Issue 
Mechanics & Industry
Volume 19, Number 6, 2018



Article Number  607  
Number of page(s)  20  
DOI  https://doi.org/10.1051/meca/2018049  
Published online  12 February 2019 
Regular Article
Analysis of couplestresses and piezoviscous effects in a layered connectingrod bearing
^{1}
Département de Génie Mécanique, Laboratoire de Mécanique et Structures (LMS), Université 8 mai 1945 Guelma,
BP 401,
Guelma (24000), Algeria
^{2}
Université de Lyon, CNRS INSALyon, LaMCoS, UMR5259,
69621
Lyon, France
^{*} email: Benyebka.BouSaid@insalyon.fr
Received:
14
October
2017
Accepted:
4
December
2018
In this work, the combined effects of couplestresses and piezoviscosity on the dynamic behavior of a compression ignition engine bigend connectingrod bearing with elastic layer are investigated using the V. K. Stokes microcontinuum theory. It is assumed that the journal (crankpin) is rigid and the bigend bearing consists of a thin compressible elastic liner fixed in an infinitely stiff housing. The governing Reynolds' equation and the viscous dissipation term appearing on the RHS of energy equation are modified using the V. K. Stokes microcontinuum theory. The nonNewtonian effect is introduced by a new material constant η, which is responsible for couplestress property, and the piezoviscosity effect by the pressure–viscosity coefficient α appearing in the wellknown Barus' law. In the proposed model, the nonlinear transient modified Reynolds equation is discretized by the finite difference method, and the resulting system of algebraic equations is solved by means of the subrelaxed successive substitutions method to obtain the fluidfilm pressure field as well as the film thickness distribution. The crankpin center trajectories for a given load diagram are determined iteratively by solving the nonlinear equilibrium equations of the journal bearing system with the improved and damped Newton–Raphson method for each time step or crankshaft rotation angle. According to the obtained results, the effects of couplestresses and piezoviscosity on the nonlinear dynamic behavior of dynamically loaded bearings with either stiff or compliant liners are significant and cannot be overlooked.
Key words: Compression ignition engine / couplestress / piezoviscous fluid / coated bearing / dynamically loaded journal bearing
© AFM, EDP Sciences 2019
1 Introduction
Nowadays, reciprocating machines such as internal combustion engines and compressors are the most important class of machinery extensively used in diverse engineering applications. Unquestionably, dynamic behavior of such machines is strongly dependent on the performance characteristics of their bearings.
The crankshaft and connectingrod bearings of these modern machines with high horsepower and high loads must be correctly designed to support large dynamic loads resulting from combustion pressure in the engine cylinder and inertia forces due to reciprocating and rotating motions of solids, which belong to the crankslider mechanism.
These loads, which are generally determined from the crankslider kinematics and dynamics, vary both in magnitude and direction during an engine cycle. Under these severe operating conditions, the behavior of the dynamically loaded journal bearing system becomes strongly nonlinear requiring a complete nonlinear transient analysis. This later involves the simultaneous solutions of the complex multiphysical fluid–solid interaction problem, governed by several nonlinear PDEs. This type of analysis is extremely essential when the engine rotates at high speeds, for example, in the case of Formula 1.
Many researchers in the fields of fluid–film lubrication and engine design have tried to formulate lubricants with new chemical compounds to enhance the dynamic behavior of rotating systems. Applications of various types of Newtonian and nonNewtonian fluids or combination of conventional mineral and syntheticbased lubricants with different polymer additives are examples of the efforts made to achieve better dynamic performance characteristics of journal bearing systems.
The rheological behavior of mineral or synthetic motor oils used as lubricants is significantly affected by the presence of various additives such as viscosity index (VI) improver polymers, which are characterized by long chains.
These polymers can be classified into two categories: hydrocarbon copolymers and polymethacrylates. In general, oils containing VI additives such as multigrade motor oils must be considered as nonNewtonian shear thinning fluids. Their viscosity decreases when the shear or strain rate to which they are subjected increases, and they are affected by the Weissenberg effect, i.e. during flow, forces appear perpendicular to the shear planes. These properties have been analyzed by a Weissenberg rheogoniometer [1]. According to Lodge [2], these forces could be proportional to the square of the rate of shear.
Rosenberg [3,4] has shown experimentally that the minimum film thickness of a journal bearing lubricated by polymerized oils is more important than that measured with the pure mineral oils having the same viscosity. In order to determine the effects of VI additives on the journal bearing behavior, Robin [5] developed a test bearing. The operating conditions for which the tests were performed are N = 3 krpm and W_{0} = 4 kN, which are the rotation velocity of the shaft and the applied load, respectively. It was found that the introduction of high concentration of polymethacrylates to the base mineral oil reduces the friction torque by about 25% without important change in the film thickness. It was also concluded that it would be interesting to use oils with high concentration of VI additives having low molecular weight such as polymethacrylates, rather than weak concentrations of additives with high molecular weight.
In the Oliver's experimental work [6], it was found that the presence of dissolved polymer in the lubricant increases the load carrying capacity of the lubricating film and decreases the friction coefficient.
In another experimental investigation, Scott and Suntiwattana [7] showed that addition of a small amount of longchained additives like some polymers such as polyisobutylene can enhance the lubricating effectiveness of conventional Newtonian lubricant.
The microstructure of these new lubricants can translate, rotate and deform independently.
It is observed from the experimental results that using the microcontinuum or micropolar theory is more suitable for the theoretical study of such lubricants. Application of classical NavierStokes and energy equations to describe their motion leads to erroneous results. Thus, many rheological models such as power law, viscoelastic, couplestress and micropolar are proposed in the technical literature.
In order to better describe the rheological behavior of this kind of lubricant, different microcontinuum theories have been developed. The Stokes' microcontinuum theory [8,9] is the simplest theory of fluids proposed in the technical literature since the 1960s, which allows the polar effects such as the presence of couplestresses and body couples in addition to the body forces and surface forces. However, it neglects the elasticity and the normal forces effects appearing during flow because of the presence of such additives.
In this theory, the isovolume couplestress fluids are characterized by two constants, namely, μ and η, whereas only one parameter appears for a Newtonian isovolume fluid, which is the dynamic viscosity μ. The new material constant η is responsible for couplestress property. In the literature, the effects of couplestresses on the behavior of journal bearings are generally studied by defining the couplestress parameter which has the dimension of length and can be thought of as a fluid property depending on the size of the polymer molecule.
In an excellent investigation, Fatu et al. [10] have studied the importance of piezoviscous and shearthinning effects on the dynamic behavior of three typical compliant connectingrod bigend bearings used for a Formula 1 engine, a compression ignition engine and a spark ignition engine. The constitutive equation relating the fluid dynamic viscosity to rate of shear that has been used by the authors to describe the nonNewtonian shear thinning behavior is similar to that proposed by Gecim [11].
To study the piezoviscous effects, the model that has been used by the authors is that suggested by Chu and Cameron, which is suitable for higher pressures [12].
They proved that for the studied cases, the nonNewtonian shear thinning effects could not be neglected, especially for highspeed engines, and the piezoviscous effects are more significant than the nonNewtonian effects and lead to increasing film thickness.
It should be noted that the same constitutive equation has been used by Paranjpe [13], and Wang et al. [14,15] in order to investigate the nonNewtonian shearthinning effects in smooth and rough dynamically loaded bearings.
In an earlier work [16], we have presented a theoretical study of stiff bigend connectingrod bearings dynamic behavior for both Diesel and gasoline engines using the couplestress fluid model. The mobility method [17,18] suggested in 1965 by Booker has been adopted to the numerical treatment of nonlinear motion equations of the rigid crankpin. The effects of different values of couplestress parameter l on the minimum film thickness, peak pressure, flow rate, power loss as well as crankpin trajectories were investigated for both engines.
In the present work, which can be considered as an extension of the above work [16], the combined effects of couplestresses and piezoviscosity on the dynamic behavior of a compression ignition engine bigend connectingrod bearing with elastic layer are investigated using the Stokes microcontinuum theory. It is assumed that the journal (crankpin) is rigid and the bigend bearing consists of a thin compressible elastic liner fixed in an infinitely stiff housing.
The damped Newton–Raphson method is used, in improved form, to predict the dynamic response of layered connectingrod bearings subjected to a load cycle instead of the mobility method since this later is not appropriate for the analysis of compliant or partially grooved journal bearings despite its rapidity to obtain dynamic responses of stiff cylindrical journal bearings [19]. However, when using the Newton–Raphson method, the inverse lubrication problem needs to be solved instead of the direct problem, i.e. dynamical loads applied on bearing are known but corresponding eccentricities of journal need to be calculated by iteration.
2 Theoretical analysis
2.1 Governing equations of couplestress fluid mechanics
The governing equations for the transient flow of an incompressible (isovolume) couplestress fluid, neglecting the body forces and body couples, can be written in the general form as [8,9] (1) (2) where , which is the total skew symmetric stress tensor, and is the Laplace operator. (3)where and , which are the strain rate tensor and the mean curvature rate vector, respectively.
These are the equations of conservation of mass, conservation of linear momentum and conservation of energy, respectively.
Although it is not necessary for the present analysis, the modified energy equation has been derived only for the purpose of giving the expression of the dissipation function appearing on the RHS of equation (3), which will be useful for calculating the power loss.
2.2 Modified Reynolds equation for piezoviscous fluids with couplestress
With the usual assumptions considered for the lubrication film, the modified Reynolds equation for twodimensional isothermal flow of piezoviscous lubricant with couplestress can be derived for the connectingrod bigend bearing represented in Figure 1 [20]. (4) where (5)where ω_{s} is the shaft (crankpin) angular velocity, ω_{b} is the bigend bearing angular velocity, is the angular velocity of crankshaft (N being the engine rotational speed in rpm), θ_{2} is the crankshaft angle, which varies from 0 to 4π, is the ratio of the crankshaftarm length to the connectingrod length and is the mean angular velocity determined from the kinematics analysis of crankslider mechanism [18].
Note that for crankshaft bearings (main bearings), equation (5) reduces to .
In the above equations, h is the lubricant film thickness, l has the dimension of length and can be regarded as a fluid property depending on the size of the high polymer molecule and as it approaches to zero, equation (4) reduces to the classical Reynolds equation for Newtonian fluid. α is the pressure–viscosity coefficient appearing in the Barus law, which gives the viscosity–pressure dependency at constant temperature: (6) where μ_{0} is the dynamic viscosity for p = 0, and α is the pressure–viscosity coefficient, which can be obtained by plotting the natural logarithm of dynamic viscosity μ versus pressure p. The slope of the graph corresponds to the value of α. The pressure–viscosity coefficient is a function of the molecular structure of the lubricant and its physical characteristics.
There are various formulae available to calculate the pressure–viscosity coefficient in the technical literature such as the Wooster's relationship [21]. Some of these equations are accurate for certain fluids and inaccurate for others.
It should be noted that the piezoviscosity effect varies between oils, and it is more considerable for naphthenic oils than paraffinic oils. Water, by contrast, shows only a small rise, almost negligible, in viscosity variation with pressure.
There are many other formulae for viscosity–pressure relationships. A short review of some of the empirical formulae for the viscosity–pressure relationships is given in reference [22].
Fig. 1 Bigend connectingrod bearing and coordinates systems. 
2.3 Boundary conditions
The boundary conditions associated to the Reynolds equation (4) may be classified as follows:
Boundary conditions related to the environment in which the system operates: (7a)
Boundary related to lubricant supply: (7c)
Boundary conditions related to lubricant flow (cavitation phenomenon): (7d)
At these conditions, we can add for aligned journal bearings the following condition: (7e)
2.4 Oil film thickness
The connectingrod housing and the crankpin are considered as infinitely stiff and thus only the thin compliant liner will deform Figure 1. In the mobile frame (X_{3},Y_{3}) related to the connectingrod, the film thickness of the undeformed bearing is a function of the radial clearance C = R_{b} − R_{s} and the crankpin center position defined by e_{X} and e_{Y} as expressed in the following equation: (8) where θ is the bearing angle (cylindrical coordinate) originating at the Xaxis.
When the film thickness h_{0} is modified with the elastic deformation of the fluid filmbearing liner interface, the film geometry becomes (9) where is the compliance operator of the bearing liner in (m/Pa). This operator gives a relation between pressure and elastic displacement and not between force and displacement.
The simplified elastic model used in equation (9) to calculate the radial deformation due to hydrodynamic pressure is more accurate when we assume that the liner thickness, t_{l}, is much smaller than the bearing radius, i.e. [23–25].
The bearing configuration considered is similar to that recently studied by Thomsen and Klit using a threenode triangular finite element method for solving Reynolds equation [26]. In this study, where only the local deformations of the bearing liner are considered, the authors have used the same elastic model to calculate the radial displacement at the fluid filmbearing liner interface. The liner is made of an almost incompressible material, namely, the PEEK (Polyetheretherketone) composite having a modulus of elasticity E of 6 GPa and a Poisson's ratio σ of 0.40.
Note that the compliance operator C, which is considered as a key parameter in EHD problems, can be determined in matrix form, called compliance matrix, using a standard finite element or boundary element analysis of the whole bearing structure.
2.5 Finite difference formulation of the modified Reynolds equation
The finite difference method is used to approach the nonlinear transient pressure equation (4) called here the modified Reynolds' equation.
The bearing surface is divided into N_{x} × N_{z} rectangular cells, i.e. the total number of nodes is (N_{x} + 1) × (N_{z} + 1). A computational grid of 61 × 21 nodal points is selected for the present investigation. This computational size is chosen as a compromise between time of calculations and accuracy.
After discretization, the modified Reynolds equation takes the following form: (10) where ; ; ; ; is the nodal oil film thickness calculated at previous time steps, i = 1, …, N_{x }+ 1 and j = 1, …, N_{z}.
In equation (10), , are the mesh sizes in the circumferential and axial directions, respectively, and is the time increment. being the crank angle increment where nsteps is the number of steps in one engine cycle, i.e. the complete load cycle divided into nsteps parts. So, the total number of data will be (nsteps+1), including the first and the last data (e.g. for △θ_{2} = 5 ° , nsteps = 144, and for △θ_{2} = 1 ° , nsteps = 720).
The nonlinear algebraic equations system (10) resulting from the spatialtemporal discretization of the nonlinear modified Reynolds equation is solved by the successive substitution method with underrelaxation coefficient ω ranging from 0 to 1 similar to the onedimensional nonlinear root finding Wegstein's method in order to determine the oil film pressure field p_{i,j}: (11) where the superscript m indicates the number of iteration for the successive substitution method.
Typical ω values selected for the cases studied in this work range from 10^{−3} to 10^{−1} depending on the nonlinear behavior of the modified Reynolds equation. However, ω can be equal to 1 when solving the linear Reynolds equation, i.e. for stiff bearing and/or isoviscous cases.
This method consists of building up a series of solutions , where the initial estimate of solution. The nodal pressures being calculated by solving the following system by the Gauss–Seidel algorithm with overrelaxation coefficient Ω_{GS} in order to accelerate the procedure convergence, especially when dealing with transient and nonlinear problems, and to fulfill the Reynolds cavitation conditions (7d) by incorporating the Christopherson algorithm [27]. During the iterative computation of fluid film pressure, the pressure vanishes (p_{cav} = 0) if the calculated pressure becomes negative: (12) where (n) and (n + 1) are the steps of Gauss–Seidel iteration.
The iterative Gauss–Seidel procedure is stopped when at each grid point (node i, j) the maximum relative error between two successive iterations fell below a tolerant error of 10^{−6}, i.e. (13)
Note that in relaxation methods, the execution time or even the numerical stability is greatly affected by the relaxation coefficient used. The optimum value for Ω_{GS} is not always predictable in advance [28].
The optimum value of Ω_{GS} is that which is able to converge to the solution for a tolerant error in a minimum number of iterations or a small computing time. In practice, it is generally determined via numerical simulations.
The stopping criterion of iterations in the successive substitution algorithm is (14) ∥n∥ being the relative least square norm.
2.6 Load balance and crankpin center orbit
When the external load acting on the bearing varies both in direction and magnitude, the journal (crankpin) center describes a trajectory within the bearing. The determination of this trajectory requires the solution of the nonlinear equilibrium equations at each time step or crank rotation angle with an iterative method. An inverse solution of the Reynolds' equation is then required.
For an aligned journal bearing, the equilibrium equations may be written when inertia forces of the crankpin are neglected as (15) where F_{X}, F_{Y }are the applied load components.
At each time, the position of the journal (crankpin) center defined by the eccentricity vector is determined when the lift force vector balances the applied load expressed in the mobile coordinate system related to the connectingrod (X_{3},Y_{3}).
Equation (15) can be rewritten as (16a) where r_{X} and r_{Y} are the components of the residual vector r (t), which is a nonlinear function of e_{X} and e_{Y}.
The damped Newton–Raphson method is used to solve the set of two nonlinear equations (16a). This method is formulated on the linearization of equilibrium equations using two variables, Taylor expansion of equations (16a) in the neighborhood of the kth trial solution, i.e. (16b)
In the relaxed Newton–Raphson method, the (k+1)th trial solution is (17) where ω_{NR} is the damping factor in the interval (0, 1).
It was found during simulations that for liners with low elasticity modulus and for piezoviscous lubricants leading to a nonlinear Reynolds equation, the use of the damped Newton–Raphson method described above with ω_{NR} = 10^{−1} is required in order to obtain the numerical convergence. On the other hand, for rigid liner and/or isoviscous lubricant, the ω_{NR} value can be taken to be equal to 1, because in such a case both Newtonian and nonNewtonian Reynolds equations have a linear behavior in terms of pressure p.
Note that the ω_{NR} value can be automatically adjusted during iterations based on the rate of solution convergence.
The corrections to are found by solving the following linear algebraic system deduced from equation (16b)" (18)
The obtained linear system is solved analytically.
The stopping criterion for the improved Newton–Raphson method by which the iterative process can be ended without any loss in solution accuracy is (19a) which represents the Laplace's norm L_{1} of the residual.
The Euclidean L_{2} norm of the residual can also be used, i.e. (19b) where ε = 10^{−}^{2} and k_{max} are the predefined convergence tolerance and the maximum number of iterations, respectively.
The partial derivatives appearing in matrix equation (18) are evaluated numerically by central finite differences, i.e.
where δ = 10^{−10} for calculations made in double precision.
2.7 Hydrodynamic characteristics
2.7.1 Side leakage flow
The side leakage flow at bearing edges is calculated by the following relation: (21) where and are the angles delimiting the active zone of the bearing.
In the above equations, h is the film thickness and is the axial mean flow velocity calculated by the following relationship obtained for piezoviscous couplestress fluid: (22)
The axial flow velocity w (x, y, z) as well as the circumferential velocity u (x, y, z) is determined from integration of field equations governing the motion of the lubricating oil in x and zdirections: (23a) (23b)
Using the following boundary conditions, (24a) (24b) we get (25a) (25b)
2.8 Power loss
The total power loss is evaluated on the active zone of bearing from (26) where (27)
In equation (27), and are the dissipation functions due to the shear stress and the couplestress effects, respectively. These two functions that appear on the second hand of modified energy equation (3) can be defined in hydrodynamic lubrication theory as (28) (29)
After integration with respect to y, we get (30) and (31)which is the power loss due to the squeezing effect.
In equation (31), and are the Cartesian components of the journal (crankpin) center velocities in X and Ydirections, respectively.
3 Results and discussion
3.1 Computation procedure
Based on the analysis described in the present paper, a computational code in MSFortran 90 was developed to study the isothermal elastohydrodynamic (IEHD) behavior of layered connectingrod bigend bearings lubricated with piezoviscous fluids with couplestress.
The prediction of instantaneous position of the shaft (crank pin) within the bush (bigend bearing) requires the solution of an inverse problem throughout the thermodynamic cycle of a reciprocating engine. In summary, the procedure of calculation needs the following five steps:
Step 1: From a position of the shaft center (e_{X}, e_{Y}), we calculate the film thickness h and the corresponding pressure p.
Step 2: The hydrodynamic load W is calculated by integrating the pressure on the bearing surface.
Step 3: The calculated load is then compared with the applied load F: if the calculated load is different from the applied load, we correct the position of the shaft, e.g. by the iterative Newton–Raphson method, and we restart the calculation.
Step 4: The iterative process is pursued until convergence: the center position of the shaft inside the bearing as well as the hydrodynamic characteristics of bearing are thus determined at each time t.
Step 5: The load cycle is repeated several times until two successive load cycles give identical shaft orbits.
The computed results include bearing center orbits, variations of the minimum film thickness, the peak pressure, the power loss and the side leakage flow versus the crank rotation angle θ_{2}.
In this section, the couplestress and the piezoviscous effects in the ungrooved connectingrod bigend bearing with thin elastic liner of the Ruston and Hornsby 6 VEBX MK III fourstroke marine diesel engine are investigated. This particular connectingrod bearing is the most analyzed bearing in the technical literature [29].
The polar and Cartesian diagrams of the dynamic load applied by the crankpin on the big end bearing of 0.127 m width during one engine cycle are presented graphically in Figure 2. The load data expressed in the mobile frame related to connectingrod are reported in Table 1 for crank step △θ_{2} = 10°. These data are interpolated using cubic splines for each 1° in order to ensure an accurate solution.
The peak load is about 208 kN occurring at θ_{2} = 10° after top dead center (ATDC). For a fourstroke engine, there are 720° crank angles in one engine cycle corresponding to two complete rotations of crankshaft. The time step used in the analysis being 1° crank angle. So, there are 720 time steps or parts in one engine cycle, viz. nsteps = 720.
Input parameters of engine, layered bigend bearing and the properties of the motor oil ISO VG 100 used in this investigation are reported in Tables 2–4.
To analyze bigend bearing, which has a 360° circumferential oil supply groove of 0.0127 m width machined at its midsection, we model it by treating each half of the bearing land as a single bearing and assuming that the magnitude of the supply pressure was negligible, i.e. p_{s}_{ }= 0. By exploiting symmetry, it was only necessary to analyze one half of the bearing land of 0.057 m width subjected to half loading. So, the load data given in Table 1 must be halved.
Fig. 2 Dynamic loading on the Ruston and Hornsby 6 VEBX connectingrod bigend bearing including gas and inertia forces. 
Diesel Ruston and Hornsby 6 VEB engine parameters data.
Input parameters (geometric characteristics and operating conditions) used to study the performance characteristics of layered connectingrod bearing.
Rheological and physical properties of the motor oil ISO VG 100 under study.
3.2 Validation
In order to verify the proposed method of solution, the specific case chosen is also the connectingrod bigend bearing of the Ruston and Hornsby engine with and without a full circumferential groove.
In Figure 3, we compare the results obtained in the frame related to the connectingrod (X_{3},Y_{3}) by the current method of solution based on the damped Newton–Raphson algorithm and the subrelaxed successive substitutions method with those calculated by a separate computer program using the mobility method of Booker described in details in reference [16]. The calculations were performed in isothermal regime for both grooved and ungrooved bearings with stiff liner using a Newtonian fluid of dynamic viscosity µ_{0} = 15 mPa · s as lubricant.
Table 5 gives another check for the correctness of the algorithm and the computer program by comparing the predicted values of the minimum oil film thickness and maximum film pressure with some results from the literature for both grooved (full circumferential groove) and ungrooved connectingrod bearings [31–35].
Good agreement is observed and we may conclude that the method of solution used in the present analysis is validated for a rigid bearing lubricated with isoviscous Newtonian fluids and operating under isothermal conditions. Note that the discrepancies between the results may be caused partly by the effect of supply pressure. Indeed, our calculations were performed for the grooved bearing configuration by setting the supply pressure in the feeding groove at zero instead of p_{s} = 0.294 MPa as stated in reference [35].
Fig. 3 Comparison of predicted crank pin center cyclic path for both grooved and ungrooved Ruston and Hornsby 6 VEB connectingrod bearings determined after two load cycles. 
Comparison of numerical results obtained in the present analysis with those provided by literature.
3.3 Parametric study
Three different coated bearing configurations of the ungrooved Hornsby & Ruston connectingrod bigend bearing are used for the isothermal analysis in order to put in evidence the effects of the piezoviscous and the presence of couplestresses in the lube oil on the dynamic responses of these bearings. In this analysis, the dynamic responses of a traditional bearing with a stiff liner of 2 mm thickness are calculated and compared with those obtained for bearings with compliant liners made from white metal (Babbitt) and polymer materials. Properties of the oil sample and elastic characteristics of the bearing liner materials are reported in Tables 3 and 4.
3.4 Couplestress effects
In Figure 4, three stationary journal (crankpin) orbits are plotted in polar diagrams corresponding to the three following configurations: a traditional bearing with stiff liner, which is considered as rigid in solid line, and two bearings with different liner materials (white metal, which is a compressible material, and polymer (PEHD), which is an incompressible material) in dash and dashdot lines, respectively. The clearance circle e/C = 1 is also plotted with dash line. For each bearing, the lubricant is considered as a piezoviscous fluid (α = 20GPa^{−1}), and the load cycle is repeated several times until two successive load cycles give identical shaft or journal orbits.
It is observed that the crankpin center orbits for both Newtonian and nonNewtonian lubricants have similar shapes even though the bearing is compliant. But the couplestresses produce more contracted trajectories of the shaft center, which result in higher minimum oil film thicknesses as depicted in Figures 5 and 6.
Although the minimum film thickness is sensitively increased by the presence of couplestresses, the maximum film pressure is not really affected for both stiff and compliant bearings as shown in Figures 7a and 7b. It is well known that taking into account couplestress effects leads to a pressure distribution increase.
For the compliant bearing with a Babbitt liner, the average values of minimum film thickness and maximum film pressure obtained over the cycle are, respectively, about 9.82 μm and 22.32 MPa for the Newtonian oil and 27.61 μm and 20.39 MPa for the oil blended with polymers (couplestress fluid model). Hence, the couplestresses result in an important increase of the minimum film thickness by about 180% and a very slight drop in peak film pressure by about 9%.
Likewise, for the compliant bearing with a liner made of polymer (PEHD), which is a more compliant material characterized by a lower elasticity modulus (E = 0.9 GPa), the couplestresses lead to a much more important increase of the minimum film thickness by about 317% and a much slighter drop in peak film pressure by about 6% even though the minimum film thickness and the peak film pressure are much lower than those obtained in the previous bearing case, i.e. the compliant bearing with a liner made of white metal (Babbitt).
Table 6 summarizes the results obtained for the three different coated bearing configurations.
Figure 8 depicts the variation of side leakage flow and power loss versus crank angle for both Newtonian (l = 0) and nonNewtonian oils (l/C = 0.3). It is shown that for the same applied dynamic load, with Newtonian fluid we obtain higher side leakage flow and power loss when compared to couplestress fluid.
For the compliant bearing with a Babbitt liner (Tab. 6), the mean values of side leakage flow and power loss for the Newtonian fluid are higher than those for the couplestressfluid by about 40 and 36%, respectively. Note that the peaks of side leakage flow and power loss occur at θ_{2} = 10^{°} ATDC.
Fig. 4 Stationary crankpin center orbits for a complete engine cycle giving two crankshaft rotations. 
Fig. 5 Global minimum oil film thickness during the complete load cycle. 
Fig. 6 Minimum oil film thickness calculated at the bearing centerline during the complete load cycle. 
Fig. 7 Maximum oil film pressure during the complete load cycle. 
Comparison of hydrodynamic characteristics for isothermal analysis.
Fig. 8 Side leakage flow and power loss during the complete load cycle. 
3.5 Effects of bearing liner elasticity
Stationary journal orbits of coated and uncoated (stiff) bearings calculated for both Newtonian and couplestress lubricants are compared in Figure 4. As can be seen, the journal orbit of the bearing coated with the liner in white metal (Babbitt) is almost identical to that of the stiff bearing. Unlike, the journal orbit of the bearing coated with PEHD liner is significantly affected by the local deformation of the bearing liner compared to the journal orbit of the bearing coated with liner in white metal characterized by an elasticity modulus greater than that of the PEHD.
From Figures 5 to 7, which describe the history of the minimum oil film thickness and the peak oil film hydrodynamic pressure predicted for both Newtonian and couplestress fluids, it can be seen that there are significant differences throughout the cycle in the minimum film thickness and in the peak film pressure between the most compliant bearing, i.e. the bearing with a liner in PEHD and the bearing with a liner in white metal.
As indicated in Table 6, the least minimum oil film thickness in one engine cycle obtained for the piezoviscous Newtonian case is 9.825 μm for the compliant bearing with a liner in white metal and 5.88 μm for the compliant bearing with a liner in PEHD (about 40% variation), and the largest peak oil film hydrodynamic pressure is about 22 MPa for the compliant bearing with a liner in white metal (Babbitt) and 16 MPa for the compliant bearing with a liner in PEHD (about 27% variation). However, the least minimum oil film thickness in one engine cycle calculated for the piezoviscous fluid with couplestresses is 27.61 μm for the compliant bearing with a liner in white metal and 24.52 μm for the compliant bearing with a liner in PEHD (i.e. about 11% variation, which is quite very small than that of the Newtonian case), and the largest peak oil film hydrodynamic pressure is about 20 MPa for the compliant bearing with a liner in white metal and 15 MPa for the compliant bearing with a liner in PEHD (i.e. about 25% variation), which is almost identical to that of the Newtonian case.
From the same figures, it can also be seen that the global least minimum oil film thicknesses calculated for the bearing with a liner in white metal (i.e. the second bearing configuration) and the bearing with a liner in PEHD (third bearing configuration) occurs at different values of the crankshaft rotation angle due to the elasticity effect of the bearing liner, whereas the peak oil film hydrodynamic pressure always is present at the same crank angle, i.e. in the vicinity of θ_{2} = 10° ATDC for which the peak load appears as depicted in Figure 2. Note that the global minimum film thickness arises at the layered bearing edges and it is always lower than that calculated in the midsection of the compliant bearing.
Figure 8 also compares the variations of side leakage flow and power loss with crank angle for the three bearings configurations (stiff and compliant bearings). The leakage flow as well as the power loss show little difference over the cycle for both Newtonian and nonNewtonian cases. Note that the power loss is calculated considering only the area with a full film. It can be concluded that there is no significant effect of the bearing compliance on the side leakage flow and the power loss. These results agree qualitatively with those obtained by Mc Ivor and Fenner [36] when the global deformations of the whole structure of the Ruston and Hornsby connectingrod bigend bearing are considered in their finite element analysis.
Due to the bearing liner deformation, the gap between the deformed bearing surface and the shaft is geometrically different than that between the rigid bearing surface and the shaft. Accordingly, the hydrodynamic pressure profile is also different for the two bearing configurations. Figure 9 shows the hydrodynamic pressure profiles along the centerline calculated for the bearing with a liner in white metal and the bearing with a liner in PEHD for some crank rotation angles. As expected, one peak pressure occurs in the circumferential direction of the bearings since only the local deformations are considered. Besides, the maximum hydrodynamic pressure for the most compliant bearing (bearing with PEHD liner) is lower than that obtained in the case of the bearing with a liner in white metal having a greater elasticity.
Fig. 9 Hydrodynamic pressure profiles in the midsection of the bearing for some crank angles . 
4 Concluding comments
The combined effects of couplestresses and piezoviscosity in dynamically loaded connectingrod bigend bearings with thin elastic liners under isothermal conditions have been undertaken. The motor oil used for lubricating such bearings was modeled as a nonNewtonian couplestress fluid in order to take into account the couplestresses in addition to the surface forces due to the presence of various polymer additives. We showed that this fluid model is characterized by a skew symmetric stress tensor which comprises two physical properties µ and η denoting the classical dynamic viscosity and the additional coefficient that specifies the couplestress character of the fluid, respectively. To these two coefficients, we have added a third one denoted α to take into account the piezoviscosity effect. As the piezoviscosity α is independent of temperature in the present work, the combined effects of couplestresses and piezoviscosity on the dynamic response of three layered connectingrod bearing configurations were only investigated using the isothermal assumption. Moreover, an improved and relaxed iterative Newton–Raphson method has been proposed for solving the equilibrium equations in order to determine the trajectories of the crankpin center within layered bearings.
The conclusions are as follows:

The pressure equation derived in this paper is more general than the classical Reynolds equation for the study of dynamically loaded bearings with elastic layers using piezoviscous fluids with couplestress as lubricants.

With the same applied dynamic loads, the couplestress fluids yield higher least minimum oil film thickness, and more contracted orbits than Newtonian fluids.

The elastic deformations effects result in an expansion of the orbit and a decrease of the maximum hydrodynamic pressure especially for layered bearings with lowelasticity modulus coatings (e.g. bearing with PEHD liner).

For the cases investigated, the couplestress effects are more significant than the piezoviscous effects.
Nomenclature
C: Bearing radial clearance, m
E: Young's modulus of the bearingliner, Pa
F_{X}, F_{Y}: Applied load components, N
F: Dynamic load applied on the bigend bearing, , N
e_{X}, e_{Y}: Displacement components of the shaft (crankpin) center, m
l: Characteristics length of polymer additives,=, m
N: Rotational speed of engine (crankshaft), rpm
p_{max}: Instantaneous maximum film pressure, Pa
p_{cav}: Cavitation pressure, Pa
Q_{z}: Side leakage flow, m^{3}/s
T: Temperature field in the fluid, K
t_{l}: Thickness of bearingliner, m
k: Thermal conductivity of the fluid, W/m · K
ℓ_{2}: Crankshaftarm length, m
ℓ_{3}: Connectingrod length, m
(O_{b}, X_{3}, Y_{3}) or (O_{b}, X_{3}, Y_{3}): Mobile frame related to the connectingrod
z: Axial coordinate measured from middle section plane of the bearing, m
v_{i}: Cartesian components of the velocity of fluid, m/s
α: Pressure–viscosity coefficient, Pa^{−1}
η: Material constant responsible for couplestresses, N · s
μ: Absolute viscosity of lubricant, Pa · s
ν: Kinematic viscosity of lubricant, m^{2}/s
σ: Poisson's ratio of the bearingliner, −
C_{p}: Lubricant specific heat, J/kg · K
θ_{3}: Angle between X_{1}axis related to engine block and connectingrod, rad
: Instantaneous angular position of the start and the end of the pressure curve, respectively, rad
ω_{2}: Angular velocity of the crankshaft, , rad/s
ω_{s}: Angular velocity of shaft (crankpin), rad/s
ω_{b}: Angular velocity of bearing, rad/s
: Average angular velocity of shaft and bearing, , rad/s
ψ: Angle between the direction of applied load F and Xdirection, rad
ρ: Mass density of lubricant, kg/m^{3}
: Line vector where the superscript T means the transpose
Abbreviations
IEHD: : Isothermal elastohydrodynamic
ISO: : International Organization for Standardization
PEHD: : Polyethylene high density
VI: : Viscosity index of lubricating oil
nsteps: : Number of time steps in a load cycle
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Cite this article as: B. Laouadi, M. Lahmar, B. Bousaïd, H. Boucherit, A. Mouassa, Analysis of couplestresses and piezoviscous effects in a layered connectingrod bearing, Mechanics & Industry 19, 607 (2018)
All Tables
Input parameters (geometric characteristics and operating conditions) used to study the performance characteristics of layered connectingrod bearing.
Comparison of numerical results obtained in the present analysis with those provided by literature.
All Figures
Fig. 1 Bigend connectingrod bearing and coordinates systems. 

In the text 
Fig. 2 Dynamic loading on the Ruston and Hornsby 6 VEBX connectingrod bigend bearing including gas and inertia forces. 

In the text 
Fig. 3 Comparison of predicted crank pin center cyclic path for both grooved and ungrooved Ruston and Hornsby 6 VEB connectingrod bearings determined after two load cycles. 

In the text 
Fig. 4 Stationary crankpin center orbits for a complete engine cycle giving two crankshaft rotations. 

In the text 
Fig. 5 Global minimum oil film thickness during the complete load cycle. 

In the text 
Fig. 6 Minimum oil film thickness calculated at the bearing centerline during the complete load cycle. 

In the text 
Fig. 7 Maximum oil film pressure during the complete load cycle. 

In the text 
Fig. 8 Side leakage flow and power loss during the complete load cycle. 

In the text 
Fig. 9 Hydrodynamic pressure profiles in the midsection of the bearing for some crank angles . 

In the text 
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