Static Failure Theories

Let us summarize the majority of stress failures in metals at room temperature as either static (ductile or brittle) or fatigue. It has been common to treat most metals as isotropic. A general three-dimensional (3D) state of stress is expressed as the stress tensor 

Note that the x, y, and z directions are arbitrary. However, it is beneficial and customary to choose them along the principal axes of the component to be analyzed. Failure theories are based on the principal stresses. Consider a general 3D state of stress on an infinitesimal cube, where all components of the stress tensor are nonzero. F rincipal stresses are the eigenvalues of the stress tensor. Mathematically, the eigenvalues are a stress tensor where the off-diagonal terms, that is, shear stresses are zero. 

Mechanical design and analysis is based on either a deterministic or probabilistic approach. The overall probabilistic approach is beyond the scope of this text, but needless to say, it includes the distribution of both the loads and material strengths. The deterministic approach is usually referred to as the factor of safety approach. The first approach to a design is often to calculate the factor of safety tj, which is in general defined as the ratio of allowable property over the applied loading. In mechanical terms, the allowable property is usually either an allowable load or strength over an applied load or stress. 

The static failure theory for brittle materials has historically been the Coulomb-Mohr failure theory. A simple definition of a brittle material is that the ultimate compressive strength is greater than the tensile strength, 5 and the yield strengths are approximately eqnal to the nltimate strengths in tension and compression, = S and Cast iron, rock, and concrete are typical materials that are analyzed by 

FIGURE 2.3 Coulomb-Mohr and modified Mohr static failure theories for brittle materials. 

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FIGURE 2.4 Static failure theories for ductile materials. 

The previous static failure theory discussions were intended to transition from the knowledge that a design engineer may bring from a background of metals or other geotechnical materials to that of polymers. Specific time-dependent failures such as creep rupture and fatigue are discussed in more detail in the apphcable, subsequent chapters. 

In many cases, a product fails when the material begins to yield plastically. In a few cases, one may tolerate a small dimensional change and permit a static load that exceeds the yield strength. Actual fracture at the ultimate strength of the material would then constitute failure. The criterion for failure may be based on normal or shear stress in either case. Impact, creep and fatigue failures are the most common mode of failures. Other modes of failure include excessive elastic deflection or buckling. The actual failure mechanism may be quite complicated each failure theory is only an attempt to explain the failure mechanism for a given class of materials. In each case a safety factor is employed to eliminate failure. 

Li, X.B. Zuo, YJ. Ma, C D. 2005. Failure Criterion of Strain Energy Density and Catastrophe Theory Analysis of Rock Subjected to Static-dynamic Coupling Loading. Chinese Journal of Rock Mechanics and Engineering 24(16) 2814-2824. 

In theory, finite element strength reduction factor method can be expressed as in the finite element static steady-state calculation, if the system is unstable, finite element calculations will not converge. Based on this principle, in the nonlinear finite element slope stability analysis, we obtain formula (1), using equation (1) to adjust the surface of the structure strength (cohesion c and friction angle reduction factor CO, so that the system reaches a critical failure state... 

The validity of the viscoelastic model (5.32) has been tested against experimental and molecular dynamics simulation results [26, 27, 28]. The detailed comparison has established that the viscoelastic model works remarkably well for wavenumbers k km, where km denotes the first peak position of the static structure factor S k). However, it has also been found that the situation is not so satisfactory for smaller wavenumbers, where the viscoelastic model is shown in some circumstances to yield even qualitatively incorrect results. This failure was attributed to the fact that the single relaxation time model (5.31) cannot describe both the short-time behavior of the memory function, dominated by the so-called binary collisions, and in particular the intermediate and long-time behavior where in the liquid range additional slow processes play an important role (see the next subsection). It is obvious that these conclusions demand a more rigorous consideration of the memory function, which lead to the development of the modern version of the kinetic theory. Nevertheless, the viscoelastic model provides a rather satisfactory account of the main features of microscopic collective density fluctuations in simple liquids at relatively large wavenumbers, and its value should not be undervalued. 

Failure, thermally activated 181-182 Ferroelectric networks 81, 159 Reid theory 203-205 Rexibility, kinetic 1, 133 -, static 1... 

Okabe method is obtained from the equilibrium of the active or passive wedges (Fig. 9a and b). In addition to the forces that exist under static conditions of the failure wedge in a dry, cohesionless backfiU, the wedge is also subjected to horizontal and vertical pseudo-static forces specified as the mass of the wedge multiplied by pseudo-static accelerations ai, = khg and av = kvg. The total active and passive thrusts on a rigid waU retaining a dry, cohesionless backfill can be presented in a form similar to what is developed for the static conditions in the Coulomb s theory ... 

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