An implicit integration method for solving non-linear problems in mechanics of structures under complex loading histories

Transcription

1 An implicit integration method for solving non-linear problems in mechanics of structures under complex loading histories M. Arzt, W. Brocks & R. Mohr GKSS Forschungszentrum Geesthacht GmbH Institutfur Werkstofforschung Max-Planck-Strafie, Geesthacht, Germany Abstract An implicit integration method is discussed. This method allows for studying nonlinear material behaviour, such as viscoplasticity. The material behaviour is described by so-called internal variables. The model applied here was developed by Lemaitre and Chaboche. The integration method is implemented in a usersubroutine in the commercial Finite-Element-Method package ABAQUS. The results of a numerical analysis of the rotation of an aircraft turbine disk are presented. Cyclic loading conditions are studied, and the results are compared to numerical results found in literature. 1 Introduction Structures under cyclic loading conditions are studied. The inelastic behaviour under cyclic loading is an interesting task since the failure of structures is often caused by repeated cycles of loading, e.g. an aircraft turbine disk, Dambrine & Mascarell [7]. To simulate the material behaviour in case of cyclic loading, multiaxial and out-of-phase loading conditions, non-linear constitutive relations have to be applied, Benallal & Ben Cheikh [3] and Ohno [15]. At present, constitutive equations described by internal variables are the common approach to simulate the phenomena of isotropic and kinematic hardening, Nouailhas [13] and Wang & Ohno [17]. The viscoplastic model applied is based on the works of Chaboche [4], [5] and Lemaitre & Chaboche [11]. These constitutive relations are developed to simulate cyclic loading. The model of Chaboche, implemented in the commercial FEM package ABAQUS is rate-independent, plasticity behaviour, Aravas [1].

2 372 Computational Methods and Experimental Measurements The model applied in this paper is rate-dependent, to simulate viscoplastic behaviour. This model is implemented in a user-subroutine of ABAQUS. The unknowns of the problem are defined in the second paragraph. The constitutive relations are strongly non-linear. Therefore, the numerical analysis of structures is complex and the integration methods are difficult to develop. The user-subroutine is based on an implicit integration method, Hornberger & Stamm [8] and Olschewski [14]. An expansion in a Taylor series of the constitutive equations is carried out. An Euler scheme is applied too. The increments of the unknowns are a superposition of the rates at the begin of the increment and at the end of the increment. A weighting factor permits to chose an explicit integration method or an implicit integration method, and the mid-point rule as well. The solution of the linearized system is determined using a public domain solver, the second task. The third task is to determine the so-called Jacobi matrix, or the tangent operator, in order to reduce the number of iterations, and to increase the time increments. The required processing time and memory for storage, to solve these problems, is still important. Even today, there are few complex computations carried out in industry, although the performances of computers is increasing, and strategies are developed to carry out these computations, Arzt et al. [2], Cognard & Ladeveze [6] and Ladeveze & Rougee [9]. These methods allow to reduce the processing time and to decrease the storage capacities required, Ladeveze [10] and Lesne & Savalle[12]. 2 Material Behaviour Here, the unknowns of the problem are defined, stress, back stress, isotropic hardening and the inelastic strain, Chaboche [4], [5] and Lemaitre & Chaboche [11]. The mechanical behaviour is described by so-called internal variables, the equivalent plastic strain, p, and the kinematic hardening variable, a, in order to simulate cyclic phenomena. The expansion or contraction, and translation of the yield surface are represented by the corresponding variables, the isotropic hardening function, R, and the back stress, x, respectively. Isotropic and kinematic hardening are comined, and kinematic hardening is non-linear. The potential of the free energy, where the density is />, is taken as pv = i De*V 4- I caaa + h(p). (1) 2 o The elastic stiffness or Hooke matrix is D. Kinematic hardening constants are c and a. For the isotropic hardening, the function h(p) is chosen as the potential: The asymptotic value of the isotropic hardening function is Q, eqn. (5), and b is an exponent.

3 Computational Methods and Experimental Measurements 373 The state equations are obtained by differentiating the potential of the free energy (1), using (2): a = Be*', (3) 2 x = - caa, (4) The constitutive equations are derived from the dissipation potential: (5) Where n stands for the hardening exponent, and K is the coefficient of resistance. The yield function /(<r, x, R) is defined as: /(<r,x,fl) = J2(<r-x)-/Z-*, (7) and the second invariant or the equivalent von Mises stress is. where cr*** is the deviatoric part of the stress. The yield strength at zero plastic strain is k. A function, the MacCauley bracket, is defined as: (y) = yify>0 and (j/> = 0 if y < 0. Derivation of the dissipation potential leads to the constitutive equations, Lemaitre & Chaboche [11]: R = b(q-r)p. (10) The second term, the relaxation term, of the kinematic hardening is introduced to decrease the effect of the plastic strain. The function of the relaxation term, < (p), is The constants $00 and w are material parameters. The third term describes the static recovery, the parameters are the constant d and the exponent r. This version of the model is a rather simple one. More kinematic hardening variables can be introduced, in order to describe the phenomenon of hysteresis.

4 374 Computational Methods and Experimental Measurements 3 Implicit Integration Method An implicit integration method, Hornberger & Stamm [8], Olschewski [14] and Qi [16], is developed. The constitutive equations, (8), (9) and (10), are taken in the following expression: x = h(cr, x, R) R = r(<r,x, R). (11) (12) The aim is to determine the increments of the inelastic strain, Ae'", the back stress, Ax, and isotropic hardening function, AA, for a given time increment, A(, by the rates at the begin of the increment, t, and the end of the increment, t + At. A superposition, of the rates at the begin and the end of the increment, is applied: (13) Ac'" = (14) The superposition includes the explicit, 0 = 0, and implicit, 0=1, integration method by Euler, and the mid-point rule, 0 = 1/2, as well, since the parameter can be choosen in 0 < 0 < 1. The rates at the end of the increment are unknown. An expansion of the constitutive equations using a Taylor series at the begin of the increment leads to a linearized form: <9g <9g AcrH da t ^ 9x dh c9h X t + At = h t + ^~ ACTt da dr dr ACT 4 da Ax 4- t Ax-f t Ax + t ^g 9h 5r (15) (16) AB (17) (18) (19) The system of equations to be solved leads to 4 unknowns, the increments of the stress, ACT, back stress, Ax, isotropic hardening function, A/2, and the inelastic strain, Ae'*\ To solve the system, the elastic behaviour and the usual decomposition of the total strain are taken = D(Ac- (20) Therefore, the numerical calculations are reduced to solve a system of the size of 13 unknowns. In case of a three dimensional problem, there are 6 components of the stress, 6 components of the back stress, and one of the isotropic hardening. The system to be solved is written in the following form: MAy = (21)

5 Computational Methods and Experimental Measurements 375 The matrix M is of the following structure M = The components of the matrix are: M21 M22 11*23 ni (22) <9g, M. = {l-,a^j gb ah gb oa The right hand side vector is given i - Ae The components are determined by the constitutive equations, (8), (9) and (10), and the rates of the previous time step, applying the superposition, (14), (15) and (16): 63 (23) 63 = The vector of the unknowns is: H- (1 - ACT Ax (24) The system is solved using subroutines developed by Cleve Moler, University of New Mexico, Argonne National Laboratory. Once, the vector of the unknowns is determined, the inelastic strain increment, eqn. (20), is calculated. Then the stress, back stress, isotropic hardening and the inelastic strain are updated. The last task is to define the Jacobi matrix. Therefore, the system, eqn. (21), is taken and transformed to obtain a form like: ACT = KnAe ^n,0- (25) The matrix, K^, is determined for each integration point, n. And then, the matrix, K, of the global problem is assembled in order to calculate the equilibrium solution. The system, (21), is solved analytically. First, the increments of the back stress, Ax, and isotropic hardening function, AH, are determined, since these are independent of the strain increment, Ac. These increments are a function of the increment of the stress, ACT.

6 376 Computational Methods and Experimental Measurements The Jacobi matrix is of the following structure: K-I = -[MH - MizK^ where the matrices are: Kr = m33m22 - m23m32 and M and the vector is Finally, the Jacobi matrix, Kn, is obtained. 4 Numerical Simulation As a numerical simulation is carried out the problem of a rotating aircraft turbine disk, Lesne & Savalle, [12]. The problem is quasi-static and isothermal conditions are assumed. The mechanical loading is the centrifugal force. The problem is axisymmetric. The diameter of the disk is 300 mm, and the diameter of the whole is 75 mm. The thickness of the disk at the bore is 50 mm, and at the neck the thickness is 15 mm. Infig.1 the mesh of the turbine disk is shown. The number of three-node elements is 571, and the number of nodes is 335. The plane of symmetry is the plane 1-3. The total number of degree of freedom is about 629. Figure 1: mesh of the aircraft turbine disk The temperature field is homogeneous, and the temperature is of 550 C. The material is the nickel-based alloy INCONEL 718, the data is given in tab. 1, Benallal & Ben Cheikh, [3]. The data allows for studying cyclic mean-stress relaxation. Table 1: parameters of Chaboche model E = MPa v 0.3 k = 646 MPa Q = -185 MPa c = 500 d = 0 1/s n b a r K MPa MPa p = 8190 kg/nf

7 Computational Methods and Experimental Measurements 377 A loading of 500 cycles is simulated. The shape of thefirstthree loading cycles is presented infig.2. The minimum rotation speed is 1500 r/min, and the maximum rotation speed is r/min. In contrast to Lesne & Savalle, [12], where the maximum rotation speed is r/min. In fact of the possibility to study cyclic mean-stress relaxation, the maximum rotation speed is higher, in order to obtain important inelastic strains time t in [sec] Figure 2: cycles of loading The processing time on an IBM Rise/System 6000 computer is 7 hs, CPU. The integration parameter is chosen equal to 1/2. The results are compared to data determined by Lesne and Savalle, [12]. In fig. 3 the stress versus inelastic strain for a node at the bore and at the neck is presented. An important variation of the stress is noticed. The inelastic strain magnitude is of 0.6 % at the bore. In fact of the mean-stress relaxation, and the higher level of the maximum rotation speed, the stresses and inelastic strains are of the same magnitude. bore, [12] neck, [12] bore, authors neck, authors inelastic strain 33^ in [%] Figure 3: hoop stress versus inelastic hoop strain, at the bore and at the neck

8 378 Computational Methods and Experimental Measurements The hoop stress at the beginning of maximum rotation speed for thefirstand tenth cycle, and for cycles 100 and 500 are given infigs.4, 5, 6 and 7, respectively. The stress distribution versus the number of cycles is observed. The minimum level of the hoop stress decreases, and the maximum level of the hoop stress increases. The distribution is shown, to give an idea of the zones where plastic flow occurs. Remark a high level of the stress at the bore and at the neck. Important stress gradients are observed at the exterior part of the disk. The stable stress distribution is nearly reached at cycle 100. Figure 4: hoop stress at beginning of maximum rotation speed for cycle [MPa] 969 Figure 5: hoop stress at beginning of maximum rotation speed for cycle 10 Figure 6: hoop stress at beginning of maximum rotation speed for cycle 100 [MPa] Figure 7: hoop stress at beginning of maximum rotation speed for cycle 500

9 Computational Methods and Experimental Measurements Conclusions The performances of the method is shown. Numerical results correspond to the results obtained by Lesne & Savalle [12]. Accuracy is sufficient. The aim is to take into account anisotropy, an isothermal conditions and damage as well. The increment size will be controlled by procedures in order to accelerate convergence and decrease processing time. Acknowledgements The authors would like to thank Rainer Sievert for providing experimental data, which allowed us to verify the subroutine. References 1. Aravas, N., On the Numerical Integration of a Class of Pressure-Dependent Plasticity Models, InternationalJournal for Numerical Methods in Engineering, John Wiley & Sons Ltd, vol. 24, pp , Arzt, M., Cognard, J.-Y. & Ladeveze, P., A Large Time Increment Strategy for the Analysis of Viscoplastic Structures Under Complex Loading Histories, Proceedings of the International Seminar on Multiaxial Plasticity, eds. Benallal A., Billardon R. & Marquis D., Laboratoire de Mecanique et Technologic, Ecole Normale Superieure de Cachan, France, pp. 434^460, Benallal, A. & Ben Cheikh, A., Constitutive Equations for Anisothermal Elasto-Viscoplasticity, Constitutive Laws for Engineering Materials: Theory and Applications, eds. Desai, C.S. et al, Elsevier Science Publishing Co Inc, pp , Chaboche, J.-L., Cyclic Viscoplastic Constitutive Equations, Part I: A Thermodynamically Consistent Formulation, Journal of Applied Mechanics, Transactions of the ASME, New York, vol. 60, pp , december Chaboche, J.-L., Cyclic Viscoplastic Constitutive Equations, Part II: Stored Energy Comparison Between Models and Experiments, Journal of Applied Mechanics, Transactions of the ASME, New York, vol. 60, pp , december Cognard, J.-Y. & Ladeveze, P., A Parallel Computer Implementation for Elastoplastic Calculations with the Large Time Increment Method, Nonlinear Engineering Computations, Swansea, pp. 1-10, 1991

10 380 Computational Methods and Experimental Measurements 7. Dambrine, B. & Mascarell, J. P., About the Interest of Using Unified Viscoplastic Models in Engine Hot Components Life Prediction, Proceedings of the International Seminar on High Temperature Fracture Mechanisms and Mechanics, eds. Bensussan, P. & Mascarell, J. P., Mechanical Engineering Publications, London, pp. 1-15, Hornberger, K. & Stamm, T, An Implicit Integration Algorithm with a Projection Method for Viscoplastic Constitutive Equations, International Journal for Numerical Methods in Engineering, John Wiley & Sons Ltd, vol. 28, pp , Ladeveze, P. & Rougee, P., Plasticite et viscoplasticite sous chargement cyclique : propri6tes et calcul du cycle limite, Comptes Rendus de I'Academic des Sciences, Gauthier-Villars, Paris, tome 301, serie II, no. 13, pp , Ladeveze, P., La methode a grand increment de temps pour r analyse de structures a comportement non lineaire decrit par variables intermes, Comptes Rendus de VAcademic des Sciences, Gauthier-Villars, Paris, tome 309, serie II, pp , Lemaitre, J. & Chaboche, J.-L., Mechanics of Solid Materials, Cambridge University Press, UK, Lesne, M. P. & Savalle, S., An Efficient Cycles Jump Technique for Viscoplastic Structure Calculations Involving Large Number of Cycles, Proceedings of the Second International Conference on Computational Plasticity, eds. Owen, D. R. J.; Onate, E. & Hinton E., Pineridge Press Ltd, Swansea, pp , Nouailhas, D., Unified Modelling of Cyclic Viscoplasticity: Application to Austenitic Stainless Steels, International Journal of Plasticity, Pergamon Press pic, vol. 5, pp , Olschewski, J., Plastizitdtstheorie II, Vorlesung an der Technischen Universitat Berlin, BAM /4, Bundesanstalt fiir Materialforschung und -priifung, November Ohno, N., Constitutive Modelling of Cyclic Plasticity with Emphasis on Ratchetting, InternationalJournal of Mechanical Science, Elsevier Science Ltd, vol. 40, nos. 2-3, pp , Qi, W., Entwicklung eines allgemeinen FE-Bausteins fur viskoplastische Materialmodelle auf der Basis der fur thermisch-mechanische Belastungsvorgdnge erweiterten Integrationsmethode nach Tanaka, Diplomarbeit, Technische Universitat Berlin, Institut fur Mechanik, Berlin, Wang, J.-D. & Ohno, N., Two Equivalent Forms of Non-linear Kinematic Hardening: Application to Non-isothermal Plasticity, International Journal of Plasticity, Pergamon Press pic, vol. 7, pp , 1991