Жарықтардың таралуын күшейту факторы және материалының қатайғыштығы

Stress intensity factor at which a crack's propagation increases drastically

In materials science, fracture toughness is the critical stress intensity factor of a sharp crack where propagation of the crack suddenly becomes rapid and unlimited. A component's thickness affects the constraint conditions at the tip of a crack with thin components having plane stress conditions and thick components having plane strain conditions. Plane strain conditions give the lowest fracture toughness value which is a material property. The critical value of stress intensity factor in mode I loading measured under plane strain conditions is known as the plane strain fracture toughness, denoted When a test fails to meet the thickness and other test requirements that are in place to ensure plane strain conditions, the fracture toughness value produced is given the designation Fracture toughness is a quantitative way of expressing a material's resistance to crack propagation and standard values for a given material are generally available. Slow self sustaining crack propagation known as stress corrosion cracking, can occur in a corrosive environment above the threshold and below Small increments of crack extension can also occur during fatigue crack growth, which after repeated loading cycles, can gradually grow a crack until final failure occurs by exceeding the fracture toughness.

Material typeMaterialKIc (MPa · m1/2)MetalAluminum 14–28Aluminum alloy (7075) 20 35Inconel 718 73 87Maraging steel (200 Grade) 175Steel alloy (4340) 50Titanium alloy 84–107CeramicAluminum oxide 3–5Silicon carbide 3–5Soda lime glass 0.7–0.8Concrete 0.2–1.4 PolymerPolymethyl methacrylate 0.7–1.60Polystyrene 0.7–1.1 CompositeMullite fibre composite 1.8–3.3Silica aerogels 0.0008–0.0048
Fracture toughness varies by approximately 4 orders of magnitude across materials. Metals hold the highest values of fracture toughness. Cracks cannot easily propagate in tough materials, making metals highly resistant to cracking under stress and gives their stress–strain curve a large zone of plastic flow. Ceramics have a lower fracture toughness but show an exceptional improvement in the stress fracture that is attributed to their 1.5 orders of magnitude strength increase, relative to metals. The fracture toughness of composites, made by combining engineering ceramics with engineering polymers, greatly exceeds the individual fracture toughness of the constituent materials.

The presence of grains in a material can also affect its toughness by affecting the way cracks propagate. In front of a crack, a plastic zone can be present as the material yields. Beyond that region, the material remains elastic. The conditions for fracture are the most favorable at the boundary between this plastic and elastic zone, and thus cracks often initiate by the cleavage of a grain at that location. At low temperatures, where the material can become completely brittle, such as in a body centered cubic (BCC) metal, the plastic zone shrinks away, and only the elastic zone exists. In this state, the crack will propagate by successive cleavage of the grains. At these low temperatures, the yield strength is high, but the fracture strain and crack tip radius of curvature are low, leading to a low toughness. At higher temperatures, the yield strength decreases, and leads to the formation of the plastic zone. Cleavage is likely to initiate at the elastic plastic zone boundary, and then link back to the main crack tip. This is usually a mixture of cleavages of grains, and ductile fracture of grains known as fibrous linkages. The percentage of fibrous linkages increase as temperature increases until the linkup is entirely fibrous linkages. In this state, even though yield strength is lower, the presence of ductile fracture and a higher crack tip radius of curvature results in a higher toughness.

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. 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 ASTM standard E1820 for the measurement of fracture toughness recommends three coupon types for fracture toughness testing, the single edge bending coupon [SE(B)], the compact tension coupon [C(T)] and the disk shaped compact tension coupon [DC(T)]. Each specimen configuration is characterized by three dimensions, namely the crack length (a), the thickness (B) and the width (W). The values of these dimensions are 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 three point flexural test configuration. For the same characteristic dimensions, compact configuration takes a lesser amount of material compared to three point flexural test.

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 the longitudinal, transverse and short transverse directions, where the 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.

The specimen showing stable crack growth shows an 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 cleavage 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. The most accurate way of evaluating K R curve is taking presence of plasticity into account depending on the relative size of the 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 a 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 through the relationship given in the ASTM standard. 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

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

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 studies show that the size dependence is less detected in the experimental data for the Secant method.

The tear test (e. g. Kahn tear test) provides a semi quantitative measure of toughness in terms of tear resistance. This type of test requires a smaller specimen, and can, therefore, be used for a wider range of product forms. The tear test can also be used for very ductile aluminium alloys (e. g. 1100, 3003), where linear elastic fracture mechanics do not apply.

Many ceramics with polycrystalline structures develop large cracks that propagate along the boundaries between grains, rather than through the individual crystals themselves since the toughness of the grain boundaries is much lower than that of the crystals. The orientation of the grain boundary facets and residual stress cause the crack to advance in a complex, tortuous manner that is difficult to analyze. Simply calculating the additional surface energy associated with the increased grain boundary surface area due to this tortuosity is not accurate, as some of the energy to create the crack surface comes from the residual stress.