The Theoretical Measure of the Ductility of Failure for All Isotropic Materials in All States of Stress1

2016 ◽  
Vol 83 (6) ◽  
Author(s):  
Richard M. Christensen
Keyword(s):  
Stress State ◽  
Brittle Failure ◽  
Failure Stress ◽  
Isotropic Materials ◽  
Failure Theory

The ductile/brittle failure theory for homogeneous and isotropic materials is extended to give a rational and mathematically rigorous measure for the ductility of failure. This new failure number methodology is completely developed and proved to be valid and general. It applies to all isotropic materials as subjected to any and all states of stress. Not only does the failure theory predict the safety or failure for any given stress state, it then projects the quantitative ductility level for the failure stress state. Many important examples are given with detailed interpretations of the results and with guides for general usage.

Evaluation of Ductile/Brittle Failure Theory and Derivation of the Ductile/Brittle Transition Temperature

2015 ◽  
Vol 83 (2) ◽  
Author(s):  
Richard M. Christensen
Keyword(s):  
Transition Temperature ◽  
Brittle Failure ◽  
Evaluation Process ◽  
Isotropic Materials ◽  
Brittle Transition ◽  
Failure Theory ◽  
Ductile Brittle Transition Temperature ◽  
Brittle Transition Temperature ◽  
Materials Failure

A recently developed ductile/brittle theory of materials failure is evaluated. The failure theory applies to all homogeneous and isotropic materials. The determination of the ductile/brittle transition is an integral and essential part of the failure theory. The evaluation process emphasizes and examines all aspects of the ductile versus the brittle nature of failure, including the ductile limit and the brittle limit of materials' types. The failure theory is proved to be extraordinarily versatile and comprehensive. It even allows derivation of the associated ductile/brittle transition temperature. This too applies to all homogeneous and isotropic materials and not just some subclass of materials' types. This evaluation program completes the development of the failure theory.

scholarly journals A Comprehensive Theory of Yielding and Failure for Isotropic Materials

2007 ◽  
Vol 129 (2) ◽  
pp. 173-181 ◽  
Author(s):  
Richard M. Christensen
Keyword(s):  
Stress State ◽  
Brittle Failure ◽  
Three Dimensional ◽  
Present Theory ◽  
Full Density ◽  
Isotropic Materials ◽  
Comprehensive Theory ◽  
General Material ◽  
Ductile Behavior ◽  
Limiting Case

A theory of yielding and failure for homogeneous and isotropic materials is given. The theory is calibrated by two independent, measurable properties and from those it predicts possible failure for any given state of stress. It also differentiates between ductile yielding and brittle failure. The explicit ductile-brittle criterion depends not only upon the material specification through the two properties, but also and equally importantly depends upon the type of imposed stress state. The Mises criterion is a special (limiting) case of the present theory. A close examination of this case shows that the Mises material idealization does not necessarily imply ductile behavior under all conditions, only under most conditions. When the first invariant of the yield/failure stress state is sufficiently large relative to the distortional part, brittle failure will be expected to occur. For general material types, it is shown that it is possible to have a state of spreading plastic flow, but as the elastic-plastic boundary advances, the conditions for yielding on it can change over to conditions for brittle failure because of the evolving stress state. The general theory is of a three-dimensional form and it applies to full density materials for which the yield/failure strength in uniaxial tension is less than or at most equal to the magnitude of that in uniaxial compression.

The Failure Theory for Isotropic Materials: Proof and Completion

2019 ◽  
Vol 87 (5) ◽  
Author(s):  
Richard M. Christensen
Keyword(s):  
Brittle Failure ◽  
Ductile Failure ◽  
Isotropic Materials ◽  
Failure Theory ◽  
History Of ◽  
Failure Properties ◽  
Complete Form ◽  
The Many ◽  
Entire Theory

Abstract This work represents the completion of the many developments in recent years on failure theory for homogeneous and isotropic materials. Presented here is the resulting failure formalism in final and technically complete form. Significant further results are also given for the verification of the failure formalism. The scope of this paper goes from the history of misguided failure theory investigations right up to the present final tested forms ready for applications. For every predicted failure level in terms of the stresses, there is an accompanying ductility level. This ranges from brittle failure up to fully ductile failure. The entire theory is calibrated by only two specified parameters (failure properties). Nothing else is needed. The seemingly interminable, actually centuries long search for the missing theory of failure has finally been brought to a resolute and successful conclusion.

Proof of the Splitting Mode of Compressive Brittle Failure and Integration With General Failure Theory

2019 ◽  
Vol 86 (9) ◽  
Author(s):  
Richard M. Christensen
Keyword(s):  
Uniaxial Compression ◽  
Special Interest ◽  
Brittle Failure ◽  
Difficult Problem ◽  
Theoretical Treatment ◽  
Isotropic Materials ◽  
Failure Theory ◽  
Mode Of Failure ◽  
Technical Capability

The problem of special interest is the nature of the mode of failure in uniaxial compression at the brittle limit. This problem is known by observation to undergo a splitting mode of failure. The present work gives a full theoretical treatment and proof for this mode of failure. The general failure theory of Christensen for isotropic materials provides the basis for the derivation. The solution demonstrates the depth of technical capability that is required from the failure theory to treat such a classically difficult problem.

scholarly journals An independent derivation and verification of the voids nucleation failure mechanism: significance for materials failure

2019 ◽  
Vol 475 (2222) ◽  
pp. 20180755 ◽  
Author(s):  
Richard Christensen ◽  
Zhi Li ◽  
Huajian Gao
Keyword(s):  
Stress State ◽  
Failure Mechanism ◽  
Density Functional ◽  
Failure Modes ◽  
Shear Bands ◽  
Failure Criteria ◽  
Atomic Scale ◽  
Ideal Strength ◽  
Failure Theory ◽  
The Ideal

Independent derivations are given for the failure criteria of the purely dilatational stress state involving voids nucleation failure as well as for the purely distortional stress state involving shear bands failure. The results are consistent with those from a recently derived failure theory and they further substantiate the failure theory. The voids nucleation mechanism is compared with the ideal theoretical strength of isotropic materials as derived by density functional theory and two other atomic-scale methods. It is found that a cross-over occurs from the voids nucleation failure mechanism to the ideal strength limitation as the tensile to compressive strengths ratio, T / C , increases toward a value of unity. All the results are consistent with the failure modes transition results from the general failure theory.

Failure Mechanics—Part II: The Central and Decisive Role of Graphene in Defining the Elastic and Failure Properties for all Isotropic Materials

2014 ◽  
Vol 81 (11) ◽  
Author(s):  
Richard M. Christensen
Keyword(s):  
Elasticity Theory ◽  
Three Dimensional ◽  
Decisive Role ◽  
Common Belief ◽  
Failure Mechanics ◽  
Isotropic Materials ◽  
Failure Theory ◽  
Failure Properties ◽  
2D Elasticity

Continuing from Part I (Christensen, 2014, “Failure Mechanics—Part I: The Coordination Between Elasticity Theory and Failure Theory for all Isotropic Materials,” ASME J. Appl. Mech., 81(8), p. 081001), the relationship between elastic energy and failure specification is further developed. Part I established the coordination of failure theory with elasticity theory, but subject to one overriding assumption: that the values of the involved Poisson's ratios always be non-negative. The present work derives the physical proof that, contrary to fairly common belief, Poisson's ratio must always be non-negative. It can never be negative for homogeneous and isotropic materials. This is accomplished by first probing the reduced two-dimensional (2D) elasticity problem appropriate to graphene, then generalizing to three-dimensional (3D) conditions. The nanomechanics analysis of graphene provides the key to the entire development. Other aspects of failure theory are also examined and concluded positively. Failure theory as unified with elasticity theory is thus completed, finalized, and fundamentally validated.

Failure Mechanics—Part I: The Coordination Between Elasticity Theory and Failure Theory for all Isotropic Materials

2014 ◽  
Vol 81 (8) ◽  
Author(s):  
Richard M. Christensen
Keyword(s):  
Mechanical Behavior ◽  
Elasticity Theory ◽  
Classical Theory ◽  
Theory Of Elasticity ◽  
Poisson's Ratio ◽  
Failure Mechanics ◽  
Isotropic Materials ◽  
Intimate Connection ◽  
Failure Theory ◽  
Theory Failure

Failure mechanics is comprised of the failure theory for homogeneous and isotropic materials along with all of its implications and applications. The present failure theory is found to have an intimate connection with elasticity behavior even though plasticity may also transpire. This becomes apparent and useful when the classical theory of elasticity is renormalized to give a simpler and more transparent (but still exact) formalism. The connection or coordination between elasticity and failure then explicitly occurs through the use of the renormalized Poisson's ratio to characterize the ductility of failure. With this unification of failure theory and elasticity theory, failure mechanics can be extended to explain other anomalous aspects of mechanical behavior and prepare it for applications.

scholarly journals Regularities of crustal faulting and tectonophysical indicators of fault metastability

2018 ◽  
Vol 9 (3) ◽  
pp. 629-652
Author(s):  
Yu. L. Rebetsky
Keyword(s):  
Stress State ◽  
Fault Zone ◽  
Seismic Data ◽  
Brittle Failure ◽  
Stress Gradient ◽  
Large Earthquake ◽  
Fault Zones ◽  
Time Interval ◽  
Entire Area ◽  
Specific Distribution

The problem of forecasting seismic hazards is discussed. The stress state data characterizing various aspects of brittle failure are reviewed in detail. It is shown that the most convenient tool for analyzing such data is the Mohr stress diagram and the Coulomb criterion. Noted is the role of a fluid in not only reducing the normal stresses responsible for brittle failure, but also predetermining the major processes in fault zones. In each fault body, a node can be distinguished as a fault part wherein the main structural and material transformations take place. The node contains narrow elongated zones of modification of mylonites, from protomylonites to ultramylonites and blastomylonites, that are related to the localization of continuous and discontinuous shear deformation. Due to the metamorphic processes, fault zones are less strong than the surrounding consolidated blocks of the crust. A theoretical analysis of the mechanism of displacements along the discontinuities of different scale ranks shows differences in their manifestation. Tectonic and seismic displacements along the rupture occupy the entire area at once, while displacements along the fault zone occur in stages along its extent and follow the ‘rolling-carpet’ principle that is also typical of intra-crystal dislocations. The stress state in the vicinity of ruptures and faults has different characteristic features. Based on the seismological and tectonophysical data on earthquake focal parameters and discontinuities, it is possible to identify two or three ranks of stresses, which differ in the laws predetermining their mutual relationships. Actually, this conclusion contradicts the hypothesis of self-similarity of discontinuities in their continuous range, from a dislocation to a fault zone, which length amounts to tens of kilometers. Besides, it imposes a restriction on the use of statistical analysis of seismic data. The seismic data show that in the source of a large earthquake, displacement develops as a running band (‘rolling-carpet’ principle). In the source of a weak earthquake, it occupies the entire earthquake focal area at once. The differences in the types of shearing in the sources of weak and strong earthquakes are related to the relationships between three dynamic parameters of the medium: velocity of seismic wave propagation, rate of rupture propagation, and displacement rate of the sides of the fracture. Using tectonophysical methods, the stress state was reconstructed for the seismically active regions of the planet and the sources of the mega-earthquakes of the 21st century. Based on the reconstructions, the mean strength and stress values were calculated, and the specific features of the stress fields were revealed. It is established that the strongest regional earthquakes ‘avoided’ the areas with increased effective isotropic pressure. The sizes of the sources of the strongest earthquakes were controlled by the size of the region with decreased effective pressure. The sites, wherefrom the earthquake were initiated, were often located in the zones of the highest stress gradients. These regularities support the term “metastability of the state of fault zone” (introduced to seismology from the physics of the states of matter) and justify it by a specific distribution pattern of stress values prior to the mega-earthquake. Based on the tectonophysical definition of the metastable state of faults, the important role is outlined for a stress gradient zone that represents a location wherein a trigger earthquake occurs. The ‘maturity’ of the zone with increased stress gradient values is, in essence, a characteristic of the time interval of metastability of the fault zone.

scholarly journals A criterion for the fragmentation of bubbly magma based on brittle failure theory

Nature ◽  
10.1038/45210 ◽  
1999 ◽  
Vol 402 (6762) ◽  
pp. 648-650 ◽  
Author(s):  
Youxue Zhang
Keyword(s):  
Brittle Failure ◽  
Failure Theory
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