The Failure Theory for Isotropic Materials

2013 ◽  
pp. 30-49 ◽  
Keyword(s):  
Isotropic Materials ◽  
Failure Theory

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.

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.

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.

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.

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.

Materials Failure Theory and Applications, Change Is Coming

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Richard M. Christensen
Keyword(s):  
Failure Criteria ◽  
The State ◽  
Isotropic Materials ◽  
Current State ◽  
Failure Theory ◽  
The Future ◽  
Materials Failure ◽  
The Many ◽  
State Of The Field

This overview/survey assesses the state of the discipline for the failure of homogeneous and isotropic materials. It starts with a quick review of the many historical but unsuccessful failure investigations. Then, it outlines the dysfunctional current state of the field for failure criteria. Finally, it converges toward the technical prospects that can and very likely will bring much needed change and progress in the future.

scholarly journals Homogeneous and Isotropic Materials Failure Theory: A Critical Appraisal

2021 ◽  
pp. 1-6
Author(s):  
Richard M. Christensen
Keyword(s):  
Critical Appraisal ◽  
Isotropic Materials ◽  
Failure Theory ◽  
Critical Detail ◽  
Materials Failure

Abstract The historical status of failure theory is surveyed and found to be close to chaotic. Abandoning that source, the constructive associations and operations that must be required in order to form a viable theory of materials failure are examined in critical detail. The consequent failure theory has been established and its future is discussed.

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