Richard M. Christensen
Brief Biography

Failure Theory for
Materials Science and Engineering

Professor Research Emeritus Aeronautics and Astronautics

Senior Scientist Retired Materials Science &


Three dimensional failure criteria are given for various materials classes. These include both isotropic and anisotropic material symmetries, and are applicable for macroscopic homogeneity. In the isotropic materials form, the properly calibrated failure criteria can distinguish ductile from brittle failure for specific stress states. Although most of the results are relevant to quasi-static failure, some are for time dependent failure conditions as well as for fatigue conditions.



  1. Purpose and Conditions - Attention is given to many failure related matters, but especially to the physical and mathematical basis for the failure criteria under examination.
  2. Yield and Failure Criteria for Isotropic Materials - Historical and modern failure criteria for isotropic materials are outlined and discussed. A recently developed yield and failure formalism is given which is completely calibrated by the two failure properties in uniaxial tension and compression. It necessarily involves an inherent transition from ductile to brittle failure mechanisms across the range of materials types.
    Manuscripts of Published Papers
  3. Failure Criteria for Anisotropic Fiber Composite Materials - A physically based failure formulation is given for aligned fiber composite materials. Two coordinated failure criteria are derived, one for the fiber controlled mode and one for the matrix controlled mode. The targeted applications are to carbon fiber, polymeric matrix (or equivalent) types of systems.
  4. Cumulative Damage Leading to Fatigue and Creep Failure for General Materials - Four different cumulative damage models are compared. All four models are calibrated by constant amplitude failure data bases and do not contain any adjustable parameters. Only the flaw growth theory/model shows a consistent and realistic life prediction capability for variable amplitude conditions.
    Manuscript of Published Paper
  5. Failure of Fiber Composite Laminates: Progressive Damage and Polynomial Invariants - Two distinctly different failure methods are given. The first is that of progressive damage where lamina level failure criteria are used to predict the sequence of damage/failures within the laminate. The second method is that of polynomial invariants which takes the scale of the laminate itself and its symmetry properties as the fundamental basis for failure characterization. The two methods are carefully and thoroughly compared.
  6. Critical Experimental and Theoretical Tests for Failure Criteria - The evaluation procedure for isotropic material failure criteria involves assessing their theoretical basis followed by comparisons with critical data sets (ductile and brittle). All of the long standing commonly used failure forms are found to be fundamentally incapable of functioning as general three dimensional failure criteria. All but one (Mises) cannot even serve for just a single, restricted materials type. Only the recently developed, two property failure theory outlined in Section II correlates well with all the data cases and has complete generality, going from the ductile limit to the brittle limit. Some failure criteria for fiber composites are also evaluated.
  7. The Ductile/Brittle Transition, Gaging Ductility Levels - Under the heading Organizing Principle the isotropic material failure criterion is put into the form that is best for examining ductility and brittleness matters. Starting with the ductile/brittle transition, a failure number index is derived that gages the level of ductility (and brittleness) in failure. The failure number, Fn, is expressed in terms of the uniaxial strength ratio, T/C, and the nondimensionalized first invariant of the stress state at failure. Many examples with detailed explanations are given.
  8. Fracture Mechanics - A sharply focused examination is taken of the two major failure disciplines: fracture mechanics and failure criteria. First, the historical development of the field of fracture mechanics is summarized. Then, typical but revealing examples of the use of each approach are given. Finally, the importance and usefulness of the two failure theories are assessed, both individually and relative to each other.
  9. Micromechanics Failure Analysis - The role of micromechanics in synthesizing macroscopic failure criteria is assessed. Micromechanics is seen as having a special capability beyond just that of predicting micron scale failure. Three examples are given that illustrate how particular failure modes at the micron scale bridge over to provide needed and sometimes critical information for macroscopic failure. Two of the examples are appropriate to carbon fiber composites, while the third one is for particulate composites.
  10. Defining Yield Stress and Failure Stress (Strength) - Historically there have been many different and confusing designations for the yield stress and strength. But for failure criteria purposes it is necessary to specify clear and rigorous definitions of these properties. The yield stress is determined to be given by the maximum of the second derivative of the stress strain curve. Since this is difficult to implement, an easy to use approximation to it is found. The strength property is determined to involve certain integrations over the stress strain curve, and it also is very easy to use with data.
  11. Nanomechanics of Graphene - Graphene is the planar form of carbon atoms at only one atom of thickness. Its hexagonal coordination permits characterization of the corresponding in-plane isotropic, elastic properties. These properties are derived using a nondimensional nanoscale variable that differentiates between bond stretching and bond bending (distortion) effects. The resulting elastic properties and the ensuing failure consequences are found to have a profound dependence upon the nanoscale variable.
  12. Are the Elements Ductile or Brittle: A Nanoscale Evalutation – A failure mode analysis is given for those elements in the Periodic Table that form solids at ambient conditions. The nanoscale evaluation distinguishes between the bond bending and the bond stretching mechanisms of resistance and leads to a rank order list of the ductility’s for the most commonly considered elements. Some indications are also given to gauge the quantitative levels of ductility (or brittleness) for the elements.
  13. Probabilistic Failure and Probabilistic Life Prediction - The special characteristics of power law relations are used to derive the Weibull form of probability distribution functions, with no need to resort to the overly simplistic and misleading “chain of links” argument. Secondly, in a totally different direction, power law forms are used to derive the theory of kinetic crack growth culminating in macroscopic failure. Finally, the two formalisms are merged to yield a probabilistic treatment of materials failure. Critical tests of the overall theory are conducted using the largest database in existence for creep rupture failure (static fatigue).
  14. Completion and Closure on Failure Criteria for Unidirectional Fiber Composites - Building upon previous work in Sections III and IX the failure criterion for unidirectional fiber composites is examined using a sensitivity analysis as applied to its transverse, matrix controlled failure properties. A new and general relationship is found between these three properties thereby reducing the number of independent properties needed to calibrate the theory to five. A broad but detailed evaluation of the resulting failure criterion is given. The results are conclusive and final.
  15. Failure Mechanics: The Coordination Between Elasticity Theory and Failure Theory - 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 appraised in its complementary relationship to fracture mechanics and to all of the other historical mechanics related disciplines.
  16. Failure Mechanics: The Central and Decisive Role of Graphene in Defining the Elastic and Failure Properties - Continuing from Part I, [1], 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 elasticity problem appropriate to graphene, then generalizing to three dimensional conditions. The nanomechanics analysis of graphene provides the key to the entire development.
  17. Why Carbon Based Materials are so Exceptional and Vital - The element carbon in its many different and distinctive materials forms usually has superior properties compared with other materials. The mechanical properties of stiffness and strength for the different forms of carbon are examined. Ductility is also considered. The materials forms include diamond, graphene, graphite, fullerenes, nanotubes, carbon fibers and carbon fiber/ polymeric matrix composites. The range of behaviors are surveyed and in some critical cases studied in closest detail, including at the nanoscale.
  18. A New Theory of Strain Hardening and its Consequences for Yield Stress and Failure Stress - A new theory of strain hardening is developed. Important in its own right, the strain hardening solution is also of decisive use in rigorously defining the historically broad concepts of yield stress and failure stress. Under ideal conditions yield stress is found to represent a 3rd order transition. Failure stress is an explicit “failure of function” criterion rather than just being the primitive notion of breaking into pieces. Computational extensions and opportunities are discussed.
  19. Evaluation of Ductile/Brittle Failure Theory, Derivation of the Ductile/Brittle Transition Temperature - 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 proven 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, not just to some sub-class of materials types. This evaluation program completes the development of the failure theory.


Failure Surface Graphics


Recent Additons
Updated Jan. 11, 2016

Failure Characterization

Key Junctures


Navigating the Website
Understanding the Discipline

The First Failure Criterion

General Matters


New Book on Failure

How Do Mises & Tresca Fit In


Is It Stress or Strain

A Basic Failure Mechanism


Can Atomic/Nano Scale
Failure Events Predict
Macroscopic Failure

The Ductile-Brittle Problem



Failure Theory Applications


The Brittle Limit

Unsolved Problems


Which is Fundamental:
T&C or S&D

Physical Ductility of Elements


Letter to WWFE

WWFE-II Results


A Call to Service

Timoshenko Medal



Copyright© 2016
Richard M. Christensen

Looking Ahead