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| '''Failure theory''' is the science of predicting the conditions under which solid materials fail under the action of 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, 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. Though failure theory has been in development for over 200 years, its level of acceptability is yet to reach that of continuum mechanics.
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| 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 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 yield.
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| == Material failure ==
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| In [[materials science]], '''material failure''' is the loss of load carrying capacity of a material unit. This definition ''per se'' introduces 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.
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| == Types of material failure ==
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| Material failure can be distinguished in two broader categories depending on the scale in which the material is examined:
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| === Microscopic failure ===
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| Microscopic material failure is defined in terms of crack propagation and initiation. 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''.
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| === Macroscopic failure ===
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| 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:
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| * Stress or strain failure
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| * Energy type failure ([[S-criterion]], [[T-criterion]])
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| * Damage failure
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| * Empirical failure.
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| 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 behaviour at one level is considered as a collective of its behaviour at a sublevel. An efficient deformation and failure model should be consistent at every level.
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| == Brittle material failure criteria ==
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| Failure of brittle materials can be determined using several approaches:
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| * Phenomenological failure criteria
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| * [[fracture mechanics|Linear elastic fracture mechanics]]
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| * [[fracture mechanics|elastic-plastic fracture mechanics]]
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| * Energy-based methods
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| * [[Cohesive zone model|Cohesive zone methods]]
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| === Phenomenological failure criteria ===
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| 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
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| :<math>
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| \sigma_c < \sigma_3 < \sigma_1 < \sigma_t \,
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| </math>
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| Note that the convention that tension is positive has been used in the above expression.
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| The '''maximum strain criterion''' has a similar form except that the principal strains are compared with experimentally determined uniaxial strains at failure, i.e.,
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| :<math>
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| \varepsilon_c < \varepsilon_3 < \varepsilon_1 < \varepsilon_t \,
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| </math>
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| The maximum principal stress and strain criteria continue to be widely used in spite of severe shortcomings.
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| 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. For brittle materials, some popular failure criteria are
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| * criteria based on invariants of the [[Cauchy stress tensor]]
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| * the [[yield surface|Tresca or maximum shear stress]] failure criterion
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| * the [[von Mises yield criterion|von Mises]] or maximum elastic distortional energy criterion
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| * the [[Mohr-Coulomb theory|Mohr-Coulomb failure criterion]] for cohesive-frictional solids
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| * the [[Drucker Prager|Drucker-Prager failure criterion]] for pressure-dependent solids
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| * the [[Bresler Pister yield criterion|Bresler-Pister failure criterion]] for concrete
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| * the [[Willam-Warnke yield criterion|Willam-Warnke failure criterion]] for concrete
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| * the [[Hankinson's equation|Hankinson criterion]], an empirical failure criterion that is used for orthotropic materials such as wood.
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| * the [[Hill yield criteria]] for anisotropic solids
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| * the [[Tsai-Wu failure criterion]] for anisotropic composites
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| * the [[Johnson–Holmquist damage model]] for high-rate deformations of isotropic solids
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| * the [[Hoek-Brown failure criterion]] for rock masses
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| * the [[Critical state soil mechanics|Cam-Clay failure theory]] for soils
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| === Linear elastic fracture mechanics ===
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| {{main|Fracture mechanics}}
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| 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 theory 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
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| :<math>
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| \sigma = \sqrt{\cfrac{2 E \gamma}{\pi a}}
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| </math>
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| 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
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| :<math> | |
| K_{\rm Ic} = Y\sigma_c\sqrt{\pi a}
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| </math>
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| 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.
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| 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.
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| === Energy-based methods ===
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| {{Main|Fracture mechanics}}
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| 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
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| :<math>
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| G_I := \cfrac{P}{2t}~\cfrac{du}{da}
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| </math>
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| 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'''.
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| The [[fracture toughness]] and the critical strain energy release rate for [[plane stress]] are related by
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| :<math>
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| G_{\rm Ic} = \cfrac{1}{E}~K_{\rm Ic}^2
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| </math>
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| 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.
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| == Ductile material failure criteria ==
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| {{Main|Yield (engineering)}}
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| Criteria used to predict the failure of ductile materials are usually called [[yield (engineering)|yield]] criteria. Commonly used failure criteria for ductile materials are:
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| * the [[Yield surface|Tresca]] or maximum shear stress criterion.
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| * the [[von Mises yield criterion]] or distortional strain energy density criterion.
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| * the [[Gurson yield criterion]] for pressure-dependent metals.
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| * the [[Hosford yield criterion]] for metals.
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| * the [[Hill yield criteria]].
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| * various criteria based on the invariants of the Cauchy stress tensor.
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| | |
| 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:
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| * the [[Johnson-Cook plasticity model|Johnson-Cook model]]
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| * the [[Steinberg-Guinan plasticity model|Steinberg-Guinan model]]
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| * the [[Zerilli-Armstrong plasticity model|Zerilli-Armstrong model]]
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| * the [[Mechanical threshold stress plasticity model|Mechanical threshold stress model]]
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| * the [[Preston-Tonks-Wallace plasticity model|Preston-Tonks-Wallace model]]
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| | |
| 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.
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| == See also ==
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| {{Mechanical failure modes}}
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| {{Continuum mechanics|cTopic=[[Solid mechanics]]}}
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| * [[Fracture mechanics]]
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| * [[Fracture]]
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| * [[Stress intensity factor]]
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| * [[Yield (engineering)]]
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| * [[Yield surface]]
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| * [[Plasticity (physics)]]
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| * [[Structural failure]]
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| * [[Strength of materials]]
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| * [[Ultimate failure]]
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| * [[Damage Mechanics]]
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| * [[Size effect on structural strength]]
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| * [[Concrete fracture analysis]]
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| == References ==
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| <references/>
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| == External links ==
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| {{DEFAULTSORT:Failure Theory (Material)}}
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| [[Category:Mechanical failure]]
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| [[Category:Plasticity]]
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| [[Category:Solid mechanics]]
| |
| [[Category:Mechanics]]
| |
| [[Category:Materials science]]
| |
| [[Category:Materials degradation]]
| |
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