Failure theory (material)

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Failure theory is the science of predicting the conditions under which solid materials lose their strength under the action of external loads. The failure of a material is usually classified into brittle failure (fracture) or ductile failure (yield). Depending on the conditions (such as temperature, state of stress, loading rate) most materials can fail in a brittle or ductile manner or both. However, for most practical situations, a material may be classified as either brittle or ductile. Though failure theory has been in development for over 200 years, its level of acceptability is yet to reach that of continuum mechanics.

In mathematical terms, failure theory is expressed in the form of various failure criteria which are valid for specific materials. Failure criteria are functions in stress or strain space which separate "failed" states from "unfailed" states. A precise physical definition of a "failed" state is not easily quantified and several working definitions are in use in the engineering community. Quite often, phenomenological failure criteria of the same form are used to predict brittle failure and ductile yield.

1 Brittle material failure criteria
2 Ductile material failure criteria
3 References
4 See also
5 External links

[edit] Brittle material failure criteria

Failure of brittle materials can be determined using several approaches:

Phenomenological failure criteria
Linear elastic fracture mechanics
elastic-plastic fracture mechanics
Energy-based methods
Cohesive zone methods

[edit] Phenomenological failure criteria

Criteria that can be used to determine whether a material has failed in a brittle manner include

maximum stress criteria
maximum strain criteria
Mohr-Coulomb failure criterion
Tresca or maximum shear stress failure criterion
von Mises or maximum elastic distortional energy criterion
Drucker-Prager failure criterion
Hill's anisotropic failure criteria
Tsai-Wu failure criterion
criteria based on invariants of the Cauchy stress tensor

[edit] Linear elastic fracture mechanics

The approach taken in linear elastic fracture mechanics is to estimate the amount of energy needed to grow a preexisting crack in a brittle material. The earliest fracture mechanics approach for unstable crack growth is Griffiths' theory ^[1]. When applied to the mode I opening of a crack, Griffiths' theory predicts that the critical stress ( $σ$ ) needed to propagate the crack is given by

$\sigma = \sqrt{\cfrac{2 E \gamma}{\pi~a}}$

where $E$ is the Young's modulus of the material, $γ$ is the surface energy per unit area of the crack, and $2 a$ is the crack length. The quantity $\sigma\sqrt{\pi a}$ is postulated as a material parameter called the 'fracture toughness. The mode I fracture toughness is defined as

$K_{Ic} = \sigma_c\sqrt{\pi a}$

and is determined experimentally. Similar quantities $K I I c$ and $K I I I c$ can be determined for mode II and model III loading conditions.

The state of stress around cracks of various shapes can be expressed in terms of their stress intensity factors. Linear elastic fracture mechanics predicts that a crack will extend when the stress intensity factor at the crack tip is greater than the fracture toughness of the material. Therefore the critical applied stress can also be determined once the stress intensity factor at a crack tip is known.

See the article on fracture mechanics for more detail.

[edit] Energy-based methods

The linear elastic fracture mechanics method is difficult to apply for anisotropic materials (such as composites) or for situations where the loading or the geometry are complex. The strain energy release rate ^[2] approach has proved quite useful for such situations. The strain energy release rate for a mode I crack which runs through the thickness of a plate is defined as

$G_I := \cfrac{P}{2t}~\cfrac{du}{da}$

where $P$ is the applied load, $t$ is the thickness of the plate, $u$ is the displacement at the point of application of the load due to crack growth, and $2 a$ is the length of the crack. The crack is expected to propagate when the strain energy release rate exceeds a critical value $G I c$ - called the critical strain energy release rate.

The fracture toughness and the critical stain energy release rate are related by

$G_{Ic} = \cfrac{1}{E}~K_{Ic}^2$

where $E$ is the Young's modulus. If an initial crack size is known, then a critical stress can be determined using the strain energy release rate criterion.

See the article on fracture mechanics for more detail.

[edit] Ductile material failure criteria

Criteria used to predict the failure of ductile materials are usually called yield criteria. Commonly used failure criteria for ductile materials are:

the Tresca or maximum shear stress criterion.
the von Mises yield criterion or distortional strain energy density criterion.
the Hill yield criteria.
various criteria based on the invariants of the Cauchy stress tensor.

See the article on yield for further details on yield criteria.

There is another important aspect to ductile materials - the prediction of the ultimate failure strength of a ductile material. Several models for predicting the ultimate strength have been used my the engineering community with varying levels of success. For metals, such failure criteria are usually expressed in terms of a combination of porosity and strain to failure or in terms of a damage parameter.