upper T Baseline right-parenthesis equals 0 period"/>(2.242)
It is clear from (2.242), (2.243) and (2.245) that
From (2.215)1, (2.244) and (2.246)
It is noted from (2.211) that the quantities ε¯A+12ε^Ah and ε¯T+12ε^Th appearing in (2.242), (2.243) and (2.246) are the in-plane strains on the mid-plane x3=12h (i.e. the neutral plane) which are zero for the loading case under consideration.
2.18.3.2 Plane Strain Bending
Plane strain bending conditions are characterised by a zero transverse strain everywhere in the beam so that ε¯T=ε^T=0. It is also assumed that σA=σt=ΔT=0. It then follows that the relations (2.235), (2.236), (2.240) and (2.241) reduce to
indicating the value of the transverse bending moment that must be applied to the beam to ensure plane strain conditions. It is noted from (2.250) that E˜A may be interpreted as a plane strain bending modulus.
This completes the description of the fundamental theoretical concepts that will be required by subsequent chapters of this book.
References
1 1. Eringen, A.C. (1967). Mechanics of Continua. New York - London - Sydney: John Wiley & Sons, Inc.
2 2. Dunne, F.P.E. and Petrinic, N. (2005). Introduction to Computational Plasticity. Oxford University Press.
3 3. Love, A.E.H. (1944). A Treatise on the Mathematical Theory of Elasticity. Chapter XI, 4thed.
3 Maxwell’s Far-field Methodology Applied to the Prediction of Effective Properties of Multiphase Isotropic Particulate Composites
Overview:The far-field methodology developed by Maxwell, when estimating the effective electrical conductivity of isotropic particulate composites, is used to estimate effective thermoelastic properties of multiphase isotropic composites. In particular, Maxwell’s methodology applied to the analogous thermal conduction problem is described, extending the approach to multiphase spherical particles having different sizes and properties. The methodology is also used to estimate the effective bulk and shear moduli, and the thermal expansion coefficients, of multiphase isotropic particulate composites. Results correspond with expressions derived in the literature, and coincide with, or lie between, variational bounds for all volume fractions. These characteristics, relating to isotropic effective properties, indicate that results obtained using the methodology, are not necessarily restricted to low volume fractions, as originally suggested by Maxwell. It is concluded that Maxwell’s methodology is a unifying optimum technique to estimate the properties of multiphase isotropic particulate composites, because
