Properties for Design of Composite Structures. Neil McCartney. Читать онлайн. Newlib. NEWLIB.NET

Автор: Neil McCartney
Издательство: John Wiley & Sons Limited
Серия:
Жанр произведения: Техническая литература
Год издания: 0
isbn: 9781118789780
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theta plus left-parenthesis StartFraction 3 kappa Subscript m Baseline Over kappa Subscript eff Baseline plus 2 kappa Subscript m Baseline EndFraction minus 1 right-parenthesis StartFraction alpha b cubed cosine theta Over r squared EndFraction comma b less-than r less-than infinity period"/>(3.9)

      If the cluster in Figure 3.1(a) is represented accurately by the effective medium shown in Figure 3.1(b), then, at large distances from the cluster, the temperature distributions (3.8) and (3.9) should be identical, leading to the fourth step, where the perturbation terms in relations (3.8) and (3.9) are equated, so that

      which is a ‘mixtures’ relation for the quantity 1/(κ+2κm). On using (3.1), the effective thermal conductivity may be estimated using

      For multiphase composites, Hashin and Shtrikman [5, Equations (3.21)–(3.23)] derived bounds for magnetic permeability, pointing out that they are analogous to bounds for effective thermal conductivity. Their conductivity bounds may be expressed in the following simpler form, having the same structure as the result (3.10) derived using Maxwell’s methodology

      where κmin is the lowest value of conductivities for all phases, whereas κmax is the highest value.

      3.3 Bulk Modulus and Thermal Expansion Coefficient

      3.3.1 Spherical Particle Embedded in Infinite Matrix Subject to Pressure and Thermal Loading

      u Subscript r Baseline equals f left-parenthesis r right-parenthesis comma u Subscript theta Baseline equals 0 comma u Subscript phi Baseline equals 0 period(3.13)

      The corresponding strain field obtained from (2.142) is then given by

      epsilon Subscript r r Baseline equals StartFraction partial-differential u Subscript r Baseline Over partial-differential r EndFraction comma epsilon Subscript theta theta Baseline equals epsilon Subscript phi phi Baseline equals StartFraction u Subscript r Baseline Over r EndFraction comma epsilon Subscript r theta Baseline equals epsilon Subscript r phi Baseline equals epsilon Subscript theta phi Baseline equals 0 period(3.14)

      The stress field follows from stress-strain relations expressed in the form (see (2.160) for the Cartesian equivalent)

      where λ and μ are Lamé’s constants and where α is now the linear coefficient of thermal expansion. On using the equilibrium equations (2.130)–(2.133), it can be shown that, within the spherical particle of radius a, the resulting bounded displacement and stress fields are given by

      u Subscript r Superscript p Baseline zero width space zero width space equals alpha Subscript p Baseline upper Delta upper T r minus StartFraction p 0 Over 3 k Subscript p Baseline EndFraction r comma u Subscript theta Superscript p Baseline equals u Subscript phi Superscript p Baseline equals 0 comma(3.16)

      sigma Subscript r r Superscript p Baseline equals sigma Subscript theta theta Superscript p Baseline equals sigma Subscript phi phi Superscript p Baseline equals minus p 0 comma sigma Subscript r theta Superscript p Baseline equals sigma Subscript r phi Superscript p Baseline equals sigma Subscript 


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