For high dynamic excitation, e.g. by earthquakes, the vibrations of steel structures lead to inelastic material behavior. Hystereses, developing under cyclic loading, are responsible for the dissipation of energy. Additionally, stress concentration at small defects results in the nucleation and the growth of microvoids which is referred to as damage, here especially as ultra low cycle fatigue. The material damage influences the stiffness of a structure and its response to dynamic excitation. With increasing load the voids can coalesce and form a macrocrack which destroys the structural integrity and peril the overall safety.
 A material model is proposed which describes the evolution and distribution of inelastic strains and isotropic ductile damage for mild construction steel by means of a set of internal variables. Viscoplasticity as well as isotropic and kinematic hardening are taken into account. The evolution of isotropic hardening is related to the growth of a strain memory surface which accounts for the strain amplitude history of the material. Under tension isotropic ductile damage develops for significant inelastic strains [1].
 The material model is implemented in the frameworks of the finite element method with displacement based ansatz functions. The equation of motion is solved with the Newmark method. To overcome the phenomenon of vanishing dissipation energy in case of mesh refinement due to strain localization a nonlocal extension in the form of an implicit gradient formulation is applied.
The presented model is used to analyse 3D structures subjected to seismic excitation

Dinkler, Dieter

Heinrich, Sven

Kowalsky, Ursula

English

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XII International Conference on Computational Plasticity. Fundamentals and ApplicationsCOMPLAS XIIE. On˜ate, D.R.J. Owen, D. Peric and B. Sua´rez (Eds)EVOLUTION OF NONLOCAL DAMAGE IN STEEL UNDERCYCLIC STRAININGSven Heinrich∗, Ursula Kowalsky∗ and Dieter Dinkler∗∗ Institute for Structural AnalysisTechnische Universita¨t BraunschweigBeethovenstr. 51, 38106 Braunschweig, Germanye-mail: s.heinrich@tu-braunschweig.de, web page: http://www.statik.tu-bs.deKey words: Continuum damage mechanics, ductile damage, strain localization, vis-coplasticity, finite element analysis, strain memory surfaceAbstract. For high dynamic excitation, e.g. by earthquakes, the vibrations of steelstructures lead to inelastic material behavior. Hystereses, developing under cyclic loading,are responsible for the dissipation of energy. Additionally, stress concentration at smalldefects results in the nucleation and the growth of microvoids which is referred to asdamage, here especially as ultra low cycle fatigue. The material damage influences thestiffness of a structure and its response to dynamic excitation. With increasing load thevoids can coalesce and form a macrocrack which destroys the structural integrity and perilthe overall safety.A material model is proposed which describes the evolution and distribution of inelasticstrains and isotropic ductile damage for mild construction steel by means of a set ofinternal variables. Viscoplasticity as well as isotropic and kinematic hardening are takeninto account. The evolution of isotropic hardening is related to the growth of a strainmemory surface which accounts for the strain amplitude history of the material. Undertension isotropic ductile damage develops for significant inelastic strains [1].The material model is implemented in the frameworks of the finite element methodwith displacement based ansatz functions. The equation of motion is solved with theNewmark method. To overcome the phenomenon of vanishing dissipation energy in caseof mesh refinement due to strain localization a nonlocal extension in the form of an implicitgradient formulation is applied.The presented model is used to analyse 3D structures subjected to seismic excitation.1 INTRODUCTIONThe energy induced to steel structures by earthquakes is dissipated mainly by inelasticdeformations at predefined regions within the structure. The repeated straining of the1Evolution of nonlocal damage in steel under cyclic straining  1057S. Heinrich, U. Kowalsky and D. Dinklermaterial may lead to failure of structural members and a subsequent collapse of thecomplete structure.Before macrocracks become visible it has been observed that the stiffness and strengthof low alloy steel is deteriorated by microcracks [2, 3] which evolve from voids in themicrostructure and are referred to as damage [4]. For monotonic loading with considerablestrains ductile damage develops and in case of earthquakes the cyclic ductile damageevolution is often referred to as ultra low cycle fatigue.This paper focuses on a method to predict isotropic ductile damage for cyclic straining.For that purpose, a continuum damage model is developed in the framework of thermody-namics taking into account the evolution of isotropic and kinematic hardening, softening,and damage.Under cyclic loading the material is subjected to alternating stress states. The de-ployment of damage under tension and compression is different as pressure may closethe voids in the material matrix. For the evolution of damage under compression var-ious approaches exist[5, 6, 7]. Within the present work, under hydrostatic compressionthe evolution of damage is inactive and the effect of damage on the material stiffness isreduced.To account for the relationship between hardening and the inelastic strain amplitudehistory a strain memory surface is included in the model. It evolves when a strain state isapproached which exceeds the previous maximum strain level. The evolution of isotropichardening depends on the size of the strain memory surface.Damage and softening are always accompanied by localization of deformations in smallprocess zones whose size is characteristic for the material. For FE-analyses the element sizein the damaged region serves as size of the process zone and results are mesh-dependent.As a remedy an implicit gradient enhanced formulation [8, 9] for a non-local damagevariable is introduced which includes an additional material-dependent parameter to affectthe size of the process zone.2 MATERIAL MODELThe mathematical description is achieved by a set of ordinary differential equations offirst order with respect to time. The current state of the material is described by internalvariables and related evolution equations, which depend on the effective stress and strainquantitiesσ˜ =11− h · D¯ · σ , ε˜ = (1− h · D¯) · ε (1)in accordance to the principle of energy equivalence [10]. The crack closure parameter htakes into account that some but not every microcrack closes under compression (h < 1.0)while for tension (h = 1.0) all voids reduce the net area [4]. D¯ is devoted to the nonlocaldamage variable. The material model for the description of the coupled viscoplastic anddamage behavior is given in 3D.21058S. Heinrich, U. Kowalsky and D. Dinkler2.1 Viscoplastic material with hardeningAssuming small strains an additive decomposition of elastic and inelastic strain rate isadmissible with the elastic strain rate based on Hooke’s lawε˙ = ε˙el + ε˙in , ε˙el =ddt(E˜−1 : σ). (2)The inelastic viscoplastic deformations follow the approach by Chaboche and Rousselier[11, 12], yielding˙˜εin =∂σ˜ex∂σ˜p˙ (3)withp˙ = ε˙0〈σ˜exσp〉n. (4)The effective over-stress σ˜ex governs the viscoplastic material behavior and is determinedby a modified yield criterion based on Gurson [13] and Tvergaard and Needleman [14]σ˜ex = σ˜eq − σY −K =√3J2(σ˜ −X)d + D¯ (J1(σ˜ −X))2 − σY −K (5)with the initial yield strength σY . The first and the second invariantJ1(σ˜ −X) = 13tr(σ˜ −X) , J2(σ˜ −X)d = 12tr((σ˜ −X)d · (σ˜ −X)d) (6)are computed with respect to the effective active stress tensor σ˜−X and the effective activestress deviator (σ˜−X)d, respectively. The kinematic hardening X develops according toX˙ = a(23C∂σ˜eq∂σ˜−X)p˙ . (7)The yield function considers isotropic hardening K for which the evolution equation isgiven byK˙ = b(Q−K)p˙ . (8)The parameters a and b describe the velocity of hardening evolution while the maximumvalues of hardening are defined by the parameters C and Q, respectively.Figure 1 depicts the v. Mises yield cylinder and the contraction of the yield surfacecombined with a formation of caps due to damage.2.2 Strain Memory SurfaceExperiments have revealed that cyclic hardening depends on the strain amplitude and iscoupled to the maximum strain level. In particular the evolution of hardening is differentfor decreasing and increasing strain amplitudes. A strain memory surface MM =23(ε˜in − β)(ε˜in − β)− q2 ≤ 0 (9)31059S. Heinrich, U. Kowalsky and D. DinklerD=0(v.Misesyield cylinder)D>0(yield surface forcumulative damage)σ2σ1σ3Figure 1: Evolution of the yield surface with emerging damageis proposed to capture the strain loading history. It depends on the variables β and q,where β describes the center of the surface and q specifies its radius in the strain space.Their evolution is given byβ˙ = (1− c) ˙˜εin,q˙ = c (nF )T nM p˙}M ≥ 0 , β˙ = 0,q˙ = 0}M < 0 (10)where nF and nM are the normals to the yield surface and the strain memory surface,respectively.For a virgin material without hardening effects Figure 2 shows the evolution of thestrain memory surface in the two-dimensional principal strain plane. A significant largeone-dimensional load step shall result in the stress state A and the inelastic strain ∆ε˜inA inthe direction of nF,A. The center of the strain memory surface ∆βA evolves correspondingto the inelastic strain tensor. For the first inelastic step nM is supposed to be equal tonF and the change in size of the surface ∆qA is equal to half the value of ∆ε˜inA . Furtherinelastic strain evolution inside the surface does not influence its properties. For themultiaxial stress state given at point B the inelastic strain develops outside the surface(M > 0) and denotes a new level of strain amplitude. Since nF is not equal to nM theenlargement ∆qB is less than half the absolute value of inelastic strain.The approaches to describe the saturation value for isotropic hardening Q range fromlinear function [15] to exponential functions [16] with additional parameters. Within thiswork, the evolution of the saturation value of isotropic hardening Q is assumed to be alinear functionQ˙ = QK · q˙ (11)41060S. Heinrich, U. Kowalsky and D. DinklerεinΔqAstrain�memorysurface�MAΔβAΔinεAIIεinnM,BBIyieldsurface�FσIIσIBnF,BA nF,AεinΔqAΔβBIIεinIΔqBnM,BnF,BnF=nMA nM,AΔqB| βBΔ |ΔinεB| |ΔinεBMBΔqA| βAΔ || |ΔinεA~ ~~~~~~~Figure 2: Evolution of the strain memory surfacewith the parameter QK .2.3 Damage EvolutionThe onset of damage is determined by a damage threshold surface as experimentshave shown that the nucleation of cavities does not occur for cyclic loading with smallamplitudes. In the model presented here damage evolution is prohibited if the equivalentplastic strain is smaller than a parameter εinDε˜in,+eq − εinD ≤ 0 −→ D˙ = 0 ,> 0 −→ D˙ > 0 . (12)Furthermore it is assumed that damage does not grow for compression states. The twoterms of the damage evolution equationD˙ = (c1 + c2e−c3 p+)p˙+ + c4(c5 − D¯)〈tr(∂σ˜ex∂σ˜p˙+)〉(13)consider transient and saturation behavior in damage evolution as well as damage due tovolumetric yielding and depend on the damage parameters c1 - c5.The nonlocal damage variable D¯ is obtained from structural FE-analysis by an implicitgradient enhanced formulation following Equation (15). The material behavior and thegoverning equations are described in detail in [1].Stresses and internal variables are determined from the material equations and thedisplacement field by the implicit Euler approach with first order accuracy. Moreover alocal iteration scheme improves the convergence of the global FE-analysis.51061S. Heinrich, U. Kowalsky and D. Dinkler3 DISCRETISATION IN TIME AND SPACEFor the numerical space-time domain analyses the weak form of the equation of motioncan be obtained from the principle of virtual work∫Ωδu · ρu¨dΩ +∫Ωδu ·Cu˙dΩ +∫Ωδε : σdΩ =∫Ωδu · pdΩ +∫∂Ωδu · t¯d∂Ω +∫Ωδu · ρu¨gdΩ . (14)Assuming linear kinematics Equation (14) is discretized in space with the finite-elementmethod, while the generalized Newmark method is applied for the integration in time.Here, the parameters are set to α = 12and β = 14which implicitly approaches the statevariables by a linear evolution.The softening of the material accumulates in process zones. For local damage modelsthe mesh size serves as width for the process zone and results become mesh-dependent. Innonlocal models an additional material-specific parameter called internal length lc definesthe size of the process zone and serves as a numerical parameter regularizing the solution.Here, an implicit gradient formulation is applied [8, 9, 17], which governs the spatialdistribution of locally developed damage D, see Equation (13), to the nonlocal substituteD¯ by the partial differential equationD¯ − l2c∇2D¯ = D . (15)The weak form of the partial differential equation for the nonlocal damage field D¯ resultsfrom weighted residua∫Ω(δD¯(D¯ −D)+∇δD¯ l2c ∇D¯) dΩ− l2c ∫∂ΩδD¯ n∇D¯ d∂Ω = 0 (16)with the virtual nonlocal damage δD¯ and is discretized in space by the finite-elementmethod. It is assumed that the nonlocal distribution is time-independent.For 3D analyses triquadratic twentyseven-node brick elements are used for the spatialdiscretization of the displacement and the damage field.4 STRUCTURAL ANALYSISThe following examples demonstrate the ability of the model to describe the evolution ofdamage for steel constructions under different loading conditions. The material-dependentmodel parameters are determined from experimental data via genetic algorithms (see [18]for further information).The accuracy of the material model is evaluated by means of recalculations of experi-ments on compact tension specimens [19] under monotonic loading.The distribution of damage for the CT-specimen is shown in Figure 3 for the momentthe opening displacement ∆u at the left boundary reaches 5 mm. Three quarters of the61062S. Heinrich, U. Kowalsky and D. Dinklerspecimen are depicted. It can be seen that damage starts inside the specimen due to themultiaxial stress state in the middle while at the boundaries a rearrangement of stressesoccurs. The results indicate that the developed damage model is able to reproduce theexperimental investigations correctly.∆Figure 3: Distribution of damage for a CT-specimen under constant loading [18]In a second application, the model is applied to a cantilever with an IPE 80 profile.The cantilever of 5m height is subjected to the north-south component of the Kobeearthquake from 1995. The earthquake causes inelastic cyclic straining in the bottompart of the cantilever resulting in material damage evolution at the base of the cantileverwhich ist shown in Figure 4. It can be seen that damage occurs in the flanges whichare subjected to high alternating tension and compression. Evidently the shear bandsaccumulate damage and weaken the structure.Figure 4: IPE 80, distribution of damage after seismic loading [18]71063S. Heinrich, U. Kowalsky and D. Dinkler5 CONCLUSIONSA nonlocal damage model for structural steel regarding viscoplastic material behavior,isotropic and kinematic hardening as well as isotropic damage is developed in order toestimate the amount of damage in steel structures subjected to inelastic cyclic strainingas it may occur under seismic excitation. For the description of damage evolution, ductiledamage is taken into account whereas damage evolution is only active for tension and sig-nificant inelastic strain. The phenomenon of strain localization which always accompaniessoftening and damage is described by a nonlocal implicit gradient enhanced formulation.It is shown, that the proposed model can be employed to assess the damage state ofsteel structures for different loading conditions. The applicability of the presented modelto other kinds of steel or other cyclic loading conditions persists. Increasing computingpower and parallel computing enhance the application of continuum damage analyses ofcomplex three-dimensional structures.REFERENCES[1] Kowalsky, U. and Meyer, J. and Heinrich, S. and Dinkler, D. A nonlocal damagemodel for mild steel under inelastic cyclic straining. Comp. Mat. Sc. (2012) 63:28-34.[2] Hommel, J.-H. and Meschke, G. A hybrid modeling concept for ultra low cycle fatigueof metallic structures based on micropore damage and unit cell models, Int. Journalof Fatigue (2010), 32:1885-1894.[3] Yang, L. and Sun, B. and Guo, Y. Damage Evaluation Based on Electrical ResistanceMeasurements, Key Engineering Materials (2008) 385-387:589–592.[4] Lemaitre, J. A Course on Damage Mechanics. Springer, (1992)[5] Pirondi, A. and Bonora, N. Modeling ductile damage under fully reversed cycling,Computational Material Science (2003) 26:129–141.[6] Mediavilla, J. and Peerlings R. and Geers, M. A nonlocal triaxiality-dependent ductiledamage model for finite strain plasticity, Computer Methods in Applied Mechanicsand Engineering (2006) 195:4617–4634.[7] Kanvinde, A. and Deierlein, G. Cyclic Void Growth Model to Assess Ductile FractureInitiation in Structural Steels due to Ultra Low Cycle Fatigue, Journal of EngineeringMechanics (2007) 133:701–712.[8] Peerlings, R. and de Borst, R. and Brekelmans, W. and Geers, M. Gradient EnhancedDamage Modelling of Concrete Fracture, Mechanics of Cohesive-Frictional Material(1998) 3:323–342.81064S. Heinrich, U. Kowalsky and D. Dinkler[9] Engelen, R. and Geers, M. and Baaijens, F. Nonlocal implicit gradient-enhancedelasto-plasticity for the modelling of softening behavior, International Journal ofPlasticity (2003) 19:403–433.[10] Cordebois, J. and Sidoroff, F. Damage Induced Elastic Anisotropy, Proceedings ofthe Euromech Colloquium (1982) 115:761–774.[11] Chaboche, J. and Rousselier, G. On the Plastic and Viscoplastic Constitutive Equa-tions - Part I: Rules Developed with Internal Variable Concept, Journal of PressureVessel Technology (1983) 105:153–158.[12] Chaboche, J. and Rousselier, G. On the Plastic and Viscoplastic Constitutive Equa-tions - Part II: Application of Internal Variable Concept to the 316 Stainless Steel,Journal of Pressure Vessel Technology (1983) 105:159–164.[13] Gurson, A. Continuum Theory of Ductile Rupture by Void Nucleation and Growth:Part I - Yield Criteria and Flow Rules for Porous Media, Journal of EngineeringMaterials and Technology (1977) 99:2–15.[14] Tvergaard, V. and Needleman, A. Analysis of the cup-cone fracture in a round tensilebar, Archives of Mechanics (1984) 32:157–169.[15] Ambroziak, A. and Klosowski, P. The elasto-viscoplastic Chaboche model, TASKQuarterly (2006) 10:49–61.[16] Taleb, L. and Cailletaud, G. An updated version of the multimechanism model forcyclic plasticity, International Journal of Plasticity (2010) 26:859-874[17] Kowalsky, U. and Zu¨mendorf, T. and Dinkler, D. Random fluctuations of materialbehaviour in FE-damage-analysis, Computational Materials Science (2007) 39:8–16.[18] Velde, J. 3D Nonlocal Damage Modeling for Steel Structures under Earthquake Load-ing. Institut fu¨r Statik, TU Braunschweig, 2010.[19] Sester, M. and Mohrmann, R. and Riedel, H. Application of a mechanism-basedcreep damage model to creep crack growth in a 12% chromium steel, Materials atHigh Temperatures (1998) 15: 265-27091065

Evolution of nonlocal damage in steel under cyclic straining

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Evolution of nonlocal damage in steel under cyclic straining

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