Modeling Cylindrical Inhomogeneity of Finite Length with Steigmann–Ogden Interface

Lidiia Nazarenko,Holm Altenbach,Henryk Stolarski

doi:10.3390/technologies8040078

Lidiia Nazarenko, Holm Altenbach + Show 1 more

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https://doi.org/10.3390/technologies8040078

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Abstract

A mathematical model employing the concept of energy-equivalent inhomogeneity is applied to analyze short cylindrical fiber composites with interfaces described by the Steigmann–Ogden material surface model. Real inhomogeneity consists of a cylindrical fiber of finite length, and its surface possessing different properties is replaced by a homogeneous, energy-equivalent cylinder. The properties of the energy-equivalent fiber, incorporating properties of the original fiber and its interface, are determined on the basis of Hill’s energy equivalence principle. Closed-form expressions for components of the stiffness tensor of equivalent fiber have been developed and, in the limit, shown to compare well with the results available in the literature for infinite fibers with the Steigmann–Ogden interface model. Dependence of those components on the radius, length of the cylindrical fiber, and surface parameters is included in these expressions. The effective stiffness tensor of the short-fiber composites with so-defined equivalent cylindrical fibers can be determined by any homogenization method developed without accounting for interface.

Highlights

Interphases between inhomogeneities and the matrix may have a very pronounced influence on the effective behavior of entire composites
In [42], it is shown that, for cylindrical approximation of short fibers described by Gurtin–Murdoch surface expressed in Equations (3)–(9), if MS (x) = 0, the stiffness tensors Ceq have transversely isotropic symmetry, characterized by 5 independent constants, and have the following form: Ceq = C1 + ĈT, (13)
A mathematical model employing the concept of the equivalent inhomogeneity (EEI) [19,41,42,43] has been generalized to introduce the surface effects described by the Steigmann–Ogden model [22,23] derived within the strain gradient elasticity [35]

Summary

Introduction

Interphases between inhomogeneities and the matrix may have a very pronounced influence on the effective behavior of entire composites. Generalization of the Gurtin–Murdoch model was proposed by Steigmann and Ogden [22,23], who introduced the resistance of the surface to both stretching and bending Their development is based on the Kirchhoff–Love shell kinematics and, as such, implies that the surface energy in the Steigmann–Ogden model includes both the surface membrane strain tensor and the surface curvature tensor. The energy-equivalent inhomogeneity (EEI) approach, recently presented in [19,41,42,43], is applied to short fibers modeled as cylindrical inhomogeneity of finite length with a Steigmann–Ogden model of interface.

General Considerations

Steigmann–Ogden Surface Model and Associated Elastic Energy

Evaluation of the Surface Energy Related to the Bending

Constitutive Tensor of the Energy-Equivalent Cylinder

Comparison with the Existing Results for the Cylinder of Infinite Length with

Conclusions