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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in the x3-direction. When the plate is uniaxially loaded in the x2-direction, the parameter νt is the Poisson’s ratio determining the transverse through-thickness deformation in the x3-direction.

      It is useful, first, to show the form of the stress-strain equations (2.196) when the material is transverse isotropic about the x3-axis, so that they may be used when considering the properties of unidirectional plies in a laminate where the fibres are aligned in the x3-direction of the ply, and so that use can be made of analysis given in the previous section. It follows from (2.196) that when the material is transverse isotropic about the x3-axis, the stress-strain relations are of the form

      As S11=1/υT, S12=−νt/υT and S66=1/μt it follows from (2.189) that for a transverse isotropic solid the following condition must be satisfied:

      In Chapter 4 considering fibre-reinforced materials, stress-strain relations are required for the cylindrical polar coordinates (r,θ,z) corresponding to the relations (2.197), which are given by

      table attributes columnalign left end attributes row cell epsilon subscript r r end subscript equals 1 over E subscript text T end text end subscript sigma subscript r r end subscript minus nu subscript text t end text end subscript over E subscript text T end text end subscript sigma subscript theta theta end subscript minus nu subscript text A end text end subscript over E subscript text A end text end subscript sigma subscript z z end subscript plus alpha subscript text T end text end subscript capital delta T comma epsilon subscript r z end subscript equals fraction numerator sigma subscript r z end subscript over denominator 2 mu subscript text A end text end subscript end fraction comma end cell row cell epsilon subscript theta theta end subscript equals negative nu subscript text t end text end subscript over E subscript text T end text end subscript sigma subscript text rr end text end subscript plus 1 over E subscript text T end text end subscript sigma subscript theta theta end subscript minus nu subscript text A end text end subscript over E subscript text A end text end subscript sigma subscript z z end subscript plus alpha subscript text T end text end subscript capital delta T comma epsilon subscript theta z end subscript equals fraction numerator sigma subscript theta z end subscript over denominator 2 mu subscript text A end text end subscript end fraction comma end cell row cell epsilon subscript z z end subscript equals negative nu subscript text A end text end subscript over E subscript text A end text end subscript sigma subscript r r end subscript minus nu subscript text A end text end subscript over E subscript text A end text end subscript sigma subscript theta theta end subscript plus 1 over E subscript text A end text end subscript sigma subscript z z end subscript plus alpha subscript text A end text end subscript capital delta T comma epsilon subscript r theta end subscript equals fraction numerator sigma subscript r theta end subscript over denominator 2 mu subscript text t end text end subscript end fraction. end cell end table(2.199)

      When the fibres are aligned in a direction parallel to the x1-axis, as required in Chapters 6 and 7 concerning laminates and their plies, the transverse isotropic stress-strain relations, resulting from the orthotropic form (2.196), are given by

      where, again, the relation (2.198) must be satisfied.

      For plane strain conditions such that ε11≡0, it follows from (2.200) that

      table attributes columnalign left end attributes row cell sigma subscript 11 equals nu subscript text A end text end subscript left parenthesis sigma subscript 22 plus sigma subscript 33 right parenthesis minus E subscript text A end text end subscript alpha subscript text A end text end subscript capital delta T comma end cell row cell epsilon subscript 22 plus epsilon subscript 33 equals left parenthesis fraction numerator 1 minus nu subscript text t end text end subscript over denominator E subscript text T end text end subscript end fraction minus fraction numerator 2 nu subscript text A end text end subscript superscript 2 over denominator E subscript text A end text end subscript end fraction right parenthesis left parenthesis sigma subscript 22 plus sigma subscript 33 right parenthesis plus 2 left parenthesis alpha subscript text T end text end subscript plus nu subscript text A end text end subscript alpha subscript text A end text end subscript right parenthesis capital delta T. end cell end table(2.201)

      When ΔT=0, the term ε22+ε33 is the change in volume per unit volume ΔV/V for the plane strain conditions under discussion when an equiaxial transverse stress σ is applied such that σ2=σ3=σ. It then follows that a plane strain bulk modulus kT can be defined by

      1 over k subscript text T end text end subscript equals fraction numerator 2 left parenthesis 1 minus nu subscript text t end text end subscript right parenthesis over denominator E subscript text T end text end subscript end fraction minus fraction numerator 4 nu subscript text A end text end subscript 


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