Failure theory (material) - AbsoluteAstronomy.com

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

Fracture

A fracture is the separation of an object or material into two, or more, pieces under the action of stress.The word fracture is often applied to bones of living creatures , or to crystals or crystalline materials, such as gemstones or metal...

) or ductile failure (yield

Yield (engineering)

The yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to deform plastically. Prior to the yield point the material will deform elastically and will return to its original shape when the applied stress is removed...

). 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.

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.

Material failure

In materials science

Materials science

Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. This scientific field investigates the relationship between the structure of materials at atomic or molecular scales and their macroscopic properties. It incorporates...

, 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

Microscopic

The microscopic scale is the scale of size or length used to describe objects smaller than those that can easily be seen by the naked eye and which require a lens or microscope to see them clearly.-History:...

, to macroscopic

Macroscopic

The macroscopic scale is the length scale on which objects or processes are of a size which is measurable and observable by the naked eye.When applied to phenomena and abstract objects, the macroscopic scale describes existence in the world as we perceive it, often in contrast to experiences or...

. 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

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 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 bhanu 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

Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modelled as a continuous mass rather than as discrete particles...

and classical fracture mechanics

Fracture mechanics

Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.In...

. 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 Needleman

Alan Needleman

Alan Needleman was born in 1944 in Philadelphia, PA and is currently the Florence Pirce Grant University Professor of Mechanics of Solids and Structures at Brown University in Providence, RI. Professor Needleman received his B.S. from the University of Pennsylvania in 1966, a M.S. and Ph.D. from...

, is known as GTN. Another approach, proposed by Rousselier, is based on continuum damage mechanics (CDM) and thermodynamics

Thermodynamics

Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

. 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 presents a classification of macroscopic failure criteria in four categories:

Stress or strain failure
Energy type failure (S-criterion, T-criterion
T-criterion
The T-failure criterion is a set of failure criteria, that can be used to predict both brittle and ductile failure. These criteria were designed as a replacement for the von Mises yield criterion which predicts the unphysical result that pure hydrostatic tensile loading of metals never leads to...

)
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 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.

Brittle material failure criteria

Failure of brittle materials can be determined using several approaches:

Phenomenological failure criteria
Linear elastic fracture mechanics
Fracture mechanics
Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.In...
elastic-plastic fracture mechanics
Fracture mechanics
Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture.In...
Energy-based methods
Cohesive zone methods

Phenomenological failure criteria

The failure criteria that were developed for brittle solids were the maximum stress/strain criteria. The maximum stress criterion assumes that a material fails when the maximum principal stress

in a material element exceeds the uniaxial tensile strength of the material. Alternatively, the material will fail if the minimum principal stress

is less than the uniaxial compressive strength of the material. If the uniaxial tensile strength of the material is

and the uniaxial compressive strength is

, then the safe region for the material is assumed to be

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 experimentall determined uniaxial strains at failure, i.e.,

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. For brittle materials, some popular failure criteria are

criteria based on invariants of the Cauchy stress tensor
the Tresca or maximum shear stress
Yield surface
A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of inside the yield surface is elastic. When the stress state lies on the surface the material is said to have reached its yield point and the...

failure criterion
the von Mises or maximum elastic distortional energy criterion
the Mohr-Coulomb failure criterion
Mohr-Coulomb theory
Mohr–Coulomb theory is a mathematical model describing the response of brittle materials such as concrete, or rubble piles, to shear stress as well as normal stress. Most of the classical engineering materials somehow follow this rule in at least a portion of their shear failure envelope...

for cohesive-frictional solids
the Drucker-Prager failure criterion for pressure-dependent solids
the Bresler-Pister failure criterion
Bresler Pister yield criterion
The Bresler-Pister yield criterion is a function that was originally devised to predict the strength of concrete under multiaxial stress states...

for concrete
the Willam-Warnke failure criterion
Willam-Warnke yield criterion
The Willam-Warnke yield criterion is a function that is used to predict when failure will occur in concrete and other cohesive-frictional materials such as rock, soil, and ceramics...

for concrete
the Hankinson criterion
Hankinson's equation
Hankinson's equation is a mathematical relationship for predicting the off-axis uniaxial compressive strength of wood. The formula can also be used to compute the fiber stress or the stress wave velocity at the elastic limit as a function of grain angle in wood...

, an empirical failure criterion that is used for orthotropic materials such as wood.
the Hill yield criteria
Hill yield criteria
Rodney Hill has developed several yield criteria for anisotropic plastic deformations. The earliest version was a straightforward extension of the von Mises yield criterion and had a quadratic form. This model was later generalized by allowing for an exponent m...

for anisotropic solids
the Tsai-Wu failure criterion
Tsai-Wu failure criterion
The Tsai-Wu failure criterion is a phenomenological failure theory which is widely used for anisotropic composite materials which have different strengths in tension and compression...

for anisotropic composites
the Johnson–Holmquist damage model
Johnson–Holmquist damage model
In solid mechanics, the Johnson–Holmquist damage model is used to model the mechanical behavior of damaged brittle materials, such as ceramics, rocks, and concrete, over a range of strain rates...

for high-rate deformations of isotropic solids
the Hoek-Brown failure criterion for rock masses

Linear elastic fracture mechanics

The approach taken in linear elastic fracture mechanics

Fracture mechanics

is to estimate the amount of energy needed to grow a preexisting crack in a brittle material. The earliest fracture mechanics

Fracture mechanics

approach for unstable crack growth is Griffiths' theory . When applied to the mode I

Fracture mechanics

opening of a crack, Griffiths' theory predicts that the critical stress (

) needed to propagate the crack is given by

where

is the Young's modulus of the material,

is the surface energy per unit area of the crack, and

is the crack length for edge cracks or

is the crack length for plane cracks. The quantity

is postulated as a material parameter called the fracture toughness. The mode I fracture toughness

Fracture toughness

In materials science, fracture toughness is a property which describes the ability of a material containing a crack to resist fracture, and is one of the most important properties of any material for virtually all design applications. The fracture toughness of a material is determined from the...

for plane strain is defined as

where

is a critical value of the far field stress and

is a dimensionless factor that depends on the geometry, material properties, and loading condition. The quantity

is related to the stress intensity factor

Stress Intensity Factor

The stress intensity factor, K, is used in fracture mechanics to predict the stress state near the tip of a crack caused by a remote load or residual stresses. It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion...

and is determined experimentally. Similar quantities

and

can be determined for mode II

Fracture mechanics

and model III

Fracture mechanics

loading conditions.

The state of stress around cracks of various shapes can be expressed in terms of their stress intensity factor

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

The linear elastic fracture mechanics method is difficult to apply for anisotropic materials (such as composites

Composite material

Composite materials, often shortened to composites or called composition materials, are engineered or naturally occurring materials made from two or more constituent materials with significantly different physical or chemical properties which remain separate and distinct at the macroscopic or...

) or for situations where the loading or the geometry are complex. The strain energy release rate

Strain energy release rate

Strain energy release rate is the energy dissipated during fracture per unit of newly created fracture surface area. This quantity is central to fracture mechanics because the energy that must be supplied to a crack tip for it to grow must be balanced by the amount of energy dissipated due to the...

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

where

is the applied load,

is the thickness of the plate,

is the displacement at the point of application of the load due to crack growth, and

is the crack length for edge cracks or

is the crack length for plane cracks. The crack is expected to propagate when the strain energy release rate exceeds a critical value

- called the critical strain energy release rate.

The fracture toughness

Fracture toughness

and the critical strain energy release rate for plane stress are related by

where

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 criteria

Criteria used to predict the failure of ductile materials are usually called yield

Yield (engineering)

criteria. Commonly used failure criteria for ductile materials are:

the Tresca
Yield surface
A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of inside the yield surface is elastic. When the stress state lies on the surface the material is said to have reached its yield point and the...

or maximum shear stress criterion.
the von Mises yield criterion or distortional strain energy density criterion.
the Gurson yield criterion for pressure-dependent metals.
the Hosford yield criterion
Hosford yield criterion
The Hosford yield criterion is a function that is used to determine whether a material has undergone plastic yielding under the action of stress.- Hosford yield criterion for isotropic plasticity :...

for metals.
the Hill yield criteria
Hill yield criteria
Rodney Hill has developed several yield criteria for anisotropic plastic deformations. The earliest version was a straightforward extension of the von Mises yield criterion and had a quadratic form. This model was later generalized by allowing for an exponent m...

.
various criteria based on the invariants of the Cauchy stress tensor.

The yield surface

Yield surface

A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of inside the yield surface is elastic. When the stress state lies on the surface the material is said to have reached its yield point and the...

of a ductile material usually changes as the material experiences increased deformation

Deformation (mechanics)

Deformation in continuum mechanics is the transformation of a body from a reference configuration to a current configuration. A configuration is a set containing the positions of all particles of the body...

. 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, kinematic hardening, and viscoplasticity

Viscoplasticity

Viscoplasticity is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied...

. Some such models are:

the Johnson-Cook model
the Steinberg-Guinan model
the Zerilli-Armstrong model
the Mechanical threshold stress model
the Preston-Tonks-Wallace model

There is another important aspect to ductile materials - the prediction of the ultimate failure strength

Tensile strength

Ultimate tensile strength , often shortened to tensile strength or ultimate strength, is the maximum stress that a material can withstand while being stretched or pulled before necking, which is when the specimen's cross-section starts to significantly contract...

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 parameter.

Material failure

Types of material failure

Microscopic failure

Macroscopic failure

Brittle material failure criteria

Phenomenological failure criteria

Linear elastic fracture mechanics

Energy-based methods

Ductile material failure criteria

See also