Constitutive equation of friction based on the subloading-surface concept

The subloading-friction model is capable of describing static friction, the smooth transition from static to kinetic friction and the recovery to static friction after sliding stops or sliding velocity decreases. This causes a negative rate sensitivity (i.e. a decrease in friction resistance with increasing sliding velocity). A generalized subloading-friction model is formulated in this article by incorporating the concept of overstress for viscoplastic sliding velocity into the subloading-friction model to describe not only negative rate sensitivity but also positive rate sensitivity (i.e. an increase in friction resistance with increasing sliding velocity) at a general sliding velocity ranging from quasi-static to impact sliding. The validity of the model is verified by numerical experiments and comparisons with test data obtained from friction tests using a lubricated steel specimen.


Introduction
The rate-and-state friction model (e.g. [1][2][3][4]), the basic idea of which is the dependence of the shear stress or friction coefficient on the rate of sliding and some state (hyperelastic-based plasticity) based on the multiplicative decomposition of the deformation gradient is formulated rigorously, as done by Hashiguchi & Yamakawa [27] for monotonic loading behaviour and by Hashiguchi [28] for general loading behaviour including cyclic loading behaviour. It is desirable to formulate a constitutive equation with a high generality and the universality, while any unconventional model, i.e. a cyclic plasticity model other than the subloading-surface model, has not been extended to multiplicative finite strain theory. (vi) The crystal plasticity analysis, in which slip analyses in numerous slip systems are required, is realized rationally with high efficiency because a yield judgement is not required and the stress is pulled back automatically to the yield surface [28,29]. Consequently, the subloading-surface model is valid for the description of irreversible mechanical behaviour in solids for rate-independent and rate-dependent behaviours ranging from the micro-to the macro-level for pressure-independent and pressure-dependent materials for finite deformation, combining both physical relevance and numerical efficiency.
We now concentrate our discussion on the friction phenomena in solids. With the exception of an isolated body floating in a vacuum, all bodies in the natural world are in contact with other bodies and thus subject to friction. The theoretical basis for the description of friction phenomena was established by the friction model based on the concept of the subloading surface, called the subloading-friction model [30,31]. The simplest formulation of the subloading-friction model also provides the basis for crystal plasticity analysis [28,29]. The subloading-friction model assumes that the plastic sliding velocity develops gradually as the contact stress (traction) approaches the friction-yield surface, based on the concept of the subloading surface [20,21,29]. It describes the friction resistance such that it first reaches a peak, i.e. static friction, then gradually decreases to a minimum, i.e. kinetic friction, and it recovers as sliding stops or sliding velocity decreases. The model is based on the fact that the friction coefficient decreases by plastic sliding, which breaks the adhesion between surface asperities on the contact surface, but it recovers with time, which causes the reconstruction of adhesion. Thus, negative rate sensitivity (i.e. a decrease in friction resistance with increasing sliding velocity) is described; such sensitivity is observed in dry friction without lubrication. By contrast, the positive rate sensitivity (i.e. an increase in the friction resistance with increasing sliding velocity) is observed in the sliding between lubricated surfaces or between soft solids, e.g. indium, Teflon and various polymers, which is often called the fluid friction or the wet friction.
Dry friction and fluid friction exhibit high and low friction resistances and large and small differences between static friction and kinetic friction, respectively. Furthermore, the stick-slip phenomenon is often induced by dry friction [32], in which friction resistance fluctuates upwards and downwards intensely and intermittently. It should be avoided in machine elements such as gears and bearings in order to induce smooth movement. By contrast, fluid friction with a positive rate sensitivity does not cause the stick-slip phenomenon, but its theoretical prediction is of importance for mechanical designs of lubricated machinery elements such as gears and bearings in which low friction resistance is desirable and of wheel tyres travelling on wet roads in which high friction resistance is required for high braking performance, and so forth. This paper begins by giving a detailed physical interpretation of the subloading-friction model, which is then extended to the generalized subloading-friction model. The generalized model is capable of describing not only the negative rate sensitivity but also the positive rate sensitivity for an arbitrary sliding velocity ranging from quasi-static to impact sliding by introducing the concept of overstress [33,34] for the elasto-viscoplastic deformation behaviour. Numerical experiments for several settings of material parameters are presented to clarify the effects of the material parameters on friction resistance and the physical meanings of these parameters contained in the subloading-overstress friction model. Furthermore, friction tests using a lubricated steel specimen are performed to examine the validity of the model. These test results are reproduced in simulations using the subloading-overstress friction model. Consequently, it is verified that the subloading-overstress friction model is capable of describing friction phenomena with positive rate sensitivity in addition to negative rate sensitivity for an arbitrary sliding velocity ranging from quasi-static to impact sliding.

Physical interpretation and basic formulation of the subloading-friction model
The subloading-friction model [30,31] for negative rate sensitivity observed in dry friction is reviewed below. We add concise physical interpretations along with the basic mathematical formulation in preparation for the generalization of the model to describe not only negative but also positive rate sensitivities for an arbitrary sliding velocity.

(a) Definitions of basic mechanical variables
The sliding velocity vectorv, which is defined as the relative velocity of the counter (slave) body to the main (master) body, is orthogonally decomposed into the normal sliding velocity vectorv n and the tangential sliding velocity vectorv t to the contact surfacē v =v n +v t = −v n n +v t t v , I is the second-order identity tensor possessing the components of Kronecker's delta δ ij (δ ij = 1 for i = j and δ ij = 0 for i = j), n is the unit outward-normal vector of the main body, and The minus sign is added to the definition ofv n so that friction increases as the counter body approaches the main body. The symbolic notations u ⊗ vw = u(v • w) and (Tv) i = T ir v r with Einstein's summation convention being used for arbitrary vectors u, v, w and arbitrary secondorder tensor T, and denotes the magnitude (i.e. norm). An observed velocity of a material particle depends on the velocities of the observer and is thus not an objective quantity. On the other hand, the observed velocities of material particles of the main and the counter bodies on a contact surface are affected identically by the velocity of the observer, noting that these material particles possess the same position. Consequently, their relative velocity, i.e. the sliding velocitȳ v, is an objective quantity, which is independent of the observer.
Here, it is further assumed thatv is additively decomposed into an elastic sliding velocityv e and a plastic sliding velocityv p , i.e.v =v e +v p . (2.4) The elastic sliding velocity is infinitesimal because it is induced by the elastic deformations of asperities on the contact surface; it is uniquely related to the rate of contact stress. On the other hand, the plastic sliding velocity is induced by the mutual slips between the surfaces of the main and the counter bodies and thus it is finite usually. Then, the direction of the plastic sliding velocity is given by the plastic potential function of the contact stress and thus depends on the contact stress but is independent of the rate of contact stress, as will be formulated in §2c(iii). Hence, the rate of contact stress is not determined uniquely by the sliding velocity, making it difficult to carry out the analysis if the elastic sliding velocity is not given. Therefore, we incorporate the elastic sliding velocity analogously to the elastic strain rate in ordinary elastoplastic constitutive equations. Furthermore,v n andv t split additively into elastic and plastic parts as follows: The contact stress (traction) vector f acting on the main body orthogonally splits into the normal traction vector f n and the tangential traction vector f t , and The minus sign is added to the definition of f n such that friction is generated when the normal contact stress applied to the main body is compressive.

(b) Elastic sliding velocity
Assuming that a solid does not split under moderate deformations, a hyperelastic constitutive relation possessing the elastic potential energy function is adopted for the elastic part of deformation in the exact finite strain theory [27]. By contrast, surface asperities on the main body adhere to different surface asperities on the counter body one after another during a sliding process and thus the hyperelastic relation would not hold in the friction phenomenon. Then, recalling the hypoelastic relation in terms of the Cauchy stress and the elastic strain rate (the symmetric part of the elastic velocity gradient) for the elastic deformation of solids [35], let the elastic sliding velocity be formulated as follows: where ( • ) stands for the corotational rate [29]. The corotational time-derivative • f of contact stress in equation (2.11) is orthogonally decomposed into normal and tangential parts to the contact surface as 16) The second-order tensor ω c is the spin of the contact surface. The objectivity of the corotational contact stress rate  f is formulated such that the influence of rigid-body rotation, ω c f, is excluded from the material-time derivative • f. The second-order tensor C e is the contact elastic modulus tensor given by C e = α n n ⊗ n + α t (I − n ⊗ n), C e−1 = 1 α n n ⊗ n + 1 α t (I − n ⊗ n), (2.18) where α n and α t are the normal and tangential contact elastic constants, respectively, in the contact surface. These values are set large because the elastic sliding is caused by elastic deformations of the surface asperities. Actually, we may choose α n , α t = 10 2 ∼ 10 5 GPa mm −1 for common metals as the penalty parameters.

(c) Formulation of plastic sliding velocity
In this section, the plastic sliding velocity is formulated on the basis of the concept of the subloading surface [20,29].
(i) Normal friction-yield and subloading-friction surfaces Let the friction-yield surface be given by 20) where μ is the hardening/softening function describing the expansion/contraction of the frictionyield surface. The friction-yield stress function f (f) for the Coulomb friction law is given by for which μ specifies the coefficient of friction. Equation (2.21) will be adopted for actual calculations in later sections. The abrupt transition from the elastic to the plastic sliding state is described if the interior of the friction-yield surface is assumed to be a purely elastic domain. However, the plastic sliding velocity is induced even by the rate of contact stress inside the friction-yield surface and it develops gradually as contact stress approaches the friction-yield surface, thereby exhibiting the smooth elastic-plastic transition. Then, to describe the plastic sliding velocity induced by the rate of contact stress inside the friction-yield surface, we incorporate the following postulate based on experimental facts [22,29].
Fundamental postulate of unconventional elastoplasticity (subloading-surface concept). The contact stress approaches the friction-yield surface when the plastic sliding velocity is induced but it recedes from the friction-yield surface when only the elastic sliding velocity is induced.
In this context, it is first required to incorporate the three-dimensional measure which describes the approaching degree of the contact stress to the friction-yield surface, renamed the normal friction-yield surface, in order to formulate the plastic sliding velocity based on the subloadingsurface concept. Then, let the following subloading-friction surface, which always passes through the current contact stress and maintains a similar shape and orientation to the normal frictionyield surface, be introduced: where r (0 ≤ r ≤ 1) is the ratio of the size of the subloading-friction surface to that of the normal friction-yield surface and is called the normal friction-yield ratio, playing the role of the measure of the approaching degree of contact stress to the normal friction-yield surface. For the Coulombtype friction-yield surface, the normal friction-yield ratio is given by the ratio of the size of the cross-section of the subloading-friction surface to that of the normal friction-yield surface on the constant normal contact stress plane as shown in figure 1. On the basis of the above-mentioned fundamental postulate of elastoplastic sliding, the rate of the normal friction-yield ratio r must satisfy the following conditions:    An explicit example of the functionŪ(r) is whereũ is a material constant. Equations (2.25) with equation (2.27) can be analytically integrated in the case of a monotonic sliding process as whereū p = v p dt is the accumulated plastic sliding displacement, and r 0 andū p 0 are the initial values of r andū p , respectively. The analytical integration would be beneficial in implicit numerical calculations [22], while further work is required for the formulation of the implicit calculation method, i.e. a return-mapping projection based on the subloading-friction model. Here, it should be noted that numerical calculations can be performed with high efficiency by the forward-Euler method based on the subloading-friction model which possesses an automatic-controlling function to attract the contact stress to the normal friction-yield surface.
There exist the following functions other than the one in equation (2.27) which satisfy the conditions in equation (2.26), but unfortunately analytical integration does not exist for them: The three functional forms forŪ(r) in equations (2.27)  and 100 cot π 2 r < −300 ln r < 100 The cotangent function in equation (2.27) will be adopted in subsequent analyses for the following reasons.
(1) It possesses the largest negative value in the range r > 1 among these three kinds of functionŪ(r). Then, it provides the most intense controlling function to pull back the contact stress to the normal friction-yield surface when the stress jumps out from that surface in numerical calculations. Here, note that the evolution rule of the normal frictionyield ratio r is given by 25), so that r becomes negative, i.e.
• r < 0, and thus the contact stress is pulled back to the normal friction-yield surface if it goes over that surface, i.e. r > 1, leading toŪ(r) < 0. This fact will be explained in detail in §2g with the illustration in figure 4.
(2) It can be analytically integrated as shown in equation (2.28) and it thereby provides an advantage in the implicit numerical calculation. )

(ii) Evolution of the friction coefficient
The following characteristics for the variation of the friction coefficient are deduced from the hypothesis of adhesion in dry friction [36].
(1) The friction coefficient decreases with plastic sliding because the adhesion of surface asperities on the contact surface is broken by plastic sliding. (2) The friction coefficient recovers with time because the adhesion of surface asperities is reconstructed as time elapses.
Then, the evolution rule for the friction coefficient is assumed as follows: where μ s and μ k (μ s ≥ μ ≥ μ k ) are material constants designating the maximum and minimum values of μ for static friction and kinetic friction, respectively. κ(> 0) is a material constant specifying the rate of decrease of μ in the plastic sliding process, and ξ (> 0) is a material constant specifying the rate of recovery of μ as time elapses.

(iii) Plastic sliding velocity
The time differentiation of the subloading-friction surface in equation (2.22) reads Now, we assume that the plastic sliding velocity is induced in the direction of the projection of the outward-normal vector of the subloading-friction surface to the contact surface, which may be called the tangent-associated flow rule, i.e. For the Coulomb friction-yield function described in equation (2.21), substituting   (2.43) from which the magnitude of the plastic sliding velocity in terms of the sliding velocity, denoted • Λ, is derived as (2.44) The corotational rate of contact stress is obtained from equations (2.4), (2.14) 2 , (2.35) and (2.44) as follows: The loading criterion for the plastic sliding velocity is given bȳ v p = 0 : noting m p + (∂f (f)/∂f) • C e t n > 0 in the denominator of the positive plastic multiplier in terms of the sliding velocity in equation (2.44). Here, note that the infinite plastic relaxation is generated so that the contact stress infinitely decreases if the denominator decreases to zero as known from equation (2.45).

(f) Calculation of the normal friction-yield ratio
The normal friction-yield ratio r can be calculated using one of the following two methods.
(1) We calculate r using the following equation derived from the subloading-friction surface in equation (2.22) for both of the elastoplastic sliding and elastic sliding processes after the contact stress vector f and the hardening variable μ are updated: Method (2) would be superior to method (1) because the normal friction-yield ratio is calculated directly from the plastic sliding velocity.

(g) Basic features of the subloading-friction model
The fundamental aspects of friction phenomena are described appropriately by the subloadingfriction model as follows.
(1) The static friction as a peak and the decrease in the kinetic friction are described. (2) The negative rate sensitivity is described because the friction coefficient decreases by plastic sliding and the recovery of the friction coefficient requires an elapse of time. (3) The smooth transition from the elastic to the plastic sliding state is described, fulfilling the smoothness condition [37,38] because the plastic sliding velocity is not induced suddenly but induced gradually as the contact stress approaches the normal friction-yield surface so that a smooth contact stress versus sliding displacement curve is depicted, while the elastic sliding is usually quite small. Here, note that the smoothness of the transition is irrelevant to the elastic rigidity (magnitude of elastic sliding), as known from the existence of the smooth transition from the rigid to plastic transition.

Generalized subloading-friction model: subloading-overstress friction model
In the subloading-friction model described in the preceding sections, the sliding velocity is postulated to consist of elastic and plastic sliding velocities. This leads to a negative rate sensitivity (i.e. a decrease in friction resistance with increasing sliding velocity) because the adhesion of surface asperities is broken more quickly at higher sliding velocities and the recovery of adhesion requires time. In what follows, we consider the generalization of the subloadingfriction model to describe not only negative but also positive rate sensitivities by extending the notion of the overstress, which has been adopted in the description of viscoplastic deformation behaviour.
(a) Formulation of the generalized subloading-friction model the overstress from the yield surface in the overstress model [33,34]. In other words, although the creep model does not possess a loading criterion for the creep strain rate, the overstress model possesses the loading criterion for the viscoplastic strain rate. Here, it should be noted that a loading criterion for the viscoplastic strain rate as well as for the plastic strain rate in the ordinary elastoplastic constitutive equation is required for the realistic description of the rate-dependent elastoplastic deformation such that the viscoplastic strain rate is not induced when the stress is reduced from the yield surface. The subloading-friction model can then be generalized so as to describe both negative and positive rate sensitivities appropriately by incorporating the concept of overstress. First, let the viscoplastic sliding velocity be given as Hence, by substituting equations (2.14) and (3. where η v and n are material constants. denotes the Macaulay bracket, i.e. s < 0 : s = 0 and s ≥ 0 : s = s (s: an arbitrary scalar). Note that the loading criterion for the viscoplastic sliding velocity is furnished by introducing the Macaulay bracket. The surface that passes through the current contact stress and that is similar to the normal friction-yield surface (similar on the constant normal contact stress plane for the Coulomb friction-yield surface) is called the dynamic loading friction surface. r is newly defined as the ratio of the size of the dynamic loading friction surface to that of the normal friction-yield surface and thus is called the dynamic loading frictionyield ratio. Here, note that r ≤ 1 holds in the quasi-static sliding process but r > 1 holds in the dynamic sliding process. Variations of internal variables are induced generally by irreversible (inelastic) sliding and hence the plastic sliding velocityv p in the evolution rules of the internal variables in the elastoplastic sliding should be replaced by the viscoplastic sliding velocityv vp in the evolution rules of the internal variables in the elasto-viscoplastic sliding. Then, from equation (2.32), the evolution rule for the coefficient of friction μ is generalized to (3.4) where κ and ξ are the material constants defined for equation (2.32). Note here that equation (3.3) is rewritten in incremental forms, where dū =vdt is the increment of sliding displacement. In quasi-static sliding, equation (3.3) is reduced to (3.6) which leads to fulfilling the normal friction-yield condition without the overstress. In impact sliding, by contrast, which describe unrealistically the elastic sliding behaviour with an infinite friction strength, so that these equations are inapplicable to sliding phenomena at high sliding velocities. We then modify equation (3.3) as follows: (3.9) in which the viscoplastic sliding velocity becomes infinite as the dynamic loading friction-yield ratio r approaches the value of the material parameterr(>1), i.e. r →r. Then, we callr the limit dynamic loading friction-yield ratio.
The viscoplastic sliding velocity is induced suddenly at the moment that the contact stress reaches the normal friction-yield surface, i.e. when the dynamic loading friction-yield ratio becomes unity (r = 1), so that a smooth elastic-viscoplastic transition cannot be described by equation (3.9). Then, we further modify these equations as follows: 3.10) by incorporating the variable r s (0 ≤ r s ≤ 1), called the subloading-friction-yield ratio, which is calculated using equations (2.25) and (2.48) by replacing the plastic sliding velocityv p with the viscoplastic sliding velocityv vp , i.e. and withŪ (r s ) =ũ cot π 2 r s . (3.13) Thus, the viscoplastic sliding velocity is induced by the overstress f (f) − r s μ from the subloadingfriction surface r − r s = 0, i.e. f (f) = r s μ, (3.14) so that a smooth elastic-viscoplastic transition is described.
Simulations reproducing test data are difficult when using equation (3.10) with a viscoplastic term in the power form but, as will be described in §5, high accuracy simulations can be performed by the use of the exponential function as which will be used in subsequent sections. The generalized subloading-friction model proposed above is referred to as the subloading-overstress friction model. The response of the subloading-overstress friction model for fluid friction is schematically shown in figure 5. The elastic sliding is depicted with exaggeration for a concise explanation, and the normalized overstress divided by the normal contact stress f n is written simply as the term 'overstress' in this figure. The dynamic loading friction-yield ratio r coincides with the subloading-friction-yield ratio r s , i.e. r = r s in quasi-static sliding and increases above r s with increasing sliding velocity. However, r does not rise abover, i.e. r ≤r, while the equality r =r is realized only in impact sliding.   (1) In the quasi-static sliding process (dμ/dt ∼ = 0, dū vp /dt ∼ = 0, dū /dt ∼ = 0, df /dt ∼ = 0) in which the terms other than the second terms on the right-hand sides are negligible in equations (3.16) and (3.17), we have μ = μ s from equation (3.16) and r = r s from equation (3.17), resulting in f t /f n = r s μ s due to equation (2.22) with equation (2.21). Therefore, the contact stress moves, satisfying the subloading-surface in the quasi-static sliding process, so that the original subloading-friction model is reproduced in that process.
In other words, the subloading-overstress model is generalized so as to contain the subloading-friction model formulated for the dry friction in §2. (2) In the fast sliding process (dμ/dt → ∞, dū vp /dt → ∞, dū /dt → ∞, df /dt → ∞) for which the creep part of the second term on the right-hand side is negligible in equation (3.16), the friction coefficient μ decreases with the viscoplastic sliding displacementū vp ≡ v vp dt after reaching a peak.
Friction resistance varies with the destruction of the adhesion of surface asperities and the shearing of the viscous fluid lying between contact surfaces. The former is dominant in dry friction whereas the latter is dominant in fluid friction because the surfaces are in direct contact with each other in dry friction but are in indirect contact via the viscous medium in fluid friction. Then, adhesion of surface asperities is stronger in dry friction than in fluid friction. We thus infer the following differences between the responses of dry and fluid friction in the monotonic linear sliding process at constant sliding velocities (figure 6).  (3) The friction resistance is obviously greater in dry friction than in fluid friction, and the difference between a peak and a bottom contact stress ratio is larger in dry friction than in fluid friction. (4) The sliding displacement in the transition from the peak to the bottom contact stress ratio is smaller in dry friction than in fluid friction because the contact is direct in the former but indirect via fluid in the latter. (5) The friction coefficient μ decreases with sliding displacement, as described in (1). Then, dry friction exhibits a negative rate sensitivity because the contact stress ratio f t /f n is given by the friction coefficient itself. On the other hand, fluid friction exhibits a positive rate sensitivity because the contact stress ratio is given by the friction coefficient plus the overstress, while the overstress increases with sliding velocity inducing a higher viscous resistance. (6) The ratio f t /(μ s f n ) approaches unity (i.e. r = 1: normal sliding-yield state) in dry friction but becomes greater than unity (i.e. r > 1: over normal sliding-yield state) in fluid friction.
In addition, note that it is not required to use equation (2.43) and/or equation (2.45) in the subloading-friction model even for the calculation of dry friction behaviour. In fact, the dry friction behaviour is generated in the quasi-static sliding process in the subloading-overstress friction model. Consequently, we need only use equation (

Numerical experiments
The basic mechanical properties of the subloading-overstress friction model formulated in the preceding section are examined below in numerical experiments using the constitutive equation     6. The effect of the sliding velocity on the peak (maximum) and bottom (minimum) contact stress ratios in the contact stress ratio versus sliding displacement curves, which are read from figure 8, are plotted in figure 9. Here, the peak and bottom contact stress ratios increase with sliding velocity in a certain velocity range, whereas they converge to certain values outside this range.

Comparison with test data
To examine the validity of the generalized subloading-overstress model, a lubricated friction test was performed. A schematic diagram of the test apparatus is shown in figure 10.   which is placed between the tools, is subjected to the constant normal load (5 kN) and is pulled up at constant velocity. The pulling force is measured by the load cell (maximum load: 100 kN) attached to the upper part of the test plate. The test plate is made of galvannealed steel sheet with a friction area of width 30 mm, height 300 mm and thickness 0.7 mm. The tool steel SKD-11 of width 40 mm, height 30 mm and thickness 20 mm is used for the tools which grasp the test plate. Therefore, the friction contact area for the normal contact stress is 900 mm 2 and therefore the normal contact stress is 5.56 MPa, whereas the friction contact area for the tangential contact stress is 1800 mm 2 . The friction surfaces were polished and coated with anti-rust oil prior to the tests. The pulling-up velocity of the test plate is set at five levels: 1, 10, 50, 100, 200 mm min −1 . The friction test was adopted supposing the press forming of thin sheet metal in the metal forming process.
The measured relationship between the contact stress ratio f t /f n and the tangential sliding displacementū t is shown in figure 11a. The contact stress ratio first peaks and then gradually falls to a stationary value, exhibiting a positive rate sensitivity, i.e. larger contact stress ratios at higher sliding velocities.   Based on figure 11a, a comparison of the calculated and test results for the influence of the sliding velocity on the peak (maximum) and bottom (minimum) values in the contact stress ratio versus sliding displacement curves is shown in figure 12, where the close coincidence can be observed.

Concluding remarks
A generalized subloading-friction model, i.e. the subloading-overstress friction model, was formulated and the validity was verified in numerical experiments and by comparison with test data. The model is able to describe the following basic behaviours of friction phenomena.
(1) Peak and subsequent minimum friction resistance at a constant sliding velocity and the recovery of friction resistance as sliding stops or the sliding velocity decreases. (2) Both negative and positive rate sensitivities in the unified form, while only the former has been described by the subloading-friction model [30,31]. (3) Sliding behaviour at general sliding velocity ranging from quasi-static to impact sliding.
The generalized subloading-friction model is formulated using the general friction-yield surface but the Coulomb friction-yield surface was adopted in the explicit analyses to explain the fundamental behaviour of the model concisely. The subloading-overstress friction model can be readily extended to describe the effect of the normal contact stress and the anisotropy on the friction resistance by incorporating the more general friction-yield surface which is the nonlinear function of the normal contact stress [30,31] and takes account of the orthotropic and rotational anisotropies [42,43].
Crystal plasticity is widely analysed by inappropriate methods using a creep model [40,41,44]. The subloading-overstress friction model, regarding the tangential contact stress and the tangential contact yield stress as the resolved shear stress and the critical shear stress, respectively, provides a rigorous analysis with high efficiency for rate-dependent crystal plasticity in addition to rate-independent crystal plasticity [28,29]. Details of the results of the calculation of crystal plasticity are forthcoming. Furthermore, the subloading-friction model can be applied to the prediction of continental slip-type earthquakes to prevent earthquake disasters in the near future.