A simple model for friction detachment at an interface of finite size mimicking Fineberg’s experiments on uneven loading

УДК 539.421

A simple model for friction detachment at an interface of finite size mimicking Fineberg&s experiments on uneven loading

A. Papangelo1, B. Stingl2, N.P. Hoffmann23, and M. Ciavarella1

This work presents a model and simulation results for the friction detachment of a finite sized interface, following previous results on the phenomenon by Ben-David and Fineberg, namely "experiments demonstrating that the ratio of shear to normal force needed to move contacting bodies can, instead, vary systematically with controllable changes in the external loading configuration". In particular, we extend a previous one-dimensional simulation model by Bar Sinai with colleagues to a quasi 2D model to allow for a tilting of one of the contacting blocks. While Bar Sinai with colleagues postulate that the presence of "slow fronts" of detachment (an order of magnitude lower than the usual Rayleigh fronts as in crack propagation) is due to a strengthening term in the friction law, which is not always measured in unlubricated contacts, we find slow fronts also with a purely weakening law.

The basic laws of dry friction seem simple and have apparently been understood centuries ago by the Italian scientist Leonardo and the French engineers Coulomb and Amontons: the frictional force between two bodies in static contact is proportional to the normal force. The constant of proportionality is called "coefficient of (static) friction". This surprisingly simple law, independent on the macroscopic shape of the body and the apparent area of contact, has been interpreted first by Bowden and Tabor [1] as due to the fact that the contact occurs only on microscopic asperities and the pressure locally is sufficient to cause yield — friction is therefore due to asperity junction plastic failure, and the friction coefficient is the ratio between shear strength and hardness, i.e. about 3 times the yield strength of the softer material (this is the so called "ploughing term", and there can be also an adhesion contribution). Later on, statistical models explained that the average pressure on each asperity tends to be constant, and only the number of contacting asperities changes with increasing normal force. Greenwood and Williamson [2] confirmed the proportionality of contact area and normal force with such a statistical approach. Hence, what happens on the asperity scale is crucial to friction.

When sliding starts, thermal effects are activated, both at bulk and asperity scales, and this is usually one of the factors called upon to explain the rate-dependence of friction as part of the difference between static and dynamic friction coefficient (see [3]). Yet, for many scopes, the Coulomb law is still very much in use, and more complicated laws are often developed empirically trying to account for the influence of other factors, such as speed, normal load, surface conditions, temperature, etc. (see [4]). To summarize, while the basic law is still useful for very qualitative results, the understanding of frictional processes is still very far from even remotely completed.

Dieterich [5, 6] and Ruina [7] suggested a rate- and state-dependent friction law to explain a large amount of laboratory data on rock friction. The state variable quantifies the contact state between sliding surfaces or the internal structure of the gouge layer between sliding surfaces. Various empirical forms exist which consider interface strength as a dynamic entity that is inherently related to both fast processes such as detachment/reattachment, and the slow process of contact area rejuvenation. This can be put as an extension of the Bowden-Tabor model, which states the solid friction force F needed to slide one solid against the other at relative velocity V reads: F = ct£, with

© Papangelo A., Stingl B., Hoffmann N.P., Ciavarella M., 2014

a) Geometric aging: asperities creep plastically, and 2 increases quasi-logarithmically with contact duration <5, the "geometric age". When motion starts, contacts get gradually destroyed, after age <, and replaced by new ones. When the interface sits still, it ages (strengthens); when it slides, it rejuvenates (and hence weakens, according to this first criterion). This incidentally causes a static friction peak by loading at constant velocity.

b) Velocity dependence of the sliding strength a: for many interfaces increasing strength is found (similar to lubrication, but in dry friction it is a rheology resulting from thermally activated "depinning" dissipative events occurring within the adhesive joint).

It is this second term which shows crucially important in explaining the transient from rest to motion. Until recently, there was only indirect measurement of all these effects of contact area size at asperity scale, and no detailed information about the space-time evolution of the contact area and stresses along the interface.

Recently, an interesting series of experiments has been conducted by the group of Jay Fineberg in Israel, based on a new measurement technique [8]. In their experiments, they use two blocks of transparent material (polymethyl methacrylate, PMMA), permitting real-time visualization of the net contact area that forms the interface separating two blocks of like material. The underlying idea is that by illuminating with a very small angle to the surface, only the areas in contact can transmit the light across the interface, and hence, the change in contact area can be detected optically. Hence, sliding is seen indirectly as a "change of contact area", but notice that, for a given normal load, we would expect the contact area to stay globally constant. These studies are showing a complex pattern of propagating fronts at the interface, where the fast fronts (subsonic and intersonic) are very similar to those expected from classical theories of fracture mechanics. The third type of front, an order of magnitude slower, was found to be the dominant mechanism for the rupture of the interface, since sliding occurs only when one such slow front traverses the entire interface. Indeed, slow fronts are responsible for the most significant redistribution of stress (due to the local change of contact area), whereas fast fronts are too fast to alter the state of the interface. The resulting apparent friction coefficient depends very much on the details of the dynamical processes of these alternating slow and fast fronts, and these in turn strongly depend on the loading and geometrical conditions. In more recent experiments (Ben-David and Fineberg [9]), Fineberg&s group varied the stress profiles

across the interface by a small tilting of the top block (one or two hundredths of a degree). They found a large variation on the apparent friction coefficient with different loading states and normal forces, and they observed different details of the dynamics of the rupture leading to relative motion.

The only other experiment of the type of Fineberg in Israel has been attempted in Japan, by the group of Ken Nakano [10]. With a simple serial block model they did explain how the apparent static friction coefficient can be less than the nominal static friction coefficient due to the residual strain in the slider, and a difference between static and dynamic friction coefficient. However, a careful look at the experiment shows that they seem to have different results on the contact area reduction during sliding — Nakano&s group has a large 30 %, whereas Fineberg group had only just 10 %, despite the experiments seem on the same material, polymethyl methacrylate. In addition we note that authors of [10] do not report a dependency of the static friction coefficient on the applied normal stress distribution. However, comparing Figs. 4, 6 and 7 in their publication a slight dependency can be observed.

The slow fronts have been tentatively explained by a Burridge-Knopoff spring-block type model [11], introducing some appropriate statistical distribution of the stiffness and rupture thresholds of the surface contacts, as well as time delays to reattach broken contacts. They found that for moderately flexible surface contacts, an excess stress develops at the tip of the arrested Rayleigh fronts, which can trigger a slow detachment front, representing the motion of individual blocks, whose properties determine the front speed.

In a recent 2D block model [12] with standard Coulomb friction (but with difference in a static and dynamic friction coefficient) a few results obtained by the experiments are reproduced. In particular, that front speed depends strongly on both the instantaneous stresses and the friction coefficients, through a non-trivial scaling parameter. However: (i) they do not seem to find the slow fronts— the slowest front was about a fourth of Rayleigh speed— still an order of magnitude faster than Fineberg&s slow fronts (ii) and they have a different scaling of speed vs excess stress, less universal. On the other hand, there is no real reduction of apparent friction. Another interesting recent paper is that of Scheibert and Dysthe [13], which shows with a minimal quasistatic 1D model, how a very small "tilting" in Fineberg&s experiments can affect the kinematics of the transition from static friction to stick-slip motion. The analysis in a sense corresponds to that predicted decades ago by Cattaneo and Mindlin [14-16] using Amon-tons& law of friction. Here, however, the "full stick" solution is considered to be uniform shear distribution (avoiding to discuss edge effects), whereas the pressure is "tilted"

by the effect of the tangential force, which is applied to a certain height of the specimen. Below a certain height (and in particular, for the tangential force aligned with the interface), no precursor is observed.

Recently, the block model [11] has been extended to a continuous model by Bouchbinder and coworkers [17-19], who have obtained a set of interesting results. Confirming the findings [11], with a mesoscopic model of friction containing evolution laws for the contact area and for the "elastic" contribution term in friction, they seem to explain most of the findings by Fineberg. In particular, it was shown that Fineberg&s slow fronts are an intrinsic and robust property of simple non-monotonic rate-and-state friction laws [20, 21]. In other words, they occur with strenghtening friction laws, associated with a new velocity scale cmin, intrinsically determined by the friction law, below which steady state rupture cannot propagate. Ruptures hence occur in a continuum of states, from cmin to elastic wave-speeds, as found experimentally.

In the present paper, we shall attempt another approach to the explanation of the slow fronts and in general of the Fineberg&s results—paying attention to the effect which was considered most relevant in the latest contributions from Finberg&s group, namely the tilting of the pad. Our research was motivated by studies on the sticking to sliding transition in disc brakes [22], where the influence of the tilting on the detachment process was observed. In this work we present numerical simulations using our disc brake model using polymethyl methacrylate parameters.

We consider an elastic block between two rigid plates, as schematically described in Fig. 1. This model set up is a generic representation of a sliding system with dry friction and was even used for some numerical investigations on the brake pad-disk contact [22]. (This is why some terminology is taken from the brake system field).

Here we summarize the key features of the model:

- rigid backing plate,

- rigid brake disc,

- plane strain condition,

- isotropic and linear elastic lining material,

- free edges,

Driving velocity Fig. 1. The model of a sliding pad between two rigid plates

- small deformations,

- absence of wear.

Following [18] we integrated the equation of motion for the x direction along y to obtain a reduction of the problem from 2D to 1D where we took the rigid body motion of the backing plate into account (see the appendix for the complete derivation). The final equation of motion is: d2u (x, t)

+ Ga( t)

+ Tf (x, t) - c — (x, t), dt

where u(x, t) is an average, in the y direction, of the displacement of the lining material in the x direction. On the left hand side one can find the inertial term, while the right hand side terms express the contribution of the stresses a xx and a ; Tf is the frictional stress and c represents a numerical damping in the system.

The frictional stress was obtained using Bouchbinder&s

model [16-19] with the frictional stress

/ i 1 x ~ , . ~ , „ a a I vr

Tf """ """ ™

Tf = Te + Tv = Te + nv* A sign(vr) ln

the evolution equations for the elastic frictional stress

= .0 A t& , ~dt ~ Vr D

and for the contact age 9 d£ = 1 9| vr| J | T |

The real area of contact

A(a w, 9) = A0

is given by the contact age and the pressure dependent unaged contact area

In the above equations vr is the relative velocity between the elastic material and brake disk, a jj is the normal pressure at the interface, while the others quantities are listed and described in the Table 1.

To study the detachment process we apply the normal force with a linear increase from 0 to Fmax in the first 10 s; after some rest of 10 s to give to the system the possibility to reach the equilibrium, we increase the velocity of the brake disc with a linear ramp from 0 to vmax in the time interval from 20 to 22 s; the velocity of the disc and the force are then kept constant. These loading trajectories are shown in Fig. 2.

After each simulation we computed the static friction coefficient from the interface stresses by

Description Value

Elastic shear modulus G, MPa 3100

Poisson&s modulus v 0.35

Density p, kg/m3 1200

Mean size of contacts D, jjm 0.5

Shear strength Tc, MPa 130

Hardness a H, MPa 540

Viscous friction coefficient n, (MPa • s)/jm 27

Reference scale of velocity v*, jm/s 0.1

Coefficient b 0.075

Reference temporal scale , s З.ЗЗ • 10-4

Interfacial stiffness-effective height of the asperities ratio j0/ h ,MPa/ jm 300

Pad height H, mm 15

Pad length L, mm 200

Pad depth p, mm 120

Mean size of contacts D, jjm

Shear strength Tc, MPa

Hardness а H, MPa

Viscous friction coefficient n, (MPa • s)/^m

Reference scale of velocity v*, jm/s

Coefficient b

Reference temporal scale ф0, s

Interfacial stiffness-effective

Pad depth p, mm

fs = max

p |xf( X, y = 0)dx

З.ЗЗ • 10

P Ia yy (* y = 0)d*

where L is the length of the pad and p is its depth, respectively 200 and 120 mm.

The sign of the eccentricity is taken coherent with the x-axis and in the following sections it is computed as 2e/L.

varying the eccentricity, the normal force and the brake disc velocity. All the simulated sticking to sliding transitions share some common features: the detachment process starts from one edge and transverses through the whole interface. This detachment front can generally be subdivided into three parts.

As an example we take the detachment process obtained when the normal force is applied in the middle of the backing plate (i.e. e = 0). As shown in Fig. 3 the detachment process starts with a slow precursor originating at the left which accelerates to the middle front at approximately -45 mm and slows down again at approximately +55 mm to the slower right front. However, depending on the parameters, the left and the right front do not always exist, while the middle detachment fronts is a robust feature of the system. As shown in the following, we will discuss the dependency of the static friction coefficient on the external loading and its relation to the interface dynamics sketched above.

Figure 4 shows the increase of the friction coefficient moving the normal force from the leading to the trailing edge. The difference between the highest and the lowest value of the static friction coefficient with respect to the mean value is around the 0.5 %. While this variation is small with respect to the observations by Fineberg&s group [9], we observe the same trend: the friction coefficient is higher for a tilting to the trailing edge.

Varying the normal force, the static friction coefficient seems to saturate to a limit value for high forces as shown in Fig. 5 where we collected the value of the static friction coefficient for zero eccentricity.

Connecting the variation of the friction coefficient with the interface dynamics two interesting relations were found.

We ran simulations using the parameters of the sketched geometry and for polymethyl methacrylate as listed in the Table 1 taken from [18]. With the model and the loading described above we ran a large numerical parameter study

Vua, 10 m/s

Fig. 2. The loading trajectories on normal force (a) and driving velocity (b)

-100 -80 -60 -40 -20 0 20 40 60 80 100 Space, mm

Fig. 3. Detachment front originating from the left edge: A slow precursor exists for the first 55 mm (left front), followed by the faster intermediate (middle) front. Closer to the trailing edge from +55 mm on the slower right front can be observed. The area 1 indicates points with a velocity above 40 % of the maximum velocity reached during this detachment process; the velocity of the points 2 is below this arbitrarily chosen but characteristic threshold which has been introduced to visualize the detachment front. On the bottom right hand corner the dynamical process without labels and in greyscale is reported. (For this particular case e = 0)

Fig. 4. Dependency of the static friction coefficient on the eccentricity. F = 1000 N, ^bd = 0.1 mm/s, H = 15 mm

Fig. 7. Static friction coefficient versus the length of the left precursor. The events shown are the same as in Fig. 6

Fig. 5. The static friction coefficient as a function of the normal force for zero eccentricity

Fig. 6. Static friction coefficient as a function of the left precursor velocity. The detachment events shown here share a normal force of 1000 N but were observed for different eccentricities

Fig. 8. Velocity of the left (1), middle (2) and right front (3) versus the normal force, compared to the shear wave speed (5) and the speed of the slow fronts (4) reported in [17]. e = 0

Fig. 9. Linear relationship between the inverse of the left precursor velocity and the normal force

Figure 6 reports a linear dependency between the static friction coefficient and the velocity of the left precursor front. In addition to the velocity, the static friction also depends linearly on the length of the left precursor, as shown in Fig. 7. The latter linear relation coincides with the results found by Otsuki and Matsukawa [23]. However, the underlying physics are different, as for [23], the linear instability of the quasistatic precursor leads to a rapid front while in our system the whole detachment process is quasistatic as discussed below.

This overall behaviour was found for the value of the normal force equal to 1000 and 5000 N, while for lower normal forces the detachment process does not include the left precursor front.

The order of magnitude of the velocity of the detachment fronts is close to the slow fronts reported in the literature, except for the left precursor fronts that seem even slower, see Fig. 8. Looking to the normal force dependency

of the front velocities the right front differs from the other two: While for the middle fronts and for the left hand side fronts the front velocity diminishes increasing the magnitude of the normal force, the right hand side fronts seem to maintain a constant average velocity for all the values of eccentricity. The lower detachment velocity could be related to the more pronounced influence of the friction when the normal force is higher. We plotted the reciprocal of the velocity of the left hand side fronts against the value of the normal force. The result is reported in Fig. 9 and shows a clear linear relationship between the above mentioned variables. To sum up, the velocity of the precursors seems to be in inverse proportion with respect to the normal force applied.

The left, middle and right detachment fronts increase their velocity when the driving velocity vbd is increased. This kind of relationship is also reported in [23]. Plotting the velocity of the fronts with respect to the driving velocFig. 10. Dependency of the left (1), middle (2) and right front velocity (3) on the driving velocity for 500 N and zero eccentricity; 4 cslow, 5 C shear

ity in a double-logarithmic diagram the curve approaches a line as shown in Fig. 10 for a normal force of 500 N and zero eccentricity. This common feature suggests that our detachment fronts are quasistatic, just as the precursors observed in [23].

We test this hypothesis by suddenly stopping the disc during the detachment phase. Figure 11 shows the velocity of the friction material along the sample against the time. At 20.15 s we suddenly stopped the disc and all the dynamics at the interface stopped synchronously. These results support the idea that the fronts are indeed quasi-static and do not possess an intricate inner or local dynamics.

We finally close our discussion with a numerical experiment on the contribution of the velocity strengthening part of the friction law. We eliminated the strengthening branch from the friction law setting the viscous friction coefficient ^ equal to 0 and ran simulations measuring the velocity of the detachment fronts. Figure 12 compares the velocity of the fronts with and without strengthening. As one can see the velocity of the fronts seems to be not influenced by the strengthening branch. This is an interesting result as the velocity of the slow fronts was reported to depend on the minimum of the steady state friction coefficient. However, eliminating the strengthening branch this minimum vanishes. Potentially the tilting degree of freedom in our model—which is generic for any sliding system of finite length—is a crucial contribution to the detachment process. The importance of this friction in107

f ____4_

i i V I 1 1 1 1 i i i

Eccentricity

Fig. 12. Influence of the velocity strengthening branch in the friction law on the velocities of the left, middle and right front, plotted here against the eccentricity: 1 — vfleft with strengthening,

duced torque resulting in a tilted pressure distribution in the interface has already been discussed in [13].

The velocity strengthening term does, however, have a marked effect on the overall system level static friction coefficient. Figure 13 shows the static friction coefficient as a function of the driving velocity. Plotted in a semi-logarithmic diagram the static friction coefficient increases linearly with the disc velocity. We believe that this behaviour is linked to the strengthening branch of the static friction coefficient. To check this hypothesis we set the viscous friction coefficient to 0 and again evaluated the behaviour of the static friction coefficient with respect to the driving velocity (see Fig. 14). In this case a higher driving velocity results in a lower static friction coefficient until a limit value is reached. This kind of behaviour can be due to the effect of the aging. When the driving velocity is low the detachment takes longer than in the case with high driving velocity. Thus the aging of the contact area becomes important and yields to an increase of the static friction coefficient.

-50 0 50

Space, mm

Fig. 11. Velocity field of the pad for a sudden stop of the disc during the detachment process at 20.15 s. e = 0

Fig. 13. Dependency of the static friction coefficient on the driving velocity with the velocity strengthening contribution in the friction law

Fig. 14. Dependency of the static friction coefficient on the driving velocity without the velocity strengthening contribution in the friction law

To our knowledge this work is the first to give numerical confirmation of Fineberg&s experimental results [9] on the effect of asymmetric normal loading. While the observed variation of the friction coefficient is rather small we were able to identify clear relationships between the precursor fronts and the macroscopic friction coefficient. Especially the influence of the precursor length coincides with previous results [23], although the underlying mechanism may be of different nature. Nevertheless Fineberg&s experimental results [9] show that the precursor length decreases when the static friction coefficient increases which is indeed opposite in our results. The origin of this contrary finding might be in the loading geometry: Fineberg applies a homogeneous shear load along the interface while Otsuki considers edge loading. The coincidence of our results— obtained with a homogeneous loading—with Otsuki&s edge loading system is probably due to our simplified continuum mechanics description as discussed below.

The detachment fronts found in our studies are quasi-static and independent on the minimum of the steady state friction coefficient. Indeed, from our numerical investigations, the front velocity is not influenced by the strengthening part of the friction law, even for eccentricity equal to zero, for which our model is more similar to that used by Bouchbinder [18]. Nevertheless, our model was able to predict the intensively discussed slow fronts semiquantiti-vely. However, we stress that this result may be due to the strong simplifications in our model which were introduced by the one dimensional approximation of the two dimensional continuum. This leads to a lack of vertical dynamics as the only vertical degree of freedom is given by the rigid body motion of the backing plate. We believe that the approximation error is especially large at the edges, e.g. our model does not allow for lift off at the edges of the inter-face—a process that might have a tremendous effect on the nucleation of detachment fronts. Extension of the present model to capture full 2D continuum dynamics of the pad is thus a topic of ongoing and future studies.

Acknowledgement

The authors are thankful to Y. Bar-Sinai for his helpful comments on the reduction from 2D to 1D.

References

Sons, 1995. - 336 p.

Res. - 1983. - V. 88. - P. 10359-10370.

Appendix. The equation of motion in the x direction

We have a two dimensional spatial problem as all the fields of displacements, velocities etc. are in general functions of the spatial coordinates x and y and of the time. In

this appendix we show how we reduced the problem to a one dimensional problem in which the field of displacements and of the velocity of the elastic material are only function of the x direction (for the symbols used refers to the Table 1).

In general the small displacement field of the elastic material is described by these equations:

ux (x У, t)

u(x, y, t) =

Uy ( x, y, t) 0

The elastic material is subjected to the kinematic boundary condition at the bottom

ux0(x> t)

u(x, 0, t) =

as the displacements in the y and in the z direction are not allowed along the interface. Using the y coordinate of the rigid backing plate ybp(t) and the tilt angle a(t) (see Fig. 1) we know the displacement field at the bottom of the backing plate, that for small deformation is: 0

u(x, H, t) =

ybp(t) - H + a(t)x 0

where H is the thickness of the elastic material in the unstrained condition.

Both the edges are free, so, considering the stress tensor, we can write for the left-hand side

"-1" CTxy CTxz "-1" "0"

x, y, t)|x=-Lj2 0 = = yy ayz 0 = 0

and for the right-hand side

"1 " CTxx CTxy CTxz "1 " "0 "

x, y, t)lx=L2 0 = = °yy °yz 0 = 0

For the linear elastic and isotropic material in plane strain condition we can write the Hooke&s law as follow:

where the strains are defined as in the linearized theory

Using the first two rows of Eq. (A6) we obtain v

Thus the equation of motion is

d2ux xx ( x, y, t) + da xy( x, y,t)

at2 dx dy

d2Uy xy(x, У, t) + 3ayy(^ y, t)

L dt2 J dx dy

where p is the density of the elastic material that we consider as a constant.

Now we integrate the x component of the equation of motion in the y direction over the pad from y = 0 to y = ybp (t) - H + a(t)x = y. With this strategy we are claiming that the equations will not be satisfied in every x and y coordinate but only on average in the y direction. The following derivation and the further hypothesis will make this

argumentation clearer. The integration yields

jpiix (x, y, t)dy =

= y dQxx (x, y, t) dy + y xy (x, y, 0 dy =

= 0dxlr-vPx^ y, t) + yy (^ У, t) |dy +

+ (x, ybp (t) - H + a(t)x, t) -a^ (x, 0, t) = = f d2ix(x, y, t) ^yy(x, y,t) d +

+ ^xy (x, ybp (t) - H + a(t)x, t) -CTxy (x, 0, t), (A10)

where we have used Eqs. (A7) and (A8). The above equation can be written as follows

A = B + C + D + E, (A11)

where each term in Eq. (A11) points to the corresponding term in Eq. (A10).

Let&s assume that the dependency along the y direction of the x displacement can be neglected and define the average x displacement

u(x, t) =-—-—— i ux (x, y, t)dy. (A12)

ybp(t) + a(t )x 0

From the small displacement hypothesis we can assume that ybp (t) + a(t) x = H, so we can directly write

A = J pitx(x, y, t)dy = pH^^, (A13)

B = ^H d2u(*,t) (A14)

as the upper bound of integration has lost its dependency on x and t.

For the term C, we use the Hooke&s law and Eqs. (A7) and obtain

v У da (x, y, t) , v 2G

C = 1 -v;

_H d2ux(x, y, t). v 2G(1 -v)

xv I-xV Y& Jdy +---0 Эх2 1 -v 1 -2v

H д Uy (x, y, t) v

I y dy =

dUy ( x, y, t) dx

where we have used the boundary condition at the top and the average displacement as defined in Eq. (A12).

For the evaluation of the term D we use the Hooke&s law (A6), the strain definition (A7) and the boundary condition (A3) to obtain D = o^y(x, H, t) = 2Gexy (x, H, t) =

+ a (t )

For the remaining derivative we need to approximate the y dependency of the x displacement which has been averaged out in Eq. (A12). So we define

ux(x, y, t) = a(x, t)y + b(x, t). (A17)

From the upper boundary condition (A3) we obtain

Then the linear approximation has to fulfill Eq. (A 12) hence

(x t) = -1 J (ay - aH)dy = H,

and finally

ux = --?£ ( y-H )•

(A15) Hence substituting Eq. (A21) in Eq. (A16)

axy (x, H, t) = Gl 2u( x, t)

+ a(t)

(A19) (A20)

(A21) (A22)

Plugging all the terms A, B, C, D, E in Eq. (A11) we obtain the equation of motion of the elastic material that now depends only on the x direction

d 2u( x, t)

+ Ga(t ) | 1 +

t2 +

dx2 2Gu ( x, t ) H

+ Tf(x, t) - c—, (A23) dt

where the term E has become the frictional stress Tf (x, t), and we added a numerical damping c du/dt that takes into account all the damping in the system.

Поступила в редакцию 17.02.2014 г.

Сведения об авторах

Papangelo Antonio, Ph.D. Student, Politecnico di Bari, Italy, ant.papangelo@gmail.com

Stingl Bernhard, Researcher Assistant, Hamburg University of Technology, Germany, bernhard.stingl@tuhh.de

Hoffmann Norbert P., Prof., Hamburg University of Technology, Germany, Imperial College London, UK, norbert.hoffmann@tuhh.de Ciavarella Michele, Prof., Politecnico di Bari, Italy, engineeringchallenges@gmail.com