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Towards a canonical elastoplastic damage model
Emgineeriug FrocrunMechanics Vol. 48,No. 2, pp. 151466,1994
Pergamon
Copyright0 1994.ElsevierScienceLtd
0013-7944f93MMDS2 . I
Printedin GreatBritain.All riahtsreserved ool3-7944/!M -s7.00+ 0.00
TOWARDS A CANONICAL ELASTOPLASTIC DAMAGE MODEL SALAH EL-DIN F. TAHER,t MOHAMMED H. BALUCH and ALI H. AL-GADHIB Department of Civil Engineering, Ring Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Ahatraet-Fundamental aspects of elastoplastic damage are outlined. Time-independent isotropic damage is considered in order to study material degradation. By splitting the total strain tensor into its components of elastic damage and plastic damage and using recoverable energy equivalence, three distinct modes of behavior are particularized. For each mode of behavior, a suitable damage variable is culled. An in-depth analysis of this formulation reveals a certain incongruity in the assumptions postulated in some of the previously proposed models. The suggested generalized concepts are supported by experimental evidence.
1. INTRODUCTION THE CRACKING of materials results from creation, propagation and coalescence of microcracks. For materials characterized by ductile behavior, Chaboche [l] outlined four different levels of cracking: (1) crack nucleation, (2) microcrack initiation, (3) macrocrack initiation and (4) breaking up. On the other hand, Bazant et al. [2] highlighted failure of two other types of structural materials: (1) failure due to the initiation of macroscopic crack growth, and (2) failure due to the growth of large stable microscopic cracks. The behavior, in general, is dependent upon the interaction between prominent modes of the irreversible changes of the microstructure, i.e. slip and microcracking. Slip on preferred crystallographic planes is promoted by shear stresses available for moving and stacking dislocations (line defects) into preferential configuration [3]. Response dominated by slip in shear planes is perceived as ductile. Ortiz [4] pointed out the importance of noting that both cracking and plastic flow of concrete (as a general material) exhibit a variety of typical features that are not contained within the classical theories of either fracture mechanics or plasticity. Whereas plastic strain does not significantly reduce the elastic moduli, microcracking causes both inelastic strain and a reduction of the elastic moduli [5-7]. Microcracking in the cleavage mode occurs on planes perpendicular to the highly tensioned directions and gives indication that microcracks are actually not randomly oriented but exhibit a prevalent orientation, raising the issue of stress-induced anisotropy of incremental elastic moduli [8]. Response characterized by microcracking in the cleavage mode has typically been classified as brittle [3]. The extension of microcracks interacts with the plasticity of the material, producing an effect known as elastoplastic coupling [9-l 31. Different combinations of continuum damage and plasticity theories have been proposed, aiming progressively at reducing the number of assumptions and increasing the degree of rationality [4, 14-321. Reviewing previous work on energy-based coupled elastoplastic damage theories, Ju [20] made use of the strain split in elastoplastic damage formulation and developed a sound thermodynamic constitutive model which is further reviewed in this presentation. The objective of this study is to conceptualize isotropic elastoplastic damage behavior rather than to develop a new model. In this paper, the concepts of generalized damage variables, generalized material degradation paths and generalized effective stresses based on the hypothesis of strain equivalence are introduced. Insofar as isotropic damage is concerned, the generalized damage variable can be obtained from uniaxial relations [16,33]. Therefore, the proposed concepts are applied to interpret experimental behavior of concrete in uniaxial compression as well as copper 99.9% in uniaxial tension. In addition, the ramification of the generalized damage variable concept Wn leave from Faculty of Engineering, Tanta University, Tanta. Egypt. 151
IS2
S. E.-D. F. TAHER et ui.
with reference to the existing ductile damage model of Lemaitre [16] is discussed. Finally, an extension to Mazars’ model [33] for concrete is proposed.
2. GENERALIZED DAMAGE VARIABLES Damage can be defined as a collection of permanent microstructural changes concerning material the~orn~~ni~l properties (e.g. stiffness, strength, anisotropy, etc.) brought about in a material by a set of irreversible physical microcracking processes resulting from the application of thermomechanical loadings [34];.Among the various definitions of the damage variable is the ratio of damaged surface area over total (nominal) surface area at a local material point [35-371. This definition further led to alternative forms as the change in the elastic compliances (stiffnesses) [1E-18,33,3744]. A broad definition of the generalized damage variables can he adopted as follows. “If a material has n generalized degrading properties, zi, i = 1, n, then at any time, t, the damage variable associated with any property zi, d,(r), is given by dJt)
zi(t)
= 1 -ZFp
in which z,(tt,) is the value of the ith property at its threshold time tc at which its degradation takes place.” The time can be unde~t~ as the real time for viscous damage models, intrinsic time for end~~o~cd~~ models, or pseudo-time for rate-inde~ndent models. The generalized damage variables have the following properties: (1) They are non-decreasing in the process of thermomechanical loading. (2) Zero values of the damage variables correspond to undamaged state before or at time tti. (3) Critical values ef are maximum values taking place at times t$ and need not define rupture as in Kachanov’s sense [4s]. A rupture criterion is required to interrelate the generalized damage variables. (4) &, QI- 00, 11, $ E 10,dzi1. For moss materials, d;, 2 0. (5) Rate values, &,, are equal to zero through unloadings since unloading is an elastic process.
3. GENERALIZED MATERIAL DEGRADATION PATDS Rock-like materials exhibit a variety of irreversible changes. Response of such materials is characterized by strain softening in the post peak region. This ~hara~te~sti~ is still debated as to whether it is a material or structural property [45-511.A typical stress-strain relation is shown in Fig. 1 for a material which is assumed to be purely elastic initially and for which five unloading paths are described schematically based on the recovered energy density ol. Path 1. Elastic unloading with no permanent deformation and full energy recovery. It represents a typical form of nonlinear elasticity and damage concept is trivial. For this path
Path 2. Perfect plastic unloading with neither deformation nor energy recovery (as that used in the “Bounding surface theory” [52] and in the “Sub-loading surface” models [53]). In this case o,=o,
c =c$‘,
&f’=o.
(2b)
This type is not to be confused with the conventional elastoplastic behavior where unloading takes place parallel to the original elastic modulus.
153
Towards a canonical elastoplastic damage model
-
Unlording
Fig. I. Generalized material degradation paths.
Path 3. Ductile unloading with flow stress degradation [54]. It represents a typical form of elastoplasticity for which c = c(E3) + &) - IL(3):~,:43), 0, = $0 :@) e -2 e c$
=
(w
E,‘:a.
Path 4. Brittle unloading with stiffness degradation. All microcracks are assumed to close upon unloading and permanent deformation is zero. It represents a typical form of secant type model w, = ia :&” = fQ:(l ~=~jf))=E’(t):a,
- @‘)E,:@ $‘=O
dL4’= 1 - E;: E(t).
(W
Path 5. Generalized damage [54] giving, for most materials, recoverable between the previous two cases. It represents damageelastoplastic 0, = fa :Cp) =@:(l 6 = Ef.7 + 60) p , e
energy intermediate coupling
-@)E,&’
cr= E-‘(t):a
dy’ = 1 - E;’ : E(r).
(W
Strictly speaking, path (5) is capable of capturing the features of path (2) if ds5)approaches -co. In this case E(f) = (1 - di’)) E. = + 00, which is consistent with Fig. 1. In addition, path (5) can coalesce to path (3) by setting dt5)= 0 and to path (4) by letting di5)= 1 - E;‘:a@c -I. In
154
S. E.-D. F. TAHER et al.
retrospect, it is important to underline that a particular solid is per se neither brittle nor ductile (contrary to the numerous models developed on these bases). 4. GENERALIZED
DECOMPOSITION
OF STRAIN TENSOR
The formal split of the total strain tensor, E, in the case of generalized damage for isothermal process into the elastic damage and plastic damage components is assumed at the outset [20], i.e. (Fig. 2a) c(t) = c&) +6&f).
(3)
The time here is recognized as pseudo-time introduced herein to have a common scale for the three phase of behavior; total, elastic and plastic damage. For this decomposition the stress tensor G may be correlated, at any time t, to the total strain tensor and its components as follows: a(t) = A(t):+)
= A(t):(&)
+ s(t))
(4a)
a(t) = E(t):&(t)
(4b)
a(t) = a, + P(t&&),
W)
where A, E and P are fourth order tensors whose initial values are A,, E. and PO, respectively and 0, is the stress tensor at the onset of plastic deformation at time t;. Equations (4) are introduced in similarity to Hook’s law (ati = &Q,) but the latter was introduced to model elastic behavior solely. Plots of eqs (4) are sketched in Fig. 2 in which A(t), E(t) and P(f) can be interpreted as secant moduli that can be easily expressed in terms of the generalized total, elastic and plastic damage variables d,, de and d,, respectively, and as given by (1), i.e.
A(t) = (1 - d,(tNAo
Pa)
E(t) = (1-
VW
d&NE,
P(t) = (1 - d,(t))P,.
Fig. 2. Generalized decomposition of the total strain.
w
Towards a canonical e~stopl~tic damage model
155
Unloading in stress-total strain space follows the generalized damaged path (Fig. 2a) with slope E(t), while in stress-eiastic strain space follows a quasi sub-brittle path (Fig. Zb) and a quasi sub-perfect piastic path in stress-plastic strain space (Fig. 2~).The proposed concept ofcomponent damage variables leads to interesting features, which are summarized as follows. Elastic damage may only appear beyond a certain threshold limit; i.e. time r,dassociated with d,, tt2 0. Tbe elastic d~age threshold has been reported hy many investigators [15-17,20-22,33,39,55]. By contrast, Krajcinovic and Fonseka [.56]and Loland [36] adopted the notion, for concrete, of initial damage due to inherent flaws or cracks due to nonhomogeneous shrinkage during curing. Initiation of plastic deformation of certain materials may take place simultaneously with loading application. In this case a, (which represents a yield limit) is zero for tj = 0 and hence A0# &. This can be verified from the behavior of materials such as concrete. Plastic damage may start as early as plastic defo~ation takes place, i.e. tj> r;. Lack of ex~~rnent~ evidence makes unreIiable any assump~on that plastic defo~ation onset follows elastic damage (ti > tt). However, Simo and Ju [21,22] and Ju [20] used effective stress quantities based on an elastic damage variable in their fo~ulations asking that plastic defo~ation is subsequent to damage. Elastic and plastic damage variables are independent in the sense that they need not evolve at the same time or at the same rate. Therefore, the Helmholtz (total) free energy, $, cannot be simply partitioned using a single damage parameter, i.e. jl@,, 49d) = (1 - ~~~*~~~, Q) # (1 - ~)~~~{~~)+ Iraq)}
(6)
as postulated by Ju [20]. In eq. (61, q, d denote a suitabie set of plastic variables and damage variable, res~ctively. $(c:,, 4, d) is a locally averaged ~orno~n~~~ free energy function of damage material, @‘(cc,q) signifies the total potential energy function of an undamaged (virgin) material. I,@~) and J/i(q) are the uncoupled elastic and plastic potential energy functions, respectively. Partitioning of this sort is valid only if g and q are related to the stress vector, CT,by the same modulus as given in eq. (4a). The more generalized expression of the Helmholtz functional as a weighted func~on should he +&, 4, &d,) = (1- ~~~~~~~~~ f (1- d~~~~(q~,
(7)
in which dq is the damage variable conjugate to q. Linear hardening models are described by setting dp= 0, whereas d, = 0 represents the case of linear elasticity, 5. GENERALIZES ~~~C~E
S’I’RE$B~ CQNCEPT
Based on the hypothesis of strain equivalence introduced by LemaitreE57J: “The strain associated with a damage state under the applied stress is covalent to the strain associated with its undama~d state under the applied effective stress.” Thus, substituting eqs (5) into eqs (4), the strain equivalence can be expressed as follows: CT
c=A;":~_~,-------=A,“:$,
(?
z+_?.1 -da
in which 8, $@,ap and &,are the total, elastic, plastic and initial plastic effective stresses, respectively. Figure 3 shows the equivalence of total strain and its components, For an element defined by normal II and subjected to force E, its effective area Scan be express in terms of the total area, S, through the total damage variable d,, i.e. ,?-= Sfl -d,),
(9)
156
S. E.-D. F. TAHER
ef al.
Elastic
0
t
--IIl-
-
-1-e
(l-d,)-’ I
+
a
m
+
Plastic
Phstic
Effective rprcc
Physical rpwc
Fig. 3. Generalized
and the effective traction vector nz by the same damage variable
effective stress concept.
= 6 * a) is related to the total traction vector _T(_T= 0 - a)
_T=L
s
E S(1
T
(10)
-c&)=1-
Equations (9) and (10) are consistent with the effective stress concept originated by Rabotnov [SS]. Consequently, eq. (8a) is sufficient for formulating constitutive relations. On the other hand, for a ductile unloading path; d, = 0, 0 = b, # d; A0 need not be equal to &, and uniqueness of the effective stress is lost. However, if for this case A,, = I$,, then f = 6 - 6, - A,‘:a(l/(l
-d,)
- 1) = A,‘:a(d,/(l
-d,))
= d,A,‘:b,
thus 6 = A,/d,(P;:c7P) = a/(1 -d,); therefore uniqueness is achieved since 6 is related to eP. A very important special case arises when d, = d, = dP = d, thus leading definitely to a unique effective stress d = d, = r7,,= a/( 1 - d) and Ju’s formulation [20] is recovered. However, for full representation of both the stress and strain histories, only two of the relations shown in eqs (8) are sufficient. This is achieved by using eqs
Towards a canonical etastoplastic damage model
157
(8) in eq. (3), yielding the form of the dependency relationship amongst the three effective stresses
as 5 6 =-=:Ao:
1 -da
= A,:[E,i:8e+
+$++$-]:B)([ P,‘:(ci, - r&o)].
A,,:($$-:u~)
(11)
In other words, elastoplastic damage follows constraint uncoupling. 6. DEMONSTRATION OF ELASTOPLASTIC DAMAGE UNCOUPLING In this section, existing experimental evidence for both a brittle material (concrete) and a ductile material (copper 99.9%) will be reviewed and interpreted according to the proposed generalized damage concepts. 6.1. Concrete Popovics [59] proposed the following stress-total strain relation for plain concrete subjected to uniaxial compression:
0
0
m-
f&i
5
-= 5”
L
m-l-t-
m’
(12)
;
0 Y where a, and Q are peak stress and strain, respectively, and BZis a parameter dependent on a,. Adopting a value of m = 2 for a, = 17.2MPa (2500 psi), eq. (12) becomes (Fig. 4a)
0 -=5
2;0
Id
(13) t: 2’ ; 0Y In addition, use is also made of a plastic-total strain (c~- &) relation by Kansan and Iirsa [6]: 5”
l+
($=0.145($+0.127@.
(14)
The moduli and damage variable conjugate to the total strain are obtained by making use of eq. (13):
=-A :
and
Ol+?;y=l+?J
tlW
S. E.-D. F. TAHER et al.
(4
1.2 r
1
0
2
4
3
5
6
Normalized total strain (E/E,) (b) 1.2
1.0 2 2
0.8
z r 9 =&
OA
: 5 v1
04. 0.2
rl 0.2
0
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Elastic strain/peak total strain (E&) (c) 1.2
0.6
0.4
0.2
0
1
2
Plastic strain/peak
3
4
5
6
total strain (E#,)
Fig. 4. (a) Stress-total strain relationship for concrete under uniaxial compression. (b) Stress-lastic strain relationship for concrete under uniaxial compression. (c) Stress-plastic strain relationship for concrete under uniaxial compression.
Towards a canonical aIastopl~~c damage model The
159
elastic strain component is ~lc~lated using equs (3) and (141, yielding (Fig. 4b)
The rn~u1~ and dama~ variable conjugate to the elastic strain ~orn~o~ent are
2 =--X~=1.14Ao 0.873 2
OW
and
estimate that & # [email protected] assumption of this lotion (f7a) snbs~~tiates the irn~~at equality is quite common in the literature where A*is construed to be the elastic (young’s) modulus. Similarly, the modub and damage variable conjugate to the plastic strain component {Fig. 4cJ are
=Lxk7.87A 0.127 2
0
and
0.127 [li(~CB127+L145(’ ~xaminatioa of the evolution of damage variables for concrete in impression as well as other auxiliary q~ntiti~ reveals the fo~lo~ng. Onset of plastic defo~ation starts sirn~~~eo~sly with load a~licatio~~ This inclusion is also in accordance with Spooner and Dougill [60]. Elastic strain ~~t~bntion to the total starts at a higher rate than the plastic strain, as shown in Fig. 5. The ~nt~b~tions become equal at a total strain ratio C/C,,= 2.55, corres~nding to a stress ratio a/o, = 0.67, following which the plastic strain contribution becomes greater than the elastic strain. The maximum elastic strain ratio (r&J is 1.30 at c/ty = 3 and a/~~ = 0.6. Su uentiy, the elastic strain decreases and this phenomenon of strain ~tro~~io~ is also
S. E.-D. F. TAHER et al.
160
.’
-
Total strain
- - +-*-..*
Elrstic-damrgo rtrain Plastic-drmago strain
2
.*’ _a
4
3
Total strain/peak
.
.*’ ..* .** .* .*
/
1
f .* .*
6
5
strain (E/E,)
Fig. 5. Strain components for concrete under uniaxial compression.
illustrated in Q - ce space (Fig. 4). In contrast, the elastic strain keeps on decreasing until the total strain is equal to the plastic strain at L/L, = 6, a/a, = 0.32. (3) The total damage variable, da, is a continuous function and its limit as c/cU+NI yields unity while its value at E/C,= 6 is 0.97. The trend of da along with d, and dp is shown in Fig. 6. (4) The elastic damage variable, de, is a singular function at C/L, = 6, but it is continuous in the domain 6/cUE IO,61 though it has an inflection point at L/c. = 3.91. At this value of the total strain ratio, the elastic damage parameter is interpreted to be critical since it should be nondecreasing and this value (d:) is found to be 0.82. The other parameters corresponding to the inflection point of the elastic damage curve are found to be: ce/cu = 1.19, s/c,, = 2.72, a/a, = 0.48, da = 0.93 and d, = 0.98. These values may be considered critical values for complete damage.
Criticrl limit
0.2
i I-/’
0
- - - -
Total Ehstic
-
Plastic
- -
I’ 8’ I 1
I 2
Normalized
I 3
I 4
I 5
I 6
total rtrain
Fig. 6. Evolution of generalized damage variables for concrete u, der uniaxial compression.
Towards a canonical elastoplastic damage model
161
(5) As depicted in Fig. 6, the onset of elastic damage is associated with a threshold total strain ratio c/tu = 0.17 and for which a/a,=0.34, c,lc,=O.14, ~&,=0.03, d,=O.O3 and d,=O.19. At this stress level, the plastic strain as well as the total damage variable are negligible. This result using damage concepts confirms the validity of the assumption of linear elastic behavior for e/a, < 0.3 as used in elastoplastic computational models (e.g. [al]). In such a case, E,, may be reduced to A,, without violation of the generalized damage concepts as presented. However, this is valid only for m = 2, while a higher discrepancy is found for other values of m. (6) The plastic damage variable, 4, is a continuous function and follows an asymptotic trend as defined by lim,,, d, = 1.0. It grows at a much higher rate than both d, and d,. This gives indications that: (a) damage causes more degradation in the plastic stiffness than both the elastic and overall stiffness of concrete; (b) onset of plastic damage commences prior to elastic damage; and (c) linear hardening or perfect plasticity are inappropriate approximations for elasto-plasticity as applied to concrete. (7) A similar phenomenological model predicting damage for concrete in tension can be developed using appropriate experimental data. 6.2. Copper (99.9%) A complete set of experimental data for copper (99.9%) has been provided by Lemaitre [15, 16,621. It is a very ductile material and shows elongation almost equal to its original length at rupture. The elastic strain is negligible when compared to the plastic or total strain. Figure 7a shows the stress-total strain relationship which can be decomposed into stress-elastic strain and stress-plastic strain relationships as shown in Figs 7b and c. Data fitting using a Ramberg-Osgood hardening law of the form a = kcilM (where k and M are coefficients) gives poor correlation. However, if the law is used in the form a = a, + kcj’M (as originally assumed by Ramberg and Osgood [63]), it gives a good fit but not better than a cubic polynomial. Decomposing the total strain into its components, elastic and plastic damage, then monitoring the degradation of the initial moduli, the generalized damage variables are obtained. Figure 8 shows the evolution of the three damage variables d,, d, and d, associated with the total, elastic and plastic stiffness degradation, respectively. The following phenomenological implications are deduced: (1) The total damage variable is zero up to yield (very small domain), then increases abruptly to evolve asymptotically to unity. (2) The plastic damage variable threshold is intermediate between thresholds of other damage variables. (3) The elastic damage variable evolution may be approximated by linear relation as given by Lemaitre [16, 17, 621. (4) At high values of strain near rupture, the elastic stiffness degrades more than the plastic stiffness. This is evidenced by the higher rate of evolution of the elastic damage variable at later stages. (5) Unlike concrete, the total damage variable is not intermediate to other damage variables. This is attributed to the fact that the initial total slope A, is very steep (almost vertical) compared to E, and PO (Figs 7a-c). 7. ON LEMAITRE’S
DUCTILE DAMAGE MODEL
Ju [20] criticized Lemaitre’s ductile damage model and stated: “The fundamental problem of the ductile plastic damage formulation advocated by Lemaitre [15-l 71 is the non-optimal choice of the locally averaged free energy potential. In particular, damage is associated only with elastic strains and the damage energy release rate is shown to be the elastic strain energy expressed by Lemaitre [I 61. This treatment amounts to uncoupled plasticity and damage processes, thus in a sense contradicting experimental evidence that plastic variables also contribute to the initiation and growth of microcracks”.
(4 600 -
Strets - tocal rmin relationship (8ftsr J. Lcmritre, 1985)
100 -& “0 <
0
I
I
I
I
I
I
I
I
I
I
I
10
20
30
40
50
60
70
SO
90
100
110
I 2.2
1 2.4
100
110
Total strain (x100)
(b) Strom - elastic strain relationship
600
500 F
I 0.2
0
I 0.4
I 0.6
I 0.8
I 1.0
I 1.2
I 1.4
I 1.6
I 1.8
I 2.0
Elastic strain (x100)
(cl Smoss - plastic strain relationship
0
10
20
30
40
50
Plastic
60
70
80
90
strain (x100)
Fig. 7. (a) Stress-total strain relationship for copper 99.9% under uniaxial tension (after J. Lemaitre[ 161).(b) Stress-elastic strain relationship for copper 99.9%. (c) Stress-plastic strain relationship for copper 99.9%. 162
163
Towards a canonical elastoplastic damage model
Evolution of dwna~e vuiabla d.
e
0.8
5
0.6
a 2 2 1 n
0.4
9-
/ /-
d
8
0.2 0
..* .*
.*3.’
I’ / 3
0
.* .*
c-.* .-
1
I
10
20
..-a 30
.*-
.*
Totrl jg,rtjc
- - - - Phrtic
.*.*
40
. . ... .. .
I
I
I
I
I
I
I
50
60
70
80
90
100
110
Total strain (EXP2) Fig. 8. Generalized damage variables for copper 99.9%.
In Lemaitre’s formulation following.
of a damage model for ductile fracture [16], he considered the
(1) The von Mises equivalent stress for plasticity beg= (f S : S)‘j2, with S being the deviatoric stress tensor. hardening law of the special form crc4= kp’l”, p = (@‘: ~p)‘/~;k and M being material parameters; p the equivalent strain and &’ is the Euler-Almansi plastic strain tensor in large deformation theory. (3) Existence of a potential of dissipation ‘p*. (4) Considered only elastic damage and defined it as the degradation in the elastic (Young’s) modulus E, i.e. D in the formulation of Lemaitre [16] is simply 4 in this paper. Lemaitre derived the following rate equation for the evolution of (elastic) damage:
(2) Ramberg-Osgood
where v is Poisson’s ratio, cH is the hydrostatic pressure = 1/3@(a). S, and so are temperature and material dependent parameters. (‘) implies derivative with respect to time. Subsequent to eq. (19), Lemaitre proceeded to use the damage modified form of the Ramberg-Osgood law to express o, in terms of the equivalent strain p. However, by virtue of the approximations made in the model that the damage variable is conjugate to the elastic processes and appears only in the elastic free energy and also that it is found experimentally from degradation of Young’s modulus, it is essential that for consistency, the non-damage modified Ramberg-Osgood relationship be used, yielding for the evolution of damage the rate equation
Equation (20) is different from that derived by Lemaitre by the term (1 - d,)2 appearing in the denominator on the right hand side. Integration of eq. (20) for proportional loading or even uniaxial cases will yield a nonlinear expression for de, which should be linear as shown by Lemaitre [16] from experiments. This indicates that the assumed form for the potential of dissipation cp* may need to be revised.
S. E.-D. F. TAHER ei al.
164
8. E~~NSION
OF NAZARS’ NODAL FOR CONCRETE
Nazars [33] develop a model which id~i~s the behavior of unite as brittle. It required two damage variables in tension and compression, then weighted together to produce a singie damage behavior which was further used in the m~ified form of Hook’s law. Following the same approach with the help of the generalized damage variables and the notion of equivalent strain;
where j = t, c for tension and companion, respectively, and
where the wei~tiug parameters ajdepend on the state of stress with a, = 1 and ac = 0 in pure tension whereas a, = 0 and cr,= 1 in pure compression. For the general case, the sum of a, and a, must always be unity, Finally, the tensorial constitutive equations can be expressed by mo~fying Hook’s law for the three phases of behavior as follows: d = C& = c,, (1 - d&k, where C,, is the conventional initial stress-strain matrix but employing the initial moduli Aa, A’@ and Pl, for k = a, e and p, ~s~tively, and Ed= c. 9. CONCLUSION
The overall ~ha~or of mate~als exhibiti~ isotropic elastoplastic damage processes under various loading conditions cannot be properly chara~ter~d by a single damage variable as has been the adopted practice. The use of the suggested generalize damage variables approach underlines constraint uncoupling between different response phases classified as total, elastic and plastic. The correct expression of the d~ompo~ Helmholtz total free energy is given. Models cast in this form can be readily implemented into the finite element computational framework, with the damage variables being known explicitly in terms of the total strain. In addition, the concept of general~d damage variables as ~re~nted herein allows for unloading along a general degraded path in contrast to the more restrictive cases of either purely brittle or, purely ductile path as per existing models. The practical use of this concept was shown in the modifi~tion of Mazars” model for concrete. ~c~~ow~e~e~nf-~e authors wish to acknowl~ge the support of the ~c~rtment of Civil En~n~~ng, U~versity of Petroleum and Minerals, Dhahran, Saudi Arabia in the pursuit of this work.
[l] J. L. Chaboche, Fracture rn~b~~
King Fahd
and damage mechanics: ~rnp~~en~~ty of approaches, pp. 309-321 (~987). [2] 2. P. Bazant, Y. Xi and S. G. Reid, Statistical size effect in quasi-b~ttie structures: I. Is Weibull theory app~i~ble? J. Engng Me&., AXE 117(1I), 2609-2622 (1991). [3] D. Krajcinovic, Mechanics of solids with a pro~ive~y dete~orating stratus, in ~pplicutiono~~ructure ~eeh~ic.~ fo Ce~e~t~t~ ignites (Editedby S. P. Shah), pp. 453-477. NATO-ARW, No~hw~te~ Unive~ity, IL, U.S.A. 4-7 September (1984). [4] M. &tiz, A constitutive theory for the inetastic behavior of concrete. Meek. Mater. 4,67-93 (1985). [S] T. C. I-&u, F. L?. Slate, 0. M. Sturman and G. Winter. ~~~m~ing of plain concrete and the shape of the stress-strain curve. 3. Am. Caner. Inst. 60, 227-239 (1963).
Towards a canonical eiastoplastic damage model
165
[6] D. Karsan and J. 0. Jirsa, Behavior of concrete under compressive loading. J. struct. Engng. Diu., ASCE 95, 2543 (1969). [7] J. Wastiels, Behavior of concrete under multiaxial stresses--a
review. Cement Concr. Res. 9, 35 (1979). [8] Z. P. Bazant and C. L. Shieh, Hysteretic fracturing endochronic theory for concrete. J. Engng Mech., ASCE 106(S), 929-950 (I 980). [9] T. Hueckel, On plastic flow of granular and rocklike materials with variable elastic moduli. Bull. Polish Acad. Sci. u(8) 405 (I 975). [10) T. Hueckel, Coupling of elastic and plastic deformations of bulk solids. Meclrtrnics II, 227 (1976). strain space formutation of plasticity. Int. J. nun-&n. i1rj Y. F, Dafalias. EIasto-ntastic counoling within a the~~ynamic Mech. 12, 32jf1977). _ WI Y. F. Dafalias, Il’iushin’s postulate and resulting the~odynamic conditions on elasto-plastic coupling. Znf. J. Solids Structures 13, 239 (1977). fl31 Y. F. Dafalias, Restrictions on the continuous description of elasto-plastic coupling for concrete within thermodynamics, in Inelastic Behavior of Pressure Vessels and Piping Components (Edited by T. Y. Chang and E. Krempl), p. 29. Series PVP-PB-028, ASME, New York (1978). P41 Z. P. Bazant and S. S. Kim, Plastic-fracturing theory for concrete. J. Engng Mech., ASCE llS(3) 407-428 (1979). Wl J. Lemaitre, Coupled elasto-plasticity and damage constitutive equations. Invited lecture at FENOMECHM, Stuttgart (RFA); Comput. Merh. appl. Mech. 51, 31-48 (I985). [I61 J. Lemaitre, A continuous damage mechanics model for ductile fracture. J. Engng Muter. Tech&. IM,83%89 (1985). [a J. Lemaitre, Plasticity and damage under random loading, in Proc. iOth U.S. ~ut~ona~ Congress ofAppliedMeehffnjcs (Edited by J. P. Lamb) (pp. 12%134), Austin, TX. ASME, New York (I&20 June 1986). WI G. Frantziskonis and C. S. Desai, Constitutive model with strain softening. Inr. J. Solids Struefures 2x6). 733-756 (1987). v91 A. Dragon and Z. Morz, A continuum theory for plastic-brittle behavior of rock and concrete. In?. J. Engng Sci. 17, 121-137 (1979). PO1 I. W. Ju, On energy-based coupled elastoplastic damage theories: constitutive modeling and computational aSptXtS. Int. J. Soli& Structures Z!!(7), 803-833 (1989). IN J. C. Simo and J. W. Ju, Strain- and stress-based continuum damage models, I. Formulation. inf. J. Solids Structures 23(7), 821-840 (t987). damage models, II. Computational aspects. Znt. J. Solids PA J. C. Simo and J. W. Ju, Strain- and stress-based continue Structures 23(7), 84 I-869 (1987). ~31 D. J. Stevens and T. Krauthammer, Nonlocal continuum damage/plasticity model for impulse loaded RC beams. J. struct. Engng., ASCE
llS(9),
2329-2347
(1989).
~41 S. Yazdani and H. L. Schreyer, Combined plasticity and damage mechanics model for plain concrete. 1. Engng Mech. 116(7), 3435-1450 (1990). P51 J. C. Simo, Strain softening and dissipation: a unification of approaches, in Cracking and Damage, Strain Localization and Size E$‘kct (Edited by J. Mazars and Z. P. Bazant), pp. -61. Elsevier Applied Science, Barking (1989). I261 H. L. Schreyer, Formulations for nonlocal softening in a finite zone with anisotropic damage, in Cracking and Damage, Strain ~ocu~izur~onand Size Effect (Edited by J. Mazars and Z. P. Bazant). Elsevier Applied Science, Barking (I 989). ~71 J. P. Cordebois and F. Sidoroff, Endommagement anisotrope en elasticite et plasticite. J. ~ec~~que Theor. Appl. (Speciai Issue) (1982). 1281A. Dragon, Plasticity and ductile fracture damage: study of void growth in metals. &gng Fracture Mech. 21,875-885 (1985). A. Dragon and A. Chihab, On finite damage: ductile fracture-damage evolution. Mech. Mater. 4, 95-106 (1985). tz; J. Lemaitre and J. Dufailly, Modellisation et identification de l’en-dommagement plastique des metaux. 3 eme Congres Franc& de Mecanique, Grenoble, France (I 977). [311 C. L. Chow and J. Wang, An anisotropic theory of continuum damage mechanics for ductile fracture. Engng Fracture Mech. 27, 547-558 (1987). [321 C. L. Chow and J. Wang, An anisotropic continuum damage theory and its application to ductile crack initiation, in Proc. ASME Winter Annual Meeting (Edited by A. S. D. Wang and G. K. Haritos) (pp. l-9). Boston, MA (f 3- 18 December 1987). 1331J. Mazars, Application de la mecanique de l’endommagement au comportemention lineaire et a la rupture du beton de structure. These de Doctorat d’Etat, Universite Paris, France (1984). [341 R. Talreja, A continuum mechanics characterization of damage in composite materials. Proc. R. Sot. Land., Ser. A 399, 195-215 (1985). [351 J. Janson and J. Hult, Fracture mechanics and damage mechanics: a combined approach. J. Mecanique Appt. 1,69-84 (1977). I361 K. E. Loland, Mathematical modelling of deformational and fracture properties of concrete based on damage mechanical principles-application on concrete with and without addition of silica fume. Dr. Ing. Thesis, University of Trondheim, Norway (i 981). [37l M. Lorrain and K. E. Loland, Damage theory applied to concrete, in Fracture Mec~unics ofconcrere (Edited by F. H. Wittmann), pp. 34-369. Elsevier, Amsterdam (1983). [38J J. Mazars, Mechanical damage and fracture of concrete structures, in Advances in Fracture Research (Fracture 81), Volume 4, pp. 1499-1506. Proceedings I.C.F.S. Cannes, France. Pergamon Press, Oxford (1982). r391 J. Mazars, Description of the behavior of composite concretes under complex loading through continuum damage mechanics, in Proc. 10th. U.S. National Congress qfApp/ied Mechanics (Edited by J. P. Lamb) (pp. 135-139). Austin, TX. ASME, New York (16-20 June 1986). WI L. Resende and J. B. Martin, Damage constitutive model for granular materials. Comput. Meth. uppl. Mech. Engng 42, I-18 (1984). [411 L. Resende, A damage mechanics constitutive theory for the inelastic behavior of concrete. Comput. Meth. appt. Me&. Engng 60, 57-93 (1987). 1421 P. tadeveze, On an anisotropic damage theory. Proc. of the CNRS Internatiunfft Colloquium on Failure Criteria ef Srrucrural Media. Internai Report No. 34, L.M.T., Cachan, Villars de Lans, France (1983).
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(431 J. Mazars and J. Lemaitre, Application of continuous damage mechanics to strain and fracture behavior of concrete, in Application of Fracture Mechanics to Cementitious Composites (Edited by S. P. Shah) pp. 375-388. NATO-ARW, Northwestern University, (4-7 September 1984). [44] C. Saouridis and J. Mazars, A multiscale approach to distributed damage and its usefulness for capturing structural size effect, in Cracking and Damage, Strain Localization and Size E&ct (Edited by J. Mazars and Z. P. Bazant), pp. 391403. Elsevier Applied Science, Barking (1989). [45] L. M. Kachanov, On the creep fracture time. Izu. Akad. Nauk SSR, Otd. Tekh. Nauk 8, 2631 (1958) [in Russian]. [46] H. E. Read and G. A. Hegemier, Strain softening of rock, soil and concrete-a review article. Mech. Mater. 3,271-294 (1984). [47] J. S. Sandler, Strain softening for static and dynamic problems. ASME Winter Annunl Meeting, Symp. on Constitutive Equations: Micro, Macro and Computational Aspects. CEQ, New Orleans (December 1984).
[48] F. H. Wu and L. B. Freund, Deformation trapping due to thermoplastic instability in one-dimensional wave propagation. J. Mech. Phys. So/ia!s 32, 119-132 (1984). [49] J. G. M. Van Mier, Strain softening of concrete under multiaxial loading conditions. Doctoral Dissertation, Eindhoven University of Technology, The Netherlands (1984). [50] G. Frantziskonis, Progressive damage and constitutive behavior of geomaterials including analysis and implementation. Ph.D. dissertation, Department of Civil Engineering, Univeristy of Arizona, Tucson, AZ (1986). [51] P. E. Roelfstra, Summary of the discussion on: nonlocal models, localization limiters, and size effect, part 2, and general discussion, in Cracking and Damage, Strain Localization and Size Ejects (Edited by J. Mazars and Z. P. Bazant), pp. 539-546. Elsevier Applied Science, Barking (1989). [52] Y. F. Dafalias, The concept and application of the bounding surface in plasticity theory, in Proc. IUTAM Conf on Physical Non-linearities in Structural Analysis (Edited by J. Hult and I. Lemaitre). Springer, Berlin (1981). [53] K. Hashiguchi, Subloading surface model in unconventional plasticity. Inr. J. Solids Structures, 25(S), 917-945 (1989). [54] M. Elites and J. Planas, Material models, fracture mechanics of concrete structures from theory to application, in Report of the Technical Committee 90-FMA, Fracture Mechanics to Concrete-Application, RILEM (Edited by L. Elfgran), pp. 1666. Chapman & Hall (1989). [55] J. Mazars, Mechanical damage and fracture of concrete structures. 5th Int. Conf. on Fracture, Cannes (1980). [56] D. Krajcinovic and G. U. Fonseka, The continuous damage theory of brittle material-parts I, II. J. appi. Mech. 48, 809-815, 816-824 (1981). [57] J. Lemaitre, Evaluation of dissipation and damage in metals. Proc. I.C.M. Volume 1, Kyoto, Japan (1983). [58] Y. N. Rabotnov, Creep rupture. Proc. XII International Congress in Applied Mechanics, Stanford, CA. Springer, Berlin (1988). [59] S. Popovics, A numerical approach to the complete stress-strain curve of concrete. Cement. Concr. Res. 3, 583-599 (1973). [60] D. C. Spooner and J. W. Dougill, A quantitative assessment of damage sustained in concrete during compressive loading. Mag. Concr. Res. 27, 92, 151-160 (1975). [61] D. R. J. Owen, J. A. Figueiras and F. Damjanic, Finite element analysis of reinforced and prestressed concrete structures including thermal loading. Comput. Meth. appl. Mech. Engng 41, 323-366 (1983). [62] J. Lemaitre, A Course on Damage Mechanics. Springer, Berlin (1992).
[63] W. Ramberg and W. R. Osgood, Description of stress-strain curves by three parameters. NASA TN902 (1943). (Received 24 May 1993)
Pergamon
Copyright0 1994.ElsevierScienceLtd
0013-7944f93MMDS2 . I
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TOWARDS A CANONICAL ELASTOPLASTIC DAMAGE MODEL SALAH EL-DIN F. TAHER,t MOHAMMED H. BALUCH and ALI H. AL-GADHIB Department of Civil Engineering, Ring Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Ahatraet-Fundamental aspects of elastoplastic damage are outlined. Time-independent isotropic damage is considered in order to study material degradation. By splitting the total strain tensor into its components of elastic damage and plastic damage and using recoverable energy equivalence, three distinct modes of behavior are particularized. For each mode of behavior, a suitable damage variable is culled. An in-depth analysis of this formulation reveals a certain incongruity in the assumptions postulated in some of the previously proposed models. The suggested generalized concepts are supported by experimental evidence.
1. INTRODUCTION THE CRACKING of materials results from creation, propagation and coalescence of microcracks. For materials characterized by ductile behavior, Chaboche [l] outlined four different levels of cracking: (1) crack nucleation, (2) microcrack initiation, (3) macrocrack initiation and (4) breaking up. On the other hand, Bazant et al. [2] highlighted failure of two other types of structural materials: (1) failure due to the initiation of macroscopic crack growth, and (2) failure due to the growth of large stable microscopic cracks. The behavior, in general, is dependent upon the interaction between prominent modes of the irreversible changes of the microstructure, i.e. slip and microcracking. Slip on preferred crystallographic planes is promoted by shear stresses available for moving and stacking dislocations (line defects) into preferential configuration [3]. Response dominated by slip in shear planes is perceived as ductile. Ortiz [4] pointed out the importance of noting that both cracking and plastic flow of concrete (as a general material) exhibit a variety of typical features that are not contained within the classical theories of either fracture mechanics or plasticity. Whereas plastic strain does not significantly reduce the elastic moduli, microcracking causes both inelastic strain and a reduction of the elastic moduli [5-7]. Microcracking in the cleavage mode occurs on planes perpendicular to the highly tensioned directions and gives indication that microcracks are actually not randomly oriented but exhibit a prevalent orientation, raising the issue of stress-induced anisotropy of incremental elastic moduli [8]. Response characterized by microcracking in the cleavage mode has typically been classified as brittle [3]. The extension of microcracks interacts with the plasticity of the material, producing an effect known as elastoplastic coupling [9-l 31. Different combinations of continuum damage and plasticity theories have been proposed, aiming progressively at reducing the number of assumptions and increasing the degree of rationality [4, 14-321. Reviewing previous work on energy-based coupled elastoplastic damage theories, Ju [20] made use of the strain split in elastoplastic damage formulation and developed a sound thermodynamic constitutive model which is further reviewed in this presentation. The objective of this study is to conceptualize isotropic elastoplastic damage behavior rather than to develop a new model. In this paper, the concepts of generalized damage variables, generalized material degradation paths and generalized effective stresses based on the hypothesis of strain equivalence are introduced. Insofar as isotropic damage is concerned, the generalized damage variable can be obtained from uniaxial relations [16,33]. Therefore, the proposed concepts are applied to interpret experimental behavior of concrete in uniaxial compression as well as copper 99.9% in uniaxial tension. In addition, the ramification of the generalized damage variable concept Wn leave from Faculty of Engineering, Tanta University, Tanta. Egypt. 151
IS2
S. E.-D. F. TAHER et ui.
with reference to the existing ductile damage model of Lemaitre [16] is discussed. Finally, an extension to Mazars’ model [33] for concrete is proposed.
2. GENERALIZED DAMAGE VARIABLES Damage can be defined as a collection of permanent microstructural changes concerning material the~orn~~ni~l properties (e.g. stiffness, strength, anisotropy, etc.) brought about in a material by a set of irreversible physical microcracking processes resulting from the application of thermomechanical loadings [34];.Among the various definitions of the damage variable is the ratio of damaged surface area over total (nominal) surface area at a local material point [35-371. This definition further led to alternative forms as the change in the elastic compliances (stiffnesses) [1E-18,33,3744]. A broad definition of the generalized damage variables can he adopted as follows. “If a material has n generalized degrading properties, zi, i = 1, n, then at any time, t, the damage variable associated with any property zi, d,(r), is given by dJt)
zi(t)
= 1 -ZFp
in which z,(tt,) is the value of the ith property at its threshold time tc at which its degradation takes place.” The time can be unde~t~ as the real time for viscous damage models, intrinsic time for end~~o~cd~~ models, or pseudo-time for rate-inde~ndent models. The generalized damage variables have the following properties: (1) They are non-decreasing in the process of thermomechanical loading. (2) Zero values of the damage variables correspond to undamaged state before or at time tti. (3) Critical values ef are maximum values taking place at times t$ and need not define rupture as in Kachanov’s sense [4s]. A rupture criterion is required to interrelate the generalized damage variables. (4) &, QI- 00, 11, $ E 10,dzi1. For moss materials, d;, 2 0. (5) Rate values, &,, are equal to zero through unloadings since unloading is an elastic process.
3. GENERALIZED MATERIAL DEGRADATION PATDS Rock-like materials exhibit a variety of irreversible changes. Response of such materials is characterized by strain softening in the post peak region. This ~hara~te~sti~ is still debated as to whether it is a material or structural property [45-511.A typical stress-strain relation is shown in Fig. 1 for a material which is assumed to be purely elastic initially and for which five unloading paths are described schematically based on the recovered energy density ol. Path 1. Elastic unloading with no permanent deformation and full energy recovery. It represents a typical form of nonlinear elasticity and damage concept is trivial. For this path
Path 2. Perfect plastic unloading with neither deformation nor energy recovery (as that used in the “Bounding surface theory” [52] and in the “Sub-loading surface” models [53]). In this case o,=o,
c =c$‘,
&f’=o.
(2b)
This type is not to be confused with the conventional elastoplastic behavior where unloading takes place parallel to the original elastic modulus.
153
Towards a canonical elastoplastic damage model
-
Unlording
Fig. I. Generalized material degradation paths.
Path 3. Ductile unloading with flow stress degradation [54]. It represents a typical form of elastoplasticity for which c = c(E3) + &) - IL(3):~,:43), 0, = $0 :@) e -2 e c$
=
(w
E,‘:a.
Path 4. Brittle unloading with stiffness degradation. All microcracks are assumed to close upon unloading and permanent deformation is zero. It represents a typical form of secant type model w, = ia :&” = fQ:(l ~=~jf))=E’(t):a,
- @‘)E,:@ $‘=O
dL4’= 1 - E;: E(t).
(W
Path 5. Generalized damage [54] giving, for most materials, recoverable between the previous two cases. It represents damageelastoplastic 0, = fa :Cp) =@:(l 6 = Ef.7 + 60) p , e
energy intermediate coupling
-@)E,&’
cr= E-‘(t):a
dy’ = 1 - E;’ : E(r).
(W
Strictly speaking, path (5) is capable of capturing the features of path (2) if ds5)approaches -co. In this case E(f) = (1 - di’)) E. = + 00, which is consistent with Fig. 1. In addition, path (5) can coalesce to path (3) by setting dt5)= 0 and to path (4) by letting di5)= 1 - E;‘:a@c -I. In
154
S. E.-D. F. TAHER et al.
retrospect, it is important to underline that a particular solid is per se neither brittle nor ductile (contrary to the numerous models developed on these bases). 4. GENERALIZED
DECOMPOSITION
OF STRAIN TENSOR
The formal split of the total strain tensor, E, in the case of generalized damage for isothermal process into the elastic damage and plastic damage components is assumed at the outset [20], i.e. (Fig. 2a) c(t) = c&) +6&f).
(3)
The time here is recognized as pseudo-time introduced herein to have a common scale for the three phase of behavior; total, elastic and plastic damage. For this decomposition the stress tensor G may be correlated, at any time t, to the total strain tensor and its components as follows: a(t) = A(t):+)
= A(t):(&)
+ s(t))
(4a)
a(t) = E(t):&(t)
(4b)
a(t) = a, + P(t&&),
W)
where A, E and P are fourth order tensors whose initial values are A,, E. and PO, respectively and 0, is the stress tensor at the onset of plastic deformation at time t;. Equations (4) are introduced in similarity to Hook’s law (ati = &Q,) but the latter was introduced to model elastic behavior solely. Plots of eqs (4) are sketched in Fig. 2 in which A(t), E(t) and P(f) can be interpreted as secant moduli that can be easily expressed in terms of the generalized total, elastic and plastic damage variables d,, de and d,, respectively, and as given by (1), i.e.
A(t) = (1 - d,(tNAo
Pa)
E(t) = (1-
VW
d&NE,
P(t) = (1 - d,(t))P,.
Fig. 2. Generalized decomposition of the total strain.
w
Towards a canonical e~stopl~tic damage model
155
Unloading in stress-total strain space follows the generalized damaged path (Fig. 2a) with slope E(t), while in stress-eiastic strain space follows a quasi sub-brittle path (Fig. Zb) and a quasi sub-perfect piastic path in stress-plastic strain space (Fig. 2~).The proposed concept ofcomponent damage variables leads to interesting features, which are summarized as follows. Elastic damage may only appear beyond a certain threshold limit; i.e. time r,dassociated with d,, tt2 0. Tbe elastic d~age threshold has been reported hy many investigators [15-17,20-22,33,39,55]. By contrast, Krajcinovic and Fonseka [.56]and Loland [36] adopted the notion, for concrete, of initial damage due to inherent flaws or cracks due to nonhomogeneous shrinkage during curing. Initiation of plastic deformation of certain materials may take place simultaneously with loading application. In this case a, (which represents a yield limit) is zero for tj = 0 and hence A0# &. This can be verified from the behavior of materials such as concrete. Plastic damage may start as early as plastic defo~ation takes place, i.e. tj> r;. Lack of ex~~rnent~ evidence makes unreIiable any assump~on that plastic defo~ation onset follows elastic damage (ti > tt). However, Simo and Ju [21,22] and Ju [20] used effective stress quantities based on an elastic damage variable in their fo~ulations asking that plastic defo~ation is subsequent to damage. Elastic and plastic damage variables are independent in the sense that they need not evolve at the same time or at the same rate. Therefore, the Helmholtz (total) free energy, $, cannot be simply partitioned using a single damage parameter, i.e. jl@,, 49d) = (1 - ~~~*~~~, Q) # (1 - ~)~~~{~~)+ Iraq)}
(6)
as postulated by Ju [20]. In eq. (61, q, d denote a suitabie set of plastic variables and damage variable, res~ctively. $(c:,, 4, d) is a locally averaged ~orno~n~~~ free energy function of damage material, @‘(cc,q) signifies the total potential energy function of an undamaged (virgin) material. I,@~) and J/i(q) are the uncoupled elastic and plastic potential energy functions, respectively. Partitioning of this sort is valid only if g and q are related to the stress vector, CT,by the same modulus as given in eq. (4a). The more generalized expression of the Helmholtz functional as a weighted func~on should he +&, 4, &d,) = (1- ~~~~~~~~~ f (1- d~~~~(q~,
(7)
in which dq is the damage variable conjugate to q. Linear hardening models are described by setting dp= 0, whereas d, = 0 represents the case of linear elasticity, 5. GENERALIZES ~~~C~E
S’I’RE$B~ CQNCEPT
Based on the hypothesis of strain equivalence introduced by LemaitreE57J: “The strain associated with a damage state under the applied stress is covalent to the strain associated with its undama~d state under the applied effective stress.” Thus, substituting eqs (5) into eqs (4), the strain equivalence can be expressed as follows: CT
c=A;":~_~,-------=A,“:$,
(?
z+_?.1 -da
in which 8, $@,ap and &,are the total, elastic, plastic and initial plastic effective stresses, respectively. Figure 3 shows the equivalence of total strain and its components, For an element defined by normal II and subjected to force E, its effective area Scan be express in terms of the total area, S, through the total damage variable d,, i.e. ,?-= Sfl -d,),
(9)
156
S. E.-D. F. TAHER
ef al.
Elastic
0
t
--IIl-
-
-1-e
(l-d,)-’ I
+
a
m
+
Plastic
Phstic
Effective rprcc
Physical rpwc
Fig. 3. Generalized
and the effective traction vector nz by the same damage variable
effective stress concept.
= 6 * a) is related to the total traction vector _T(_T= 0 - a)
_T=L
s
E S(1
T
(10)
-c&)=1-
Equations (9) and (10) are consistent with the effective stress concept originated by Rabotnov [SS]. Consequently, eq. (8a) is sufficient for formulating constitutive relations. On the other hand, for a ductile unloading path; d, = 0, 0 = b, # d; A0 need not be equal to &, and uniqueness of the effective stress is lost. However, if for this case A,, = I$,, then f = 6 - 6, - A,‘:a(l/(l
-d,)
- 1) = A,‘:a(d,/(l
-d,))
= d,A,‘:b,
thus 6 = A,/d,(P;:c7P) = a/(1 -d,); therefore uniqueness is achieved since 6 is related to eP. A very important special case arises when d, = d, = dP = d, thus leading definitely to a unique effective stress d = d, = r7,,= a/( 1 - d) and Ju’s formulation [20] is recovered. However, for full representation of both the stress and strain histories, only two of the relations shown in eqs (8) are sufficient. This is achieved by using eqs
Towards a canonical etastoplastic damage model
157
(8) in eq. (3), yielding the form of the dependency relationship amongst the three effective stresses
as 5 6 =-=:Ao:
1 -da
= A,:[E,i:8e+
+$++$-]:B)([ P,‘:(ci, - r&o)].
A,,:($$-:u~)
(11)
In other words, elastoplastic damage follows constraint uncoupling. 6. DEMONSTRATION OF ELASTOPLASTIC DAMAGE UNCOUPLING In this section, existing experimental evidence for both a brittle material (concrete) and a ductile material (copper 99.9%) will be reviewed and interpreted according to the proposed generalized damage concepts. 6.1. Concrete Popovics [59] proposed the following stress-total strain relation for plain concrete subjected to uniaxial compression:
0
0
m-
f&i
5
-= 5”
L
m-l-t-
m’
(12)
;
0 Y where a, and Q are peak stress and strain, respectively, and BZis a parameter dependent on a,. Adopting a value of m = 2 for a, = 17.2MPa (2500 psi), eq. (12) becomes (Fig. 4a)
0 -=5
2;0
Id
(13) t: 2’ ; 0Y In addition, use is also made of a plastic-total strain (c~- &) relation by Kansan and Iirsa [6]: 5”
l+
($=0.145($+0.127@.
(14)
The moduli and damage variable conjugate to the total strain are obtained by making use of eq. (13):
=-A :
and
Ol+?;y=l+?J
tlW
S. E.-D. F. TAHER et al.
(4
1.2 r
1
0
2
4
3
5
6
Normalized total strain (E/E,) (b) 1.2
1.0 2 2
0.8
z r 9 =&
OA
: 5 v1
04. 0.2
rl 0.2
0
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Elastic strain/peak total strain (E&) (c) 1.2
0.6
0.4
0.2
0
1
2
Plastic strain/peak
3
4
5
6
total strain (E#,)
Fig. 4. (a) Stress-total strain relationship for concrete under uniaxial compression. (b) Stress-lastic strain relationship for concrete under uniaxial compression. (c) Stress-plastic strain relationship for concrete under uniaxial compression.
Towards a canonical aIastopl~~c damage model The
159
elastic strain component is ~lc~lated using equs (3) and (141, yielding (Fig. 4b)
The rn~u1~ and dama~ variable conjugate to the elastic strain ~orn~o~ent are
2 =--X~=1.14Ao 0.873 2
OW
and
estimate that & # [email protected] assumption of this lotion (f7a) snbs~~tiates the irn~~at equality is quite common in the literature where A*is construed to be the elastic (young’s) modulus. Similarly, the modub and damage variable conjugate to the plastic strain component {Fig. 4cJ are
=Lxk7.87A 0.127 2
0
and
0.127 [li(~CB127+L145(’ ~xaminatioa of the evolution of damage variables for concrete in impression as well as other auxiliary q~ntiti~ reveals the fo~lo~ng. Onset of plastic defo~ation starts sirn~~~eo~sly with load a~licatio~~ This inclusion is also in accordance with Spooner and Dougill [60]. Elastic strain ~~t~bntion to the total starts at a higher rate than the plastic strain, as shown in Fig. 5. The ~nt~b~tions become equal at a total strain ratio C/C,,= 2.55, corres~nding to a stress ratio a/o, = 0.67, following which the plastic strain contribution becomes greater than the elastic strain. The maximum elastic strain ratio (r&J is 1.30 at c/ty = 3 and a/~~ = 0.6. Su uentiy, the elastic strain decreases and this phenomenon of strain ~tro~~io~ is also
S. E.-D. F. TAHER et al.
160
.’
-
Total strain
- - +-*-..*
Elrstic-damrgo rtrain Plastic-drmago strain
2
.*’ _a
4
3
Total strain/peak
.
.*’ ..* .** .* .*
/
1
f .* .*
6
5
strain (E/E,)
Fig. 5. Strain components for concrete under uniaxial compression.
illustrated in Q - ce space (Fig. 4). In contrast, the elastic strain keeps on decreasing until the total strain is equal to the plastic strain at L/L, = 6, a/a, = 0.32. (3) The total damage variable, da, is a continuous function and its limit as c/cU+NI yields unity while its value at E/C,= 6 is 0.97. The trend of da along with d, and dp is shown in Fig. 6. (4) The elastic damage variable, de, is a singular function at C/L, = 6, but it is continuous in the domain 6/cUE IO,61 though it has an inflection point at L/c. = 3.91. At this value of the total strain ratio, the elastic damage parameter is interpreted to be critical since it should be nondecreasing and this value (d:) is found to be 0.82. The other parameters corresponding to the inflection point of the elastic damage curve are found to be: ce/cu = 1.19, s/c,, = 2.72, a/a, = 0.48, da = 0.93 and d, = 0.98. These values may be considered critical values for complete damage.
Criticrl limit
0.2
i I-/’
0
- - - -
Total Ehstic
-
Plastic
- -
I’ 8’ I 1
I 2
Normalized
I 3
I 4
I 5
I 6
total rtrain
Fig. 6. Evolution of generalized damage variables for concrete u, der uniaxial compression.
Towards a canonical elastoplastic damage model
161
(5) As depicted in Fig. 6, the onset of elastic damage is associated with a threshold total strain ratio c/tu = 0.17 and for which a/a,=0.34, c,lc,=O.14, ~&,=0.03, d,=O.O3 and d,=O.19. At this stress level, the plastic strain as well as the total damage variable are negligible. This result using damage concepts confirms the validity of the assumption of linear elastic behavior for e/a, < 0.3 as used in elastoplastic computational models (e.g. [al]). In such a case, E,, may be reduced to A,, without violation of the generalized damage concepts as presented. However, this is valid only for m = 2, while a higher discrepancy is found for other values of m. (6) The plastic damage variable, 4, is a continuous function and follows an asymptotic trend as defined by lim,,, d, = 1.0. It grows at a much higher rate than both d, and d,. This gives indications that: (a) damage causes more degradation in the plastic stiffness than both the elastic and overall stiffness of concrete; (b) onset of plastic damage commences prior to elastic damage; and (c) linear hardening or perfect plasticity are inappropriate approximations for elasto-plasticity as applied to concrete. (7) A similar phenomenological model predicting damage for concrete in tension can be developed using appropriate experimental data. 6.2. Copper (99.9%) A complete set of experimental data for copper (99.9%) has been provided by Lemaitre [15, 16,621. It is a very ductile material and shows elongation almost equal to its original length at rupture. The elastic strain is negligible when compared to the plastic or total strain. Figure 7a shows the stress-total strain relationship which can be decomposed into stress-elastic strain and stress-plastic strain relationships as shown in Figs 7b and c. Data fitting using a Ramberg-Osgood hardening law of the form a = kcilM (where k and M are coefficients) gives poor correlation. However, if the law is used in the form a = a, + kcj’M (as originally assumed by Ramberg and Osgood [63]), it gives a good fit but not better than a cubic polynomial. Decomposing the total strain into its components, elastic and plastic damage, then monitoring the degradation of the initial moduli, the generalized damage variables are obtained. Figure 8 shows the evolution of the three damage variables d,, d, and d, associated with the total, elastic and plastic stiffness degradation, respectively. The following phenomenological implications are deduced: (1) The total damage variable is zero up to yield (very small domain), then increases abruptly to evolve asymptotically to unity. (2) The plastic damage variable threshold is intermediate between thresholds of other damage variables. (3) The elastic damage variable evolution may be approximated by linear relation as given by Lemaitre [16, 17, 621. (4) At high values of strain near rupture, the elastic stiffness degrades more than the plastic stiffness. This is evidenced by the higher rate of evolution of the elastic damage variable at later stages. (5) Unlike concrete, the total damage variable is not intermediate to other damage variables. This is attributed to the fact that the initial total slope A, is very steep (almost vertical) compared to E, and PO (Figs 7a-c). 7. ON LEMAITRE’S
DUCTILE DAMAGE MODEL
Ju [20] criticized Lemaitre’s ductile damage model and stated: “The fundamental problem of the ductile plastic damage formulation advocated by Lemaitre [15-l 71 is the non-optimal choice of the locally averaged free energy potential. In particular, damage is associated only with elastic strains and the damage energy release rate is shown to be the elastic strain energy expressed by Lemaitre [I 61. This treatment amounts to uncoupled plasticity and damage processes, thus in a sense contradicting experimental evidence that plastic variables also contribute to the initiation and growth of microcracks”.
(4 600 -
Strets - tocal rmin relationship (8ftsr J. Lcmritre, 1985)
100 -& “0 <
0
I
I
I
I
I
I
I
I
I
I
I
10
20
30
40
50
60
70
SO
90
100
110
I 2.2
1 2.4
100
110
Total strain (x100)
(b) Strom - elastic strain relationship
600
500 F
I 0.2
0
I 0.4
I 0.6
I 0.8
I 1.0
I 1.2
I 1.4
I 1.6
I 1.8
I 2.0
Elastic strain (x100)
(cl Smoss - plastic strain relationship
0
10
20
30
40
50
Plastic
60
70
80
90
strain (x100)
Fig. 7. (a) Stress-total strain relationship for copper 99.9% under uniaxial tension (after J. Lemaitre[ 161).(b) Stress-elastic strain relationship for copper 99.9%. (c) Stress-plastic strain relationship for copper 99.9%. 162
163
Towards a canonical elastoplastic damage model
Evolution of dwna~e vuiabla d.
e
0.8
5
0.6
a 2 2 1 n
0.4
9-
/ /-
d
8
0.2 0
..* .*
.*3.’
I’ / 3
0
.* .*
c-.* .-
1
I
10
20
..-a 30
.*-
.*
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- - - - Phrtic
.*.*
40
. . ... .. .
I
I
I
I
I
I
I
50
60
70
80
90
100
110
Total strain (EXP2) Fig. 8. Generalized damage variables for copper 99.9%.
In Lemaitre’s formulation following.
of a damage model for ductile fracture [16], he considered the
(1) The von Mises equivalent stress for plasticity beg= (f S : S)‘j2, with S being the deviatoric stress tensor. hardening law of the special form crc4= kp’l”, p = (@‘: ~p)‘/~;k and M being material parameters; p the equivalent strain and &’ is the Euler-Almansi plastic strain tensor in large deformation theory. (3) Existence of a potential of dissipation ‘p*. (4) Considered only elastic damage and defined it as the degradation in the elastic (Young’s) modulus E, i.e. D in the formulation of Lemaitre [16] is simply 4 in this paper. Lemaitre derived the following rate equation for the evolution of (elastic) damage:
(2) Ramberg-Osgood
where v is Poisson’s ratio, cH is the hydrostatic pressure = 1/3@(a). S, and so are temperature and material dependent parameters. (‘) implies derivative with respect to time. Subsequent to eq. (19), Lemaitre proceeded to use the damage modified form of the Ramberg-Osgood law to express o, in terms of the equivalent strain p. However, by virtue of the approximations made in the model that the damage variable is conjugate to the elastic processes and appears only in the elastic free energy and also that it is found experimentally from degradation of Young’s modulus, it is essential that for consistency, the non-damage modified Ramberg-Osgood relationship be used, yielding for the evolution of damage the rate equation
Equation (20) is different from that derived by Lemaitre by the term (1 - d,)2 appearing in the denominator on the right hand side. Integration of eq. (20) for proportional loading or even uniaxial cases will yield a nonlinear expression for de, which should be linear as shown by Lemaitre [16] from experiments. This indicates that the assumed form for the potential of dissipation cp* may need to be revised.
S. E.-D. F. TAHER ei al.
164
8. E~~NSION
OF NAZARS’ NODAL FOR CONCRETE
Nazars [33] develop a model which id~i~s the behavior of unite as brittle. It required two damage variables in tension and compression, then weighted together to produce a singie damage behavior which was further used in the m~ified form of Hook’s law. Following the same approach with the help of the generalized damage variables and the notion of equivalent strain;
where j = t, c for tension and companion, respectively, and
where the wei~tiug parameters ajdepend on the state of stress with a, = 1 and ac = 0 in pure tension whereas a, = 0 and cr,= 1 in pure compression. For the general case, the sum of a, and a, must always be unity, Finally, the tensorial constitutive equations can be expressed by mo~fying Hook’s law for the three phases of behavior as follows: d = C& = c,, (1 - d&k, where C,, is the conventional initial stress-strain matrix but employing the initial moduli Aa, A’@ and Pl, for k = a, e and p, ~s~tively, and Ed= c. 9. CONCLUSION
The overall ~ha~or of mate~als exhibiti~ isotropic elastoplastic damage processes under various loading conditions cannot be properly chara~ter~d by a single damage variable as has been the adopted practice. The use of the suggested generalize damage variables approach underlines constraint uncoupling between different response phases classified as total, elastic and plastic. The correct expression of the d~ompo~ Helmholtz total free energy is given. Models cast in this form can be readily implemented into the finite element computational framework, with the damage variables being known explicitly in terms of the total strain. In addition, the concept of general~d damage variables as ~re~nted herein allows for unloading along a general degraded path in contrast to the more restrictive cases of either purely brittle or, purely ductile path as per existing models. The practical use of this concept was shown in the modifi~tion of Mazars” model for concrete. ~c~~ow~e~e~nf-~e authors wish to acknowl~ge the support of the ~c~rtment of Civil En~n~~ng, U~versity of Petroleum and Minerals, Dhahran, Saudi Arabia in the pursuit of this work.
[l] J. L. Chaboche, Fracture rn~b~~
King Fahd
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