Schütt, J.

Wagner, Werner

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Universita¨t Karlsruhe (TH)Institut fu¨r BaustatikA 3D-Plasticity Modelfor the Description of Concreteand its 3D-FE-ImplementationJ. Schu¨tt, W. WagnerMitteilung 2(2003)BAUSTATIKUniversita¨t Karlsruhe (TH)Institut fu¨r BaustatikA 3D-Plasticity Modelfor the Description of Concreteand its 3D-FE-ImplementationJ. Schu¨tt, W. WagnerMitteilung 2(2003)BAUSTATIKc©Prof. Dr.–Ing. W. Wagner Telefon: (0721) 608–2280Institut fu¨r Baustatik Telefax: (0721) 608–6015Universita¨t Karlsruhe E–mail: bs@.uni-karlsruhe.dePostfach 6980 Internet: http://www.bs.uni-karlsruhe.de76128 KarlsruheA 3D-Plasticity Model for the Description ofConcrete and its 3D-FE-ImplementationJ. Schu¨tt, W. WagnerInstitut fu¨r Baustatik, Universita¨t Karlsruhe,Kaiserstr. 12, D-76131 Karlsruhe, GermanyAbstractA three-dimensional material law is introduced which is based on a multi-surface plasticity theory for small strains. Diﬀerent yield criteria of theDrucker-Prager type in combination with a spherical segment to include all-side-compression are used. The special formulation of the yield-function leadsonly to a small number of necessary parameters which can be ﬁxed withinuniaxial experimental tests. The parameters for the description of the con-crete behaviour are: the energy release rate, the ultimate tensile strength, thecompression strength and two ﬁtting-parameters. The material model is im-plemented into a reﬁned geometrically non-linear isoparametric hexahedralelement with special formulations to reduce locking eﬀects.Keywords: Concrete behaviour, Multi-surface plasticity, Material modelling,Numerical simulation1. IntroductionA realistic consideration of the structural behaviour of concrete constructionsusing the ﬁnite element method requires a three-dimensional description of1the material and the element formulation. Three-dimensional ﬁnite elementsare helpful to model the exact geometry of structures. Especially for bound-aries, junction areas or similar this may be essential. To receive the realisticmaterial behavior – like stresses and strains – a three-dimensional materialmodel is indispensable. Many approaches for multiaxial models can be foundin the literature which are based on the theory of plasticity but most of themare used with two-dimensional elements like plates or shells e.g. Hofstetterand Mang [1], Menrath [2] and many others. The model introduced herediscusses the numerical description of pure, not reinforced concrete and isimplemented in a hexahedral element.2. Basic Ideas of the Numerical Model2.1 The Composed Yield FunctionsThe introduced material law is based on a multi-surface plasticity theory forsmall strains. As all signiﬁcant mechanisms of concrete occur within smallstrains, an additive split into elastic and inelastic partsE = Eel + Epl (1)does not represent any restrictions. For the following formulation a divisionof the second Piola-Kirchhoﬀ stress-tensor into a hydrostatic and a deviatoricpart is necessaryS = SH + SD , SD = [II− 13(1⊗ 1)]S . (2)The used material-law is a combination of three conical yield functions of the2Drucker-Prager-type, see Figure 1fj(S, κj) = |SD|+ αjI1(S)−√23yj(κj) = 0 , j = 1, 2, 3 . (3)SD denotes the deviatoric stress tensor, I1 the ﬁrst invariant of the stress ten-sor S. α is a hardening-parameter, the yield stress y depends on the internalvariable κ. The so called inverted cone f3 is only used for numerical stabilityat hydrostatic tension. Furthermore a spherical surface with radius R andorigin L for the compression area is part within the considered modelf(S, κ) =√|SD|2 + 19(I1 − L(κ))2 −R(κ) = 0 . (4)2.2 Parameters of the ModelIt is possible to employ the results of well known uniaxial experimental testsfor diﬀerent concretes to choose the parameters of the yield functions for arealistic description of concrete behaviour. Using the tensile strength fctmand the energy release during cracking gf the exponential softening, whichoccurs during tests in the tensile case, is formulated asy1(κ1) = β1 fctm exp(− κ1κu1)with κu1 =gffctm. (5)The parameters α1 and β1 for the tension part of the yield surface are deﬁnedwith the tensile strength fctm and the compressive strength fcmα1 =√23γ1fcm − fctmγ1fcm + fctm, β1 =2γ1fcmγ1fcm + fctm. (6)The ﬁtting parameter γ1 can be ﬁxed by comparing with the two-dimensional3tests of Kupfer or Ottosen which may be found e.g. in Hofstetter and Mang[1].Considering uniaxial compression in experiments one observes that at aroundone third of the compressive strength square hardening begins. After themaximum strength square softening occurs. This behaviour is approximatedbyy2(κ2) =β213fcm(1 + 4κ2κe+ 2(κ2κe)2), κ2 < κeβ2 fcm(1−(κ2 − κeκu2 − κe)2), κe ≤ κ2 < κu2 .(7)With the energy release rate gc it is possible to formulateκe =4fcm3Ecm, κu2 =3gc2fcm+ κe . (8)The values α2 and β2α2 =√23γ2 − 12γ2 − 1 , β2 =γ22γ2 − 1 (9)depend only on γ2, which may be found in the same way as γ1. The conditionof a smooth transition between the sphere f4 and the rest of the model deﬁnesthe origin L and the radius R asL(κ2) = −(√54α2 + 2)γ2√23y2(κ2) , (10)R(κ2) = −(√23+ 6α22)γ2√23y2(κ2) . (11)2.3 Return-mapping algorithmThe numerical application is based on a local iteration to satisfy the principle4of maximum dissipation with the yield-condition f as a constrain. Thus, animplicit Euler-backward algorithm is chosen to integrate the rate independentevolution equations for the inelastic strains and the internal variable.An implicit time integration results in the classical return-mapping algorithmfor the actual stress state, see Simo and Hughes [3]. Starting at the time n anelastic predictor leads to the so called trial stress state Str. Than the plasticcorrector is used to fulﬁl the Kuhn-Tucker-conditions at the end of the timestep n+ 1 for all yield-functionsfj,n+1(Sn+1, κn+1) ≤ 0 , ∆λj,n+1 ≥ 0 , fj,n+1∆λj,n+1 = 0 . (12)In Eq (12) the increment of the plastic parameter for time step n to n+ 1 isdenoted by ∆λn+1.The reversed return mapping direction of the inverted cone leads to modiﬁedconditionsfj,n+1(Sn+1, κn+1) ≥ 0 , ∆λj,n+1 ≤ 0 , fj,n+1∆λj,n+1 = 0 . (13)From all admitted yield surfaces (m) only a few number (z) will be activatedfor an arbitrary deformation state. To combine z active surfaces Koiter [5]gives a modiﬁed ﬂow ruleE˙pl =z∑i=1λ˙i ∂Sfi with z ≤ m . (14)This has to be taken in account in the constitutive equation within the elastic-predictor and plastic-corrector procedure.As an example a description of the return-mapping algorithm for one singleconical surface is given. For the other cones and the sphere similar formula-tions occur.5At the ﬁrst step of the algorithm an elastic increase of the strains is assumedStr = Cel : (En+1 − Epln ) . (15)If the yield conditionf tr = |SD,tr|+ α I1(Str)− k(κn)≤ 0 : elastic state≥ 0 : plastic state(16)is violated, a local Newton iteration is used to get the incremental update ofthe plastic parameters ∆λn+1 and κn+1. After updating stresses and strainsSn+1 = Str −∆λn+1 Cel : (nn+1 + α 1) (17)Epln+1 = ∆λn+1 ∂Sfn+1 (18)the elastoplastic tangent modulus for a conical part of the surface at thetime-step n+1 can be formulatedCepn+1 =dSn+1dEn+1=[Θ − (Θ : ∂Sf)⊗ (Θ : ∂Sf)∂Sf : Θ : ∂Sf − ∂κk]n+1(19)with the relations∂Sfn+1 = [n+ α 1]n+1 ∂2SSfn+1 =[1|SD|([II− 13(1⊗ 1)]− n⊗ n)]n+1n =SD|SD| Θ n+1 =[C−1 +∆λ ∂2SSf]−1n+1.(20)At the end of each time-step the moduli of all of the z active parts of theyield-surface are coupled.The Newton-Raphson method assures the quadratic convergence within theequilibrium iterations, see e.g. Simo and Hughes [3], Kahn and Huang [4].62.4 ImplementationIn contrast to many other works in which a three-dimensional model is im-plemented into a 2D ﬁnite element formulation, see e.g. Menrath [2] or Hof-stetter and Mang [1], here a three-dimensional element is used. Therefore itis possible to take advantage of the 3D-description of the material for a real-istic simulation in a multiaxial state. The model is implemented in a reﬁnedgeometrically non-linear isoparametric hexahedral element with linear shapefunctions for the displacements. Assumed natural strains for shear stainsand strains in the third direction as well as enhanced assumed strains areintroduced to reduce locking eﬀects. A detailed description of the elementformulation can be found in Klinkel et al. [6].3. Speciﬁcation of the Yield SurfaceThe example of one cube-like element under simple loading is investigatedto describe the correct behaviour of the numerical concrete model. Thestresses acting in the ﬁrst direction are set to the value of S11 = σ , whilethe axial stresses are set to the value of S22 = S33 = F σ in which F isa factor. With these values the ﬁrst invariant reads I1(F ) = (1 + 2F ) σand the norm of the deviatoric part of the second Piola-Kirchhoﬀ stresses|SD(F )| = √2/3 |1 − F |σ. Figure 2 shows the yield surface for the valuesγ1 = 3.0, γ2 = 1.2 and the ratio fcm/fctm = 10.0, which approximate thematerial behaviour of concrete well. If a linear kinematic relation is assumedall loading paths will be linear with the constant gradient of the loading path7δ = |SD(F )|/I1. Five intersection points characterize the yield surface: P1and P5 lie on the I1-axis, while P2, P3 and P4 are the intersections of the singleyield surfaces f1, f2 and f4. Additionally the sign of σ and the necessary valueof F to reach the points are depicted in Table 1. The maximum stress beforesoftening occurs isσmax =√32|SD||1− F | =βifci|1− F |( 1δαi− 1 + 1). (21)This relation is only valid for the conical surfaces fi with i = 1, 2.3.1 Tension AreaWithin this area only surface f2 is active, thus failure is characterized by ex-ponential softening described in Eq (5). The area is limited by the points P1,hydrostatic tension, and P2, where the path reaches f2 beforehand. Figure 3shows stress-strain-curves for possible F . The hydrostatic tension state withF = 1.0 will activate the inverted cone f3.3.2 Mixed AreaThe yield surfaces for tension f1 and compression f2 will both be activatedwithin the mixed area. Firstly the square hardening mechanism of f2 occurs,then exponential softening tension failure of f1 steps in and determines thecurve. The boundaries of this area lie between points P2 and P3.3.3 Compression AreaTo achieve failure in longitudinal direction the stresses σ get the opposite8sign. The area between P3 and P4 is characterized by the yield cone f2whereas the area between P4 and P5 is described by the sphere. Since bothareas follow the square hardening / softening law the stress-strain-curves willbe similar and are shown together in Figure 3. The diﬀerence between thesurfaces is obvious. In opposite to the sphere diﬀerent loading cases limitedby the cone have the same intersection point with the σ = 0 axis.4. ConclusionsA three-dimensional model for concrete which is based on a multi-surfaceplasticity-theory for small strains has been introduced. The yield-surface isa combination of three conditions of Drucker-Prager type and one sphericalpart for the compression case. It is possible to get the necessary parame-ters for the three-dimensional formulation from uniaxial experimental tests.To take the advantage of the realistic description of the model it is imple-mented into a three-dimensional ﬁnite element formulation. An isoparamet-ric geometrically non-linear hexahedral element with special interpolationsof the shear-strains to reduce locking-eﬀects is used. The numerical three-dimensional model reproduces the phenomenological behaviour of concretein a multiaxial state very well.References[1] Hofstetter G and Mang HA. Computational Mechanics of Concrete Struc-tures. Viehweg & Sohn, Braunschweig Wiesbaden, 1995.9[2] Menrath H. Numerische Simulation des nichtlinearen Tragverhaltensvon Stahlverbundtra¨gern. PhD thesis, Universita¨t Stuttgart, Institut fu¨rBaustatik, 1999. Bericht-Nr. 29(1999).[3] Simo JC and Hughes TJR. Computational Inelasticity, volume 7, Me-chanics and Materials. Springer-Verlag, Interdisciplinary Applied Math-ematics, New York, 1998.[4] Kahn AS and Huang S. Continuum Theorie of Plasticity. John Wiley &Sons, Inc., New York, 1995.[5] Koiter WT. Stress-strain relations, uniqueness and variational theoremsfor elastic-plastic materials with a singular yield surface. The QuarterlyJournal of Mechanics and Applied Mathematics 1953; 11:350–354.[6] Klinkel S, Gruttmann F and Wagner W. A continuum based three-dimensional shell element for laminated structures. Computers and Struc-tures 1999; 71:43–62.10Figures and TablesFigure 1: The combined yield surfaceFigure 2: Exact geometry of the yield surface11Points√32|SD| I1√32δ σ F σmaxP1 0 3σmax 0 + 1 2029fctmP2137 |σmax| −57σmax −135 + −67 7043fctmP3218 |σmax| −188 σmax −78 + −138 16043 fctmP434 |σmax| −64σmax −12 − −14 85fcmP5 0 −3σmax 0 − −1 2.1809fcmTable 1: Points on the yield surfaceFigure 3: Diﬀerent areas of the yield surface12

A 3D-Plasticity Model for the Description of Concrete and its 3D-FE-Implementation

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A 3D-Plasticity Model for the Description of Concrete and its 3D-FE-Implementation

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