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Anisotropic pseudogap in the half-filling 2-d Hubbard model at finite T

a r X i v :c o n d -m a t /9810393v 2 [c o n d -m a t .s t r -e l ] 3 N o v 1998Anisotropic pseudogap in the half-?lling 2-d Hubbard model at ?nite T

Taiichiro Saikawa and Alvaro Ferraz

Laborat′o rio de Supercondutividade,

Centro Internacional de F′?sica da Mat′e ria Condensada,

Universidade de Bras′?lia,

CEP 70919-970Bras′?lia-DF,Brazil

(October 23,1998)

We have studied the pseudogap formation in the single-particle spectra of the half-?lling two-

dimensional Hubbard https://www.doczj.com/doc/8815583476.html,ing a Green’s function with the one-loop self-energy correction of the

spin and charge ?uctuations,we have numerically calculated the self-energy,the spectral function,

and the density of states in the weak-coupling regime at ?nite temperature.Pseudogap formations

have been observed in both the density of states and the spectral function at the Fermi level.

The pseudogap in the spectral function is explained by the non-Fermi-liquid-like nature of the self-

energy.The anomalous behavior in the self-energy is caused by both the strong antiferromagnetic

spin ?uctuation and the nesting condition on the non-interacting Fermi surface.In the present

approximation,we ?nd a logarithmic singularity in the integrand of the imaginary part of the self-

energy.Concerning the energy dependence of the spectral function and the self-energy,two theorems

are proved.They give a necessary condition in the self-energy to produce the pseudogap at the Fermi

level.The pseudogap in the spectral function is highly momentum dependent on the Fermi surface.

It opens initially in the (±π,0),(0,±π)regions as the normal state pseudogap observed in the high-

T c superconductors and if the interaction is increased,it spreads to other Fermi surface sectors.

The anisotropy of the pseudogap is produced by the low-energy enhancement of the spin excitation

around Q =(π,π)and the ?atness of the band dispersion around the saddle point.

PACS numbers:71.10.Fd,71.20.-b

I.INTRODUCTION The recent series of angle-resolved-photoemission experiments in high-T c superconductors have brought a large number of new insights concerning the low energy single-particle states of these materials.The experimental ob-servations include a rather wide ?at-band region around (π,0)in the e?ective band dispersion [1–3]and also the anisotropy of the normal state pseudogap [4–9]which is consistent with the d -wave symmetry superconducting gap.Many theoretical approaches and computer simulations have been performed to investigate the pseudogap.Naturally,our ultimate objective is to explain the existing experimental data.However,the main purpose of this work is not to give a direct explanation of the experiments,but rather,to study the pseudogap formation in the Hubbard model which is often used to describe the high-T c materials.To avoid confusion,we should make clear what kinds of pseudogap we are dealing with in this paper.In reality,in the literature this term has been used to represent several di?erent features.What it is common in all cases is the fact that the pseudogap indicates a suppression or disappearance of the spectral intensity at the Fermi level.We can list the following three di?erent cases.The ?rst one is a disappearance of the spectral intensity due to the downward shift of the band dispersion around (π,0)region [10].For example,some of the quantum Monte Carlo (QMC)simulations show such behavior in the strong coupling regime and at very low dopings [11–13].This phenomenon is quite interesting since it looks as if it violates the Luttinger theorem on the Fermi surface.The second case is an e?ective pseudogap in the spectral function between a strong quasiparticle peak and a weak satellite peak.It is possible to ?nd this situation even in a normal Fermi liquid.The third one is a suppression of the single-particle

peak at the Fermi level with a two peak structure in its place.The pseudogap of our interest is this latter case.It should be clearly distinguishable from the second case.One physical origin of this kind of pseudogap is of course the fact that it is a precursor of the spin density wave (SDW)transition which takes place in the 2-d Hubbard model for zero doping at T =0.We are aware of some models which take into account precursor e?ects of the superconducting transition to explain the pseudogap behavior observed in the normal phase of high-T c materials [14–17].We don’t consider the superconducting transition e?ects in this paper.

In our work,we provide an explanation of the origin of the pseudogap formation in the 2-d Hubbard model,at half-?lling,in the weak coupling regime.To calculate the spectrum we use a single-particle Green’s function which has a paramagnon-like one-loop self-energy correction for both charge and spin channels.At ?rst we show the results

of our numerical calculations of the density of states(DOS),the spectral function,and the self-energy.The analysis of the real and imaginary parts of the self-energy plays a crucial role in the understanding of the microscopic origin of the possible structures of the spectra.Following this,we discuss the detailed origin of the pseudogap formation within our formulation.

We found pseudogap formations at the Fermi level in both the spectral function and the DOS.The pseudogap in the spectral function is explained by the non-Fermi-liquid behavior of the self-energy,i.e.,its imaginary part has a negative peak and its real part has a positive slope.We show that both the strong antiferromagnetic spin?uctuation and the nesting condition on the non-interacting Fermi surface are the origins of the anomalous behaviors in the self-energy. We also generally argue the relation between the spectral function and the self-energy at the Fermi level.We show, for example,that if the spectral function has a pseudogap,the real part of the self-energy has a positive slope at the Fermi level.An auxiliary relation is used to present the argument.This relation derived from the Kramers-Kronig transformation is proved in the appendix.

The numerically obtained pseudogap shows a strong momentum dependence.For a certain choice of the parameters, the pseudogap appears around(π,0)but not at(π/2,π/2).We show that the anisotropic pseudogap comes from the low-energy enhancement of the spin excitation around Q=(π,π)and the?atness of the band dispersion.

II.MODEL

In the momentum representation the Hubbard model,with a nearest-neighbor hopping,on a two-dimensional square lattice can be written as

H= k,σ(εk?μ)a?kσa kσ+U

χc(q,νm)

2

3

+

(2.4)

iνm?(εk+q?εk)

withνm=2mπT and f(x)is the Fermi distribution function de?ned by f(x)=1/[exp(x/T)+1].

III.NUMERICAL RESULTS

To calculate the spectra obtained from the above Green’s function,we made the analytic continuation of the energy variables of the self-energy and the Green’s function.In this way we obtained the retarded self-energy and the retarded Green’s function.As a result,the imaginary part of the self-energy becomes,

ImΣR(k,ω)=?U2 q 32ImχR c(q,εk+q?ω)?ImχR0(q,εk+q?ω)

×[f(εk+q)+n(εk+q?ω)](3.1) where n(x)is the Bose distribution function de?ned as n(x)=1/[exp(x/T)?1].‘R’denotes the retarded function. The corresponding real part can be obtained from the imaginary part by means of the Kramers-Kronig transformation. Using those schemes,we have numerically calculated the susceptibilities,the self-energy,and the Green’s function. The momentum summation in Imχ0(q,ν)was reduced to a contour integral in the Brillouin zone and roughly4000 line-elements have been numerically summed up along the contour.The momentum summation in the imaginary part of the self-energy and the DOS have been done on a120×120mesh in the Brillouin zone.All parameters used in the calculation are within the Stoner instability condition in RPA.

A.Density of States

In Fig.1we show the U dependence of the DOS N(ω)= k A(k,ω)for T/t=0.5where the spectral function A(k,ω)is de?ned by

1

A(k,ω)=?

ImΣR(k,ω)

π

imaginary part(Fig.3(c))shows a negative peak.This tendency becomes stronger as we decrease the temperature. Moreover,the Green’s function has three poles which correspond to the solutions ofω?εk F=ReΣR(k F,ω)for T/t=0.22as shown in Fig.3(b).Two of the poles are associated with two peaks of A(k F,ω)in Fig.3(a)and the other pole is linked to the pseudogap.Those features in the spectral function and the self-energy clearly indicate the destruction of the Fermi-liquid quasiparticle states.We will argue in more detail on the relation between the pseudogap and the anomalous behavior in the self-energy in the next section.

This same trend in the spectral function has already been shown in the so-called two-particle self-consistent(TPSC) approach by Vilk and Tremblay[22,23].In the FLEX approach[21],however,there is no pseudogap in the spectral function although their self-energy shows a similar non-Fermi-liquid behavior that both our result and TPSC approach have shown.The detailed analysis and comparisons between those approaches can be seen in Ref.[23].

For the weak coupling regime at half-?lling,the?nite temperature pseudogap formation in A(k,ω)in a QMC simulation[24]was attributed to a?nite size e?ect.Making a di?erent analysis of the QMC simulation data,the pseudogap formation was con?rmed by Cre?eld et al[20].The latter authors have used a singular-value-decomposition method instead of the maximum-entropy method to obtain the spectral function from the calculated?nite temperature Green’s function.They obtained a clear pseudogap opening even with a12×12system size.

IV.PSEUDOGAP AND THE NON-FERMI-LIQUID BEHA VIOR IN THE SELF-ENERGY

In this section we discuss the origin of the non-Fermi-liquid behavior in the imaginary part of the self-energy.This section is organized in three subsections.Firstly we analyze the imaginary part of the self-energy within the present paramagnon theory.A log singularity is found in the integrand of the imaginary part of the self-energy.Next,we examine a possibility of having the same tendency of the imaginary part of the self-energy using a model susceptibility which does not produce the log singularity.Finally,we discuss a general relation between the self-energy and the spectral function around the Fermi level.The argument in the?nal subsection does not depend on the models used and on the origin of the pseudogap.

A.Log singularity in the integrand of the imaginary part of the self-energy

We can now explain the origin of the non-Fermi-liquid negative peak behavior in ImΣ(k,ω)in the present formu-lation.The point is how to take into account the strong enhancement of the spin?uctuation around q=(π,π)≡Q. Let’s consider the integrand of the spin component of the self-energy at k=k F.The integrand is

I k F(q,ω)=?

3

1?U ReχR0(Q,?ω) 2+ U ImχR0(Q,?ω) 2(4.2) with

ImχR0(Q,ν)=π

2 tanh

νπ dν′ImχR0(Q,ν′)

1?[x/(4t)]2 is the density of states for the non-interacting bandεk where K(k)is the complete elliptic integral of the?rst kind.In the same way,we also have

f(εk F+Q)+n(εk F+Q?ω)=?1

2T .(4.5)

Note that we have used the nesting conditionεk F+Q=?εk F=0on the Fermi surface.Then,we obtain

I k F(Q,ω)=?3

1?U ReχR0(Q,?ω) 2+ U ImχR0(Q,?ω) 2.(4.6)

Forω~0,by the cancellation of theω/T dependence in tanh and coth,we see that the numerator always has a log singularity since

2 tanh

?ω2T ~1|ω| .(4.7)

By approximating the denominator with its value atω=0,the behavior of the integrand for q=Q is dominated by the log singularity,

I k F(Q,ω)~?3U2

1?U ReχR0(Q,0) 2.(4.8)

In lowering the temperature or by increasing U,1?U ReχR0(Q,0)becomes smaller and the contribution from the region around Q=(π,π)dominates the other region’s contribution and it causes the larger peak of ImΣ(k F,ω)at ω=0.

In Fig.4we plot the temperature dependence of I k F(Q,ω)for U/t=2.0using Eq.(4.6).Fig.4(a)shows the low-energy enhancement of Imχs(Q,ν).The log divergence atω=0exists for any?nite values of T.I k F(Q,ω) shows a remarkable enhancement at the same time as Imχs(Q,ν)has a sharp increase.Clearly,one can see that this enhancement of I k F(Q,ω)aroundω=0causes the non-Fermi-liquid,i.e.,the negative peak structure in the imaginary part of the self-energy.

B.No log-singularity case

In the last subsection we have found a logarithmic behavior in the integrand of the imaginary part of the self-energy. This has a strong in?uence in producing the non-Fermi-liquid behavior in the self-energy.In general,however,the logarithmic van Hove singularity in the DOS will be smeared to some extent by any?nite interaction between electrons.One might suppose that the non-Fermi-liquid-like behavior we have seen in the last section is an artifact of our approximation.In this subsection,we consider this question by introducing a model susceptibility which does not possess the log-singularity behavior.Again we take into account only the spin components and neglect other channels in the self-energy.

From ImχR0(Q,ν)given by Eq.(4.3),we see that the logarithmic behavior comes from the non-interacting density of statesρ0(ω).To consider the case thatρ0(ω)does not have any divergence atω=0,we simply assume thatρ0(ω) is?nite atω=0.Then,from Eq.(4.3),we can approximate the imaginary part ofχ0(Q,ν)as ImχR0(Q,ν)=cνaroundν=0where c is a positive constant.Since we would like to see the low-energy behavior of the integrand of ImΣ,it is su?cient to approximate the spin susceptibility aroundν=0.Furthermore,approximating Reχ0(Q,ν)by its value atν=0,we obtain the imaginary part of the model susceptibility written as

Imχ?R s(Q,ν)=

2U2

(c/2)ωcoth(ω/(2T))

numerical data,but the accuracy is not important here.As U/t becomes large,Imχ?R s(Q,ν)shows a low-energy

Stoner enhancement.In Fig.5(b)and(c),we show the U evolution of I?k

F (Q,ω).For small U/t,we see theω

dependence similar to the Fermi-liquid-likeω2behavior aroundω=0although I?k

F (Q,ω)is the integrand of the

imaginary part of the self-energy.As U/t increases,this behavior disappears and?nally a very sharp negative peak structure develops atω=0.

Thus,even if the density of states does not have a log singularity,the strong negative peak appears in the integrand of the imaginary part of the self-energy as the parameter set approaches the Stoner instability condition.

C.Necessary condition for the pseudogap formation

As our numerical results have shown,the non-Fermi-liquid behavior of the self-energy and the pseudogap are related to each other.In all cases,as we have seen in our results,a pseudogap in A(k F,ω)accompanies the positive slope of the real part of the self-energy.In this subsection,we generally argue the relation between the self-energy and the spectral function at the Fermi level.We prove two theorems which hold between the self-energy and the spectral function.These theorems determine which conditions in the self-energy are necessary in order to have a pseudogap in the spectral function.Through the argument,we assume that:(i)the imaginary part of the(retarded)self-energy is always negative and?nite,(ii)atω=0,we have(εk F?μ)+ReΣR(k F,0)=0.We leave the chemical potentialμto keep the generality.Those assumptions are physically reasonable.

Next,we enunciate the two theorems,after that,we give the proofs of them.

Theorem I—If the imaginary part of the self-energy has a maximum at the Fermi level,the spectral function has a maximum at the Fermi level.

One easily sees that the conventional Fermi liquid satis?es the theorem I.

Theorem II—If the spectral function has a minimum at the Fermi level,the real part of the self-energy has a positive slope at the Fermi level.

Note that if the spectral function has a hollow at the Fermi level,we can say that the spectral function has a two peak structure around the hollow since the spectral sum over energy is?nite.Hence,to discuss the existence of the pseudogap,it is su?cient to observe if the spectral function has a maximum or minimum at the Fermi level.

Proof of theorem I—Ifωis slightly di?erent from0,from the de?nition of the spectral function,we have

A(k F,ω)=1

|ImΣR(k F,ω)|+[ω?(εk F?μ)?ReΣR(k F,ω)]2/|ImΣR(k F,ω)|

.(4.11)

Ifω=0,by the assumption(ii),we obtain

A(k F,0)=

1

[Re G R(k,ω)]2+[Im G R(k,ω)]2

(4.13) and

ImΣR(k,ω)=

Im G R(k,ω)

1

|ImΣR(k F,ω)|=

.(4.16)

|Im G R(k F,0)|

Since the spectral function has a minimum atω=0,we see|Im G R(k F,0)|<|Im G R(k F,ω)|forωnear zero.Thus, we have

ImΣR(k F,0)

We easily see that the reverses of the two theorems do not always hold.For example,when ReΣR(k F,ω)has a positive slope,A(k F,ω)may have a single peak at the Fermi level.However,we can never have a simultaneous maximum in ImΣR(k F,ω)and a pseudogap in A(k F,ω)at the Fermi level.The argument we gave here does hold very generally and it does not depend on the models and the origin of the pseudogap.

The reverse of the lemma is not always true either.However,if we include an additional condition in the initial assumption such as the imaginary part of the self-energy has always a minimum or a maximum at the Fermi level,the revere of the lemma is satis?ed.To prove this let’s assume this new condition.From the lemma,ReΣ(k F,ω)always has a(positive or negative)slope at the Fermi level.Thus,the maximum(minimum)of ImΣR(k F,ω)is automatically linked to the negative(positive)slope of ReΣR(k F,ω).The reverse of the lemma follows from this.

By combining the reverse of the lemma and the theorem I,we immediately?nd that when ReΣR(k F,ω)has a negative slope atω=0,the spectral function has a single peak at the Fermi level.In other words,as far as the Fermi-liquid-like negative slope remains in ReΣR(k F,ω)atω=0,the pseudogap is never produced in the spectral function at the Fermi level.

V.BAND DISPERSION AND ANISOTROPIC PSEUDOGAP

In Fig.6(a)we show the band dispersion extracted from the numerical data of the spectral function A(k,ω)for several momentum directions.The dot indicates the peak of the spectral function.The peaks roughly follow the original non-interacting band dispersion except around(π,0).Clear(pseudo)gap structure can be seen around(π,0). In contrast,around(π/2,π/2),there is no such pseudogap structure.This exempli?es the anisotropy of the pseudogap on the Fermi surface.We show the structures of the spectral functions around(π,0)in Fig.6(b).We chose k=(π,0), (0.9π,0),and(0.85π,0)along the k x axis.By varying k,the symmetric two peak structure at k=(π,0)gradually changes into a single peak structure with a satellite peak.The satellite peak corresponds to the shadow band of the SDW band dispersion.In our earlier work[30],another sign of similar shadow band formation around k=(π/2,π/2) and(π,0)has been observed in the dispersion spectrum for a weakly doped Hubbard model.

In Fig.7we compare the spectral functions and the self-energies for k F=(π,0)and(π/2,π/2).ReΣR(k F,ω)at k F=(π,0)in Fig.7(b)has a large positive slope aroundω=0.ReΣR(k F,ω)at k F=(π/2,π/2)also has a positive slope,but it is still weak.ImΣR(k F,ω)at k F=(π,0)in Fig.7(c)shows a sharp negative peak atω=0.In contrast, ImΣR(k F,ω)at k F=(π/2,π/2)has a weak minimum atω=0.These remarkable momentum dependences of the real and imaginary parts aroundω=0determines the opening of the pseudogap.

For larger values of U/t,we have obtained a pseudogap in A(k,ω)also at k F=(π/2,π/2).In Fig.8,we show the spectral function for U/t=3.0and at T/t=0.5.We chose the higher temperature to avoid the Stoner instability. The pseudogap at(π/2,π/2)is weaker than the one at(π,0).Comparing with the spectral function for U/t=2.0 shown in Fig.7(a),the intensity of A(k F,ω)in Fig.8is weak and its width are rather wide.This indicates band broadening e?ects due to the larger value of U/t.

To understand the reason for the anisotropy of the pseudogap,it is su?cient to analyze the imaginary parts of the self-energy atω=0for k F=(π,0)and(π/2,π/2).There are two factors:the di?erence of the e?ective region of the q summation and the di?erent behaviors of the bosonic excitation energy in the imaginary part of the self-energy.The ?rst one is a kind of the selection rule of the momenta of the fermion-boson excitation and the latter is the restriction of energy on its excitation.We explain them as follows.

The distribution function term f(εk F+q)+n(εk F+q?ω)produces an important restriction to the q summation in ImΣ(k F,0).In Fig.9the approximated summation area in the(q x,q y)-plane is shown by the shaded region in which q satis?es the condition|f(εk

+q)+n(εk F+q)|>0.5for T/t=0.22.We see that for both k F=(π,0)and(π/2,π/2)

F

the region around q=Q contributes to the q summation.However,the shaded area around Q for k F=(π,0)is greater than that for(π/2,π/2).Thus,ImΣ(k F,ω)for k F=(π,0)has a larger contribution from the spin excitation around Q=(π,π).

The bosonic energyεb(q)≡εk F+q?ωin the expression of ImΣ(k F,ω)for k F=(π,0)behaves in a totally di?erent way from that for k F=(π/2,π/2).To see the behavior forω~0and around q~Q,we can write εb(Q+δq)~?εk F+δq,whereδq is a small vector to represents deviation from Q.Thus,we see that the behavior of εb(Q+δq)is determined by the band dispersionεk around k=k F.Hence,the bosonic energy contribution to ImΣfor k F=(π,0)is quite concentrated within a small region around0energy near the saddle point(?at-band)dispersion near(π,0).In contrast,the bosonic energy for k F=(π/2,π/2)can easily deviate from0as q moves in the summation region near Q sinceεk F has a linear slope around k F=(π/2,π/2).Thus,the low energy enhancement in the spin ?uctuation around Q makes a stronger contribution to ImΣ(k F,ω)at k F=(π,0)than that at k F=(π/2,π/2).As a result,the self-energy has a remarkable momentum dependence,and the anisotropy of the pseudogap is explained. There are some QMC simulations of the spectral functions in the weak-coupling regime at half-?lling[20,24,31]. With the exception of the results from Cre?eld et al.(Ref.[20]),a clear evidence of the anisotropic pseudogap in the spectral function is lacking in the QMC simulations.We think that this is also due to the cluster size e?ect and the di?erence in method in extracting the information of the spectral function from the Green’s function as emphasized in Ref.[20].We know of no other example of the anisotropic pseudogap in the t-U Hubbard model.However,in FLEX calculations,for the doped Hubbard model[32,33],a clear momentum dependence in the imaginary part of the self-energy atω=0near the Fermi momentum was obtained.They found that|ImΣ(k F,0)|has a large value along the Fermi surface which becomes larger and larger as k F approaches the(π,0)region.This tendency is consistent with our results at half-?lling shown in Fig.7(c).

VI.CONCLUDING REMARKS

In this work we have studied the pseudogap formation in the two-dimensional Hubbard model at half-?lling.We obtain a pseudogap formation in the density of states and in the spectral function.The pseudogap of the spectral function is produced when two conditions are satis?ed:(i)a strong anti-ferromagnetic spin?uctuation,(ii)a nesting condition on the Fermi surface.The pseudogap on the Fermi energy is highly anisotropic and its associated symmetry is similar to the d-wave symmetry.The anisotropy is determined by the?atness of the band dispersion.

We emphasize that the pseudogap formation discussed is highly dependent on the speci?c character(perfect nesting) of the free band dispersion or the hopping term in the square lattice.Thus,the pseudogap is physically di?erent from a gap formation such as the Mott insulator transition which takes place in a strong coupling regime.Our pseudogap can take place even in the weak coupling regime.

We have focused on the undoped single-band Hubbard model with nearest neighbor hopping.Unfortunately,the model we have used may be quite simple to make a quantitative comparison with the experimental data.Nevertheless, it is possible to apply our argument in the pseudogap originated,for example,in the Hubbard model with the next nearest neighbor hopping(t-t′-U model).In this model,the nesting condition associated with the momentum Q holds only around the(π,0)region(and,of course,also at the other symmetric three parts in the Brillouin zone).Hence, the pseudogap will open in this region.However,the region in the k x=±k y direction on the Fermi surface does not satisfy the nesting condition.Thus,we can predict that the electronic states around this diagonal region will remain in the Fermi liquid regime.Our argument is consistent with the physical picture of the so-called hot and cold quasiparticles by B.Stojkovi′c and Pines[26].Investigation with more realistic models will be done in future works. Very recently,the information of the self-energy has been extracted from the angle-resolved photoemission data [34].In the work,it has been obtained that the observed normal state pseudogap accompanies a sharp negative peak in ImΣR and a positive slope in ReΣR.The tendency of their data is qualitatively in good agreement with our results. In the present work we have used a paramagnon-theory self-energy to calculate the electronic states.This treatment has a restriction due to the Stoner criterion in the RPA susceptibility.However,we believe that already in the level of this scheme,it contains important ingredients for the pseudogap formation and the partial destruction of the Fermi-liquid quasiparticles at the Fermi level.In particular,the anisotropic pseudogap formation in the spectral function in the calculated band dispersion indicates the coexistence of the Fermi-liquid-like quasiparticles and the SDW-like quasiparticles at the Fermi level.

ACKNOWLEDGMENTS

We would like to acknowledge useful discussions with S.L.Garavelli and P.E.de Brito.This work was supported by the Conselho Nacional de Desenvolvimento Cient′??co e Tecnol′o gico-CNPq and by the Financiadora de Estudos e Projetos-FINEP.Most part of the numerical calculations of this work have been done with the supercomputing system at the Institute for Materials Research,Tohoku University,Japan.T.S.thanks the Material Science Group at IMR Tohoku University for the use of the supercomputing facilities.

APPENDIX:A RELATION IN THE KRAMERS-KRONIG TRANSFORMATION

In this appendix we give a simple proof of a relation obtained from the Kramers-Kronig transformation.We apply this relation to the real and imaginary parts of the self-energy in the main text of the present paper.The relation holds for any smooth functions.Suppose the following Kramers-Kronig relation for such two functions g(x)and h(x),

g(x)=P

x′?x

.(A1)

Then,the relation can be mentioned as follows.If h(x)has a maximum(minimum)at a certain point x=x0,g(x) has a positive(negative)slope at x0.We assume that h(x)has a peak at x=x0and it can be expanded around x0 as h(x)~h(x0)+(γh/2)(x?x0)2.This is reasonable as far as h(x)is a smooth function.Here,γh is the second derivative of h(x)at x=x0.Ifγh>0(<0),then the peak is a minimum(maximum).

Let’s observe the di?erence between g(x0+ε)and g(x0)withεbeing a small shift from x0.We obtain that

g(x0+ε)?g(x0)=P

(x′?x0)2?ε(x′?x0)

.(A2)

We easily see that the functionε/[(x′?x0)2?ε(x′?x0)]decreases rapidly as x′deviates from x0and x0+ε.We can approximate the integration using the expansion of h(x)around x0as

g(x0+ε)?g(x0)~P

(x′?x0)2?ε(x′?x0)

.(A3)

where x1

g(x0+ε)?g(x0)

.(A4)

Thus,forγh>0(γh<0)at x=x0,the slope of g(x0)is positive(negative).

By applying the same argument to the opposite transformation de?ned as

h(x)=?P

x′?x

,(A5)

we?nd that if g(x)has a maximum(minimum)at x=x0,the slope of h(x)at x=x0is negative(positive).

In the above derivation,we assumed that the functions of our interests can be approximated with a quadratic form around a peak point we have interests.One can show the same relation in the case that the function behaves as h(x)∝|x?x0|around x=x0.This is what happens if h(x)is the imaginary part of the self-energy of a marginal Fermi liquid[35].Even if the peak is aδ-function,the relation we argued is still applicable although the slope becomes in?nite.One example of this can be seen in the mean-?eld Green’s function for the spin-density-wave states.

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FIG.1.U dependence of the density of states N(ω)at T/t=0.5.

FIG.2.Temperature dependence of the density of states N(ω)for U/t=2.0.Inset shows the overall structures of N(ω). FIG.3.Temperature evolution of(a)the spectral function and(b)the real and(c)imaginary parts of the self-energy for U/t=0.2and at k F=(π,0)

(Q,ω)of the imaginary part of the self-energy FIG.4.Temperature dependence of(a)Imχs(Q,ν)and(b)the integrand I k

F

for U/t=2.0.

FIG.5.U dependence of the excitation spectra of(a)the model susceptibility(see the text)Imχ?s(Q,ν)and(b),(c)the (Q,ω)of the imaginary part of the self-energy for the model susceptibility.

integrand I?k

F

FIG.6.(a)Band dispersion for several momentum directions and(b)A(k,ω)around(π,0)region calculated for U/t=2.0 and at T/t=0.22.

FIG.7.(a)Spectral function at k F=(π,0)(solid line)and(π/2,π/2)(dotted line),and the corresponding(b)real and (c)imaginary parts of the self-energy.The parameters are U/t=2.0and T/t=0.22.

FIG.8.Spectral function at k F=(π,0)(solid line)and(π/2,π/2)(dotted line)for U/t=3.0and at T/t=0.5.Note the di?erences in scale compared with Fig.7(a).

FIG.9.Main contribution area(shown with shade)to the q summation in ImΣ(k F,0)for T/t=0.22and at(a) k F=(π/2,π/2)and(b)k F=(π,0).Momenta q’s in the region satisfy|f(εk

+q)+n(εk F+q)|>0.5.

F

ω/t N (ω)

Fig.1: T. Saikawa and A. Ferraz

ω/t N (ω)

Fig.2: T. Saikawa and A. Ferraz

–2

02ω/t

00.5

1A (k F ,ω)(a)

U/t=2.0

k F =(π,0)Fig.3: T. Saikawa and A. Ferraz

T/t=0.22 T/t=0.5

–2

02

ω/t

–0.500.5R e ΣR

(k F ,ω)/t (b)

ω–εk F –202

ω/t –1–0.50I m ΣR

(k F ,ω)/t (c)

ω/t I k F (Q ,ω)/

t Fig.4: T. Saikawa and A. Ferraz

ν/t I m χs (Q ,ν)

I *k F (Q ,ω)/

t Fig.5: T. Saikawa and A. Ferraz

ω/t I *k F (Q ,ω)/t ν/t I m χ*s (Q ,ν)

k

ω/t Fig.6: T. Saikawa and A. Ferraz

ω/t A (k ,ω)

–2

02ω/t

00.5

1A (k F ,ω)(a)

U/t=2.0T/t=0.22

Fig.7: T. Saikawa and A. Ferraz

k F =(π,0) (π/2,π/2)

–6–4–20246ω/t –0.500.5R e ΣR

(k F ,ω)/t (b)–6–4–20246

ω/t –1–0.50I m ΣR

(k F ,ω)/t (c)

–6–4–20246

ω/t 00.1

0.2

0.3A (k F ,ω)U/t=3.0

T/t=0.5Fig.8: T. Saikawa and A. Ferraz

k F =(π,0) (π/2,π/2)

q x /πq y /πFig.9: T. Saikawa and A. Ferraz

q x /πq y /π

The way常见用法

The way 的用法 Ⅰ常见用法: 1)the way+ that 2)the way + in which(最为正式的用法) 3)the way + 省略(最为自然的用法) 举例:I like the way in which he talks. I like the way that he talks. I like the way he talks. Ⅱ习惯用法: 在当代美国英语中,the way用作为副词的对格,“the way+ 从句”实际上相当于一个状语从句来修饰整个句子。 1)The way =as I am talking to you just the way I’d talk to my own child. He did not do it the way his friends did. Most fruits are naturally sweet and we can eat them just the way they are—all we have to do is to clean and peel them. 2)The way= according to the way/ judging from the way The way you answer the question, you are an excellent student. The way most people look at you, you’d think trash man is a monster. 3)The way =how/ how much No one can imagine the way he missed her. 4)The way =because

用Abaqus进行压电(Piezoelectric)悬臂梁模拟入门详解_第二版

用ABAQUS 进行压电(Piezoelectric )悬臂梁模拟入门详解 作者:X.C. Li 2014.8 (第二版) 本文着重讲述在用ABAQUS 模拟压电材料时,材料常数的设置。希望对入门者有所帮助。如果发现错误请发邮件到:Lxc1975@https://www.doczj.com/doc/8815583476.html, 。 1. 问题描述 柱状体10×4×2如下图 左端固定,右端自由;上表面受均匀压力500;上、下表面电压分别为50V 、0V 。 压电材料PZT-4,选z-方向(该方向上尺寸为2)为极化方向,文献Haojiang Ding, Jian Liang : The fundamental solutions for transversely isotropic piezoelectricity and boundary element method 给出的材料常数 111213334466111212.6, 7.78, 7.43, 11.5, 2.56, 0.5()c c c c c c c c ======-(10210N m -??); 15313312.7, 5.2, 15.1 e e e ==-=(-2C m ?); -121103300=730, =635, =8.8541910 λελεε???(1-1C V m -?) 这些常数在ABAQUS 中的输入将在本文2.3中详细说明。必须说明的是以上材料常数所对应的的本构关系: 111213121113131333444466 0 0 0 0 0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0 xx xx yy yy zz zz yz yz zx xy c c c c c c c c c c c c σεσεσεσγσγσ??????????????????=??????????????????? ???31313315150 0 0 0 0 0 0 0 0 00 0 0x y z zx xy e e E e E e E e γ??????????????????????-???????????????????????????? 15111511333131330 0 0 0 0 0 00 0 0 0 00 00 0 0 0 0xx yy x x zz y y yz z z zx xy E D e D E e e e e D E εελελγλγγ??????????????????????????=+??????????????????????????????????

The way的用法及其含义(二)

The way的用法及其含义(二) 二、the way在句中的语法作用 the way在句中可以作主语、宾语或表语: 1.作主语 The way you are doing it is completely crazy.你这个干法简直发疯。 The way she puts on that accent really irritates me. 她故意操那种口音的样子实在令我恼火。The way she behaved towards him was utterly ruthless. 她对待他真是无情至极。 Words are important, but the way a person stands, folds his or her arms or moves his or her hands can also give us information about his or her feelings. 言语固然重要,但人的站姿,抱臂的方式和手势也回告诉我们他(她)的情感。 2.作宾语 I hate the way she stared at me.我讨厌她盯我看的样子。 We like the way that her hair hangs down.我们喜欢她的头发笔直地垂下来。 You could tell she was foreign by the way she was dressed. 从她的穿著就可以看出她是外国人。 She could not hide her amusement at the way he was dancing. 她见他跳舞的姿势,忍俊不禁。 3.作表语 This is the way the accident happened.这就是事故如何发生的。 Believe it or not, that's the way it is. 信不信由你, 反正事情就是这样。 That's the way I look at it, too. 我也是这么想。 That was the way minority nationalities were treated in old China. 那就是少数民族在旧中

An anisotropic Phong BRDF model

An Anisotropic Phong BRDF Model Michael Ashikhmin Peter Shirley August13,2000 Abstract We present a BRDF model that combines several advantages of the various empirical models cur-rently in use.In particular,it has intuitive parameters,is anisotropic,conserves energy,is reciprocal,has an appropriate non-Lambertian diffuse term,and is well-suited for use in Monte Carlo renderers. 1Introduction Physically-based rendering systems describe re?ection behavior using the bidirectional re?ectance distri-bution function(BRDF).For a detailed discussion of the BRDF and its use in computer graphics see the volumes by Glassner[2].At a given point on a surface the BRDF is a function of two directions,one toward the light and one toward the viewer.The characteristics of the BRDF will determine what“type”of material the viewer thinks the displayed object is composed of,so the choice of BRDF model and its parameters is important.We present a new BRDF model that is motivated by practical issues.A full rationalization for the model and comparison with previous models is provided in a seperate technical report[1]. The BRDF model described in the paper is inspired by the models of Ward[8],Schlick[6],and Neimann and Neumann[5].However,it has several desirable properties not previously captured by a single model. In particular,it 1.obeys energy conservation and reciprocity laws, 2.allows anisotropic re?ection,giving the streaky appearance seen on brushed metals, 3.is controlled by intuitive parameters, 4.accounts for Fresnel behavior,where specularity increases as the incident angle goes down, 5.has a non-constant diffuse term,so the diffuse component decreases as the incident angle goes down, 6.is well-suited to Monte Carlo methods. The model is a classical sum of a“diffuse”term and a“specular”term. ρ(k1,k2)=ρs(k1,k2)+ρd(k1,k2).(1) For metals,the diffuse componentρd is set to zero.For“polished”surfaces,such as smooth plastics,there is both a diffuse and specular appearance and neither term is zero.For purely diffuse surfaces,either the traditional Lambertian(constant)BRDF can be used,or the new model with low specular exponents can be used for slightly more visual realism near grazing angles.The model is controlled by four paramters: ?R s:a color(spectrum or RGB)that speci?es the specular re?ectance at normal incidence.

(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...

“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.

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

ansysworkbench设置材料属性

(所用材料为 45号钢,其参数为密度7890 kg∕m^-3,杨氏模量为 2.09*10^11,波动比为0.269。.) 在engineering data或任意分析模块内,都行。我仅以静力学分析模块简单的说一下。 1.双击下图engineering data或右击点edit A 1寿Sta?c Structural (ANSYS) 2夕Engineering Data “ J 3 ? GeOmetry 亨J 斗* MCclel ^T Z 5淘5e?>T J ?C? Soluton^T Z 70 ReSIJIu 層J Static StructUral (ANSYS) 3場StrUCtUral Stee-I□t??FatigUe Data atzero mean Str亡com&s from 19θ? A5MESPV Code f Section 8p Diy 2r Tab∣e 5-110.1 *CIidC hereto add a new material 4. 新建,输入45 2.通过VieW打开OUtline和PrOPertieS选项,点击下图 3.会出现下面的图,点A* EngInee∏r∣? DAta

OUtiine Of SehematiC A2: Engineering Deta UnlaXIat TeSt Ddta Biaxial TeSt Data Hr Test Date √αlur∏etπc Test Datd I 田 HyPereIaSbC 田 Plasticity [±J Llfe (±J Strenath 6. 出现下图 3 t ?> Strycturel SUel □ ? Fatigue Data atzero mean stress ClanIeifrom 1993 ASMEBFV COd¢, SertiQn¢, DiV 2H TabIe S-Ilo.1 4 ■?關> 书 □ ? CIiCk hereto add a πe?-∣, material 5. 左键双击击 toolbox 内的 denSity 禾口 isotropicelasticity PhY^jcal Properties ----------- 1 f?^l DenSll? J OrthOtrOPiC SeCant COeffiCiertt ISOtrOPiC TnStantanCOUS COeff r OrthOtrOPIC InStartaneQUS Co ( Constant DarnPIng COef z fiaent Damping FaCtOr{βj COntelltS OfEngi(IeeringDdta 上 空 S.. DMCriPbon Material I5QtrOPiC SECant COefflClent Of $5 Anisotropic El astidty 曰 Experimentai StressStrain Datd

The way的用法及其含义(一)

The way的用法及其含义(一) 有这样一个句子:In 1770 the room was completed the way she wanted. 1770年,这间琥珀屋按照她的要求完成了。 the way在句中的语法作用是什么?其意义如何?在阅读时,学生经常会碰到一些含有the way 的句子,如:No one knows the way he invented the machine. He did not do the experiment the way his teacher told him.等等。他们对the way 的用法和含义比较模糊。在这几个句子中,the way之后的部分都是定语从句。第一句的意思是,“没人知道他是怎样发明这台机器的。”the way的意思相当于how;第二句的意思是,“他没有按照老师说的那样做实验。”the way 的意思相当于as。在In 1770 the room was completed the way she wanted.这句话中,the way也是as的含义。随着现代英语的发展,the way的用法已越来越普遍了。下面,我们从the way的语法作用和意义等方面做一考查和分析: 一、the way作先行词,后接定语从句 以下3种表达都是正确的。例如:“我喜欢她笑的样子。” 1. the way+ in which +从句 I like the way in which she smiles. 2. the way+ that +从句 I like the way that she smiles. 3. the way + 从句(省略了in which或that) I like the way she smiles. 又如:“火灾如何发生的,有好几种说法。” 1. There were several theories about the way in which the fire started. 2. There were several theories about the way that the fire started.

ansysworkbench设置材料属性

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Outline of Schematic A2: Engineering Data Uniaxiat Test D&ta Biaxial Test Data Test Data Volumetnc Test Data I 田 Hyperelasbc 田 Plasticity R1 Life (±J Strength 6. 出现下图 3 Structural Sled □ ? Fatigue Data atzero mean stress comej from 1993 ASMEBfV Code, Section?, Div 2H Table S-110.1 4 言關> 45 □ * Click hereto add a new material 5. 左键双击击 toolbox 内的 density 禾口 isotropicelasticity Phy^jcal Properties ---------------------- i Densib/ ---- Orthotropic Secant Coefficient Isotropic Instantaneous Coeff Orthotropic Instantaneous Cot Constant Damping Coeffiaent Damping Factor Contents of Engineering Data 上 空 S.. DtEcripbon Material Inotropic Secant Co efficient of $5 Anisotropic El astidty 曰 Experimental Stressstrain Datd

way 的用法

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" What is the difference between the time accurate solution of Navier-Stokes equations and the DNS solution? " I'm not sure I understand this question correctly, but as far as I am aware DNS (Direct Numerical Simulation) is defined as a time accurate solution of the Navier-Stokes equations. Perhaps you mean "What is the difference between laminar & turbulent DNS?" ? Believing this to be so from reading the rest of your message I wrote the following: When the DNS is of turbulence rather than a laminar flow the turbulence requires initialization in some way. The same is true of LES (LES is always turbulent, because laminar LES = DNS by definition). I have found little reported work on the proceedures used to accomplish this turbulence initialization. I can only speak for the LES code I use (and the generations of code that preceded it). The turbulence is initialized by setting an initial flow field that has a random fluctation velocity component added to the inital mean velocity. ----------------- For example: U_initial_cell = U_initial_mean + (Random_number * U_initial_mean * 0.20) ( Random_number has a value in the range -1 to +1 ) This function sets the initial cell velocity to that of the initial_mean with a tolerance of 20% (i.e. + or - 20%). So if the U_initial_mean was 1.0 then the initial velocity of the cell could be anywhere between 0.8 and 1.2 depending on the Random_number (an intrinsic computer function). ----------------- The value for initial fluctuation (20% etc) is a fairly arbitrary value just required to `kick-start' the turbulence. Once the simulation has been kick-started and run sufficiently long enough for the correct energy cascade to be observed (by monitoring k.e. of the flow) the statistics data from that point onward is o.k to be used for results. This accumulation of statistical data is one of the reasons why LES/DNS turbulent simulations require so much more time to run. Providing the same random_number is used on the same cells during initialization the computations of two DNS cases will be exactly* the same when all other conditions (boundary, geometry, etc) are equivalent. *exactly is defined as Phi(x,t)_simulation_A = Phi(x,t)_simulation_B

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各向异性相干长度(anisotropiccoherencelength)百科小物理当今社会是一个高速发展的信息社会。生活在信息社会,就要不断地接触或获取信息。如何获取信息呢?阅读便是其 中一个重要的途径。据有人不完全统计,当今社会需要的各种信息约有80%以上直接或间接地来自于图书文献。这就说 明阅读在当今社会的重要性。还在等什么,快来看看这篇各向异性相干长度(anisotropiccoherencelength)百科小物 理吧~ 各向异性相干长度(anisotropiccoherencelength) 各向异性相干长度(anisotropiccoherencelength) 在主轴坐标系中,按各向异性GL理论,此时相干长度定义 为 `xi^2(T)=hbar^2//2m^**|alpha|` 这里$m^**=(m_1^**m_2^**m_3^**)^{1/3}$,1,2,3对应于x,y,z方向的三个分量。若磁场沿z-轴方向,例如对层状结构氧化物超导体即沿晶轴c的方向,则在m1*m2*=mab*时,对应于ab晶面的 $xi_{ab}(T)=hbar//(2m_{ab}^**|alpha(T)|)^{1/2}$,也 可写成ab2=0Hc2∥,这里0是磁通量子,Hc2∥为平行于c 轴时的第二临界磁场,0为真空磁导率,而 $xi_c=xi_{ab}(m_{ab}^**//m_c^**)^{1/2}$。在Tc附近,

它们与温度T的关系为 $xi_{ab}(T)=xi_{ab}(0)(1-T//T_c)^{-1/2}$ $xi_c(T)=xi_c(0)(1-T//T_c)^{-1/2}$ 这篇各向异性相干长度(anisotropiccoherencelength)百科小物理,你推荐给朋友了么?

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t h e-w a y-的用法

The way 的用法 "the way+从句"结构在英语教科书中出现的频率较高, the way 是先行词, 其后是定语从句.它有三种表达形式:1) the way+that 2)the way+ in which 3)the way + 从句(省略了that或in which),在通常情况下, 用in which 引导的定语从句最为正式,用that的次之,而省略了关系代词that 或 in which 的, 反而显得更自然,最为常用.如下面三句话所示,其意义相同. I like the way in which he talks. I like the way that he talks. I like the way he talks. 一.在当代美国英语中,the way用作为副词的对格,"the way+从句"实际上相当于一个状语从句来修饰全句. the way=as 1)I'm talking to you just the way I'd talk to a boy of my own. 我和你说话就象和自己孩子说话一样. 2)He did not do it the way his friend did. 他没有象他朋友那样去做此事. 3)Most fruits are naturally sweet and we can eat them just the way they are ----all we have to do is clean or peel them . 大部分水果天然甜润,可以直接食用,我们只需要把他们清洗一下或去皮.

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.doczj.com/doc/8815583476.html, (G.Z.Voyiadjis),rabual1@https://www.doczj.com/doc/8815583476.html, (R.K.Abu Al-Rub). International Journal of Plasticity 23(2007) 1874–1900

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