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{{Short description|Science of predicting if, when, and how a given material will fail under loading}}
{{more citations needed|date=November 2014}}
{{Continuum mechanics|cTopic=solid}}
{{Mechanical failure modes}}

'''Material failure theory''' is an interdisciplinary field of [[materials science]] and [[solid mechanics]] which attempts to [[Failure analysis|predict the conditions]] under which solid [[material]]s fail under the action of [[Structural load|external loads]]. The failure of a material is usually classified into [[brittle]] failure ([[fracture]]) or [[ductile]] failure ([[yield (engineering)|yield]]). Depending on the conditions (such as [[temperature]], state of [[Stress (mechanics)|stress]], loading rate) most materials can fail in a brittle or ductile manner or both. However, for most practical situations, a material may be classified as either brittle or ductile.

In mathematical terms, failure theory is expressed in the form of various failure criteria which are valid for specific materials.  Failure criteria are functions in [[Stress–strain analysis|stress or strain space]] which separate "failed" states from "unfailed" states.  A precise physical definition of a "failed" state is not easily quantified and several working definitions are in use in the engineering community.  Quite often, phenomenological failure criteria of the same form are used to predict brittle failure and ductile yields.

== Material failure ==
In [[materials science]], '''material failure''' is the loss of load carrying capacity of a material unit. This definition introduces to the fact that material failure can be examined in different scales, from [[microscopic]], to [[macroscopic]]. In structural problems, where the structural response may be beyond the initiation of nonlinear material behaviour, material failure is of profound importance for the determination of the integrity of the structure. On the other hand, due to the lack of globally accepted [[fracture]] criteria, the determination of the structure's damage, due to material failure, is still under intensive research.

== Types of material failure ==
Material failure can be distinguished in two broader categories depending on the scale in which the material is examined:

=== Microscopic failure ===
Microscopic material failure is defined in terms of crack initiation and propagation. Such methodologies are useful for gaining insight in the cracking of specimens and simple structures under well defined global load distributions. Microscopic failure considers the initiation and propagation of a crack. Failure criteria in this case are related to microscopic fracture. Some of the most popular failure models in this area are the micromechanical failure models, which combine the advantages of [[continuum mechanics]] and classical [[fracture mechanics]].<ref>Besson J., Steglich D., Brocks W. (2003), Modelling of plain strain ductile rupture, ''International Journal of Plasticity'', 19.</ref> Such models are based on the concept that during [[plastic deformation]], microvoids nucleate and grow until a local plastic neck or fracture of the intervoid matrix occurs, which causes the coalescence of neighbouring voids. Such a model, proposed by Gurson and extended by Tvergaard and [[Alan Needleman|Needleman]], is known as GTN. Another approach, proposed by Rousselier, is based on continuum [[damage mechanics]] (CDM) and [[thermodynamics]]. Both models form a modification of the von Mises yield potential by introducing a scalar damage quantity, which represents the void volume fraction of cavities, the porosity ''f''.

=== Macroscopic failure ===
Macroscopic material failure is defined in terms of load carrying capacity or energy storage capacity, equivalently. Li<ref>Li, Q.M. (2001), Strain energy density failure criterion, ''International Journal of Solids and Structures'' '''38''', pp. 6997–7013.</ref> presents a classification of macroscopic failure criteria in four categories:
* Stress or strain failure
* Energy type failure (S-criterion, [[T-criterion]])
* Damage failure
* Empirical failure

Five general levels are considered, at which the meaning of deformation and failure is interpreted differently: the structural element scale, the macroscopic scale where macroscopic stress and strain are defined, the mesoscale which is represented by a typical void, the microscale and the atomic scale. The material behavior at one level is considered as a collective of its behavior at a sub-level. An efficient deformation and failure model should be consistent at every level.

== Brittle material failure criteria ==
Failure of brittle materials can be determined using several approaches:
* Phenomenological failure criteria
* Linear elastic [[fracture mechanics]]
* Elastic-plastic fracture mechanics
* Energy-based methods
* [[Cohesive zone model|Cohesive zone methods]]

=== Phenomenological failure criteria ===
The failure criteria that were developed for brittle solids were the maximum [[stress (mechanics)|stress]]/[[finite strain theory|strain]] criteria.  The '''maximum stress criterion''' assumes that a material fails when the maximum [[principal stress]] <math>\sigma_1</math> in a material element exceeds the uniaxial tensile strength of the material.  Alternatively, the material will fail if the minimum principal stress <math>\sigma_3</math> is less than the uniaxial compressive strength of the material.  If the uniaxial tensile strength of the material is <math>\sigma_t</math> and the uniaxial compressive strength is <math>\sigma_c</math>, then the safe region for the material is assumed to be
:<math>
    \sigma_c < \sigma_3 < \sigma_1 < \sigma_t \,
 </math>
Note that the convention that tension is positive has been used in the above expression.

The '''maximum strain criterion''' has a similar form except that the principal strains are compared with experimentally determined uniaxial strains at failure, i.e.,
:<math>
    \varepsilon_c < \varepsilon_3 < \varepsilon_1 < \varepsilon_t \,
 </math>
The maximum principal stress and strain criteria continue to be widely used in spite of severe shortcomings.

Numerous other phenomenological failure criteria can be found in the engineering literature.  The degree of success of these criteria in predicting failure has been limited.  Some popular failure criteria for various type of materials are:
* criteria based on invariants of the [[Cauchy stress tensor]]
* the [[Tresca yield criterion|Tresca or maximum shear stress]] failure criterion
* the [[von Mises yield criterion|von Mises]] or maximum elastic distortional energy criterion
* the [[Mohr-Coulomb theory|Mohr-Coulomb failure criterion]] for cohesive-frictional solids
* the [[Drucker Prager|Drucker-Prager failure criterion]] for pressure-dependent solids
* the [[Bresler Pister yield criterion|Bresler-Pister failure criterion]] for concrete
* the [[Willam-Warnke yield criterion|Willam-Warnke failure criterion]] for concrete
* the [[Hankinson's equation|Hankinson criterion]], an empirical failure criterion that is used for orthotropic materials such as wood
* the [[Hill yield criteria]] for anisotropic solids
* the [[Tsai-Wu failure criterion]] for anisotropic composites
* the [[Johnson–Holmquist damage model]] for high-rate deformations of isotropic solids
* the [[Hoek-Brown failure criterion]] for rock masses
* the [[Critical state soil mechanics|Cam-Clay failure theory]] for soil

=== Linear elastic fracture mechanics ===
{{main|Fracture mechanics}}
The approach taken in [[fracture mechanics|linear elastic fracture mechanics]] is to estimate the amount of energy needed to grow a preexisting crack in a brittle material.  The earliest [[fracture mechanics]] approach for unstable crack growth is Griffiths' theory.<ref>[http://gallica.bnf.fr/ark:/12148/bpt6k560264/f178.table  Griffiths, A.A. 1920. The phenomena of rupture and flow in solids. Phil.Trans.Roy.Soc.Lond. A221, 163.]</ref>  When applied to the [[fracture mechanics|mode I]] opening of a crack, Griffiths' theory predicts that the critical stress (<math>\sigma</math>) needed to propagate the crack is given by
:<math>
   \sigma = \sqrt{\cfrac{2 E \gamma}{\pi a}}
 </math>
where <math>E</math> is the Young's modulus of the material, <math>\gamma</math> is the surface energy per unit area of the crack, and <math>a</math> is the crack length for edge cracks or <math>2a</math> is the crack length for plane cracks.  The quantity <math>\sigma\sqrt{\pi a}</math> is postulated as a material parameter called the '''fracture toughness'''.  The mode I [[fracture toughness]] for [[plane strain]] is defined as
:<math>
   K_{\rm Ic} = Y\sigma_c\sqrt{\pi a} 
 </math>
where <math>\sigma_c</math> is a critical value of the far field stress and <math>Y</math> is a dimensionless factor that depends on the geometry, material properties, and loading condition.  The quantity <math>K_{\rm Ic}</math> is related to the [[stress intensity factor]] and is determined experimentally.  Similar quantities <math>K_{\rm IIc}</math> and <math>K_{\rm IIIc}</math> can be determined for [[fracture mechanics|mode II]] and [[fracture mechanics|model III]] loading conditions.

The state of stress around cracks of various shapes can be expressed in terms of their [[stress intensity factor]]s.  Linear elastic fracture mechanics predicts that a crack will extend when the stress intensity factor at the crack tip is greater than the fracture toughness of the material.  Therefore, the critical applied stress can also be determined once the stress intensity factor at a crack tip is known.

=== Energy-based methods ===
{{Main|Energy release rate}}

The linear elastic fracture mechanics method is difficult to apply for anisotropic materials (such as [[Composite material|composites]]) or for situations where the loading or the geometry are complex.  The [[strain energy release rate]] approach has proved quite useful for such situations.  The strain energy release rate for a mode I crack which runs through the thickness of a plate is defined as
:<math>
   G_I = \cfrac{P}{2t}~\cfrac{du}{da}
 </math>
where <math>P</math> is the applied load, <math>t</math> is the thickness of the plate, <math>u</math> is the displacement at the point of application of the load due to crack growth, and <math>a</math> is the crack length for edge cracks or <math>2a</math> is the crack length for plane cracks.  The crack is expected to propagate when the strain energy release rate exceeds a critical value <math>G_{\rm Ic}</math> - called the '''critical strain energy release rate'''.

The [[fracture toughness]] and the critical strain energy release rate for [[plane stress]] are related by
:<math>
   G_{\rm Ic} = \cfrac{1}{E}~K_{\rm Ic}^2 
 </math>
where <math>E</math> is the Young's modulus.  If an initial crack size is known, then a critical stress can be determined using the strain energy release rate criterion.

== Ductile material failure (yield) criteria ==
{{unreferenced section|date=June 2013}}
A yield criterion often expressed as yield surface, or yield locus, is a hypothesis concerning the limit of elasticity under any combination of stresses. There are two interpretations of yield criterion: one is purely mathematical in taking a statistical approach while other models attempt to provide a justification based on established physical principles. Since stress and strain are [[tensor]] qualities they can be described on the basis of three principal directions, in the case of stress these are denoted by <math>\sigma_1 \,\!</math>, <math>\sigma_2 \,\!</math>, and <math>\sigma_3 \,\!</math>.

The following represent the most common yield criterion as applied to an isotropic material (uniform properties in all directions). Other equations have been proposed or are used in specialist situations.

===Isotropic yield criteria===
'''Maximum principal stress theory''' – by [[William Rankine]] (1850). Yield occurs when the largest principal stress exceeds the uniaxial tensile yield strength. Although this criterion allows for a quick and easy comparison with experimental data it is rarely suitable for design purposes. This theory gives good predictions for brittle materials.
{{block indent|<math>\sigma_1 \le \sigma_y \,\!</math>}}

'''Maximum principal strain theory''' – by St.Venant. Yield occurs when the maximum principal [[strain (materials science)|strain]] reaches the strain corresponding to the yield point during a simple tensile test. In terms of the principal stresses this is determined by the equation:
{{block indent|<math>\sigma_1 - \nu\left(\sigma_2 + \sigma_3\right) \le \sigma_y. \,\!</math>}}

'''Maximum [[shear stress]] theory''' – Also known as the [[Tresca yield criterion]], after the French scientist [[Henri Tresca]]. This assumes that yield occurs when the shear stress <math>\tau\!</math> exceeds the shear yield strength <math>\tau_y\!</math>:
{{block indent|<math>\tau = \frac{\sigma_1 - \sigma_3}{2} \le \tau_y. \,\!</math>}}

'''Total strain energy theory''' – This theory assumes that the stored energy associated with elastic deformation at the point of yield is independent of the specific stress tensor. Thus yield occurs when the strain energy per unit volume is greater than the strain energy at the elastic limit in simple tension. For a 3-dimensional stress state this is given by:
{{block indent|<math>\sigma_{1}^2 + \sigma_{2}^2 + \sigma_{3}^2 - 2 \nu \left(\sigma_1 \sigma_2 + \sigma_2 \sigma_3 + \sigma_1 \sigma_3\right) \le \sigma_y^2. \,\! </math>}}

'''Maximum distortion energy theory''' ([[von Mises yield criterion]]) also referred to as '''octahedral shear stress theory'''.<ref>{{Cite web |last=sdcadmin |date=2022-05-05 |title=What is von Mises Stress? |url=https://sdcverifier.com/articles/what-is-von-mises-stress/ |access-date=2022-11-03 |website=SDC Verifier |language=en-US}}</ref> – This theory proposes that the total strain energy can be separated into two components: the ''volumetric'' ([[hydrostatic]]) strain energy and the ''shape'' (distortion or [[Shearing (physics)|shear]]) strain energy. It is proposed that yield occurs when the distortion component exceeds that at the yield point for a simple tensile test. This theory is also known as the [[von Mises yield criterion]].

The [[yield surface]]s corresponding to these criteria have a range of forms.  However, most isotropic yield criteria correspond to [[convex polytope|convex]] yield surfaces.

===Anisotropic yield criteria===
When a metal is subjected to large plastic deformations the grain sizes and orientations change in the direction of deformation. As a result, the plastic yield behavior of the material shows directional dependency.  Under such circumstances, the isotropic yield criteria such as the von Mises yield criterion are unable to predict the yield behavior accurately. Several anisotropic yield criteria have been developed to deal with such situations.
Some of the more popular anisotropic yield criteria are:
* [[Hill yield criteria|Hill's quadratic yield criterion]]
* [[Hill yield criteria|Generalized Hill yield criterion]]
* [[Hosford yield criterion]]

=== Yield surface ===
{{Main|Yield surface}}
The [[yield surface]] of a ductile material usually changes as the material experiences increased [[deformation (mechanics)|deformation]].  Models for the evolution of the yield surface with increasing strain, temperature, and strain rate are used in conjunction with the above failure criteria for [[isotropic hardening plasticity|isotropic hardening]], [[kinematic hardening plasticity|kinematic hardening]], and [[viscoplasticity]].  Some such models are:
* the [[Johnson-Cook plasticity model|Johnson-Cook model]]
* the [[Steinberg-Guinan plasticity model|Steinberg-Guinan model]]
* the [[Zerilli-Armstrong plasticity model|Zerilli-Armstrong model]]
* the [[Mechanical threshold stress plasticity model|Mechanical threshold stress model]]
* the [[Preston-Tonks-Wallace plasticity model|Preston-Tonks-Wallace model]]

There is another important aspect to ductile materials - the prediction of the [[tensile strength|ultimate failure strength]] of a ductile material.  Several models for predicting the ultimate strength have been used by the engineering community with varying levels of success.  For metals, such failure criteria are usually expressed in terms of a combination of porosity and strain to failure or in terms of a [[damage (mechanics)|damage]] parameter.

== See also ==
* [[Fracture mechanics]]
* [[Fracture]]
* [[Stress intensity factor]]
* [[Yield (engineering)]]
* [[Yield surface]]
* [[Plasticity (physics)]]
* [[Structural failure]]
* [[Strength of materials]]
* [[Ultimate failure]]
* [[Damage mechanics]]
* [[Size effect on structural strength]]
* [[Concrete fracture analysis]]

== References ==
{{reflist}}

{{Topics in continuum mechanics}}

[[Category:Mechanical failure]]
[[Category:Plasticity (physics)]]
[[Category:Solid mechanics]]
[[Category:Mechanics]]
[[Category:Materials science]]
[[Category:Materials degradation]]
[[Category:Fracture mechanics]]