ICCCM 2017 5th International Conference on Computational Contact Mechanics 5-7 July 2017, Lecce - Italy
Identification solutions with contact problems via the solution of the Cauchy Problem Thouraya Nouri Baranger1, Stéphane Andrieux2 1 Université Claude Bernard Lyon 1, LMC2 EA7427, F69622 Villeubanne, France E-mail:
[email protected] 2
ONERA, Palaiseau, France E-mail:
[email protected] Keywords:
Contact mechanics, Cauchy problem, Inverse problems.
This communication is aimed at exploiting displacement data measured at a part of a stress-free boundary of a solid, or couple of solids by digital cameras, for the solution of some identification problems encountered in contact (possibly with fiction) mechanics. Roughly speaking, the general applications are the identification of actual contact boundary conditions on unreachable parts of the boundary of solids and then the identification of involved physical parameters such as friction coefficients. Advances in the development of Digital cameras and Image Correlation techniques (DIC) now make it possible to have measurement means for full field surface displacements that are cheap and relatively easy to manage, and more importantly, leading to very large amounts of information [1]. Nevertheless, the use of these surface data is still largely restricted either to qualitative estimation or to quantitative analysis based on a plane mechanical state or on homogeneous-through-the-thickness assumptions [2]. Clearly, problems involving contact conditions fall out of this set of assumptions and cannot then be addressed without a more sophisticated approach. Then aiming at a true 3D quantitative imaging process, the problem of reconstruction of the fields inside the solid from images obtained on parts of its boundary has to be addressed. One approach in dealing with this problem in mechanics is to first reformulate it within the continuous framework, taking advantage of the fact that the amount and spatial density of the information obtained using the digital image correlation techniques makes it possible to consider that the complete displacement field is available on a part of the boundary and is not reduced on it to pinpoint data only. Then it is possible secondly to formulate it as a Cauchy or Data Completion Problem on a domain Ω, taking into account the fact that an overspecified data pair is given on a part denoted by Γm of its boundary: the capture of displacement fields via the DIC on a stress-free boundary with unit normal n gives access to the Dirichlet and Neumann pair (U, σ.n) on Γm. The Cauchy Problem is an archetypal ill-posed problem ([3], [4]). In previous works ([5],[6],[7]) , the authors proposed a variational method for solving such problems initially for linear operators (linear elasticity) and then for a rather wide large class of nonlinear ones (namely, operator obtained within the convex framework of Standard Generalized Materials [8]). The method decomposes into two steps. First, two auxiliary usual well-posed problems Pi , i = 1,2 are defined, each one using one only of the overspecified boundary data on Γm and a given normal stress vector field η on the remaining part of the boundary of the solid. Thus, being clear that, if η is such that the two solutions of these problems are equal, then the Cauchy problem is solved by the common value of the displacement field because it meets on Γm the Dirichlet and the Neumann condition as well, the second step is to build an adequate gap between the solution of P1 and P2 as a functional of the unknown η and to minimize it. In this communication, the method will be applied firstly to the identification of the contact zones and the friction coefficient for a linearly elastic solid in contact with another one. Here, although the boundary conditions over the possible contact and friction area are nonlinear, the Cauchy Problem is a linear one because the unknown is taken as the actual value of these conditions. Beyond the identification of the displacement and normal stress vector fields over the
contact area, a simple and original algorithm is described for the identification of the Coulomb friction coefficient. The second application is an extension of the method to the nonlinear case where possible contact areas are taken into account between the solids under scrutiny, that is when the contact area in considered as an internal boundary of the domain Ω. Because the Cauchy Problem is now nonlinear, and even if the splitting into the two (nonlinear) problems P1 and P2 is straightforward, a new gap has to be designed. It relies on the notion of Implicit Standard Generalized Materials [9] or more generally on the concept of Bregman divergence that has been recently identified as a general way for building adequate gap functionals [10].
References [1] A. Sutton, J.H. S. Orteu, Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications. Springer, USA (2009) [2] S. Avril, M. Bonnet, A. Bretelle, M. Grédiac, F. Hild, P. Ienny, F. Latourte, D. Lemosses, A. Pagano, E. Pagnacco, F. Pierron, “Overview of identification methods of mechanical parameters based on full-fields measurements” Exp.Mech. 48, 381–402 (2008). [3] J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations. Dover, NewYork (1923) [4] F. Ben Belgacem. “Why is the Cauchy problem severely ill-posed? ” Inverse Problems 23(2), 823–836 (2007) [5] S. Andrieux,S., T.N. Baranger, “An energy error-based method for the solution of the Cauchy problem in 3D linear elasticity” Comput. Methods Appl. Mech. Eng. 197, 902–920 (2008) [6] S. Andrieux, T.N. Baranger, “Solution of non linear Cauchy problem for hyperelastic solids” Inverse Problems, 31(11),115003–115022 (2015) [7] S. Andrieux, T.N. Baranger, “On the determination of missing boundary data for solids with nonlinear material behaviors, using displacement fields measured on a part of their boundaries” J. Mech. Phys. Solids, Volume 97, Pages 140–155 (2016). [8] B. Halphen, Q.S. Nguyen, “Sur les matériaux standards généralisés” J. Mécanique, 14, 39–63 (1975) [9] G. de Saxcé, Z.Q. Feng, “New inequation and functional for contact with friction: the implicit standard material approach”, Mech. Struct. and Mach. 19(3), 301-325 (1991) [10] S. Andrieux, “Inverse problems and experiments: a fruitful symbiosis”, AerospaceLab Journal, 12, (2016)
