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Multi-scale analysis of multiple damage mechanisms coupled with inelastic behavior of composite materials
Pergamon
Mechanics Research Communications, Vol. 27, No. 3, pp. 295-300, 2000 Copyright © 2000 Elsevier Science Ltd Printed in the USA. All rights reserved 0093-6413/00IS-see front matter
PII: S0093-6413(00)00095-1
MULTI-SCALE ANALYSIS OF MULTIPLE DAMAGE MECHANISMS COUPLED WITH INELASTIC BEHAVIOR OF COMPOSITE MATERIALS George Z. Voyiadjis Department of Civil and Environmental Engineering, Louisiana State University Baton Rouge, LA 70803,USA Babur Deliktas Department of Civil Engineering, Mustafa Kemal Univeristy, Hatay, Turkey (Received 12 October 1999; accepted for print 1 March 2000)
Introduction When subjected to loads materials exhibit defects that lead in some cases to specific pattem formations due to plastic flow, damage, and fracture. Experimental observations indicate that the failure mechanisms of heterogeneous materials occur at localized zones of damage where interaction and coalescence of microcracks take place, which in turn it leads to the degradation of the global stiffness. Due to localization of damage the length scale of damage distribution decreases. As damage localizes over a narrow region of the continuum the characteristic length governing the variation of damage falls far below the scale where state variables of strain and damage can be regarded as quantities that could be used to describe the response of the continuum. This leads to the case where the wavelength of the damage distribution is predicted to be much smaller than the size of the material heterogeneities [1]. The deficiency of classical continua to capture such scale effects due to localization of damage makes it necessary to look for alternative strategies for the solution of the problem such as micro-polar continua, Cosserat continua, and non local approaches. In the case of the non local approach, the common procedure is to introduce the non local terms either through an integral equation or through a gradient approach. Aifantis [2] suggested the gradient approach in order to describe plastic instabilities including dislocation patterning and spatial characteristics of shear bands. The gradient methods proposed by Lasry and Belytschko [3] provide an alternative to the nonlocal integral equations. In this approach the equations remain local in a finite element sense and linearization is implemented directly. The gradient term in plasticity models is introduced through the yield function [2]. Gradients in combination with continuum isotropic damage has been presented by Peerlings et. al. [4] and for anisotropic damage by Kuhl et. al.[5]. Their anisotropic damage evolution is based on the mieroplane model with enhancement by introducing additional gradient terms in the constitutive equations. Recently Lacy et. al. [6] proposed a mesoscale gradients approach in order to obtain more precise characterization of evolution of damage which is not statistically homogenous and strictly dependent on material inhomogeneity at the mesoscale. Lacy et. al [6] also discussed the deficiency of the maeroscale strain and damage gradient theories in capturing mesoscale length scale effects. In this work an attempt is made to introduce damage and plasticity internal state variables at both the macro and mesoscale levels in order to provide sufficient details of defects and their interaction to properly (i.e. physically) characterize the material behavior. This will provide an adequate characterization of these defects in terms of size, orientation, distribution, spacing, interaction among defects, etc. In order to achieve this the bridging of multiple scales need to be addressed and 295
296
G.Z. VOYIADJIS and B. DELIKTAS
implemented properly. The gradient of these internal state variables will also be incorporated in order to address the nonlocal effects. An attempt to expand this bridging to micro=scale level is computationally prohibitive for the immediate future. Definitions of RVE and Sub-RVEs The internal state variables will be categorized into two groups. The first one is statistically homogeneous at the representative volume element (RVE). The suitably chosen damage variables and corresponding response functions ( a , e , ~ and C) that are statistically homogeneous within the observation window of the RVE are used in this work. The definition of such RVE is detailed in the work by Nemat Nasser and Hori [7]. According to this work, a brief summary about the definition of the RVE is introduced here. A cubical element with dimension, LRVE, is needed in order to fulfill the following conditions d
LRV~-
(( 1
L~
,
--
~L tgXk RVE ( C~
(1)
where d is a characteristic size of the micro-constituents, Lc is the heterogeneity correlation length, L is the characteristic macroscopic structural dimension, tl ° is the mean field stress and (x, ,x 2,x 3) are the components of the Cartesian coordinates. The RVE implied here is the matrix with a single fiber in the middle of the RVE (Figure 1). Effective Homogenous Continuum
~
~
Matrix
v
v
o
Figure 1 Schematic Representation of the RVE In the literature this RVE is the necessary minimum observation window that is used for the determination of the statistically homogenous elastic stiffness. The other one can only be statistically homogenous at a subvolume of the RVE. For an RVE made of two phase materials the defect in each constituent and in the interphase (debonding) cannot be categorized as statistically homogenous for the RVE unless a very low order measure of these defects is used to characterize damage or plasticity. The subvolume characterization of damage and plasticity at a level below the RVE allows one to adequately characterize the details of these defects. This sub RVEs definition for the composite material in the case of multi scale analysis is introduced here by defining an equivalent minimum observation window for each constituent of composite where the response function of each constituent is statistically homogenous within the equivalent RVEs (Figure 2). The sub RVE damage distribution of each constituent can be characterized at a point within the corresponding RVEs. In Figure 2 mesoscale damage parameters, (~rn,t~f,~i) may be evaluated at the point (~,, ~ 2) in the average sense so that the following relation can be written as follows
DAMAGE MECHANISMS IN INELASTIC COMPOSITES
297
~}smv= 1-:-~-[Tl~mdV~v
^f = 1 [,q~fdVsv ' ^, = ! [,q~idVsv qsv (2) ¢~sv Vsv" VsvVsv J where Vsv is the volumeof the correspondingconstituent and TI is a smoothingoperator that accountsfor discretemicrostructureand crack interaction.
Matrix ..... ........................ . . . . . . . ~
• '...
] ,.~
Interface ]
Matrix
Fiber
Interface
Fiber
,.,
-
=
_-
-
_m=l
Figure 2. Sub-RVEs for Multiscale Composite Materials The discretized systems in Figure 2 can be obtained by translating subvolume centroid coordinates with some differential distance (d~l,d~2,d~3) and the subvolume averaged damage parameters reevaluated at the new location Macro-Mesoscale Plastic and Damage Gradient Theory The components of the macroscale gradient terms and the norm of the averaged mesoscale gradient terms of the macroscale internal state variables of both plasticity and damage may be used as additional higher order intemal variables. In such an approach, the thermoelastic Helmolthz free energy may be expressed in terms of both sets of internal state variable such that ~ = ~I/(£e,T, A((~k], - - .
L ( k ) , - - - - . ~(k),
rk))
)
(3)
In equation (3) r = p (plastic) or r = d (damage). A((rk)) represents the macroscale internal state variables such that (A((k))--p,~AS, x,)')
where p , ct variables characterize the isotropic and
kinematic hardening in plasticity, and ~5, K, ]¢ variables characterize the damage and the corresponding isotropic and kinematic hardening in damage, respectively. The higher order internal state variables are introduced by averaged mesoscale gradient terms, which are a measure of the mesostructural variability within the RVE. With the higher order representation of the material inhomogeneity, it is possible to evaluate the inelastic flow, time dependent deformation, evolution of non-uniformly distributed interactive crack systems as well as the interaction of slips planes with the micro defect systems The corresponding gradient terms are defined as follows
A(F) ,k)
1 VRVE j" i ' r ' dVRvE (4a),
VA 'r,
~"~)Aim 0Aim }
VRVE
VRVE VRvE
(4b)
298
G.Z. VOYIADJIS and B. DELIKTAS
where in equations (4) r = p (plasticity) or r = d (damage). In equation (4d) the higher order mesoscale gradient term, V'~Ik~ is dependent on the macroscale variables, --~k)'A(OHowever, the random periodic boundary condition ensures that there is no net flux of mesoscale gradients across the RVE boundary. Such a constraint would effectively prevent coupling between macroscale gradient and mesoscale gradient terms. One can now express, the analytical form of the Helmholtz free energy as the quadratic form of its internal state variables 4- 1 ^ ( r ) ~ ( r )
. A(r)
.i l h ( r ) ~ [ Z A ( r )
• ~(r)
(5) 1 _ ( r ) XTX'7 A ( r )
~ k (r) + : ----''(k)
1
~d~k)VAo,) VA(k) (r)
^ ^ (r)
^ ^ (r)
In equation (5) the coefficients are dependent on the sizes and spacing of the fibers. In the case of the gradient theory, these coefficients become also dependent on the gradient of the fiber size and fiber spacing variation. The functional dependency of these coefficients can be obtained by studying the interaction problem of an inclusion embedded in an infinite homogeneous matrix subjected to a macroscopic stress rate and corresponding strain rate at infinity. The value of the thermodynamic forces can be obtained through the evolution relations of the internal state variables. However, it should be pointed out that there are two types of evolution equations that need to be developed, one at the macro and the other at the mesoscale level. The former can be obtained by assuming the physical existence of the dissipation potential at the macroscale. The later one can be obtained by a micromechanical or phenomenological approach. One may consider that the evolution equations of the internal state variables, •A~(k)9 (r) can be obtained by integrating the evolution equations of the local internal state variables at the mesoscale, that is ,~{r) ) is a more cumbersome task (k) over the domain of the RVE. However, integration of the A({rk)
~(r)is "'(k)
mathematically since at the mesoseale
a function of many different aspects of material
inhomogeneities such as interaction of defects, size of defects, spacing between them, and distribution of defects within the sub RVE. Therefore, in this work evolution equations of the macroscale internal state variables are obtained through the use of the generalized normality rule of thermodynamics. In this regard, the macroscale dissipation potential, is defined only in terms of the macroscale flux variables as follows o = o ( ~ p, A,r, ~k~,vA,.~k~,
"'(k'[I-"~'l['lh
(6)
By using the Legendre-Fenchel transformation of the dissipation potential (O), one can obtain complementary laws in the form of the evolution laws of flux variables as a function of the dual variables (r) w(r) (r) ^ ^ (r) O" = O * (o,V~k),V (k~,VVV~k~, VV(k~ )
(7)
It is possible to decouple the potential O" into the plastic and the damage dissipation potential parts as follows: N
N
N
N
i=l
i=l
o" = F ( o . E R , . E X , ) + i=l
i=l
(8)
DAMAGE MECHANISMS IN INELASTIC COMPOSITES
299
In equation (8), R and X are the isotropic and kinematic hardening forces respectively of plasticity and K and F and the isotropic and kinematic hardening forces respectively of damage. Y is the thermodynamic force conjugate to the damage variable. Evolution Equations for the Internal State Variables In order to obtain evolution equations for the internal state variables, a proper analytical form of the potentials that are defined in equation (8) need to be obtained. In this work in order to be consistent and satisfy the generalized normality rule of thermodynamics the following form of potentials, F and G are defined [8] k~ XllX 1 + k~ X2 : X2 + ~ p X3 : X3 -I-~p X4 • X 4 (9a) F = f + 2a~ 2b~ 2c 2 2d 2
G=g +
kd
rl :r, +
d
d
r2 :r2 +k r :1"3
d
:r,
(gb)
2a 2 2b 2 2c~ 3 2d 2 In equation (9a) f represents the yield function that is gradient dependent and is given as follows I
/f
f = I3_ ( t ~ - ZNX i : ~ - ~ X
[2[,
~=,
j
~,
i=,
]t
2 -- i~__lR, <0
(10)
=
)j
Similarly in equation (9b) g represents the damage function with the corresponding gradient dependent damage criterion and is defined as follows g=
Y,- I'~ : P :
Y,-~,r,
(11)
-1=0
i=l
where the fourth order tensor P describes the anisotropic nature of the damage growth and the initiation of damage. Its form is given as a function of the hardening tensor h [9]
hij = k 11 ~K 8ikl~kj+80~.V 2
)
(12)
The gradient term of ~ is defined as ~. = K-- b~V2K-e~K.VK (13) The plastic multiplier, k p and damage multiplier kd can be obtained using the consistency condition for plasticity (t" = 0) and damage ~ = 0). The plastic multiplier, ~,Pcan be expressed as follows ~,P =(M p +Lp)-I ~0f
: E(II)): I~
(14)
where ~f'E'af+
MP=~ " "~
~f "{(aP+dP ~)-~XI+(k~'+kPdP'~z~0X,"
a~ f " J + ~X-2-2:
+Ofax~ :,Ic~3~-+kpx3 } 3
}
bp3--X-2+kpX2 -2 + (15)
and Lp = a p +dPa~ + b ~ l : V + c ~ I : V V
(16)
300
G.Z. VOYIADJIS and B. DELIKTAS
Similarly the damage multiplier, ~,~can be obtained by using the consistency condition for the damage criterion 2~d = (Md +Ld)-'bg ~:0
(17)
where -+
bY ar,
"or,
" ~
[!1}'"~---~
a+ ) J
01"2
b~
+k~X 2 +
(18) bE "
~
j
and Ld _ bg b~ y bg 0"~br bY
(19)
Conclusion A multi scale gradient theory is developed here through the use of the intemal state variables. It enables one to address issues of the inhomogeneity of the material at the mesoscale and the nonlocal behavior of the material at the macroscale. It also helps interpret the collective behavior of defects such as dislocations and cracks. References [1] G. Pijaudier-Cabot. Non Local Damage. In H. B. Muhlhaus, editor, Continuum Models for Materials with Microstructure, pages 105-143. John Wiley & Sons Ltd,, 1995. [2] E. C. Aifantis. On the microstructural origin of certain inelastic models. Journal of Engineering Materials and Technology, 106:326-330, 1984. [3] D. Lasry and T. Belytschko. Localization limiters in transient problems. International Journal of Solid and Structures, 24:581-597, 1988. [4] R. H. J. Peerlings, R. de Borst, W. A. M. Brekelmans, and J. H. P. de Vree. Gradient enhanced damage for quisa-brittle materials. International of Journal of Numerical Methods in Engineering, 39:3391-3403, 1996. [5] E. Kuhl, E. Ramm, and R. de Borst. An anisotropic gradient damage model for quassi-birttle materials. Computer Method in Applied Mechanics and Engineering, 1999. will appear. [6] T.E. Laeey, D.L. McDowell and R. Yalreja, Nonlocal Concepts for Evolution of Damage, Mechanics of Materials, (in press), 2000. [7] S. Nemat-Naser, and M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, North-Holland, Amsterdam, The Netherlands, 1993. [8] G. Z. Voyiadjis and B. Deliktas, A Coupled Anisotropic Damage Model for the Inelastic Response of Composite Materials. Journal of Computer Methods in Applied Mechanics and Engineering, 2000 (in press). [9] G. Z. Voyiadjis and T. Park, "Local and Interfacial Damage Analysis of Metal Matrix Composites," International Journal of Engineering Science, Vol. 33, No. 11, pp. 1595-1621, 1995.
Mechanics Research Communications, Vol. 27, No. 3, pp. 295-300, 2000 Copyright © 2000 Elsevier Science Ltd Printed in the USA. All rights reserved 0093-6413/00IS-see front matter
PII: S0093-6413(00)00095-1
MULTI-SCALE ANALYSIS OF MULTIPLE DAMAGE MECHANISMS COUPLED WITH INELASTIC BEHAVIOR OF COMPOSITE MATERIALS George Z. Voyiadjis Department of Civil and Environmental Engineering, Louisiana State University Baton Rouge, LA 70803,USA Babur Deliktas Department of Civil Engineering, Mustafa Kemal Univeristy, Hatay, Turkey (Received 12 October 1999; accepted for print 1 March 2000)
Introduction When subjected to loads materials exhibit defects that lead in some cases to specific pattem formations due to plastic flow, damage, and fracture. Experimental observations indicate that the failure mechanisms of heterogeneous materials occur at localized zones of damage where interaction and coalescence of microcracks take place, which in turn it leads to the degradation of the global stiffness. Due to localization of damage the length scale of damage distribution decreases. As damage localizes over a narrow region of the continuum the characteristic length governing the variation of damage falls far below the scale where state variables of strain and damage can be regarded as quantities that could be used to describe the response of the continuum. This leads to the case where the wavelength of the damage distribution is predicted to be much smaller than the size of the material heterogeneities [1]. The deficiency of classical continua to capture such scale effects due to localization of damage makes it necessary to look for alternative strategies for the solution of the problem such as micro-polar continua, Cosserat continua, and non local approaches. In the case of the non local approach, the common procedure is to introduce the non local terms either through an integral equation or through a gradient approach. Aifantis [2] suggested the gradient approach in order to describe plastic instabilities including dislocation patterning and spatial characteristics of shear bands. The gradient methods proposed by Lasry and Belytschko [3] provide an alternative to the nonlocal integral equations. In this approach the equations remain local in a finite element sense and linearization is implemented directly. The gradient term in plasticity models is introduced through the yield function [2]. Gradients in combination with continuum isotropic damage has been presented by Peerlings et. al. [4] and for anisotropic damage by Kuhl et. al.[5]. Their anisotropic damage evolution is based on the mieroplane model with enhancement by introducing additional gradient terms in the constitutive equations. Recently Lacy et. al. [6] proposed a mesoscale gradients approach in order to obtain more precise characterization of evolution of damage which is not statistically homogenous and strictly dependent on material inhomogeneity at the mesoscale. Lacy et. al [6] also discussed the deficiency of the maeroscale strain and damage gradient theories in capturing mesoscale length scale effects. In this work an attempt is made to introduce damage and plasticity internal state variables at both the macro and mesoscale levels in order to provide sufficient details of defects and their interaction to properly (i.e. physically) characterize the material behavior. This will provide an adequate characterization of these defects in terms of size, orientation, distribution, spacing, interaction among defects, etc. In order to achieve this the bridging of multiple scales need to be addressed and 295
296
G.Z. VOYIADJIS and B. DELIKTAS
implemented properly. The gradient of these internal state variables will also be incorporated in order to address the nonlocal effects. An attempt to expand this bridging to micro=scale level is computationally prohibitive for the immediate future. Definitions of RVE and Sub-RVEs The internal state variables will be categorized into two groups. The first one is statistically homogeneous at the representative volume element (RVE). The suitably chosen damage variables and corresponding response functions ( a , e , ~ and C) that are statistically homogeneous within the observation window of the RVE are used in this work. The definition of such RVE is detailed in the work by Nemat Nasser and Hori [7]. According to this work, a brief summary about the definition of the RVE is introduced here. A cubical element with dimension, LRVE, is needed in order to fulfill the following conditions d
LRV~-
(( 1
L~
,
--
~L tgXk RVE ( C~
(1)
where d is a characteristic size of the micro-constituents, Lc is the heterogeneity correlation length, L is the characteristic macroscopic structural dimension, tl ° is the mean field stress and (x, ,x 2,x 3) are the components of the Cartesian coordinates. The RVE implied here is the matrix with a single fiber in the middle of the RVE (Figure 1). Effective Homogenous Continuum
~
~
Matrix
v
v
o
Figure 1 Schematic Representation of the RVE In the literature this RVE is the necessary minimum observation window that is used for the determination of the statistically homogenous elastic stiffness. The other one can only be statistically homogenous at a subvolume of the RVE. For an RVE made of two phase materials the defect in each constituent and in the interphase (debonding) cannot be categorized as statistically homogenous for the RVE unless a very low order measure of these defects is used to characterize damage or plasticity. The subvolume characterization of damage and plasticity at a level below the RVE allows one to adequately characterize the details of these defects. This sub RVEs definition for the composite material in the case of multi scale analysis is introduced here by defining an equivalent minimum observation window for each constituent of composite where the response function of each constituent is statistically homogenous within the equivalent RVEs (Figure 2). The sub RVE damage distribution of each constituent can be characterized at a point within the corresponding RVEs. In Figure 2 mesoscale damage parameters, (~rn,t~f,~i) may be evaluated at the point (~,, ~ 2) in the average sense so that the following relation can be written as follows
DAMAGE MECHANISMS IN INELASTIC COMPOSITES
297
~}smv= 1-:-~-[Tl~mdV~v
^f = 1 [,q~fdVsv ' ^, = ! [,q~idVsv qsv (2) ¢~sv Vsv" VsvVsv J where Vsv is the volumeof the correspondingconstituent and TI is a smoothingoperator that accountsfor discretemicrostructureand crack interaction.
Matrix ..... ........................ . . . . . . . ~
• '...
] ,.~
Interface ]
Matrix
Fiber
Interface
Fiber
,.,
-
=
_-
-
_m=l
Figure 2. Sub-RVEs for Multiscale Composite Materials The discretized systems in Figure 2 can be obtained by translating subvolume centroid coordinates with some differential distance (d~l,d~2,d~3) and the subvolume averaged damage parameters reevaluated at the new location Macro-Mesoscale Plastic and Damage Gradient Theory The components of the macroscale gradient terms and the norm of the averaged mesoscale gradient terms of the macroscale internal state variables of both plasticity and damage may be used as additional higher order intemal variables. In such an approach, the thermoelastic Helmolthz free energy may be expressed in terms of both sets of internal state variable such that ~ = ~I/(£e,T, A((~k], - - .
L ( k ) , - - - - . ~(k),
rk))
)
(3)
In equation (3) r = p (plastic) or r = d (damage). A((rk)) represents the macroscale internal state variables such that (A((k))--p,~AS, x,)')
where p , ct variables characterize the isotropic and
kinematic hardening in plasticity, and ~5, K, ]¢ variables characterize the damage and the corresponding isotropic and kinematic hardening in damage, respectively. The higher order internal state variables are introduced by averaged mesoscale gradient terms, which are a measure of the mesostructural variability within the RVE. With the higher order representation of the material inhomogeneity, it is possible to evaluate the inelastic flow, time dependent deformation, evolution of non-uniformly distributed interactive crack systems as well as the interaction of slips planes with the micro defect systems The corresponding gradient terms are defined as follows
A(F) ,k)
1 VRVE j" i ' r ' dVRvE (4a),
VA 'r,
~"~)Aim 0Aim }
VRVE
VRVE VRvE
(4b)
298
G.Z. VOYIADJIS and B. DELIKTAS
where in equations (4) r = p (plasticity) or r = d (damage). In equation (4d) the higher order mesoscale gradient term, V'~Ik~ is dependent on the macroscale variables, --~k)'A(OHowever, the random periodic boundary condition ensures that there is no net flux of mesoscale gradients across the RVE boundary. Such a constraint would effectively prevent coupling between macroscale gradient and mesoscale gradient terms. One can now express, the analytical form of the Helmholtz free energy as the quadratic form of its internal state variables 4- 1 ^ ( r ) ~ ( r )
. A(r)
.i l h ( r ) ~ [ Z A ( r )
• ~(r)
(5) 1 _ ( r ) XTX'7 A ( r )
~ k (r) + : ----''(k)
1
~d~k)VAo,) VA(k) (r)
^ ^ (r)
^ ^ (r)
In equation (5) the coefficients are dependent on the sizes and spacing of the fibers. In the case of the gradient theory, these coefficients become also dependent on the gradient of the fiber size and fiber spacing variation. The functional dependency of these coefficients can be obtained by studying the interaction problem of an inclusion embedded in an infinite homogeneous matrix subjected to a macroscopic stress rate and corresponding strain rate at infinity. The value of the thermodynamic forces can be obtained through the evolution relations of the internal state variables. However, it should be pointed out that there are two types of evolution equations that need to be developed, one at the macro and the other at the mesoscale level. The former can be obtained by assuming the physical existence of the dissipation potential at the macroscale. The later one can be obtained by a micromechanical or phenomenological approach. One may consider that the evolution equations of the internal state variables, •A~(k)9 (r) can be obtained by integrating the evolution equations of the local internal state variables at the mesoscale, that is ,~{r) ) is a more cumbersome task (k) over the domain of the RVE. However, integration of the A({rk)
~(r)is "'(k)
mathematically since at the mesoseale
a function of many different aspects of material
inhomogeneities such as interaction of defects, size of defects, spacing between them, and distribution of defects within the sub RVE. Therefore, in this work evolution equations of the macroscale internal state variables are obtained through the use of the generalized normality rule of thermodynamics. In this regard, the macroscale dissipation potential, is defined only in terms of the macroscale flux variables as follows o = o ( ~ p, A,r, ~k~,vA,.~k~,
"'(k'[I-"~'l['lh
(6)
By using the Legendre-Fenchel transformation of the dissipation potential (O), one can obtain complementary laws in the form of the evolution laws of flux variables as a function of the dual variables (r) w(r) (r) ^ ^ (r) O" = O * (o,V~k),V (k~,VVV~k~, VV(k~ )
(7)
It is possible to decouple the potential O" into the plastic and the damage dissipation potential parts as follows: N
N
N
N
i=l
i=l
o" = F ( o . E R , . E X , ) + i=l
i=l
(8)
DAMAGE MECHANISMS IN INELASTIC COMPOSITES
299
In equation (8), R and X are the isotropic and kinematic hardening forces respectively of plasticity and K and F and the isotropic and kinematic hardening forces respectively of damage. Y is the thermodynamic force conjugate to the damage variable. Evolution Equations for the Internal State Variables In order to obtain evolution equations for the internal state variables, a proper analytical form of the potentials that are defined in equation (8) need to be obtained. In this work in order to be consistent and satisfy the generalized normality rule of thermodynamics the following form of potentials, F and G are defined [8] k~ XllX 1 + k~ X2 : X2 + ~ p X3 : X3 -I-~p X4 • X 4 (9a) F = f + 2a~ 2b~ 2c 2 2d 2
G=g +
kd
rl :r, +
d
d
r2 :r2 +k r :1"3
d
:r,
(gb)
2a 2 2b 2 2c~ 3 2d 2 In equation (9a) f represents the yield function that is gradient dependent and is given as follows I
/f
f = I3_ ( t ~ - ZNX i : ~ - ~ X
[2[,
~=,
j
~,
i=,
]t
2 -- i~__lR, <0
(10)
=
)j
Similarly in equation (9b) g represents the damage function with the corresponding gradient dependent damage criterion and is defined as follows g=
Y,- I'~ : P :
Y,-~,r,
(11)
-1=0
i=l
where the fourth order tensor P describes the anisotropic nature of the damage growth and the initiation of damage. Its form is given as a function of the hardening tensor h [9]
hij = k 11 ~K 8ikl~kj+80~.V 2
)
(12)
The gradient term of ~ is defined as ~. = K-- b~V2K-e~K.VK (13) The plastic multiplier, k p and damage multiplier kd can be obtained using the consistency condition for plasticity (t" = 0) and damage ~ = 0). The plastic multiplier, ~,Pcan be expressed as follows ~,P =(M p +Lp)-I ~0f
: E(II)): I~
(14)
where ~f'E'af+
MP=~ " "~
~f "{(aP+dP ~)-~XI+(k~'+kPdP'~z~0X,"
a~ f " J + ~X-2-2:
+Ofax~ :,Ic~3~-+kpx3 } 3
}
bp3--X-2+kpX2 -2 + (15)
and Lp = a p +dPa~ + b ~ l : V + c ~ I : V V
(16)
300
G.Z. VOYIADJIS and B. DELIKTAS
Similarly the damage multiplier, ~,~can be obtained by using the consistency condition for the damage criterion 2~d = (Md +Ld)-'bg ~:0
(17)
where -+
bY ar,
"or,
" ~
[!1}'"~---~
a+ ) J
01"2
b~
+k~X 2 +
(18) bE "
~
j
and Ld _ bg b~ y bg 0"~br bY
(19)
Conclusion A multi scale gradient theory is developed here through the use of the intemal state variables. It enables one to address issues of the inhomogeneity of the material at the mesoscale and the nonlocal behavior of the material at the macroscale. It also helps interpret the collective behavior of defects such as dislocations and cracks. References [1] G. Pijaudier-Cabot. Non Local Damage. In H. B. Muhlhaus, editor, Continuum Models for Materials with Microstructure, pages 105-143. John Wiley & Sons Ltd,, 1995. [2] E. C. Aifantis. On the microstructural origin of certain inelastic models. Journal of Engineering Materials and Technology, 106:326-330, 1984. [3] D. Lasry and T. Belytschko. Localization limiters in transient problems. International Journal of Solid and Structures, 24:581-597, 1988. [4] R. H. J. Peerlings, R. de Borst, W. A. M. Brekelmans, and J. H. P. de Vree. Gradient enhanced damage for quisa-brittle materials. International of Journal of Numerical Methods in Engineering, 39:3391-3403, 1996. [5] E. Kuhl, E. Ramm, and R. de Borst. An anisotropic gradient damage model for quassi-birttle materials. Computer Method in Applied Mechanics and Engineering, 1999. will appear. [6] T.E. Laeey, D.L. McDowell and R. Yalreja, Nonlocal Concepts for Evolution of Damage, Mechanics of Materials, (in press), 2000. [7] S. Nemat-Naser, and M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials, North-Holland, Amsterdam, The Netherlands, 1993. [8] G. Z. Voyiadjis and B. Deliktas, A Coupled Anisotropic Damage Model for the Inelastic Response of Composite Materials. Journal of Computer Methods in Applied Mechanics and Engineering, 2000 (in press). [9] G. Z. Voyiadjis and T. Park, "Local and Interfacial Damage Analysis of Metal Matrix Composites," International Journal of Engineering Science, Vol. 33, No. 11, pp. 1595-1621, 1995.