Entire solutions of nonlocal elasticity models for composite materials
Giuseppina Autuori Patrizia Pucci
Discrete & Continuous Dynamical Systems - S 2018, 11(3): 357-377 doi: 10.3934/dcdss.2018020

Many structural materials, which are preferred for the developing of advanced constructions, are inhomogeneous ones. Composite materials have complex internal structure and properties, which make them to be more effectual in the solution of special problems required for civil and environmental engineering. As a consequence of this internal heterogeneity, they exhibit complex mechanical properties. In this work, the analysis of some features of the behavior of composite materials under different loading conditions is carried out. The dependence of nonlinear elastic response of composite materials on loading conditions is studied. Several approaches to model elastic nonlinearity such as different stiffness for particular type of loadings and nonlinear shear stress–strain relations are considered. Instead of a set of constant anisotropy coefficients, the anisotropy functions are introduced. Eventually, the combined constitutive relations are proposed to describe simultaneously two types of physical nonlinearities. The first characterizes the nonlinearity of shear stress–strain dependency and the latter determines the stress state susceptibility of material properties. Quite satisfactory correlation between the theoretical dependencies and the results of experimental studies is demonstrated, as described in [2,3] as well as in the references therein.

keywords: Existence theorems entire solutions fractional elliptic operators nonlocal elasticity models composite materials
Kirchhoff systems with nonlinear source and boundary damping terms
Giuseppina Autuori Patrizia Pucci
Communications on Pure & Applied Analysis 2010, 9(5): 1161-1188 doi: 10.3934/cpaa.2010.9.1161
In this paper we treat the question of the non--existence of global solutions, or their long time behavior, of nonlinear hyperbolic Kirchhoff systems. The main $p$--Kirchhoff operator may be affected by a perturbation which behaves like $|u|^{p-2} u$ and the systems also involve an external force $f$ and a nonlinear boundary damping $Q$. When $p=2$, we consider some problems involving a higher order dissipation term, under dynamic boundary conditions. For them we give criteria in order that $ || u(t,\cdot) ||_q\to\infty$ as $t \to\infty$ along any global solution $u=u(t,x)$, where $q$ is a parameter related to the growth of $f$ in $u$. Special subcases of $f$ and $Q$, interesting in applications, are presented in Sections 4, 5 and 6.
keywords: blow up. nonlinear source and boundary damping terms Kirchhoff systems non continuation

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