1 Fundamental concepts
Failure criteria are quantitative rules that relate the state of a material or structure to the onset of loss of function. They are intended to capture a practical threshold, such as the beginning of yielding, the start of cracking, the collapse of a component, or the point at which deformation becomes excessive for service. In engineering use, such criteria connect measurable variables to performance limits and help translate laboratory data into design decisions.
1.1 Definition of failure
In a broad sense, failure occurs when a part can no longer meet its intended purpose. This may mean complete separation, unstable crack growth, permanent distortion, or any condition that makes continued operation unsafe or ineffective. The exact meaning depends on the application: a bridge, a pressure vessel, and a plastic housing may each have different failure thresholds even if the same material is involved.
1.2 Purpose in engineering design
Failure criteria provide a common basis for design, analysis, and testing. Engineers use them to estimate allowable loads, select materials, size parts, and evaluate whether a design meets performance requirements. They also support comparison among different materials and help identify which mechanism is likely to govern under a given loading condition.
1.3 Safety factors and margins
Because real structures contain imperfections and experience uncertain conditions, designs usually include safety factors or margins. These reduce the risk that actual service conditions will exceed the predicted limit. Safety factors are chosen based on the consequences of failure, the reliability of available data, manufacturing variability, and the quality of analysis.
1.4 Limit states and serviceability
A limit state is a condition beyond which a structure no longer satisfies a specified requirement. Ultimate limit states concern collapse or fracture, while serviceability limit states concern usability, such as excessive deflection, vibration, or leakage. Failure criteria may be used for either type, depending on whether the goal is to prevent catastrophic loss or to maintain acceptable performance.
2 Classification of failure criteria
Failure criteria can be grouped according to the physical quantity they monitor. Some focus on stress or strain, others on energy, accumulated damage, or statistical likelihood of breakdown. In practice, multiple criteria may be combined because different mechanisms often compete during service.
2.1 Strength-based criteria
Strength-based criteria compare a local stress or strain measure with a limiting value obtained from testing or experience. They are widely used in simple design calculations because they are easy to apply and interpret. These criteria are especially common when one failure mode dominates.
2.2 Deformation-based criteria
Deformation-based rules use strain, displacement, rotation, or curvature as the controlling measure. They are useful when performance depends more on shape change than on stress alone, as in flexible components, thin shells, or structures that must maintain clearance and alignment.
2.3 Energy-based criteria
Energy-based approaches judge failure by the amount of mechanical energy available for crack growth, yielding, or damage accumulation. They are particularly valuable in fracture mechanics, where crack extension depends on whether sufficient energy can be released from the system.
2.4 Damage-based criteria
Damage-based criteria track the progressive degradation of stiffness, strength, or internal structure. They are often used for fatigue, creep, composites, and other materials in which failure develops gradually rather than at a single sharply defined point.
2.5 Probabilistic criteria
Probabilistic methods treat failure as a matter of likelihood rather than a fixed threshold. They account for scatter in material properties, flaws, manufacturing variation, and uncertain loading. These methods are useful in reliability analysis and in applications where absolute deterministic limits are unrealistic.
3 Stress-based failure theories
Stress-based theories relate failure to the internal forces acting within a material. They are among the oldest and most widely known criteria, and many are derived from idealized tests on simple specimens. Their usefulness depends strongly on the material class and the dominant mode of loading.
3.1 Maximum principal stress criterion
This criterion assumes failure begins when the largest tensile principal stress reaches a critical value. It is often associated with brittle materials, which tend to break when a tensile stress exceeds their tensile strength. The method is simple but may be inaccurate when shear or compressive effects are important.
3.2 Maximum shear stress criterion
Maximum shear stress criteria assume yielding or failure begins when the greatest shear stress in the material reaches a limiting level. These criteria are especially relevant to ductile metals, where plastic flow is driven largely by shear deformation.
3.2.1 Tresca criterion
The Tresca criterion states that yielding occurs when the maximum shear stress equals the shear stress measured at yield in a uniaxial tensile test. It is conservative and easy to apply. For many metals, it gives results close to more refined criteria while remaining straightforward in manual calculations.
3.2.2 Comparison with von Mises criterion
The von Mises criterion is based on distortion energy rather than maximum shear stress, but both are used for ductile yielding. Tresca usually predicts slightly lower allowable stress, making it more conservative. Von Mises is smoother mathematically and often fits experimental data better in multiaxial loading.
3.3 Distortion energy criterion
Distortion energy theories assume that yielding starts when the energy associated with shape change reaches a critical value. They separate the effect of hydrostatic pressure from the part of stress that actually distorts the material. This makes them well suited to ductile metals.
3.3.1 von Mises yield criterion
The von Mises criterion states that yielding occurs when an equivalent stress, computed from the principal stresses, reaches the uniaxial yield strength. It is one of the most commonly used yield rules in metal plasticity. Its smooth stress surface makes it especially convenient for numerical simulation.
3.3.2 Applications to ductile metals
This criterion is widely applied to steels, aluminum alloys, and other ductile metals under complex loading. It is used in design checks, forming analysis, and finite element models. In these applications, it helps predict the onset of plastic flow and plastic collapse.
3.4 Normal stress criteria for brittle materials
Brittle materials often fail with little plastic deformation, so normal tensile stress is frequently the controlling quantity. Such criteria assume that cracks open when tensile stress becomes too large, while compressive strength may be much higher than tensile strength.
3.4.1 Rankine criterion
The Rankine criterion states that failure occurs when the maximum principal tensile stress reaches the tensile strength of the material. It is commonly used for brittle solids such as cast iron, concrete in tension, and some ceramics in simplified analysis.
3.4.2 Coulomb–Mohr criterion
The Coulomb–Mohr criterion accounts for different tensile and compressive strengths by using a linear stress relation. It is often applied to brittle materials whose behavior depends on both normal stress and shear stress. The method is more flexible than a pure tensile-stress rule, though still idealized.
4 Strain-based failure criteria
Strain-based criteria focus on deformation rather than force per unit area. They are useful when local shape change governs performance, when plastic flow is extensive, or when the applied load is not the most direct indicator of failure.
4.1 Maximum principal strain criterion
This criterion assumes failure begins when the largest principal strain reaches a limiting value. It is simple to interpret and can be useful for brittle or weakly ductile materials. Because strain is directly related to deformation, it can sometimes correlate better with visible damage than stress-based measures.
4.2 Shear strain criteria
Shear strain criteria use the distortion part of deformation as the failure indicator. They are relevant when sliding and shape change dominate, particularly in ductile deformation. In plasticity analysis, these criteria can be linked to the development of permanent form change.
4.3 Use in ductile deformation analysis
For ductile materials, strain criteria are often used to describe forming limits, necking, and local plastic instability. They may be more informative than stress criteria when a component undergoes large deformation before fracture. This is especially true in sheet metal forming and crash analysis.
4.4 Strain limits in design
Designers sometimes impose strain limits to prevent excessive distortion, loss of fit, or local damage. These limits may be based on allowable elongation, curvature, or equivalent plastic strain. Strain limits are particularly useful where function depends on geometry rather than on immediate material rupture.
5 Energy and fracture-based criteria
Energy-based and fracture-mechanics criteria address the conditions under which cracks form or grow. They are central to the analysis of structural integrity, especially in materials that contain preexisting flaws or develop cracks during service.
5.1 Strain energy density
Strain energy density is the elastic energy stored per unit volume of material. Failure may be associated with a critical level of stored energy, especially in brittle fracture or localized damage. This approach can be useful when no single stress component captures the observed behavior well.
5.2 Griffith fracture criterion
The Griffith criterion links crack growth to the balance between released elastic energy and the energy required to create new crack surfaces. It is fundamental to brittle fracture theory. The criterion explains why a small crack can remain stable until a critical size or stress is reached.
5.3 Linear elastic fracture mechanics
Linear elastic fracture mechanics, or LEFM, describes crack behavior when the material around the crack tip is mostly elastic. It provides tools for predicting unstable fracture from the size of a flaw, the applied stress, and the geometry of the component.
5.3.1 Stress intensity factor
The stress intensity factor measures the severity of the stress field near a crack tip. When it reaches a critical value, crack extension may become unstable. This quantity is widely used because it captures both load and flaw size in a single parameter.
5.3.2 Critical crack size
For a given stress level and geometry, there is often a maximum crack size that a component can tolerate safely. If a flaw exceeds this size, fracture may occur suddenly. This concept is important in inspection planning and damage tolerance assessment.
5.4 Energy release rate
The energy release rate is the amount of potential energy available for crack advance per unit new crack area. When this rate reaches the material’s resistance, crack growth becomes possible. It is a central quantity in fracture mechanics and is closely related to the stress intensity approach.
5.5 Toughness and crack resistance
Toughness describes a material’s ability to resist crack growth and absorb energy before breaking. High toughness usually means better tolerance of flaws and higher resistance to unstable fracture. Crack resistance may also increase during growth in some materials, which can stabilize crack extension.
6 Failure criteria for specific materials
Different material classes respond differently to stress, strain, temperature, and time. As a result, no single criterion fits all materials. Engineering practice therefore selects methods that match the characteristic behavior of each class.
6.1 Ductile metals
Ductile metals usually show significant plastic deformation before fracture. Their failure analysis often separates yielding, plastic collapse, and final rupture.
6.1.1 Yielding and plastic collapse
Yielding begins when the material first undergoes permanent deformation. With increasing load, the entire section may plastically deform until collapse occurs. Criteria such as von Mises and Tresca are often used to predict these stages.
6.1.2 Ductile fracture
Ductile fracture typically follows substantial plastic deformation and is often associated with void growth, coalescence, and necking. Stress triaxiality and strain accumulation are important factors. The fracture process may be gradual, but final separation can still occur abruptly.
6.2 Brittle materials
Brittle materials have limited plasticity and usually fail by crack growth. Their strength can be strongly affected by flaws and surface defects.
6.2.1 Crack initiation
Crack initiation occurs when local tensile stress becomes sufficient to open an existing defect or microcrack. For brittle solids, this threshold may be close to the nominal strength. Surface finish and specimen size can strongly influence the result.
6.2.2 Rapid fracture
Once a crack becomes unstable, fracture may proceed very quickly. Brittle rupture often gives little warning and may propagate across a section almost instantaneously. This behavior makes flaw control and inspection particularly important.
6.3 Polymers
Polymers often exhibit time-dependent and temperature-sensitive behavior. Their failure may involve yielding, crazing, softening, or environmental attack, depending on the chemistry and loading conditions.
6.3.1 Time-dependent yielding
Many polymers deform in a viscoelastic or viscoplastic manner, so their apparent yield point depends on loading rate and temperature. Slow loading can reduce the observed strength, while faster loading may increase it. This makes test conditions especially important.
6.3.2 Environmental stress cracking
Some polymers fail by cracking under stress in the presence of certain chemicals. This mode may occur at stresses below the nominal strength and can be difficult to predict without specific testing. It reflects the combined influence of mechanical load and environment.
6.4 Ceramics and composites
Ceramics and composite materials often fail in a localized or progressive fashion rather than through uniform yielding. Their structure strongly influences the applicable criteria.
6.4.1 Fiber failure
In fiber-reinforced composites, fiber rupture can be a major load-carrying failure mode. Because fibers often provide most of the stiffness and strength in one direction, fiber failure can sharply reduce performance.
6.4.2 Matrix cracking
Matrix cracking refers to breakage in the surrounding material that binds or supports reinforcement. It may not cause immediate collapse, but it can reduce stiffness and open paths for further damage. It often precedes more serious failure.
6.4.3 Delamination
Delamination is separation between layers in laminated composites. It can grow under bending, impact, or cyclic loading and may be difficult to detect early. Fracture-based and interlaminar criteria are commonly used to assess this problem.
7 Time-dependent failure
Some failure processes depend strongly on duration as well as load level. Under repeated cycles or sustained stress, damage may accumulate slowly until a threshold is reached. This makes time an essential variable in reliability assessment.
7.1 Fatigue criteria
Fatigue failure results from cyclic loading and may occur at stresses well below monotonic strength. It often begins with microscopic crack initiation followed by gradual crack growth.
7.1.1 S-N approach
The S-N approach relates stress amplitude to the number of cycles to failure. It is widely used for high-cycle fatigue and provides a practical design tool when loading is repetitive and the elastic response dominates.
7.1.2 Strain-life approach
The strain-life approach relates cyclic strain amplitude to fatigue life and is useful when local plasticity occurs. It is especially valuable for low-cycle fatigue, where deformation is large and damage accumulates quickly.
7.1.3 Crack growth laws
Crack growth laws describe how an existing crack advances with each load cycle. They are central to damage-tolerant design because they permit life prediction from measurable crack size and loading history.
7.2 Creep rupture criteria
Creep is slow, permanent deformation under sustained stress, usually at elevated temperature. Creep rupture occurs when this process leads to loss of load-carrying capacity after a period of time.
7.2.1 Stress rupture
Stress rupture criteria estimate the time to failure under constant load and temperature. They are important for components that operate for long periods in heat, such as turbines, boilers, and high-temperature piping.
7.2.2 Larson–Miller parameter
The Larson–Miller parameter is a time-temperature superposition tool used to correlate creep rupture data. It combines stress, temperature, and rupture life into a single empirical relation that aids extrapolation from test results.
7.3 Viscoelastic and viscoplastic effects
Viscoelastic materials show time-dependent elastic response, while viscoplastic materials show time-dependent permanent deformation. Both behaviors can alter apparent strength and failure thresholds. Accurate criteria therefore need to account for loading rate, duration, and temperature.
8 Multiaxial and complex loading
Real components often experience several stresses at once, rather than simple uniaxial loading. Failure criteria must then combine effects from multiple directions and loading histories.
8.1 Combined stress states
Combined stress states arise when tension, compression, shear, and bending act together. Equivalent stress or equivalent strain measures are often used to reduce these states to a single failure parameter. The chosen measure depends on the material and failure mode.
8.2 Nonproportional loading
Nonproportional loading occurs when the direction or ratio of principal stresses changes during a load cycle. This can intensify damage, especially in metals that harden or soften depending on the deformation path. Specialized criteria are often needed for such histories.
8.3 Hydrostatic pressure effects
Hydrostatic pressure affects failure differently in different materials. Ductile metals are often relatively insensitive to it during yielding, while brittle materials and polymers may be strongly influenced by pressure. This distinction is important in high-pressure and confinement problems.
8.4 Anisotropic material behavior
Anisotropic materials have direction-dependent properties. Their failure cannot be captured well by isotropic criteria because strength, stiffness, and crack resistance vary with orientation. Composite laminates and textured metals are common examples.
9 Experimental determination
Failure criteria are only useful when their parameters are determined from reliable experiments. Test methods must reproduce the relevant stress state, temperature, and loading rate as closely as practical.
9.1 Mechanical testing methods
Common tests include tensile, compression, shear, bend, impact, fatigue, and creep experiments. Fracture tests and crack-growth specimens are used when crack behavior is of interest. Each method highlights a different aspect of failure response.
9.2 Calibration from test data
Calibration converts raw measurements into criterion parameters such as yield stress, fracture toughness, or fatigue constants. Good calibration requires adequate sample numbers and careful interpretation of scatter. The derived values are often specific to the test conditions used.
9.3 Specimen geometry and boundary conditions
Specimen shape and support conditions can influence the observed failure mode. Stress concentrations, friction, size, and constraint effects may alter results. For this reason, standardized geometries are often used to improve comparability.
9.4 Validation and uncertainty
After calibration, criteria should be checked against independent data. Validation assesses whether the model predicts behavior outside the original test set. Uncertainty arises from material variability, measurement error, and incomplete representation of real service conditions.
10 Computational implementation
Modern design practice often implements failure criteria within numerical simulation tools. These methods allow engineers to study complex geometries, load histories, and interaction among multiple damage modes.
10.1 Finite element analysis
Finite element analysis divides a component into small elements and computes local stress, strain, and deformation. Failure criteria can then be applied at each point to identify critical regions. This approach is widely used in structural and materials engineering.
10.2 Constitutive modeling
Constitutive models describe how a material responds to loading over time. They provide the stress-strain relations needed for failure prediction. Good models must represent elasticity, plasticity, damage, and rate dependence when these effects matter.
10.3 Failure indices
Failure indices are dimensionless quantities that indicate proximity to a limit. A value near unity often suggests imminent failure under the chosen criterion. They are convenient for comparing safety across a component and for scanning large numerical models.
10.4 Damage evolution models
Damage evolution models describe how properties degrade after the onset of cracking, yielding, or microstructural deterioration. They are useful when failure is progressive rather than instantaneous. Such models can capture stiffness loss, softening, and eventual separation.
10.5 Numerical convergence and mesh sensitivity
Computational failure predictions may depend on element size, discretization, and solver settings. Crack localization and softening can produce mesh sensitivity if not handled carefully. Convergence checks are therefore essential to ensure that predicted failure is physically meaningful.
11 Applications in engineering practice
Failure criteria are used throughout engineering to ensure safety, economy, and reliability. They help bridge the gap between material behavior and practical design rules.
11.1 Structural design
In structural design, failure criteria determine allowable stresses, required thicknesses, and load limits. They are used for beams, shells, pressure vessels, supports, and other load-bearing systems. Proper selection of the criterion is crucial for matching the dominant mode of failure.
11.2 Component life prediction
Engineers use failure criteria to estimate service life under cyclic, thermal, or sustained loading. This is especially important for parts that cannot be frequently replaced. Life prediction methods support maintenance planning and inspection scheduling.
11.3 Materials selection
Different materials are preferred depending on whether strength, toughness, stiffness, or durability is the main concern. Failure criteria help compare candidate materials under the expected loading regime. They also reveal whether a more ductile or more flaw-tolerant material is needed.
11.4 Quality control and standards
Manufacturing and inspection often rely on failure criteria to set acceptance limits. Standards may specify test procedures, allowable defect sizes, or performance thresholds. These rules help ensure consistent product quality and reduce the chance of unexpected breakdown.
12 Limitations and assumptions
Failure criteria are useful simplifications, but they are not complete descriptions of real materials. Their predictions depend on assumptions that may hold only over a limited range of conditions.
12.1 Idealized material behavior
Many criteria assume homogeneity, isotropy, and simple constitutive behavior. Real materials may contain inclusions, pores, residual stresses, or directional structure. As a result, idealized formulas can miss important local effects.
12.2 Scale effects
Strength and fracture behavior often depend on specimen size. Larger parts are more likely to contain a critical flaw, and stress gradients may differ from those in small test samples. This makes direct extrapolation from laboratory coupons to full-scale structures uncertain.
12.3 Environmental influences
Temperature, moisture, oxidation, chemicals, and radiation can all alter mechanical response. A criterion calibrated in one environment may not remain valid in another. Environmental effects are especially important for polymers, high-temperature metals, and exposed structures.
12.4 Model uncertainty
No criterion can perfectly represent all damage processes. Uncertainty enters through material scatter, measurement limitations, and incomplete knowledge of the governing mechanism. Engineers address this through conservatism, validation, and probabilistic methods where appropriate.