Solvabilty of generalized Hammerstein integral equations on unbounded domains, with sign-changing kernels

In this work we study an Hammerstein generalized integral equation u(t)=∫_{-∞}^{+∞}k(t,s) f(s,u(s),u′(s),...,u^{(m)}(s))ds, where k:ℝ²→ℝ is a W^{m,∞}(ℝ²), m∈ℕ, kernel function and f:ℝ^{m+2}→ℝ is a L¹-Carathéodory function. To the best of our knowledge, this paper is the first one to consider discontinuous nonlinearities with derivatives dependence, without monotone or asymptotic assumptions, on the whole real line. Our method is applied to a fourth order nonlinear boundary value problem, which models moderately large deflections of infinite nonlinear beams resting on elastic foundations under localized external loads.

Predictions of J integral and tensile strength of clay/epoxy nanocomposites material using phase field model

We predict macroscopic fracture related material parameters of fully exfoliated clay/epoxy nano- composites based on their fine scale features. Fracture is modeled by a phase field approach which is implemented as user subroutines UEL and UMAT in the commercial finite element software Abaqus. The phase field model replaces the sharp discontinuities with a scalar damage field representing the diffuse crack topology through controlling the amount of diffusion by a regularization parameter. Two different constitutive models for the matrix and the clay platelets are used; the nonlinear coupled system con- sisting of the equilibrium equation and a diffusion-type equation governing the phase field evolution are solved via a NewtoneRaphson approach. In order to predict the tensile strength and fracture toughness of the clay/epoxy composites we evaluated the J integral for different specimens with varying cracks. The effect of different geometry and material parameters, such as the clay weight ratio (wt.%) and the aspect ratio of clay platelets are studied.