Subloading-friction model with saturation of tangential contact stress

The incorporation of the saturation of the tangential contact stress with the increase of the normal contact stress is required for the analysis of the friction phenomenon of solids and structures subjected to a high normal contact stress, which cannot be described by the Coulomb friction condition, in which the tangential contact stress increases linearly with the increase of the normal contact stress. In this article, the subloading-friction model, which is capable of describing the smooth elastic—plastic transition, the static—kinetic transition, and the recovery of the static friction during the cease of sliding, is extended to describe this property. Further, some numerical examples are shown, and the validity of the present model will be verified by the simulation of the test data on the linear sliding of metals.


Introduction
All bodies in the natural world are exposed to the friction phenomena, contacting with the other bodies. Therefore, it is indispensable to analyze the friction phenomena rigorously in addition to the deformation behaviors of the contact bodies themselves.
Here, it should be noticed that the friction is the typical irreversible, i.e., plastic phenomenon, so that it should be formulated within the framework of the elastoplasticity theory. Then, the various friction models in the elastoplasticity have been formulated as the rigid-plasticity [1,2] and the perfect-plasticity [3][4][5][6][7][8][9][10]. However, these past formulations fall within the conventional plasticity assuming that the interior of the sliding-yield surface is the purely-elastic domain, so that the accumulation of plastic sliding during the cyclic loading of the tangential contact stress as seen in the loosening of the bolt and nut cannot be predicted. In addition, the simple friction model [11] falling within the framework of the creep model without the sliding yield surface was formulated, but the creep model is inapplicable to the general sliding velocity, since the creep sliding is induced in any low stress level as delineated by Hashiguchi [12,13], and thus it was applied only to the simulation of the sliding behavior at high sliding velocity [14]. Eventually, it is irrelevant to the usual sliding behavior including the quasi-static sliding.
Further, various friction models regardless to the elastoplasticity have been proposed hitherto so that they are limited to the one-dimensional sliding behavior represented by the rate-and-state model [15][16][17][18][19][20][21] and the fundamentally-irrational models [22][23][24][25][26][27] involving the time itself in order to describe the recovery of the friction coefficient caused by the cease of sliding have been proposed, which result in the loss of the objectivity as the constitutive relation [28,29]. The loss of the objectivity is evident from the fact that the evaluation of elapsed time is accompanied with the n f , t ambiguity in the judgment about when the sliding commences and ceases, especially in the state that the sliding velocity varies in a low-velocity region. Besides, the Coulomb friction equation with the nonhardening friction yield surface enclosing the purely-elastic domain is adopted widely in the commercial finite element method (FEM) software, e.g., Abaqus, Marc, etc., and explained extensively in the well-known monographs (e.g., Refs. [30,31]). It is merely capable of describing the constant friction coefficient independent of the sliding displacement.
The subloading-friction model (Hashiguchi et al. [32,33]) is capable of describing the smooth elasticplastic transition, the reduction of friction coefficient from the static to the kinetic friction, the recovery to the static friction during the cease of sliding, etc. It has been applied to the metal friction [32,33], the metalto-soil friction [34] and the stick-slip behavior [35]. Further, it has been extended to describe the orthotropic anisotropy [36,37] and the viscoplastic sliding behavior for the lubricated (wet) friction [38]. Now, it should be recognized that the tangential contact stress increases but saturates with the increase of normal contact stress. This property is of crucial importance for the predictions of the friction behaviors in the fastening of bolts and nuts, the wedge driving or penetration, the compression of solid body, the strip rolling processing of metal plates, the earthquake occurrence caused by the sliding phenomenon between continental plates and oceanic plates, etc., in which high normal contact stresses are applied. However, it cannot be described by the Coulomb friction condition [12,13], in which the tangential contact stress increases linearly with the normal contact stress. Then, the formulations of the subloading-friction model taking account of this property have been proposed by Hashiguchi et al. [32,33] and Ozaki et al. [39]. However, the former www.Springer.com/journal/40544 | Friction formulations [32,33] adopting the teardrop-type or the parabola-shaped sliding yield surface involve the physical irrationality that the tangential contact stress decreases with the increase of the normal contact stress in a high normal contact stress region, and the latter formulation [39] is irrelevant to solid materials as metals but limited to the rubbers.
In this article, the rigorous subloading-friction model incorporating the sliding-yield stress function, in which the tangential contact stress increases but saturates with the increase of the normal contact stress, will be formulated, and the numerical examples for the monotonic and the reciprocal sliding under several levels of normal contact stress with various stationary time of sliding will be shown, adopting the present model. Further, the validity of the present model will be verified by the simulation of the test data [14] on the linear sliding between metals.

Sliding displacement and contact stress
The sliding displacement vector u , which is defined as the relative sliding displacement of the counter (slave) body to the main (master) body, is orthogonally decomposed into the normal sliding displacement vector n u and the tangential sliding displacement vector t u to the contact surface as Eq. (1) [13,40]: n is the unit outward-normal vector of the surface of main body, and I denotes the second-order identity tensor. The minus sign is added for n u to be positive when the counter body approaches the main body.
The sliding displacement vector u can be exactly decomposed into the elastic sliding displacement e u and the plastic (irreversible) sliding displacement p u in the additive form even for the finite sliding displacement, i.e., The contact stress vector f acting on the main body is additively decomposed into the normal contact stress vector n f and the tangential contact stress vector t f as Eq. (8): The minus sign is added for n f to be positive when the compressive stress acts to the main body by the counter body.

Hyperelastic sliding behavior
The hyperelastic-based plastic constitutive relation is adopted in order to formulate the rigorous constitutive equation for sliding phenomenon, while the elastic sliding displacement is quite small because it is induced by the elastic deformation of the surface asperities which are infinitesimally small compared with the contact surface. Then, let the contact stress Then, the elastic sliding work w e done during the elastic sliding is uniquely determined by the elastic  (12) Let the following simplest elastic sliding energy function e ( )  u in the quadratic form be adopted.
where the second-order tensor E designates the elastic contact tangent stiffness modulus fulfilling the symmetry  (14) Assuming the isotropy on the contact surface, i.e. the independence of frictional property to a sliding direction on the contact surface and introducing the normalized rectangular coordinate system  1 2 3 ( , , ) e e e 1 2 ( , , ) e e n fixed to the contact surface, the elastic contact tangent stiffness modulus tensor E is given as Eq. (15): I n n n n e e e e n n E I n n n n e e e e n n (15) where n  and t  are the normal and tangential elastic contact moduli, respectively. Equation (14) with Eq. (15) leads to

Elastoplastic sliding velocity
The plastic sliding velocity is formulated based on the subloading surface concept and the tangentialassociated flow rule in this section.

Sliding normal-yield and sliding subloading surfaces
Firstly, let the sliding normal-yield surface and the sliding subloading surface, which passes through the current contact stress and is similar to the sliding normal-yield surface, be given by (18) where  is the sliding hardening/softening function, and r is the sliding normal-yield ratio, i.e., the ratio of the size of the sliding-subloading surface to that of the sliding normal-yield surface as shown in Fig. 1.
Here, Eq. (19) holds for the isotropic yield stress function. ( , ) ( , ) (19) Then, the consistency condition for the sliding subloading surface in Eq. (18) is given by (20) The evolution rule of the sliding hardening/softening function  is given as Eq. (21) (Fig. 2): where m  is the material constant designating the friction sliding resistance per unit apparent contact area, and r a is the ratio of the real contact area R a to the apparent contact area A , a called the real contact area ratio. i.e., Then, the following relation would hold from the time-derivative of Eq. (22) with Eq. (21).
On the other hand, the relation    a n 1 exp( ) r b f (b is the material constant) proposed for rubbers by Ozaki et al. [39] is irrelevant to the plastic sliding history, so that it would not hold in metals.
The rate of the sliding normal-yield ratio r is given by with 0 ( 1) where u  is the material constant (cf. Hashiguchi [12,13]). The contact stress is automatically attracted to the sliding normal-yield surface in the plastic sliding process, and it is pulled back to that surface even when it goes over the surface in numerical calculation because of 0 r   for 1 r  from Eq. (25) with Eq. (26), as shown in Fig. 3.

Plastic sliding velocity and elastoplastic sliding velocity
The plastic sliding velocity, the sliding velocity, and the contact stress rate are formulated in the following.
The substitution of Eqs. (21) and (25) into Eq. (20) leads to (27) Now, assume the tangential-associated flow rule [12]: (30) where   and t n are the magnitude and the direction, respectively, of the plastic sliding velocity.

Sliding-yield surface with saturation of tangential contact stress
Now, we adopt the following sliding-yield stress function  (38) leading to the sliding normal-yield surface ( leading to n n t n n n for 0 0 for where n c is the material constant with the inverse dimension of stress. Equation (39) represents the sliding normal-yield surface with the fusiform shape, which expands from the origin to the positive direction of the normal contact stress n f in the three-dimensional stress space t1 t2 n ( , , ). value  for a larger value of n c , as shown in Fig. 4(a). The sliding normal-yield and the sliding subloading surfaces satisfying Eqs. (40) and (41) are shown in Fig. 4(b).
noting Eq. (19). Further, it follows from Eqs. (19) and (29) that The substitutions of Eqs. (44)-(49) into Eqs. (34)-(36) lead to u  is determined to describe the increasing rate of the contact stress for the plastic sliding rate. n c is determined to designate the dependence of the ratio of the tangential vs. normal contact stress on the normal contact stress.

Numerical experiments
The principal mechanical responses of the present subloading-friction model will be examined by performing the numerical experiments in this section. The material parameters are chosen with the two levels of the static sliding hardening functions s  as: Firstly, the tangential contact stress paths at five levels of constant normal contact stress are shown in Fig. 5, f2 2 n n t n n n 2 n t 1 1 2 2 n t f1 1 f2 2 p t n t f1 1 f2 2 n n t n n n 2 n t 1 1 2 2 n t f1 1 f2 2 www.Springer.com/journal/40544 | Friction in which the sliding normal-yield surface for the static and the kinetic isotropic hardening functions are depicted by red and black curves, respectively. The maximum contact tangential stress is larger for the larger static sliding hardening function. Next, the variations of tangential contact stress at the five levels of normal contact stress in the monotonic sliding followed by the reverse sliding for the different static sliding hardening functions are shown in Fig. 6.
The variations of tangential contact stress with the five levels of stationary sliding time just after the unloading for the different static sliding hardening functions in the monotonic and the reciprocal sliding are shown in Figs. 7 and 8, respectively. More remarkable recoveries of the tangential contact stress by the longer stationary sliding time are shown in these figures.
The variations of the tangential contact stress in the pulsating sliding with three levels of stationary sliding time just after the unloading to the zero tangential contact stress is shown in Fig. 9. The recovery of the tangential contact stress with the increase of sliding time is shown, while it decreases gradually with the increase of number of cycles.

Comparison with experiments
The simulation of the test data for the linear sliding behavior for the boric acid lubricated A16111-T4 and tool steel interface in Ref. [14] is shown in Fig. 10 to verify the applicability of the present model to the description of real friction phenomenon. The linear sliding behavior at the constant normal contact stress is the most basic behavior among various sliding behaviors, but strangely, the available test data is quite little as the authors could find only the test data from Ref. [14]. Fortunately, this test data is suitable for the verification of the present model, since the rather high normal contact stress up to 600 MPa is applied in this data. The relations of the contact tangential stress vs. the sliding displacement for the five levels of the constant normal contact stresses are represented, where the values of the material parameters are chosen as 3 3 n t   www.Springer.com/journal/40544 | Friction while the tangential sliding velocity t  u is 1 mm•s −1 in the test data.
The variations of the tangential contact stress are closely simulated by the present friction model, as shown in Fig. 10.
The comparison for the relations of the tangential contact stress vs. the normal contact stress for four levels of the sliding displacement is shown in Fig. 11, which is depicted from the test and the calculated values. The fact that the ratio of the tangential contact stress to the normal contact stress is not constant but decreases gradually with the increase of the normal contact stress is simulated closely. The incorporation of the present model is required to describe the friction behavior in the test data accurately. A test datum specifying an explicit sliding displacement less than 1 mm is not shown in the figure of Ref. [14]. Then, only the calculated result for the displacement 0.5 mm is shown by the dashed curve in Fig. 11.
The quite close simulation of the test result [14] is attained by the present model. Then, the applicability of the present friction model for the prediction of real friction behavior between metals would be verified. Besides, the simulation of the test data was also performed by Gearing et al. [14]. However, their friction model [11] used for the simulation is Fig. 10 Comparison of the simulation results and test data of boric acid lubricated A16111-T4/tool steel interface in Ref. [14] for tangential contact stress vs. sliding displacement relations at six levels of normal contact stress, where the calculated results are shown by solid lines.

Fig. 11
Comparison of the simulation results and test data of boric acid lubricated A16111-T4/tool steel interface in Ref. [14] for tangential contact stress vs. normal contact stress relations in four levels of sliding displacement, where the calculated results are shown by solid lines. The dashed curve shows the calculated result for the sliding displacement = 0.5 mm.
physically impertinent belonging to the creep model, which is approximately applicable to the sliding behavior at high rate but inapplicable to the sliding behavior at the moderate rate including the quasi-static sliding behavior as delineated in Hashiguchi [12,13], while the sliding velocity in the test data in Ref. [14] is rather high as t  u = 1 mm•s −1 . On the other hand, the subloading-friction model is concerned with the sliding behavior in the general sliding velocity.
The tangential contact stress lowers to a constant value after it exhibits the peak value, as shown in Figs. 6-8, and 10. Therefore, it would be independent of the sliding displacement for the sliding displacement larger than 5 mm, as shown in Fig. 11. The inconsistent tendency in the test results for the sliding displacement = 5, 10, and 20 mm in this figure would show the difficulty of the precise measurement of the tangential contact stress for a large sliding displacement in tests under a constant normal contact stress.

Concluding remarks
The subloading-friction model (Hashiguchi et al. | https://mc03.manuscriptcentral.com/friction [32,33], etc.) is extended in this article. The extended model is summarized to be capable of describing the following principal properties.
1) The smooth transition from the elastic to the plastic transition leads to the continuous variation of the elastoplastic stiffness modulus, which is of the importance for the numerical calculation since the loading criterion, i.e., the judgment whether the plastic sliding induced is not required.
2) The contact stress is automatically pulled back to the normal sliding-yield surface when it goes out from that surface, leading to the highly efficient numerical calculation.
3) The friction decreases from the static to the kinetic friction.
4) The friction resistance recovers during the stationary state of sliding.
5) The tangential contact stress saturates with the increase of the normal contact, so that the tangential contact stress does not exceed the shear strength of the contacting solids.
Property 5 was not furnished in the past subloadingfriction model. These principal mechanical characteristics of the proposed model is represented by the numerical experiments for the linear monotonic and the reciprocal sliding with the cease of sliding in the unloaded state. Further, the validity of the present model is verified by the simulation of the test data [14] for metals. The present model will contribute to the development of the boundary value problems for the deformation/sliding of solids and structures subjected to a high normal contact stress.
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Koichi HASHIGUCHI. He received his Ph.D. degree from Tokyo Institute of Technology, Japan, in 1976. After that he served as assistant professor, associate professor, and professor at Kyushu University, Japan. He is currently an emeritus professor at Kyushu University, member of the Engineering Academy of Japan, and the technical adviser of MSC software Ltd., Japan. His research area covers the elasto-plastic deformation and the friction phenomena of solids, proposing the subloading surface model.