The aim of the present book is the formulation, mathematical study and numerical treatment of static and dynamic problems in mechanics and engineering sciences involving nonconvex and nonsmooth energy functions, or nonmonotone and multivalued stress-strain laws. Such problems lead to a new type of variational forms, the hemivariational inequalities, which also lead to multivalued differential or integral equations. Innovative numerical methods are presented for the treament of realistic engineering problems. This book is the first to deal with variational theory of engineering problems…mehr
The aim of the present book is the formulation, mathematical study and numerical treatment of static and dynamic problems in mechanics and engineering sciences involving nonconvex and nonsmooth energy functions, or nonmonotone and multivalued stress-strain laws. Such problems lead to a new type of variational forms, the hemivariational inequalities, which also lead to multivalued differential or integral equations. Innovative numerical methods are presented for the treament of realistic engineering problems. This book is the first to deal with variational theory of engineering problems involving nonmonotone multivalue realations, their mechanical foundation, their mathematical study (existence and certain approximation results) and the corresponding eigenvalue and optimal control problems. All the numerical applications give innovative answers to as yet unsolved or partially solved engineering problems, e.g. the adhesive contact in cracks, the delamination problem, the sawtooth stress-strain laws in composites, the shear connectors in composite beams, the semirigid connections in steel structures, the adhesive grasping in robotics, etc. The book closes with the consideration of hemivariational inequalities for fractal type geometries and with the neural network approach to the numerical treatment of hemivariational inequalities.
I Introductory Topics.- 1 Elements of Nonsmooth Analysis.- 1.1 Convexity and Sub differential.- 1.2 Generalized Gradient and Related Calculus.- 1.3 Minimization Problems. Duality of Convex Functionals.- 1.4 Miscellanea: Fans, Quasidifferentials, Codifferentials.- II Mechanical Theory.- 2 Nonsmooth Mechanics I.- 2.1 Convex Superpotentials.- 2.2 Nonconvex Superpotentials.- 2.3 Boundary Conditions Expressed via Convex Superpotentials.- 2.4 Boundary Conditions Expressed via Nonconvex Superpotentials.- 2.5 Extensions to Function Spaces.- 3 Nonsmooth Mechanics II.- 3.1 Material Laws Expressed via Convex Superpotentials. An Overview.- 3.2 Material Laws Expressed via Nonconvex Superpotentials I.- 3.3 Material Laws Expressed via Nonconvex Superpotentials II.- 3.4 Loading and Unloading Problems. The Advantage of the Use of Superpotentials.- 3.5 Material Laws and Boundary Conditions Expressed via Fans, Quasi differentials and Codifferentials.- 4 Hemivariational Inequalities.- 4.1 The Derivation of Hemivariational Inequalities in Mechanical Problems.- 4.2 Hemivariational and Variational-Hemivariational Inequalities.- 4.3 Substationarity Problems for the Potential or the Complementary Energy.- 4.4 Loading and Unloading Problems, Eigenvalue Problems for Hemivariational Inequalities and Dynamic Problems.- 4.5 On the F-superpotential and the V-superpotential. Quasidiffer-entiability in Mechanics.- 5 Multivalued Boundary Integral Equations.- 5.1 The Indirect and the Direct Method for Nonmonotone Boundary Conditions.- 5.2 Complement for Adhesively Bonded Cracks.- III Mathematical Theory.- 6 Static Hemivariational Inequalities.- 6.1 Coercive Hemivariational Inequalities.- 6.2 Semicoercive Hemivariational Inequalities.- 6.3 On the Substationarity of the Energy.- 6.4 Variational Hemivariational Inequalities.- 6.5 Applications to Engineering Problems.- 7 Eigenvalue and Dynamic Problems.- 7.1 On the Eigenvalue Problem for Hemivariational Inequalities.- 7.2 Dynamic Hemivariational Inequalities.- 7.3 Applications to Engineering Problems: Von Kármán Plates and Thermoelasticity.- 8 Optimal Control and Identification Problems.- 8.1 Formulation of the Problem.- 8.2 Mathematical Study of the Optimal Control Problem Governed by Hemivariational Inequalities.- 8.3 Applications to Engineering Problems.- IV Numerical Applications.- 9 On the Numerical Treatment of Hemivariational Inequalities.- 9.1 The First Numerical Attempts and the Questions of Stability and Uniqueness.- 9.2 The Microspring Approximation Method of the Decreasing Branch.- 9.3 The Method of Decreasing Branch Approximation by Monotone Laws.- 9.4 Application I: Cleavage in Laminated Composites and the Non-monotone Unilateral Contact Problem.- 9.5 Application II: The Nonmonotone Friction Problem and the Combined Unilateral Contact Problem with Nonmonotone Friction.- 10 On the Approximation of Hemivariational Inequalities by Variational Inequalities.- 10.1 General Formulation of the Method.- 10.2 Application III: Nonmonotone Friction Interface Conditions with Debonding.- 10.3 Application IV:Adhesive Joints in Structural Mechanics.- 10.4 Application V: Comparison with the Path Following Method.- 10.5 Application VI: Nonmonotone Stress-Strain Laws. The Sawtooth Behaviour of Composites.- 10.6 Application VII: Shear Connectors in Composite Beams.- 11 The Method of Substationary Point Search.- 11.1 General Formulation of the Method.- 11.2 On the Numerical Implementation of the Algorithm.- 11.3 Application VIII: Delamination and Adhesive Joints in Structural Mechanics.- 11.4 Application IX: Semirigid Connections in Steel Structures.- 12 On a Decomposition Method into Two Convex Problems.- 12.1 General Formulation of the Method.- 12.2 Application X: The Stamp Problem and the Interfacial Debonding in Composites.- 13 Dynamic Hemivariational Inequalities and Crack Problems.- 13.1 Application XI: Numerical Treatment of Dynamic Hemivariational Inequalities.- 13.2 Application XII: The Unilateral Contact and Nonmonotone Friction Problem in Cracks.- 13.3 Application XIII: Fracture of Cracks Repaired by an Adhesive Material.- 14 Applications of the Theory of Hemivariational Inequalities in Robotics.- 14.1 Application XIV: Adhesive Grasping Problem in Robotics.- 14.2 Application XV: On the Optimal Control of the Adhesive Grasping Problem in Robotics.- 15 Addenda: Hemivariational Inequalities, Fractals and Neural Networks.- 15.1 Fractals in Mechanics. An Introduction.- 15.2 Application XVI: Hemivariational Inequalities for Fractal Interfaces.- 15.3 The Neural Network Approach to Hemivariational Inequalities.- 15.4 Application XVII: D.C.B. Specimen Modelling. The Neural Network Approach.- 15.5 Application XVIII: The Inverse Delamination Problem as a Supervised Learning Problem for a Neural Network. Extensions.- References.
I Introductory Topics.- 1 Elements of Nonsmooth Analysis.- 1.1 Convexity and Sub differential.- 1.2 Generalized Gradient and Related Calculus.- 1.3 Minimization Problems. Duality of Convex Functionals.- 1.4 Miscellanea: Fans, Quasidifferentials, Codifferentials.- II Mechanical Theory.- 2 Nonsmooth Mechanics I.- 2.1 Convex Superpotentials.- 2.2 Nonconvex Superpotentials.- 2.3 Boundary Conditions Expressed via Convex Superpotentials.- 2.4 Boundary Conditions Expressed via Nonconvex Superpotentials.- 2.5 Extensions to Function Spaces.- 3 Nonsmooth Mechanics II.- 3.1 Material Laws Expressed via Convex Superpotentials. An Overview.- 3.2 Material Laws Expressed via Nonconvex Superpotentials I.- 3.3 Material Laws Expressed via Nonconvex Superpotentials II.- 3.4 Loading and Unloading Problems. The Advantage of the Use of Superpotentials.- 3.5 Material Laws and Boundary Conditions Expressed via Fans, Quasi differentials and Codifferentials.- 4 Hemivariational Inequalities.- 4.1 The Derivation of Hemivariational Inequalities in Mechanical Problems.- 4.2 Hemivariational and Variational-Hemivariational Inequalities.- 4.3 Substationarity Problems for the Potential or the Complementary Energy.- 4.4 Loading and Unloading Problems, Eigenvalue Problems for Hemivariational Inequalities and Dynamic Problems.- 4.5 On the F-superpotential and the V-superpotential. Quasidiffer-entiability in Mechanics.- 5 Multivalued Boundary Integral Equations.- 5.1 The Indirect and the Direct Method for Nonmonotone Boundary Conditions.- 5.2 Complement for Adhesively Bonded Cracks.- III Mathematical Theory.- 6 Static Hemivariational Inequalities.- 6.1 Coercive Hemivariational Inequalities.- 6.2 Semicoercive Hemivariational Inequalities.- 6.3 On the Substationarity of the Energy.- 6.4 Variational Hemivariational Inequalities.- 6.5 Applications to Engineering Problems.- 7 Eigenvalue and Dynamic Problems.- 7.1 On the Eigenvalue Problem for Hemivariational Inequalities.- 7.2 Dynamic Hemivariational Inequalities.- 7.3 Applications to Engineering Problems: Von Kármán Plates and Thermoelasticity.- 8 Optimal Control and Identification Problems.- 8.1 Formulation of the Problem.- 8.2 Mathematical Study of the Optimal Control Problem Governed by Hemivariational Inequalities.- 8.3 Applications to Engineering Problems.- IV Numerical Applications.- 9 On the Numerical Treatment of Hemivariational Inequalities.- 9.1 The First Numerical Attempts and the Questions of Stability and Uniqueness.- 9.2 The Microspring Approximation Method of the Decreasing Branch.- 9.3 The Method of Decreasing Branch Approximation by Monotone Laws.- 9.4 Application I: Cleavage in Laminated Composites and the Non-monotone Unilateral Contact Problem.- 9.5 Application II: The Nonmonotone Friction Problem and the Combined Unilateral Contact Problem with Nonmonotone Friction.- 10 On the Approximation of Hemivariational Inequalities by Variational Inequalities.- 10.1 General Formulation of the Method.- 10.2 Application III: Nonmonotone Friction Interface Conditions with Debonding.- 10.3 Application IV:Adhesive Joints in Structural Mechanics.- 10.4 Application V: Comparison with the Path Following Method.- 10.5 Application VI: Nonmonotone Stress-Strain Laws. The Sawtooth Behaviour of Composites.- 10.6 Application VII: Shear Connectors in Composite Beams.- 11 The Method of Substationary Point Search.- 11.1 General Formulation of the Method.- 11.2 On the Numerical Implementation of the Algorithm.- 11.3 Application VIII: Delamination and Adhesive Joints in Structural Mechanics.- 11.4 Application IX: Semirigid Connections in Steel Structures.- 12 On a Decomposition Method into Two Convex Problems.- 12.1 General Formulation of the Method.- 12.2 Application X: The Stamp Problem and the Interfacial Debonding in Composites.- 13 Dynamic Hemivariational Inequalities and Crack Problems.- 13.1 Application XI: Numerical Treatment of Dynamic Hemivariational Inequalities.- 13.2 Application XII: The Unilateral Contact and Nonmonotone Friction Problem in Cracks.- 13.3 Application XIII: Fracture of Cracks Repaired by an Adhesive Material.- 14 Applications of the Theory of Hemivariational Inequalities in Robotics.- 14.1 Application XIV: Adhesive Grasping Problem in Robotics.- 14.2 Application XV: On the Optimal Control of the Adhesive Grasping Problem in Robotics.- 15 Addenda: Hemivariational Inequalities, Fractals and Neural Networks.- 15.1 Fractals in Mechanics. An Introduction.- 15.2 Application XVI: Hemivariational Inequalities for Fractal Interfaces.- 15.3 The Neural Network Approach to Hemivariational Inequalities.- 15.4 Application XVII: D.C.B. Specimen Modelling. The Neural Network Approach.- 15.5 Application XVIII: The Inverse Delamination Problem as a Supervised Learning Problem for a Neural Network. Extensions.- References.
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