The work presented hereafter deals with an important issue arising in frictional contact problems involving flexible bodies: the occurrence of more than one solution, with an emphasis on the quasi-static incremental problem in the presence of rectilinear obstacle and in two dimensions. The conditions for the existence of multiple solutions to the quasi-static incremental problem, with an intrinsic combinatorial character, are presented for several criteria. A simplified criterion is proposed that avoids the exponential character of the problem. An algorithm is proposed for the computation of all the solutions of the incremental problem and to verify the sharpness of the frictional coefficient estimates corresponding to the several criteria. The contributions may be summarized as follows: (i) new simplified (sufficient) criterion for uniqueness of solution based on the solution of an optimization problem, avoiding the exponential character of the (necessary and sufficient) complete criterion of Alart or of the (sufficient) criterion due to Andersson; the proposed criterion assumes that the onset of multiplicity is associated with a mode involving sliding in the whole contact candidate region. (ii) The use of the suggested algorithm to compute all the solutions of the quasi-static incremental problem (for a given loading) in a finite element version of Klarbring's two degree of freedom model. For some lumped mass examples, all the solutions were calculated and their dependencies on some parameters were discussed. The conditions under which a problem may have multiple solutions were also discussed for some lumped models.

下文中提出的工作解决了涉及柔性体的摩擦接触问题中出现的一个重要问题：一个以上解决方案的出现，重点是在存在直线障碍物和二维情况下的准静态增量问题。针对几种标准，提出了具有固有组合特性的拟静态增量问题的多个解的存在条件。提出了一种简化的标准，可以避免问题的指数特征。提出了一种算法，用于计算增量问题的所有解，并验证与多个标准相对应的摩擦系数估计的清晰度。贡献可总结如下：（i）基于优化问题的解的新的简化的（充分）准则，用于解决问题的唯一性，避免了Alart的（必要和充分）完全准则或由于Andersson导致的（充分）准则的指数特征; 提出的标准假设多重性的开始与涉及在整个接触候选区域中滑动的模式相关。（二）在Klarbring的两个自由度模型的有限元版本中，使用建议的算法来计算准静态增量问题的所有解（对于给定的载荷）。对于一些集总质量示例，计算了所有解，并讨论了它们对某些参数的依赖性。对于某些集总模型，还讨论了问题可能具有多种解决方案的条件。