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**Hemivariational inequalities. Applications in mechanics and engineering.**
*(English)*
Zbl 0826.73002

Berlin: Springer-Verlag. xvi, 451 p. (1993).

The scope of the book is the study of problems in mechanics and engineering science whose variational formulations are hemivariational inequalities. These variational forms involve nonconvex energy functions and express the principle of of virtual works in its inequality form. In most cases the nonconvex energy functions are nonsmooth and, therefore, the methods of nonsmooth analysis are employed for the mathematical study and the numerical treatment of the hemivariational inequalities.

The book is divided into four parts: The “Introductory topics” (chapter 1), the “Mechanical theory” (chapters 2 to 5), the “Mathematical theory” (chapters 6 to 8), and the “Numerical applications” (chapters 9 to 15). In Part I we give the necessary mathematical background concerning convexity and subdifferential, generalized gradient and duality, elements of the theory of fans and quasidifferentiability. Part II deals with the mechanical aspects of the theory of hemivariational inequalities. In this part we define the notions of convex and nonconvex superpotentials and, by means of these notions, we introduce boundary conditions and material laws expressed through convex and nonconvex superpotentials.

Part III deals with the mathematical theory of hemivariational and variational-hemivariational inequalities, as well as with their exact relation to substationarity problems. Moreover, the eigenvalue problem for hemivariational inequalities is studied along with dynamic hemivariational inequalities arising in the theory of von Kármán laminated plates and thermoelasticity. The mathematical part of the book concludes with the formulation and study of the optimal control problem of systems governed by hemivariational inequalities.

Part IV is devoted to numerical applications and takes the largest part of the book. We present there numerical applications related to real engineering problems. The chapters of this part of the book are fairly independent from the rest of the book, since they describe numerical techniques and point to concrete engineering applications. Finally, in the last chapter we study hemivariational inequalities defined on fractal geometries, and we attempt to adapt the numerical techniques for hemivariational inequalities to a neurocomputing environment.

The book is divided into four parts: The “Introductory topics” (chapter 1), the “Mechanical theory” (chapters 2 to 5), the “Mathematical theory” (chapters 6 to 8), and the “Numerical applications” (chapters 9 to 15). In Part I we give the necessary mathematical background concerning convexity and subdifferential, generalized gradient and duality, elements of the theory of fans and quasidifferentiability. Part II deals with the mechanical aspects of the theory of hemivariational inequalities. In this part we define the notions of convex and nonconvex superpotentials and, by means of these notions, we introduce boundary conditions and material laws expressed through convex and nonconvex superpotentials.

Part III deals with the mathematical theory of hemivariational and variational-hemivariational inequalities, as well as with their exact relation to substationarity problems. Moreover, the eigenvalue problem for hemivariational inequalities is studied along with dynamic hemivariational inequalities arising in the theory of von Kármán laminated plates and thermoelasticity. The mathematical part of the book concludes with the formulation and study of the optimal control problem of systems governed by hemivariational inequalities.

Part IV is devoted to numerical applications and takes the largest part of the book. We present there numerical applications related to real engineering problems. The chapters of this part of the book are fairly independent from the rest of the book, since they describe numerical techniques and point to concrete engineering applications. Finally, in the last chapter we study hemivariational inequalities defined on fractal geometries, and we attempt to adapt the numerical techniques for hemivariational inequalities to a neurocomputing environment.

### MSC:

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74P10 | Optimization of other properties in solid mechanics |

49J40 | Variational inequalities |