Hill’s criterion

Hill [15] has developed a generalisation of the von Mises criterion for anisotropic materials. Anisotropy is defined with respect to specific axes fixed within the material which, in the case of orthotropic materials, are mutually perpendicular. Then, a 1-2-3 axes set can be chosen to align with the directions of orthotropy and the yield criterion defined with respect to the stresses in this axis set. This precludes the use of principal stresses as the principal directions do not in general coincide with the directions of orthotropy. Therefore, Hill s criterion is a generalisation of Equation (12.9)... 

Figure 3 shows that both Hill s criterion and the ESIS stress criterion have been fulfil during our EWF tests indicating that the fracture process occurred in plane stress conditions and therefore the application of the EWF theory is validated. 

Figure 4. a) Typical a i x vs L plot, employed to verify the plane-stress solicitation conditions in accordance with Hill s criterion and stress criterion of ESIS protocol (i.e. MDPE-C). (b) Typical plot of Wp vs. L for DDENT specimens of... 

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] ... 

Tresca and R. von Mises criteria are for isotropic materials. In 1948, Rodney Hill provided a quadratic yield criterion for anisotropic materials. A special case of this criterion is von Mises criterion. In 1979, Hill proposed a non-quadratic yield criterion. Later on several other criteria were proposed including Hill s 1993 criterion. Rodney HiU (1921-2011) was bom in Yorkshire, England and has tremendous contribution in the theory of plasticity. 

Hill s yield criterion for anisotropic materials (see also Ref. 5, Chapter 10) is based on the following equation... 

Hence, in the present chapter the Hill and Marsch equations fractal analogs are obtained, which has shown, that cross-linked epoxy polymers microhardness is defined by their structure only, characterized by its fractal dimension. Tabor s criterion is only fulfilled for Euclidean (or close to them) solids. The cross-linking degree enhancement results to loosely packed matrix loosening and corresponding reduction of cross-linked epoxy polymers microhardness. The similar results obtained for linear polyethylene and nanocomposites on its basis, filled with organoclay [16]. 

Thus, the fractal analogues of the Hill and Marsch equations obtained above have shown that the microhardness of crosslinked epoxy polymers is defined only by their structure, characterised by its fractal dimension. Tabor s criterion is correct for Euclidean (or close to Euclidean) solids only. The degree of increase in crosslinking results in loosening of the loosely packed matrix and to a corresponding reduction of the microhardness of crosslinked epoxy polymers [60]. 

Figure 3. Plots of net maximum stress (ama ) versus L, in order to verify the plane-stress solicitation conditions in accordance with Hill s theory (aniax< 1.15 Cy) and stress criterion ofESIS protocol (0.9 < < 1.1) (a)DDENT A-...

The plot in Figure 4(a) shows that Hill s plasticity criterion [13] was not fulfill for the DDENT specimens employed, since the maximum stress determined in the simples was always higher than 1.15ay, an indication that plane stress conditions may not be achieved. However, ESIS protocol has established an alternate stress criterion that considers valid ligament lengths when the following condition is met [1] ... 

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. 

Criterion 0 Tsai-Hill Interlaminar Failure Surface-Ply with S =8.2 Experimental Failure... 

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. 

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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