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## Dale Chimenti, Stanislav Rokhlin, and Peter Nagy

Print publication date: 2011

Print ISBN-13: 9780195079609

Published to Oxford Scholarship Online: November 2020

DOI: 10.1093/oso/9780195079609.001.0001

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# Fundamentals of Composite Elastic Properties

Chapter:
1 (p.3) Fundamentals of Composite Elastic Properties
Source:
Physical Ultrasonics of Composites
Publisher:
Oxford University Press
DOI:10.1093/oso/9780195079609.003.0005

In this chapter, we review the mechanical behavior of composites considered from a macroscopic perspective, i.e., the microscopic heterogeneity of the material is ignored in this treatment. Our objective is to provide an overview of the basic composite constitutive behavior and to set the notation for the subsequent chapters. To establish this framework, we draw on concepts from continuum mechanics and elasticity, both of which are also covered by specialized books on these topics. The results in this chapter are important for us because they provide the theoretical framework for all the elastic wave phenomena we describe in detail in the subsequent chapters. Stress in a solid body is measured in force per unit area; there are normal stresses, acting along a normal to the infinitesimal element of the area, and shear (tangential) stresses, acting in the plane of the element. Let us assume that an infinitesimal traction force dT acts on an infinitesimal surface element dA = ndA, where n denotes the normal unit vector of the surface element. In index notation, the stress tensor is then defined through . . . dTi = σijdAj. (1.1) . . . The sign convention for the stress tensor in a Cartesian coordinate system is shown in Fig. 1.1. The choice of coordinate system is arbitrary, but for the sake of simplicity and concreteness, let us develop the relationships in a Cartesian system. They can all be generalized at a later time. Only the stress components acting on the surface elements with positive normal vectors are shown for clarity. On the surface elements with negative normal vectors, the stress directions are opposite. Conventionally, the first index indicates the normal of the surface the stress component is acting upon and the second index indicates the direction of the resulting traction force (however, we will show shortly that equilibrium conditions require that the stress tensor be symmetric, therefore the order of the indices is only of academic importance). For example, σ11 is the normal stress acting on the x2, x3 plane in the x1 direction, σ12 is the shear stress acting on the same plane in the x2 direction, and so forth.

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