The TsaiWu failure criterion [1] is widely used for orthotropic composite materials under plane stress due to its simplicity and low number of material constants. Unfortunately, this failure criterion cannot be applied with rigour because there is no simple way to determine one of the required material constants: the interaction term . Tsai [2] suggested a procedure to narrow the range of admissible values for based on a series of general failure mode interaction assumptions. In fact, Tsai [2] graphically rationalised the range of admissible values for of some composite laminae, but he never identified analytical expressions to find the upper and lower limits. Based on that same procedure and the same failure mode interaction assumptions, this paper provides the expressions for the upper and lower limits for of any composite lamina. These new expressions also lay the foundation to find that the TsaiWu failure criterion is not suitable for certain composite materials whose properties meet specific relationships.
In order to estimate the failure of a composite material, understanding test results is as important as understanding the failure criterion. Consequently, a new methodology is also proposed to fit the TsaiWu failure envelope to bidirectional laminate test results.
Tsai and Wu [1] presented a general theory of strength for anisotropic materials, which is equivalent to Eq.(1) for specially orthotropic materials under plane stress.

(1) 
For the failure envelope to be physically meaningful, the failure envelope is expected to intercept each stress axis, surrounding the zero stress state, and be closed (not openended like a hyperboloid). In order to meet these stability conditions, Tsai and Wu [1] imposed the following inequalities:

(2) 
The following expressions relate the TsaiWu failure criterion terms to material strength parameters to be determined experimentally. Regarding the sign convention, it was assumed that normal stress is positive in tension and negative in compression. All ultimate strength parameters are in absolute value.

The only remaining term to be defined is the interaction term , which is related to the interaction of two stress components. This term can be determined by an infinite number of combined stresses. Different experimental tests have been proposed [1], [3], [4] to find but they are difficult to perform and, usually, empirical expressions are used instead.
Tsai [2] described the failure mode interaction assumptions for unidirectional lamina that are illustrated in Figure 1. Those are the same assumptions that Tsai and Melo [5] used to find the range of admissible values for of some composite laminae and recommend a value of as a good approximation for all materials. The assumptions try to be representative of the most common failure mode interactions but in no way exclusive.
Figure 1: Failure mode interactions (after Tsai [2]).
In the assumptions: 
If there is no shear stress (), the envelope of the TsaiWu failure criterion intersects the and axes at the same points as the maximum stress failure criterion envelope does. These points are called anchor points because, the shape of the TsaiWu failure criterion envelope changes with the interaction term but it always passes through these points. As noted by Tsai [2], the admissible ranges of the slopes at the anchor points of the failure envelope corresponds to the shaded areas in Figure 2.
Figure 2: Admissible ranges of the slopes at the anchor points of the TsaiWu failure envelope (after Tsai [2]). 
The interaction term for the TsaiWu failure criterion is usually treated as an empiricial constant with the following definition:

(6) 
where for the failure surface to be closed (this condition can be derived from Eq.(2). Now, in addition to the condition for a closed failure envelope, new upper and lower limits for can be derived from the slope at the anchor points of the previous assumptions.
Assumption 1a

where it can be proven that
Assumption 2a

where it can be proven that
Assumption 3

where it can be proven that
Assumption 4

where it can be proven that
Tsai [2] determined graphically the range of admissible values for for some composite laminae by plotting as a function of the slope at each of the anchor points. Figure 3 shows the range of admissible values for for the same composite laminae but determined with Eq.(7) to (14). The ranges of admissible values found look alike the ones shown in Tsai and Melo [5] but something went unnoticed for the Kevlar/epoxy lamina. For this specific lamina, the upper limit () is less than the lower limit (), which means that it is not possible to meet failure mode interaction assumptions 1a and 4 together.
Figure 3: Admissible ranges of the interaction term based on assumptions 1a, 2a, 3 and 4. 
In Section 3.3, it was proved that there are certain materials whose TsaiWu failure envelope cannot satisfy the failure mode interaction asssumptions 1a, 2a, 3 and 4 together. Table 1 summarises the requirements to ensure that the lower limits are always less than the upper limits of the interaction term under any pair of the failure mode interaction asssumptions from the set 1a, 2a, 3 and 4.
✓  
✓  ✓  ✓  ✓  
✓  ✓  ✓  ✓  
✓  ✓  ✓  ✓ 
A common testing procedure [6] is to find the strength of unidirectional lamina from bidirectional laminate and the strength of unidirectional lamina from unidirectional laminate. Instead of having four simple stress states to define a failure envelope, these test methods result in two combined stress states and two simple stress states. The following procedure is proposed in order to define the strength terms and of the TsaiWu failure criterion such that the failure envelope passes through up to four specific combined stress states and meets the failure mode interaction assumptions:

(15) 
For example, the TsaiWu failure envelope was found for a unidirectional high strength carbon fibre / epoxy prepeg lamina with four differente methods. Figure 4 shows the TsaiWu failure envelopes when it is generated whith the most commonly recommended values for (0 and 0.5), the interaction term that best fits the failure envelope to the failure mode interaction assumptions, and the interaction term that best fits the failure envelope to the failure mode interaction assumptions and also passes through two composed stress states corresponding to bidirectional laminate tests that have the same longitudinal failure strain.
Figure 4: Example: TsaiWu failure envelopes (=0) in stress space (up) and strain space (down). 
Tsai [2] presented the idea of finding the range of admissible values for the interaction term from general failure mode interaction assumptions. Applying the same idea, this paper provides closed form expressions to find the interaction term for some of the most common failure assumptions found in composite materials. If no test data is available for a particular material, these expressions may be employed to find a range of admissible values for the interaction term . Using the recommended value for all materials is discouraged in favour of applying this procedure for each particular material.
The expressions to find the interaction term based on general failure mode interaction assumptions also show that the TsaiWu failure criterion cannot satisfy all the assumptions unless the material strength parameters meet certain relationships. Therefore, the suitability of the TsaiWu failure criterion has been narrowed to certain materials.
Last, regarding the strength parameters that are used to define the TsaiWu failure envelope, it has been noted that the strength parameters of a composite material can be determined from unidirectional and bidirectional laminates. A procedure has been proposed to use the actual combined stress state at failure of these specimens to define the TsaiWu failure envelope.
This paper and the research behind is a joint effort between Ingeniacity and the University of Surrey. Ingeniacity is a company specialised in interdisciplinary and innovative engineering. Composite materials in the naval, automotive and energy sectors are one of the main areas of expertise of Ingeniacity.
Ultimate tensile strength in fibre direction 1.
Ultimate compressive strength in fibre direction 1.
Ultimate tensile strength in direction 2.
Ultimate compressive strength in direction 2.
Stress in direction when the tensile stress in direction reaches failure.
Stress in direction when the compressive stress in direction reaches failure.
Ultimate shear strength.
[1] S. W. Tsai and E. M. Wu, «Journal of Composite Materials,» A general theory of strength for anisotropic materials, vol. 5, no. 1, pp. 58–80, 1971.
DOI: 10.1177/002199837100500106.
[2] S. W. Tsai, «Think Composites,» Theory of Composite Design, 1992, ISBN: 0961809035.
[3] I. I. Gol’denblat and V. A. Kopnov, «Polymer Mechanics,» Strength of glassreinforced plastics in the complex stress state, vol. 1, pp. 54–59, 1965.
[4] E. Ashkenaz, «Polymer Mechanics,» Problems of the anisotropy of strength, vol. 1, 60–70, 1965. DOI: 10.1007/BF00860686.
[5] S. W. Tsai and J. D. D. Melo, «Stanford Aeronautics & Astronautics,» Composite Materials Design and Testing: Unlocking mystery with invariants, 2015, ISBN: 9780986084508.
[6] Composite Materials Handbook, «SAE International,» CMH17, Volume 1: Polymer Matrix Composites: Guidelines for Characterization of Structural Materials, 2017, ISBN: 9780768091458.
Published on 20/06/24
Accepted on 29/08/23
Submitted on 23/05/23
Volume 08  COMUNICACIONES MATCOMP21 (2022) Y MATCOMP23 (2023), Issue Núm. 5  Materiales y Estructuras, 2024
Licence: Other