Variational Principles in Structural Mechanics

Application of theorem of minimum potential energy to a complex structure: Part II: three-dimensional analysis

The solution of elastostatic problems and the principles of minimum potential energy, minimum complementary work, under fuzzy boundary conditions

The effective modulus of rock as a composite material

Cohesive zone size of microcracks in brittle materials

Thermal stresses in microstretch elastic plates

On the derivation of well posed boundary value problems in structural mechanics. : Kerr, AD Princeton Univ., New Jersey, USA 5F,18R Int. J. Solids, Struct. V12, N1, 1976, P1–11

On the derivation of well posed boundary value problems in structural mechanics

In this paper by the use of “convex analysis” the general nonlinear unilateral static problem of structural mechanics is formulated. By means of the “subdifferentiability” concept the unilateral nonlinear boundary conditions and the unilateral nonlinear constitutive laws are formulated as “subdifferential relations”, permitting the derivation of the variational inequalities of virtual and complementary virtual work. Thus the minimum potential and complementary energy theorems are proved. These minimum theorems are used to formulate the analysis as a nonlinear programming problem. Numerical examples on structures with nonlinear unilateral constitutive laws and nonlinear unilateral boundary conditions illustrate the theory.

An approximate solution for determining stresses in cross-ply composite tubes subjected to a circumferential temperature gradient is presented. The solution is based on the principle of complementary virtual work (PCVW) in conjunction with a Ritz approximation on the stress field and accounts for the temperature dependence of material properties. The PCVW method is compared with a planar elasticity solution using temperature-independent material properties and a Navier approach. The net effect of including temperature-dependent material properties is that the peak absolute values of the stresses are reduced. The dependence of the stresses on the circumferential location is also reduced in comparison with the case of temperature-independent properties.

Chapter 8 - Basic Equations and Solution Procedure

Corresponding relationships between certain incremental equilibrium equations and compatibility equations can be derived for structures undergoing gross deformation. Investigation of these relationships leads to the development of a new principle of virtual work and a form of energy called stress energy. The concept of stress energy has been developed by considering the potential energy of the forces applied to the structure or mechanism and its relationship to Libove's from of complementary energy and Charlton's virtual work method has been discussed.

Structural and solid mechanics problems can show complex global behavior and, in many cases, without an analytical solution. Numerical methods, such as the Finite Element Method (FEM), are traditionally used to study these systems. The Cellular Automata (CA) is a model based on the idea that the microscopic behavior of a system can be reproduced by smaller systems with simpler local rules. As this approach goes from a microscale to a macroscale, this technique enables the study of microscopic effects in macroscopic objects in a natural way. This paper studies the convergence of a hybrid technique of CA and FEM by developing a computational module capable of simulating solid- and structural-mechanics problems. Four update schemes are studied, evaluating the convergence, with different ordinations and degrees of discretization. Keywords: Finite Elements Methods, Cellular Automata, solid and structural mechanics.

Time-dependent stress variations in symmetrically viscoelastic composite laminates under uniaxial tensile load

On discrete and discretized non-linear elastic structures in unilateral contact (stability, uniqueness and variational principles)

A cross section of the rod is taken as object of investigation. The freedom of the section in free or constraint case is analyzed and the definition of virtual displacement of the section is given, which can be expressed by a variational operation. Assuming the variational and partial differential operations has commutativity, based on the hypothesis about surface constraint subjected to the rod, the freedom of the section on constraint surface is discussed and the equations satisfied by virtual displacements of the section are given. Combining D'Alembert principle and the principle of virtual work, D'Alembert-Lagrange principle is established. When constitutive equation of material of the rod is linear, the principle can be transformed to Euler-Lagrange form. From the principle, a dynamical equation in various forms such as Kirchhoff, Lagrange, Nielsen and Appell equation can be derived. For the case when a rod is subjected to a surface or a nonholonomic constraint, Lagrange equation with undetermined multipliers is obtained. Integral variational principle of dynamics of a super-thin elastic rod is also established, from which Hamilton principle formulation is obtained when the material of the rod is linear. Finally, canonical variables to describe the state of the section and Hamilton function are defined, and Hamilton canonical equation is derived. The analytical methods of dynamical modeling of a super-thin elastic rod have been constructed, which can serve as a theoretical framework of analytical dynamics of a super-thin elastic rod with two independent variables.