CONTINUUM DAMAGE-HEALING MECHANICS WITH APPLICATION TO SELF...
Transcript of CONTINUUM DAMAGE-HEALING MECHANICS WITH APPLICATION TO SELF...
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
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CONTINUUM DAMAGE-HEALING MECHANICS
WITH APPLICATION TO SELF-HEALING COMPOSITES
EVER J. BARBERO1 Mechanical and Aerospace Engineering,
West Virginia University, Morgantown,
WV 26506-6106
FABRIZIO GRECO AND PAOLO LONETTI Department of Structural Engineering, University of Calabria,
87036, Arcavacata di Rende, Cosenza, Italy
ABSTRACT: The general behavior of self-healing materials is modeled including both irreversible and healing processes. A constitutive model, based on a continuum thermodynamic framework, is proposed to predict the general response of self-healing materials. The self-healing materials response produces a reduction in size of micro cracks and voids, opposite to damage. The constitutive model, developed in the mesoscale, is based on the proposed Continuum Damage-Healing Mechanics (CDHM) casted in a consistent thermodynamic framework that automatically satisfies the thermodynamic restrictions. The degradation and healing evolution variables are obtained introducing proper dissipation potentials, which are motivated by physically based assumptions. An efficient three-step operator slip algorithm, including healing variables, is discussed in order to accurately integrate the coupled elastoplastic-damage-healing constitutive equations. Material parameters are identified by means of simple and effective analytical procedures. Results are shown in order to demonstrate the numerical modeling of healing behavior for damaged polymeric matrix composite. Healed and not healed cases are discussed in order show the model capability and to describe the main governing characteristics concerning the healed systems evolution. KEYWORDS: continuum damage-healing mechanics, self-healing composites, internal variable methods.
1 * Author to whom the correspondence should be addressed: Email: [email protected]
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1 INTRODUCTION
Structural material behavior is dominated by irreversible processes development, such
as damage and residual strain release, which reduce the structural integrity and service life.
Damage and irreversible deformation phenomena affect the integrity of the material,
by creation and coalescence of micro cracks, fiber breaks, fiber matrix debond, etc. The
evolution of internal defects produces structural degradation, with consequent stiffness and
strength reduction (Dvorak 2000). Once micro-cracks development reaches the critical state,
no further stress redistribution occurs, and the material rapidly reaches the failure condition
(Aboudi 1991, Pindera 1992; Piggott et al. 2000; Herakovich 1999).
During the last decade, modeling of dissipative phenomena have received much
attention and several numerical models have been developed, which describe in various ways
the inelastic response of materials. Continuum theories in a thermodynamic framework
describe material degradation as stiffness and strength reduction by means of microscopic or
macroscopic variables (Murakami 1987; Chaboche 1988; Chow and Wang 1987, Voyiadjis
and Deliktas 2000; Ladeveze and Le Dantec 1992). In particular, elastoplastic theories
describe the slips of the material at micro scale, whereas the Continuum Damage Mechanics
(CDM) provides a macroscopic representation of the micro cracks and voids distribution in
terms of stiffness reduction. Contrarily to dissipative phenomena, recent experimental
observations and procedures have shown the possibility of healing several classes of materials
(Kessler and White 2001; Ando et al. 2002a-b; Brown et al. 2002; Miao et al., 1995). The
healing effects can be caused by chemical, physical or biological phenomena leading to a
progressive reduction of internal material defects. Experimental evidence reveals that
materials can be repaired or healed in various ways and consequently the structure can be
rehabilitated.
A brief literature review reveals that different healing processes have been analyzed,
mainly from a phenomenological point of view, such as geological rocks densification (Miao
and Wang, 1994), autogenous healing of concrete or ceramic materials (Ramm and
Biscoping, 1998; Jacobsen et al., 1996; Jacobsen and Sellevold, 1996; Ando, 2002 a-b),
microcracks skeleton regeneration in biological system and so on. Numerical modeling of
these processes has not been sufficiently investigated. Different models related to biological
healing behavior for bone remodeling or wound skin regeneration have been developed for
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relatively simple cases (Adam, 1999; Simpson, 2000), but to the author’s knowledge, only a
constitutive model for compaction of crushed rock salt has been proposed in a rigorous
thermodynamic framework (Miao et. 1995).
Recently, a novel material processing technique was reported by (White et al. 2001;
Kessler and White 2001; Brown, et al.; 2002), which describes the ability of composite
materials with polymeric matrix to heal autonomically. Such materials are able to reduce
material degradation with the aid of a healing agent by means of chemical interactions.
Healing processes can be considered opposite to damage. Consequently, the system can be
rehabilitated and the integrity of the material is recovered to a certain extent.
In polymer-matrix composites, the autonomic healing procedure occurs as follows.
Healing agents stored in microcapsules are uniformly dispersed in the matrix material. Once a
crack breaks a microcapsule, the healing agent is released and distributed by capillary action.
The healing agent then contacts the catalyst, which is uniformly distributed in the matrix, and
adhesive bonding takes place. The efficiency of the repair depends on the catalyst and the
microcapsules density. Therefore, the microcraks evolution and the degradation processes can
be controlled and, consequently, the material is self-repaired. More details about the
technique and material characteristic of the healing agent can be found in the literature (White
et al. 2001; Kessler and White 2001; Brown et al., 2002, Barbero and Lonetti 2003, Barbero
et al. 2004).
The main purpose of the present paper is to generalize CDM including healing
processes and consequently Continuum Damage-Healing Mechanics (CDHM) is proposed.
The model is developed in a consistent thermodynamic framework and it is based on the
method of internal variables. The proposed constitutive model is quite general and capable of
simulating different healing processes. The constitutive equations are obtained by a
phenomenological thermodynamic approach using the method of internal variables. Then, an
application to composite healing behavior is proposed and a numerical model is developed to
predict damage and irreversible deformation processes for a self-healing fiber-reinforced
lamina.
Damage and inelastic mechanisms have been discussed previously and experimentally
validated (Barbero and DeVivo 2001, Barbero & Lonetti 2001, 2002; Lonetti et. al. 2003),
whereas the coupled healing-damage and irreversible deformations constitutive model is the
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main contribution of this paper. The proposed model predicts the distributed damage and the
unrecoverable deformation in a mesoscale lamina representation, which refers to a single
lamina. The orthotropic nature of composite lamina leads to a tensorial description of the
healing variables, which control void and micro cracks healing along different directions. An
effective damaged-healed configuration is introduced, in which the body is considered
without discontinuities. In order to describe the internal variable evolution, different
dissipation potentials for damage, healing, and irreversible deformations (plasticity, residual
strain recovery, etc.), are introduced. Coupling among various dissipation phenomena could
be further refined as shown by Abu Al-Rub and Voyiadjis (2003). In the context of convex
analysis, and supported by experimental observations, different limit thresholds related to
damage, plasticity, and healing domains are derived. One basic assumption of the model,
motivated by experimental observations, is that the healing agent acts when a sufficiently
large micro crack density affects the material. In the model this occurs when the damage
domain reaches the critical surface and at the same time the healing thermodynamic forces
reach the healing domain. Healing is tracked by means of a thermodynamic potential that
describes the evolution of healing agent.
The constitutive relationships and evolution equations define a non-linear differential
problem, which is solved by means of a proper numerical algorithm. The main equations are
integrated by Euler-backward technique, which is a stable numerical procedure to determine
the actual solution by an incremental/iterative method. In particular, an elastic-predictor and
Damage-Healing-Plasticity corrector integration scheme is used to solve the incremental non-
linear constitutive equations.
Damage and plasticity potentials are identified by means of simple but effective
procedures described in Barbero and DeVivo (2001), Barbero and Lonetti (2001, 2002),
Lonetti et al. (2003). These are based on available data, which can easily obtained by standard
experimental procedures. In addition, a similar identification procedure is proposed here to
identify the healing potential. Due to lack of experimental data, the effect of allowable healing
values is investigated by conducting a parametric study. A sensitivity analysis in terms of
healing variables is presented in order to show the suitability of the model to predict
mechanical behavior of self-healing material systems. Results are also shown in order to
validate the numerical model with available experimental data for damaged polymeric matrix
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composite. Healed and not healed cases are discussed in order show the capability of model to
describe the possible evolution of the self-healing composite system.
2 THERMODYNAMIC FORMULATION
The proposed formulation is based on generalized thermodynamics (Coleman and
Gurtin, 1967; Lubliner, 1972), in which internal variables are introduced in the
thermodynamic constitutive relationships to describe the inelastic processes at the current
material state. The constitutive equations are thermodynamically consistent with the Clausius-
Duhem inequality
( ): 0sT TT
ρ ψ− + − ⋅∇ ≥&& &σ ε q (1)
where σ and ε are the stress and strain tensors, , , , Tsρ ψ and q are mass density,
specific Helmholtz free energy (HFE), specific entropy, temperature, and heat flux,
respectively. A purely mechanical theory and infinitesimal deformations are assumed in the
proposed model. Without loss of generality the additive strain decomposition is considered
consistently with the small deformation hypotheses, by which the total deformation strain is a
linear function of both elastic and plastic term, i.e. e pε ε ε= + . Moreover, a set of internal
variables are introduced to describe both degradation and healing effects
( ) %( ) ( ), , , , , ,pd d d p p p h h hpδ µ= ∈ ℑ = ∈ ℑ = ∈ ℑϕ ϕ ϕ ϕ ε ϕ ϕD H (2)
where ( ), ,pD Hε and ( ), ,pδ µ are the tensorial and scalar variables related to
damage (d), plasticity (p) and healing (h) mechanisms and iℑ with i=(d, p, h) are the
corresponding domain spaces. It is worth noting that chemical effects related to the healing
processes are only introduced from a mesoscopic point of view by mean of internal variables,
which describe the stiffness variation during the evolution phenomena without considering
any diffusion processes.
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The actual thermodynamic state can be described by the HFE,
: d p hψ × × × →C ℑ ℑ ℑ , which is a function of both observable and internal variables
( ), , ,e d p hψ ψ= ε ϕ ϕ ϕ (3)
with e ∈ε C being the admissible elastic deformation set. Substituting Eq.(3) into Eq.(1) and
introducing the thermodynamic associated driving forces, the following constitutive equations
hold
, , , .d p he d p h
ψ ψ ψ ψρ ρ ρ ρ∂ ∂ ∂ ∂= = − = − =
∂ ∂ ∂ ∂σ
ε ϕ ϕ ϕV V V (4)
with ( ) ( ) ( ), , , and ,d d D p p H H HRγ σ φ= = =V V Y V V V V Y , whereσ is the stress in the
effective configuration and , D HY Y are the thermodynamic forces related to damage and
healing, respectively. The dissipation potential Ξ is a positive defined function
mechanical dissipation
0d d p p h hΞ = + − ≥/& & &: ϕ : ϕ : ϕV V V (5)
where the minus sign of the hV term is due to the un-dissipative nature of the healing
process.”
The Helmholtz Free Energy is assumed to be separable as follows
( ) ( ) ( ), , , , , , , ,e d p h e p D H pψ ψ ε δ µ= = Φ + Πϕ ϕ ϕ ε ε (6)
where :Φ × × × →d h pC ℑ ℑ ℑ is the elastic deformation function, which depends on both
damage and healing tensor components; whereas :Π → is a scalar function, which
expresses the evolution of inelastic variables. The functional Φ depends on the actual value
of the internal variables
( ) ( ) ( ) ( )1, , , : , :2
e p p pΦ = − −ε ε ε ε ε εD H E D H (7)
where E is the fourth order damaged-healed stiffness tensor. The irreversible nature of healing
and damage processes leads to a monotonically increasing evolution function. Without loss of
generality, they are expressed in the following uncoupled forms
( ) ( ) ( ) ( ), , d p hp pδ µ δ µΠ = Π + Π + Π (8)
in which
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( ) ( )
( )
11
0 0
11
00
d1 1 2 1 2
2
p 2 p p1 1
= exp , , ,
1= c p , p ,c2
d d d d d dd
ppp
pp
d c c c c cc
p dpp
δδ
δ δ
ψ δδ δ δ δδ
ψ +
∂ Π = − ⋅ − ∈ ℑ ∈ ∂
∂Π = − ⋅ ∈ ℑ ∈
∂
∫
∫ (9)
The damage and plasticity potentials and d pΠ Π described by Barbero and Lonetti
(2001, 2002), Lonetti et al. (2003), are used in order to obtain a good correspondence between
numerical and experimental data. Next, in lack of experimental data, we assume a healing
potential similar to the damage potential but in the corresponding thermodynamic space
( )11
0 0
H h h1 2 1 2
2= exp , , c ,c ,h h h
hd c c c
µµ
µ µ
ψ µµ µ µ µµ
∂ Π = − ⋅ − ∈ ℑ ∈ ∂ ∫ (10)
This expression is motivated by intrinsic conditions of the phenomenon, in accordance with
the experimental evidence, which shows that for the self healing composite the process is
basically primed by the damage evolution. Future developments are needed in order to
completely validate Eq. (10).
The material constants ( )1 2 1 21, , , ,pd d h hc c c c c introduced in Eqs. (9)-(10) are identified in
terms of available data as shown in Section 6. The complementary laws related to dissipation
process can be expressed by homogeneous and convex potentials in terms of the associate
thermodynamic forces (Hansen and Schreyer, 1994)
( ) ( ) ( ) ( ), ,V V V V V Vd p h d d p p h h∗ ∗ ∗ ∗ϒ = ϒ + ϒ + ϒ (11)
The principle of maximum dissipation in the effective reference frame defines an
equilibrium thermodynamic solution, which corresponds to a constrained optimization
problem. The Lagrangian multiplier method (LMM) can be used to solve the problem, where
the functional
( ) ( ) ( ) ( )V V V* * *& & &p d hP D Hλ λ λ∏ = − Ξ + ϒ + ϒ + ϒP D H (12)
depends on ( ), , p d hλ λ λ& & & which are the Lagrangian multipliers related to plasticity, damage,
and healing, respectively. In order to extremize ∏ , the following necessary conditions must
be satisfied
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0, 0, 0D P H
∂∏ ∂∏ ∂∏= = =
∂ ∂ ∂V V V (13)
which correspond to plastic strain rate, damage, and healing evolution laws. The kinematic
internal variables grow along the direction normal to the corresponding potential surface * * *
, , , P D HV V V& & && & &pp P D D H HD Hϕ λ ϕ λ ϕ λ
∂ϒ ∂ϒ ∂ϒ= = =
∂ ∂ ∂ (14)
3 DAMAGE AND HEALING REPRESENTATION
When distributed damage controls the mechanical behavior, many materials, including
polymer matrix composites, usually exhibit a quasi brittle macroscopic behavior. For
example, experimental observations on polymer-matrix composite prior to failure show a
continuous distribution of microcracks in the matrix. During loading, the total energy of the
system is dissipated mainly into new surface formation, whereas a minor fraction is used to
nucleate existing microcracks.
The healing behavior is generated by distributed agents, which can be assumed to be
uniformly distributed in the body. For example, in composite materials the microencapsulated
healing agent is embedded along with a catalyst into a polymeric matrix. The healing occurs
when the crack path in the matrix reaches the microcapsule and, by rupture, the healing agent
locally triggers a chemical polymerization reaction, leading to crack healing. Since the
healing agent is continuously distributed, it can be analytically described by a continuous
function. The same model can be applied to different materials such as crushed rock salt or
bone.
Damage and healing phenomena are described at the mesoscale by internal state
variables, which represent microcracks formation and healing, respectively. Two sets of
tensor and scalar valued variables are introduced in the constitutive equations. In particular, D
and H tensors describe the area change produced by microcracks and healing evolution,
respectively, and scalar valued variables δ and µ control the evolution phenomena. Lacking
experimental observations to justify a more complex behavior, the evolution of the damage
and healing surfaces are assumed to be isotropic.
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In composite materials, microcraks and voids have preferential growth directions
which coincide with the material directions. Therefore the principal directions of the damage
and healing tensors are assumed to coincide with the material coordinates. In the principal
reference frame, they are expressed by the following equations
3 3
1 1
i i i i i ii i
d h= =
= Ä = Äå åD n n H n n (15)
where Ä represents the dyadic product; whereas di, hi and ni are the eigenvalues and the
eigenvectors of the tensors D and H, respectively.
Damage and healing tensors represent the net area change due to three-dimensional
voids and micro-cracks distributions developed during the loading history. In the context of
continuum mechanics, physical interpretation of damage and healing can be shown in Fig. 1.
A representative volume element of arbitrary orientation is shown in different configurations
(C0, CDH, FC ) initial, actual (damaged-healed), and effective, respectively. Moreover, FC∗
and DHC∗ represent the effective and damage-healed configurations free of elastic
deformation, respectively. In Fig. 1, Fe represents the elastic deformation gradient. The
deformation gradients DHχ and ∗DHχ describe the following transformations: : FDH DHC C→χ
and * : FDH DHC C∗∗ →χ , where DHχ and *
DHχ have the eigenvectors coinciding with in and i∗n ,
respectively. The deformation of an arbitrary segment dxi to idx between damaged-healed
and effective configurations is expressed by introducing a transformation tensor as
i DH= idx dxχ , with i=1,2,3 (16)
From Nanson’s Theorem (Ogden 1983) specialised to the principal reference system in , a
generic area element is transformed by the following equations
% ( ) ( ) [ ]( ) ( )1/ 2 1/ 21 1 det2 2
T
DH DH DH DH DHdS dS− −= × = ⋅ × ⋅ =n dx dy dx dy nχ χ χ χ χ (17)
where ( )× denotes vector product. The area reduction along the principal directions can be
expressed in terms of the eigenvalues of H and D tensors as
( )( ) %1 1 i ii i i id h dS dS− + = n n with i=1,2,3 (18)
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Here di and hi are the eigenvalues of the damage and healing tensors along different
planes and define the net area change due to degradation or healing phenomena. From an
irreversible thermodynamic point of view, the evolution is based on a positive unilateral
variation. From Eq.(17) and (18), the transformation tensor DHχ in the principal reference
frame assumes the following expression
( )( )( )( )( )( )
( )( )( )( )( )( )
( )( )( )( )( )( )
11
22
33
2 2 3 3
1 1
1 1 3 3
2 2
1 1 2 2
3 3
1 1 1 11 1
1 1 1 11 1
1 1 1 11 1
DH
DH
DH
d h d hd h
d h d hd h
d h d hd h
χ
χ
χ
− + − +=
− +
− + − +=
− +
− + − +=
− +
(19)
The effective stress σ represents the stress associated to FC by the same loading
related to the CDH configuration and corresponds to a fictitious first Piola-Kirchhoff tensor
referred to FC :
[ ] 1det 1/2 1/2−= =σ χ χ σ χ σDH DH DH Μ -1 (20) where M is the effective damage tensor and corresponds to the stress tensor transformation
between CF and CDH . In view of Eq. (19) and Eq.(20), M is a diagonal fourth order tensor
2 3 2 3 1 3 1 3 1 2 1 21 1 2 2 3 3; ; ; ; ;
2 2 2
D D H H D D H H D D H HD H D H D Hdiag
Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω = Ω Ω Ω Ω Ω Ω
M (21)
with
( ) ( )1 , 1 ,D Hi i i id hΩ = − Ω = − i=1,2,3 (22)
where σ is represented in contracted ( Barbero,1999), whereas the effective stress is
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( ) ( )( )
( ) ( )( )
( ) ( )( )
( ) [ ] [ ]
( ) [ ] [ ]
( )
11
22
33
11 22
11 33
1111 11
1 1
2222 22
2 2
3333 33
3 3
1 1 122 212 121 2 1 2
11 132213 131 3 1 3
23
1det 1 1
1det 1 1
1det 1 1
1det 1 1 1 1
1det 1 1 1 1
1det
DHDH
DHDH
DHDH
DH DHDH
DH DHDH
DH
d h
d h
d h
d d h h
d d h h
σσ χ σ
σσ χ σ
σσ χ σ
σσ χ σ χ
σσ χ σ χ
σ
= =− +
= =− +
= =− +
= =− − + +
= =− − + +
=
χ
χ
χ
χ
χ
χ [ ] [ ]22 33
11 2322 232 3 2 31 1 1 1DH DH d d h h
σχ σ χ =− − + +
(23)
In view of Eq.(21), and according with the Principle of Equivalent Elastic Energy
(Cordebois & Sidoroff 1977), the stiffness tensor is defined by the following expression
( , ) : : T=E D H M E M (24)
or in components form:
( )( )
( )
211 12 131 1 1 2 1 2 1 3 1 3
222 232 2 2 3 2 3
233 3 3
44 2 3 2 3
55 1 3 1 3
66 1 2 1 2
, , 1,3
0 02
0 0 , , 4,62
0 02
0
D H D D H H D D H H
D H D D H Hij
D H
D D H H
D D H H
ij
D D H H
ij
E E E
E E E i j
E
E
EE i j
E
E
Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω = Ω Ω Ω Ω Ω Ω =
Ω Ω Ω Ω Ω Ω
Ω Ω Ω Ω = =
Ω Ω Ω Ω
= 1,3 4,6 or 4,6 1,3 i and j i and j= = = =
(25)
4 CONSTITUTIVE EQUATIONS
The internal variables used in the thermodynamic constitutive equations are listed in
Table 1 together with their associated driving forces. The Helmholtz Free Energy potential is
expressed by the following expression
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( ) ( ) 1 1 22
21 2 11
2
1 : : exp2
1 exp , 2
p p d d dd
p h h hh
c c cc
c p c c cc
δψ δ
µ µ
= − − + − +
+ + −
ε ε ε εE
(26)
whereas the thermodynamic forces are defined by Eq.(4)
( ): pp
ψρ ∂= − = −
∂σ ε ε
εE (27)
= :σ σ-1M (28)
( ) ( )1 : :2
p pψρ ∂ ∂ = − = − − − ∂ ∂ D EY
D Dε ε ε ε (29)
( ) ( )ψρ ∂ ∂ = = − − ∂ ∂
1 : :2
p pEYH H
H ε ε ε ε (30)
1pR c p
pψρ ∂
= − = −∂
(31)
12
exp 1ddc c
ψ δγ ρδ
∂ = − = ⋅ − ∂ (32)
12
exp 1hhc c
ψ µφ ρµ
∂ = = ⋅ − ∂ (33)
The healing thermodynamic forces concept can be illustrated for a simple tensile stress
cycle by considering the balance of dissipated energy. A generic stress-strain curve is shown
in Fig. 2, in which damage and healing are assumed to increase with increasing stress. The
total energy dissipated during a cycle is the area 0ABCE . The recovery energy due to healing
effects and the dissipation energy due to damage and plasticity are defined by CDE , BDE
and OAFE areas, respectively. The healing phenomena is generated by internal energy
production, which is obtained by spending chemical energy stored in the healing agent or
provided externally by a sinterization process or biological bone repair. The healing process
increases the stiffness of the material. In view of Eq.(5), for a uniaxial stress state, damage
and healing energies are dissipated during BCD and CD paths respectively and they are
mathematically written as
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( )
( ) ( )
0
0
dD D
2h=0
hH H
2d=const
1 1Y dD C2 2 1- d
C 1 1Y dH =21- d 2 1- h
σ
σ
Ξ = + ≥
Ξ = − ≤
∫
∫
110
11
0
= -2
2
2
( 34)
Eqs. ( 34) represent areas BCDE and DCE in which uncoupled damage and healing growth
is assumed. Damage energy production DΞ describes material degradation and it is a strictly
positive definite function. The healing phenomena generate internal energy production, which
is opposite to the damage dissipation.
In view of Eqs. (27)-(33) under the hypothesis of decoupling between different
processes and according with the Clausius-Duhem inequality, the thermodynamic dissipation
function has to be necessarily positive
%:
: 0
: 0
pP
D d
H h- -
σ ε
γ δ
φ µ
Ξ = ⋅ ≥
Ξ = ⋅ ≥
Ξ ⋅ ≤
& &
&&
& &
+ R p 0
Y D +
= Y H
(35)
where , and P D HΞ Ξ Ξ are the dissipation functions related to plasticity, damage, and healing
processes, respectively, with P D HΞ = Ξ + Ξ + Ξ being the total dissipation function. The total
dissipation is always positive because healing phenomena are activated only when the
magnitude of the microcracks distribution is significant (White et al. 2001) and the efficiency
of the healing mechanism less than 100%. A 100% healing efficiency would correspond to
perfect healing, where all damage dissipation would be recovered (i.e. D HΞ = Ξ ).
According to the method of local state, the evolution laws can be derived from
dissipation potentials, whose existence is postulated a priori. Damage and plasticity potentials
previously proposed by Barbero and DeVivo (2001), Barbero and Lonetti (2001, 2002),
Lonetti et al. (2003), have shown good correspondence between experimental data and
numerical results. These are
( ) ( )
( )( )
12
0
2 21 2 1 2 1 21 2 11 22 12
2 2 24 5 644 55 66 0
: :
( ) 2
d D D D
d p
f
f g f f f f f
f f f R p R
γ δ γ
σ σ σ σ σ σ
σ σ σ
= − −
= = + + + + +
+ + + − −
σ σ
Y J Y
(36)
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where 0γ and 0R are the plasticity and damage thresholds, DJ is a fourth order damage
characteristic tensor and if are material parameters. For healing, an evolution potential
similar to the damage one is proposed
( ) ( )1
20: :H H H Hf φ µ φ= − −Y J Y (37)
where 0φ is the healing threshold and HJ is a fourth order healing tensor. The previous
assumptions will be clarified Section 6. From Eq.(14), the evolution vectors are assumed to
develop along the normal direction of the corresponding potential surface
( )
%
11 1,
22 2
33 3
1 2 6 511 1 12 66 55
,2 1 422 2 12 44
11
0 0: 1 0 0:
0 0
1
2 2 2 2
2 2 20
d
p
D DD Dd d d d d
D DVD D D d
D D
p p p pp V
J Yf
J YJ Y
f f f f fpg
f f f fsym
δλ λ λ
σ σ σ σλ λ
σ σ σ
− − = = ∇ = = ⋅ ∆ −
+ + = = ∇ =
+ +
&& & &&
&
&& &&
&
J YD
Y J Yϕ
ϕε
( )
( ) ( ) ( )
11 1,
22 2
33 3
2 2 2, , , , , , ,11 1 22 2 33 3
11
0 0: 1 0 0:
0 0
with
H
H HH HH H H H H
H HVH H H H
H H
D H D H D H D H D H D H D H
J Yf
J YJ Y
J Y J Y J Y
µλ λ λ
− − = = ∇ = = ⋅ ∆
∆ = + +
& & & &&&
J YH
Y J Yϕ
(38)
4.1 INELASTIC AND HEALING DOMAIN
Next, we briefly summarize the main expressions from previous plasticity and damage
formulations (Barbero and DeVivo 2000, Barbero and Lonetti 2001, 2002; Lonetti et al.
2003). Damage and plasticity were based on experimental observations of acoustic emissions,
which indicate marked damage and plasticity thresholds (Liu et al. 1997; Gong et al. 2000).
The initial threshold values are represented by γ 0 and 0R for damage and plasticity,
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
15
respectively. An anisotropic damage criterion for polymeric composite materials is written in
terms of tensorial parameters
( ) ( ) ( )11 22d
0g γ δ γ= + ⋅ − −D D D D DY : J :Y H Y (39)
where, :g +′ →d dℑ . Substituting Eqs. (29) and (27) in Eq. (39), the damage domain in the
stress space has the same shape of the Tsai-Wu surface, which is a widely accepted failure
surface (Barbero 1999). The procedure for identification of the DJ and DH tensors is based on
comparison of Eqs. (39) and (32) with the Tsai-Wu surface. The plasticity domain is identical
to the plasticity potential (second of Eq. (36)), but written in the effective Damage-Healing
configuration in order to recover the coupling between different modes (damage, healing,
plasticity). In Eq. (36), ( )R p is the isotropic evolution function and fi are material parameters
that depend on experimental values obtained from testing a single composite lamina.
Analogously to damage processes, a healing domain is introduced that is similar in expression
to the damage one, but written in different thermodynamic force space
( ) ( ) ( )11 22
0: :H H H H H Hg φ µ φ= + ⋅ − −Y J Y H Y (40)
where HJ and HH are tensor valued variables that define the healing shape surface and φ (µ)
is the healing evolution function (Eq. (33)). The healing surface is motivated by experimental
evidence. In particular, healing phenomena start when significant micro cracks distribution is
observed. Subsequently, the material is rehabilitated with a finite efficiency depending on
microcapsule and catalyst density. Therefore, 0φ controls the beginning of healing, and the
healing surface defines the limit space related to possible healing production. Moreover, the
analogy with the gd surface is motivated by experimental observations that show how healing
phenomena depend on micro cracks and voids distribution. Therefore, the proposed model
predicts healing evolution by introducing a healing surface, which is obviously similar to the
damage surface. Healing processes start at those points in which a considerable damage
values is observed. In the model this occurs when the healing thermodynamic forces reach
the corresponding healing surface.
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
16
4.2 EVOLUTION EQUATIONS
The kinematic evolution laws are derived using the principle of maximum dissipation
in a consistent and generalized thermodynamic approach. The solution is obtained by solving
a non-linear system equations using an incremental, iterative process. For a generic
thermodynamic state, the Kuhn-Tucker Optimality (KTO) conditions must be satisfied
( ) ( ) ( ) ( ) ( ) ( )0, 0, 0D D D D P P P P H H H Hg dg g dg g dg= = = = = =V V V V V V (41)
Substituting Eqs.(27)-(33) and (38) in (41), a system of linear equations in the
unknown quantities , and d p hl l l& & & is obtained
[ ] [ ]
11 12 13 11
21 22 23 22
31 32 33 33
, ,
d
p
h
a a a ba a a ba a a b
λλλ
+ = = = =
0,
&
&
&
λA b
A bl (42)
with
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
17
°
°
° °
11 ,,
112 ,
13 ,
11
121 ,
1 122 ,
,
,
,
[ ] ,
D
H
D
d D dd d
D
d Dp
D p
d Dh
D
d D
D
pd
pp
p
g ga f f
ga g
ga f
gb d
ga f
ga g
Y
Y
Y
YDY
Y MY
YY
YY
MD
M M σ
ε
Η
εε
σσ
σσ
gg
g d
e
-
-
- -
æ ö¶ ¶ ¶ ¶ ÷ç= Ñ + Ñ ÷ç ÷÷çç ¶ ¶ ¶¶è øæ ö¶ ¶ ÷ç= Ñ ÷ç ÷÷çç¶ ¶è øæ ö¶ ¶ ÷ç= Ñ ÷ç ÷÷çç ¶¶è øæ ö¶ ¶ ÷ç= ÷ç ÷÷çç ¶¶è ø
¶ ¶= Ѷ¶
¶ ¶= Ñ +¶¶
s
°
°
°
,
123 ,
122
33 ,,
132 ,
31 ,
33
,
[ ] ,
,
,
,
,
H
H
D
pp
R
ph
p
h H hh h
H
h Hp
H p
h Hd
H
h
g R gR p
ga f
gb d
g ga f f
ga g
ga f
gb
Y
Y
Y
MH
M
YHY
Y MY
YDY
Y
σ
σσ
σ εεσ
ε
ff
f m
-
-
-
æ ö¶ ¶ ÷ç Ñ ÷ç ÷÷çç ¶ ¶è ø
¶ ¶= Ѷ¶
¶ ¶=¶¶
æ ö¶ ¶ ¶ ¶ ÷ç= Ñ + Ñ ÷ç ÷÷çç ¶ ¶ ¶¶è øæ ö¶ ¶ ÷ç= Ñ ÷ç ÷÷çç¶ ¶è øæ ö¶ ¶ ÷ç= Ñ ÷ç ÷÷çç ¶¶è ø
¶=¶
.H
H dY εε
æ ö¶ ÷ç ÷ç ÷÷çç ¶è ø
(43)
Moreover, substituting Eqs. (43) into Eq. (42), it is possible to derive the incremental
relationships
[ ] [ ], ,
1 1, ,
, ,
D D
H H
d D
Dd dd dp
p p p p
h hd hh H
H
g
f fdgdiag g d diag g A d d
df f g
λλ ελ
− −
∂ ∂ ∂ ∂ ∇ ∇ ∂ ∂ = ∇ = ∇ = ∂∂ ∇ ∇ ∂ ∂ ∂ ∂
Y Yp
Y Y
YYdD
dε MdH
YY
σ σ
εσ εεσ
ε
ℵℵℵ
(44)
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
18
where iℵ , with j=d,p,h, represent the stiffness tensor contributions related to damage,
plasticity, and healing mechanisms. Using Eqs. (24), (27), (28), the incremental stress in the
effective configuration can be expressed in the following form 1
1 1 1: : : : : :−
− − − = + = + + σ σ σ σ σd dd d d d d d
d dM MM M D H MD H
, (45)
where
% %( ):= −σ ε εp
E , : :d dd d dd d
= +M MM D HD H
. (46)
Substituting Eqs. (46) and (24) in Eq.(45), the incremental stress-strain relationship in
the actual configuration is written as
:edphTd d=σ εE (47)
with
( )1 1
: :
: : : : : : : : :
epdh ep dhT T T
ep pT
elasto plasticity
dh d h d hT
damage healing
d d d dd d d d
−
− −
−
= +
= −
= + − +
E E E
E M E M I
M M M ME M E M D HD H D H
ℵ
ℵ ℵ ℵ ℵ
(48)
where E epT and Edh
T represent the elastoplastic and healing-damage contributions.
5 INTEGRATION PROCEDURE
The solution is obtained by an incremental-iterative procedure based on a return-
mapping algorithm (Ju 1989; Luccioni et al. 1996; Crisfield 1991). In particular, a predictor-
corrector scheme is used. The initial deformation increment is considered perfectly elastic or
elastic-damaged, so that the stress variation is a function of the initial elastic-damaged
stiffness tensor, as 1i i eTE−= + ∆σ σ ε . The total deformation increment is divided into an
elastic and a plastic terms. Subsequently, the stiffness matrix is transformed by degradation
and/or healing variables. The solution at step n+1 is subjected to the following evolution
restrictions for damage, plasticity and healing effects
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19
( ) ( )( ) ( )
( )
1 1 1 1
1 11 1
1 1 1
0 , 0 , 0
0 , 0 , 0
0 , 0
λ λ γ γ
λ λ
λ λ φ
+ + + +
+ ++ +
+ + +
≥ ⋅ ≤ ≤
≥ ⋅ ≤ ≤
≥ ⋅ ≤
& &
& &
& &
σ σ
d d d D d Dn n n n
p p p pn nn n
h h h H h Hn n n
g g
g R g R
g g
Y Y
Y Y( )1, 0φ + ≤n
(49)
which correspond to Kuhn-Tucker Optimality conditions. The initial Lagrangian Multipliers
solution obtained by Eq. (42) corresponds to a thermodynamic state that does not necessarily
satisfy Eq. (49). Therefore, an iterative procedure is needed to solve the non-linear problem.
Using the constitutive equations, the surface domain at the i+1-th iteration can be expressed
to the first order by Taylor expansion and the nonlinear system is reduced to the following
linearized equations
1
0λλλ
+
∆ ∇ ∆ + = ∆
d d d
d p p
h h h
i
g gg gg g
(50)
in which iλ∆ are the unknown quantities (see Appendix I). Between the n and n+1 steps, the
kinematic and thermodynamic forces are updated by the following incremental relationships
++ += + 11 1
k kn n nσ σ ∆σ
( )d h
d hg f fλ λ∆ λ
+− −
+ − ∂ ∂ ∂ ∂ ∂ = − ∆ + ∆ + ∆ − ∂ ∂ ∂ ∂∂
σ ε εσ
M E M EE M
D D H H
11 1
1 1
kp
k p pn
n
11
1 1
kdk k d
n n n nn
fλ+
++ +
∂= + ∆ = + ∆
∂D D D D
D
( )1n+1 n+1 n+1σ −= M D σ
% % % %1
( ) ( 1) ( )1 1 1 1 λε ε ∆ε ε
σ
kpp p k p k p k pn n n n
n
g+
++ + + +
∂= + = + ∆
∂
11
1 1 λ+
++ +
∂= + ∆ = + ∆
∂
khk k h
n n n nn
fH H H HH
11 1 1 1
k k k hn n n nµ µ µ µ λ+
+ + + += + ∆ = + ∆
11 1 1 1
k k k dn n n nδ δ δ δ λ+
+ + + += + ∆ = + ∆
11 1 1 1
k k p pn n n np p p p λ+
+ + + += + ∆ = + ∆ (51)
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20
6 IDENTIFICATION OF MATERIAL PARAMETERS
The material parameters in the constitutive equations are determined in terms of
experimentally observed material behavior. The identification is done by solving a non-linear
system of equations obtained by comparing the healing domain (Eq.(40)) and a classical Tsai-
Wu surface (Barbero, 1999) in stress space. The identification of the healing parameters is
shown here, whereas damage and plasticity parameter identification is described by Lonetti et
al. (2003), Barbero and Lonetti (2001, 2002).
As shown by experimental observations, healing starts only when a significant
microcrack distribution occurs in the matrix. Moreover, it is well known that healing
processes are generated by microcraks evolution. Therefore, the basic idea is to assume a
healing surface similar in expression to the damage one. The healing potentials described by
Eqs. (40) and (37) involve two characteristic tensors JH and HH that define the domain shape
and the evolution of kinematic variables Eq. (33). The healing scalar function φ represents
isotropic growth of the healing domain, where the scalar 0φ corresponds to the initial healing
threshold. Damage and healing surfaces in the corresponding thermodynamic forces spaces
are shown in Fig. 3, with healing and damage hardening given by ( )0 0, φ φ γ γ+ + .
Identification of the characteristic healing tensors is provided by the following non-
linear system of equations. Considering a uniaxial ultimate stress state for
tension/compression loading ( )1 1 ,t cFσ = and substituting Eq. (30) in (40), with 1
C E−
= the
following equations holds
( )( ) ( ) ( ) ( )
12
1 11 112 2211 1 11 12 3 2 3
1 1 1 1
11 1 1 1
H Hi j
j j j j
C CJ F H Fd h d h
+ ⋅ = − + − +
, j=t,c (52)
in which ( )φ φ+ =0 1 is assumed to match the Tsai-Wu criterion at failure, t and c stand for
tension and compression, respectively, ( )1 1,j jd h represent the damage and healing parameters
at failure. Equation (52) represents a non-linear system from which 11 11 and H HJ H are
determined in terms of experimental data ( )1 1 1, ,j jF d h with j=t,c.
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21
Analogously, for transverse tension perpendicular to the fiber orientation, the following
equation holds at failure load ( )2 2tFσ =
( )( ) ( ) ( ) ( )
12
1 22 222 2222 2 2 22 3 2 3
2 2 2 2
11 1 1 1
t tt t t t
C CJ F H Fd h d h
+ ⋅ = − + − +
(53)
which provide a relationship between the transverse components 22 22 and H HJ H of the
characteristic healing tensor. Along the in-plane and out-of-plane shear directions
( )4 4 5 5 6 6, ,F F Fσ σ σ= = = , the healing surface projected at failure into the stress space leads
to the following equations
( ) ( )
( ) ( )
112
22 266 666 611 22 1 2
2 21 2 1 2 1 2 1 2 1 21 2
112
22 255 5533 5 3 511 1
2 21 3 1 3 1 3 1 3 1 31 3
1H H H H
D D H H H H D D H HH Hs s s s s s s s s ss s
H HH H
D D H H H H D D H HH Hs s s s s s s s s ss s
C F C FJ J H H
J C F H C FJ H
+ + + = Ω Ω Ω Ω Ω Ω Ω Ω Ω ΩΩ Ω
+ + + Ω Ω Ω Ω Ω Ω Ω Ω Ω ΩΩ Ω
( ) ( )
112
22 244 4433 322 4 2 4
2 22 3 2 3 2 3 2 3 2 32 3
1
1H HH H
D D H H H H D D H HH Hs s s s s s s s s ss s
J HJ C F H C F
=
+ + + = Ω Ω Ω Ω Ω Ω Ω Ω Ω ΩΩ Ω
(54)
in which the components of the integrity tensor are ( ) ( )1 and 1H Djs js is jsh dΩ = − Ω = − with
j=1,3. Here djs, hjs, represent the damage and healing at shear failure. Since shear strength are
independent of the shear stress. Eqs. (54) have to be sign independent. Therefore, the linear
terms have to be zero
1 2
1 2
0H H
H Hs s
H H+ =
Ω Ω, 31
1 3
0HH
H Hs s
HH+ =
Ω Ω, 32
2 3
0,HH
H Hs s
HH+ =
Ω Ω (55)
which introduce relationships between the in-plane and out-of-plane H components.
Moreover, introducing scalar parameters jsHr with ( )12,13,23j = , Eq.(55) can be written as
122 1= − sHH r H , 13
3 1= − sHH r H 232 3= − sHH r H (56)
with
( )( )
( )( )
( )( )
12 13 232 3 3
1 1 2
1 1 1; ;
1 1 1s s s
sH sH sHs s s
h h hr r r
h h h− − −
= = =− − −
(57)
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22
The jsHr terms represent the scalar ratio between principal healing eigenvalues.
Physically, they represent the availability of healing agent and mathematically, the ultimate
shape of the healing domain. The stress-strain relationships at failure in the effective reference
frame CDH is described by Eq. (28). In particular, the shear components can be written as
%6 12
12 126
ult
sD sHult
Gk k
σγ
= , %5 13
13 135
ult
sD sHult
Gk k
σγ
= , %4 23
23 234
ult
sD sHult
Gk k
σγ
= (58)
with
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )
12 121 2 1 2
13 131 3 1 3
23 232 3 2 3
k 1 1 , k 1 1
k 1 1 , k 1 1
k 1 1 , k 1 1
sD s s sH s s
sD s s sH s s
sD s s sH s s
d d h h
d d h h
d d h h
= − − = + +
= − − = + +
= − − = + +
(59)
For example, k ijsH with ( )12,13,23j = represent the ratio between damaged-healed
stiffness at failure and damaged (not healed) stiffness at failure /Healed Damagediz RG G . These
parameters are related to the density of microcapsules and catalyst. Numerical evaluation is
obtained by simple shear tests, which yield the product of healing and damage values at
failure, simply as the ratio between ultimate to virgin shear modulus (Eqs. (58)). Introducing
Eqs. (59) in Eqs. (54) the following equations hold
121222 66 211
612 12 12 12 12
131333 55 211
513 13 13 13 13
232333 44 222
423 23 23 23 23
1
1
1
HHsHsH
sH sH sH sH sD
HHsHsH
sH sH sH sH sD
HHsHsH
sH sH sH sH sD
J rJ r C Fk k r k k
J rJ r C Fk k r k k
J rJ r C Fk k r k k
+ =
+ =
+ =
(60)
Finally, Eqs. (53), (60) and Eq. (56) describe a non-linear system of equations in the unknown
quantities jsr ( )12,13,23j = and ( )2 3 22 33, , ,H H H HH H J J , from which healing characteristic tensors
are completely identified in terms of experimental data.
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23
7 RESULTS AND DISCUSSION
The identification procedure for damage and plasticity at the mesoscale level requires
experimental data based on the single composite lamina behavior (Barbero and Lonetti 2001,
2002, Lonetti et al. 2003). The strength values for different directions (longitudinal,
transverse, in-plane/out-of-plane shear) and the critical damage values yield a non-linear
system in which the unknown parameters are the damage characteristic tensors components,
and D Dij iJ H . Basically, the same procedure is proposed here to identify the healing domain.
The central assumption is that the gh-function has the same analytical expression of the
damage one. Therefore, the surface shape is mainly controlled by healing critical values and
the characteristic tensors and H HJ H .
From experimental point of view, damage develops into micro cracks. Subsequently,
these cracks reach the microcapsules and the healing agent is released. Numerically, the
process is described by a healing domain, which controls the onset of healing. The evolution
of such domain is defined by the normality rule, triggered when the driving thermodynamic
healing forces reach the gh-surface.
Healing and damage are shown schematically in Fig. 4, in which gd with 0 1γ γ+ =
and gh with 0φ = represent the damage and healing surfaces at failure and at the beginning of
the process, respectively. The healing threshold 0φ defines when the process starts and it is
dependent on the density of microcapsules and catalyst. Moreover, the healing critical
eigenvalues h1c, h2c, h3c, represent the maximum allowable values of healing, which are to be
obtained by experimental procedures. From experimental point of view, microcapsule and
catalyst density control two different phenomena: the beginning and the efficiency of the
healing process. The critical healing values and the healing threshold controls both
phenomena. From numerical point of view, the latter controls the beginning of the process,
whereas the former define the maximum size of the healing surface and the allowable values
of the healing tensor. Only numerical results in term of sensitivity analysis are shown in this
work, but an experimental investigation has been initiated (Barbero et al. 2004) to proper
define both values.
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
24
Lacking experimental data for the healing process, the model has been used to
demonstrate the effect of healing on Carbon-Epoxy T300-5208 (Herakovic 1998) for which
the damage behavior is well documented. The material parameters determined by the
identification procedure described in Section 6 are shown in Table 2. In Figs. 5-7, the solid
line represents the actual behavior without healing and the dotted live the predicted behavior
with the healing turned off by setting a high value of healing threshold 0φ . The material
properties are shown in Table 1. The damage evolution parameters 1 2 0, , ,D Dc c γ and damage
characteristic tensors JD, HD are identified in terms of available data for a single lamina as
explained in Barbero and Lonetti (2001), using the material properties reported in that same
reference. Subsequently, the healing phenomenon is evaluated assuming
( )1 20.15, 0.1 5, 0.4h hoc c E φ= = − = as reference values.
The in-plane shear stress-strain curve, as shown in Fig. 5-6-7, highlights the stiffness
improvement due to the healing effects. The sensitivity to healing of T300-5208 vs. healing
threshold 0φ is shown in Fig. 5 for an in-plane shear test under monotonic loading. The curve
labelled “no healing” represents the prediction using actual material property data. It predicts
accurately the experimental data because the T300-5208 material had no healing agent in it.
The addition of healing clearly increases the shear stiffness and strength of the material. The
sensitivity of T300-5208 vs. evolution parameter 1hc is shown in Fig. 6. The sensitivity of
T300-5208 vs. evolution parameter 2hc is shown in Fig. 7. The parameters 1
hc and 2hc control
the evolution (hardening) of the healing domain. Increasing 1hc makes it harder to heal the
material because the domain grows rapidly. This translates into less healing in Fig. 6. The
absolute value of 2hc controls the exponential decay in Eq. (10) with larger absolute values
resulting in more healing, as shown in Fig. 7.
8 CONCLUSIONS
Continuum Damage Mechanics has been extended for the first time to incorporate
healing process into what is called Continuum Damage Healing Mechanics (CDHM).
Furthermore, the theory has been used to develop a specific model for fiber-reinforced
polymer matrix composites experiencing damage, plasticity, and healing. Expressions are
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
25
given for the various domains, potentials, and evolution equations based on insight gained
from experimental observations. A procedure for identifying the healing parameters is
outlined. The procedure for integration of the evolution equations is given. Finally, the
applicability of the theory and particular composite model is demonstrated by performing a
parametric study of the effect of healing evolution parameters on the shear response of a
material for which the non-healing response is well known.
Appendix I
Equation Section (Next)
For a generic (i+1) load step and k iteration, the damage, plasticity, and healing functions at
the first term of the Taylor expansion are written as
( ) ( )
( ) ( )
( ) ( )
1 11 1 1 1 1
1 1
1 11 11 1 1
1 1
1 11 1 1 1 1
1 1
k kd dd d Dk Dk k k
i i i i iDi i
k kp pk kp p k ki ii i i
i ik kh h
h h H k H k k ki i i i iH
i i
g gg g
g gg g R RR
g gg g
γ γγ
φ φφ
+ ++ + + + +
+ +
+ ++ ++ + +
+ +
+ ++ + + + +
+ +
∂ ∂≅ + − + −
∂ ∂
∂ ∂≅ + − + −
∂∂
∂ ∂≅ + − + −
∂ ∂
Y YY
Y YY
σ σσ
(A1)
Using Eqs.(27)-(33), the thermodynamic forces ( ), , , , ,D HRγ φY Yσ can be expressed
as function of internal kinematic variables ( ), , , , ,p pδ µεD H as
( ) ( ) ( ) ( )1 1 ( 1) ( ) 11 1 1 1 1 1 1 1
1 1 1
k k kD D HD Dk Dk k k p k p k k k
i i i i i i i ipi i i
Y Y Y+ + + ++ + + + + + + +
+ + +
∂ ∂ ∂∆ − = − + − + −
∂ ∂ ∂Y Y Y D D H H
D Hε ε
ε(A2)
( ) ( )1 11 1 1 1
1
kk k k ki i i i
i
γγ γ γ δ δδ
+ ++ + + +
+
∂∆ − = −
∂ (A3)
( ) ( ) ( ) ( ) ( )( 1) ( ) 1 ( 1) ( ) 1 11 1 1 1 1 1 1 1 1
1 1 1
k k kk k k p k p k k k k k
i i i i i i i i ipi i i
σ+ − + + ++ + + + + + + + +
+ + +
∂ ∂ ∂− = − + − + −
∂ ∂ ∂
σ σσ σ ε εε
M D D D H HD H
(A4)
( ) ( )1 11 1 1 1
1
kk k k ki i i i
i
RR R p pp
+ ++ + + +
+
∂− = −
∂ (A5)
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26
( ) ( ) ( ) ( )1 1 ( 1) ( ) 11 1 1 1 1 1 1 1
1 1 1
k k kH H HH H k H k k k p k p k k k
i i i i i i i ipi i i
+ + + ++ + + + + + + +
+ + +
∂ ∂ ∂∆ − = − + − + −
∂ ∂ ∂Y Y YY Y Y H H D DH D
ε εε
(A6)
( ) ( )1 11 1 1 1
1
kk k k ki i i i
i
φφ φ φ µ µµ
+ ++ + + +
+
∂∆ − = −
∂ (A7)
Moreover, the incremental expressions of the kinematic variables using equations (38)
are expressed as
( ) ( )1 11 1 1 1
1
, kd
k k d k k di i i iD
i
fλ δ δ λ+ ++ + + +
+
∂− = ∆ − = −∆
∂D D
Y (A8)
% %( ) ( )( 1) ( ) ( 1) ( )1 1 1 1
1
, kpp k p k p k k p
i i i ii
g p pλ λ+ +
+ + + +
+
∂− = ∆ − = −∆
∂ε ε
σ (A9)
( ) ( )1 11 1 1 1
1
, kh
k k h k k hi i i iH
i
fλ µ µ λ+ ++ + + +
+
∂− = ∆ − = −∆
∂H H
Y (A10)
Finally, introducing Eq.(A8)-(A10) and (A2)-(A7) into Eqs.(A1), a system of equations is
obtained, in which iλ∆ are the unknown quantities
( )
( )
( )
11 1 1
11 1 1
11 1 1
, ,
, ,
, ,
k k kd d dd p d h d d p h
i d p hi i i
k k kp p pp p d h p d p h
i d p hi i ik k kh h h
h p d h h d p hi d p h
i i i
g g gg g
g g gg g
g g gg g
λ λ λ λ λ λλ λ λ
λ λ λ λ λ λλ λ λ
λ λ λ λ λ λλ λ λ
++ + +
++ + +
++ + +
∂ ∂ ∂≅ + ∆ + ∆ + ∆
∂ ∂ ∂
∂ ∂ ∂≅ + ∆ + ∆ + ∆
∂ ∂ ∂
∂ ∂ ∂≅ + ∆ + ∆ + ∆
∂ ∂ ∂
(A11)
where
1
d d D d d
d D D
d d D p
p D p
d h D f
h D H
g g f g
g g g
g g f
γγ δλ
λ
λ
−
∂ ∂ ∂ ∂ ∂ ∂= − ∂ ∂ ∂∂ ∂ ∂
∂ ∂ ∂ ∂= ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂= ∂∂ ∂ ∂
YDY Y
Y MY
YY Y
ε σ
Η
(A12)
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
27
1
1 1
1
[ ]
[ ]
p p d
d D
p p p p
p p
p p h
h h
g g f
g g g g RR p
g g f
λ
λ
λ
−
− −
−
∂ ∂ ∂ ∂=
∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂
= − ∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂
=∂∂ ∂∂
σσ
σεσ σ
σσ
MD Y
M M
MH Y
(A13)
°1
,
,
,
h h H h h
h H h
h h H p
p H p
h h H d
d H d
g g f g
g g g
g g f
ff ml
l
l
-
æ ö¶ ¶ ¶ ¶ ¶ ¶ ÷ç= - ÷ç ÷ç ÷¶ ¶ ¶¶ ¶ ¶è øæ ö¶ ¶ ¶ ¶ ÷ç= ÷ç ÷ç ÷¶ ¶ ¶è ø¶æ ö¶ ¶ ¶ ¶ ÷ç= ÷ç ÷ç ÷¶¶ ¶ ¶è ø
YHY Y
Y MY
YDY Y
ε σ (A14)
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32
LIST OF FIGURES
Fig. 1. Clockwise form top left: undamaged, damaged-healed, effective, effective without
elastic deformations, damaged–healed without elastic deformation.
Fig. 2. Stress-strain curve: Elasto-plastic-damage and elasto-plastic-damage-healing.
Fig.3. Schematic healing-damage domain and thresholds
Fig .4. Healing and damage surfaces.
Fig. 5. T300-5208 Sensitivity of healing vs. healing threshold 0φ for an in-plane shear
monotonic loading test with damage.
Fig. 6. T300-5208 Sensitivity of healing to healing evolution parameter 1
1
h
dc
c for an in-plane
shear monotonic loading test with damage.
Fig.7. T300-5208 Sensitivity of healing to healing evolution parameter 2
2
h
dc
c for an in-plane
shear monotonic loading test with damage.
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
33
Table 1. Observable, Kinematic, and Conjugate Variables.
Internal state variables Quantity Observable variables
Kinematic variables Thermodynamic
forces
Strain ε
Temperature T
Damage D YD
Damage evolution δ γ
Plastic strain pε σ
Hardening p R
Healing H HY
Healing evolution µ φ
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
34
Table 2. Healing parameters.
Parameters T300-5208
11HJ , 22
HJ 0.3885e-15, 0.3522e-11
1HH , 2
HH 0.4255e-7, -0.1046e-6
1 2, , h hc c φ 0.15, -0.1e+05, 0.4
h1c, h2c=h3c 0.1, 0.5
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
F
χdx3
dx1 dx2OO**
CDH*
n dA * *
*
t dA* *
OO
CFt dA*
*
n dA * *
*
*
χdx3
0dx1
A
X
dx1
e
Y
0
Bdx20
O
n0 dA0 Fdx3
dx2O
n dA
ZC0C
CDH
dx1
dx3
O dx2
n dA
CF
DH
DH
F e-1 F e-1F e
*
*
X
Y
ZC
B
A
dx2*
*dx3
*dx1
X
Y
Z
X
Y
Z Z
Y
X
C
A
B
*
*
C
BA
C*
*A
*B
0
0
0
Fig. 1. Clockwise form top left: undamaged, damaged-healed, effective, effective without elastic deformations, damaged–healed without
elastic deformation.
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
Pε
A
σ
E
F Eedp
ε
edp-h0EB
CE
D
O
Fig. 2. Stress-strain curve: Elasto-plastic-damage and elasto-plastic-damage-healing.
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
γ 0
=T.W.Dg
0γ +γ=1d.
Y11D
Y22D
g
0φ
.h
H
YH22
Y11
φ+φ=10
H
Fig 3. Schematic healing-damage domain and thresholds.
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
.
Hg0
Healing production
=T.W.Dg
,Y22YD22H
,YY11D
11H
φγ +γ=1
0
0
Fig 4.Healing and damage surfaces
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
0
10
20
30
40
50
60
70
80
90
0 0.2 0.4 0.6 0.8 1 1.2
Experimental
no Healing
0.6
0.7
1
1.2
σ xy [Mpa]
ε xy
Fig. 5. T300-5208 Sensitivity of healing vs. healing threshold 0φ for an in-plane shear monotonic loading test with damage.
0φ
σXY
σXY
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
0
10
20
30
40
50
60
70
80
0 0.2 0.4 0.6 0.8 1 1.2
Experimental
no Healing
6.67E-05
0.667
6.67
8
σxy [Mpa]
εxy
Fig. 6. T300-5208 Sensitivity of healing to healing evolution parameter 1
1
h
dc
c for an in-plane shear monotonic loading test with damage.
1
1
h
dc
c
σXY
σXY
International Journal of Damage Mechanics, vol. 14-January 2005, pp. 51-81.
0
10
20
30
40
50
60
70
80
90
100
0 0.2 0.4 0.6 0.8 1 1.2
Experimental
no Healing
0.0025
0.005
0.075
0.1
σxy [Mpa]
εxy
Fig.7. T300-5208 Sensitivity of healing to healing evolution parameter 2
2
h
dc
c for an in-plane shear monotonic loading test with damage.
2
2
h
dc
c
σXY
σXY