Stressed Composite Structures

The mechanics of structures with initial stresses is a traditional part of structural mechanics. It is closely related to the important problem of stability of structures. The basic concepts of elastic stability of structures go back to works by Euler (1759) and Bryan (1889). Later, it was found that the problem of deformation of solids with initial stresses is related to variational principles and nonlinear problems in elasticity; see Trefftz (1933), Marguerre (1938), Prager (1947), Hill (1958), Washuzu (1982). Historical detail up to the 1940s can be found in the book by Timoshenko (1953). Observing the basic concepts of the traditional mechanics of stressed structures, we agree that these are suitable for uniform structural elements (plates, beams, and so on) made of homogeneous materials, but not for complex structures (such as a network plate or a lattice mast) or structures made of composite materials (such as fiber reinforced or textile materials). Many concepts of the classicaltheory, such as a cross section or neutral plane (axis), correspond to no mechanical objects if we consider an inhomogeneous structure. As a result, we come to the conclusion that it would be useful to have a theory of thin inhomogeneous structures developed on the basis of 3-D elasticity theory with no simplifying assumptions (with no a priori hypothesis).

1. Introduction in the Homogenization Method as Applied to Stressed Composite Materials. - 1. 1. Linear Model of a Stressed Elastic Body. - 1. 2. Homogenization Method in the Mechanics of Composites. The Basic Approaches. - 1. 3. Homogenization Method in the Mechanics of Stressed Composites. - 2. Stressed Composite Plates and Membranes. - 2. 1. Stressed Plates (2-D Model). - 2. 2. Stressed Plates (Transition from 3-D to 2-D Model, In-Plane Initial 27 Stresses). - 2. 3. Stressed Plates (Transition from 3-D to 2-D Model, Moments of Initial Stresses). - 2. 4. 2-D Boundary Conditions Derived from the 3-D Boundary Problem. - 2. 5. 3-D and 2-D Energy Forms for a Stressed Plate and a Stability Criterion for a Plate. - 2. 6. Membrane (2-D model). - 2. 7. Membrane (Transition from 3-D to 2-D Model). - 2. 8. Plates with no Initial Stresses. Computing of Resultant Initial Stresses. - 3. Stressed Composite Beams, Rods and Strings. - 3. 1. Stressed Beam (1-D model). - 3. 2. Stressed Beams (Transition from 3-D to 1-D Model, Initial Axial Stresses). - 3. 3. Stressed Beams (Transition from 3-D to 1-D Model, Moments of Initial Stresses). - 3. 4. 1-D boundary Conditions Derived from 3-D Elasticity Problem. - 3. 5. 3-D and 1-D Energy Forms for a Stressed Beam and a Stability Criterion for a Beam. - 3. 6. Strings (1-D Model). - 3. 7. Strings (Transition from a 3-D to a 1-D Model). - 3. 8. Beams with no initial stresses. Computing of resultant initial stresses and moments. - 4. Calculation and Estimation of Homogenized Stiffnesses of Plate-Like and Beam-Like Composite Structures. - 4. 1. Variational Principles for Stiffnesses of Nonhomogeneous Plates. - 4. 2. Variational Principles for Stiffnesses of Nonhomogeneous Beams. - 4. 3. The Homogenization Method Modified for Lattice Plats. - 4. 4. The Homogenization Method Modified for Lattice Beams. - 4. 5. Review of Software Suitable for Homogenization Procedures. - Appendix A. Plates and Beams in Different Coordinate Systems. - A1. Homogenized Stiffnesses of a Plate in Different Coordinate Systems. - A2. Homogenized Stiffnesses of a Beam in Different Coordinate Systems. - Appendix B. Critical Loads for Some Composite Plates. - References.