Fracture mechanics - Citizendia

Fracture mechanics is a method for predicting failure of a structure containing a crack. 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. Solid mechanics is the branch of Mechanics, Physics, and Mathematics that concerns the behavior of solid matter under external actions (e Solid mechanics is the branch of Mechanics, Physics, and Mathematics that concerns the behavior of solid matter under external actions (e

In modern materials science, fracture mechanics is an important tool in improving the mechanical performance of materials and components. Materials Science or Materials Engineering is an interdisciplinary field involving the properties of matter and its applications to various areas of Science and It applies the physics of stress and strain, in particular the theories of elasticity and plasticity, to the microscopic crystallographic defects found in real materials in order to predict the macroscopic mechanical failure of bodies. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Stress is a measure of the average amount of Force exerted per unit Area. A material is said to be elastic if it deforms under stress (e Crystalline solids have a very regular atomic structure that is the local positions of atoms with respect to each other are repeated at the atomic scale Fractography is widely used with fracture mechanics to understand the causes of failures and also verify the theoretical failure predictions with real life failures. Fractography is the study of fracture surfaces of materials Fractographic methods are routinely used to determine the cause of failure in engineering structures especially in product

The need for fracture mechanics

Tay Bridge Disaster (1879)

In many cases, failure of engineering structures through fracture can be fatal; one example is that of the Tay Rail Bridge disaster (right). The Tay Bridge (sometimes unofficially the Tay Rail Bridge) is a Railway Bridge approximately two and a quarter miles (three and a half kilometres Often disasters occur because engineering structures contain cracks - arising either during production or during service (e. g. from fatigue). For instance, growth of cracks in pressure vessels due to crack propagation could cause a fatal explosion. If failure were ever to happen, we would rather it were by yield or by leak before break. The yield strength or yield point of a Material is defined in Engineering and Materials science as the stress at which a material

Since cracks can lower the strength of the structure beyond that due to loss of load-bearing area a material property, above and beyond conventional strength, is needed to describe the fracture resistance of engineering materials. This is the reason for the need for fracture mechanics - the evaluation of the strength of cracked structures.

The history of fracture mechanics

Griffith's energy relation

An edge crack (flaw) of length

a

in a material.

Fracture Mechanics was invented during World War I by English aeronautical engineer, A.A.Griffith, to explain the failure of brittle materials^[1]. Alan Arnold Griffith ( 13 June 1893 &ndash 13 Oct 1963) was an English engineer who among many other contributions is best known Griffith's work was motivated by a couple of facts:

The stress needed to fracture bulk glass is around 100 MPa. Stress is a measure of the average amount of Force exerted per unit Area. A fracture is the (local separation of an object or material into two or more pieces under the action of stress. Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many
The theoretical stress needed for breaking atomic bonds is approximately 10,000 MPa.

A theory was needed to reconcile these conflicting observations. Also, experiments on glass fibers that Griffith himself conducted suggested that the fracture stress increases as the fiber diameter decreases. Hence the uniaxial tensile strength, which had been used extensively to predict material failure before Griffith, could not be a specimen-independent material property. Griffith suggested that the low fracture strength observed in experiments, as well as the size-dependence of strength, was due to the presence of microscopic flaws in the bulk material.

To verify the flaw hypothesis, Griffith introduced an artificial flaw in his experimental specimens. The artificial flaw was in the form of a surface crack which was much larger than other flaws in a specimen. The experiments showed that the product of the square root of the flaw length ( $a$ ) and the stress at fracture ( $σ f$ ) was nearly constant, i. e. ,

$\sigma_f\sqrt{a} \approx C ~.$

An explanation of this relation in terms of linear elasticity theory is problematic. Linear elasticity theory predicts that stress (and hence the strain) at the tip of a sharp flaw in a linear elastic material is infinite. To avoid that problem, Griffith developed a thermodynamic approach to explain the relation that he observed. In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning "

The growth of a crack requires the creation of two new surfaces and hence an increase in the surface energy. Surface energy quantifies the disruption of intermolecular bonds that occurs when a surface is created Griffith found an expression for the constant $C$ in terms of the surface energy of the crack by solving the elasticity problem of a finite crack in an elastic plate. Briefly, the approach was

compute the potential energy stored in a perfect specimen under an uniaxial tensile load. Potential energy can be thought of as Energy stored within a physical system
fix the boundary so that the applied load does no work and then introduce a crack into the specimen.

The crack relaxes the stress and hence reduces the elastic energy near the crack faces. The elastic energy is the Energy which causes or is released by the elastic distortion of a solid or a fluid On the other hand, the crack increases the total surface energy of the specimen. Surface energy quantifies the disruption of intermolecular bonds that occurs when a surface is created The next step was to

compute the change in the free energy (surface energy - elastic energy) as a function of the crack length. Free energy may refer to In science: Thermodynamic free energy, the energy in a physical system that can be converted to do work in particular

Failure occurs when the free energy attains a peak value at a critical crack length, beyond which the free energy decreases by increasing the crack length, i. Free energy may refer to In science: Thermodynamic free energy, the energy in a physical system that can be converted to do work in particular Free energy may refer to In science: Thermodynamic free energy, the energy in a physical system that can be converted to do work in particular e. by causing fracture. A fracture is the (local separation of an object or material into two or more pieces under the action of stress. Using this procedure, Griffith found that

$C = \sqrt{\cfrac{2E\gamma}{\pi}}$

where $E$ is the Young's modulus of the material and $γ$ is the surface energy density of the material. Surface energy quantifies the disruption of intermolecular bonds that occurs when a surface is created Assuming $E = 62$ GPa and $γ = 1$ J/m² gives excellent agreement of Griffith's predicted fracture stress with experimental results for glass.

Irwin's modification of Griffith's energy relation

The plastic zone around a crack tip in a ductile material.

Griffith's work was largely ignored by the engineering community until the early 1950s. The reasons for this appear to be (a) in the actual structural materials the level of energy needed to cause fracture is orders of magnitude higher than the corresponding surface energy, and (b) in structural materials there are always some inelastic deformations around the crack front that would make the assumption of linear elastic medium with infinite stresses at the crack tip highly unrealistic. F. Erdogan (2000)^[2]

Griffith's theory provides excellent agreement with experimental data for brittle materials such as glass. Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many For ductile materials such as steel, though the relation $\sigma_y\sqrt{a} = C$ still holds, the surface energy ( $γ$ ) predicted by Griffith's theory is usually unrealistically high. Ductility is a mechanical property used to describe the extent to which materials can be deformed plastically or "stretched" into "wires" without Steel is an Alloy consisting mostly of Iron, with a Carbon content between 0 Surface energy quantifies the disruption of intermolecular bonds that occurs when a surface is created A group working under G. R. Irwin ^[3] at the U. Dr George Rankin Irwin ( February 26 1907 &ndash October 9 1998) was an American scientist in the field of Fracture mechanics S. Naval Research Laboratory (NRL) during World War II realized that plasticity must play a significant role in the fracture of ductile materials. Ductility is a mechanical property used to describe the extent to which materials can be deformed plastically or "stretched" into "wires" without

In ductile materials (and even in materials that appear to be brittle ^[4]), a plastic zone develops at the tip of the crack. Ductility is a mechanical property used to describe the extent to which materials can be deformed plastically or "stretched" into "wires" without Plastic is the general common term for a wide range of synthetic or semisynthetic organic solid materials suitable for the manufacture of industrial products As the applied load increases, the plastic zone increases in size until the crack grows and the material behind the crack tip unloads. Plastic is the general common term for a wide range of synthetic or semisynthetic organic solid materials suitable for the manufacture of industrial products The plastic loading and unloading cycle near the crack tip leads to the dissipation of energy as heat. In Physics, dissipation embodies the concept of a Dynamical system where important mechanical modes such as Waves or Oscillations lose Energy In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Physics, heat, symbolized by Q, is Energy transferred from one body or system to another due to a difference in Temperature Hence, a dissipative term has to be added to the energy balance relation devised by Griffith for brittle materials. In physical terms, additional energy is needed for crack growth in ductile materials when compared to brittle materials.

Irwin's strategy was to partition the energy into two parts:

the stored elastic strain energy which is released as a crack grows. This is the thermodynamic driving force for fracture.
the dissipated energy which includes plastic dissipation and the surface energy (and any other dissipative forces that may be at work). In Physics, dissipation embodies the concept of a Dynamical system where important mechanical modes such as Waves or Oscillations lose Energy Plastic is the general common term for a wide range of synthetic or semisynthetic organic solid materials suitable for the manufacture of industrial products Surface energy quantifies the disruption of intermolecular bonds that occurs when a surface is created The dissipated energy provides the thermodynamic resistance to fracture. Then the total energy dissipated is $G = 2γ + G p$ where $γ$ is the surface energy and $G p$ is the plastic dissipation (and dissipation from other sources) per unit area of crack growth.

The modified version of Griffith's energy criterion can then be written as

$\sigma_f\sqrt{a} = \sqrt{\cfrac{E~G}{\pi}} ~.$

For brittle materials such as glass, the surface energy term dominates and $G \approx \gamma = 1$ J/m². For ductile materials such as steel, the plastic dissipation term dominates and $G \approx G_p = 1000$ J/m². For polymers close to the glass transition temperature, we have intermediate values of $G \approx 1-1000$ J/m². A polymer is a large Molecule ( Macromolecule) composed of repeating Structural units typically connected by Covalent Chemical bonds The glass transition temperature, T g is the temperature at which an Amorphous solid, such as Glass or a Polymer, becomes brittle

Stress intensity factor

Another significant achievement of Irwin and his colleagues was to a method of calculating the amount of energy available for fracture in terms of the asymptotic stress and displacement fields around a crack front in a linear elastic solid ^[3]. Dr George Rankin Irwin ( February 26 1907 &ndash October 9 1998) was an American scientist in the field of Fracture mechanics This asymptotic expression for the stress field around a crack tip is

$\sigma_{ij} \approx \left(\cfrac{K}{\sqrt{2\pi r}}\right)~f_{ij}(\theta)$

where $σ i j$ are the Cauchy stresses, $r$ is the distance from the crack tip, $θ$ is the angle with respect to the plane of the crack, and $f i j$ are functions that are independent of the crack geometry and loading conditions. Irwin called the quantity $K$ the stress intensity factor. Stress Intensity Factor, K is used in Fracture mechanics to more accurately predict the stress state ("stress intensity" near the tip of a crack caused Since the quantity $f i j$ is dimensionless, the stress intensity factor can be expressed in units of $\text{Pa-}\sqrt{\text{m}}$ .

Strain energy release rate

Irwin was the first to observe that if the size of the plastic zone around a a crack is small compared to the size of the crack, the energy required to grow the crack will not be critically dependent on the state of stress at the crack tip. Dr George Rankin Irwin ( February 26 1907 &ndash October 9 1998) was an American scientist in the field of Fracture mechanics ^[2] In other words, a purely elastic solution may be used to calculate the amount of energy available for fracture.

The energy release rate for crack growth or strain energy release rate may then be calculated the change in elastic strain energy per unit area of crack growth, i. e. ,

$G := -\left[\cfrac{\partial U}{\partial a}\right]_P = -\left[\cfrac{\partial U}{\partial a}\right]_u$

where $U$ is the elastic energy of the system and $a$ is the crack length. Either the load $P$ or the displacement $u$ can be kept fixed while evaluating the above expressions.

Irwin showed that for a mode I crack the strain energy release rate and the stress intensity factor are related by

$G = G_I = \begin{cases} \cfrac{K_I^2}{E} & \text{plane stress} \\ \cfrac{(1-\nu^2) K_I^2}{E} & \text{plane strain} \end{cases}$

where $E$ is the Young's modulus, $ν$ is the Poisson's ratio, and $K I$ is the stress intensity factor in mode I. A fracture is the (local separation of an object or material into two or more pieces under the action of stress. Stress Intensity Factor, K is used in Fracture mechanics to more accurately predict the stress state ("stress intensity" near the tip of a crack caused In Solid mechanics, Young's modulus (E is a measure of the Stiffness of an isotropic elastic material Poisson's ratio ( ν) named after Simeon Poisson, is the ratio of the relative contraction strain, or transverse strain (normal to Stress Intensity Factor, K is used in Fracture mechanics to more accurately predict the stress state ("stress intensity" near the tip of a crack caused A fracture is the (local separation of an object or material into two or more pieces under the action of stress. Irwin also showed that the strain energy release rate of a planar crack in a linear elastic body can be expressed in terms of the mode I, mode II, and mode III stress intensity factors for the most general loading conditions. A fracture is the (local separation of an object or material into two or more pieces under the action of stress. A fracture is the (local separation of an object or material into two or more pieces under the action of stress. A fracture is the (local separation of an object or material into two or more pieces under the action of stress.

Next, Irwin adopted the additional assumption that the size and shape of the energy dissipation zone remains approximately constant during brittle fracture. This assumption suggests that the energy needed to create a unit fracture surface is a constant that depends only on the material. This new material property was given the name fracture toughness and designated $G I c$ . 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 Today, it is the related quantity K_Ic which is called the fracture toughness and is now universally accepted as the defining material property in linear elastic fracture mechanics. 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

Limitations of linear elastic fracture mechanics

$The S.S. Schenectady split apart by brittle fracture while in harbor (1944)$

The S. S. Schenectady split apart by brittle fracture while in harbor (1944)

But a problem arose for the NRL researchers because naval materials, e. A fracture is the (local separation of an object or material into two or more pieces under the action of stress. g. , ship-plate steel, are not perfectly elastic but undergo significant plastic deformation at the tip of a crack. In Materials science, deformation is a change in the shape or size of an object due to an applied force. One basic assumption in Irwin's linear elastic fracture mechanics is that the size of the plastic zone is small compared to the crack length. However, this assumption is quite restrictive for certain types of failure in structural steels though such steels can be prone to brittle fracture, which has led to a number of catastrophic failures. .

Linear-elastic fracture mechanics is of limited practical use for structural steels for another more practical reason. Fracture toughness testing is very expensive and engineers believe that sufficient information for selection of steels can be obtained from the simpler and cheaper Charpy impact test. The Charpy impact test, also known as the Charpy v-notch test, is a standardized high strain -rate test which determines the amount of Energy

Elastic-plastic fracture mechanics

Vertical stabilizer, which separated from the aircraft leading to a fatal crash(2001)

Most engineering materials show some inelastic behavior under operating conditions that involve large loads. The vertical stabilizers, or fins, of Aircraft, Missiles or Bombs are typically found on the aft end of the Fuselage or body In such materials the assumptions of linear elastic fracture mechanics may not hold, that is,

the plastic zone at a crack tip may have a size of the same order of magnitude as the crack size
the size and shape of the plastic zone may change as the applied load is increased and also as the crack length increases.

Therefore a more general theory of crack growth is needed for elastic-plastic materials that can account for:

the local conditions for initial crack growth which include the nucleation, growth, and coalescence of voids or decohesion at a crack tip.
a global energy balance criterion for further crack growth and unstable fracture.

R-curve

An early attempt in the direction of elastic-plastic fracture mechanics was Irwin's crack extension resistance curve or R-curve. Dr George Rankin Irwin ( February 26 1907 &ndash October 9 1998) was an American scientist in the field of Fracture mechanics This curve acknowledges the fact that the resistance to fracture increases with growing crack size in elastic-plastic materials. The R-curve is a plot of the total energy dissipation rate as a function of the crack size and can be used to examine the processes of slow stable crack growth and unstable fracture. However, the R-curve was not widely used in applications until the early 1970s. The main reasons appear to be that the R-curve depends on the geometry of the specimen and the crack driving force may be difficult to calculate ^[2].

J-integral

In the mid-1960s J. R. Rice (then at Brown University) and G. Brown University is a highly esteemed private University located in Providence, Rhode Island and is a member of the Ivy League. P. Cherepanov independently developed a new toughness measure to describe the case where there is sufficient crack-tip deformation that the part no longer obeys the linear-elastic approximation. Rice's analysis, which assumes non-linear elastic (or monotonic deformation-theory plastic) deformation ahead of the crack tip, is designated the J integral^[5]. Plastic is the general common term for a wide range of synthetic or semisynthetic organic solid materials suitable for the manufacture of industrial products The J- Integral represents a way to calculate the Strain energy release rate, or work ( Energy) per unit fracture surface area in a material This analysis is limited to situations where plastic deformation at the crack tip does not extend to the furthest edge of the loaded part. It also demands that the assumed non-linear elastic behavior of the material is a reasonable approximation in shape and magnitude to the real material's load response. The elastic-plastic failure parameter is designated J_Ic and is conventionally converted to K_Ic using Equation (3. 1) of the Appendix to this article. Also note that the J integral approach reduces to the Griffith theory for linear-elastic behavior.

Fully plastic fracture mechanics

If the alloy is so tough that the yielded region ahead of the crack extends to the far edge of the specimen before fracture, the crack is no longer an effective stress concentrator. Instead, the presence of the crack merely serves to reduce the load-bearing area. In this regime the failure stress is conventionally assumed to be the average of the yield and ultimate strengths of the alloy.

Engineering applications of fracture mechanics

The following information is needed for a fracture mechanics prediction of failure:

Applied load
Residual stress
Size and shape of the part
Size, shape, location, and orientation of the crack

Usually not all of this information is available and conservative assumptions have to be made.

Occasionally post-mortem fracture-mechanics analyses are carried out. In the absence of an extreme overload, the causes are either insufficient toughness (K_Ic) or an excessively large crack that was not detected during routine inspection.

Short summary

Arising from the manufacturing process, interior and surface flaws are found in all metal structures. Not all such flaws are unstable under service conditions. Fracture mechanics is the analysis of flaws to discover those that are safe (that is, do not grow) and those that are liable to propagate as cracks and so cause failure of the flawed structure. Structural failure refers to loss of the load -carrying capacity of a component or member within a Structure or of the structure itself Fracture mechanics as a subject for critical study has barely been around for a century and thus is relatively new. There is a high demand for engineers with fracture mechanics expertise - particularly in this day and age where engineering failure is considered 'shocking' amongst the general public.

Appendix: mathematical relations

Griffith's crack theory: strain energy release rate

For the simple case of a thin rectangular plate with a crack perpendicular to the load Griffith’s theory becomes:

$G = \frac{\pi \sigma^2 a}{E}\,$ (1. 1)

where $G$ is the strain energy release rate, $σ$ is the applied stress, $a$ is half the crack length, and $E$ is the Young’s modulus. An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically (i The strain energy release rate can otherwise be understood as: the rate at which energy is absorbed by growth of the crack.

However, we also have that:

$G_c = \frac{\pi \sigma_f^2 a}{E}\,$ (1. 2)

If $G$ ≥ $G c$ , this is the criterion for which the crack will begin to propagate.

Irwin's modified Griffith crack theory: fracture toughness

Eventually a modification of Griffith’s solids theory emerged from this work; a term called stress intensity replaced strain energy release rate and a term called fracture toughness replaced surface weakness energy. Stress Intensity Factor, K is used in Fracture mechanics to more accurately predict the stress state ("stress intensity" near the tip of a crack caused 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 Both of these terms are simply related to the energy terms that Griffith used:

$K_I = \sigma \sqrt{\pi a}\,$ (2. 1)

and

$K_c = \sqrt{E G_c}\,$ (for plane stress) (2. Stress is a measure of the average amount of Force exerted per unit Area. 2)

$K_c = \sqrt{\frac{E G_c}{1 - \nu^2}}\,$ (for plane strain) (2. 3)

where K_I is the stress intensity, K_c the fracture toughness, and $ν$ is Poisson’s ratio. Stress Intensity Factor, K is used in Fracture mechanics to more accurately predict the stress state ("stress intensity" near the tip of a crack caused 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 Poisson's ratio ( ν) named after Simeon Poisson, is the ratio of the relative contraction strain, or transverse strain (normal to It is important to recognize the fact that fracture parameter K_c has different values when measured under plane stress and plane strain

Fracture occurs when $K_I \geq K_c$ . For the special case of plane strain deformation, $K c$ becomes $K I c$ and is considered a material property. The subscript I arises because of the different ways of loading a material to enable a crack to propagate. It refers to so-called "mode I" loading as opposed to mode II or III:

$The three fracture modes.$

The three fracture modes.

There are three ways of applying a force to enable a crack to propagate:

Mode I crack – Opening mode (a tensile stress normal to the plane of the crack)
Mode II crack – Sliding mode (a shear stress acting parallel to the plane of the crack and perpendicular to the crack front)
Mode III crack – Tearing mode (a shear stress acting parallel to the plane of the crack and parallel to the crack front)

We must note that the expression for $K I$ in equation 2. Stress is a measure of the average amount of Force exerted per unit Area. A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material 1 will be different for geometries other than the center cracked plate, as discussed in the article on stress intensity. Stress Intensity Factor, K is used in Fracture mechanics to more accurately predict the stress state ("stress intensity" near the tip of a crack caused Consequently, it is necessary to introduce a dimensionless correction factor, Y, in order to characterize the geometry. In Dimensional analysis, a dimensionless quantity (or more precisely a quantity with the dimensions of 1) is a Quantity without any Physical units We thus have:

$K_I = Y \sigma \sqrt{\pi a}\,$ (2. 4)

where Y is a function of the crack length and width of sheet given by:

$Y \left ( \frac{a}{W} \right ) = \sqrt{\sec\left ( \frac{\pi a}{W} \right )}\,$ (2. 5)

for a sheet of finite width W containing a through-thickness crack of length 2a, or

$Y \left ( \frac{a}{W} \right ) = 1.12 - \frac{0.41}{\sqrt \pi} \frac{a}{W} + \frac{18.7}{\sqrt \pi} \left ( \frac{a}{W} \right )^2 - \cdots\,$ (2. 6)

for a sheet of finite width W containing a through-thickness edge crack of length a

Elastic-plastic fracture mechanics theory

Since engineers became accustomed to using K_Ic to characterise fracture toughness, a relation has been used to reduce J_Ic to it:

$K_{Ic} = \sqrt{E^* J_{Ic}}\,$ where

E * = E

for plane strain and $E^* = \frac{E}{1 - \nu^2}$ for plane stress (3. 1)

The remainder of the mathematics employed in this approach is interesting, but is probably better summarised in external pages due to its complex nature (refer to the Useful Websites section).

References

^ Griffith, A.A. 1920. The theory of rupture and flow in solids. Phil.Trans.Roy.Soc.Lond. A221, pp. 163-198.
^ ^a ^b ^c E. Erdogan (2000) Fracture Mechanics, International Journal of Solids and Structures, 27, pp. 171-183.
^ ^a ^b Irwin G (1957), Analysis of stresses and strains near the end of a crack traversing a plate, Journal of Applied Mechanics 24, 361–364.
^ Orowan, E. , 1948. Fracture and strength of solids. Reports on Progress in Physics XII, 185-232.
^ Rice,J.R. 1968. A path independent integral and the approximate analysis of strain concentration by notches and cracks. Trans. ASME: J. Appl. Mech. 35, 379-386

C. P. Buckley, "Material Failure", Lecture Notes (2005), University of Oxford
T. The University of Oxford (informally "Oxford University" or simply "Oxford" located in the city of Oxford, Oxfordshire, England is the L. Anderson, "Fracture Mechanics: Fundamentals and Applications" (1995) CRC Press.

External links

eFunda - Fracture Mechanics
UMIST - Charpy Impact Test
Brown University Engineering - Mathematical Relations
Fracture Mechanics Notes by Prof. A fracture is the (local separation of an object or material into two or more pieces under the action of stress. 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 Fracture mechanics --> Stress corrosion cracking ( SCC) is the unexpected sudden failure of normally Ductile metals or tough Thermoplastics Stress Intensity Factor, K is used in Fracture mechanics to more accurately predict the stress state ("stress intensity" near the tip of a crack caused The strain energy release rate (or energy release rate is the Energy dissipated during Fracture per unit of newly created fracture surface area Alan Zehnder (from Cornell University)
Nonlinear Fracture Mechanics Notes by Prof. John Hutchinson (from Harvard University)
Notes on Fracture of Thin Films and Multilayers by Prof. John Hutchinson (from Harvard University)
Mixed mode cracking in layered materials by Profs. John Hutchinson and Zhigang Suo (from Harvard University)
Fracture Mechanics by Prof. Piet Schreurs (from TU Eindhoven, Netherlands)
Introduction to Fracture Mechanics by Dr. C. H. Wang (DSTO - Australia)
Fracture mechanics course notes by Prof. Rui Huang (from Univ. of Texas at Austin)

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