Tensor derivative

Abstract: Foreword to the Classics Edition Preface Conventions Part I. Geometrical Prerequisites for Three-Dimensional Continuum Mechanics: 1. Vectors, determinants, and motivation for tensors 2. Cartesian tensors Part II. Problems in Continuum Mechanics: 3. Viscous fluids 4. Foundations in elasticity 5. Some examples of static oroblems in elasticity 6. Introduction to dynamic problems in elasticity Part III. Water Waves: 7. Formulation of the theory of surface waves in an inviscid fluid 8. Solution in the linear theory 9. Group speed and group velocity 10. Nonlinear effects Part IV. Variational Methods and Extremum Principles: 11. Calculus of variations 12. Characterization of Eigenvalues and equilibrium states as extrema Bibliography Hints and answers Index.

Abstract: Basic mechanics Variational formulations, work and energy theorems Theory of beams (strength of materials) Torsion of beams Theory of thin plates Bending of thin plates in polar coordinates Two-dimensional problems in Cartesian coordinates Two-dimensional problems in polar coordinates Thermo-elasticity Elastic stability Theory of thin shells Elasto-plasticity Elasto-viscoplasticity Nonlinear continuum mechanics Nonlinear elasticity Finite-strain elasto-plasticity Cyclic plasticity Damage mechanics Strain locolization Micro-mechanics of materials Cylindrical coordinates Cardan's formulae Matrices for the representation of second- and fourth-order tensors Zero-stress constraints.

Abstract: In this paper, we extend some classes of structured matrices to higher-order tensors. We discuss their relationships with positive semi-definite tensors and some other structured tensors. We show that every principal sub-tensor of such a structured tensor is still a structured tensor in the same class, with a lower dimension. The potential links of such structured tensors with optimization, nonlinear equations, nonlinear complementarity problems, variational inequalities and the non-negative tensor theory are also discussed.