Admissible values for the interaction term of the Tsai-Wu failure theory under plane stress and fitting of the failure envelope to test data

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==ADMISSIBLE VALUES FOR THE INTERACTION TERM OF THE TSAI-WU FAILURE THEORY UNDER PLANE STRESS AND FITTING OF THE FAILURE ENVELOPE TO TEST DATA==

'''O.A. Pellejer<sup>a,b</sup>, A. Fernandez<sup>a</sup>, L. Cui<sup>b</sup>'''
-->

==Abstract==

The Tsai-Wu failure criterion is widely used for orthotropic composite materials under plane stress due to its simplicity and low number of material constants. However, there is no consensus on how to find the interaction term. Based on general failure mode interaction assumptions, this paper provides closed form expressions to find the interaction term. In addition, the suitability of the Tsai-Wu’s failure criterion is narrowed to materials whose properties meet specific relationships. Last, it is shown how the Tsai-Wu failure envelope can be defined to pass through four biaxial stress states that best represent the actual testing conditions. All these findings conform an easy-to-apply methodology to define the Tsai-Wu failure envelope that best fits specific assumptions and test data.

==1 Introduction==

The Tsai-Wu 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 <math display="inline">F_{12}</math>. Tsai [2] suggested a procedure to narrow the range of admissible values for <math display="inline">F_{12}</math> based on a series of general failure mode interaction assumptions. In fact, Tsai [2] graphically rationalised the range of admissible values for <math display="inline">F_{12}</math> 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 <math display="inline">F_{12}</math> of any composite lamina. These new expressions also lay the foundation to find that the Tsai-Wu 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 Tsai-Wu failure envelope to bidirectional laminate test results.

==2 Tsai-Wu failure theory under plane stress==

Tsai and Wu [1] presented a general theory of strength for anisotropic materials, which is equivalent to Eq.[[#eq-1|(1)]] for specially orthotropic materials under plane stress.

<span id="eq-1"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>F_{1}\sigma _{1}+F_{2}\sigma _{2}+F_{11}\sigma _{1}^{2}+F_{22}\sigma _{2}^{2}+F_{66}\tau _{12}^{2}+2F_{12}\sigma _{1}\sigma _{2} = 1 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (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 open-ended like a hyperboloid). In order to meet these stability conditions, Tsai and Wu [1] imposed the following inequalities:

<span id="eq-2"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>F_{11} > 0 </math>
|-
| style="text-align: center;" | <math> F_{22} > 0 </math>
|-
| style="text-align: center;" | <math> F_{11}F_{22}-F_{12}^2 \ge 0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
|}

The following expressions relate the Tsai-Wu 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.

<span id="eq-3"></span>
<span id="eq-4"></span>
<span id="eq-5"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>F_i = \frac{1}{\left(\sigma _i^T\right)_{ult}}-\frac{1}{\left(\sigma _i^C\right)_{ult}} i = 1,2 </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
|-
| style="text-align: center;" | <math> F_{ii} = \frac{1}{\left(\sigma _i^T\right)_{ult}  \left(\sigma _i^C\right)_{ult}} i = 1,2 </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
|-
| style="text-align: center;" | <math> F_{66} = \frac{1}{\left(\tau _{12}\right)_{ult}^2}  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
|}
|}

The only remaining term to be defined is the interaction term <math display="inline">F_{12}</math>, which is related to the interaction of two stress components. This terms can be determined by an infinite number of combined stresses. Different experimental tests have been proposed [1], [3], [4] to find <math display="inline">F_{12}</math> but they are difficult to perform and, usually, empirical expressions are used instead.

==3 Unidirectional lamina==

===3.1 Failure mode interaction===

Tsai [2] described the failure mode interaction assumptions for unidirectional lamina that are illustrated in Figure [[#img-1|1]]. Those are the same assumptions that Tsai and Melo [5] used to find the range of admissible values for <math display="inline">F_{12}</math> of some composite laminae and recommend a value of <math display="inline">F_{12}^* = -0.5</math> 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.

<div id='img-1'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Pellejer_115108660-InteractionModes.png|269px|Failure mode interactions (after Tsai1992). 

f(σ₁,σ₂) = F₁σ₁+F₂σ₂+F₁₁σ₁²+F₂₂σ₂²+2F₁₂σ₁σ₂ 

In the assumptions: X,X',Y,Y'≫σ₁,σ₂≈0]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 1:''' Failure mode interactions (after Tsai1992). 

<math>f(\sigma _1,\sigma _2) = F_{1}\sigma _{1}+F_{2}\sigma _{2}+F_{11}\sigma _{1}^{2}+F_{22}\sigma _{2}^{2}+2F_{12}\sigma _{1}\sigma _{2}</math> 

In the assumptions: <math>\mathrm{X},\mathrm{X'},\mathrm{Y},\mathrm{Y'}\gg \sigma _1,\sigma _2 \approx 0</math>
|}

===3.2 Failure envelope slope at the anchor points===

If there is no shear stress (<math display="inline">\tau _{12}=0</math>), the envelope of the Tsai-Wu failure criterion intersects the <math display="inline">\sigma _1</math> and <math display="inline">\sigma _2</math> axes at the same points as the maximum stress failure criterion envelope does. These points are called anchor points because, the shape of the Tsai-Wu failure criterion envelope changes with the interaction term <math display="inline">F_{12}</math> 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 [[#img-2|2]].

<div id='img-2'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Pellejer_115108660-SlopesAtAnchors.png|420px|Admissible ranges of the slopes at the anchor points of the Tsai-Wu failure envelope (after Tsai1992).]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 2:''' Admissible ranges of the slopes at the anchor points of the Tsai-Wu failure envelope (after Tsai [2]).
|}

===3.3 <span id='lb-3.3'></span>Range of admissible values for F₁₂===

The interaction term <math display="inline">F_{12}</math> for the Tsai-Wu failure criterion is usually treated as an empiricial constant with the following definition:

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>F_{12} = \frac{F_{12}^{*}}{\sqrt{\mathrm{X} \mathrm{X'} \mathrm{Y} \mathrm{Y'}}} = F_{12}^{*}\sqrt{F_{11}F_{22}} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
|}

where <math display="inline">F_{12}^{*} \in [-1,1]</math> for the failure surface to be closed (this condition can be derived from Eq.[[#eq-2|(2)]]. Now, in addition to the condition for a closed failure envelope, new upper and lower limits for <math display="inline">F_{12}^{*}</math> can be derived from the slope at the anchor points of the previous assumptions.

'''Assumption 1a'''

<span id="eq-7"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\frac{\partial \sigma _2}{\partial \sigma _1}\left(\mathrm{X},0\right)= -\infty \quad \longrightarrow \quad  F_{12\mathrm{,1aL}}^{*} = \frac{\displaystyle -F_2}{\displaystyle 2 \sqrt{F_{11}F_{22}} \mathrm{X}} </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
|-
| style="text-align: center;" | <math> \frac{\partial \sigma _2}{\partial \sigma _1}\left(\mathrm{X},0\right)= -\frac{\mathrm{Y}}{\mathrm{X}} \quad \longrightarrow \quad  F_{12\mathrm{,1aU}}^{*} = \frac{\displaystyle -(2 F_{11} \mathrm{X}+F_1)+ F_2 \frac{\mathrm{Y}}{\mathrm{X}}}{\displaystyle -2 \sqrt{F_{11}F_{22}} \mathrm{Y}} </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
|}
|}

where it can be proven that <math display="inline">F_{12\mathrm{,1aL}}^{*} < F_{12\mathrm{,1aU}}^{*}</math>

'''Assumption 2a'''

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\frac{\partial \sigma _2}{\partial \sigma _1}\left(\mathrm{X'},0\right)= \frac{\mathrm{Y}}{\mathrm{X'}} \quad \longrightarrow \quad  F_{12\mathrm{,2aL}}^{*} = \frac{\displaystyle -2 F_{11} \mathrm{X'}+F_1+F_2\frac{\mathrm{Y}}{\mathrm{X'}}}{\displaystyle 2 \sqrt{F_{11}F_{22}} \mathrm{Y}} </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
|-
| style="text-align: center;" | <math> \frac{\partial \sigma _2}{\partial \sigma _1}\left(\mathrm{X'},0\right)= +\infty \quad \longrightarrow \quad  F_{12\mathrm{,2aU}}^{*} = \frac{\displaystyle F_2}{\displaystyle 2 \sqrt{F_{11}F_{22}} \mathrm{X'}} </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
|}
|}

where it can be proven that <math display="inline">F_{12\mathrm{,2aL}}^{*} < F_{12\mathrm{,2aU}}^{*}</math>

'''Assumption 3'''

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\frac{\partial \sigma _2}{\partial \sigma _1}\left(0,\mathrm{Y}\right)= \frac{\mathrm{Y}}{\mathrm{X'}} \quad \longrightarrow \quad  F_{12\mathrm{,3L}}^{*} = \frac{\displaystyle \frac{\mathrm{Y}}{\mathrm{X'}}(2 F_{22} \mathrm{Y}+F_2)+F_1}{\displaystyle -2 \sqrt{F_{11}F_{22}} \mathrm{Y}} </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
|-
| style="text-align: center;" | <math> \frac{\partial \sigma _2}{\partial \sigma _1}\left(0,\mathrm{Y}\right)= 0 \quad \longrightarrow \quad  F_{12\mathrm{,3U}}^{*} = \frac{\displaystyle -F_1}{\displaystyle 2 \sqrt{F_{11}F_{22}}\mathrm{Y}} </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
|}
|}

where it can be proven that <math display="inline">F_{12\mathrm{,3L}}^{*} < F_{12\mathrm{,3U}}^{*}</math>

'''Assumption 4'''

<span id="eq-14"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\frac{\partial \sigma _2}{\partial \sigma _1}\left(0,\mathrm{Y'}\right)= \frac{\mathrm{Y'}}{\mathrm{X}} \quad \longrightarrow \quad F_{12\mathrm{,4L}}^{*} = \frac{\displaystyle \frac{\mathrm{Y'}}{\mathrm{X}}(-2 F_{22} \mathrm{Y'}+F_2)+F_1}{\displaystyle 2 \sqrt{F_{11}F_{22}} \mathrm{Y'}}</math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
|-
| style="text-align: center;" | <math> \frac{\partial \sigma _2}{\partial \sigma _1}\left(0,\mathrm{Y'}\right)= 0 \quad \longrightarrow \quad F_{12\mathrm{,4U}}^{*} = \frac{\displaystyle F1}{\displaystyle 2 \sqrt{F_{11}F_{22}} \mathrm{Y'})}  </math>
| style="width: 5px;text-align: right;white-space: nowrap;" | (14)
|}
|}

where it can be proven that <math display="inline">F_{12\mathrm{,4L}}^{*} < F_{12\mathrm{,4U}}^{*}</math>

Tsai [2] determined graphically the range of admissible values for <math display="inline">F_{12}</math> for some composite laminae by plotting <math display="inline">F_{12}</math> as a function of the slope at each of the anchor points. Figure [[#img-3|3]] shows the range of admissible values for <math display="inline">F_{12}</math> for the same composite laminae but determined with Eq.[[#eq-7|(7)]] to [[#eq-14|(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 (<math display="inline">F_{12\mathrm{,4U}}^{*} = -0.48</math>) is less than the lower limit (<math display="inline">F_{12\mathrm{,1aL}}^{*} = -0.33</math>), which means that it is not possible to meet failure mode interaction assumptions 1a and 4 together.

<div id='img-3'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Pellejer_115108660-ComparisonF12.png|492px|Admissible ranges of the interaction term F₁₂<sup>*</sup> 

based on assumptions 1a, 2a, 3 and 4.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 3:''' Admissible ranges of the interaction term <math>F_{12}^{*}</math> 

based on assumptions 1a, 2a, 3 and 4.
|}

===3.4 Suitability of the Tsai-Wu’s failure criterion===

In Section [[#lb-3.3|3.3]], it was proved that there are certain materials whose Tsai-Wu failure envelope cannot satisfy the failure mode interaction asssumptions 1a, 2a, 3 and 4 together. Table [[#table-1|1]] summarises the requirements to ensure that the lower limits are always less than the upper limits of the interaction term <math display="inline">F_{12}^*</math> under any pair of the failure mode interaction asssumptions from the set 1a, 2a, 3 and  4.


{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
|+ style="font-size: 75%;" |<span id='table-1'></span>Table. 1 Requirements for the upper limits of <math>F_{12}^{*}</math> 

to be less than the lower limits.
|-
| colspan='1' style="border-right: 2px solid;border-left: 2px solid;border-right: 2px solid;" | 
| style="border-left: 2px solid;border-right: 2px solid;" | <math>F_{12\mathrm{,1aU}}^{*}</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>F_{12\mathrm{,2aU}}^{*}</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>F_{12\mathrm{,3U}}^{*}</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>F_{12\mathrm{,4U}}^{*}</math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">F_{12\mathrm{,1aL}}^{*}</math>  
| style="border-left: 2px solid;border-right: 2px solid;" | ✓
| style="border-left: 2px solid;border-right: 2px solid;" | <math>\mathrm{Y}\leq \mathrm{Y'}</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>\displaystyle \frac{\mathrm{X}}{\mathrm{X'}} \geq \frac{\mathrm{Y}}{\mathrm{Y'}}</math>
| style="border-left: 2px solid;border-right: 2px solid;" | <math>\displaystyle \frac{\mathrm{X}}{\mathrm{X'}} \leq \frac{\mathrm{Y'}}{\mathrm{Y}}</math>
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">F_{12\mathrm{,2aL}}^{*}</math>  
| style="border-left: 2px solid;border-right: 2px solid;" | ✓
| style="border-left: 2px solid;border-right: 2px solid;" | ✓
| style="border-left: 2px solid;border-right: 2px solid;" | ✓
| style="border-left: 2px solid;border-right: 2px solid;" | ✓
|- style="border-top: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">F_{12\mathrm{,3L}}^{*}</math>  
| style="border-left: 2px solid;border-right: 2px solid;" | ✓
| style="border-left: 2px solid;border-right: 2px solid;" | ✓
| style="border-left: 2px solid;border-right: 2px solid;" | ✓
| style="border-left: 2px solid;border-right: 2px solid;" | ✓
|- style="border-top: 2px solid;border-bottom: 2px solid;"
| style="border-left: 2px solid;border-right: 2px solid;" | <math display="inline">F_{12\mathrm{,4L}}^{*}</math>  
| style="border-left: 2px solid;border-right: 2px solid;" | ✓
| style="border-left: 2px solid;border-right: 2px solid;" | ✓
| style="border-left: 2px solid;border-right: 2px solid;" | ✓
| style="border-left: 2px solid;border-right: 2px solid;" | ✓

|}

==4 Test results for anchor points==

A common testing procedure [6] is to find the strength of unidirectional <math display="inline">0^\mathrm{o} </math> lamina from bidirectional <math display="inline">[90/0]_{ns}</math> laminate and the strength of unidirectional <math display="inline">90^\mathrm{o} </math> lamina from unidirectional <math display="inline">[90]_{ns}</math> 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 <math display="inline">F_i</math> and <math display="inline">F_{ij}</math> of the Tsai-Wu failure criterion such that the failure envelope passes through up to four specific combined stress states and meets the failure mode interaction assumptions:

<ol>

<li>Find the strength terms <math display="inline">F_i</math> and <math display="inline">F_{ij}</math> of the Tsai-Wu failure criterion as usual. </li>
<li>Find the interaction term <math display="inline">F_{12}^*</math> that best fits the failure envelope to your failure mode interaction assumptions (see Section [[#lb-3.3|3.3]]). </li>
<li>Find the actual stress states of the lamina at failure when testing the laminate specimen.  </li>
<li> Solve the linear system to find the terms of the strength tensors <math display="inline">F_i</math> and <math display="inline">F_{ij}</math> that force the envelope to pass through the actual stress states at failure:

<span id="eq-15"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>

\begin{bmatrix}

\left(\sigma _1^T\right)_{ult} & \left(\sigma _2^{T_1}\right)_{ult} & \left(\sigma _1^T\right)_{ult}^2 & \left(\sigma _2^{T_1}\right)_{ult}^2 \\                 \left(\sigma _1^C\right)_{ult} & \left(\sigma _2^{C_1}\right)_{ult} & \left(\sigma _1^C\right)_{ult}^2 & \left(\sigma _2^{C_1}\right)_{ult}^2 \\                 \left(\sigma _1^{T_2}\right)_{ult} & \left(\sigma _2^T\right)_{ult} & \left(\sigma _1^{T_2}\right)_{ult}^2 & \left(\sigma _2^T\right)_{ult}^2 \\                 \left(\sigma _1^{C_2}\right)_{ult} & \left(\sigma _2^C\right)_{ult} & \left(\sigma _1^{C_2}\right)_{ult}^2 & \left(\sigma _2^C\right)_{ult}^2 \\\end{bmatrix} \begin{bmatrix}

F_1 \\F_2 \\F_{11} \\F_{22} \\\end{bmatrix}</math>
|-
| style="text-align: center;" | <math> \qquad \qquad = \begin{bmatrix}

1-2F_{12}\left(\sigma _1^T\right)_{ult}  \left(\sigma _2^{T_1}\right)_{ult}\\         1-2F_{12}\left(\sigma _1^C\right)_{ult}  \left(\sigma _2^{C_1}\right)_{ult}\\         1-2F_{12}\left(\sigma _1^{T_2}\right)_{ult}  \left(\sigma _2^T\right)_{ult}\\         1-2F_{12}\left(\sigma _1^{C_2}\right)_{ult} \left(\sigma _2^C\right)_{ult}\\\end{bmatrix} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (15)
|}</li>
<li>Find the intercepts of the Tsai-Wu failure envelope with the coordinate axes in the stress space.   </li>
<li>Recalculate the range of admissible values for the interaction term <math display="inline">F_{12}^*</math>. </li>
<li>Verify the error of the new Tsai-Wu failure envelope at the points corresponding to the actual stress states at failure. If needed, iterate from step [[#4 Test results for anchor points|4]] until the error is considered to be satisfactorily small. For most cases, this procedure should easily converge in very few steps. </li>

</ol>

For example, the Tsai-Wu failure envelope was found for a unidirectional high strength carbon fibre / epoxy prepeg lamina with four differente methods. Figure [[#img-4|4]] shows the Tsai-Wu failure envelopes when it is generated whith the most commonly recommended values for <math display="inline">F_{12}^*</math> (0 and -0.5), the interaction term <math display="inline">F_{12}^*=-0.285</math> that best fits the failure envelope to the failure mode interaction assumptions, and the interaction term <math display="inline">F_{12}^*=-0.297</math> 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.

<div id='img-4'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Draft_Pellejer_115108660-ExampleUD.png|328px|Example: Tsai-Wu failure envelopes (τ₁₂=0) 

in stress space (top) and strain space (bottom).]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 4:''' Example: Tsai-Wu failure envelopes (<math>\tau _{12}</math>=0) 

in stress space (up) and strain space (down).
|}

==5 Conclusions==

Tsai [2] presented the idea of finding the range of admissible values for the interaction term <math display="inline">F_{12}</math> from general failure mode interaction assumptions. Applying the same idea, this paper provides closed form expressions to find the interaction term <math display="inline">F_{12}</math> 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 <math display="inline">F_{12}</math>. Using the recommended value <math display="inline">F_{12}^{*} = -0.5</math> for all materials is discouraged in favour of applying this procedure for each particular material.

The expressions to find the interaction term <math display="inline">F_{12}</math> based on general failure mode interaction assumptions also show that the Tsai-Wu failure criterion cannot satisfy all the assumptions unless the material strength parameters meet certain relationships. Therefore, the suitability of the Tsai-Wu failure criterion has been narrowed to certain materials.

Last, regarding the strength parameters that are used to define the Tsai-Wu 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 Tsai-Wu failure envelope.

==Acknowledgement==

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.

==Nomenclature==

<br />$\left(\sigma_i^T\right)_{ult}$ Ultimate tensile strength in direction <math display="inline">i</math>.
<br />$\left(\sigma_i^C\right)_{ult}$ Ultimate compressive strength in direction <math display="inline">i</math>.
<br />$\left(\tau_{12}\right)_{ult}$ Ultimate shear strength.
<br />$\LTS = \left(\sigma_1^T\right)_{ult}$ Ultimate tensile strength in fibre direction 1.
<br />$\LCS = \left(\sigma_1^C\right)_{ult}$ Ultimate compressive strength in fibre direction 1.
<br />$\TTS = \left(\sigma_2^T\right)_{ult}$ Ultimate tensile strength in direction 2.
<br />$\TCS = \left(\sigma_2^C\right)_{ult}$ Ultimate compressive strength in direction 2.
<br />$\SSS = \left(\tau_{12}\right)_{ult}$ Ultimate shear strength in plane 12.
<br />$\left(\sigma_i^{T_j}\right)_{ult}$ Stress in direction <math display="inline">i</math> when the tensile stress in direction <math display="inline">j</math> reaches failure.
<br />$\left(\sigma_i^{C_j}\right)_{ult}$ Stress in direction <math display="inline">i</math> when the compressive stress in direction <math display="inline">j</math> reaches failure.

===BIBLIOGRAPHY===