## Mechanics of composites

A while ago, I wrote a simple document for undergraduates in order to explain that composite materials can fail in different ways. This was created as a high level document which could be used to find useful references with regards to failure modes, basic failure criteria and damage propagation models. I wanted to share this with you in case you are new in this field or just if you simply want to learn some basics of composites!

A composite can be defined as a material which is composed of two or more constituents of different chemical properties, with resultant properties different to those of the individual components. They usually consist of a continuous phase (matrix) and a distributed phase (reinforcement). These reinforcements can be fibrous, particulate or lamellar and they are usually stiff and strong, so that they are responsible for providing the stiffness and the strength of the composite. On the other hand, the matrix provides shear strength, toughness and resistance to the environment.

Fibre reinforced composites are considered as the strongest and sometimes also the stiffest, due to:

- Alignment of molecules or structural elements.
- Very fine structures.
- Elimination of defects.
- Unique structures.
- Statistical factors.

With regards to fibre reinforced composite materials, their main failure modes are:

- Fibre failure induced by tension in fibre direction.
- Fibre failure induced by compression in fibre direction.
- Matrix fracture induced by tension.
- Matrix fracture induced by compression.
- Delamination

It is remarkable that fibre failure typically caused composite failure, whereas matrix failure may not cause the same drastic effect.

### 1 FIBRE FAILURE

**Fibre tensile failure.** This failure mode usually leads to catastrophic failure, releasing a big amount of energy [1]. This type of failure is usually brittle.

**Fibre compressive failure.** Fibre reinforced composites can fail due to microbuckling of its fibres induced by compression loads. Apart from that, a phenomenon known as kinking can also be the result of compression loads. It consists of a local shear deformation of the matrix along a band [1-2].

### 2 MATRIX FAILURE

**Matrix tensile failure.** This failure mode usually results in a perpendicular fracture surface to the applied load [1].

**Matrix compressive failure.** Failure occurs at an angle with the loading direction. Therefore, this failure mode is produced by shear failure rather than compressive one [1].

### 3 FAILURE CRITERIA

The aim of this section is to provide a general overview of the most common failure criteria for composite materials. Note that there is no official classification of the failure criteria [1] [3], so the one presented here can be different if compared to other publications.

#### 3.1 Polynomial failure criteria

These criteria predict failure by interpolating between a few experimental points. They do not differentiate between failure modes. They can be expressed as follow:

According to these theories, failure occurs when the function f is equal to one.

The most common failure criteria based on this polynomial form are listed below:

- Tsai-Hill [4].
- Tsai-Wu [5].
- Hoffman [6].

#### 3.2 Non-interactive failure criteria

A non-interactive failure criterion is that one which only takes into account the effect of one stress or strain component for each failure condition. IT does not consider any interaction between the different components.

The two general non-interactive failure criteria are:

- Maximum Stress Theory.
- Maximum Strain Theory.

However, it is important to highlight that if a biaxial loading case is considered, the Maximum Strain Theory predicts failure with expressions that contain two stress components. Thus, some authors prefer to consider this theory as an interactive failure criterion.

#### 3.3 Failure criteria which differentiates between failure modes

Each theory is composed of various equations in order to define failure for different modes. Some of them are based on the Maximum Stress and the Maximum Strain theories, differentiating between fibre failure in tension/compression and matrix failure in tension/compression. Others include failure due to shear stresses as a separated failure mode.

The most common are:

- Hashin and Rotem [7].
- Hashin [8], which is a modification of the one above [7].

With regards to delamination, it is worth stating that it is usually studied using cohesive element approaches.

#### 4 DEGRADATION OF MATERIAL PROPERTIES

In their original formulation, most of the failure criteria do not include the effect of progressive damage in the material.

However, further modifications of those theories have allowed several authors to include these effects, predicting how the material properties will suffer degradation due to damage.

Most of the theories are based on the fracture energy per unit area of the material. That parameter is typically used for computing a damage variable which is then included in the calculations of stresses [9]. Those damage variables have values between zero (no damage) and one (fully damaged).

#### REFERENCES

[1] Pinho, S. (2005) *Modelling failure of laminated composites using physially-based failure models. *Imperial College London, London.

[2] Hull, D. and Clyne, T. W. (1981) *An introduction to composite materials.* Cambridge: Cambridge University Press.

[3] Paris, F. (2001) *NASA/CR-2001-210661 A study of failure criteria of fibrous composite materials*. Virginia: National Aeronautics and Space Agency.

[4] Tsai, S. W. (1965) *NASA/CR-224 Strength characterisation of composite materials.* Virginia: National Aeronautics and Space Agency.

[5] Tsai, S. W. and Wu, E. M. (1971) ‘A General Theory of Strength for Anisotropic Materials’, *Journal of Composite Materials, *5, pp. 58-80.

[6] Hoffman,O. (1697) ‘The brittle strength of orthotropic materials’, *Journal of Composite Materials, *1, pp. 200-206.

[7] Hashin Z. and Rotem, A. (1973) ‘A fatigue failure criterion for fibre reinforced materials’, *Journal of Composite Materials, *7, pp. 448-464.

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

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