# Material failure theory

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Failure theory is the science of predicting the conditions under which solid materials fail 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.

## Material failure

In materials science, material failure is the loss of load carrying capacity of a material unit. This definition per se introduces the fact that material failure can be examined in different scales, from microscopic, to macroscopic. In structural problems, where the structural response may be beyond the initiation of nonlinear material behaviour, material failure is of profound importance for the determination of the integrity of the structure. On the other hand, due to the lack of globally accepted fracture criteria, the determination of the structure's damage, due to material failure, is still under intensive research.

## Types of material failure

Material failure can be distinguished in two broader categories depending on the scale in which the material is examined:

### Microscopic failure

Microscopic material failure is defined in terms of crack propagation and initiation. Such methodologies are useful for gaining insight in the cracking of specimens and simple structures under well defined global load distributions. Microscopic failure considers the initiation and propagation of a crack. Failure criteria in this case are related to microscopic fracture. Some of the most popular failure models in this area are the micromechanical failure models, which combine the advantages of continuum mechanics and classical fracture mechanics.[1] Such models are based on the concept that during plastic deformation, microvoids nucleate and grow until a local plastic neck or fracture of the intervoid matrix occurs, which causes the coalescence of neighbouring voids. Such a model, proposed by Gurson and extended by Tvergaard and Needleman, is known as GTN. Another approach, proposed by Rousselier, is based on continuum damage mechanics (CDM) and thermodynamics. Both models form a modification of the von Mises yield potential by introducing a scalar damage quantity, which represents the void volume fraction of cavities, the porosity f.

### Macroscopic failure

Macroscopic material failure is defined in terms of load carrying capacity or energy storage capacity, equivalently. Li[2] presents a classification of macroscopic failure criteria in four categories:

• Stress or strain failure
• Energy type failure (S-criterion, T-criterion)
• Damage failure
• Empirical failure.

Five general levels are considered, at which the meaning of deformation and failure is interpreted differently: the structural element scale, the macroscopic scale where macroscopic stress and strain are defined, the mesoscale which is represented by a typical void, the microscale and the atomic scale. The material behavior at one level is considered as a collective of its behavior at a sub-level. An efficient deformation and failure model should be consistent at every level.

## Brittle material failure criteria

Failure of brittle materials can be determined using several approaches:

### Phenomenological failure criteria

The failure criteria that were developed for brittle solids were the maximum stress/strain criteria. The maximum stress criterion assumes that a material fails when the maximum principal stress $\sigma_1$ in a material element exceeds the uniaxial tensile strength of the material. Alternatively, the material will fail if the minimum principal stress $\sigma_3$ is less than the uniaxial compressive strength of the material. If the uniaxial tensile strength of the material is $\sigma_t$ and the uniaxial compressive strength is $\sigma_c$, then the safe region for the material is assumed to be

$\sigma_c < \sigma_3 < \sigma_1 < \sigma_t \,$

Note that the convention that tension is positive has been used in the above expression.

The maximum strain criterion has a similar form except that the principal strains are compared with experimentally determined uniaxial strains at failure, i.e.,

$\varepsilon_c < \varepsilon_3 < \varepsilon_1 < \varepsilon_t \,$

The maximum principal stress and strain criteria continue to be widely used in spite of severe shortcomings.

Numerous other phenomenological failure criteria can be found in the engineering literature. The degree of success of these criteria in predicting failure has been limited. For brittle materials, some popular failure criteria are

### Linear elastic fracture mechanics

Main article: 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.[3] When applied to the mode I opening of a crack, Griffiths' theory predicts that the critical stress ($\sigma$) 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, $\gamma$ is the surface energy per unit area of the crack, and $a$ is the crack length for edge cracks or $2a$ is the crack length for plane cracks. The quantity $\sigma\sqrt{\pi a}$ is postulated as a material parameter called the fracture toughness. The mode I fracture toughness for plane strain is defined as

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

where $\sigma_c$ is a critical value of the far field stress and $Y$ is a dimensionless factor that depends on the geometry, material properties, and loading condition. The quantity $K_{\rm Ic}$ is related to the stress intensity factor and is determined experimentally. Similar quantities $K_{\rm IIc}$ and $K_{\rm IIIc}$ 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.

### Energy-based methods

Main article: Fracture mechanics

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 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 $a$ is the crack length for edge cracks or $2a$ is the crack length for plane cracks. The crack is expected to propagate when the strain energy release rate exceeds a critical value $G_{\rm Ic}$ - called the critical strain energy release rate.

The fracture toughness and the critical strain energy release rate for plane stress are related by

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

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.

## Ductile material failure criteria

Main article: Yield (engineering)

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

The yield surface of a ductile material usually changes as the material experiences increased deformation. Models for the evolution of the yield surface with increasing strain, temperature, and strain rate are used in conjunction with the above failure criteria for isotropic hardening, kinematic hardening, and viscoplasticity. Some such models are:

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 by 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.