Tsai/Hill criterion

In practice the second term in the above equation is found to be small relative to the others and so it is often ignored and the reduced form of the Tsai-Hill Criterion becomes... 

If the fibres are aligned at 15° to the jc-direction, calculate what tensile value of Ox will cause failure according to (i) the Maximum Stress Criterion (ii) the Maximum Strain Criterion and (iii) the Tsai-Hill Criterion. The thickness of the composite is 1 mm. 

Solving this gives X = 169. Hence a stress of Ox = 169 MN/m would cause failure. It is more difficult with the Tsai-Hill criterion to identify the nature of the failure ie tensile, compression or shear. Also, it is generally found that for fibre angles in the regions 5°-15 and 40 -90 , the Tsai-Hill criterion predictions are very close to the other predictions. For angles between 15° and 40° the Tsai-Hill tends to predict more conservative (lower) stresses to cause failure. 

A single ply glass/epoxy composite has the properties Usted below. If the fibres are aligned at 30° to the x-direction, determine the value of in-plane stresses, a, which would cause failure according to (a) the Maximum Stress criterion (b) the Maximum Strain criterion and (c) the Tsai-Hill criterion. 

Hence once again a stress of Ojt = 92 MN/m is the limiting condition, (iii) Tsai-Hill Criterion Letting X be the multiplier for... 

For the Tsai-Hill Criterion, letting X = multiplier on stress a, then solving... 

To account for different strengths in tension and compression, Hoffman added linear terms to Hill s equation (the basis for the Tsai-Hill criterion) [2-23] ... 

The Tsai-Hill criterion governs failure of a lamina (the strength-analysis procedure could, of course, involve another criterion). 

The most common approaches and criteria available for life prediction of composite laminates under multiaxial faligue have been briefly presented in the introduction to this section dedicated to continuous fiber reinforced composites. Among them, the criteria formulated by Fawaz and EUyin [76,77] and Smith and Pascoe [75], and a polynomial formulation based on the Tsai—Hill criterion [68], have already been... 

Figure 7.20 Accuracy of the modified Tsai—Hill criterion in the hfe estimation for global multiaxial data [1].

Figure 17.52 The angular dependence of fracture strength of a cfrp unidirectional Type I carbon fiber lamina Vf=0.5). Source Reprinted from Chamis CC, Sinclair JH, Exp Mech, 17, 339-346, 1977 and Tsai-Hill criterion presented by Hull, Hull D, Clyne DW, Cambridge University Press, Cambridge, 1996.

For the composites with continuous fibres, a great number of criteria are available, such as those of deformation or of the maximal effort, and the Tsai/Hill criterion or TsaiAVu criterion. In accordance with these criteria, the composite materials will be broken when one of the mechanical components reaches a limit value (experimentally determined, in the majority of cases). The Tsai/Hill criterion defines the upper limit of the effort, above which the fracture occur ... 

For in-plane failure in a lamina, the Tsai-Hill criterion was used with classical lamination theory to predict failure load. The Tsai-Hill criterion is given by... 

Since the in-plane stresses (Cn, CT22 12) ibe surface ply near the free edge are not constant, the average stresses over a distance of 2t from the free edge were used in the Tsai-Hill criterion for failure prediction. Further, since the surface plies are partially free from constraints (the lamination effect), the in-plane shear strength should be lower than that measured with [ 45]2s specimen. Thus, we took the value S = 14.4 ksi (for AS4/3501-6) reported in most literature. For T300/5208 graphite/epoxy composite we found S = 8.2 ksi was suitable. 

Accompanying the free-edge interlaminar stress analysis was the use of reduced lamina moduli to account for the lamina matrix cracking predicted by the Tsai-Hill criterion. Caution must be exercised in this reduction of elastic constants. When a certain elastic constant is reduced, in order to satisfy constraint conditions on the elastic constants, the remaining constants should also be adjusted. For example, due to transverse matrix failure, E2 was reduced to a small value (e.g. 100 psi). In order to meet the constraint condition like V23 < E2/E3, V23 should also be reduced. 

For the fiber-dominated laminate (0 = 0°) and highly matrix- dominated laminates (0 > 30°), failure initiates from the interior of the laminate and the classical Tsai-Hill criterion is accurate. 

The failure loads predicted using classical Tsai-Hill criterion are 93.4 ksi for [ 30/90]g laminate and 48 ksi for [ 45/90]s laminate. These predicted failure loads are much higher than the test data. Thus, failure in these laminates must have initiated from the free edges. 

Experimental and predicted failure stresses using Tsai-Hill criterion for different... 

These data are the average values of at least four specimens tested for each laminate. Except for the [0/90/45/-45]s laminate, the other three laminates showed open mode delamination at loads significantly lower than the respective ultimate loads. Except for [0/90/45/-45]s laminate, the Tsai-Hill criterion fails to predict the ultimate stress. 

From the experimental results for ic/4 laminates, we note that if delamination does not occur (e.g. [0/90/45/-45]s) then Tsai-Hill criterion can predict laminate strength. If delamination occurs before total laminate failure, then the laminate strength is lower than that predicted by Tsai-Hill criterion. 

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