Depending on the conditions (such as temperature, state of stress, loading rate) most materials can fail in a brittle or ductile manner or both.
This definition introduces to the fact that material failure can be examined in different scales, from microscopic, to macroscopic.
Such methodologies are useful for gaining insight in the cracking of specimens and simple structures under well defined global load distributions.
[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 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: 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.
, then the safe region for the material is assumed to be Note that the convention that tension is positive has been used in the above expression.
Numerous other phenomenological failure criteria can be found in the engineering literature.
Some popular failure criteria for various type of materials are: 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 (
is a dimensionless factor that depends on the geometry, material properties, and loading condition.
can be determined for mode II and model III loading conditions.
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 where
is the displacement at the point of application of the load due to crack growth, and
The crack is expected to propagate when the strain energy release rate exceeds a critical value
The fracture toughness and the critical strain energy release rate for plane stress are related by where
If an initial crack size is known, then a critical stress can be determined using the strain energy release rate criterion.
There are two interpretations of yield criterion: one is purely mathematical in taking a statistical approach while other models attempt to provide a justification based on established physical principles.
The following represent the most common yield criterion as applied to an isotropic material (uniform properties in all directions).
Although this criterion allows for a quick and easy comparison with experimental data it is rarely suitable for design purposes.
: Total strain energy theory – This theory assumes that the stored energy associated with elastic deformation at the point of yield is independent of the specific stress tensor.
As a result, the plastic yield behavior of the material shows directional dependency.
Several anisotropic yield criteria have been developed to deal with such situations.
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.
Several models for predicting the ultimate strength have been used by the engineering community with varying levels of success.