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On nonlocal regularization in one dimensional finite strain elasticity and plasticity

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The purpose of this article is to explore in details the theoretical and numerical aspects of the behavior of spatial trusses, undergoing large elastic and/or elastoplastic strains. Two nonlocal formulations are proposed in order to regularize the
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  ORIGINAL PAPER L. Driemeier    C. Comi    S. P. B. Proenc¸a On nonlocal regularization in one dimensional finite strainelasticity and plasticity  Received: 19 July 2004 / Accepted: 28 October 2004 / Published online: 25 January 2005   Springer-Verlag 2005 Abstract  The purpose of this article is to explore indetails the theoretical and numerical aspects of thebehavior of spatial trusses, undergoing large elastic and/or elastoplastic strains. Two nonlocal formulations areproposed in order to regularize the problem, avoidingthe mesh dependence of the numerical response. Theclassical example of a simple bar in tension is chosen toexplore the various features of the proposed models andto highlight the interplay between material and geo-metrical nonlinearity in the localization. Keywords  Localization    Plasticity    Large strains 1 Introduction Classical local continuum theories are based on theassumption that the material behavior at a point onlydepends on the value of a set of state and internalvariables at that point. In other words, classical con-tinuum theories do not incorporate a material lengthscale. These theories are able to interpret the materialbehavior in large number of applications. However,when the material response becomes highly non homo-geneous (localized), local theories become inadequate.Localization phenomena can be induced by geome-try, boundary conditions, material heterogeneity or localdefects in the presence of softening materials and/or verylarge deformation. Localization zones were firstobserved in experimental tests of traction on metallicspecimen. In this case, the plastic deformations con-centrate, at the macroscopic level, in shear bands orLu ¨der bands. Various experimental evidences of locali-zation in metals are available in the literature [1, 2, 3].With local models, the width of the localization zonestends to zero, with the nowadays well known numericalconsequences in terms of pathological mesh dependence.Mathematically, the (initial) boundary value problembecomes ill-posed [4–6] and the mathematical descrip-tion no longer represents the physical reality. Thisproblem is reported in the literature when materialnonlinearity, such as damage or plasticity, is considered.Well-posedness of the (initial) boundary value prob-lem can be restored by using regularization techniques,which provide accurate numerical solutions. Nonlocalgradient models [7–10], nonlocal integral models [11–14]and micropolar models [15] have been formulated andeffectively used.Extension of these inelastic models to the large straincontext have also been studied in the literature, see e.g.[16–20], but always the regularization is performed onthe inelastic softening response. Thus, as pointed out in[16], the boundary value problem may still becomeill-posed due to geometric effects at very large strains.This work focuses on localization phenomena in trussstructures due to geometrical and/or material nonlin-earity and it is organized as follow. Kinematics back-ground and constitutive relations when large strains areconsidered are briefly reviewed. Localization is under-stood as a bifurcation into a harmonic mode of van-ishing wavelength and critical condition are identified bythe existence of stationary waves in a wave propagationanalysis [16].Two nonlocal models are proposed, both based onthe nonlocal stretch variable, defined as weighted aver-age of its local value over a certain neighborhood.The effectiveness of the proposed models is numeri-cally checked on a one-dimensional bar in tension. Thisclassical example analytically solved in [21] has beenextensively used in the literature to demonstrate the Comput Mech (2005) 36: 34–44DOI 10.1007/s00466-004-0640-7L. Driemeier ( & )Department of Mechatronics and Mechanical SystemsEngineering, University of Sa˜o Paulo,Sa ˜o Paulo – SP – 05508-900, BrazilE-mail: drimeie@usp.brC. ComiDepartment of Structural Engineering, Politecnico di Milano,P.zza Leonardo da Vinci, 32, 20133 Milan, ItalyS. P. B. Proenc¸aDepartment of Structural Engineering University of Sa ˜oPaulo – Sa˜o Carlos – SP – 13560-570, Brazil  shortcomings of local models and the regularizationproperties of enhanced formulations. 2 Local model review 2.1 Kinematics backgroundThe evolution of a one-dimensional body undergoinglarge deformation is considered. The current position of this body is described by a function of the Lagrangiancoordinate  X   and time  t  ,  x ¼ / ð  X  ; t  Þ ð 1 Þ The displacement  u ð  X  ; t  Þ  is given by the difference fromthe current  ð  x Þ  and the srcinal  ð  X  Þ  position.The deformation gradient at a given point is definedby,  F  ð  X  ; t  Þ¼ @  / ð  X  ; t  Þ @   X   ¼ 1 þ u ;  X  ð  X  ; t  Þ ð 2 Þ where  u ;  X   is the derivative of the displacement  u  withrespect to the Lagrangian coordinate  X  . Note that in thisone dimensional case the deformation gradient is simplya scalar, there is no need of polar decomposition intostretch and rigid rotation, and the (left or right) stretchcoincides with  F  .Considering a bar stretched from its initial length  L  toa current length  ‘ , one can also define the stretch ratio  k which represents the mean value of the deformationgradient over the bar k ¼  ‘  L  ð 3 Þ In the linear elastic range, there is a relation betweeninitial and current cross-section area,  A  and  a , respec-tively, a ¼  A k  2 m ð 4 Þ which depends on the stretch ratio  k  and the Poissoncoefficient  m . For  m ¼ 0 the area of the cross section ispreserved for any value of stretching, while for  m ¼ 0 : 5the volume of the bar is constant. For values 0  <  m <  0 : 5, area and volume are modified by the deformation.There are many possibilities in generating a strainmeasure from  F  , see e.g. [22], and the following equa-tions, e m  ¼ 1 m ð  F  m  1 Þ  if   m 6¼ 0 ,ln  F   if   m ¼ 0   ð 5 Þ define the one dimensional form case of the particular strain Lagrangian family . The stress measure associatedwith each strain measure can be obtained imposing thatthe internal power  _ w i be the same whatever stress andstrain measure be adopted, and equal to the product of the nominal stress  r  N   ¼  P  A  by the linear strain rate  _ e  N   ¼  _  F  ð m ¼ 1 Þ , q 0  _ w i ¼ r m _ e m  ¼  P  A _  F  ;  ð 6 Þ or, r m  ¼ r  N   F  1  m :  ð 7 Þ where  q 0  is the initial mass density and  P   is the axialload. Table 1 shows the relations between  m  and themost common strain/stress measures.The strain rates can be derived from Eq. (5) as _ e m  ¼  F  m  1 _  F   ¼  F  m d  ;  ð 8 Þ with d   ¼  F   1 _  F   ð 9 Þ being called instantaneous strain rate. The conjugatestress rate reads, _ r m  ¼  _ r  N   F  1  m þð 1  m Þ  F   m r  N   _  F  ;  ð 10 Þ 2.2 Constitutive relationsThe relations between strains and stresses can be ex-pressed through a material constitutive law, which is notnecessarily linear, stated as r m  ¼ r m ð e m Þ ;  ð 11 Þ Equation (11) can be time differentiated _ r m  ¼ D m _ e m ;  ð 12 Þ Box 1 : Constitutive model for finite strain elastoplasticity where  D m  ¼ d r m d e m is the tangent modulus. Table 1  Conjugated stress–strain measures. m  e m  r m Almansi  ) 2  12  1   F   2    r  N   F  3 Logarithmic 0  e ¼ ln  F   r ¼ r  N   F  Linear 1  e  N   ¼  F    1  r  N   ¼  P  =  A Green 2  12  F  2  1    r  N   F   1 1.  Additive decomposition of the strain rate: _ e ¼  _  F  e  F  e  1 þ  _  F   p   F   p   1 ¼  _ e e þ _ e  p  2.  Linear elastic relation: _ r ¼  E  _ e e 3.  Yield surface and hardening law:  f  ð r ;  R Þ¼j r j  R ð e  p  Þ _  R ¼  _ c h  e  p  ;  R ð Þ  h  e  p  ;  R ð Þ¼ @   R @  c 4.  Flow rule: _ e  p  ¼  _ c  sign ð r Þ 5.  Loading-unloading conditions:  f    0  _ c  0  _ c  f   ¼ 035  Substituting Eq. (12), particularized to linear stress-strain measures, into Eq. (10) one has, _ r m  ¼  F  1  m D  N  _ e  N   þð 1  m Þ  F   m r  N  _ e  N   ¼ D m _ e m  ð 13 Þ where, D m  ¼  F  2 ð 1  m Þ D  N   þð 1  m Þ  F  1  2 m r  N  ;  ð 14 Þ Equation (14) shows that, for small stretching values, i.e.  F   ffi 1, the difference between constitutive moduli D m  ¼ D  N   þð 1  m Þ r  N   is negligible only if   D  N   >>  r  N  .When plasticity is considered, the most commonlyused strain measure is the Hencky’s logarithmic strain.For the particular isotropic case considered here, thelogarithmic strain  e  is conjugated to Kirchhoff stress  r (see [22]), and the constitutive model in the materialversion can be expressed as summarized in Box 1.The classical multiplicative decomposition of thedeformation gradient into elastic and plastic parts isassumed,  F   ¼  F  e  F   p  , leading to the additive decompo-sition of the logarithmic strain rates (Box 1, item 1).Detailed formulation of elastoplasticity with largestrains is out of the scope of this work. For furtherexplanations, see [23] for one- or [24] for three-dimensional case. 3 Problem statement Localization instabilities leading to ill-posedness of theinitial boundary value problem can be evidenced instatics through a bifurcation analysis or in dynamicsthroughawavepropagation analysis [21,16]. Waveswithzero or imaginary velocity do not propagate, leading to alocalized effect, in a infinitely small neighborhood and tofailure without energy dissipation. Accordingly, locali-zation inception in a continuum corresponds to theexistence of imaginary or stationary waves [4].3.1 Wave propagation analysis for linear elastic modelIn the total lagrangian formulation, conservation of momentum in rate form, for a bar of initially uniformarea  A , can be written as  A  _ r  N  ð Þ ;  X  ¼ q 0  A €_ u  ð 15 Þ in terms of nominal stress.From Eq. (10), the relation between the rate of nominal stress  _ r  N   and the rate of the general stress  _ r m  is _ r  N   ¼  _ r m  F  m  1 þð m  1 Þ r  N   F   1 _  F   ð 16 Þ which, from equations (12), (7) and (8), can be rewrittenas _ r  N   ¼  F  m  2  F  m D m þð m  1 Þ r m ½  _  F   ð 17 Þ Consider an initial homogeneous state throughoutthe bar, and consider also the propagation of a har-monic wave defined by, v ð  X  Þ¼  _ u ð  X  Þ¼ v 0 e  i n ð  X   ct  Þ  ð 18 Þ where  v 0  is the amplitude of the wave,  n  is the wavenumber,  i  is the imaginary constant such that  i 2 ¼ 1and  c  is the phase velocity. Substituting Eq. (18) and (2)into Eq. (17) and its result into Eq. (15), one obtains   F  m  2  F  m D m þð m  1 Þ r m ½ þ q 0 c 2   n 2 v 0 e  i n ð  X   ct  Þ  ¼ 0 ð 19 Þ which gives a velocity of wave propagation c 2 ¼  F  m  2 q 0  F  m D m þð m  1 Þ r m ½  ð 20 Þ Stationary waves are found when  c  vanishes  F  m D m þð m  1 Þ r m  ¼ 0  ð 21 Þ When condition (23) is attained, stationary waves of anywavelength, and in particular of arbitrary small wave-length ( n !1Þ ;  are available; therefore the initialboundary problem is no longer hyperbolic and becomesill-posed.In the linear hyperelastic case one can set  r m  ¼  E  e m and from (23), with  D m  ¼  E  , one can compute the crit-ical deformation gradient for localization, for differentthe strain definitions given by (5). These values are re-ported in Table 2.It is interesting to point out from Eq. (21) that forGreen’s strain definition the localization occurs whenthe bar is compressed, while for Logarithmic andAlmansi’s strain localization is induced in tension. Forthe particular case of logarithmic strains, localizationoccurs when, r ¼  E   ð 22 Þ It is worth noting that this value of stress can hardly beattained in structural materials. This result will becompared with the results obtained for elastoplasticmodel in Sect. 3.2. Remark 1.  The critical condition  F   ¼ e  obtained for thelogarithmic strain (see Table 2) expresses in the onedimensional case the more general Legendre-Hadamardconditions established in [25] for Hencky’s logarithmicstrain. Remark 2.  In the one-dimensional case, the violation of the stability criterion [26, 27, 28] exactly coincides withthe onset of the localization phenomena. Loss of sta-bility corresponds to non-positive second order work Table 2  Critical deformation gradient strain m  F  Logarithmic 0 ELinear 1  9 = Green 2  ffiffiffiffiffiffiffiffi 1 = 3 p  36  d 2 W    ¼  _ r _ e  0  ð 23 Þ This product becomes non-positive when, in a uniaxialtension or compression test, the slope of the axial stress-strain curve is zero or negative ( softening  behavior).Using the nominal stress and linear strain definitions,inception of instability occurs when _ r  N  _ e  N   ¼ 0  ð 24 Þ Algebraic manipulations of Eq. (24) leads to the sameexpression (21) found through the wave propagationanalysis. Remark 3:  All the above results have been obtainedusing a Total Lagrangian formulation. Alternatively,one can use an Updated Lagrangian formulation inwhich the basic strain and stress measures are the log-arithmic strain  e  and the Cauchy (or true) stress  r c ,defined as the ratio between the axial load and the cur-rent area  r c  ¼  P  = a . Using the Updated Lagrangianformulation, the family of stress conjugated with the  family of deformation  defined by Eq. (5) is obtained fromenergy conservation in the updated configuration,  q being the current mass density q  _ w i ¼ r m _ e m  ¼  P a F   1 _  F  ;  ð 25 Þ The conservation of momentum is written as, a r c ð Þ ;  x ¼ q  a  € u  ð 26 Þ in terms of Cauchy stress and of the updated Lagrangiancoordinate  x .In this case, the wave propagation analysis developedin this Section leads to a result dependent on the Poissoncoefficient  m . For example, for logarithmic strain conju-gated to Cauchy stress, the localization condition isgiven by2 mr c  ¼  E   ð 27 Þ and for  m ¼ 0 there is no localization.3.2 Wave propagation analysis for an elastoplasticmodelWhen plasticity is considered, the linearization of gov-erning relations is performed under the assumption that _  f   ¼ 0 at each point. From Box 1, one has, _  f   ¼  _ r  sign ð r Þ h _ c ¼ 0  ð 28 Þ with _ r ¼  E   _ e  _ e  p  ð Þ ;  _ e  p  ¼  _ c sign ð r Þ ð 29 Þ From this relation, using Eq. (8), one can computethe plastic multiplier _ c ¼  EF   1 sign ð r Þ  E  þ h  _ u ;  X   ð 30 Þ Moreover, in view of relation between  r  and  r  N  ,Eq. (8), one has _ r  N   ¼  _ r  F   1  r  F   2 _  F   ð 31 Þ Substituting Eqs. (30) and (29) into Eq. (31) one canfind, _ r  N   ¼  E   r ð Þ  F   2 _ u ;  X     E  2  F   2  E  þ h  _ u ;  X   ð 32 Þ Again, the propagation of a harmonic wave definedin Eq. (18) is considered. Substituting Eq. (32) and (18)in the equilibrium (15), one can compute the wavevelocity as c 2 ¼  F   2 q 0  E   r ð Þ  E  2  E  þ h    ð 33 Þ and the stationary condition is given by r ¼  hE  ð  E  þ h Þ ð 34 Þ Differently from the linear elastic case, for small valuesof   h , as in the case of metals, localization can occur at alow level of stress, which can be reached in the material.When small strains are assumed condition (33)becomes (see [21]), c 2 ¼  1 q 0 hE  E  þ h    ð 35 Þ and, for hardening materials ( h  >  0 Þ  there is nolocalization. 4 Nonlocal model In order to regularize the response, which becomesunstable due to geometric and/or material nonlinearities,two nonlocal formulations are developed. Nonlocalformulations of elastic models have been first proposedin [29, 30, 31].A nonlocal model can be formulated introducing anonlocal variable   Y   defined as a weighted average of thecorresponding local measure  Y   taken over the neigh-boring material points of the body. Accordingly,  Y X  ð Þ¼ Z   W X    S  j jð Þ Y S  ð Þ d S   ð 36 Þ where  W X    S  j jð Þ is a weight function, depending on thedistance  r  ¼  X    S  j j between the source point  X   and theneighbor point  S  .In the literature there are various proposal of non-local models which differ for the choice of the nonlocalvariable   Y   and/or for the equations affected by nonlo-cality. In particular,  Y   can be a (strain-related) statevariable or an internal variable (like damage of equiva-lent plastic strain); a comparative assessment of differentnon-local approaches in the case of small strain damagemodels can be found in [14]. Recently, also nonlocaldisplacements has been proposed for the regularizationin the presence of damage [32]. 37  A possible choice for  W X    S  j jð Þ  is the Gaussianweight function, as proposed in [12, 13], W X    S  j jð Þ¼  1 W   0  X  ð Þ exp    X    S  j j 2 2 # 2  !  with W   0 ð  X  Þ¼ Z   exp    X    S  j j 2 2 # 2  ! d S   ð 37 Þ The parameter  #  represents a characteristic length thatintroduces an internal scale in the continuum model aspointed out in [31]. This parameter defines the dimen-sion of the neighborhood that affects the nonlocalfunction. The relation between this characteristic lengthand the width of the localization band will be discussednext for the two proposed models. In both the proposedmodels nonlocality is introduced by defining a nonlocaldeformation gradient    F   as in Eq. (36) but the choice of the eqs. where  F   is replaced by its wheighted averagecounterpart    F   is different.4.1 Model 1In this model nonlocality is introduced in the dynamicequilibrium eq. (15) modifying the relation betweennominal and Kirchhoff stress in the following nonlocalform: r  N   ¼ r   F   1 ð 38 Þ or in rate form: _ r  N   ¼  _ r   F   1  r   F   2  _  F   ð 39 Þ where _  F   ¼  dd t  Z   W   ð  X    S  Þ  F  ð S  Þ d S   ¼ Z   W   ð g Þ _ u ð  X   þ g Þ ;  X  þ g d g ð 40 Þ for   g ¼  X    S  .The other governing equations (compatibility andconstitutive law) are left locally defined in terms of   F  : The same procedure of wave propagation analysisdescribed in Sect. 3 leads to the following localizationconditions, for the elastic and elastoplastic casesrespectively,  E     W   ð n Þ r ¼ 0 and  hE     W   ð n Þ r  h þ  E  ð Þ  E  þ h  ¼ 0  ð 41 Þ where  W    ¼ Z   W   ð g Þ e  i ng d g  ð 42 Þ is the Fourier Transform of the weight function. Atdifference from the local model, the wave number entersinto the conditions for the existence of stationary waves.Since   W    ! 0 for  n !1 , in the elastic case one obtainsthe condition  E  ¼ 0, therefore the problem remains well-posed for any stress level. For the elasto-plastic case, as n !1 ;  the localization condition tends to the one of thesmall strain case defined in Eq. (35) and the problem isregularized for any  h  >  0. Stationary waves of zerowavelength can still be encountered if the material has asoftening behavior ( h  0 Þ :  This behavior is expected,since the regularization adopted focused on the geo-metric nonlinearity only, while the plastic activation isstill governed by the local yield function.Condition (41)b also gives the critical values of wavenumber  n cr   W   ð n cr  Þ¼  hE  r  h þ  E  ð Þ ð 43 Þ Choosing as the weight function the normalized Gaussfunction, as defined in (37), the Fourier transform turnsout to be  W    n ð Þ¼ exp   n 2 # 2 2    ð 44 Þ In the hardening case  h  >  0 ;  the smallest feasiblewavelength of bifurcated modes  K cr   ¼  2 pn cr  can then beobtained substituting this expression of    W    n ð Þ  intoEq. (43) K cr   ¼  2 pn cr  ¼  ffiffiffi 2 p   p #   ln  hE  r  h þ  E  ð Þ     0 : 5 ð 45 Þ This critical length is directly related to the materiallength  #  and defines the width of the localization zone. Itis a function of the stress  r  and of the hardening mod-ulus  h , its evolution with the equivalent plastic strain  c  inthe simple case of linear hardening (  f   ¼ r  r  y   h c Þ  isshown in Fig. 1 for different values of the material length # . For the local model on the contrary, the criticalwavelength is undetermined and thus  K cr   ¼ 0 ( n ¼1Þ isa feasible value and the problem is ill-posed. plastic strain 0.5 0.75 1 1.25 1.5 1.75 22.557.51012.51517.520     c    r     i     t     i    c    a     l    w    a    v    e     l    e    g     t     h     [    m    m     ] Fig. 1  Evolution of the critical length with plastic strain (from above # ¼ 1 mm ;  0.7 mm, 0.4 mm)38
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