[Portrait] J.R.Barber
Jon.R. and Beverly S.Holt Professor of Engineering
Arthur F.Thurnau Professor of Mechanical Engineering and Applied Mechanics

E-mail:jbarber@umich.edu

Address:
J.R.Barber,
Department of Mechanical Engineering
University of Michigan
2350 Hayward Street
Ann Arbor MI  48109-2125

Campus Address:
3440 GG Brown / 2125

Office Phone: (734) 936-0406

FAX: (734) 615-6647

Brief Biography

I graduated in Mechanical Sciences from the University of Cambridge in 1963 and joined British Rail, who later sponsored my research at Cambridge between 1965 and 1968 on the subject of thermal effects in braking systems. In 1969 I became a Lecturer and later Reader in Solid Mechanics at the University of Newcastle upon Tyne, U.K. I moved to the University of Michigan, Department of Mechanical Engineering in 1981. My current research interests are in solid mechanics with particular reference to contact mechanics, tribology, elasticity and thermoelasticity. I am a Chartered Engineer in the U.K., a Fellow of the Institution of Mechanical Engineers and have engaged extensively in consulting work in the field of stress analysis for engineering design. I am author of three books and numerous articles in the fields of Elasticity, Thermoelasticity, Contact Mechanics, Tribology, Heat Conduction and Elastodynamics. I am a member of the editorial boards of the International Journal of Mechanical Sciences and the Journal of Thermal Stresses, Acta Tribologica and co-editor of the Springer book series Solid Mechanics and its Applications.

Research

My research focuses mostly on those aspects of solid mechanics pertaining to the contact of deformable bodies. Some specific topics are:-

(i) The Coulomb friction law

The Coulomb friction law is widely used in engineering analysis because of its simplicity, but it raises challenging questions of existence, uniqueness and stability of solution that are particularly relevant when the law is implemented in a finite element solution. A critical coefficient of friction can be identified below which the problem is well-posed, but above this, a loaded frictional system may experience sudden changes of state that can only be captured by an elastodynamic formulation [1] [2]. There are obvious implications here for earthquake events. Also, frictional systems can become 'wedged' [3], meaning that they remain in a stressed state after all loads are removed — a state analogous to residual stress in plastically deformed structures. Wedging is an essential requirement for screwed fasteners to function, but can be a failure mechanism in an automated assembly procedure.

We have established tests based on the stiffness matrix to determine when these conditions can occur, and to characterize the nature of the resulting behaviour of the system [4]. Frictional vibrations have long been known to occur in many physical systems, but traditional explanations have depended on the friction coefficient being a function of sliding speed. Our work shows that instabilities (including 'stick-slip' vibrations) can arise with a constant coefficient of friction [2].

(ii) Frictional systems under periodic loading

If an elastic body in frictional contact is subjected to loads below the threshold needed for sliding, some regions of the contact area will generally experience microslip, the amplitude of which is limited by the elastic deformation of the components. Under periodic loading, reversed slip [fretting] in microslip regions damages the surface and can result in fretting fatigue failure. If the loads are applied to a frictional contact and then removed, the system does not generally revert to its original state. The instantaneous state retains some memory of the loading history, so for example, fretting damage, might depend on the order in which a set of bolts were tightened during assembly. We have shown that the condition for this not to occur — i.e. for frictional slip and dissipation to depend only on the periodic loads — is that there be no coupling between shear tractions and relative normal displacements [5]. For more details, click here.

(iii) Asymptotic arguments

A notable achievement in solid mechanics in the twentieth century was the realization that brittle fracture could be characterized by stress-intensity factors, and hence predicted based on the measured fracture toughness of the material. This depends on a generalization of the 'small-scale plasticity' argument — i.e. that whatever complex [and certainly not linear-elastic] processes occur at a crack tip leading to fracture, their magnitude must be completely determined by the properties of surrounding elastic field, and this is adequately described by a classical square-root singular asymptotic elastic field provided the non-linear region is small compared with the salient dimensions of the body. Less generally realized is the fact that this argument can be applied to any situation where a small 'process zone' is embedded in a larger body. For example, we used this method to determine the size of the crack-tip contact zone in fairly general interface crack problems [6].

Suppose two contacting bodies are pressed together under a mean load and welded together over the corresponding contact area. If the resulting monolith is now sbjected to periodic loading, resulting in periodic stress-intensity factors at the edge of the welded region, the severity of any fretting events in the original contact problem must be capable of characterization using these factors. This approach has proved effective in predicting the stress levels for infinite fretting life [7].

(iv) Contact of rough surfaces

All engineering surfaces are rough on the microscale and surface roughness has important consequences for many physical processes including for example the mechanism of friction and the electrical contact resistance between contacting surfaces [8]. Surface roughness often exhibits quasi-fractal properties over many length scales and the resulting multiscale contact problem is computationally challenging [9]. We have recently developed a very efficient algorithm for the prediction of the effective stiffness of rough surface contact, including the effect of roughness on the pull-off force associated with van der Waals adhesion forces [10]. For more details, click here.

(v) Thermoelastic contact

clutch plate after a single engagement Contacting bodies can experience non-uniform temperatures as a result of frictional heat generation at the interface or heat flow across it. In such cases, thermoelastic deformation of the contacting bodies modifies the contact pressure distribution and can lead to a rich variety of phenomena including localization and dynamic instabilities. These effects are of considerable technological importance, including, for example, non-uniform contact pressure, high local temperatures and vibrations in clutches and braking systems: a phenomenon known as Frictionally-excited Thermoelastic Instability (TEI) [11], [12]. The figure on the left shows a transmission clutch plate after a single engagement. The dark areas correspond to regions in which high local temperatures have been experienced. sinusoidal perturbations in continuous casting

The mathematical aspects of thermoelastic contact problems are of considerable interest and are challenging. Contact mechanics is conventionally defined by the Signorini inequalities precluding tensile contact tractions and interpenetration of material, but combination of these boundary conditions with simple thermal conditions leads to an ill-posed, coupled thermoelastic problem which exhibits counter examples to both existence and uniqueness of the steady state [13]. Existence problems can be resolved by using more sophisticated boundary conditions - for example, recognizing that the inevitable roughness of the surfaces will impose a thermal contact resistance that depends upon contact pressure. Interaction between thermoelastic deformation and a pressure dependent thermal contact resistance can be unstable, leading to non-uniform contact pressure [14]. The figure on the right shows a section cut from an interrupted continuous casting process. The sinusoidal perturbation in the solidification boundary was caused by thermoelastic instability associated with the mould/casting contact interface.

References

[1]. Hanbum Cho and J.R.Barber, Dynamic behavior and stability of simple frictional systems, Mathematical and Computer Modeling, Vol. 28 (1998), pp. 37-53.

[2] Hanbum Cho and J. R. Barber, Stability of the three-dimensional Coulomb friction law, Proceedings of the Royal Society (London), Vol. A 455 (1999), pp. 839-861.

[3] S. Kim, Y. H. Jang and J. R. Barber, Wedging of frictional elastic systems, Facta Universitatis, Mechanical Engineering, Vol. 17 (2019), pp. 141--148. doi: 10.22190/FUME190131019K

[4] Young Ju Ahn, Enrico Bertocchi, J.R.Barber, Shakedown of coupled two-dimensional discrete frictional systems, Journal of the Mechanics and Physics of Solids, Vol.56 (2008), pp.3433--3440.

[5] L-E. Andersson, J. R. Barber and A. R. S. Ponter, Existence and uniqueness of attractors in frictional systems with uncoupled tangential displacements and normal tractions, International Journal of Solids and Structures, Vol. 51 (2014), pp. 3710--3714.doi: 10.1016/j.ijsolstr.2014.07.004

[6] D.A.Hills and J.R.Barber, Interface cracks, International Journal of Mechanical Sciences, Vol. 35 (1993), pp. 27-37.

[7] D.A.Hills, A.Thaitirarot, J.R.Barber, and D.Dini, Correlation of fretting fatigue experimental results using an asymptotic approach, International Journal of Fatigue, Vol. 43 (2012) pp. 62–75. doi: 10.1016/j.ijfatigue.2012.02.006

[8] J.R.Barber, Bounds on the electrical resistance between contacting elastic rough bodies, Proceedings of the Royal Society (London), Vol. A 459 (2003), pp. 53-66.

[9] M.Ciavarella, G.Demelio, J.R.Barber and Yong Hoon Jang, Linear elastic contact of the Weierstrass profile, Proceedings of the Royal Society (London), Vol. A 456 (2000), pp. 387-405.

[10] J. Joe, A. Wang and J. R. Barber, Load-displacement relation and gap distribution between rough surfaces: partial differential equations approach, Journal of the Mechanics and Physics of Solids, Vol. 180, (2023), 105397, doi: 10.1016/j.jmps.2023.105397

[11] Yun-Bo Yi, J.R.Barber and P.Zagrodzki, Eigenvalue Solution of Thermoelastic Instability Problems using Fourier Reduction, Proceedings of the Royal Society (London), Vol. A 456 (2000), pp. 2799-2821.

[12] Yun-Bo Yi, J.R.Barber and D.L.Hartsock, Thermoelastic instabilities in automotive disc Brakes - Finite element analysis and experimental verification, Proc. 3rd Contact Mechanics International Symposium, Peniche, Portugal, June 2001.

[13] J.R. Barber, J. Dundurs and M. Comninou, Stability considerations in thermoelastic contact, ASME Journal of Applied Mechanics, Vol. 47 (1980), pp. 871-874.

[14] Ronggang Zhang and J.R.Barber, Effect of material properties on the stability of static thermoelastic contact, ASME Journal of Applied Mechanics, Vol. 57 (1990), pp. 365-369.

Solution of Elasticity Problems

I have developed Maple and Mathematica files for the solution of boundary-value problems in Elasticity in conjunction with my book `Elasticity'. To explore this resource, start by clicking on either Programming in Maple or Programming in Mathematica and then on to `Catalogue of Maple files' or `Catalogue of Mathematica files'. If you have never used these methods to solve problems, you will surprised how effective they are. You will however need to have Mathematica or Maple installed on your computer system.

Other selected publications

Click here for a full list of publications.

Ph.D. Graduates