Chloe Zhang
Elastic-plastic fracture mechanics (EPFM) is a theory that characterises crack tip stresses and strains. It is used in situations of greater plasticity, where the crack-tip plastic zone is comparable in size to the crack length or specimen dimensions. EPFM is particularly relevant when dealing with ductile materials, where the Linear Elastic Fracture Mechanics (LEFM) approach is no longer adequate to describe crack behaviour. EPFM accounts for the plastic deformation at the crack tip, which increases the toughness of the material. The J-Integral concept, first proposed by Rice in 1968, is a key parameter used in EPFM to describe the state of the crack tip and ensure similitude in experimental conditions.
| Characteristics | Values |
|---|---|
| Definition | A discipline that deals with the study of fracture behaviour in materials that exhibit both elastic and plastic deformation |
| Use case | Applicable in situations of greater plasticity, where the crack-tip plastic zone is comparable in size to the crack length or specimen dimensions |
| Applications | Used in engineering design and analysis to predict the fracture behaviour of materials under various loading conditions, including the design of pressure vessels and piping systems, aerospace and automotive components |
| Parameters | J-integral, crack tip opening displacement (CTOD), and plastic zone size and shape effects on fracture |
| Crack growth | Occurs when parameters such as stress corrosion stress intensity and small flaws exceed certain critical values |
| Fracture mechanics solutions | Modifications are made to account for the plastic deformation at the crack tip |
| Plastic zone | The region around the crack tip where the material has yielded and undergone plastic deformation, impacting the fracture behaviour of a material |
| Crack tip | In theory, the stress at the crack tip tends to infinity, considered a stress singularity |
| Linear elastic fracture mechanics (LEFM) | Applicable to a wide range of relatively brittle plastics; assumes small-scale yielding where the size of the plastic zone is small compared to the crack length |
| Non-linear elastic materials | J-integral is derived from the extension of linear fracture mechanics to non-linear elastic materials |
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What You'll Learn
Crack growth in ductile and semi-ductile structural metals
Elastic-plastic fracture mechanics (EPFM) is used in situations of greater plasticity, where the crack-tip plastic zone is comparable in size to the crack length or specimen dimensions. This is particularly relevant in ductile materials, where the Linear Elastic Fracture Mechanics (LEFM) approach is inadequate to describe crack behaviour. Ductile materials, including most metals, exhibit a relatively large plastic zone at the tip of the crack, where the material yields and deforms, absorbing energy and increasing toughness.
Ductile crack growth is a well-studied phenomenon, particularly in metals. The occurrence of flaws is common during the processing, fabrication, or service life of a material or component, and these flaws can manifest as cracks, voids, or metallurgical inclusions. The study of ductile crack growth is essential for understanding and improving the structural integrity of advanced thin sheet metals in engineering applications.
The mechanisms of fatigue-crack propagation are important to examine, especially in ductile materials like metals. The process of fatigue-crack growth involves a competition between intrinsic mechanisms that promote crack growth and extrinsic mechanisms that impede it. The susceptibility of ductile materials to cyclic degradation has implications for their structural applications and lifetime predictions.
Nonlinear fracture mechanics, such as the failure assessment diagram approach, are often better suited to characterise modern ductile materials. Codes such as BS 7910 and API-RP-579 provide guidance for assessing the acceptability of flaws in metallic structures and fitness for service, respectively. These codes include procedures for determining extreme and long-term loads, conducting fracture analysis, and studying fatigue crack growth to predict the life of a structure.
The transition from ductile to brittle fracture behaviour is also an important consideration in structural metals. This transition is influenced by factors such as temperature and strain rate, as illustrated in the study of Tvergaard and Needleman (1993). Understanding this transition is crucial for predicting and preventing potential failures in structures, especially in applications like pipelines, where ductile behaviour is preferred to avoid brittle fractures.
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J-integral methods for crack extension
Elastic plastic fracture mechanics (EPFM) is a field of mechanics concerned with the study of the propagation of cracks in materials. It is used when the size of the plastic zone at the crack tip is too large, and standard fracture mechanics are no longer applicable. This is often the case with ductile materials, where the plastic zone at the crack tip is comparable in size to the crack length or specimen dimensions.
The J-integral is a method used to characterise crack extension in EPFM. It was developed in 1967 by G. P. Cherepanov and independently in 1968 by James R. Rice. The J-integral represents a way to calculate the strain energy release rate, or work (energy) per unit fracture surface area, in a material. It is defined as:
> {\displaystyle J_{i}:=\lim _{\varepsilon \rightarrow 0}\int _{\Gamma _{\varepsilon }}\left(W(\Gamma )n_{i}-n_{j}\sigma _{jk}~{\cfrac {\partial u_{k}(\Gamma,x_{i})}{\partial x_{i}}}\right)\,d\Gamma}
Where {\displaystyle \varepsilon} is a small region around the crack tip. This integral is zero when the boundary {\displaystyle \Gamma} is closed and encloses a region with no singularities. The J-integral is path-independent in plastic materials when there is no non-proportional loading. It is also path-independent for isotropic, perfectly brittle, linear elastic materials when the crack extends straight ahead with respect to its original orientation.
The J-integral can be used to determine a conservative scalar fracture characteristic for the onset of crack extension, even for non-conventional systems such as highly plastically deforming materials. It can be measured directly, either optically or by using a clip gauge in the crack tip. This allows for the definition of a critical value of the J-integral, Jc, which is required for crack extension.
The J-integral has been widely used to describe fracture in EPFM, where there is a relatively large plastic zone at the crack tip. It provides a way to quantify fracture characteristics, which is essential for mitigating structural component failure and enhancing the longevity of modern materials.
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Linear elastic fracture mechanics (LEFM)
Elastic plastic fracture mechanics (EPFM) is used in situations of greater plasticity, where the crack-tip plastic zone is comparable in size to the crack length or specimen dimensions. It is particularly useful in cases where the crack is growing in a completely plastic strain field, such as at the root of a notch.
LEFM considers the crack to grow once the strain energy release rate (SERR) at the crack tip reaches a critical value. The critical stress intensity factor, found in the plane strain condition, is accepted as the defining property in LEFM. The stress at the crack tip where the radius is nearly zero would theoretically tend to infinity, considered a stress singularity. However, in real-world applications, there must be some mechanism or property of the material that prevents such a crack from propagating spontaneously.
LEFM proposes that two cracks loaded to the same K (stress intensity factor) or G (energy release rate), in different geometries of the same polymer under the same environmental conditions, will behave identically. Its strength lies in the availability of Y solutions for various geometries and computational methods to derive others. The energy release rate, G, is the effective forward force, per unit length, acting on the crack front. If a crack front propagates, it absorbs an energy Gc per unit area of material separated.
LEFM has been used to study the FRP-concrete interface debonding. A typical LEFM model for the FRP-concrete interface was developed by Au and Büyüköztürk (2006). They proposed a tri-layer interface fracture energy model, which is a direct application of classical interface fracture models in bi-layered beams. However, it is important to note that the transverse shear effect is not considered in this model, which could lead to an underestimation of the energy release rates if shear forces are present.
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Fracture toughness
The size of the plastic zone at the crack tip affects the fracture toughness of a material. As the size of the plastic zone increases, there is more energy absorbed, and the material exhibits higher ductility and fracture toughness. In ductile materials, the Linear Elastic Fracture Mechanics (LEFM) approach may not be adequate to describe crack behaviour, and the Elastic-Plastic Fracture Mechanics (EPFM) approach is used instead. The plastic deformation at the crack tip effectively blunts the crack and increases toughness.
The presence of grains in a material can also affect its toughness by influencing crack propagation. At low temperatures, the plastic zone shrinks away, and the material becomes completely brittle, with lower toughness. At higher temperatures, the plastic zone forms and cleavage is likely to initiate at the elastic-plastic zone boundary. The transition from brittle to ductile fracture is influenced by factors such as grain size, strain rate, and microstructural factors.
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Crack tip opening displacement (CTOD)
Elastic plastic fracture mechanics (EPFM) is a theory that attempts to explain the behaviour of cracks in situations of greater plasticity. This is particularly useful in the case of ductile materials, where the plastic zone at the crack tip is relatively large, and the Linear Elastic Fracture Mechanics (LEFM) approach is inadequate.
CTOD testing is usually performed on materials that undergo plastic deformation before failure. The testing material resembles the original, with dimensions reduced proportionally. The specimen is placed on a work table, with a notch created at the centre, and a load is applied to resemble the expected load. The crack should be generated such that the defect length is about half the depth.
The crack tip plastically deforms until a critical point, after which a cleavage crack is initiated, leading to partial or complete failure. The displacement at the crack mouth is measured, and CTOD is inferred assuming the specimen halves are rigid and rotate about a hinge point. This can be done using a specially designed clip-on gauge, which measures CTOD over a gauge length of 5mm.
CTOD is a useful parameter as it is easy to measure and has more physical meaning than other techniques. It is a single parameter that accommodates crack tip plasticity and can be used to estimate the displacement at the physical crack tip.
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Frequently asked questions
Elastic plastic fracture mechanics (EPFM) is a theory that uses the J-Integral concept to characterise crack tip stresses and strains. It arose from the desire to use fracture mechanics in situations of greater plasticity, where the crack-tip plastic zone is comparable in size to the crack length or specimen dimensions.
Linear elastic fracture mechanics (LEFM) is used when the plastic zone at the tip of the crack is small relative to the crack length. It is applicable to a wide range of relatively brittle plastics. EPFM, on the other hand, is used when the plastic zone is large, and is applicable to materials that exhibit higher fracture toughness and ductility.
The J-Integral concept was first proposed by Rice in 1968 as a path-independent integral for characterising crack tip stresses and strains. It is widely used to describe fractures that involve a relatively large plastic zone at the crack tip.
