Backout factors
Dr. Adam's follows up a previous column, in which he discussed the advantages of testing a crossply laminate and then "backing out" the unidirectional composite strength, with an expanded examination of backout factors that can be applied to other types of laminates.
In my previous column (see "Editor's Pick," at right) I discussed the advantages of testing a crossply laminate and then "backing out" the unidirectional composite strength. This included both tensile and compression testing.
A crossply laminate was defined as a composite with all plies oriented at 0° or 90°; that is, any one of the [0/90] family of laminates, with arbitrary numbers and distributions of 0° and 90° plies. However, it was indicated that one could back out unidirectional strength from other, more general laminates, but not as advantageously.
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The concept of using backout factors has been discussed repeatedly since the mid1980s^{1,2} and, in 1992, the Suppliers of Advanced Composite Materials Assn. (SACMA) introduced "recommended methods" for testing crossply laminates in both tension^{3} and compression^{4} and then backing out the unidirectional composite strength. A simple formula was presented in both documents for calculating the unidirectional composite strength from measured crossply strength, by multiplying the latter by the following factor,
BF = 2E_{11}/(E_{11} + E_{22}) (1)
where E_{11} and E_{22} represent the measured axial and transverse stiffnesses of the unidirectional composite. This unnamed factor was designated as F in the SACMA documents, but has now commonly come to be called the backout factor, since the unidirectional composite strength is "backedout" of the crossply strength experimental data. Thus, in Eq. 1, I have used the more descriptive BF.
Actually, the backout factor given in Eq. 1 is a very simplified ruleofmixtures approximation of the rigorous laminated plate theory equation, neglecting all Poisson's ratio effects. As written, it applies only to [0/90]_{ns} laminates, i.e., crossply laminates containing equal numbers of 0° and 90° plies. But note that it is actually just the ratio of the axial stiffness of the unidirectional plies to that of the crossply laminate, a simple approximation that could be used for other laminates in the [0/90] family as well.
The general laminated plate theory equation for any balanced, symmetric laminate is^{5},
BF = (Q°_{11}A_{22}  Q°_{12}A_{12}) t/ (A_{11}A_{22}  A2_{12}) (2)
where Q°_{ij} are terms in the reduced (plane stress) stiffness matrix for a 0° ply, the A_{ij}
(The subscript i and j represent combinations of "1" and "2" as indicated in Eg. 2.) This equation can be evaluated readily using a relatively simple computer program^{5}.
Of course, Eq. 2 can be simplified for any specific laminate. For example, for a [0/90]_{ns} crossply laminate it can be reduced to:
BF = [1/2E_{11}(E_{11}+ E_{12})  (v_{12} E_{12})2] / [1/4(E_{11} + E_{22})2  (v_{12} E_{22})2] (3)
where E_{11} and E_{22} are the axial and transverse stiffnesses of the 0° plies and v_{12} is the major Poisson's ratio. These are all readily measured properties.
Similar equations can be obtained for other types of laminates. However, the equations for laminates that contain other than 0° and 90° plies do become more complicated because of the shear coupling that exists. Note that Eq. 3 contains no shear or shearcoupling terms. That is, the shear stiffness properties of the unidirectional ply or of the laminate do not enter into the calculation. This is a distinct advantage of the [0/90] family of laminates since classical lamination theory is a linear analysis and shear stressshear strain properties can be very nonlinear. Influences of material nonlinearity are discussed in detail by Welsh and Adams^{5}. Fortunately, the influences usually are not severe. Likewise, for most composites the backout factor is not strongly influenced by even moderate uncertainties in the values of E_{22} and n_{12}. For these reasons, a [0/90]_{ns} crossply laminate is the logical choice for use in backing out unidirectional composite properties.
The question then arises, "Is the behavior of a [0/90]_{ns} laminate test specimen representative of actual composite laminates used in design?" To answer this question, Adams and Welsh^{6} compression tested a variety of laminate configurations fabricated from the same carbon/epoxy composite material. Most of these laminates were in the [0], [0/90], [0/±60], [0/±30/90] and [0/±45/90] families, with wide variations in the ratio of 0° plies to offaxis plies. They also gathered from the published literature similar experimental data generated by others. The corresponding backout factors were then calculated.
Fig. 1 is a plot of the backedout unidirectional composite axial compressive strength of these widely varying laminates versus the calculated backout factor. Each data point represents the average of multiple individual specimen tests, typically five or more. The plot is divided into General Laminates and Special Laminates, the division being at a backout factor of approximately 2.5, corresponding to a [0/±45/90]_{ns} (quasiisotropic) laminate. Note that a backout factor of 1 corresponds to a unidirectional composite. The unidirectional composite data presented in Fig. 1 were obtained using special test techniques, i.e., thicknesstapered specimens and minisandwich specimens^{6}. The large number of data points at a backout factor of approximately 1.8 are for [0/90]_{ns} laminates.
The Special Laminates have progressively higher backout factors, representing laminates having increasingly higher ratios of offaxis plies to 0° plies. Note that the corresponding backedout unidirectional composite axial strengths become increasingly higher as well. Values as high as 3,000 MPa (435 ksi) are indicated in Fig. 1. Experimental evidence suggests that the large number of offaxis plies surrounding the few 0° plies tends to restrict the fibers in the 0° plies from microbuckling^{6}. That is, the true compressive strength of the fibers is being approached. Unfortunately this is not achieved in commonly used (general) laminates, where microbuckling is very evident in failed compression specimens.
General laminates used in design typically range between the extremes of unidirectional composites (the most highly orthotropic composite possible) and quasiisotropic composites. In design, there is seldom a reason to go outside this range, into the realm of special laminates, since the composite axial stiffness becomes progressively lower. Note that the experimental data in the range of general can be represented reasonably a horizontal line, as shown in Fig. 1. That is, for all general (practical) laminates, the backedout 0° ply axial compressive strength is essentially constant, with [0/90]_{ns} laminates in the middle of the range. This is the design value of compressive strength for practical laminates.
Fig. 1 is for compressive strength and for only one composite material. Although some other materials have been similarly analyzed, and tensile tests have been conducted as well, obviously much more testing should be done. Nevertheless, even what has been presented here should be a very convincing indication that backout factors can be used, and that a [0/90]_{ns} laminate is a good choice when doing so.
References:
^{ 1 }Rawlinson, R.A., "The Use of Crossply and Angleply Composite Test Specimens to Generate Improved Material Property Data," Proceedings of the 36th International SAMPE Symposium, April 1991, pp. 10581068.
^{ 2 }HartSmith, L.J., "Backing Out Equivalent Unidirectional Lamina Strengths from Tests on CrossPlied Laminates," Proceedings of the 37th International SAMPE Symposium, March 1992, pp. 977990.
^{ 3 }SACMA SRM 994, "SACMA Recommended Test Method for Tensile Properties of Oriented CrossPlied FiberResin Composites," Suppliers of Advanced Composite Materials Assn. (Arlington, Va.), April 1994 (first issued as SRM 992 in 1992).
^{ 4 }SACMA SRM 694, "SACMA Recommended Test Method for Compressive Properties of Oriented CrossPlied FiberResin Composites," Suppliers of Advanced Composite Materials Assn. (Arlington, Va.), April 1994 (first issued as SRM 692 in 1992).
^{ 5 }Welsh, J.S., and Adams, D.F., "Testing of AnglePly Laminates to Obtain Unidirectional Composite Compression Strengths," Composites, Part A, Vol. 28A, 1997, pp. 387396.
^{ 6 }Adams, D.F., and Welsh, J.S., "The Wyoming Combined Loading Compression (CLC) Test Method," Journal of Composites Technology & Research, Vol. 19, No. 3, 1997, pp. 123133.
