Maximum Stress Strength of Material

This maximum stress approach, although usually adequate when one principal stress dominates, may not be valid when the structure undergoes general multi-axial loadings. To address this issue, many failure criteria were proposed. Most of them are based on principal stresses, strains, or strain energy. By overlaying an "effective" strength of material and the loading conditions, one can determine the effectiveness of the structural elements.

Applied Stress Failure Criterion

Still, in reality, many more factors are to be considered. For example, the material is never flawless; the assembly may not be perfect; the loading may not be as designed; the environment may be harsh; the maintenance may be poor; and the service life may have to be very long. Traditionally these concerns were (hopefully) offset by a single safety factor; that engineers employ like a second nature. Unfortunately, history showed that many structures failed way below their designed capacity.

From investigating fallen structures, engineers found that most failure began with cracks. These cracks may be caused by material defects (dislocation, impurities...), discontinuities in assembly and/or design (sharp corners, grooves, nicks, voids...), harsh environments (thermal stress, corrosion...) and damages in service (impact, fatigue, unexpected loads...). Most microscopic cracks are arrested inside the material but it takes one run-away crack to destroy the whole structure.

To analyze the relationship among stresses, cracks, and fracture toughness, Fracture Mechanics was introduced. The first milestone was set by Griffith in his famous 1920 paper that quantitatively relates the flaw size to the fracture stresses. However, Griffith's approach is too primitive for engineering applications and is only good for brittle materials.

	Applied Stress
Fracture Toughness		Flaw Size

For ductile materials, the milestone did not come about until Irwin developed the concept of strain energy release rate, , in 1950s. is defined as the rate of change in potential energy near the crack area for a linear elastic material.

When the strain energy release rate reaches the critical value, , the crack will grow. Later, the strain energy release rate was replaced by the stress intensity factor K with a similar approach by other researchers.

After the fundamentals of fracture mechanics were established around 1960, scientists began to concentrate on the plasticity of the crack tips. In 1968, Rice modeled the plastic deformation as nonlinear elastic behavior and extended the method of energy release rate to nonlinear materials. He showed that the energy release rate can be expressed as a path-independent line integral, called the J integral. Rice's theory has since dominated the development of fracture mechanics in Unite States. Meanwhile, Wells proposed a parameter called crack tip opening displacement (CTOD), which led the fracture mechanics research in Europe.

Thereafter, many experiments were conducted to verify the accuracy of the models of fracture mechanics. Significant efforts were devoted to converting theories of fracture mechanics to fracture design guidelines.

Recent trends of fracture research include dynamic and time-dependent fracture on nonlinear materials, fracture mechanics of microstructures, and models related to local, global, and geometry-dependent fractures. Unlike existing major theories with a single-parameter approach (, K, J, or CTOD), these recent research trends usually require more than one parameter to describe the behavior of the crack growth, which is beyond the scope of this text.