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**Please note:- Underlined text is already available in wikipedia page for fracture toughness

General Aim
Fracture toughness tests are performed to quantify the resistance of a material to failure by cracking. Such tests result in either a single valued measure of fracture toughness or in a resistance curve. Resistance curves are plots where fracture toughness parameters(K,J etc) are plotted against parameters characterizing the propagation of crack. The resistance curve or the single valued fracture toughness is obtained based on the mechanism and stability of fracture. For unstable fracture the toughness of the material decreases/ stays the same as the crack grows, resulting in a single valued fracture toughness. On other hand stable crack growth is characterized by a rising resistance curve. Fracture toughness is a critical mechanical property for engineering applications. There are several types of test used to measure fracture toughness of materials, which generally utilise a notched specimen in one of various configurations. A widely utilized standardized test method is the Charpy impact test whereby a sample with a V-notch or a U-notch is subjected to impact from behind the notch. Also widely used are crack displacement tests such as three point beam bending tests with thin cracks preset into test specimens before applying load.

The immense real-world consequence of failure of materials by fracture has motivated a variety of organization throughout the world to publish standardized procedures for fracture toughness measurements, namely ASTM, BSI, ISO,JSME. No single standard as informed through these organization is universal, so careful inspection of the standards are required before it can be applied to a specimen.

Prior to Testing
There are few common points need to considered before commencing with the testing of any specimen. The design and orientation of test specimen is one of the main points of scrutiny.

ASTM standard describes five specimen configurations, namely compact specimen, disk-shaped compact specimen, single edge notched bend specimen(SENB), middle tension and arc-shaped specimen. Each specimen configuration is characterized by three dimensions, namely the crack length(a), the thickness(B) and the width(W). The values of these dimension is determined by the demand of the particular test that is being performed on the specimen. The vast majority of the tests are carried out on either compact or SENB configuration. For the same characteristic dimensions compact configuration takes lesser amount of material compared to SENB.

Orientation of fracture is important because of the inherent non-isotropic nature of most engineering materials. Due to this there may be "planes of weakness" within the material, and crack growth along this plane may be easier compared to other direction. Due to this importance ASTM has devised a standardized way of reporting the crack orientation with respect to forging axis. The letters L, T and S are used to denote longitudinal, transverse and short transverse direction, where longitudinal direction coincides with forging axis. The orientation is defined with two letters the first one being the direction of principal tensile stress and the second one is the direction of crack propagation. Generally speaking the lower bound of the toughness of a material is obtained in the orientation where the crack grows in the direction of forging axis.

It is imperative to carefully introduce a crack into the specimen to test. To apply the results of the toughness tests to the theory we need an infinitely sharp crack prior to loading. The machined cracks do not meet this criteria, i.e. they have a finite opening at the crack tip. The most effective way of introducing a sufficiently sharp crack is through cyclic loading. Fatigue cracks are initiated at the tip of machined cracks and allowed to extend till the crack length reaches it's desired value. The cyclic loading is to be controlled carefully so as to not affect the actual toughness of the material through strain-hardening. This is done through choosing the cyclic loads which produces far smaller plastic zone compared to plastic zone of the main fracture. For example, according to ASTM E399 standard the maximum stress intensity Kmax should be no larger than 0.6KIC during the initial stage and less than 0.8KIC when crack approaches it's final size.

In certain cases grooves are machined into the sides of a fracture toughness specimen so that the thickness of the specimen is reduced to minimum of 80% of the original thickness along the intended path of crack extensions. The reason is to maintain a straight crack front during R-curve test.

The four main standardized tests are described below with KIc and KR tests valid for linear-elastic fracture mechanics(LEFM) while J and JR tests valid for elastic-plastic fracture mechanics(PEFM)

Determination of plane strain fracture toughness, KIc
When a material....Plastic zone

Plane-strain fracture toughness testing
When performing a fracture toughness test, the most common test specimen configurations are the single edge notch bend (SENB or three-point bend), and the compact tension (CT) specimens. From the above discussion, it is clear that an accurate determination of the plane-strain fracture toughness requires a specimen whose thickness exceeds some critical thickness (B). Testing has shown that plane-strain conditions generally prevail when:

$$B,a\geq2.5\left(\frac{K_{IC}}{\sigma_{YS}}\right)^2$$

Where: B is the minimum thickness that produces a condition where plastic strain energy at the crack tip in minimal

KIC is the fracture toughness of the material

$$\sigma_{YS}$$ is the yield stress of material

The test is performed by loading steadily at a rate such that KI increases steadily from 0.55 to 2.75 (MPa$$\sqrt{m}$$)/s. The load and the crack mouth opening displacement(CMOD) is noted and the test is continued till the maximum load is reached. The critical load, PQ is calculated through specialized data reduction method from the load vs CMOD plot. Then we calculate a provisional toughness KQ given as $$K_Q=\frac{P_Q}{\sqrt{W}B}f(a/W,...)$$. The function$$f(a/W,...)$$ is a dimensionless function of a/W. This function is given in polynomial form in E 399 standard, Function for compact test geometry can be found here. This provisional toughness value is recognized as the valid one when the validity requirements are met, which are

$$min(B,a)>2.5\left(\frac{K_{Q}}{\sigma_{YS}}\right)^2$$and $$P_{max}\leq 1.1P_Q$$

When a material of unknown fracture toughness is tested, a specimen of full material section thickness is tested or the specimen is sized based on a prediction of the fracture toughness. If the fracture toughness value resulting from the test does not satisfy the requirement of the above equation, the test must be repeated using a thicker specimen. In addition to this thickness calculation, test specifications have several other requirements that must be met (such as the size of the shear lips) before a test can be said to have resulted in a KIC value.

When a test fails to meet the thickness and other test requirement that are in place to insure plane-strain condition, the fracture toughness values produced is given the designation KC. Sometimes it is not possible to produce a specimen that meets the thickness requirement. For example, when a relatively thin plate product with high toughness is being tested, it might not be possible to produce a thicker specimen with plain-strain conditions at the crack tip.

Determination of fracture toughness R-curve, K-R
The specimen showing stable crack growth shows a increasing trend in fracture toughness as the crack length increases(ductile crack extension). This plot of fracture toughness vs crack length is called the resistance(R)-curve. ASTM E561 outlines a procedure for determining toughness vs crack growth curves in materials. This standard does not have a constraint over the minimum thickness of the material and hence can be used for thin sheets however the requirements for LEFM must be fulfilled for the test to be valid. The criteria for LEFM essentially states that in-plane dimension has to be large compared to the plastic zone. There is a misconception about the effect of thickness on the shape of R curve. It is hinted that for the same material thicker section fails by plane strain fracture and shows a single-valued fracture toughness, the thinner section fails by plane stress fracture and shows the rising R-curve. However the main factor that controls the slope of R curve is the fracture morphology not the thickness. In some material section thickness changes the fracture morphology from ductile tearing to clevage from thin to thick section, in which case the thickness alone dictates the slope of R-curve. There are cases where even plane strain fracture ensues in rising R-curve due to "microvoid coalescence" being the mode of failure.

Negligible plastic zone
The most accurate way of evaluating K-R curve is taking presence of plasticity into account depending on the relative size of plastic zone. For the case of negligible plasticity the load vs displacement curve is obtained from the test and on each point the compliance is found. The compliance is reciprocal of the slope of the curve that will be followed if the specimen is unloaded at certain point, which can be given as the ratio of displacement to load for LEFM. The compliance is used to determine the instantaneous crack length,a through the relationship given in ASTM standard. The crack length can also be measured through optical means. Which is followed by KI calculation using the formula$$K_I=\frac{P}{\sqrt{W}B} f(a/W,...)$$-{1]

Modification considering plasticity
The stress intensity should be corrected by calculating an effective crack length. ASTM standard suggests two alternative approaches. The first method is named Irwin's plastic zone correction. Irwin's approach describes the effective crack length to be

$$a_{eff}=a+\frac{1}{2\pi}\left(\frac{K}{\sigma_{YS}}\right)^2$$where $$\sigma_{YS}$$= Uni-axial yield strength of the material. ---[2]

The Irwin's approach leads to an iterative solution as K itself is a function of crack length.

The other method, namely secant method, uses the compliance-crack length equation given by ASTM standard to calculate effective crack length from an effective compliance. Compliance at any point in Load vs displacement curve is essentially the reciprocal of the slope of the curve that ensues if the specimen is unloaded at that point. Now the unloading curve returns to the origin for linear elastic material but not for elastic plastic material as there is a permanent deformation. The effective compliance at a point for the elastic plastic case is taken as the slope of the line joining the point and origin( i.e the compliance if the material was an elastic one). This effective compliance is used to get an effective crack growth and the rest of the calculation follows the equation

$$K_I=\frac{P}{\sqrt{W}B} f(a_{eff}/W,...)$$

The choice of plasticity correction is factored on the size of plastic zone. ASTM standard covering resistance curve suggests using Irwin's method is acceptable for small plastic zone and recommends using Secant method when crack-tip plasticity is more prominent. Also since the ASTM E 561 standard does not contain requirements on the specimen size or maximum allowable crack extension, thus the size independence of the resistance curve is not guaranteed. Few study shows that the size dependence is less detected in the experimental data for Secant method.

Determination of JIC
Strain energy release rate per unit fracture surface area is calculated by J-integral method which is a contour path integral around the crack tip where the path begins and ends on either crack surfaces. J-toughness value signifies the resistance of the material in terms of amount of stress energy required for a crack to grow. JIC toughness value is measured for elastic-plastic materials. Now the single valued JIC is determined as the toughness near the onset of the ductile crack extension( effect of strain hardening is not important). The test is performed with multiple specimen loading each of the specimen to various levels and unloading. This gives the crack mouth opening compliance which is to be used to get crack length with the help of relationships given in ASTM standard E 1820, which covers the J-integral testing. Another way of measuring crack growth is to mark the specimen with heat tinting or fatigue cracking. The specimen is eventually broken apart and the crack extension is measured with the help of the marks.

The test thus performed yields several Load vs Crack Mouth Opening Displacement(CMOD) curve, which are used to calculate J as following:-

$$J=J_{el}+J_{pl}$$

The linear elastic J is calculated using

$$J_{el}=\frac{K^2\left(1-\nu^2\right)}{E}$$and K is determined from $$K_I=\frac{P}{\sqrt{WBB_N}} f(a/W,...)$$where BN is the net thickness for side-grooved specimen and equal to B for not side-grooved specimen

The elastic plastic J is calculated using

$$J_pl=\frac{\eta A_{pl}}{B_Nb_o}$$

Where $$\eta$$=2 for SENB specimen

bo is initial ligament length given by the difference between width and initial crack length

APl is the plastic area under the load-displacement curve.

Specialized data reduction technique is used to get an provisional JQ. The value is accepted if the following criteria is met

$$min(B,b_o)\geq\frac{25J_Q}{\sigma_Y}$$