722 research outputs found
Existence and uniqueness of convex monotone positive solutions for boundary value problems of an elastic beam equation with a parameter
The purpose of this paper is to investigate the existence and uniqueness of convex monotone positive solutions for a boundary value problem of an elastic beam equation with a parameter. The proofs of the main results rely on a fixed point theorem and some properties of eigenvalue problems for a class of general mixed monotone operators. The results can guarantee the existence of a unique convex monotone positive solution and can be applied to construct two iterative sequences for approximating it. Moreover, we present some pleasant properties of convex monotone positive solutions for the boundary value problem dependent on the parameter. Finally, an example is given to illustrate the main results
Analytical Solutions to General Anti-Plane Shear Problems In Finite Elasticity
This paper presents a pure complementary energy variational method for
solving anti-plane shear problem in finite elasticity. Based on the canonical
duality-triality theory developed by the author, the nonlinear/nonconex partial
differential equation for the large deformation problem is converted into an
algebraic equation in dual space, which can, in principle, be solved to obtain
a complete set of stress solutions. Therefore, a general analytical solution
form of the deformation is obtained subjected to a compatibility condition.
Applications are illustrated by examples with both convex and nonconvex stored
strain energies governed by quadratic-exponential and power-law material
models, respectively. Results show that the nonconvex variational problem could
have multiple solutions at each material point, the complementary gap function
and the triality theory can be used to identify both global and local extremal
solutions, while the popular (poly-, quasi-, and rank-one) convexities provide
only local minimal criteria, the Legendre-Hadamard condition does not guarantee
uniqueness of solutions. This paper demonstrates again that the pure
complementary energy principle and the triality theory play important roles in
finite deformation theory and nonconvex analysis.Comment: 23 pages, 4 figures. Mathematics and Mechanics of Solids, 201
STUDY ON VIBRATION RESPONSE OF A NON-UNIFORM BEAM WITH NONLINEAR BOUNDARY CONDITION
Forced vibration of non-uniform beam with nonlinear boundary condition is studied in this paper by proposing an iterative model combining Adomian Decomposition Method and modal analysis. An exponentially tapered beam with a hardening nonlinearity spring boundary is simulated as a case study. The model accuracy is proved by comparing iteration results and analysis solutions with linear and weakly nonlinear boundary conditions. Sin-weep nonlinear frequency spectrum is then obtained by the proposed model. The influence of boundary nonlinearity on the vibration response of non-uniform beam is analyzed. And the effect of different excitation amplitudes on nonlinearity in the vibration response is studied. The mathematical model and numerical solutions proposed in this paper can be used to solve and analysis broad vibration problems on general non-uniform beams with different nonlinear boundary conditionsunder various excitations
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