## About Hashin´s failure criteria in FEA

When using Finite Element Analysis (FEA) for studying composite materials, one of the most used failure criterion is the one which was proposed by Hashin in 1980. This theory is included in all the main FEA packages and, probably, you are more than familiar with this particular model. However, what you might not know is that the failure criteria that you are defining is not exactly the Hashin’s one. If you want to know why, this is your place.

Since the available failure criteria at that point presented some inconsistencies, in 1980 Hashin developed a new criteria which differentiated between failure modes. His theory considered four different ways in which the material could fail:

Where σ_{A}^{+} is the tensile failure stress in fibre direction; σ_{A}^{–} is the compressive failure stress in fibre direction; σ_{T}^{+} is the tensile failure stress transverse to fibre direction; σ_{T}^{–} is the compressive failure stress transverse to fibre direction; τ_{T} is the transverse failure shear; τ_{A} is the axial failure shear.

As it can be observed, this failure criteria does not consider damage evolution. Therefore, if this approach is implemented according to the original Hashin’s theory, FEA will only provide indices which contains information about the failure. However, the stresses will not show any difference when compared to a model where no failure criteria has been defined.

The main point arises when you take a detailed look at the material properties which have to be defined in the FEA software. There is one specific parameter that has to be specified in order to take into account the damage in our model: **the fracture energy per unit area**. But, why is this property necessary if it is not defined in Hashin´s criteria? And, how does it affect our numerical model? Well, the thing is that nowadays FEA packages include a modification of this famous failure theory.

In 2007, Lapczyk and Hurtado proposed a modification of the Hashin’s criteria in which damage evolution is taken into account thanks to the previously mentioned fracture energy. In this occasion, the damage variables are increased using an equivalent displacement for each failure mode. With regards to the constitutive model, the effect of damage is considered by reducing the stiffness coefficients.

Where d_{f}, d_{m}, and d_{s} are damage variables for fibre, matrix and shear failure modes, respectively. Those parameters are calculated using the fracture energy per unit are and the equivalent displacement for each mode, based on the characteristic length of the elements. (For more information, refer to reference [2]).

Hence, based on Hashin’s failure criteria, these authors rewrote the different failure modes as follows:

This last form is the one that is widely used by FE analysts nowadays. I hope this post has allowed you to understand why damage can be taken into account in Hashin when using FEA software even when the original theory does not say anything about it. As I have said in previous posts, it’s not difficult to learn how to use Finite Element packages to obtain the desired results. However, it is the knowledge of what the programme is really doing what will make you a great analyst.

[1] Hashin, Z. (1980) ‘Failure Criteria for Unidirectional Fiber Composites’, *Transactions of the ASME. Journal of Applied Mechanics, *45 (2), pp. 329-334.

[2] Lapczyk, I. and Hurtado, J.A. (2007) ‘Progressive Damage Modeling in Fiber-Reinforced Materials’, *Composites Part A, *38, pp. 2333-2341.