Neil McCartney

Properties for Design of Composite Structures

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m squared upper V 1 plus n squared upper V 2 right-parenthesis upper Delta upper T comma 3rd Row epsilon prime Subscript 22 Baseline equals left-parenthesis n squared upper S 11 plus m squared upper S 12 right-parenthesis sigma 11 plus left-parenthesis n squared upper S 12 plus m squared upper S 22 right-parenthesis sigma 22 plus left-parenthesis n squared upper S 13 plus m squared upper S 23 right-parenthesis sigma 33 4th Row minus m n upper S 66 sigma 12 plus left-parenthesis n squared upper V 1 plus m squared upper V 2 right-parenthesis upper Delta upper T comma 5th Row epsilon prime Subscript 33 Baseline equals upper S 13 sigma 11 plus upper S 23 sigma 22 plus upper S 33 sigma 33 plus upper V 3 upper Delta upper T comma 6th Row epsilon prime Subscript 23 Baseline equals one-half m upper S 44 sigma 23 minus one-half n upper S 55 sigma 13 equals epsilon prime Subscript 32 Baseline comma 7th Row epsilon prime Subscript 13 Baseline equals one-half n upper S 44 sigma 23 plus one-half m upper S 55 sigma 13 equals epsilon prime Subscript 31 Baseline comma 8th Row epsilon prime Subscript 12 Baseline equals m n left-bracket left-parenthesis upper S 12 minus upper S 11 right-parenthesis sigma 11 plus left-parenthesis upper S 22 minus upper S 12 right-parenthesis sigma 22 plus left-parenthesis upper S 23 minus upper S 13 right-parenthesis sigma 33 right-bracket 9th Row plus one-half left-parenthesis m squared minus n squared right-parenthesis upper S 66 sigma 12 minus m n left-parenthesis upper V 1 minus upper V 2 right-parenthesis upper Delta upper T equals epsilon prime Subscript 21 Baseline period EndLayout"/>(2.182)

      Substitution of (2.178) into (2.182) leads to the relations

      Start 6 By 1 Matrix 1st Row epsilon prime Subscript 11 Baseline 2nd Row epsilon prime Subscript 22 Baseline 3rd Row epsilon prime Subscript 33 Baseline 4th Row 2 epsilon prime Subscript 23 Baseline 5th Row 2 epsilon prime Subscript 13 Baseline 6th Row 2 epsilon prime Subscript 12 Baseline EndMatrix equals Start 6 By 6 Matrix 1st Row 1st Column upper S prime Subscript 11 Baseline 2nd Column upper S prime Subscript 12 Baseline 3rd Column upper S prime Subscript 13 Baseline 4th Column 0 5th Column 0 6th Column upper S prime Subscript 16 Baseline 2nd Row 1st Column upper S prime Subscript 12 Baseline 2nd Column upper S prime Subscript 22 Baseline 3rd Column upper S prime Subscript 23 Baseline 4th Column 0 5th Column 0 6th Column upper S prime Subscript 26 Baseline 3rd Row 1st Column upper S prime Subscript 13 Baseline 2nd Column upper S prime Subscript 23 Baseline 3rd Column upper S prime Subscript 33 Baseline 4th Column 0 5th Column 0 6th Column upper S prime Subscript 36 Baseline 4th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column upper S prime Subscript 44 Baseline 5th Column upper S prime Subscript 45 Baseline 6th Column 0 5th Row 1st Column 0 2nd Column 0 3rd Column 0 4th Column upper S prime Subscript 45 Baseline 5th Column upper S prime Subscript 55 Baseline 6th Column 0 6th Row 1st Column upper S prime Subscript 16 Baseline 2nd Column upper S prime Subscript 26 Baseline 3rd Column upper S prime Subscript 36 Baseline 4th Column 0 5th Column 0 6th Column upper S prime Subscript 66 Baseline EndMatrix Start 6 By 1 Matrix 1st Row sigma prime Subscript 11 Baseline 2nd Row sigma prime Subscript 22 Baseline 3rd Row sigma prime Subscript 33 Baseline 4th Row sigma prime Subscript 23 Baseline 5th Row sigma prime Subscript 13 Baseline 6th Row sigma prime Subscript 12 Baseline EndMatrix plus Start 6 By 1 Matrix 1st Row upper V prime Subscript 1 Baseline 2nd Row upper V prime Subscript 2 Baseline 3rd Row upper V prime Subscript 3 Baseline 4th Row 0 5th Row 0 6th Row upper V prime Subscript 6 Baseline EndMatrix upper Delta upper T comma(2.183)

      where

      StartLayout 1st Row upper S prime Subscript 44 Baseline equals m squared upper S 44 plus n squared upper S 55 comma 2nd Row upper S prime Subscript 45 Baseline equals m n left-parenthesis upper S 44 minus upper S 55 right-parenthesis comma 3rd Row upper S prime Subscript 55 Baseline equals n squared upper S 44 plus m squared upper S 55 comma EndLayout(2.185)

      and where

      2.17.1 Transverse Isotropic and Isotropic Solids

      When considering unidirectionally reinforced fibre composites, as will be the case in Chapter 4, the effective composite properties are often assumed to be isotropic in the plane that is normal to the fibre direction taken here to be the x3-direction as coordinate rotations considered previously have been about the x3-axis. It is now assumed that S11=S22, S44=S55 and S13=S23. As m2 + n2 = 1 and

m Superscript 4 Baseline plus n Superscript 4 Baseline equals 1 minus 2 m squared n squared comma left-parenthesis m squared minus n squared right-parenthesis squared equals m Superscript 4 Baseline plus n Superscript 4 Baseline minus 2 m squared n squared equals 1 minus 4 m squared n squared comma

      it then follows from (2.184)–(2.186) that

      StartLayout 1st Row upper S prime Subscript 11 Baseline equals upper S prime Subscript 22 Baseline equals upper S 11 plus left-parenthesis upper S 66 minus 2 upper S 11 plus 2 upper S 12 right-parenthesis m squared n squared comma 2nd Row upper S prime Subscript 12 Baseline equals upper S 12 minus left-parenthesis upper S 66 minus 2 upper S 11 plus 2 upper S 12 right-parenthesis m squared n squared comma zero width space zero width space zero width space zero width space 3rd Row upper S prime Subscript 13 Baseline equals upper S prime Subscript 23 Baseline equals upper S 13 comma zero width space zero width space zero width space upper S prime Subscript 33 Baseline equals upper S 33 comma 4th Row upper S prime Subscript </p> </div><hr> <div class= Скачать книгу