existing micromechanical models [19, 20, 271, crack distribution is assumed to be random in a volume element, which implies that the damage material remains isotropic regardless of the loading applied. The new idea is to define a relationship between crack size and orientation for a given load. Three families of normal orthogonal microcracks are defined. These normals correspond to the three axes of the global coordinate system. To characterise the damage state, it is then necessary to determine three parameters per family relative to the microstructure: the mean radius E; of the cracks with normal n’;, the maximum radius amax, i of the cracks with normal n’i, the number No, i of cracks with normal n’i per unit volume. The damage parameter yi, in each direction is defined by the density of microcracks with normal n’i :
7=x
Ti. i=l
The microcracks with different orientations grow independently of one another as long y does not exceed a critical value. The anisotropy induced by loading is therefore directly expressed by the damage parameters. The volume distribution N (a,) Gi) of the radii of the cracks oriented along +$ is taken as exponential [ 14, 17, 22-241. This exponential distribution is assumed to be preserved over time. For each family of cracks, the microcracks are assumed to be nucleated when the stress state is such that the Griffith criterion is satisfied for the largest microcrack (a max,i). Th e o th er microcracks of the same family are then assumed to propagate at an appropriate rate to preserve the exponential distribution of the crack radii related to this direction (there is a criterion for each direction i). In each family of cracks, the distribution is characterised by a function N (ai; Gi), where N (ai; ?&) da; is the number of cracks per unit volume with normal n’i, and where the crack radius belongs to the interval [ai, a; + da;]. The continuous function N is discretised and is given analytically in a; : (2)
The total concentration JV of cracks with normal sn’; whose radius is greater than a; is: (3) The damage parameter then becomes:
J
i=l
The number of cracks per unit volume in direction i (i = 1, 2, 3) is assumed constant. The total number of cracks per unit volume is denoted No = 5 Na,i. i=l Assuming amax, i >> &, “i; is approximated by Yi z 6 No,; $. We then define N& by: yi = NGi akaxi.
y;= .! 1997, no 1
. N (a;, 2;) a: da;
67
(1)
This formulation
is used for programming.
(5)
68
C. Vinet,
II.3 - Growth
law
The microcrack growth mechanism is analysed by applying the theory of dynamic linear fracture mechanics. For brittle material, this theory consists of nucleation, propagation and coalescence. l Microcrack nucleation occurs in an element when the stress state satisfies the generalised Griffith criterion for the largest microcrack of the element [28, 291. The criterion is established by relating crack nucleation to crack orientation. l Damage growth as an increase in crack radius according to crack orientation is modelled by a growth law expressing the propagation of a crack under dynamic loading conditions. l Coalescence occurs when the propagating microcracks intersect each other and pulverise the element. The assumption of no interaction between cracks then no longer applies. A reasonable pulverisation criterion can be established: pulverisation occurs when the mean radius reaches half the mean distance separating two cracks. This condition defines a critical crack density value above which the material is considered as pulverised. The variation of the microcrack nucleation, propagation and coalescence mechanisms within a volume element is independent of their variation within the other elements. 11.3.1 - Crack nucleation model The nucleation criterion used is the generalised Griffith criterion as developed by Margolin [ 171 and modified by Dienes [28, 291. Dienes calculates crack stability by adding to Griffith’s criterion an energy term corresponding to the loss of energy of a closed crack. This loss of energy results from friction of the two edges of the crack (shear). This criterion is applied to each family of cracks. Nucleation occurs if:
fpv-S]>O where W is the strain energy and S is the surface energy released when the crack opens. The strain energy under mode I crack opening (tensible normal stress) is different that under modes II or III. The improved Griffith criterion depends on the crack orientation. Three criteria relative to each family of cracks (same plane) are defined for each element. They are applied completely independently for each family. Each criterion is first expressed in the local crack coordinate system then reformulated in the global coordinate system. Where i, is the normal to the family of cracks, for i, j, k = 1,2, 3 and i # j # k: l For ??kk > 0, the energy release rate under mode I crack opening is written:
G+k = 4 (’ - u2) amax 7iE
1
(7)
P. Priou
0 For ??kk 5 0, the energy release rate is written (in this case, only modes II and III are active): G- = k
8 (l - u”) 7i-E (2 - ZI) %nax,k
[j/~-%+~~r:b12 (8)
is the maximum radius of the family of cracks orthogonal to vector k, r. is the cohesion stress, and h is the friction coefficient. Defining G&,, k, G,,,, k as the maximum of (Gz and 0) and (G, and 0) respectively, the microcracks perpendicular to vector c are assumed to have nucleated if G&, k or G,, k exceed the critical energy release rate’ Gc: ’ Kfc (1 - v”) Gc=2T= E ’ The relationship between crack radius and orientation is defined by the formulation of the different criteria. The unilateral character of the behaviour (different behaviour under tension and under compression) is taken into account.
where
amax,
k
11.3.2 - Microcrack growth If one of the crack families verifies the improved Griffith criterion, the material damage grows. The microcrack growth law is obtained from the theory of dynamic fracture mechanisms [30, 311. The classical solution for the energy release rate G (t) under mode I crack opening is given, for a fast crack propagation, by the equation: - a =I.CT
~GC w .
(10)
For the solution to be valid, any material speed must be smaller than the speed of sound e in the material [25]. This assumption is verified for a. Generalising this equation for the three crack growth modes, empirical constants are introduced to control the maximum crack tip propagation rate [23, 241. We have: For k = 1,3 (for each crack family) 0 if G&,
nm(‘icmax,k
=
k 5 Gc and G,,,
k < Gc
(g-J)
if GL,,, Ic > Gc
if G&,
k 2 Gc and G,,,
k > Gc (11)
Aerospace
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Micromechanical Damage Model Taking Loading-Induced Anisotropy into Account/ Modt?le d’endommagement microme’canique prenant en compte l’anisotropie induite par le chargement
where nf , n2+ , n, nz are constants to be determined. Quantities nt and n, directly control the maximum growth rate for the different opening modes, whereas exponents nz and n; moderate the crack growth rate until Gc is sufficiently exceeded. The material damage growth can be monitored by numerically integrating equation (11) rewritten for each crack family and then evaluating the crack density within the material. 11.3.3 - Microcrack coalescence Coalescence occurs when the propagating microThis results in cracks intercept each other. pulverisation of the material and a significant loss of strength. The pulverisation criterion is based on Rajendran’s definition [23, 241. Where NO is the total number of microcracks per unit volume, the mean 1 distance d between cracks is d = iV, 3. We denote as Gnax the maximum of mean radii Ek (k = 1, 2, 3). Pulverisation occurs when a,,, reaches half the mean distance d. This coalescence condition defines a critical crack density for pulverisation 11231:
The material is therefore considered pulverised when y reaches rp in a stress state in which at least one of the main stresses is compressive. Under tension, the material lose its strength and under compresion, it obeys the Mohr-Coulomb law. The pressure is expressed by:
P=C-g
P EZ
E; 2 0 E; < 0
modulus
theory
The microcrack presence in the body involves an additional deformation and a lost of the mechanical properties. This phenomenon is taken into account by relating the local strains due to microcracks to the macroscopic strains using a homogenisation technique. 11.4.1 - Relationships between microscopic and macroscopic quantities Consider a solid of volume V assumed macroscopically homogeneous containing microcracks on a microscopic scale. The microscopic quantities are g(z) for stresses, E (x) for strains and u(z) for displacements. The corresponding macroscopic quantities are denoted C, E, U respectively. We defined 0 (t) as the volume of the solid undamaged by microcracks and 7 as the outward unit normal vector to the surface. The outer surface of the solid 1997, no 1
is denoted S, and the crack surface S; (t). The mean value of f on R is denoted 7. If surface loads T; (z) are applied to Se, the macroscopic stress tensor .&, constant on R, is defined by Ti (z) = Cij.rj (z). The local field LJ~~(zz) which results is statistically admissible if it verifies: div oij = 0 0110 CT;?. pi = Ti (z) onS, (14) aij . rj = 0 on Si (t) 1 By writing the variational problem corresponding to problem (14) and using Green’s formulation, it is shown that: The macroscopic strain Eij is defined by: Ezj = $
’I se
(Ui . ri + U, .ri) ds.
The microscopic strain “ii (z) is kinematically admissible with the local field ui (z) that fluctuates around the mean of U; (x) on S,. The hypothesis of macrohomogeneity [ 151 results in: (ui (z) - Ui (x)) J SC This yields (17):
ri (z) ds = 0
(16)
(13)
where E; : is the elastic volume strain and KP : is the pulverised material bulk modulus. The pulverised material bulk modulus is considered equal to the bulk modulus of the damaged material just before pulverisation occurs. II.4 - Effective
69
= Eij (Lx) - $
.i St (t)
(pi . pi + U? . ri) ds (17)
Applying the divergence theorem and denoting n’ the crack normal (on Si (t) , n’ = -?) yields the expression for the macroscopic strain in a volume element V containing microcracks:
+
J’
(ni . uj + nj . ui) dS
1 (18)
(n; . uj + nj . u;) dS
1
s;(t)
We set: (19)
Tensor a! is the macroscopic manifestation of the displacement discontinuities across the crack surface.
70
C. Vinet, P. Priou
By definition of the constitutive law, we have: Ei.j = CFjkl Fkl where CZ:Al is the undamaged material. Then:
compliance
strains of the material in presence of NV cracks with radius a: (20)
matrix
of
the + 16 (1 - u”)
Eij = C,p,,, zIcl + aij
3E
(21)
where ??kt = Ckl is the mean of macroscopic stress tensor. This yields the expression for the damage or effective elasticity modulii:
da!ij Cijkl = C,Q,I,l+ ~ d3kl
+
8 (1+
4
Na3 max (Zkl nk nl, 0) ni nj
Na3
3E
(22)
TO calculate Cijkl, it is necessary to determine aij or, more specifically, to know the displacement fields on the surface of each crack.
11.4.2 - Calculating
the jields [3]
In an initialy isotropic material, Margolin assumes a uniform distribution of the cracks with the same radius and same orientation. The expression of the solution fields is obtained under static load by calculating the crack displacement jumps (in a local coordinate system) based on the theory of cracks in an infinite medium isotropic (Bell [32]). This approximation is possible because of the small size of the crack (crack radius much smaller than the element edge). The solutions of the problems lead to distribute the discontinuities of the displacement jumps on the crack of a constant form, whatever the boundary conditions imposed [ 181. These displacement jumps are written:
More generally, by using the principle of superposition of solutions (because of the linearity of the solutions), he obtains the following expression in the case of coplanar cracks whose radius is statistically described by function N (a): Eij = CzPjklakl + 16 (1 - u”) Na m8X (8kl n& nl, 0) 3E
2E]-Ua;cz[, i=n,t,s (23)
where n, t, s are the normal and tangential components. These results apply to the dynamics if the material strain rate is much lower than the speed of sound in the material [25]. For coplanar cracks with the same radius a, the integral of oij on each crack is the same. This means that for NV cracks, where N is the number of cracks per unit volume and V is the volume element, we can write:
N w = -2
(ni . uj + nj . ui) dS
1
ni
nJ
+ 8(1+v)3E
Na
x (ail, nj nk + -jo kninl,-2n;nj??klnknl) (26)
were
[u;](x)=K;xLz2,
coplanar
Na=
.I’
N (CL)a3 da
This amounts to modifying three of the diagonal terms of the compliance matrix. Margolin applies this result to coplanar or isotropic crack distribution. In the case of the model presented, there are three different crack orientations. The strain tensor is written in the same way as (26) in each local crack coordinate system for a distribution N (ai, Zi) defined by (2). The expression for the global strain tensor in the coordinate system of the material is given by:
(24)
where c is the crack surface area. By investigating separately the influence of the stress field components at infinity on the displacement fields formulated by Bell on the crack surface, Margolin obtains the expression for the macroscopic
where Pi is the coordinate system Eij (ak, NC,+) is orthogonal k with
transformation matrix from local i to the global coodinate system and the strain tensor relative to cracks distribution Nk (ak, nk). Aerospace
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Micromechanical Damage Model Taking Loading-Induced Anisotropy into Account/ Modkle d’endommagement microme’canique preenant en compte l’anisotr-opie induite par le chavgement
The coefficients of the damaged compliance matrix C are given in the (0, X, Y, 2) coordinate system by: (1 - v”)
if a,, > 0
16 Naz (1 - v”) c YYYY - cy”YYY+ 3E
if Oyv > 0
+ +
if O,, > 0
c zzzz - cLz
C zrzz - CL
+ s
(1 - v”)
C XY”Y - cyzy + “‘ii
u, (Ti&+K$
C YZYZ = qzyz + x(;;
4 (~+7icg)
C %Z%Z- CL
+ fgqMa;+Na,) (28)
71
surface, consuming energy by material damage. It is therefore interesting to analyse the energy dissipated over time by damage in the ceramic. Since the material damage is due to the growth of microcracks, to calculate the dissipated energy it is necessary to analyse the energy balance of the microscopic mechanisms of the material, considering [33]: the microcrack nucleation energy, the variation of the internal energy when the microcrack occurs, the energy dissipated to overcome the material resistance during microcrack extension, and the kinetic energy of the moving parts. The elementary energy consumed by a microcrack of length 2a to extend a distance Aa is equal to: dW = 2?rGa& (30) where G is the energy release rate or extension force for a microcrack of radius a, already defined by (9) and (10). The elementary energy consumed by NV bedded cracks of radius ai (i = 1 to NV) to grow by Au; is therefore written: ivv dWT = C 2 T G (ai) a; aa, (31) i=l
Only the diagonal coefficients of the elasticity matrix are modified. The matrix therefore remains symmetric. [C] = -v/E
-v/E
- u/E
l/E+az
-v/E
- v/E
-v/E
l/E
[l/E+cul
A+=2a4(l-V2)
0
0
0
l/G+w
0
0
0
l/G
xrqcr + au (29)
Taking the loading-induced damage anisotropy into account requires processing the crack based on its radius and direction before obtaining the global behaviour of the element by homogenisation. The equation of state (Eq. 26) depends on the loading. Under tension, cracks open with a displacement proportional to the normal average stress. Under compression, they close and only the crack sliding are considered to calculate damage. The unilateral effect is taken into account in the damage compliance matrix (Eq. 28). energy
When a small projectile impacts the centre of a ceramic plate, the ceramic slows down the projectile and distributes a load on a large area of the rear 1997, no 1
+ ~,‘,) 2-u
! 2 (a$ f z$) 8 (1 71”) WV ZZ G, + E 2-V
+ 013 l/G+w
2 (&
-2
g’kli +
7iE
0
II.5 - Dissipated
The energy consumption depends on the crack radius. For an exponential distribution on the radii of cracks orthogonal to & setting:
Gc ‘nn: [ 1 l-
G+
At
1
IIAt
1
if alik, > 0
A- _ 2 8 (1 - v”) - 7r TE (2 - I/)
x [~~-7r~+i~“~*]~hnf 16 (1 - v”) E (2 - u) Xn,Cr
n2 l__ [ 1 -
Gc G-
At
else (32)
yields:
N0 ye a No Te
a
-za2
-Q
da
a a2 da
if ??kk > 0
else (33)
72
C. Vinet, P. Priou
The elementary energy consumed by NV cracks with the same orientation (orthogonal to i) with an exponential distribution of radii corresponds to:
A+2No$=A+
7%
if akk > 0
A-2No~2=A-
!!!T!I!&S 3 a
ifakk
dWT = <34) where amax is the maximum radius of the exponential distribution of cracks, Z is the mean radius of the exponential distribution of the cracks, NC is the number of cracks per unit volume, N,* = No $$ . In the case of the model, the elementary energy consumed by damage is equal at time t to the sum of the variations of the energy dissipated by crack growth for the three families of cracks. The total elementary energy consumed at time t is therefore expressed as follows: W(t)
= W (t - 1) + 2
dW (u;)
of the same high-strength material (maraging steel). The sample, with a mechanical strength less than that of steel, is placed between the two bars. Wave propagation is considered one-dimensional and nondispersive in the inlet and outlet bars. The bars preserve elastic behaviour throughout the test. When the projectile propulsed by a gas gun impacts the inlet bar, a compression wave is generated and propagates through the inlet bar. When the incident compression wave reaches the inlet bar/sample interface, it divides into a tensile wave reflected in the inlet bar and a compression wave transmitted through the sample. The wave transmitted through the sample is partially transferred to the outlet bar. A knowledge of the history of elasic strains in the inlet and outlet bars obtained using strain gauges, allows determination of the microscopic strain history in the sample.
Gas gun I
(35)
Inlet bar
H H
Outlet bar ‘B w Sample
‘A
Projectile
I
i=l
where W (t) denotes the elementary energy consumed at time t, W (t - 1) the elementary energy consumed at time t - 1 and dW (ui) the energy consumed for the growth of the family of microcracks orthogonal to ?&. In our model, the calculation of the energy dissipated by damage does not include the microcrack nucleation energy, since the material is assumed to be precracked and the number of microcracks per unit volume is assumed to remain constant over time. III - DYNAMIC PARAMETER CHARACTERISATION Mechanical characterisation of ceramic materials in the area of dynamic stresses based on simple tests allows definition of the physical data and parameters of the model. The validity of the results is checked by comparing the numerical simulations of the tests made using DYNA3D [34] with experimental data. The SiC/SiAIYON composite was dynamically characterised under compression using CREA Hopkinson bars. Because of its intrinsic nature, ceramic material does not allow reliable direct analysis of tests using the theories applied for metals. However, the curves of the incident, reflected and transmitted strains versus time at different strain rates give an idea of the dynamic behaviour of the material. III.1
- Hopkinson’s
bar test
III.l.l - Description of the test facility The tests are conducted at room temperature. The test facility includes a projectile, an inlet bar and an outlet bar, all with the same diameter and made
Scheme
1. - Test apparatus.
111.1.2 - Description of the test conducted The test specimen was a cylinder with a diameter of 7 mm and a height of 8 mm. The projectile speed was 17 m/s. At the end of the test, the sample was pulverised. Fracture of the sample occurred as the wave increased in the incident bar. The only conclusion that can be drawn is that the macroscopic fracture stress of the sample was exceeded. From the reflected signal, it is deduced that the strain rate reached -1430 s-l. From the transmitted signal, it is deduced that the mean fracture stress was 3480 MPa. The transmitted and reflected signals indicate brittle fracture and therefore little damage. The steady state and uniform stress hypotheses were not satisfied. There was no plastic behaviour. Conventional analysis of the test was therefore not possible. III.2 - Modelling
of the test
The parameters (urnax, a, n;, nz, Cr . . .) were determined from the literature and from microscopic observations. Resetting was carried out based on the tests. A comparison with a brittle model defined from the fracture stresses obtained during the tests was carried out and the influence of the mesh was analysed. 111.2.1 - Geometry, mesh and boundary conditions The two models were compared for the same mesh. Only a quarter of the problem was modelled, since symmetry conditions were imposed. The contact between solids was modelled by sliding surfaces Aerospace
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Micromechanical Damage Model Taking Loading-Induced Anisotropy into Account/ Mod2le d’endommagement microme’canique prenant en compte l’anisotropie induite par le chargement
obeying the penalisation method. The geometry was defined as follows (Table 1).
0.002
_
,$r)lq j py
O.OOl_
Table
1. - Geometry
Ii
s ‘2=
parameters.
2 E 2
73
f 0.000
._ , . . . 1;’
\
L
i -O.OOl\ -0.002 I
I
0.0000
i-;--
\$&.+i :’ ‘I
0.0002
0 I I I 0.0004 0.0006 Time (s)
I I 0.0008
I 1 0.001
0.0002L
0.0000 .g -0.0002~
The mesh included a total of 5372 elements distributed as follows: 468 for the projectile, 2388 for each bar and 128 for the test specimen. 111.2.2 - Properties of the material The constitutive equation used for maraging steel is an isotropic elasticity law. The behaviour of the specimen was modelled either by the damage model presented or by a macroscopic brittle model. The fracture criterion of the latter was a stress criterion: If the main stresses exceed the tension and compression limits (Tables 2, 3, 4) obtained during the tests, fracture occurs. The material properties (p, E, V, Ki,, CL)of SiC/SiAIYON were obtained from Table
2. - Parameters
and material
properties.
Table
3. - Fracture
limits
(T,, (tension)
(MPa)
for the brittle
material.
art
3480
Table
4. - Parameters
(traction) 320
for the damage
model.
2 -0.0006
_
-o.oooa0.0000
I
0.0002 I
0.0004 /
I Time
Fig. 1. - Hopkinson’s test data.
bar test, comparison
0.0006 I (s)
I
’ 0.0008
between
0.001
computation
and
static tests conducted by Aerospatiale in Aquitaine. The others were taken from the literature [23, 24, 30, 351 or from micrographs. It was necessary to identify the value of the maximum radius and growth parameter nz. The results are given in Figure 1. III.3 - Discussion
SiC/SiAlYON
Properties
E -0.0004~
(MPa)
Consistently with the test data, the specimen modelled by the damage law was completely pulverised. The results given in Figure I also show good correlation between numerical simulation with damage and test data. The brittle model causes too sudden a fracture (transmitted wave) and the jump observed in the reflected wave is not restored by calculation. It is therefore necessary to take the material damage into account before pulverisation. Numerical simulation is used to estimate the damage growth over time. In particular, it is observed that the material damage is due both to the growth of cracks orthogonal to the axis of symmetry of the bar and to the growth of cracks perpendicular to the specimen section, due to a tensile wave propagating through the section of the test specimen. This observation is physically confirmed during another Hopkinson bar test at a rate of 12 m/s. An additional study is proposed to illustrate the modelisation of the moving cracking. Two simple tests of pure tension and pure compression in the direction z are modelled at different strain rates. Different results are presented: the crar versus E,, curves, the influence
74
C. Vinet, P. Priou
0.10
-
,
0.08 -
,---
IF 0 2 0.06 5
-0.06 0.02 -
0.000
I 0.7
0.2
Strain Fig. 2. - a. Comparison 8 = 7.5 1o+s -- &O = 4.0 lo+*
stress/strain s-1.
under
Time Fig. 2. - d. Variation damage initialisation
compression.
of the crack
.
,
(ps)
radius
under
compression,
after
s-l, O.lC I-
0.08 brittle
fracture
2 0.06 il? E
,g 0.04 z H 0.02
0.00 r 0.0 d 0.c 101s Strain
b. Comparison stress/strain E0 = 7.5 1o+a s-t. - - EO = 4.0 lo+2 s-1.
Fig. 2. -
under
tension.
Fig. 2. - e. Variation of the crack initialisation. E0 = 7.5 1o+a SK’, - - &O = 4.0 lo+2 s-1.
l
o.o* 0.00
damage damage
0.04
0.02 maximum
of the Young’s
EO = 7.5 10+3 s-1. - - - 8 = 4.0 lo+* s-1.
0.6
0.8
(p.)
radius
under
tension
after damage
0.06 radius
0.08
(cm)
modulus
versus
crack
growth
under compression:
After damage initialisation in the material, the cracks grow proportionally to the loading, but the shear moduli are alone affected. The Young’s modulus remains unchanged until the microcrack density exceeds the critical value. The element is therefore pulverised and it obeys the Mohr Coulomb law (Fig. 2 a, c, d). l
Fig. 2. - c. Variation
0.4 Time
of the crack growth on the Young’s modulus under compression, and the crack extend versus time. We locally observe:
1.0;
.G 0.8) +z L 0.6i !$ 0.4 + w 0.2*
0.2
under tension:
The behaviour is similar to brittle fracture (Fig 2 b). From the beginning of damage, cracks grow very Aerospace
Science
and Technology
Micromechanical Modkle
Damage Model Taking Loading-Induced
d’endommagement
microme’canique
prenant
quickly (Fig. 2 e) until the macroscopic crack is reached and therefore the element fracture. The influence of the strain rate is taken into account but it would remain to study the difference of damage variation during tensile loading and unloading.
IV - CONCLUSION Fracture damage of ceramics is influenced by their microstructure. The damage law presented, based on description of the microscopic mechanisms, is used to analyse and isolate the microcracking process and represent the material stress state associated with growth of the microcracks. The new feature of this model is that it takes into account the anisotropy induced by loading. The definition of families of microcracks in privileged directions allows the damage growth to be related to the loading, and the detection of damaged areas of the material and their origin. The numerical model gives a better understanding of the behaviour of SiC/SiAlYON during dynamic impact and estimation of the energy dissipated by the damage. The model was successfully validated by a simulated Hopkinson’s bar test. Other experiments, such as to planar impacts, are planned to confirm the suitability of the model and its parameters.
REFERENCES [l] Allix Ladevkze, Le Dantec, Vittecoq. - Damage Mechanics for Composite Haminates Under Complex Loading, Yielding, Damage and Failure of Anisotropic Solids, EGFS, Mechanical Engineering Publications, 1990, 551-569. [2] Allix, Engrand, Ladeveze, Perret. - Une nouvelle approche des composites par la me’canique i’endommagement, Cachan 1993.
de
[3] Chaboche. - La mCcanique de l’endommagement et son application aux prCvisions de durt?ede vie de structures, La Recherche Ae’rospatiale, 1987, no 4, 37-54. [4] Chaboche. - Continuum Damage Mechanics, Part I and Part II, J. Of applied Mechanics, 1988, 55, 59-72. [5] Chaboche. - On the Description of Damage Induced Anisotropy and Active/Passive Effect., A.S.M.E. 1990, Winter Annual Meeting, Dallas. [6] Chaboche. - Une nouvelle condition unilatkrale pour dCcrire le comportement des matCriaux avec dommage anisotrope, C. R. Acad. Sci. Paris, 1992, 314 sCrieII, 1992, 1395-1401. [7] Chaboche. - A Continuum Damage Theory with Anisotropic and Unilateral Damage. La Recherche Ae’rospatiale, 1995, no 2, 139-147. [S] Le Maitre. - A Course on Damage Mechanics, Springer, Verlag, Berlin, 92. [9] Le Maitre, Chaboche. - M&anique des Mate’riaux solides, Dunod 85. [lo] Murakami. - A Continuum Mechanics Theory of Anisotropic Damage, in Yielding, Damage, and Failure of Anisotropic Materials, EGFS, edited by Boehler,
Mechanical Engineering Publications, London, 1990, 571-588. 1997. no 1
Anisotropy
en compte
l’anisotropie
into Account/ induite
par le clzavgement
75
[ 1I] Kachanov. - On the Creep Rupture Time, Izv. Akad. Nauk SSSR, Otd. Tekhn. Nauk, 1958, 8, 26-31. 1121Bui, Van, Stolz. - M6canique des solides anklastiques, relations entre les grandeurs macroscopiques et microscopiques pour un solide Clastique-fragile ayant des zones endommagCes, C. R. Acad. Sci. Paris, 1981, 292, no 12, sCrie II, 863-867. [13] Kachanov. - A Microcrack Model of Rock Inelasticity. Part II : Propagation of Microcracks, Mechanics of Materials, 1982, 1, 29-41. [14] Seaman et al. - Computational Models for Ductile and Britlle Fracture, J. of Applied Physics, 1976, 47, No. 11. [15] Marigo. - MCcanique des solides anelastiques : Formulation d’une loi d’endommagement d’un matCriau Clastique, C. R. Acad. Sci. Paris, 292, s&e III, 1981, 1309-1312. [16] Taylor, Chen, Kuszmaul. - Microcrack-Induced Damage Accumulation in Brittle Rock Under Dynamic Loading. Computer methods in applied mechanics and engine’ering 55, 1986, 301-320. [ 171 Margolin. - Microphysical Models for Inelastic Material Response, Int. J. Engng. Sci., 1984, 22, 1171-1179. [18] Andrieux, Bamberger, Marigo. - Un modble de matCriau microfissurC pour les b&tons et les roches, J. Me’canique The’orique et Applique’e, 1986, 5, no 3, 471-513. [ 191 Rajendran-Kroupa. - Impact Damage Models for Ceramics Materials, J. Appl. Phys., 1989, 66, 35603565. [20] Nicholas, Rajendran. - Failure Models/Material characterization at High Strain Rates, in High Velocio Impact Dynamics, edited by Zukas 1990, 268-295. [21] Baptiste. - Etude de l’injuence de la distribution de microstructures sur le comportement et l’endommagement de mate’riaux composites & l’aide de techniques de passage du microscopique au macroscopique,
Ecole CREA-EDF-INRIA, ModClisation et analyse qualitative de l’endommagement des matCriaux et des structures, Clamart, 12/90, 97-117. [22] Addessio, Johnson. - A Constitutive Model for the Dynamic Response of Brittle Materials, J. Appl. Phys., 1 April 1990, 67, 7. 1231 Rajendran. - High Strain Rate Behaviour of Metals, Ceramics, and Concrete AD-A252 979. Wright Laboratory WL-TR-924006, April 1992. [24] Rajendran. - Modeling the Impact Behaviour of AD85 Ceramic Under Multi-Axial Loading, ARL-TR- 137, May 1993. [25] Margolin. - Elastic Moduli of a Cracked Body, Znt. J. Fract., 1983, 22, 65-79. [26] Budiansky, O’Connell. - Elastic Moduli of a Cracked Solid, Int. J. Solids Structures, 1976, 12, 81-97. [27] Nemat-Nasser, Horii. - Overall Moduli of Solids with Microcraks : Load-Induced Anisotropy, J. Mech. Phys. Solids, 1983, 31, No. 2, 15.5-171. [28] Dienes. - On the Stability of Shear Cracks and the Calculation of Compressive Strengh, J. Geophys. Res., 1983, 88, No. B2, 1173-1179. [29] Dienes. - Comments on a Generalized Gr$j?th Criterion for Crack Propagation by L. G. Margolin. A
76
technical note. Eng. Fracture Mechanics, V3 No. 23, 1986, 615-617. [30] Kanninen, Popelar. - Advanced Fracture Mechanics. Oxford University Press, New York, 1985. [31] Freund. - Crack Propagation in an Elastic Solid Subjected to General Loading-l. Constant Rate of Extension, J. Mech. Solids, 1972, 20, 129 to 140. [32] Bell. - Stress from Arbitrary Loads on a Circular Crack. International Journal of Fracture 15, 1979, 85-104.
C. Vinet, P. Priou
[33] Dumitru-Zgura. - The Energy Balance of the Material Breaking Under the Action of the High PressureWater Jet, Journal de Physique IV, Colloque C8, suppl. au Journal de Physique III, Sept. 94, 4. [34] Hallquist J. O., Whirley R. G. - DYNA3D USER’S MANUAL, UCZD 19.592, ReV 5, May 1989, Lawrence Livermore National Laboratory. [35] Riou. - Impact Damage on Silicon Carbide: First Results, Journal de Physique IV, Colloque CX, supplement au Journal de Physique. ZZZ,4, 1994.
Aerospace
Science and Technology
Micromechanical Damage Model Taking Loading-Induced Anisotropy into Account C. Vinet (l), P. Priou 12) (I) Bordeaux-I
University, Airospatiale @) Akrospatiale
(Space Division), (Space Division),
Manuscript
Vinet C., Priou P., Aerospace Abstract
Science and Technology,
12, 1995; accepted February
France. France. 22, 1996.
1997, no 1, 65-76.
Micromechanical
model - Ceramics - Anisotrop
damage constitutive
law - Dynamics.
ModGe d’endommagement micromkcanique prenant en compte I’anisotropie induite par le chargement. Cet article presente un modirle d’endommagement construit a partir d’une thtorie destinee aux materiaux fragiles comportant des microfissures. Le modele est implante dans le code 3D de dynamique rapide DYNA3D. L’etude phenomenologique montre l’importance de prendre en compte les effets des micromecanismes pour modeliser la rupture macroscopique du materiau. Le modele developpe relie l’endommagement a des parambtres microscopiques (rayon de microfissures, densite de fissures...) et prend en compte le caractitre anisotrope de l’endommagement induit par le chargement, en reliant a des orientations privilegiees l’evolution des microfissures. Le caractbre unilateral (difference de comportement en traction et en compression) est consider& lors du critere de croissance des microfissures. La degradation progressive des proprietes du materiau, induite par la presence des microfissures, est obtenue a partir de la theorie des modules effectifs de Margolin et le materiau est pulverise lorsque la densite de microfissures depasse une valeur critique. Un calcul de l’energie dissipee par endommagement est propose. Le modele, applique a une ceramique SiC/SiAlyon est valid6 en comparant les resultats des essais aux barres d’Hopkinson et les simulations numeriques. Une comparaison entre un modble fragile macroscopique et le modele d’endommagement present& montre la capacite de ce dernier a predire les effets des microfissures sur le comportement a la rupture des ceramiques. Mots-elks
Aerospace
received December
78130 Les Mureaux, 78130 Les Mureaux,
This paper deals with a micromechanical damage model, based on a constitutive theory for brittle materials weakened by microcracks. The model is implemented in the DYNA3D three-dimensional explicit finite element code. The phenomenological study shows the importance of taking micromechanical effects into account to model macroscopic failure of the material. The constitutive model relates damage to microscopic parameters (size of microcracks, cracks density etc.) and takes loading-induced anisotropy damage into account by correlating microcrack growth to preferential orientations. The unilateral character (behaviour difference between tension and compression) is treated by the microcrack growth criterion. The progressive reduction in material stiffness due to the presence of microcracks is modelled using Margolin’s effective modulus expressions, and the material is pulverised if the microcrack density exceeds a critical value. Determination of the energy dissipated by damage is proposed. The constitutive model applied to SiC/SiAlYON ceramics is validated by a comparaison of the results between a Hopkinson’s Bar Test and numerical simulation. Comparing the macroscopic brittle model results with the damage model results shows the ability of the second to predict microcrack effects on the dynamic failure behaviour of ceramics. Keywords:
RikumC
66, route de Vemeuil, 66, route de Vwneuil,
Science and Technology,
: Modele micromecanique
0034-1223,
97/01/$
7.00/O
- Gramique
Gauthier-Villars
- Loi endommagement
anisotrope - Dynamiques.
66
C. Vinet, P. Priou
NOTATIONS macroscopic strain tensor mean strain tensor local strain tensor effective compliance tensor of the microcracked material Cauchy’s macroscopic stress tensor mean stress tensor local stress tensor mean radius of the cracks with normal n’i (i = 1,3) maximum radius of the cracks with normal ?zi (i = 1,3) number of cracks with normal n’i per unit volume damage parameter in direction i normal to the crack in direction i statistical distribution of the radii of the cracks with normal ?Li;in the element crack radius Poisson’s ratio of the undamaged material Young’s modulus of the undamaged material microcrack growth rate of the crack with radius a Rayleigh wave speed critical energy release rate toughness
I - INTRODUCTION Ceramics can exhibit inelastic strain under compressive, tensile and shear loading of the grains. Under mechanical loading, they rapidly incur diffuse degrading, called damage. This results in a gradual loss of their physical properties. The presence of microcracks and the type of loading determine the growth of the damage. The shape, distribution and spacing of the microcracks all affect growth of these microcracks and consequently the physical quantities involved. The state of damage is described by one or more appropriate internal variables. Several types of models can be distinguished. The phenomenological approach of continuum damage mechanics [l-lo] relies on the original Kachanov model of damage variables [1 l] and on the concept of the effective stress. It’s a macroscopic view of the damage process which depends on the microstructural kinematics. The theory of continuum damage mechanics is developed based on the general concepts of the thermodynamics of irreversible processes with internal variables. The thermodynamic potential defines the material damage state and the potential of dissipation describes the evolution of the irreversible damage. This approach requires the perfect knowing of the damage topology in order to determine the internal variables (damage tensor) and the kinematics laws governing the
damage evolution. The microscopic models [ 12-241 establish a relationship between the damage parameter (crack density) to the microstructure (geometry, grain and pore distribution...). These models tend to protect physical phenomena but for programming efficiency, it’s necessary to assume several simplifying hypotheses on crack form, orientation and distribution in the material. The damage state (chronology and distribution) not known accurately, a microphysical approach of the problem, based on microscopic descriptions and measurements of the material, was preferred over an empirical or phenomenological approach to analyse the microcracking process and simulate the material stress state associated with microcrack initiation. The properties and behaviour of a volume element of the material were obtained by integrating the microscopic results on the macroscopic volume element using mathematical homogenisation techniques. Rajendran [23, 241 propose a 3D constitutive model based on microcrack growth and pore collapse. Inelastic strains are introduced to account for microcrack opening and sliding. The pore compaction is modelled but the crack closing under compression is not considered. The formulation is based on Margolin’s effective modulus theory [25] and Budiansky-OConnell’s theory [26]. Margolin gives the expression for the effective isotropic stiffness tensor for randomly oriented, penny-shaped non-interacting microcracks of different sizes. This model appears to be well design for porous ceramics subjected to multiaxial loading. However, the fact that the damage is isotropic remains a problem. The damage model developed and implemented in DYNA3D is based on Rajendran model but the description of the microstructure is modified to introduce anisotropic damage depending on the loading and preferential microcrack directions. The emphasis was placed on determining the statistical crack distribution to allow loading-induced damage anisotropy to be taken into account. This paper describes this distribution as well as adaptation of the growth law and calculation of the damaged elastic moduli to this distribution. Calculation of the energy dissipated by the damage is proposed. The model is then used to predict macroscopic fracture of an SiC/SialYON test specimen subjected to a Hopkinson’s bar test. The numerical results are compared with the experimental data. The predefinition of three privileged orientations of microcracks proves conclusive in the case of the test presented but it does not allow to model the anisotropy induced by complex loading. The damage material symmetries are predefined by the choice of the global axes and consequently the failure planes must be known in order to have a good model. Aerospace
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Micromechanical Damage Model Taking Loading-Induced Anisotropy into Account/ Mod2le d’endommagement microme’canique prenant en compte l’anisotropie induite par le chargement
II - DESCRIPTION
where N (ai; ?Li)) is the statistical distribution of the radii of cracks with normal n’i in the element. The total microcrack density is given by:
OF THE MODEL
11.1 - General Since the ceramic investigated has an initial porosity of only 2% and plastification does not occur during the tests, closing of the pores and plastic strains were neglected. The strain process applied is therefore a brittle process for which the elastic strains are assumed infinitesimal and the anelastic strains are due only to the presence of microcracks. The macroscopic strains are defined by:
where E$ is the elastic strain tensor of the undamaged matrix, E;~ is the inelastic strain tensor accounting for the microcracks. The elastic strains of the matrix and the microcrack strains are proportional to the stress fields Ckl :
The elements of the effective compliance matrix C’ijkl are obtained from the effective modulus theory based on Margolin’s theory [25]. II.2 - Statistical
crack distribution
The model is constructed from microscopic observations of the material. Damage is described based on statistical distribution of the microcracks. The cracks are assumed penny-shaped. The interaction between cracks is neglected. This assumption remains verified for [25]: Na3 << I where N is the number of cracks per unit volume and a is the crack radius. In Rajendran’s model [23, 241 as well as in most existing micromechanical models [19, 20, 271, crack distribution is assumed to be random in a volume element, which implies that the damage material remains isotropic regardless of the loading applied. The new idea is to define a relationship between crack size and orientation for a given load. Three families of normal orthogonal microcracks are defined. These normals correspond to the three axes of the global coordinate system. To characterise the damage state, it is then necessary to determine three parameters per family relative to the microstructure: the mean radius E; of the cracks with normal n’;, the maximum radius amax, i of the cracks with normal n’i, the number No, i of cracks with normal n’i per unit volume. The damage parameter yi, in each direction is defined by the density of microcracks with normal n’i :
7=x
Ti. i=l
The microcracks with different orientations grow independently of one another as long y does not exceed a critical value. The anisotropy induced by loading is therefore directly expressed by the damage parameters. The volume distribution N (a,) Gi) of the radii of the cracks oriented along +$ is taken as exponential [ 14, 17, 22-241. This exponential distribution is assumed to be preserved over time. For each family of cracks, the microcracks are assumed to be nucleated when the stress state is such that the Griffith criterion is satisfied for the largest microcrack (a max,i). Th e o th er microcracks of the same family are then assumed to propagate at an appropriate rate to preserve the exponential distribution of the crack radii related to this direction (there is a criterion for each direction i). In each family of cracks, the distribution is characterised by a function N (ai; Gi), where N (ai; ?&) da; is the number of cracks per unit volume with normal n’i, and where the crack radius belongs to the interval [ai, a; + da;]. The continuous function N is discretised and is given analytically in a; : (2)
The total concentration JV of cracks with normal sn’; whose radius is greater than a; is: (3) The damage parameter then becomes:
J
i=l
The number of cracks per unit volume in direction i (i = 1, 2, 3) is assumed constant. The total number of cracks per unit volume is denoted No = 5 Na,i. i=l Assuming amax, i >> &, “i; is approximated by Yi z 6 No,; $. We then define N& by: yi = NGi akaxi.
y;= .! 1997, no 1
. N (a;, 2;) a: da;
67
(1)
This formulation
is used for programming.
(5)
68
C. Vinet,
II.3 - Growth
law
The microcrack growth mechanism is analysed by applying the theory of dynamic linear fracture mechanics. For brittle material, this theory consists of nucleation, propagation and coalescence. l Microcrack nucleation occurs in an element when the stress state satisfies the generalised Griffith criterion for the largest microcrack of the element [28, 291. The criterion is established by relating crack nucleation to crack orientation. l Damage growth as an increase in crack radius according to crack orientation is modelled by a growth law expressing the propagation of a crack under dynamic loading conditions. l Coalescence occurs when the propagating microcracks intersect each other and pulverise the element. The assumption of no interaction between cracks then no longer applies. A reasonable pulverisation criterion can be established: pulverisation occurs when the mean radius reaches half the mean distance separating two cracks. This condition defines a critical crack density value above which the material is considered as pulverised. The variation of the microcrack nucleation, propagation and coalescence mechanisms within a volume element is independent of their variation within the other elements. 11.3.1 - Crack nucleation model The nucleation criterion used is the generalised Griffith criterion as developed by Margolin [ 171 and modified by Dienes [28, 291. Dienes calculates crack stability by adding to Griffith’s criterion an energy term corresponding to the loss of energy of a closed crack. This loss of energy results from friction of the two edges of the crack (shear). This criterion is applied to each family of cracks. Nucleation occurs if:
fpv-S]>O where W is the strain energy and S is the surface energy released when the crack opens. The strain energy under mode I crack opening (tensible normal stress) is different that under modes II or III. The improved Griffith criterion depends on the crack orientation. Three criteria relative to each family of cracks (same plane) are defined for each element. They are applied completely independently for each family. Each criterion is first expressed in the local crack coordinate system then reformulated in the global coordinate system. Where i, is the normal to the family of cracks, for i, j, k = 1,2, 3 and i # j # k: l For ??kk > 0, the energy release rate under mode I crack opening is written:
G+k = 4 (’ - u2) amax 7iE
1
(7)
P. Priou
0 For ??kk 5 0, the energy release rate is written (in this case, only modes II and III are active): G- = k
8 (l - u”) 7i-E (2 - ZI) %nax,k
[j/~-%+~~r:b12 (8)
is the maximum radius of the family of cracks orthogonal to vector k, r. is the cohesion stress, and h is the friction coefficient. Defining G&,, k, G,,,, k as the maximum of (Gz and 0) and (G, and 0) respectively, the microcracks perpendicular to vector c are assumed to have nucleated if G&, k or G,, k exceed the critical energy release rate’ Gc: ’ Kfc (1 - v”) Gc=2T= E ’ The relationship between crack radius and orientation is defined by the formulation of the different criteria. The unilateral character of the behaviour (different behaviour under tension and under compression) is taken into account.
where
amax,
k
11.3.2 - Microcrack growth If one of the crack families verifies the improved Griffith criterion, the material damage grows. The microcrack growth law is obtained from the theory of dynamic fracture mechanisms [30, 311. The classical solution for the energy release rate G (t) under mode I crack opening is given, for a fast crack propagation, by the equation: - a =I.CT
~GC w .
(10)
For the solution to be valid, any material speed must be smaller than the speed of sound e in the material [25]. This assumption is verified for a. Generalising this equation for the three crack growth modes, empirical constants are introduced to control the maximum crack tip propagation rate [23, 241. We have: For k = 1,3 (for each crack family) 0 if G&,
nm(‘icmax,k
=
k 5 Gc and G,,,
k < Gc
(g-J)
if GL,,, Ic > Gc
if G&,
k 2 Gc and G,,,
k > Gc (11)
Aerospace
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Micromechanical Damage Model Taking Loading-Induced Anisotropy into Account/ Modt?le d’endommagement microme’canique prenant en compte l’anisotropie induite par le chargement
where nf , n2+ , n, nz are constants to be determined. Quantities nt and n, directly control the maximum growth rate for the different opening modes, whereas exponents nz and n; moderate the crack growth rate until Gc is sufficiently exceeded. The material damage growth can be monitored by numerically integrating equation (11) rewritten for each crack family and then evaluating the crack density within the material. 11.3.3 - Microcrack coalescence Coalescence occurs when the propagating microThis results in cracks intercept each other. pulverisation of the material and a significant loss of strength. The pulverisation criterion is based on Rajendran’s definition [23, 241. Where NO is the total number of microcracks per unit volume, the mean 1 distance d between cracks is d = iV, 3. We denote as Gnax the maximum of mean radii Ek (k = 1, 2, 3). Pulverisation occurs when a,,, reaches half the mean distance d. This coalescence condition defines a critical crack density for pulverisation 11231:
The material is therefore considered pulverised when y reaches rp in a stress state in which at least one of the main stresses is compressive. Under tension, the material lose its strength and under compresion, it obeys the Mohr-Coulomb law. The pressure is expressed by:
P=C-g
P EZ
E; 2 0 E; < 0
modulus
theory
The microcrack presence in the body involves an additional deformation and a lost of the mechanical properties. This phenomenon is taken into account by relating the local strains due to microcracks to the macroscopic strains using a homogenisation technique. 11.4.1 - Relationships between microscopic and macroscopic quantities Consider a solid of volume V assumed macroscopically homogeneous containing microcracks on a microscopic scale. The microscopic quantities are g(z) for stresses, E (x) for strains and u(z) for displacements. The corresponding macroscopic quantities are denoted C, E, U respectively. We defined 0 (t) as the volume of the solid undamaged by microcracks and 7 as the outward unit normal vector to the surface. The outer surface of the solid 1997, no 1
is denoted S, and the crack surface S; (t). The mean value of f on R is denoted 7. If surface loads T; (z) are applied to Se, the macroscopic stress tensor .&, constant on R, is defined by Ti (z) = Cij.rj (z). The local field LJ~~(zz) which results is statistically admissible if it verifies: div oij = 0 0110 CT;?. pi = Ti (z) onS, (14) aij . rj = 0 on Si (t) 1 By writing the variational problem corresponding to problem (14) and using Green’s formulation, it is shown that: The macroscopic strain Eij is defined by: Ezj = $
’I se
(Ui . ri + U, .ri) ds.
The microscopic strain “ii (z) is kinematically admissible with the local field ui (z) that fluctuates around the mean of U; (x) on S,. The hypothesis of macrohomogeneity [ 151 results in: (ui (z) - Ui (x)) J SC This yields (17):
ri (z) ds = 0
(16)
(13)
where E; : is the elastic volume strain and KP : is the pulverised material bulk modulus. The pulverised material bulk modulus is considered equal to the bulk modulus of the damaged material just before pulverisation occurs. II.4 - Effective
69
= Eij (Lx) - $
.i St (t)
(pi . pi + U? . ri) ds (17)
Applying the divergence theorem and denoting n’ the crack normal (on Si (t) , n’ = -?) yields the expression for the macroscopic strain in a volume element V containing microcracks:
+
J’
(ni . uj + nj . ui) dS
1 (18)
(n; . uj + nj . u;) dS
1
s;(t)
We set: (19)
Tensor a! is the macroscopic manifestation of the displacement discontinuities across the crack surface.
70
C. Vinet, P. Priou
By definition of the constitutive law, we have: Ei.j = CFjkl Fkl where CZ:Al is the undamaged material. Then:
compliance
strains of the material in presence of NV cracks with radius a: (20)
matrix
of
the + 16 (1 - u”)
Eij = C,p,,, zIcl + aij
3E
(21)
where ??kt = Ckl is the mean of macroscopic stress tensor. This yields the expression for the damage or effective elasticity modulii:
da!ij Cijkl = C,Q,I,l+ ~ d3kl
+
8 (1+
4
Na3 max (Zkl nk nl, 0) ni nj
Na3
3E
(22)
TO calculate Cijkl, it is necessary to determine aij or, more specifically, to know the displacement fields on the surface of each crack.
11.4.2 - Calculating
the jields [3]
In an initialy isotropic material, Margolin assumes a uniform distribution of the cracks with the same radius and same orientation. The expression of the solution fields is obtained under static load by calculating the crack displacement jumps (in a local coordinate system) based on the theory of cracks in an infinite medium isotropic (Bell [32]). This approximation is possible because of the small size of the crack (crack radius much smaller than the element edge). The solutions of the problems lead to distribute the discontinuities of the displacement jumps on the crack of a constant form, whatever the boundary conditions imposed [ 181. These displacement jumps are written:
More generally, by using the principle of superposition of solutions (because of the linearity of the solutions), he obtains the following expression in the case of coplanar cracks whose radius is statistically described by function N (a): Eij = CzPjklakl + 16 (1 - u”) Na m8X (8kl n& nl, 0) 3E
2E]-Ua;cz[, i=n,t,s (23)
where n, t, s are the normal and tangential components. These results apply to the dynamics if the material strain rate is much lower than the speed of sound in the material [25]. For coplanar cracks with the same radius a, the integral of oij on each crack is the same. This means that for NV cracks, where N is the number of cracks per unit volume and V is the volume element, we can write:
N w = -2
(ni . uj + nj . ui) dS
1
ni
nJ
+ 8(1+v)3E
Na
x (ail, nj nk + -jo kninl,-2n;nj??klnknl) (26)
were
[u;](x)=K;xLz2,
coplanar
Na=
.I’
N (CL)a3 da
This amounts to modifying three of the diagonal terms of the compliance matrix. Margolin applies this result to coplanar or isotropic crack distribution. In the case of the model presented, there are three different crack orientations. The strain tensor is written in the same way as (26) in each local crack coordinate system for a distribution N (ai, Zi) defined by (2). The expression for the global strain tensor in the coordinate system of the material is given by:
(24)
where c is the crack surface area. By investigating separately the influence of the stress field components at infinity on the displacement fields formulated by Bell on the crack surface, Margolin obtains the expression for the macroscopic
where Pi is the coordinate system Eij (ak, NC,+) is orthogonal k with
transformation matrix from local i to the global coodinate system and the strain tensor relative to cracks distribution Nk (ak, nk). Aerospace
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Micromechanical Damage Model Taking Loading-Induced Anisotropy into Account/ Modkle d’endommagement microme’canique preenant en compte l’anisotr-opie induite par le chavgement
The coefficients of the damaged compliance matrix C are given in the (0, X, Y, 2) coordinate system by: (1 - v”)
if a,, > 0
16 Naz (1 - v”) c YYYY - cy”YYY+ 3E
if Oyv > 0
+ +
if O,, > 0
c zzzz - cLz
C zrzz - CL
+ s
(1 - v”)
C XY”Y - cyzy + “‘ii
u, (Ti&+K$
C YZYZ = qzyz + x(;;
4 (~+7icg)
C %Z%Z- CL
+ fgqMa;+Na,) (28)
71
surface, consuming energy by material damage. It is therefore interesting to analyse the energy dissipated over time by damage in the ceramic. Since the material damage is due to the growth of microcracks, to calculate the dissipated energy it is necessary to analyse the energy balance of the microscopic mechanisms of the material, considering [33]: the microcrack nucleation energy, the variation of the internal energy when the microcrack occurs, the energy dissipated to overcome the material resistance during microcrack extension, and the kinetic energy of the moving parts. The elementary energy consumed by a microcrack of length 2a to extend a distance Aa is equal to: dW = 2?rGa& (30) where G is the energy release rate or extension force for a microcrack of radius a, already defined by (9) and (10). The elementary energy consumed by NV bedded cracks of radius ai (i = 1 to NV) to grow by Au; is therefore written: ivv dWT = C 2 T G (ai) a; aa, (31) i=l
Only the diagonal coefficients of the elasticity matrix are modified. The matrix therefore remains symmetric. [C] = -v/E
-v/E
- u/E
l/E+az
-v/E
- v/E
-v/E
l/E
[l/E+cul
A+=2a4(l-V2)
0
0
0
l/G+w
0
0
0
l/G
xrqcr + au (29)
Taking the loading-induced damage anisotropy into account requires processing the crack based on its radius and direction before obtaining the global behaviour of the element by homogenisation. The equation of state (Eq. 26) depends on the loading. Under tension, cracks open with a displacement proportional to the normal average stress. Under compression, they close and only the crack sliding are considered to calculate damage. The unilateral effect is taken into account in the damage compliance matrix (Eq. 28). energy
When a small projectile impacts the centre of a ceramic plate, the ceramic slows down the projectile and distributes a load on a large area of the rear 1997, no 1
+ ~,‘,) 2-u
! 2 (a$ f z$) 8 (1 71”) WV ZZ G, + E 2-V
+ 013 l/G+w
2 (&
-2
g’kli +
7iE
0
II.5 - Dissipated
The energy consumption depends on the crack radius. For an exponential distribution on the radii of cracks orthogonal to & setting:
Gc ‘nn: [ 1 l-
G+
At
1
IIAt
1
if alik, > 0
A- _ 2 8 (1 - v”) - 7r TE (2 - I/)
x [~~-7r~+i~“~*]~hnf 16 (1 - v”) E (2 - u) Xn,Cr
n2 l__ [ 1 -
Gc G-
At
else (32)
yields:
N0 ye a No Te
a
-za2
-Q
da
a a2 da
if ??kk > 0
else (33)
72
C. Vinet, P. Priou
The elementary energy consumed by NV cracks with the same orientation (orthogonal to i) with an exponential distribution of radii corresponds to:
A+2No$=A+
7%
if akk > 0
A-2No~2=A-
!!!T!I!&S 3 a
ifakk
dWT = <34) where amax is the maximum radius of the exponential distribution of cracks, Z is the mean radius of the exponential distribution of the cracks, NC is the number of cracks per unit volume, N,* = No $$ . In the case of the model, the elementary energy consumed by damage is equal at time t to the sum of the variations of the energy dissipated by crack growth for the three families of cracks. The total elementary energy consumed at time t is therefore expressed as follows: W(t)
= W (t - 1) + 2
dW (u;)
of the same high-strength material (maraging steel). The sample, with a mechanical strength less than that of steel, is placed between the two bars. Wave propagation is considered one-dimensional and nondispersive in the inlet and outlet bars. The bars preserve elastic behaviour throughout the test. When the projectile propulsed by a gas gun impacts the inlet bar, a compression wave is generated and propagates through the inlet bar. When the incident compression wave reaches the inlet bar/sample interface, it divides into a tensile wave reflected in the inlet bar and a compression wave transmitted through the sample. The wave transmitted through the sample is partially transferred to the outlet bar. A knowledge of the history of elasic strains in the inlet and outlet bars obtained using strain gauges, allows determination of the microscopic strain history in the sample.
Gas gun I
(35)
Inlet bar
H H
Outlet bar ‘B w Sample
‘A
Projectile
I
i=l
where W (t) denotes the elementary energy consumed at time t, W (t - 1) the elementary energy consumed at time t - 1 and dW (ui) the energy consumed for the growth of the family of microcracks orthogonal to ?&. In our model, the calculation of the energy dissipated by damage does not include the microcrack nucleation energy, since the material is assumed to be precracked and the number of microcracks per unit volume is assumed to remain constant over time. III - DYNAMIC PARAMETER CHARACTERISATION Mechanical characterisation of ceramic materials in the area of dynamic stresses based on simple tests allows definition of the physical data and parameters of the model. The validity of the results is checked by comparing the numerical simulations of the tests made using DYNA3D [34] with experimental data. The SiC/SiAIYON composite was dynamically characterised under compression using CREA Hopkinson bars. Because of its intrinsic nature, ceramic material does not allow reliable direct analysis of tests using the theories applied for metals. However, the curves of the incident, reflected and transmitted strains versus time at different strain rates give an idea of the dynamic behaviour of the material. III.1
- Hopkinson’s
bar test
III.l.l - Description of the test facility The tests are conducted at room temperature. The test facility includes a projectile, an inlet bar and an outlet bar, all with the same diameter and made
Scheme
1. - Test apparatus.
111.1.2 - Description of the test conducted The test specimen was a cylinder with a diameter of 7 mm and a height of 8 mm. The projectile speed was 17 m/s. At the end of the test, the sample was pulverised. Fracture of the sample occurred as the wave increased in the incident bar. The only conclusion that can be drawn is that the macroscopic fracture stress of the sample was exceeded. From the reflected signal, it is deduced that the strain rate reached -1430 s-l. From the transmitted signal, it is deduced that the mean fracture stress was 3480 MPa. The transmitted and reflected signals indicate brittle fracture and therefore little damage. The steady state and uniform stress hypotheses were not satisfied. There was no plastic behaviour. Conventional analysis of the test was therefore not possible. III.2 - Modelling
of the test
The parameters (urnax, a, n;, nz, Cr . . .) were determined from the literature and from microscopic observations. Resetting was carried out based on the tests. A comparison with a brittle model defined from the fracture stresses obtained during the tests was carried out and the influence of the mesh was analysed. 111.2.1 - Geometry, mesh and boundary conditions The two models were compared for the same mesh. Only a quarter of the problem was modelled, since symmetry conditions were imposed. The contact between solids was modelled by sliding surfaces Aerospace
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Micromechanical Damage Model Taking Loading-Induced Anisotropy into Account/ Mod2le d’endommagement microme’canique prenant en compte l’anisotropie induite par le chargement
obeying the penalisation method. The geometry was defined as follows (Table 1).
0.002
_
,$r)lq j py
O.OOl_
Table
1. - Geometry
Ii
s ‘2=
parameters.
2 E 2
73
f 0.000
._ , . . . 1;’
\
L
i -O.OOl\ -0.002 I
I
0.0000
i-;--
\$&.+i :’ ‘I
0.0002
0 I I I 0.0004 0.0006 Time (s)
I I 0.0008
I 1 0.001
0.0002L
0.0000 .g -0.0002~
The mesh included a total of 5372 elements distributed as follows: 468 for the projectile, 2388 for each bar and 128 for the test specimen. 111.2.2 - Properties of the material The constitutive equation used for maraging steel is an isotropic elasticity law. The behaviour of the specimen was modelled either by the damage model presented or by a macroscopic brittle model. The fracture criterion of the latter was a stress criterion: If the main stresses exceed the tension and compression limits (Tables 2, 3, 4) obtained during the tests, fracture occurs. The material properties (p, E, V, Ki,, CL)of SiC/SiAIYON were obtained from Table
2. - Parameters
and material
properties.
Table
3. - Fracture
limits
(T,, (tension)
(MPa)
for the brittle
material.
art
3480
Table
4. - Parameters
(traction) 320
for the damage
model.
2 -0.0006
_
-o.oooa0.0000
I
0.0002 I
0.0004 /
I Time
Fig. 1. - Hopkinson’s test data.
bar test, comparison
0.0006 I (s)
I
’ 0.0008
between
0.001
computation
and
static tests conducted by Aerospatiale in Aquitaine. The others were taken from the literature [23, 24, 30, 351 or from micrographs. It was necessary to identify the value of the maximum radius and growth parameter nz. The results are given in Figure 1. III.3 - Discussion
SiC/SiAlYON
Properties
E -0.0004~
(MPa)
Consistently with the test data, the specimen modelled by the damage law was completely pulverised. The results given in Figure I also show good correlation between numerical simulation with damage and test data. The brittle model causes too sudden a fracture (transmitted wave) and the jump observed in the reflected wave is not restored by calculation. It is therefore necessary to take the material damage into account before pulverisation. Numerical simulation is used to estimate the damage growth over time. In particular, it is observed that the material damage is due both to the growth of cracks orthogonal to the axis of symmetry of the bar and to the growth of cracks perpendicular to the specimen section, due to a tensile wave propagating through the section of the test specimen. This observation is physically confirmed during another Hopkinson bar test at a rate of 12 m/s. An additional study is proposed to illustrate the modelisation of the moving cracking. Two simple tests of pure tension and pure compression in the direction z are modelled at different strain rates. Different results are presented: the crar versus E,, curves, the influence
74
C. Vinet, P. Priou
0.10
-
,
0.08 -
,---
IF 0 2 0.06 5
-0.06 0.02 -
0.000
I 0.7
0.2
Strain Fig. 2. - a. Comparison 8 = 7.5 1o+s -- &O = 4.0 lo+*
stress/strain s-1.
under
Time Fig. 2. - d. Variation damage initialisation
compression.
of the crack
.
,
(ps)
radius
under
compression,
after
s-l, O.lC I-
0.08 brittle
fracture
2 0.06 il? E
,g 0.04 z H 0.02
0.00 r 0.0 d 0.c 101s Strain
b. Comparison stress/strain E0 = 7.5 1o+a s-t. - - EO = 4.0 lo+2 s-1.
Fig. 2. -
under
tension.
Fig. 2. - e. Variation of the crack initialisation. E0 = 7.5 1o+a SK’, - - &O = 4.0 lo+2 s-1.
l
o.o* 0.00
damage damage
0.04
0.02 maximum
of the Young’s
EO = 7.5 10+3 s-1. - - - 8 = 4.0 lo+* s-1.
0.6
0.8
(p.)
radius
under
tension
after damage
0.06 radius
0.08
(cm)
modulus
versus
crack
growth
under compression:
After damage initialisation in the material, the cracks grow proportionally to the loading, but the shear moduli are alone affected. The Young’s modulus remains unchanged until the microcrack density exceeds the critical value. The element is therefore pulverised and it obeys the Mohr Coulomb law (Fig. 2 a, c, d). l
Fig. 2. - c. Variation
0.4 Time
of the crack growth on the Young’s modulus under compression, and the crack extend versus time. We locally observe:
1.0;
.G 0.8) +z L 0.6i !$ 0.4 + w 0.2*
0.2
under tension:
The behaviour is similar to brittle fracture (Fig 2 b). From the beginning of damage, cracks grow very Aerospace
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Micromechanical Modkle
Damage Model Taking Loading-Induced
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prenant
quickly (Fig. 2 e) until the macroscopic crack is reached and therefore the element fracture. The influence of the strain rate is taken into account but it would remain to study the difference of damage variation during tensile loading and unloading.
IV - CONCLUSION Fracture damage of ceramics is influenced by their microstructure. The damage law presented, based on description of the microscopic mechanisms, is used to analyse and isolate the microcracking process and represent the material stress state associated with growth of the microcracks. The new feature of this model is that it takes into account the anisotropy induced by loading. The definition of families of microcracks in privileged directions allows the damage growth to be related to the loading, and the detection of damaged areas of the material and their origin. The numerical model gives a better understanding of the behaviour of SiC/SiAlYON during dynamic impact and estimation of the energy dissipated by the damage. The model was successfully validated by a simulated Hopkinson’s bar test. Other experiments, such as to planar impacts, are planned to confirm the suitability of the model and its parameters.
REFERENCES [l] Allix Ladevkze, Le Dantec, Vittecoq. - Damage Mechanics for Composite Haminates Under Complex Loading, Yielding, Damage and Failure of Anisotropic Solids, EGFS, Mechanical Engineering Publications, 1990, 551-569. [2] Allix, Engrand, Ladeveze, Perret. - Une nouvelle approche des composites par la me’canique i’endommagement, Cachan 1993.
de
[3] Chaboche. - La mCcanique de l’endommagement et son application aux prCvisions de durt?ede vie de structures, La Recherche Ae’rospatiale, 1987, no 4, 37-54. [4] Chaboche. - Continuum Damage Mechanics, Part I and Part II, J. Of applied Mechanics, 1988, 55, 59-72. [5] Chaboche. - On the Description of Damage Induced Anisotropy and Active/Passive Effect., A.S.M.E. 1990, Winter Annual Meeting, Dallas. [6] Chaboche. - Une nouvelle condition unilatkrale pour dCcrire le comportement des matCriaux avec dommage anisotrope, C. R. Acad. Sci. Paris, 1992, 314 sCrieII, 1992, 1395-1401. [7] Chaboche. - A Continuum Damage Theory with Anisotropic and Unilateral Damage. La Recherche Ae’rospatiale, 1995, no 2, 139-147. [S] Le Maitre. - A Course on Damage Mechanics, Springer, Verlag, Berlin, 92. [9] Le Maitre, Chaboche. - M&anique des Mate’riaux solides, Dunod 85. [lo] Murakami. - A Continuum Mechanics Theory of Anisotropic Damage, in Yielding, Damage, and Failure of Anisotropic Materials, EGFS, edited by Boehler,
Mechanical Engineering Publications, London, 1990, 571-588. 1997. no 1
Anisotropy
en compte
l’anisotropie
into Account/ induite
par le clzavgement
75
[ 1I] Kachanov. - On the Creep Rupture Time, Izv. Akad. Nauk SSSR, Otd. Tekhn. Nauk, 1958, 8, 26-31. 1121Bui, Van, Stolz. - M6canique des solides anklastiques, relations entre les grandeurs macroscopiques et microscopiques pour un solide Clastique-fragile ayant des zones endommagCes, C. R. Acad. Sci. Paris, 1981, 292, no 12, sCrie II, 863-867. [13] Kachanov. - A Microcrack Model of Rock Inelasticity. Part II : Propagation of Microcracks, Mechanics of Materials, 1982, 1, 29-41. [14] Seaman et al. - Computational Models for Ductile and Britlle Fracture, J. of Applied Physics, 1976, 47, No. 11. [15] Marigo. - MCcanique des solides anelastiques : Formulation d’une loi d’endommagement d’un matCriau Clastique, C. R. Acad. Sci. Paris, 292, s&e III, 1981, 1309-1312. [16] Taylor, Chen, Kuszmaul. - Microcrack-Induced Damage Accumulation in Brittle Rock Under Dynamic Loading. Computer methods in applied mechanics and engine’ering 55, 1986, 301-320. [ 171 Margolin. - Microphysical Models for Inelastic Material Response, Int. J. Engng. Sci., 1984, 22, 1171-1179. [18] Andrieux, Bamberger, Marigo. - Un modble de matCriau microfissurC pour les b&tons et les roches, J. Me’canique The’orique et Applique’e, 1986, 5, no 3, 471-513. [ 191 Rajendran-Kroupa. - Impact Damage Models for Ceramics Materials, J. Appl. Phys., 1989, 66, 35603565. [20] Nicholas, Rajendran. - Failure Models/Material characterization at High Strain Rates, in High Velocio Impact Dynamics, edited by Zukas 1990, 268-295. [21] Baptiste. - Etude de l’injuence de la distribution de microstructures sur le comportement et l’endommagement de mate’riaux composites & l’aide de techniques de passage du microscopique au macroscopique,
Ecole CREA-EDF-INRIA, ModClisation et analyse qualitative de l’endommagement des matCriaux et des structures, Clamart, 12/90, 97-117. [22] Addessio, Johnson. - A Constitutive Model for the Dynamic Response of Brittle Materials, J. Appl. Phys., 1 April 1990, 67, 7. 1231 Rajendran. - High Strain Rate Behaviour of Metals, Ceramics, and Concrete AD-A252 979. Wright Laboratory WL-TR-924006, April 1992. [24] Rajendran. - Modeling the Impact Behaviour of AD85 Ceramic Under Multi-Axial Loading, ARL-TR- 137, May 1993. [25] Margolin. - Elastic Moduli of a Cracked Body, Znt. J. Fract., 1983, 22, 65-79. [26] Budiansky, O’Connell. - Elastic Moduli of a Cracked Solid, Int. J. Solids Structures, 1976, 12, 81-97. [27] Nemat-Nasser, Horii. - Overall Moduli of Solids with Microcraks : Load-Induced Anisotropy, J. Mech. Phys. Solids, 1983, 31, No. 2, 15.5-171. [28] Dienes. - On the Stability of Shear Cracks and the Calculation of Compressive Strengh, J. Geophys. Res., 1983, 88, No. B2, 1173-1179. [29] Dienes. - Comments on a Generalized Gr$j?th Criterion for Crack Propagation by L. G. Margolin. A
76
technical note. Eng. Fracture Mechanics, V3 No. 23, 1986, 615-617. [30] Kanninen, Popelar. - Advanced Fracture Mechanics. Oxford University Press, New York, 1985. [31] Freund. - Crack Propagation in an Elastic Solid Subjected to General Loading-l. Constant Rate of Extension, J. Mech. Solids, 1972, 20, 129 to 140. [32] Bell. - Stress from Arbitrary Loads on a Circular Crack. International Journal of Fracture 15, 1979, 85-104.
C. Vinet, P. Priou
[33] Dumitru-Zgura. - The Energy Balance of the Material Breaking Under the Action of the High PressureWater Jet, Journal de Physique IV, Colloque C8, suppl. au Journal de Physique III, Sept. 94, 4. [34] Hallquist J. O., Whirley R. G. - DYNA3D USER’S MANUAL, UCZD 19.592, ReV 5, May 1989, Lawrence Livermore National Laboratory. [35] Riou. - Impact Damage on Silicon Carbide: First Results, Journal de Physique IV, Colloque CX, supplement au Journal de Physique. ZZZ,4, 1994.
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