Neil McCartney

Properties for Design of Composite Structures

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left-parenthesis StartFraction partial-differential u Subscript k Baseline Over partial-differential x Subscript l Baseline EndFraction plus StartFraction partial-differential u Subscript l Baseline Over partial-differential x Subscript k Baseline EndFraction right-parenthesis equals epsilon Subscript l k Baseline period"/>(2.143)

      With the assumption that the strain field is uniform in the composite, the displacement field is linear and may be written in the form

      u Subscript k Baseline equals epsilon Subscript k l Baseline x Subscript l Baseline comma(2.144)

      where it is assumed that the displacement vector is zero when x1=x2=x3=0.

      The local equation of state (2.111) is not of a form that can easily be related to experimental measurements as one of the state variables is assumed to be the specific entropy η. It is much more convenient, and much more practically useful, if the state variable η is replaced by the absolute temperature T. A local equation of state for the specific Helmoltz energy is assumed to have the following form (equivalent to (2.111) as implied by (2.67)–(2.69))

      where εij is the infinitesimal strain tensor introduced in Section 2.12 (see (2.107)). For infinitesimal deformations, and because ψ≡υ−Tη from (2.64), the following differential form of (2.145) may be derived from (2.112)

      normal d psi equals minus eta normal d upper T plus StartFraction 1 Over rho 0 EndFraction sigma Subscript i j Baseline normal d epsilon Subscript i j Baseline comma(2.146)

      where the specific entropy η and the stress tensor σij are now defined by

      For infinitesimal deformations, a linear thermoelastic response can be assumed so that the Helmholtz energy per unit volume ρ0ψ^ has the form

      where Cijkl are the elastic constants having the dimensions of stress or modulus, βij are thermoelastic coefficients and where T0 is a reference temperature. As the strain tensor is symmetric, it follows that βij=βji and that

      upper C Subscript i j k l Baseline epsilon Subscript i j Baseline epsilon Subscript k l Baseline identical-to upper C Subscript i j k l Baseline epsilon Subscript i j Baseline epsilon Subscript l k Baseline identical-to upper C Subscript i j l k Baseline epsilon Subscript i j Baseline epsilon Subscript k l Baseline comma implying upper C Subscript i j k l Baseline equals upper C Subscript i j l k Baseline comma(2.150)

      On substituting (2.148) into (2.147), it follows that

      eta equals minus beta Subscript i j Baseline epsilon Subscript i j Baseline plus f prime left-parenthesis upper T right-parenthesis comma(2.152)

      It should be noted that the stress components are zero everywhere when the strain is defined to be zero everywhere at the reference temperature T0.

      The inverse form of the linear stress-strain relations (2.153) is written as

      where the compliance tensor Sijkl is such that

      upper C Subscript i j k l Baseline upper S Subscript k l m n Baseline equals one-half left-parenthesis delta Subscript i m Baseline delta Subscript j n Baseline plus delta Subscript i n Baseline delta Subscript j m Baseline right-parenthesis comma(2.155)

      where use has been made of (2.15), and where

      alpha Subscript i j Baseline equals upper S Subscript i j k l Baseline beta Subscript k l Baseline comma(2.156)

      are anisotropic thermal expansion coefficients.

      2.14.1 Isotropic Materials

      The situation simplifies when the material is linear thermoelastic and isotropic so that the stress-strain relations are

      epsilon subscript i j end subscript equals 1 over E left square bracket left parenthesis 1 plus nu right parenthesis sigma subscript i j end subscript minus nu delta subscript i j end subscript sigma subscript k k end subscript right square bracket plus alpha capital delta T delta subscript i j end subscript comma(2.157)

      where E is Young’s modulus, ν is Poisson’s ratio, α

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