Equation Tsai-Hill

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... 

For an applied stress of 1 MN/m and letting X be the multiplier on this stress, we can determine the value of X to make the Tsai-Hill equation become equal to 1. 

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] ... 

Find the Tsai-Hill failure criterion for pure shear loading at various angles B to the principal material directions, i.e., the shear analog of Equation (2.134). 

Fuchs C, Bhattacharyya D and Fakirov S (2006) Microfibril reinforced polymer-polymer composites Application of Tsai-Hill equation to PP/PET composites. Compos Sci Technol 66 3161-3171. Fakirov S, Bhattacharyya D and Shields R J (2008) Nanofibril reinforced composites from polymer blends, Coll Surf A Physicochem Eng Aspects 313 2-8. 

One of the shortcomings of the maximum stress theory of failure is that there are no terms which account for interaction between stress components for the case of biaxial (or off-axis) loading. Another is that five independent equations must be satisfied. The Tsai-Hill theory of failure for anisotropic materials overcomes both of the above mentioned shortcomings. This theory can be expressed in terms of principal material stress components as follows ... 

In contrast to the Tsai-Hill theory, Equation 9.12 where different values for the strength terms have to be used for tension and compression, the Tsai-Wu theory accommodates tension and compression in the one equation. 

The earliest works of trying to model different length scales of damage in composites were probably those of Halpin [235, 236] and Hahn and Tsai [237]. In these models, they tried to deal with polymer cracking, fiber breakage, and interface debonding between the fiber and polymer matrix, and delamination between ply layers. Each of these different failure modes was represented by a length scale failure criterion formulated within a continuum. As such, this was an early form of a hierarchical multiscale method. Later, Halpin and Kardos [238] described the relations of the Halpin-Tsai equations with that of self-consistent methods and the micromechanics of Hill [29],... 

Halpin and Kardos [7] published a review that described the derivation of the Halpin-Tsai equation from the first-principle arguments of Hill [10,11,12], Hermans [13], and Kerner [14] that were focused on polymers that are reinforced by dispersed-phase fibers. The Halpin-Tsai Equation (5.6) ... 

---