H-半变分不等式及其在接触力学中的应用
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摘要
本文主要研究一类双曲型H-半变分不等式及其在粘弹性接触力学中的应用。
第二章主要研究如下H-半变分不等式：这是一类双曲型H-半变分不等式,通过将它嵌入到一类二阶发展包含问题中,再利用多值算子的满射性结果证明了该H-半变分不等式的解的存在性。
第三章研究了粘弹性压电体与基座间的动力学摩擦接触问题。其接触条件由通常的法向阻尼响应条件与摩擦定律来描述,接触条件是非单调的、有可能是多值的且具有Clarke次微分形式,因而导出的数学模型由关于位移的双曲型H-半变分不等式与关于电势的与时间有关的椭圆方程所组成。我们给出了该接触问题的解的存在性结果。
第四章研究了变形体与基座间的静力学摩擦接触问题。我们假定变形体是粘弹性的,且具有长效记忆特性。接触面的接触条件由法向柔度条件来描述。法向应力与法向位移间具有Clarke次微分形式的非单调关系,而接触摩擦假定切向剪应力是切向位移的非单调多值函数。同时我们也考虑了材料的损伤。材料的损伤会降低物体的承载能力,从而使机械系统的功能与安全性受到很大影响。我们导出了该问题的变分公式,它由一个抛物方程与一个H-半变分不等式所构成。我们陈述并证明了其存在性与唯一性结果。
The dissertation investigates a type of hyperbolic hemivariational inequality and its applications to viscoelastic contact mechanics.
In chapter two, we investigate the following hemivariational inequality: This is a type of hyperbolic hemivariational inequality. We prove the existence result by embedding the problem into a class of second-order evolution inclusion and by applying a surjectivity result for multivalued operators.
In chapter three, we investigate the dynamic frictional contact problem between a viscoelastic-piezoelectric body and a foundation. The contact is modeled by a general normal damped response condition and a friction law, which are nonmonotone, possibly multivalued and have the subdifferential form. The model consists of a system of the hemivariational inequality of hyperbolic type for the displacement, the time dependent elliptic equation for the electric potential. We state the existence result on the contact problem.
In chapter four, we investigate the quasistatic process of frictional contact between a deformable body and a foundation. The body is assumed to be viscoelastic with long memory. The contact is modelled with a general normal compliance condition. The dependence of the the normal stress on the normal displacement is assumed to have nonmonotone character of the subdifferential form. We model the frictional contact assuming that the tangential shear on the contact surface is given as a nonmonotone and possibly multivalued function of the tangential displacement. We also consider the damage of the material. The effect due to the damage leads to decrease the carrying capacity of the body. The effective functioning and safety of a mechanical system may be deteriorated by this decrease as the material undergoes damaged. We derive the variational formulation of the problem, which is in the form of a system coupling a parabolic equation and a hemivariational inequality. We state and prove our main existence and uniqueness result.
第二章主要研究如下H-半变分不等式：这是一类双曲型H-半变分不等式,通过将它嵌入到一类二阶发展包含问题中,再利用多值算子的满射性结果证明了该H-半变分不等式的解的存在性。
第三章研究了粘弹性压电体与基座间的动力学摩擦接触问题。其接触条件由通常的法向阻尼响应条件与摩擦定律来描述,接触条件是非单调的、有可能是多值的且具有Clarke次微分形式,因而导出的数学模型由关于位移的双曲型H-半变分不等式与关于电势的与时间有关的椭圆方程所组成。我们给出了该接触问题的解的存在性结果。
第四章研究了变形体与基座间的静力学摩擦接触问题。我们假定变形体是粘弹性的,且具有长效记忆特性。接触面的接触条件由法向柔度条件来描述。法向应力与法向位移间具有Clarke次微分形式的非单调关系,而接触摩擦假定切向剪应力是切向位移的非单调多值函数。同时我们也考虑了材料的损伤。材料的损伤会降低物体的承载能力,从而使机械系统的功能与安全性受到很大影响。我们导出了该问题的变分公式,它由一个抛物方程与一个H-半变分不等式所构成。我们陈述并证明了其存在性与唯一性结果。
The dissertation investigates a type of hyperbolic hemivariational inequality and its applications to viscoelastic contact mechanics.
In chapter two, we investigate the following hemivariational inequality: This is a type of hyperbolic hemivariational inequality. We prove the existence result by embedding the problem into a class of second-order evolution inclusion and by applying a surjectivity result for multivalued operators.
In chapter three, we investigate the dynamic frictional contact problem between a viscoelastic-piezoelectric body and a foundation. The contact is modeled by a general normal damped response condition and a friction law, which are nonmonotone, possibly multivalued and have the subdifferential form. The model consists of a system of the hemivariational inequality of hyperbolic type for the displacement, the time dependent elliptic equation for the electric potential. We state the existence result on the contact problem.
In chapter four, we investigate the quasistatic process of frictional contact between a deformable body and a foundation. The body is assumed to be viscoelastic with long memory. The contact is modelled with a general normal compliance condition. The dependence of the the normal stress on the normal displacement is assumed to have nonmonotone character of the subdifferential form. We model the frictional contact assuming that the tangential shear on the contact surface is given as a nonmonotone and possibly multivalued function of the tangential displacement. We also consider the damage of the material. The effect due to the damage leads to decrease the carrying capacity of the body. The effective functioning and safety of a mechanical system may be deteriorated by this decrease as the material undergoes damaged. We derive the variational formulation of the problem, which is in the form of a system coupling a parabolic equation and a hemivariational inequality. We state and prove our main existence and uniqueness result.
引文
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[46]S. Migorski. Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction. Appl. Anal.,2005,84:669-699.
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[2]I.R. Ionescu, Q.L. Nguyen, S. Wolf. Slip-dependent friction in dynamic elasticity. Nonlinear Anal.,2003,53:375-390.
[3]I.R. Ionescu, J.-C. Paumier. On the contact problem with slip displacement dependent friction in elastostatics. Int. J. Eng. Sci.,1996,34(4):471-491.
[4]K.L. Kuttler. Dynamic friction contact problem with general normal and friction laws. Nonlinear Anal.,1997,28:559-575.
[5]M. Sofonea, El-H. Essoufi. A piezoelectric contact problem with slip dependent coefficient of friction. Math. Model. Anal.,2004,9:229-242.
[6]B. Awbi, Essoufi, El-H. and M. Sofonea. A viscoelastic contact problem with normal damped response and friction. Annales Polonici Mathematici,2000,75:233-246.
[7]M. Sofonea, and M. Shillor. A quasistatic viscoelastic contact problem with friction. Comm.Appl. Anal.,2001,5:135-151.
[8]W. Han and M. Sofonea. Quasistatic Contact Problems in Viscoelasticity and Visco-plasticity. AMS, International Press, Providence, Rhode Island,2002.
[9]O. Chau, W. Han, and M. Sofonea. A dynamic frictional contact problem with normal damped response. Acta Appl. Math.,2002,71:159-178.
[10]J. L. Lions. Quelques methodes de resolution des problemes aux limites non lineaires. Dunod,1969.
[11]P. D. Panagiotopoulos. Hemivariational Inequalities, Applications in Mechanics and Engineering. Springer, Berlin,1993.
[12]Z. Naniewicz, P.D. Panagiotopoulos. Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, Inc., New York, Basel, Hong Kong,1995.
[13]M. Miettinen. A Parabolic Hemivariational Inequality. Nonlinear Anal. Theory Methods Appl.,1996,26:725-734.
[14]N.S. Papageorgiou. On the existence of solutions for nonlinear parabolic problems with nonmonotone discontinuities. J. Math. Anal. Appl.,1997,205:434-453.
[15]L. Gasinski. Hiperboliczne nierownosci hemiwariacyjne i ich zastosowanie w teorii optymalizacji ksztaltu. Rozprawa doktorska, Krakow,2000.
[16]P.D. Panagiotopoulos, G. Pop. On a type of hyperbolic variational hemivariational inequalities. J. Appl. Anal.,1999,5(1):95-112.
[17]Z.H. Liu. A Class of Evolution Hemivariational Inequalities. Nonlinear Analysis, 1999,36:91-100.
[18]Z.H. Liu. On quasilinear elliptic hemivariational inequalities of higher order. Acta Math. Hungar.,1999,85:1-8.
[19]Z.H. Liu. L. Simon. Existence results for evolution hemivariational inequalities. Adv. Math. (China),2001,30:47-55.
[20]Z.H. Liu. Some existence theorems for evolution hemivariational inequalities. Indian J. Pure Appl. Math.,2003,34:1165-1176.
[21]Z.H. Liu. Elliptic variational hemivariational inequalities. Appl. Math. Lett.,2003, 16:871-876.
[22]Z.H. Liu. Some convergence results for evolution hemivariational inequalities. J. Global Optim.,2004,29:85-95.
[23]Z.H. Liu. Existence results for evolution noncoercive hemivariational inequalities. J. Optim. Theory Appl.,2004,120:417-427.
[24]Z.H. Liu. Generalized quasi-variational hemi-variational inequalities. Appl. Math. Lett.,2004,17:741-745.
[25]Z.H. Liu. Browder-Tikhonov regularization of noncoercive evolution hemivar-iational inequalities. Inverse Problems,2005,21:13-20.
[26]D.Motreanu, P.D. Panagiotopoulos. A minimax approach to the eigenvalue problem of hemivariational inequalities and applications. Appl. Anal.,1995,58:53-76.
[27]L. Gasinski, N.S. Papageorgiou. Multiple solutions for semilinear hemivariational inequalies at resonance. Publ. Math. Debrecen,2001,59:121-146.
[28]L. Gasinski, N.S. Papageorgiou, Nonlinear hemivariational inequalities at resonance, Bull. Austral. Math. Soc.,60(1999):353-364.
[29]L. Gasinski, N.S. Papageorgiou. Existence of solutions and of multiple solutions for eigenvalue problem in hemivariational inequalities. Adv. Math. Sci. Appl.,2001, 11:437-464.
[30]L. Gasinski, N.S. Papageorgiou. Nonsmooth critical point theory and nonlinear boundary value problems. Boca Raton:Chapman and Hall, CRC Press,2005.
[31]S.T. Kyritsi, N.S. Papageorgiou. Pairs of Positive solutions for nonlinear elliptic equations with the p-Laplacian and a nonsmooth Pothtial. Ann. Mat. Pura Appl.,2005, 184:449-472.
[32]S.T. Kyritsi, N.S. Papageorgiou. Nonsmooth critical point theory on closed convex sets and nonlinear hemivariational inequalities. Nonlinear Anal.,2005,61:373-403.
[33]S.T. Kyritsi, N.S. Papageorgiou. Multiple solutions of constant sign for nonlinear Nonsmooth eigenvalue problems near resonance. Calc. Var. Partial Differential Equations, 2004,20:1-24.
[34]S. Hu, N.S. Papageorgiou. Handbook of Mutivalued Analysis Volume II:Applications. Dordrecht:Kluwer Academic Publishers,2000.
[35]S. Hu, N.S. Papageorgiou. Handbook of Mutivalued Analysis Volume I:Theory, Dordrecht:Kluwer Academic Publishers,1997.
[36]S. Migorski, A. Ochal. Inverse coefficient problem for elliptic hemivariational inequality. Nonsmooth Mechanics, London,2000,247-262.
[37]A. Ochal. Domain identification problem for elliptic hemivariational inequality. Topological Methods in Nonlinear Analysis,2000,16:237-249.
[38]Z.Naniewicz and P.D.Panagiotopoulos. The Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, NewYork,1994.
[39]P.D. Panagiotopoulos. Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhauser, Verlag, Basel, Boston,1985.
[40]M. Miettinen, and P.D. Panagiotopoulos. On parabolic hemivariational inequalities and applications. Nonlinear Analysis,1999,35:885-915.
[41]D.Goeleven, M.Miettinen, and P.D. Panagiotopoulos. Dynamic hemivariational Inequalities and their applications. J.Optimiz.Theory and Appl.,1999,103:567-601.
[42]D. Goeleven, D. Motreanu, Y. Dumont, and M. Rochdi. Variational and Hemivariational Inequalities:Theory, Methods and Applications.vol. I and II. Kluwer Academic Publishers, Boston, Dordrecht, London,2003.
[43]S. Migorski, A. Ochal. Hemivariational inequality for viscoelastic contact problem with slip dependent friction. Nonlinear Anal.,2005,61:135-161.
[44]S. Migorski. Evolution hemivariational inequality for a class of dynamic viscoelastic nonmonotone frictional contact problems. Comp. Math. Appl.,2006,52:677-698.
[45]S. Migorski. Boundary hemivariational inequalities of hyperbolic type and applications. J. Global Optim.,2005,31:505-533.
[46]S. Migorski. Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction. Appl. Anal.,2005,84:669-699.
[47]S. Migorski, A. Ochal. Dynamic bilateral contact problem for viscoelastic piezoelectric materials with adhesion. Nonlinear Analysis,2008,69:495-509.
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