pdf - The World of Mathematical Equations

CONTENTSIntroduction 51 Kinematics of Cosserat continuum 101.1 Strain and wryness tensors . . . . . . . . . . . . . . . . . 121.2 Tensors of curvature of microstructure . . . . . . . . . . . 161.3 Compatibility conditions for B . . . . . . . . . . . . . . . 171.4 Determination of D k along a curve . . . . . . . . . . . . . 181.5 Relative strain measures . . . . . . . . . . . . . . . . . . . 201.6 Linear and angular velocities . . . . . . . . . . . . . . . . 211.7 Rates of strain . . . . . . . . . . . . . . . . . . . . . . . . 211.8 Higher rates of strain . . . . . . . . . . . . . . . . . . . . . 242 Theory of stresses 262.1 Forces and couples . . . . . . . . . . . . . . . . . . . . . . 262.2 Euler’s dynamics laws . . . . . . . . . . . . . . . . . . . . 282.3 Stress and couple stress tensors . . . . . . . . . . . . . . 292.4 On principal stresses . . . . . . . . . . . . . . . . . . . . . 422.5 Motion equations . . . . . . . . . . . . . . . . . . . . . . . 453 Constitutive equations of the micropolar fluid 473.1 Axioms of the constitutive equations . . . . . . . . . . . . 483.2 Constitutive equations of the micropolar fluid . . . . . . . 493.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 543.4 Equations of an elastic micropolar fluid . . . . . . . . . . 554 Selected problems of statics 634.1 Twisted structure . . . . . . . . . . . . . . . . . . . . . . . 634.2 Plane problem . . . . . . . . . . . . . . . . . . . . . . . . 644.3 Equilibrium with free-boundary . . . . . . . . . . . . . . . 664.4 Equilibrium of phases of micropolar fluid . . . . . . . . . . 69