Damage mechanics is concerned with the representation, or modeling, of damage of materials that is suitable for making engineering predictions about the initiation, propagation, and fracture of materials without resorting to a microscopic description that would be too complex for practical engineering analysis.
[1] Damage mechanics illustrates the typical engineering approach to model complex phenomena.
To quote Dusan Krajcinovic, "It is often argued that the ultimate task of engineering research is to provide not so much a better insight into the examined phenomenon but to supply a rational predictive tool applicable in design.
[3] The state variables may be measurable, e.g., crack density, or inferred from the effect they have on some macroscopic property, such as stiffness, coefficient of thermal expansion, remaining life, etc.
The state variables have conjugate thermodynamic forces that motivate further damage.
In plasticity like formulations, the damage evolution is controlled by a hardening function but this requires additional phenomenological parameters that must be found through experimentation, which is expensive, time consuming, and virtually no one does.
[4] When mechanical structures are exposed to temperatures exceeding one-third of the melting temperature of the material of construction, time-dependent deformation (creep) and associated material degradation mechanisms become dominant modes of structural failure.
While these deformation and damage mechanisms originate at the microscale where discrete processes dominate, practical application of failure theories to macroscale components is most readily achieved using the formalism of continuum mechanics.
In this context, microscopic damage is idealized as a continuous state variable defined at all points within a structure.
State equations are defined which govern the time evolution of damage.
These equations may be readily integrated into finite element codes to analyze the damage evolution in complex 3D structures and calculate how long a component may safely be used before failure occurs.
L. M. Kachanov[5] and Y. N. Rabotnov[6] suggested the following evolution equations for the creep strain ε and a lumped damage state variable ω: where
The damage term ω is interpreted as a distributed loss of load bearing area which results in an increased local stress at the microscale.
The time to failure is determined by integrating the damage evolution equation from an initial undamaged state
is taken to be 1, this results in the following prediction for a structure loaded under a constant uniaxial stress
and m are found by fitting the above equation to creep rupture life data.
Correspondingly, extrapolation of the model beyond the original dataset of test data is not justified.
Ashby,[10] and B.F. Dyson,[11] who proposed mechanistically informed strain and damage evolution equations.
Extrapolation using such equations is justified if the dominant damage mechanism remains the same at the conditions of interest.
In the power-law creep regime, global deformation is controlled by glide and climb of dislocations.
If internal voids are present within the microstructure, global structural continuity requires that the voids must both elongate and expand laterally, further reducing the local section.
When cast in the damage mechanics formalism, the growth of internal voids by power-law creep can be represented by the following equations.
is the average initial void radius, and d is the grain size.
At very high temperature and/or low stresses, void growth on grain boundaries is primarily controlled by the diffusive flux of vacancies along the grain boundary.
When cast in the damage mechanics formalism, the growth of internal voids by boundary diffusion can be represented by the following equations.
Many precipitates are not thermodynamically stable and grow via diffusion when exposed to elevated temperatures.
As the precipitates coarsen, their ability to restrict dislocation motion decreases as the average spacing between particles increases, thus decreasing the required Orowan stress for bowing.
When cast into the damage mechanics formalism, precipitation coarsening and its effect on strain rate may be represented by the following equations.
Multiple damage mechanism can be combined to represent a broader range of phenomena.
For instance, if both void-growth by power-law creep and precipitate coarsening are relevant mechanisms, the following combined set of equations may be used: Note that both damage mechanisms are included in the creep strain rate equation.