The subloading-overstress friction model is formulated for the unified description of the dry and the fluid frictions which exhibit the negative and the positive rate dependences, i.e. the decrease and the increase, respectively, of friction resistance. The validity of this model will be verified by the comparisons with test data in this article

Hashiguchi, Koichi

Ozaki, Toshiyuki

English

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Unified Description of Dry and Fluid Frictions by Subloading-Overstress Friction ModelT. Ozaki and K. HashiguchiVIII International Conference on Computational Methods for Coupled Problems in Science and Engineering COUPLED PROBLEMS 2019 E. Oñate, M. Papadrakakis and B. Schrefler (Eds) UNIFIED DESCRIPTION OF DRY AND FLUID FRICTIONS  BY SUBLOADING-OVERSTRESS FRICTION MODEL TOSHIYUKI OZAKI* AND KOICHI HASHIGUCHI†* Design of Overhead Transmission line Dept. Kyushu Electric Engineering Consultants Inc. Kiyokawa 2-13-6, Fukuoka 810-0005, Japan e-mail: ozaki@dengi.co.jp MSC Software, Ltd. Shinjuku First West 8F, 1-23-7, Nishishinjuku, Shinjuku-ku, Tokyo 160-0023, Japan e-mail: e-mail: hashikoi87@gmail.com Key words: Dry and Fluid Frictions, Subloading-overstress Friction Model, Rate-sensitivity Abstract. The subloading-overstress friction model is formulated for the unified description of the dry and the fluid frictions which exhibit the negative and the positive rate dependences, i.e. the decrease and the increase, respectively, of friction resistance. The validity of this model will be verified by the comparisons with test data in this article.1 INTRODUCTION The dry friction without a lubrication exhibits the decrease of friction resistance with the sliding velocity, which is called the negative rate-sensitivity. On the other hand, the fluid friction exhibits the increase of friction resistance with the sliding velocity, which is called the positive rate-sensitivity. The generalized friction model, called the subloading-overstress friction model[1], is formulated based on the subloading surface model [2] [3] [4], which is capable of describing the dry and the fluid frictions by the unified formulation.    The validity of the subloading-overstress friction model is examined by the comparisons with test data in this article. Then, the capability of describing both the dry and the fluid frictions is verified by these comparisons. 2 SLIDING DISPLACEMENT AND CONTACT TRACTION VECTORS The sliding displacement vector u , which is defined as the sliding displacement of the counter (slave) body to the main (master) body, is orthogonally decomposed into the normal sliding displacement vector nu  and the tangential sliding displacement vector tu  to the contact surface as follows: = n tu u u                                                           (1) where 300T. Ozaki and K. Hashiguchi 2= ( ) = ( ) =(= = )n nntu  u u n n n n un nu u u I un                                        (2) n  being the unit outward-normal vector of the surface of main body and =n nu    n nu u                                                   (3)  Contact traction and sliding velocity.     The sliding displacement vector u  can be exactly decomposed into the elastic sliding displacement eu  and the viscoplastic sliding displacement pvu  in the additive form even for the finite sliding displacement, i.e. = pe vu u u                                                           (4) == p pv vtnpe e evn t u u uu u u                                                       (5) where = ( ) = ( ) =(= = )n nt ne e eee ee eu  n nu u n n u nnu u nu I u                                      (6) = ( ) = ( )(= = )pvp pv vp pppv vnvvt n  n n n nu u unu u n uu I                                         (7) setting = enn eeu    n nu u                                                   (8) The minus sign is added for neu  to be positive when the counter body approaches the main body. The plastic sliding displacement pvu  is derived by the unloading to the contact traction-free state along the hyperelastic constitutive equation which will be formulated in Section 3. The viscoplastic sliding flow rule will be formulated to fulfill =pvnu 0  in Section 4. The contact traction vector f  acting on the main body is additively decomposed into the normal traction vector nf  and the tangential traction vector tf  as follows (see Figure 2): = = fn n tt f f  f nf f t                                               (9) where fMain (Master) bodyCounter (Slave) body2e3 e n1etfnftuunu301T. Ozaki and K. Hashiguchi 3( ) ( )= =(= ) = fn nt n tff      nf n f n n n ff f f I n n f t                                      (10) ( )|| ||,    ( ,  || || 1 )== = =0|| ||( ) f ff ftntff       n fI n n f nff I n n ft t t t                  (11) The minus sign is added for nf  to be positive when the compression is applied to the main body by the counter body.  The contact traction vector f , nf  and tf  are calculated from the Cauchy stress σ  applied to the contact surface by virtue of the Cauchys fundamental theorem as follows: =( ) ( )= =( )=nt f nf n n n n n nf I n n nσσ σσ                                          (12)  Coulomb-type normal- and subloading-sliding surfaces 3 HYPERELASTIC SLIDING EQUATION Let the contact traction vector f  be given by the hyperelastic relation with the elastic sliding displacement energy function ( )e u  as follows: ( )eeufu                                                          (13) The simplest function ( )e u  is given by the quadratic form: ( ) 2/e e e  u u Eu                                                  (14) Main body( ))( f  fn nfr11e2e3 e ntn/ =t nf f Normal sliding-yield surfaceSubloading-sliding surface/ =t nf f rftfnf( )fn t t( )fff( ) ( )( )( ) )(tf ff    f ff n nf ff302T. Ozaki and K. Hashiguchi 4where the second-order symmetric tensor E  is the elastic contact tangent modulus tensor fulfilling the symmetry = TE E . The substitution of Eq. (14) into Eq. (13) leads to ef Eu                                                                (15) The inverse relation of Eq. (15) is given by =e u E f                                                             (16) The elastic contact tangent modulus tensor E  is given for the isotropy on the contact surface as follows: 1( )=1 1 )= (n tn t      n nn nE In nn n IE                                       (17) where n  and t  are the normal and tangential contact elastic moduli, respectively. Their values are quite large usually as 2 5 310 10 GPa/mm  for metals because the elastic sliding is caused by elastic deformations of the surface asperities. Equations (15) and (16) with Eq. (17) leads to =11=t nnt ne et net u ufffu                                                (18) Now, introduce the normalized rectangular coordinate system 1 2 3 1 2     ,( ,  ,  ) ( ,   )e e e e e nfixed to the contact surface, which changes with the rotation of the contact surface. The elastic sliding displacement and the contact traction are described as follows: 221 1211 2 ne e e enf f fuu u    ef e ne e nu                                           (19) Hence, Eq. (15) is described in the simple form as follows: 1 11 12 2 2 2   0    0 1/    0      0  0      0 ,       0    1/    0  0    0     0      0    1/t tt te ee ee e nnn nn nu uf ff u u ff fu u                                                                       (20)     The sliding velocity vector u  is the objective vector, since it is not an absolute velocity vector but the mutual velocity vector between surface points on the master and the counter bodies. Therefore, it is not necessary to use a corotational velocity vector but we only have to use the time derivative for the sliding velocity vector. Further, note that one does not need to adopt a corotational rate but one has only to use the time derivative for the contact traction vector f  by the fact: The contact traction f  is calculated from the hyperelastic equation with the substitution of the elastic displacement eu  which is obtained by subtracting the plastic displacement vector pvu  from the displacement vector u . 303T. Ozaki and K. Hashiguchi 54 NORMAL-SLIDING YIELD AND SUBLOADING-SLIDING SURFACES  Assume the following sliding-yield surface with the isotropic hardening/softening, which describes the sliding-yield condition. ( ) =f f                                                      (21)   is the isotropic hardening/softening function denoting the variation of the size of the sliding-yield surface. The friction-yield stress function ( )f f  for the Coulomb friction law is given by ( ) = / ntf f ff                                                      (22) for which   specifies the coefficient of friction.  Then, in order to introduce the measure of approaching degree to the sliding-yield surface, renamed the normal sliding-yield surface, let the following subloading-sliding surface passing through the current contact stress and maintaining a similarity to the normal sliding-yield surface be introduced, which plays the general measure of approaching degree of the contact stress to the normal sliding-yield surface (see Figure 2). ( ) =f rf                                                         (23) where (0 1)rr    is the ratio of the size of the subloading surface to that of the normal sliding-yield surface and called the normal sliding-yield ratio, playing the role of the measure of the approaching degree of the contact stress to the normal sliding-yield surface.      The evolution rule of the isotropic hardening/softening function   in Eq. (21) is extended as follows: 11= )( ( )pvk s       u Negative Positive                                       (24)     The viscoplastic sliding rate is given as follows: = )( 0pv t  u n                                                 (25) where   and tn  are the magnitude and direction, respectively, of the plastic sliding velocity as shown below.  ( ) ( ) ) || ||=1,  = 0(( ) ( )|| ||/t t tt tf f    f f nf fn n n     (0.26) with  ( ) ( )( ) ( )( )=( ) )(tf ff f       f ff fn n n nIf ff f         (0.27) 0    for <1 for mm mnvs r rr rr rr r                                             (28) or 0    for <exp[ ( )] 11 for mm mvs r rnr rr rr r                                      (29) 304T. Ozaki and K. Hashiguchi 6where v ,  ( 1)n   and  ( 1)mr   are the material parameters, while mr  is the maximum value of r , and  || || under for ( ) <= for= 0under (= for = = )for0pppvvsesvess srr rUrr r r r          0u uu 00uu 0                     (30) with ( ) = cot 2( )s sU ur r .                                                (31) Here, (0 )mr r r  in Eq. (23) is renamed as the dynamic sliding-loading ratio and (0 1)s sr r r   is called the static normal sliding-yield ratio because it designates the normal sliding-yield ratio which evolves under the virtual quasi-static elastoplastic sliding process. The viscoplastic sliding velocity is induced by the overstress ( ) sf r f  from the subloading friction surface: ( ) = ,  i.e. =s sf rr rf                                             (32) so that a smooth elasticviscoplastic transition is described.     The sliding rate and its inverse relation are given by Eqs. (4), (15) and (25) as follows: ==tt    u E fEu Efnn                                                       (33) which is represented in the incremental form as follows: ==ttd d dtd d dt  u E ff E u Enn                                               (34) 5 CALCULATION PROCEDURE The calculation by the subloading-overstress friction model may be performed by the following procedure. 1. vpdu  is calculated by the input of dt  into Eq. (25), and then its accumulation leads to vpu . 2. d  and sdr  are calculated by inputs of vpdu  into Eqs. (24) and (30), and their accumulations lead to   and sr . 3. The elastic sliding displacement eu  is calculated by vpe u = u u . 4. f  is calculated by = ef Eu . 5. r  is calculated by ( )= // ntf fr  .  305T. Ozaki and K. Hashiguchi 7These calculation processes are repeated for the further sliding. 6 COMPARISON WITH TEST DATA The comparison of the simulation by the subloading-overstress friction model with the test data [5] for the dry friction is shown in Figure 3. The test curve for sliding between roughly polished steel surfaces under the quite low sliding velocity 0.0002 mm/stu  is simulated well enough by the present model, where the material parameters are selected as follows: 113= 0.58,  = 0.38, = 35mm ,   = 0.001/ s=1500mm , =150,  2.0, 3= =10000N / mmmktnvsu r n      under the condition=10MPa,  =0.0002mm/sn tf u Comparison with test data [5] for dry friction.The comparison of the simulation by the present model with the test data [6] for the fluid friction is shown in Figure 4. The test plate is the galvannealed steel sheet which is sandwiched by the steel SKD-11 plates. The normal contact stress is 5.56 MPa. The friction surfaces were coated with the anti-rust oil prior to the tests. The drawing velocity of the test plate is set at the five levels 1,  10, 50, 100, 200 mm/min.  The simulation of the test result using the exponential function in Eq. (29) is shown by the solid lines, using the following values for the material constants. 306T. Ozaki and K. Hashiguchi 811= 0.1,  = 0.09, = 0.2mm ,   = 0.0009 / s= 80mm , =10,  1.4, 8= =1000MPa / mmmktnvsu r n       Comparison with test data for fluid friction. REFERENCES [1] Hashiguchi, K. (2018): Multiplicative hyperelatic-based plasticity for finite elastoplastic deformation/sliding: A comprehensive review, Arch. Compt. Meth. Eng., https://doi.org/10.1007/s11831-018-9256-5. [2] Hashiguchi, K. (1980): Constitutive equations of elastoplastic materials with elastic-plastic transition, J. Appl. Mech. (ASME), 47, 266-272. [3] Hashiguchi, K. (1989): Subloading surface model in unconventional plasticity, Int. J. Solids Structut., 25, 917-945. [4] Hashiguchi, K. (2017): Foundations of Elastoplasticity: Subloading Surface Model, Springer.[5] Ferrero, J. F. and Barrau, J. J. (1997): Study of dry friction under small displacements and near-zero sliding velocity, Wear, 209, 322-327. [6] Hashiguchi, K., Ueno, M., Kuwayama, T., Suzuki, N., Yonemura, S. and Yoshikawa, N. (2016): Constitutive equation of friction based on the subloading-surface concept, Proc. Royal Soc., London, A472:20160212, http://dx.doi.org/10.1098/rspa.2016.0212.307

Unified description of dry and fluid frictions  by subloading-overstress friction model

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Unified description of dry and fluid frictions by subloading-overstress friction model

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