A plasticity and anisotropic damage model for plain concrete
A plasticity and anisotropic damage model for
plain concrete
Umit Cicekli,George Z.Voyiadjis *,Rashid K.Abu Al-Rub
Department of Civil and Environmental Engineering,Louisiana State University,
CEBA 3508-B,Baton Rouge,LA 70803,USA
Received 23April 2006;received in ?nal revised form 29October 2006
Available online 15March 2007
Abstract
A plastic-damage constitutive model for plain concrete is developed in this work.Anisotropic damage with a plasticity yield criterion and a damage criterion are introduced to be able to ade-quately describe the plastic and damage behavior of concrete.Moreover,in order to account for dif-ferent e?ects under tensile and compressive loadings,two damage criteria are used:one for compression and a second for tension such that the total stress is decomposed into tensile and com-pressive components.Sti?ness recovery caused by crack opening/closing is also incorporated.The strain equivalence hypothesis is used in deriving the constitutive equations such that the strains in the e?ective (undamaged)and damaged con?gurations are set equal.This leads to a decoupled algo-rithm for the e?ective stress computation and the damage evolution.It is also shown that the pro-posed constitutive relations comply with the laws of thermodynamics.A detailed numerical algorithm is coded using the user subroutine UMAT and then implemented in the advanced ?nite element program ABAQUS.The numerical simulations are shown for uniaxial and biaxial tension and compression.The results show very good correlation with the experimental data.
ó2007Elsevier Ltd.All rights reserved.
Keywords:Damage mechanics;Isotropic hardening;Anisotropic damage
0749-6419/$-see front matter ó2007Elsevier Ltd.All rights reserved.doi:10.1016/j.ijplas.2007.03.006
*Corresponding author.Tel.:+12255788668;fax:+12255789176.
E-mail addresses:voyiadjis@https://www.sodocs.net/doc/7e16319654.html, (G.Z.Voyiadjis),rabual1@https://www.sodocs.net/doc/7e16319654.html, (R.K.Abu
Al-Rub).
International Journal of Plasticity 23(2007)
1874–1900
U.Cicekli et al./International Journal of Plasticity23(2007)1874–19001875 1.Introduction
Concrete is a widely used material in numerous civil engineering structures.Due to its ability to be cast on site it allows to be used in di?erent shapes in structures:arc,ellipsoid, etc.This increases the demand for use of concrete in structures.Therefore,it is crucial to understand the mechanical behavior of concrete under di?erent loadings such as compres-sion and tension,for uniaxial,biaxial,and triaxial loadings.Moreover,challenges in designing complex concrete structures have prompted the structural engineer to acquire a sound understanding of the mechanical behavior of concrete.One of the most important characteristics of concrete is its low tensile strength,particularly at low-con?ning pres-sures,which results in tensile cracking at a very low stress compared with compressive stresses.The tensile cracking reduces the sti?ness of concrete structural components. Therefore,the use of continuum damage mechanics is necessary to accurately model the degradation in the mechanical properties of concrete.However,the concrete material undergoes also some irreversible(plastic)deformations during unloading such that the continuum damage theories cannot be used alone,particularly at high-con?ning pressures. Therefore,the nonlinear material behavior of concrete can be attributed to two distinct material mechanical processes:damage(micro-cracks,micro-cavities,nucleation and coa-lescence,decohesions,grain boundary cracks,and cleavage in regions of high stress con-centration)and plasticity,which its mechanism in concrete is not completely understood up-to-date.These two degradation phenomena may be described best by theories of con-tinuum damage mechanics and plasticity.Therefore,a model that accounts for both plas-ticity and damage is necessary.In this work,a coupled plastic-damage model is thus formulated.
Plasticity theories have been used successfully in modeling the behavior of metals where the dominant mode of internal rearrangement is the slip process.Although the mathemat-ical theory of plasticity is thoroughly established,its potential usefulness for representing a wide variety of material behavior has not been yet fully explored.There are many research-ers who have used plasticity alone to characterize the concrete behavior(e.g.Chen and Chen,1975;William and Warnke,1975;Bazant,1978;Dragon and Mroz,1979;Schreyer, 1983;Chen and Buyukozturk,1985;Onate et al.,1988;Voyiadjis and Abu-Lebdeh,1994; Karabinis and Kiousis,1994;Este and Willam,1994;Menetrey and Willam,1995;Grassl et al.,2002).The main characteristic of these models is a plasticity yield surface that includes pressure sensitivity,path sensitivity,non-associative?ow rule,and work or strain hardening.However,these works failed to address the degradation of the material sti?ness due to micro-cracking.On the other hand,others have used the continuum damage theory alone to model the material nonlinear behavior such that the mechanical e?ect of the pro-gressive micro-cracking and strain softening are represented by a set of internal state vari-ables which act on the elastic behavior(i.e.decrease of the sti?ness)at the macroscopic level (e.g.Loland,1980;Ortiz and Popov,1982;Krajcinovic,1983,1985;Resende and Martin, 1984;Simo and Ju,1987a,b;Mazars and Pijaudier-Cabot,1989;Lubarda et al.,1994). However,there are several facets of concrete behavior(e.g.irreversible deformations, inelastic volumetric expansion in compression,and crack opening/closure e?ects)that can-not be represented by this method,just as plasticity,by itself,is insu?cient.Since both micro-cracking and irreversible deformations are contributing to the nonlinear response of concrete,a constitutive model should address equally the two physically distinct modes of irreversible changes and should satisfy the basic postulates of thermodynamics.
1876U.Cicekli et al./International Journal of Plasticity23(2007)1874–1900 Combinations of plasticity and damage are usually based on isotropic hardening com-bined with either isotropic(scalar)or anisotropic(tensor)damage.Isotropic damage is widely used due to its simplicity such that di?erent types of combinations with plasticity models have been proposed in the literature.One type of combination relies on stress-based plasticity formulated in the e?ective(undamaged)space(e.g.Yazdani and Schreyer,1990; Lee and Fenves,1998;Gatuingt and Pijaudier-Cabot,2002;Jason et al.,2004;Wu et al., 2006),where the e?ective stress is de?ned as the average micro-scale stress acting on the undamaged material between micro-defects.Another type is based on stress-based plastic-ity in the nominal(damaged)stress space(e.g.Bazant and Kim,1979;Ortiz,1985;Lubliner et al.,1989;Imran and Pantazopoulu,2001;Ananiev and Ozbolt,2004;Kratzig and Poll-ing,2004;Menzel et al.,2005;Bru¨nig and Ricci,2005),where the nominal stress is de?ned as the macro-scale stress acting on both damaged and undamaged material.However,it is shown by Abu Al-Rub and Voyiadjis(2004)and Voyiadjis et al.(2003,2004)that coupled plastic-damage models formulated in the e?ective space are numerically more stable and attractive.On the other hand,for better characterization of the concrete damage behavior, anisotropic damage e?ects,i.e.di?erent micro-cracking in di?erent directions,should be characterized.However,anisotropic damage in concrete is complex and a combination with plasticity and the application to structural analysis is straightforward(e.g.Yazdani and Schreyer,1990;Abu-Lebdeh and Voyiadjis,1993;Voyiadjis and Kattan,1999;Carol et al.,2001;Hansen et al.,2001),and,therefore,it has been avoided by many authors.
Consequently,with inspiration from all the previous works,a coupled anisotropic dam-age and plasticity constitutive model that can be used to predict the concrete distinct behavior in tension and compression is formulated here within the basic principles of ther-modynamics.The proposed model includes important aspects of the concrete nonlinear behavior.The model considers di?erent responses of concrete under tension and compres-sion,the e?ect of sti?ness degradation,and the sti?ness recovery due to crack closure dur-ing cyclic loading.The yield criterion that has been proposed by Lubliner et al.(1989)and later modi?ed by Lee and Fenves(1998)is adopted.Pertinent computational aspects con-cerning the algorithmic aspects and numerical implementation of the proposed constitu-tive model in the well-known?nite element code ABAQUS(2003)are presented.Some numerical applications of the model to experimental tests of concrete specimens under dif-ferent uniaxial and biaxial tension and compression loadings are provided to validate and demonstrate the capability of the proposed model.
2.Modeling anisotropic damage in concrete
In the current literature,damage in materials can be represented in many forms such as speci?c void and crack surfaces,speci?c crack and void volumes,the spacing between cracks or voids,scalar representation of damage,and general tensorial representation of damage.Generally,the physical interpretation of the damage variable is introduced as the speci?c damaged surface area(Kachonov,1958),where two cases are considered:iso-tropic(scalar)damage and anisotropic(tensor)damage density of micro-cracks and micro-voids.However,for accurate interpretation of damage in concrete,one should con-sider the anisotropic damage case.This is attributed to the evolution of micro-cracks in concrete whereas damage in metals can be satisfactorily represented by a scalar damage variable(isotropic damage)for evolution of voids.Therefore,for more reliable represen-tation of concrete damage anisotropic damage is considered in this study.
The e?ective(undamaged)con?guration is used in this study in formulating the damage constitutive equations.That is,the damaged material is modeled using the constitutive laws of the e?ective undamaged material in which the Cauchy stress tensor,r ij,can be replaced by the e?ective stress tensor, r ij(Cordebois and Sidoro?,1979;Murakami and Ohno,1981;Voyiadjis and Kattan,1999):
r ij?M ijkl r kle1Twhere M ijkl is the fourth-order damage e?ect tensor that is used to make the stress tensor symmetrical.There are di?erent de?nitions for the tensor M ijkl that could be used to sym-metrize r ij(see Voyiadjis and Park,1997;Voyiadjis and Kattan,1999).In this work the de?nition that is presented by Abu Al-Rub and Voyiadjis(2003)is adopted:
M ijkl?2?ed ijàu ijTd kltd ijed klàu klT à1e2T
where d ij is the Kronecker delta and u ij is the second-order damage tensor whose evolution will be de?ned later and it takes into consideration di?erent evolution of damage in di?er-ent directions.In the subsequence of this paper,the superimposed dash designates a var-iable in the undamaged con?guration.
The transformation from the e?ective(undamaged)con?guration to the damaged one can be done by utilizing either the strain equivalence or strain energy equivalence hypoth-eses(see Voyiadjis and Kattan,1999).However,in this work the strain equivalence hypothesis is adopted for simplicity,which basically states that the strains in the damaged con?guration and the strains in the undamaged(e?ective)con?guration are equal.There-fore,the total strain tensor e ij is set equal to the corresponding e?ective tensor e ij(i.e.
e ij? e ijT,which can be decomposed into an elastic strain e e
ij (= e e
ij
Tand a plastic strain
e p ij(= e p ijTsuch that:
e ij?e e
ij te p ij? e e
ij
t e p ij? e ije3T
It is noteworthy that the physical nature of plastic(irreversible)deformations in con-crete is not well-founded until now.Whereas the physical nature of plastic strain in metals is well-understood and can be attributed to the generation and motion of dislocations along slip planes.Therefore,in metals any additional permanent strains due to micro-cracking and void growth can be classi?ed as a damage strain.These damage strains are shown by Abu Al-Rub and Voyiadjis(2003)and Voyiadjis et al.(2003,2004)to be minimal in metals and can be simply neglected.Therefore,the plastic strain in Eq.(3) incorporates all types of irreversible deformations whether they are due to tensile micro-cracking,breaking of internal bonds during shear loading,and/or compressive con-solidation during the collapse of the micro-porous structure of the cement matrix.In the current work,it is assumed that plasticity is due to damage evolution such that damage occurs before any plastic deformations.However,this assumption needs to be validated by conducting microscopic experimental characterization of concrete damage.
Using the generalized Hook’s law,the e?ective stress is given as follows: r ij?E ijkl e e
kl
e4Twhere E ijkl is the fourth-order undamaged elastic sti?ness tensor.For isotropic linear-elas-tic materials,E ijkl is given by
E ijkl?2GI d
ijkl tKI ijkle5T
U.Cicekli et al./International Journal of Plasticity23(2007)1874–19001877
where I d
ijkl ?I ijklà1
3
d ij d kl is th
e deviatoric part o
f the fourth-order identity tensor
I ijkl?1
2ed ik d jltd il d jkT,and G?E=2e1tmTand K?E=3e1à2mTare the e?ective shear
and bulk moduli,respectively,with E being the Young’s modulus and m is the Poisson’s ratio which are obtained from the stress–strain diagram in the e?ective con?guration.
Similarly,in the damaged con?guration the stress–strain relationship in Eq.(4)can be expressed by:
r ij?E ijkl e e
kl
e6Tsuch that one can express the elastic strain from Eqs.(4)and(5)by the following relation:
e e ij ?Eà1
ijkl
r kl?Eà1
ijkl
r kle7T
where Eà1
ijkl is the inverse(or compliance tensor)of the fourth-order damaged elastic tensor
E ijkl,which are a function of the damage variable u ij.
By substituting Eq.(1)into Eq.(7),one can express the damaged elasticity tensor E ijkl in terms of the corresponding undamaged elasticity tensor E ijkl by the following relation:
E ijkl?Mà1
ijmn
E mnkle8TMoreover,combining Eqs.(3)and(7),the total strain e ij can be written in the following form:
e ij?Eà1
ijkl r klte p ij?Eà1
ijkl
r klte p ije9T
By taking the time derivative of Eq.(3),the rate of the total strain,_e ij,can be written as _e ij?_e e
ij
t_e p ije10T
where_e e
ij and_e p ij are the rate of the elastic and plastic strain tensors,respectively.
Analogous to Eq.(9),one can write the following relation in the e?ective con?guration:
_e ij?Eà1
ijkl _ r
kl
t_e p ije11T
However,since E ijkl is a function of u ij,a similar relation as Eq.(11)cannot be used. Therefore,by taking the time derivative of Eq.(9),one can write_e ij in the damaged con-?guration as follows:
_e ij?Eà1
ijkl _r klt_Eà1
ijkl
r klt_e p ije12T
Concrete has distinct behavior in tension and compression.Therefore,in order to ade-quately characterize the damage in concrete due to tensile,compressive,and/or cyclic loadings the Cauchy stress tensor(nominal or e?ective)is decomposed into a positive and negative parts using the spectral decomposition technique(e.g.Simo and Ju, 1987a,b;Krajcinovic,1996).Hereafter,the superscripts‘‘+”and‘‘à”designate,respec-tively,tensile and compressive entities.Therefore,r ij and r ij can be decomposed as follows:
r ij?rt
ij trà
ij
; r ij? rt
ij
t rà
ij
e13T
where rt
ij is the tension part and rà
ij
is the compression part of the stress state.
The stress tensors rt
ij and rà
ij
can be related to r ij by
rtkl ?Pt
klpq
r pqe14T
ràkl ??I klpqàPt
ijpq
r pq?Pà
klpq
r pqe15T
1878U.Cicekli et al./International Journal of Plasticity23(2007)1874–1900
such that Pt
ijkl tPà
ijkl
?I ijkl.The fourth-order projection tensors Pt
ijkl
and Pà
ijkl
are de?ned
as follows:
Ptijpq ?
X3
k?1
He^rekTTnekTi nekTj nekT
p
nekT
q
;Pà
klpq
?I klpqàPt
ijpq
e16T
where He^ rekTTdenotes the Heaviside step function computed at k th principal stress^rekTof r ij and nekTi is the k th corresponding unit principal direction.In the subsequent develop-ment,the superscript hat designates a principal value.
Based on the decomposition in Eq.(13),one can assume that the expression in Eq.(1) to be valid for both tension and compression,however,with decoupled damage evolution in tension and compression such that:
rtij ?Mt
ijkl
rt
kl
; rà
ij
?Mà
ijkl
rà
kl
e17T
where Mt
ijkl is the tensile damage e?ect tensor and Mà
ijkl
is the corresponding compressive
damage e?ect tensor which can be expressed using Eq.(2)in a decoupled form as a func-
tion of the tensile and compressive damage variables,ut
ij and uà
ij
,respectively,as follows:
Mt
ijkl ?2?ed ijàut
ij
Td kltd ijed klàut
kl
T à1;Mà
ijkl
?2?ed ijàuà
ij
Td kltd ijed klàuà
kl
T à1
e18T
Now,by substituting Eq.(17)into Eq.(13)2,one can express the e?ective stress tensor as the decomposition of the fourth-order damage e?ect tensor for tension and compression such that:
r ij?Mt
ijkl rt
kl
tMà
ijkl
rà
kl
e19T
By substituting Eqs.(14)and(15)into Eq.(19)and comparing the result with Eq.(1), one can obtain the following relation for the damage e?ect tensor such that:
M ijpq?Mt
ijkl Pt
klpq
tMà
ijkl
Pà
klpq
e20T
Using Eq.(16)2,the above equation can be rewritten as follows:
M ijpq?Mt
ijkl àMà
ijkl
Pt
klpq tMà
ijpq
e21T
One should notice the following:
M ijkl?Mt
ijkl tMà
ijkl
e22T
or
u ij?ut
ij tuà
ij
e23T
It is also noteworthy that the relation in Eq.(21)enhances a coupling between tensile
and compressive damage through the fourth-order projection tensor Pt
ijkl .Moreover,for
isotropic damage,Eq.(20)can be written as follows:
M ijkl?
Pt
ijkl
1àut
t
Pà
ijkl
1àuà
e24T
It can be concluded from the above expression that by adopting the decomposition of the scalar damage variable u into a positive u+part and a negative uàpart still enhances a
damage anisotropy through the spectral decomposition tensors Pt
ijkl and Pà
ijkl
.However,
this anisotropy is weak as compared to the anisotropic damage e?ect tensor presented in Eq.(21).
U.Cicekli et al./International Journal of Plasticity23(2007)1874–19001879
3.Elasto-plastic-damage model
In this section,the concrete plasticity yield criterion of Lubliner et al.(1989)which was later modi?ed by Lee and Fenves(1998)is adopted for both monotonic and cyclic load-ings.The phenomenological concrete model of Lubliner et al.(1989)and Lee and Fenves (1998)is formulated based on isotropic(scalar)sti?ness degradation.Moreover,this model adopts one loading surface that couples plasticity to isotropic damage through the e?ective plastic strain.However,in this work the model of Lee and Fenves(1998)is extended for anisotropic damage and by adopting three loading surfaces:one for plastic-ity,one for tensile damage,and one for compressive damage.The plasticity and the com-pressive damage loading surfaces are more dominate in case of shear loading and compressive crushing(i.e.modes II and III cracking)whereas the tensile damage loading surface is dominant in case of mode I cracking.
The presentation in the following sections can be used for either isotropic or anisotropic damage since the second-order damage tensor u ij degenerates to the scalar damage vari-able in case of uniaxial loading.
3.1.Uniaxial loading
In the uniaxial loading,the elastic sti?ness degradation variables are assumed as
increasing functions of the equivalent plastic strains et
eq and eà
eq
with et
eq
being the tensile
equivalent plastic strain and eà
eq being the compressive equivalent plastic strain.It should
be noted that the material behavior is controlled by both plasticity and damage so that, one cannot be considered without the other(see Fig.1).
For uniaxial tensile and compressive loading, rt
ij and rà
ij
are given as(Lee and Fenves,
1998)
rt?e1àutTE ete?e1àutTEeetàetpTe25Trà?e1àuàTE eàe?e1àuàTEeeààeàpTe26TThe rate of the equivalent(e?ective)plastic strains in compression and tension,eàep and etep,are,respectively,given as follows in case of uniaxial loading:
1880U.Cicekli et al./International Journal of Plasticity23(2007)1874–1900
_eteq ?_e p
11
;_eà
eq
?à_e p
11
e27T
such that
eàeq ?
Z t
_eà
eq
d t;et
eq
?
Z t
_et
eq
d te28T
Propagation of cracks under uniaxial loading is in the transverse direction to the stress direction.Therefore,the nucleation and propagation of cracks cause a reduction of the capacity of the load-carrying area,which causes an increase in the e?ective stress.This has little e?ect during compressive loading since cracks run parallel to the loading direc-tion.However,under a large compressive stress which causes crushing of the material,the e?ective load-carrying area is also considerably reduced.This explains the distinct behav-ior of concrete in tension and compression as shown in Fig.2.
It can be noted from Fig.2that during unloading from any point on the strain soften-ing path(i.e.post peak behavior)of the stress–strain curve,the material response seems to be weakened since the elastic sti?ness of the material is degraded due to damage evolution. Furthermore,it can be noticed from Fig.2a and b that the degradation of the elastic sti?-ness of the material is much di?erent in tension than in compression,which is more obvi-ous as the plastic strain increases.Therefore,for uniaxial loading,the damage variable can be presented by two independent damage variables u+and uà.Moreover,it can be noted that for tensile loading,damage and plasticity are initiated when the equivalent applied
stress reaches the uniaxial tensile strength ft
0as shown in Fig.2a whereas under compres-
sive loading,damage is initiated earlier than plasticity.Once the equivalent applied stress
reaches fà
0(i.e.when nonlinear behavior starts)damage is initiated,whereas plasticity
occurs once fà
u is reached.Therefore,generally ft
?ft
u
for tensile loading,but this is
not true for compressive loading(i.e.fà
0?fà
u
T.However,one may obtain fà
%fà
u
in case
of ultra-high strength concrete.
3.2.Multiaxial loading
The evolution equations for the hardening variables are extended now to multiaxial loadings.The e?ective plastic strain for multiaxial loading is given as follows(Lubliner et al.,1989;Lee and Fenves,1998):
U.Cicekli et al./International Journal of Plasticity23(2007)1874–19001881
_e teq ?r e^ r ij T^_e p max
e29T_e àeq ?àe1àr e^ r ij TT^_e p min e30T
where ^_e p max and ^_e p min are the maximum and minimum principal values of the plastic strain tensor _e p ij such that ^_e p 1>^_e p 2>^_e p 3where ^_e p max ?^_e p 1and ^_e p min ?^_e
p 3.Eqs.(29)and (30)can be written in tensor format as follows:
_j p i ?H ij ^_e p j
e31Tor equivalently _e teq 0_e àeq
8><>:9>=>;?H t0000000H à264375^_e p 1^_e p 2^_e p 38><>:9>=>;e32T
where
H t?r e^ r
ij Te33TH à?àe1àr e^ r ij TTe34TThe dimensionless parameter r e^ r
ij Tis a weight factor depending on principal stresses and is de?ned as follows (Lubliner et al.,1989):
r e^ r ij T?P 3k
?1h ^ r k i P k ?1j ^ r k
j e35Twhere h i is the Macauley bracket,and presented as h x i ?1
ej x j tx T,k ?1;2;3.Note that r e^ r
ij T?r e^r ij T.Moreover,depending on the value of r e^r ij T,–in case of uniaxial tension ^ r k P 0and r e^ r ij T?1,–in case of uniaxial compression ^ r
k 60and r e^ r ij T?03.3.Cyclic loading
It is more di?cult to address the concrete damage behavior under cyclic loading;i.e.transition from tension to compression or vise versa such that one would expect that under cyclic loading crack opening and closure may occur and,therefore,it is a challenging task to address such situations especially for anisotropic damage evolution.Experimentally,it is shown that under cyclic loading the material goes through some recovery of the elastic sti?ness as the load changes sign during the loading process.This e?ect becomes more sig-ni?cant particularly when the load changes sign during the transition from tension to com-pression such that some tensile cracks tend to close and as a result elastic sti?ness recovery occurs during compressive loading.However,in case of transition from compression to tension one may thus expect that smaller sti?ness recovery or even no recovery at all may occur.This could be attributed to the fast opening of the pre-existing cracks that had formed during the previous tensile loading.These re-opened cracks along with the new cracks formed during the compression will cause further reduction of the elastic sti?ness that the body had during the ?rst transition from tension to compression.The
1882U.Cicekli et al./International Journal of Plasticity 23(2007)1874–1900
consideration of sti?ness recovery e?ect due to crack opening/closing is therefore impor-tant in de?ning the concrete behavior under cyclic loading.Eq.(21)does not incorporate the elastic sti?ness recovery phenomenon as well as it does not incorporate any coupling between tensile damage and compressive damage and,therefore,the formulation of Lee and Fenves(1998)for cyclic loading is extended here for the anisotropic damage case.
Lee and Fenves(1998)de?ned the following isotropic damage relation that couples both tension and compression e?ects as well as the elastic sti?ness recovery during transi-tion from tension to compression loading such that:
u?1àe1às utTe1àuàTe36Twhere se06s61Tis a function of stress state and is de?ned as follows: se^ r ijT?s0te1às0Tre^ r ijTe37Twhere06s061is a constant.Any value between zero and one results in partial recovery of the elastic sti?ness.Based on Eqs.(36)and(37):
(a)when all principal stresses are positive then r=1and s=1such that Eq.(36)
becomes
u?1àe1àutTe1àuàTe38Twhich implies no sti?ness recovery during the transition from compression to tension since s is absent.
(b)when all principal stresses are negative then r=0and s?s0such that Eq.(36)
becomes
u?1àe1às0utTe1àuàTe39Twhich implies full elastic sti?ness recovery when s0?0and no recovery when s0?1.
In the following two approaches are proposed for extending the Lee and Fenves(1998)
model to the anisotropic damage case.The?rst approach is by multiplying ut
ij in Eq.(18)1
by the sti?ness recovery factor s:
Mt
ijkl ?2?ed ijàs ut
ij
Td kltd ijed klàs ut
kl
T à1e40T
such that the above expression replaces Mt
ijkl in Eq.(21)to give the total damage e?ect
tensor.
Another approach to enhance coupling between tensile damage and compressive dam-age as well as in order to incorporate the elastic sti?ness recovery during cyclic loading for the anisotropic damage case is by rewriting Eq.(36)in a tensor format as follows:
u ij?d ijàed ikàs ut
ik Ted jkàuà
jk
Te41T
which can be substituted back into Eq.(2)to get the?nal form of the damage e?ect tensor, which is shown next.
It is noteworthy that in case of full elastic sti?ness recovery(i.e.s=0),Eq.(41)reduces
to u ij?uà
ij and in case of no sti?ness recovery(i.e.s=1),Eq.(41)takes the form of
u ij ?uà
ij
tut
ik
àut
ik
uà
jk
such that both ut
ij
and uà
ij
are coupled.This means that during
the transition from tension to compression some cracks are closed or partially closed which could result in partial recovery of the material sti?ness(i.e.s>0)in the absence U.Cicekli et al./International Journal of Plasticity23(2007)1874–19001883
of damage healing mechanisms.However,during transition from compression to tension,the existing and generated cracks during compressive loading could grow more which causes further sti?ness degradation and such that no sti?ness recovery is expected.This is why the parameter s only a?ects the tensile damage variable u tij .However,one may argue that during tensile loading a minimal sti?ness recovery could occur due to geomet-rical constraints set up by the interaction between the cracks and the microstructure of concrete.This is what is suggested in the theory manuals of ABAQUS (2003)for the implemented plastic-damage (isotropic)model of Lubliner et al.(1989).However,the for-mer approach is followed in this work where elastic sti?ness recovery occurs only during the transition from tensile to compressive state of stress.
Now from Eq.(1),one can write the following:
r ij ?M à1ijkl r
kl e42T
where M à1ijkl can be written from Eq.(2)as follows:M à1ijkl ?12?ed ij àu ij Td kl td ij ed kl àu kl T e43T
By substituting Eq.(41)into Eq.(43),one gets a coupled damage e?ect tensor in terms of
u àij and u tij as follows:
M à1ijkl ?12
?ed im às u tim Ted jm àu àjm Td kl td ij ed km às u tkm Ted lm àu àlm T e44TIt can be noted that the above expression couples tensile damage and compressive damage.Moreover,it takes into account the sti?ness recovery during the transition from tension to compression.For full elastic sti?ness recovery (s =0),Eq.(44)yields Eq.(18)2such that M ijkl ?M àijkl .However,for no elastic sti?ness recovery (s =1),Eq.(44)reduces to
M à1ijkl ?12
?ed im àu tim Ted jm àu àjm Td kl td ij ed km àu tkm Ted lm àu àlm T e45T
which shows a coupling between tensile damage and compressive damage.
3.4.Plasticity yield surface
For the representation of concrete behavior under tensile and compressive loadings,a yield criterion is necessary.It is known that concrete behaves di?erently in tension and compression,thus,the plasticity yield criterion cannot be assumed to be similar.Assuming the same yield criterion for both tension and compression for concrete materials can lead to over/under estimation of plastic deformation (Lubliner et al.,1989).The yield criterion of Lubliner et al.(1989)that accounts for both tension and compression plasticity is adopted in this work.This criterion has been successful in simulating the concrete behav-ior under uniaxial,biaxial,multiaxial,and cyclic loadings (Lee and Fenves,1998and the references outlined there).This criterion is expressed here in the e?ective (undamaged)con?guration and is given as follows:f ????????3J 2p ta I 1tb ej p i TH e^ r max T^ r max àe1àa T_c àee àeq T60
e46Twhere J 2? s ij s ij =2is the second-invariant of the e?ective deviatoric stress s ij ? r ij à r kk d ij =3,I 1? r kk is the ?rst-invariant of the e?ective stress r ij ,j p i ?R t 0_j p i dt is 1884U.Cicekli et al./International Journal of Plasticity 23(2007)1874–1900
the equivalent plastic strain which is de?ned in Eq.(31),He^ r maxTis the Heaviside step function(H=1for^ r max>0and H=0for^ r max<0T,and^ r max is the maximum principal stress.
The parameters a and b are dimensionless constants which are de?ned by Lubliner et al. (1989)as follows:
a?ef b0=fà
Tà1
2ef b0=fà
Tà1
e47T
b?e1àaTcàeeà
eq
T
cteet
eq
T
àe1taTe48T
where f b0and fà
0are the initial equibiaxial and uniaxial compressive yield stresses,respec-
tively.Experimental values for f b0=fà
0lie between1.10and1.16;yielding a between0.08
and0.12.For more details about the derivation of both Eqs.(47)and(48),the reader is referred to Lubliner et al.(1989).
Since the concrete behavior in compression is more of a ductile behavior,the evolution of the compressive isotropic hardening function_càis de?ned by the following exponential law:
_cà?beQàcàT_eà
eq
e49Twhere Q and b are material constants characterizing the saturated stress and the rate of saturation,respectively,which are obtained in the e?ective con?guration of the compres-sive uniaxial stress–strain diagram.However,a linear expression is assumed for the evo-lution of the tensile hardening function_ctsuch that:
_ct?h_et
eq
e50Twhere h is a material constant obtained in the e?ective con?guration of the tensile uniaxial stress–strain diagram.
3.5.Non-associative plasticity?ow rule
The shape of the concrete loading surface at any given point in a given loading state should be obtained by using a non-associative plasticity?ow rule.This is important for realistic modeling of the volumetric expansion under compression for frictional materials such as concrete.Basically,?ow rule connects the loading surface and the stress–strain relation.When the current yield surface f is reached,the material is considered to be in plastic?ow state upon increase of the loading.In the present model,the?ow rule is given as a function of the e?ective stress r ij by
_e p ij ?_k p
o F p
o r ij
e51T
where_k p is the plastic loading factor or known as the Lagrangian plasticity multiplier, which can be obtained using the plasticity consistency condition,_f?0,such that f60;_k p P0;_k p f?0;_k p_f?0e52TThe plastic potential F p is di?erent than the yield function f(i.e.non-associated)and, therefore,the direction of the plastic?ow o F p=o r ij is not normal to f.One can adopt the Drucker–Prager function for F p such that:
U.Cicekli et al./International Journal of Plasticity23(2007)1874–19001885
F p ????????3J 2p ta p I 1e53T
where a p is the dilation constant.Then the plastic ?ow direction o F p =o r
ij is given by o F p o r
ij ?32 s ij ???????3J 2p ta p d ij e54T
3.6.Tensile and compressive damage surfaces
The anisotropic damage growth function proposed by Chow and Wang (1987)is adopted in this study.However,this function is generalized here in order to incorporate both tensile and compressive damage separately such that:g ?????????????????????????12
Y ?ij L ?ijkl Y ?ij r àK ?eu ?eq T60e55Twhere the superscript ±designates tension,+,or compression,à,K ?is the tensile or com-
pressive damage isotropic hardening function,K ??K ?0when there is no damage,K ?0is
the tensile or compressive initial damage parameter (i.e.damage threshold),L ijkl is a fourth-order symmetric tensor,and Y ij is the damage driving force that characterizes dam-age evolution and is interpreted here as the energy release rate (e.g.Voyiadjis and Kattan,1992a,b,1999;Abu Al-Rub and Voyiadjis,2003;Voyiadjis et al.,2003,2004).In order to simplify the anisotropic damage formulation,L ijkl is taken in this work as the fourth-order identity tensor I ijkl .
The rate of the equivalent damage _u ?eq is de?ned as follows (Voyiadjis and Kattan,
1992a,b,1999;Abu Al-Rub and Voyiadjis,2003;Voyiadjis et al.,2003,2004):_u ?eq ?????????????_u ?ij _u ?ij q with u ?eq ?Z t 0_u ?eq d t
e56TThe evolution of the tensile damage isotropic hardening function K +is assumed to have the following expression (Mazars and Pijaudier-Cabot,1989):
_K t?K t
B ttK 0K texp ?àB te1àK tK t0T _u teq e57T
whereas the evolution of the compressive damage isotropic hardening function K àis as-sumed to have a slightly di?erent form (Mazars and Pijaudier-Cabot,1989):_K à?K à0B àexp àB à1àK àK à0
_u àeq e58Twhere K ?0is the damage threshold which is interpreted as the area under the linear portion of the stress–strain diagram such that:
K ?0?f ?2
02E
e59TThe material constant B ?is related to the fracture energy G ?f ,which is shown in Fig.2for both tension and compression,as follows (Onate et al.,1988):
B ??G ?f E ‘f 0à12
"#à1P 0e60T1886
U.Cicekli et al./International Journal of Plasticity 23(2007)1874–1900
where‘is a characteristic length scale parameter that usually has a value close to the size of the smallest element in a?nite element mesh.Of course,a local constitutive model with strain softening(localization)cannot provide an objective description of localized damage and needs to be adjusted or regularized(see Voyiadjis et al.,2004).As a remedy,one may use a simple approach with adjustment of the damage law(which controls softening) according to the size of the?nite element,‘,in the spirit of the traditional crack-band theory(Bazant and Kim,1979).Therefore,the parameters B?are used to obtain mesh-independent results.A more sophisticated remedy is provided by a localization lim-iter based on weighted spatial averaging of the damage variable,which will be presented in a forthcoming paper by the authors.
The model response in the damage domain is characterized by the Kuhn–Tucker com-plementary conditions as follows:
g?60;_k?
d g??0;and_g?
<0)_k?
d
?0
?0)_k?
d
?0
?0)_k?
d
>0
8
><
>:
9
>=
>;()
effectiveeundamaged stateT
damage initiation
damagegrowth
8
><
>:
e61T
With the consideration of the above equation,one writes speci?c conditions for tensile and compressive stages:when gt<0there is no tensile damage and if gt>0there is ten-sile damage in the material;and when gà<0there is no compressive damage in the mate-rial and if gà>0,it means there is compressive damage.
4.Consistent thermodynamic formulation
In this section,the thermodynamic admissibility of the proposed elasto-plastic-damage model is checked following the internal variable procedure of Coleman and Gurtin(1967). The constitutive equations listed in the previous section are derived from the second law of thermodynamics,the expression of Helmholtz free energy,the additive decomposition of the total strain rate in to elastic and plastic components,the Clausius–Duhem inequality, and the maximum dissipation principle.
4.1.Thermodynamic laws
The Helmholtz free energy can be expressed in terms of a suitable set of internal state variables that characterize the elastic,plastic,and damage behavior of concrete.In this work the following internal variables are assumed to satisfactory characterize the concrete behavior both in tension and compression such that:
w?wee e
ij ;ut
ij
;uà
ij
;ut
eq
;uà
eq
;et
eq
;eà
eq
Te62T
where ut
eq and uà
eq
are the equivalent(accumulated)damage variables for tension and com-
pression,respectively,which are de?ned as u?
eq ?
R t
_u?
eq
d t.Similarly,et
eq
and eà
eq
are the
equivalent tensile and compressive plastic strains that are used here to characterize the
plasticity isotropic hardening,e?
eq ?
R t
_e?
eq
d t.
One may argue that it is enough to incorporate the damage tensor u?
ij instead of incor-
porating both u?
ij and u?
eq
.However,the second-order tensor u?
ij
introduces anisotropy
while the scalar variable u?
eq introduces isotropy characterized by additional hardening.
U.Cicekli et al./International Journal of Plasticity23(2007)1874–19001887
The Helmholtz free energy is given as a decomposition of elastic,w e ,plastic,w p ,and damage,w d ,parts such that:
w ?w e ee e ij ;u tij ;u àij Ttw p ee teq ;e àeq Ttw d eu teq ;u àeq Te63T
It can be noted from the above decomposition that damage a?ects only the elastic prop-erties and not the plastic ones.However,for more realistic description,one should intro-duce the damage variables in the plastic part of the Helmholtz free energy (see Abu Al-Rub and Voyiadjis,2003;Voyiadjis et al.,2003,2004).However,these e?ects are not signi?cant for brittle materials and can,therefore,be neglected.
The elastic free energy w e is given in term of the second-order damage tensors u ?ij as follows:
w e ?12e e ij E ijkl eu tij ;u àij Te e kl ?12r ij e e ij ?12
er tij tr àij Te e ij e64TSubstituting Eq.(17)along with Eqs.(1)and (20),considering r tij ?P tijkl r
kl and r
àij ?P àijkl r kl into Eq.(64)and making some algebraic simpli?cations,one obtains the fol-lowing relation:
w e ?12M à1ijpq r pq e e ij ?12
r ij e e ij e65Tsuch that one can relate the nominal stress to the e?ective stress through Eq.(42)with M à1ijkl given by
M à1ijpq ?M tà1ijkl P tklpq tM àà1ijkl P àklpq e66TThis suggests that the inverse of Eq.(66)yields Eq.(20)if the projection tensors P tklpq and P àklpq from the e?ective stress r
ij coincide with that from the nominal stress r ij ,which is not necessarily true for anisotropic damage.Similarly,for isotropic damage,one can write Eq.
(66)as follows:
M à1ijkl ?e1àu tTP tijkl te1àu àTP àijkl e67TThe Clausius–Duhem inequality for isothermal case is given as follows:
r ij _e ij àq _w P 0e68T
where q is the material density.Taking the time derivative of Eq.(63),the following expression can be written as:
_w ?o w e
o e ij _e e ij to w e
o u tij _u tij to w e
o u ij _u àij to w p
o e teq _e teq to w p
o e àeq _e àeq to w d
o u teq _u teq to w d
o u àeq _u àeq e69T
By plugging the above equation into the Clausius–Duhem inequality,Eq.(68),and mak-ing some simpli?cations,one can obtain the following relations for any admissible states such that:
r ij ?q o w e
o e e ij
e70Tand
r ij _e p ij tY tij _u tij tY àij _u àij àc t_e teq àc à_e àeq àK t_u teq àK à_u àeq P 0e71T
1888U.Cicekli et al./International Journal of Plasticity 23(2007)1874–1900
where the damage and plasticity conjugate forces that appear in the above expression are de?ned as follows:
Ytij ?àq
o w e
o u ij
e72T
Yàij ?àq
o w e
o uàij
e73T
Kt?q o w d
o ut
eq
e74T
Kà?q o w d
o uà
eq
e75T
ct?q o w p
o et
eq
e76T
cà?q o w p
o eà
eq
e77T
Therefore,one can rewrite the Clausius–Duhem inequality to yield the dissipation energy, P,due to plasticity,P p,and damage,P d,such that
P?P dtP p P0e78Twith
P p?r ij_e p ijàct_et
eq àcà_eà
eq
P0e79T
P d?Yt
ij _ut
ij
tYà
ij
_uà
ij
àKt_ut
eq
àKà_uà
eq
P0e80T
The rate of the internal variables associated with plastic and damage deformations are obtained by utilizing the calculus of functions of several variables with the plasticity and
damage Lagrangian multipliers,_k p and_k?
d ,such that th
e following objective function can
be de?ned:
X?Pà_k p F pà_kt
d gtà_kà
d
gàP0e81T
Using the well-known maximum dissipation principle(Simo and Honein,1990;Simo and Hughes,1998),which states that the actual state of the thermodynamic forces(r ij,
Y?ij ,c±,K±)is that which maximizes the dissipation function over all other possible admis-
sible states,hence,one can maximize the objective function X by using the necessary con-ditions as follows:
o X o r ij ?0;
o X
o Y?
ij
?0;
o X
o c?
?0;
o X
o K?
?0e82T
Substituting Eq.(81)along with Eqs.(79)and(80)into Eq.(82)yield the following thermodynamic laws:
_e p ij ?_k p
o F p
o r ij
e83T
_utij ?_kt
d
o gt
t
ij
e84TU.Cicekli et al./International Journal of Plasticity23(2007)1874–19001889
_uàij ?_kà
d
o gà
o Yà
ij
e85T
_eteq ?_k p
o F p
t
e86T
_eàeq ?_k p
o F p
o cà
e87T
_uteq ?_kt
d
o gt
o Kt
e88T
_uàeq ?_kà
d
o gà
o Kà
e89T
4.2.The Helmholtz free energy function
The elastic part of the Helmholtz free energy function,w e,as presented in Eq.(64)can be substituted into Eq.(70)to yield the following stress–strain relation:
r ij?E ijkl e e
kl ?E ijklee klàe p
kl
Te90T
Now,one can obtain expressions for the damage driving forces Y?
ij from Eqs.(64),(72),
and(73)as follows:
Y?rs ?à
1
2
e e
ij
o E ijkl
o u?
rs
e e
kl
e91T
By taking the derivative of Eq.(8)with respect to the damage parameter u?
ij
one obtains
o E ijkl o u?
rs ?
o Mà1
ijmn
o u?
rs
E mnkle92T
Now,by substituting Eq.(92)into Eq.(91),one obtains the following expression for Y?
ij
:
Y?rs ?à
1
2
e e
ij
o Mà1
ijmn
o u?
rs
E mnkl e e
kl
e93T
where from Eq.(66),one can write the following expression:
o Mà1
ijmn o u?
rs ?
o M?à1
ijpq
o u?
rs
P?
pqmn
e94T
One can also rewrite Eq.(93)in terms of the e?ective stress tensor by replacing e e
kl from Eq.
(4)as follows:
Y?rs ?à
1
2
Eà1
ijab
r ab
o Mà1
ijpq
o u?
rs
r pqe95T
The plastic part of the Helmholtz free energy function is postulated to have the follow-ing form(Abu Al-Rub and Voyiadjis,2003;Voyiadjis et al.,2003,2004):
qw p?ft
0etept
1
2
heet
eq
T2tfà
eà
eq
tQ eà
eq
t
1
b
expeàb eà
eq
T
e96T
1890U.Cicekli et al./International Journal of Plasticity23(2007)1874–1900
Substituting Eq.(96)into Eqs.(82)and(83)yields the following expressions for the plas-ticity conjugate forces c+and cà:
ct?ft
0th et
eq
e97T
cà?fà
0tQ1àexpeàb eà
eq
T
h i
e98T
such that by taking the time derivative of Eqs.(97)and(98)one can easily retrieve Eqs.
(49)and(50),respectively.
The damage part of the Helmholtz free energy function is postulated to have the follow-ing form:
qw d?K?
0?u?
eq
t
1
B?
fe1àu?
eq
Tlne1àu?
eq
Ttu?
eq
g e99T
where K?
0is the initial damage threshold de?ned in Eq.(59)and B±are material constants
which are expressed in terms of the fracture energy and an intrinsic length scale,Eq.(60).
Substituting Eq.(90)into Eqs.(74)and(75),one can easily obtain the following expres-sions for the damage driving forces K±:
K??K?
0?1à
1
B?
lne1àu?
eq
T e100T
By taking the time derivative of the above expression one retrieves the rate form of the damage hardening/softening function K±presented in Eqs.(57)and(58)such that:
_K??K ?0
B expàB?e1à
K?
K
T
_u?
eq
e101T
It is noteworthy that the expression that is presented in Eq.(58)for tensile damage is a slightly di?erent than the one shown in Eq.(101).However,in the remaining of this study, Eq.(57)is used.This is attributed to the better representation of the stress–strain diagram under tensile loading,when Eq.(58)is used instead of Eq.(101).
5.Numerical implementation
In this section,the time discretization and numerical integration procedures are pre-sented.The evolutions of the plastic and damage internal state variables can be obtained
if the Lagrangian multipliers_k p and_k?
d ar
e computed.The plasticity and damage consis-
tency conditions,Eqs.(52)and(61),are used for computing both_k p and_k?
d .This is shown
in the subsequent developments.Then,by applying the given strain increment
D e ij?eent1T
ij àeenTij and knowing the values of the stress and internal variables at the begin-
ning of the step from the previous step,eáTenT,the updated values at the end of the step,eáTent1T,are obtained.
The implemented integration scheme is divided into two sequential steps,corresponding to the plastic and damage parts of the model.In the plastic part,the plastic strain e p ij and the e?ective stress r ij at the end of the step are determined by using the classical radial return mapping algorithm(Simo and Hughes,1998)such that:
r ij? r tr
ij àE ijkl D e p
kl
? r tr
ij
àD k p E ijkl
o F p
o renT
kl
e102TU.Cicekli et al./International Journal of Plasticity23(2007)1874–19001891
where r tr
ij ? renTijtE ijkl D e kl is the trial stress tensor,which is easily evaluated from the given
strain increment.If the trial stress is not outside the yield surface,i.e.fe r tr
ij ;cenT
c
T60,the step
is elastic and one sets D k p?0, r tr
ij ? rent1T
ij
,e pent1T
ij
?e penT
ij
,c?ent1T?c?enT.However,if the trial
stress is outside the yield surface rent1T
ij ,e pent1T
ij
,and c?ent1Tare determined by computing D k p.
In the damage part,the nominal stress r ij at the end of the step is obtained from Eq.
(42)by knowing the damage variables u?
ij ,which can be calculated once D k?
d
are computed
from the damage consistency conditions.
5.1.Evolution of the plastic multiplier
From the e?ective consistency condition,one can write the following relation at n+1 step:
fent1T?fenTtD fe103Twhere
D f?o f
o r ij
D r ijt
o f
o^ r max
D^ r maxt
o f
o eà
eq
D eà
eq
t
o f
o et
eq
D et
eq
?0e104T
fent1T?
??????????????
3Jent1T
2
q
ta Ient1T
1
à bent1T^ rent1T
max
àe1àaTcàent1T?0e105T
bent1T?He^ rent1T
max
Tbent1Te106TMaking few arithmetic manipulations,one can obtain the plastic multiplier D k p from the following expression:
D k p?f tr
H
e107T
where f tr and H are given as
f tr?
???
3
2
r
k s tr
ij
kta I tr
1
à b^ r tr
max
àe1àaTcàenTe108T
H?3Gt9K a p at b Zte1àrTo f
o eà
eq
o F p
o^ r tr
min
àr
o f
o et
eq
o F p
o^ r tr
max
e109T
with
Z?
???
6
p
G
^ r tr
max
k s tr ij k
t3K a pà
???
2
3
r
G
I tr
1
k s tr ij k
e110T
o F p o^ r tr
min;max ?
???
3
2
r^ r tr
min;max
à1I tr
1
k s tr ij k
ta pe111T
o f o eà
eq ?àe1àaTQb expeàb eà
eq
T1à
h^ r max i
ct
e112T
o f o et
eq ?àh^ r max i
càe1àaTh
ectT
e113T
1892U.Cicekli et al./International Journal of Plasticity23(2007)1874–1900
It should be noted that,in the case of the Drucker–Prager potential function,a singular region occurs near the cone tip unless a p?0.The reason of the occurrence of the singular region is that any trial stress that belongs to the singular region cannot be mapped back to
a returning point on the given yield surface.
5.2.Evolution of the tensile and compressive damage multipliers
In the following,the damage multipliers,_k?
d ,ar
e obtained using the consistency condi-
tions in Eq.(61).The incremental expression for the damage consistency condition can be written as
g?ent1T?g?enTtD g??0e114Twhere g+is the damage surface function given in Eq.(55)and D gtis the increment of the tensile damage function which is expressed by
D g??o g?
o Y
ij
D Y?
ij
t
o g?
o K
D K?e115T
However,since Y?
ij is a function of r?
ij
and u?
ij
one can write the following:
D Y?
ij ?
o Y?
ij
o r?
kl
D r?
kl
t
o Y?
ij
o u?
kl
D u?
kl
e116T
where D u?
kl
is obtained from Eqs.(84)and(85)such that:
D u?
kl ?D k?
d
o g?
o Y
kl
e117T
and D r?
kl
can be obtained from Eq.(17)as follows:
D r?
kl ?
o Mà1?
klrs
o u?
mn
D u?
mn
r?
rs
tMà1?
klrs
D r?
rs
e118T
By substituting Eqs.(116)–(118)into Eq.(115)and noticing that_k??_u?
eq ?
????????????
_u?
ij
_u?
ij
q
,one
can obtain the following relation:
D g??o g?
o Y?
ij
o Y?
ij
o r?
kl
o Mà1?
klrs
o u?
mn
r?
rs
o g?
o Y?
mn
D k?
d
t
o g?
o Y?
ij
o Y?
ij
o r?
kl
Mà1?
klrs
D r?
rs
t
o g?
o Y?
ij
o Y?
ij
o u?
kl
?
o g?
o Y?
kl
D k?
d
t
o g?
o K?
o K?
o u?
eq
D k?
d
e119T
Substituting the above equation into Eq.(114),one obtains D k?
eq
by the following relation:
D k?
eq ?à
g?enT
H?
eq
e120T
where H?
eq is the tensile or compressive damage hardening/softening modulus and is given
as follows:
H?
eq ?
o g?
o Y?
ij
o Y?
ij
o r?
kl
o Mà1?
klrs
o u?
mn
r?
rs
o g?
o Y?
mn
t
o g?
o Y?
ij
o Y?
ij
o u?
kl
o g?
o Y?
kl
t
o g?
o K?
o K?
o u?
eq
e121TU.Cicekli et al./International Journal of Plasticity23(2007)1874–19001893
人教版八年级英语下册常用固定搭配总结
八下英语固定用法总结 1.Doing类 Have problems/trouble/difficulty in doing sth Mind doing sth Mind sb doing sth Finish doing sth Do one’s part in doing sth Keep on doing sth Keep doing sth Instead of doing sth Can’t stop/help doing sth Be busy doing sth Be interested in doing sth Succeed in doing sth Consider doing sth Allow doing sth 2.To do 类 Need to do sth Expect sb to do sth Agree to do sth Seem to do sth Wait for sb to do sth Used to do sth Make plans to do sth Ask sb to do sth Decide to do sth Want sb to do sth Want to do sth Learn to do sth Allow sb to do sth Tell sb to do sth Refuse to do sth Offer to do sth Try to do sth It takes some time to do sth Send sb to do sth Have time to do sth Hope to do sth Be able to do sth
旅游销售计划书
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The way常见用法
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compare用法与搭配
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动词的用法及各种搭配
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The way的用法及其含义(二)
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“穷游”泰国曼谷 策划人：刘婷婷 学号：2011211050 旅游地点：泰国曼谷 时间：9月上旬（泰国旅游淡季） 旅游必备：地图，防晒霜，墨镜，遮阳帽，日常用品，雨具等。 旅游着装——最好是T恤，牛仔裤。 旅游饮食——以当地食物为主。 旅游住宿——带上牙刷和梳子。 旅游天数：5天。 预备旅游景点有：大王宫—曼谷玉佛寺—金福寺—曼谷卧佛寺—王家田广场—泰国国家博物馆—云石寺—湄南河—绿山国家公园。 沟通语言：泰语及英语。 钱币使用：以泰铢为主。 行动预知经济费用为：3000元。 曼谷原意“天使之城”，是泰国首都，是东南亚第二大城市，有“佛庙之都”之誉，为黄袍佛国之泰国首都，是泰国，经济，文化和交通中心。曼谷位于湄南河畔，全市面积1568公里，人口600余万。市内河道纵横，货运频繁，有“东威尼斯”之称。这里的城市气息不浓，多是低矮的民居和商店。这个城市有四百多座佛教寺庙，其中最
有名的是大皇宫内的玉佛寺和在其河斜对面的黎明寺。其他景点有沙法里野生动物园等，另外曼谷亦是购的好地方之一，价廉物美且购物设施充足，所以购物亦是重点活动之一。 旅行预备工作： 其一：签证办理 办理签证的方法有：1：自己去泰国驻华使馆或领馆申请；2：通过旅行社代办签证，交一些手续费。泰国签证一般在一周之内，填写个人资料表、户口本、身份证、照片即可。 其二：机票预定： 由于是选在九月份去泰国旅游，机票比较宽松，不用提前预定就能买到。，方便实惠。由于没有通往泰国的火车，汽车又比较麻烦，费用高，且花费时间较长。所以，在选择交通工具上，只好选择飞机。 其三：钱币兑换：
泰国货币叫铢（泰国货币代码：THB），辅助货币还有撒旦和撒郎，两者的区别类似于1分币和25美分之间的区别。比如，100个撒丹等于1铢，这正如100个1分币等于1美元一样；而1个撒郎等于25个撒丹，正如25美分与25个分币一样。 注意事项： 1：去泰国旅游，吸汗的T恤和不短于七分的牛仔裤，外加一双透气的旅游鞋是首选。而且泰国的建筑大多富丽堂皇，如果穿得太花哨，相片出来后反而效果不好。此外，泰国的寺庙不允许穿无袖、露膝的衣服和凉拖入内，许多穿吊带短裙的女士因此被拦截在外，因此保守穿着仍是上选。 2：吃泰国菜时，首先如果不喜欢辣，在点菜时可向服务员说不要辣。其次不必花费钱财在酒水上，因为泰餐本身味道较重，通常是
(完整版)the的用法
定冠词the的用法： 定冠词the与指示代词this ,that同源,有“那(这)个”的意思,但较弱,可以和一个名词连用,来表示某个或某些特定的人或东西. (1)特指双方都明白的人或物 Take the medicine.把药吃了. (2)上文提到过的人或事 He bought a house.他买了幢房子. I've been to the house.我去过那幢房子. (3)指世界上独一无二的事物 the sun ,the sky ,the moon, the earth (4)单数名词连用表示一类事物 the dollar 美元 the fox 狐狸 或与形容词或分词连用,表示一类人 the rich 富人 the living 生者 (5)用在序数词和形容词最高级,及形容词等前面 Where do you live?你住在哪? I live on the second floor.我住在二楼. That's the very thing I've been looking for.那正是我要找的东西. (6)与复数名词连用,指整个群体 They are the teachers of this school.(指全体教师) They are teachers of this school.(指部分教师) (7)表示所有,相当于物主代词,用在表示身体部位的名词前 She caught me by the arm.她抓住了我的手臂. (8)用在某些有普通名词构成的国家名称,机关团体,阶级等专有名词前 the People's Republic of China 中华人民共和国 the United States 美国 (9)用在表示乐器的名词前 She plays the piano.她会弹钢琴. (10)用在姓氏的复数名词之前,表示一家人 the Greens 格林一家人(或格林夫妇) (11)用在惯用语中 in the day, in the morning... the day before yesterday, the next morning... in the sky... in the dark... in the end... on the whole, by the way...
purpose的用法与搭配
p u r p o s e的用法与搭配 Company Document number：WTUT-WT88Y-W8BBGB-BWYTT-19998
purpose的用法与搭配 用作名词，主要意思为“目的”“目标”，用法注意： 1.表示做某事的目的，通常用 the purpose of 的结构。如： What was the purpose of his visit 他来访的目的是什么？ He came here with [for] the purpose of seeing his family. 他来这里的目的是探亲。若 purpose 前用了物主代词，则通常连用介词 in。如： What is your purpose in being here 你在这儿干什么？ Her purpose in going to Japan is to look for her uncle. 她去日本的目的是找她叔叔。 以下结构也用介词 in。如： I have a purpose in making this trip to Europe. 我这次欧洲之行是有目的的。 2.表示为了某种目的，通常用for…purposes（其中的 purpose通常用复数）。如： He keeps a horse for pleasure purposes. 他为消遣而养马。 He learns Japanese for business purposes. 他学习日语是为做生意。 类似的例子有：for medical purposes（为了医学的目的），for defence purposes （为了防御之目的），for scientific purposes（为了科学的目的），English for commercial purposes（商业英语）等。 3.用于 on purpose, 意为“有意地”“故意地”。如： I came here on purpose to see you. 我是特意来看你的。
say-tell-talk-speak的用法和区别
词汇辨析 say、tell、speak、talk的区别 1、say意为“说出”“说过”，强调说话的内容，也可与to连用，say to sb.意为“对某人说”。 eg. He often says“hello”to me with a smile. 他常笑着向我问好。 I can say it in English. 我能用英语说它。 He says to me,“I like my hometown.”他对我说：“我喜欢我的家乡。” 2、tell意为“讲述”“告诉”，作及物动词时，指把一件事或一个故事讲出来，有连续诉说之意。如：tell the truth说实话，tell a story讲故事。tell也可接双宾语结构或复合宾语结构。如tell sb. sth.告诉某人某事；tell sb. about sth.告诉某人关于某事；tell sb.（not）to do sth.告诉某人（不要）去做某事。 eg.－What did your mother tell you just now? 刚才你妈妈告诉你什么了？ －She told me not to ride a bike quickly. It's too dangerous. 她告诉我不要快骑自行车，那太危险了。 Please tell me something about yourself.请告诉我关于你自己的一些事情。 3、speak的意思是“说话”，作不及物动词时，通常指说话的能力和方
式；作及物动词时，其后的宾语为某种语言。speak to sb.表示“同某人说话”。 eg. Would you like to speak at the meeting? 你要在会上发言吗？ Bob speaks Chinese quite well. 鲍勃汉语说得相当好。 Joe can speak a little Chinese. 乔能说一点儿汉语。 May I speak to Mr. Green? 我可以同格林先生通话吗？ （此句常用于打电话用语中） He is speaking to Lily. 他正在和莉莉说话。 4、talk的意思是“谈话，谈论”，指相互之间的谈话，一般用作不及物动词，与介词to或with连用，表示“与……交谈”。而谈及关于某人或某事时，后接介词of或about. eg. They are talking on the phone. 他们正在电话中交谈。 My mother is talking with my teacher. 我妈妈正在和我的老师谈话。We are talking in English.我们正用英语交谈。 What are they talking about? 他们正在谈论什么？ We talked about this problem for hours. 我们就这个问题谈了好几个小时。 检测： 用say、tell、speak、talk 的适当形式填空。 1. Excuse me .Can you ___________ me the way to the post office ?
商业计划书 亲子游 旅游 度假 周末游
商业计划书 公司名称：重庆泰兰科技有限公司公司主营：27度亲子旅行网 负责人：罗孟奇 2016年5月
目录 一、项目摘要 (3) 二、公司介绍 (3) 三、战略规划 (4) 四、管理组织结构 (5) 五、产品服务 (6) 六、市场预测分析 (7) 七、营销计划 (8) 八、发展计划及进度 (8) 九、财务分析 (8) 十、风险与退出 (11)
一、项目摘要 近年类似《爸爸去哪儿》亲子旅游视频节目的热播，以80后父母为主力拉动，亲子旅游正在成为当下的市场热点，这样一个拥有清晰的目标人群定位，以及强有力的消费能力支撑的旅游细分市场，正在迅速成为“互联网+旅游”主题模式的生力军，成为“旅游O2O”领域大家争相进入的热门市场。 重庆泰兰科技有限公司是一家旅游互联网行业连续创业公司，之前拥有运营泰爱玩旅行网的经验和用户，本次作为公司第二次创业，推出了27度旅行网，27度旅行网是专注5—10岁孩子及其家长为服务对象的新型、正规化的亲子游网站。我们的亲子游项目出发点是注重父母与小孩在旅游中获得的情感融合，亲子互动体验性强，寓教于乐，同时也发掘了家长以旅游形式促进亲子关系的新需求。公司亲子游路线主打重庆周边，报名以家庭为单位，时间在周末及节假日，为期1-2天，最大化的使家长和小孩在工作和学习之余，能够合理利用周末，达到放松和亲子的目的。 公司有专门的团队进行路线规划，找到合适的周边游目的地，再根据目的地的特点，设计亲子相关的互动体验活动，形成亲子游产品。家长以家庭为单位购买亲子游产品，由我们招募的亲子游达人带团，前往目的地进行体验。我们的亲子游产品价格控制在每人500-1500元，必须以家庭为单位（至少一个家长和一个小孩）报名， 8-15个家庭为一个团，每周发团，预计年营业额200万左右。 我们目前预计出让股权25%，融资50万元。我们希望投资人在重庆本土特色旅游及旅行O2O领域，和相关公共机构、企业合作有一定的资源或者经验，能够给我对接更多资源，并且对于公司的日常运作不要有太多束缚。 预计3年内市值达到1000万元，给投资人5倍的回报率。 二、公司介绍 重庆泰兰科技有限公司简介 重庆泰兰科技有限公司成立于2013年5月，其前身为2012年12月建立的专注于
“the way+从句”结构的意义及用法
“tｈｅwａｙ+从句”结构的意义及用法 首先让我们来看下面这个句子: Ｒead the foｌloｗｉnｇpassagｅａnd talkａbout ｉt wi ｔh your classmaｔes．Try tｏｔeｌl whaｔyｏu thiｎk ｏf Tom aｎd ｏｆｔhe ｗay the chiｌｄrenｔrｅated ｈim. 在这个句子中，ｔhe waｙ是先行词,后面是省略了关系副词that或in whｉch的定语从句。 下面我们将叙述“the ｗaｙ+从句”结构的用法。 1.tｈe wａy之后，引导定语从句的关系词是tｈat而不是hoｗ,因此，<<现代英语惯用法词典＞＞中所给出的下面两个句子是错误的：Ｔhｉs is ｔｈeｗayｈowｉtｈａppened. This is the way how he ａlwａｙs treats me. 2．在正式语体中,thaｔ可被in which所代替;在非正式语体中，ｔhat则往往省略。由此我们得到theｗay后接定语从句时的三种模式：1) tｈe ｗaｙ＋ｔhａt-从句２）the way +ｉn whiｃh－从句3) the ｗay +从句 例如：Ｔｈe way(in whiｃh ，thａt) ｔhｅｓｅｃomｒade ｓloｏｋａｔｐrobｌems is wrong．这些同志看问题的方法
不对。 Ｔｈｅｗａy(that ,ｉn ｗhiｃh)yoｕ’ｒe doinｇit is comple ｔeｌy crａzy.你这么个干法,简直发疯。 Wｅａdmiｒｅd him for thｅｗay ｉｎｗhｉch he fａｃeｓdiｆficultieｓ． Waｌlａｃe ａｎd Ｄarwiｎｇreed ｏn the way ｉｎwhi ｃh ｄｉfｆｅrｅnt forms ｏf life had ｂegｕn.华莱士和达尔文对不同类型的生物是如何起源的持相同的观点。 Thｉs is ｔｈe ｗaｙ（tｈａｔ) hｅdｉd it. I lｉkeｄｔhe waｙ(ｔhat) ｓｈeｏｒganized ｔhe ｍeetｉng． 3.theｗay(tｈat)有时可以与hｏw(作“如何”解）通用。例如： That’s tｈe ｗaｙ(tｈaｔ) shｅspoke. = Tｈat’s how shｅspoke.
purpose的用法与搭配.
purpose的用法与搭配 用作名词，主要意思为“目的”“目标”，用法注意： 1.表示做某事的目的，通常用the purpose of 的结构。如： What was the purpose of his visit? 他来访的目的是什么? He came here with [for] the purpose of seeing his family. 他来这里的目的是探亲。若purpose 前用了物主代词，则通常连用介词in。如： What is your purpose in being here? 你在这儿干什么? Her purpose in going to Japan is to look for her uncle. 她去日本的目的是找她叔叔。 以下结构也用介词in。如： I have a purpose in making this trip to Europe. 我这次欧洲之行是有目的的。 2.表示为了某种目的，通常用for…purposes（其中的purpose通常用复数）。如： He keeps a horse for pleasure purposes. 他为消遣而养马。 He learns Japanese for business purposes. 他学习日语是为做生意。 类似的例子有：for medical purposes（为了医学的目的），for defence purposes （为了防御之目的），for scientific purposes（为了科学的目的），English for commercial purposes（商业英语）等。 3.用于on purpose, 意为“有意地”“故意地”。如： I came here on purpose to see you. 我是特意来看你的。 She broke the dish on purpose just to show her anger. 她故意打破碟子以表示她的愤怒。 4.用于to little (no, some) purpose，表示“几乎徒劳（毫无成效，有一定效果）地”。如： Money has been invested in the scheme to very little purpose. 资金已投入那计划中却几无成效。 We spoke to little purpose. His mind was clearly made up already. 我们说的话不起作用，他显然早已下定决心了。
tell地用法和常见搭配
tell的用法和常见搭配 tell的中文含义是：说；告诉；讲述。 例句：Tell him to wait for a few minutes, please. 请告诉他等几分钟。 tell一般用作及物动词，常用于tell somebody to do something这个结构中，表示“要某人做某事”，如：Tell the kids to be quite, please. 请告诉孩子们保持安静。类似的结构还有ask somebody to do something。 tell还常用于tell somebody something和tell somebody about something这两个结构中。两个结构都有“告诉”的意思，它们的区别是：tell somebody something告诉某人某事（往往是不需要解释、说明的事）；tell somebody about something向某人讲述某事（往往含有解释、说明的意味）。试比较： Tell me your phone number. 告诉我你的。 Please tell me something about your school life. 请给我讲讲你的校园生活吧。 常用搭配： tell somebody to do something 告诉某人去做某事 tell somebody something 告诉某人某事 tell somebody about something 向某人讲述某事 speak, talk, say, tell的用法区别 这四个词的用法辨析是中考英语中考得最经常的同义词辨析之一。综观各省市的中考英语真题情况，我们发现，中考对这四个词的考查主要侧重于其用法差异和习惯表达方面的不同。因此，本文拟在这两个方面谈谈它们的具体用法和区别。 一、用法方面的区别 1.speak 强调单方的“说”或“讲”，一般用作不及物动词，要表示“对某人说(某事)”，可用 speak to [with] sb (about sth)。如： Please speak more slowly. 请说慢一点。 I spoke to [with] the chairman about my idea. 我跟主席说了我的想法。
全域旅游发展研究
全域旅游发展研究 本文首先对国内全域旅游研究成果进行了综述，然后对全域旅游发展机制和面临的挑战进行了探讨。最后，提出了全域旅游协调发展的三大策略。 标签：全域旅游；发展策略；挑战 一、引言 随着我国市场经济体制的不断深化，城乡二元经济结构弊端日益凸显。旅游业是解决我国城乡差距的重要途径，也是城乡产业结构优化和协同发展的抓手。然而随着社会经济的发展，我国旅游发展的城乡差异越发明显。呈现出乡村旅游地基础设施不完善、信息服务之后、统筹管理机制缺失等一系列问题。 二、文献综述 近年来，全域旅游发展已成为国内学术研究热点。诸多学者就移动互联网背景下全域旅游发展进行了深入探讨。基于系统发展理论，刘兰和郑雅慧就影响全域旅游发展的主要因素和功能机理进行了探索，成功构建了一个全域旅游优势互补发展模型，并对这个模型进行了实践检验。姜松、曹峥林，在对国外学者的研究进行了文献综述的基础上，提出了以城市为核心的全域旅游发展路径。黄震方、陆林等学者首先对区域旅游协调发展进行了界定，提出了以地方利益相关者为核心、以统筹协调发展为目标、以市场为导向的全域旅游发展模型并进行了实证案例研究。在以上研究成果基础上，本文深入分析了全域旅游统筹发展特点，探讨了全域旅游发展面临的挑战，提出了我国全域旅游协调发展策略。 三、全域旅游概念、特点界定 全域旅游是指在一定区域内，以旅游业为优势产业，通过对区域内经济社会资源尤其是旅游资源、相关产业、生态环境、公共服务、体制机制、政策法规、文明素质等进行全方位、系统化的优化提升。实现区域资源有机整合、产业融合发展、社会共建共享，以旅游业带动和促进经济社会协调发展的一种新的区域协调发展理念和模式。在全域旅游中，各行业积极融入其中，各部门齐抓共管，全城居民共同参与，充分利用目的地全部的吸引物要素，为前来旅游的游客提供全过程、全时空的体验产品，满足游客全方位体验需求。 四、全域旅游发展面临的挑战 （一）管理机制不完善 我国全域旅游发展缺乏完善的行政管理機制。全域旅游发展顶层设计缺失，政出多头各自为政，政府各部门间的协调机制缺乏并存。作为行业主管部门的地方旅游局因没有执法权，陷入“管得了正规军，管不了游击队”的尴尬局面。旅游
way 用法
表示“方式”、“方法”，注意以下用法： 1.表示用某种方法或按某种方式，通常用介词in(此介词有时可省略)。如： Do it (in) your own way. 按你自己的方法做吧。 Please do not talk (in) that way. 请不要那样说。 2.表示做某事的方式或方法，其后可接不定式或of doing sth。 如： It’s the best way of studying [to study] English. 这是学习英语的最好方法。 There are different ways to do [of doing] it. 做这事有不同的办法。 3.其后通常可直接跟一个定语从句(不用任何引导词)，也可跟由that 或in which 引导的定语从句，但是其后的从句不能由how 来引导。如： 我不喜欢他说话的态度。 正：I don’t like the way he spoke. 正：I don’t like the way that he spoke. 正：I don’t like the way in which he spoke. 误：I don’t like the way how he spoke. 4.注意以下各句the way 的用法： That’s the way (=how) he spoke. 那就是他说话的方式。 Nobody else loves you the way(=as) I do. 没有人像我这样爱你。 The way (=According as) you are studying now, you won’tmake much progress. 根据你现在学习情况来看，你不会有多大的进步。 2007年陕西省高考英语中有这样一道单项填空题： ——I think he is taking an active part insocial work. ——I agree with you_____. A、in a way B、on the way C、by the way D、in the way 此题答案选A。要想弄清为什么选A，而不选其他几项，则要弄清选项中含way的四个短语的不同意义和用法，下面我们就对此作一归纳和小结。 一、in a way的用法 表示：在一定程度上，从某方面说。如： In a way he was right.在某种程度上他是对的。注：in a way也可说成in one way。 二、on the way的用法 1、表示：即将来(去)，就要来(去)。如： Spring is on the way.春天快到了。 I'd better be on my way soon.我最好还是快点儿走。 Radio forecasts said a sixth-grade wind was on the way.无线电预报说将有六级大风。 2、表示：在路上，在行进中。如： He stopped for breakfast on the way.他中途停下吃早点。 We had some good laughs on the way.我们在路上好好笑了一阵子。 3、表示：(婴儿)尚未出生。如： She has two children with another one on the way.她有两个孩子，现在还怀着一个。 She's got five children，and another one is on the way.她已经有5个孩子了，另一个又快生了。 三、by the way的用法