Fracture
A fracture is the separation of an object or material into two or more pieces under the action of applied stress, often appearing as a crack or complete break.[1] In materials science and engineering, the study of fractures is central to fracture mechanics, a field that analyzes crack initiation, propagation, and failure to predict and prevent structural breakdowns in components like bridges, aircraft, and machinery.[2] Fractures are broadly classified into two types: brittle, where failure occurs with little or no plastic deformation, leading to sudden and catastrophic cracks; and ductile, involving significant plastic deformation before separation, often resulting in a more gradual failure mode.[3] Understanding these mechanisms is crucial for designing safer materials and assessing risks in high-stress environments, as uncontrolled fractures have contributed to numerous engineering disasters.[4]Fundamentals
Definition and Scope
Fracture is the irreversible separation of a solid material into two or more parts when subjected to applied stress that exceeds its capacity to withstand loading, resulting in breaking or cracking.[5] This process fundamentally differs from elastic deformation, which involves reversible straining, or plastic yielding, which allows permanent shape change without complete separation.[6] The theoretical foundation of fracture mechanics originated with A.A. Griffith's 1921 work on brittle fracture in glass, where he analyzed the propagation of pre-existing cracks through an energy balance approach, explaining why brittle materials fail at stresses far below their theoretical strength.[7] Griffith's criterion for the onset of brittle fracture provides a quantitative relation, expressed as: \sigma_f = \sqrt{\frac{2E\gamma}{\pi a}} where \sigma_f denotes the fracture stress, E is the Young's modulus, \gamma represents the surface energy required to create new crack surfaces, and a is the half-length of an internal crack (or the full length for an edge crack).[8] This equation highlights the critical role of flaw size in determining material strength, shifting focus from uniform material properties to defect-controlled failure. The scope of fracture studies in materials science and engineering broadly applies to solid materials such as metals, ceramics, and polymers, where failure modes range from rapid brittle separation to more gradual ductile processes.[9] Understanding fracture is essential for designing reliable components, as it informs strategies to mitigate risks in applications from structural aerospace parts to biomedical implants, distinguishing catastrophic failure from controlled deformation.[10]Basic Principles of Fracture Mechanics
Linear elastic fracture mechanics (LEFM) forms the cornerstone of modern fracture analysis, focusing on the behavior of cracks in materials that remain predominantly elastic. Originating from A.A. Griffith's pioneering work on brittle fracture in 1921, LEFM was formalized by G.R. Irwin in the 1950s to address the stress concentration and propagation criteria for cracks under linear elastic conditions.[7][11] LEFM assumes small-scale yielding, meaning the region of plastic deformation at the crack tip is much smaller than both the crack length and the overall specimen dimensions, allowing linear elasticity to govern the far-field response while capturing the intense local stresses. Additionally, the material is assumed to behave elastically away from the crack tip, with quasi-static loading and no significant time-dependent effects. These assumptions enable predictive models for crack stability and growth based on continuum mechanics.[11] Central to LEFM is the stress intensity factor K, which quantifies the magnitude of the three-dimensional stress field near the crack tip and serves as a fracture criterion. Irwin defined three primary modes of crack loading: mode I for tensile opening normal to the crack plane, mode II for in-plane sliding or shear, and mode III for out-of-plane tearing or anti-plane shear. For an infinite plate containing a central through-crack of length $2a subjected to uniform remote tensile stress \sigma perpendicular to the crack, the mode I stress intensity factor is expressed as K_I = \sigma \sqrt{\pi a}. This formulation highlights how K integrates the effects of applied stress, crack length, and geometry, with crack propagation initiating when K exceeds a material-specific critical value K_c. The near-tip stresses follow a singular form \sigma_{ij} \sim K / \sqrt{2\pi r}, where r is the radial distance from the tip, underscoring the infinite stress concentration in ideal elastic theory.[11] Complementing the stress-based approach, LEFM employs an energy balance criterion rooted in Griffith's 1921 theory, where stable crack growth requires the release of elastic strain energy to overcome the surface energy of new crack faces. The energy release rate G, defined as the decrease in potential energy per unit crack advance, governs fracture when G \geq G_c, the critical energy release rate. Irwin linked this to the stress intensity factor through G = \frac{K^2}{E'}, where E' is the effective modulus: E' = E under plane stress and E' = E / (1 - \nu^2) under plane strain, with E as Young's modulus and \nu as Poisson's ratio. This equivalence bridges stress and energy perspectives, enabling unified criteria for brittle fracture prediction.[7][11] The idealized elastic singularity at the crack tip is moderated in real materials by localized plasticity, forming a small plastic zone where stresses are capped by the yield strength \sigma_y. Irwin estimated the plane-stress plastic zone radius along the crack plane as r_p \approx \frac{1}{2\pi} \left( \frac{K}{\sigma_y} \right)^2, derived by setting the elastic \sigma_{yy} stress equal to \sigma_y and solving for the distance r where yielding begins. Under plane strain, the zone is roughly one-third smaller due to triaxiality constraints. This size correction validates LEFM applicability when r_p \ll a, ensuring the plastic enclave does not perturb the elastic K-dominated field.[11] For ductile materials exhibiting extensive plasticity, where r_p approaches or exceeds structural dimensions, LEFM's small-scale yielding assumption fails, prompting a shift to elastic-plastic fracture mechanics (EPFM). EPFM extends LEFM principles to nonlinear regimes using path-independent integrals like the J-integral, introduced by J.R. Rice in 1968, which generalizes [G](/page/G) for incremental plasticity and characterizes crack-tip driving force under large deformation. This transition is essential for metals and alloys where yielding precedes unstable fracture.[12]Material Response to Stress
Fracture Strength
Fracture strength, also known as breaking strength, refers to the maximum stress a material can endure immediately prior to fracturing under tensile loading, marking the point of complete failure.[13] This differs from yield strength, which indicates the onset of permanent plastic deformation without fracture, allowing materials—particularly ductile ones—to sustain loads beyond yielding before breaking.[5] In brittle materials, fracture strength often coincides with ultimate tensile strength, as failure occurs abruptly without significant plasticity, whereas in ductile materials, it follows necking after reaching the ultimate stress peak.[14] Microstructural features profoundly influence fracture strength, with grain size, defects, and temperature playing key roles. The Hall-Petch relation empirically captures the strengthening effect of finer grains, where fracture strength \sigma_f increases inversely with the square root of grain diameter d: \sigma_f = \sigma_0 + k d^{-1/2}, with \sigma_0 as the friction stress and k as the strengthening coefficient; this arises from increased grain boundary barriers to dislocation motion and crack propagation. Defects such as voids, inclusions, and microcracks act as stress concentrators, drastically lowering strength by facilitating premature crack initiation.[15] Elevated temperatures generally reduce fracture strength by enhancing atomic mobility, promoting dislocation glide, and accelerating diffusion-mediated processes like creep, though specific effects vary by material class.[16] For brittle materials exhibiting stochastic failure due to inherent flaw distributions, fracture strength follows a statistical description via the Weibull distribution, which models the probability of failure P_f under uniform stress \sigma over volume V:P_f = 1 - \exp\left[-\left(\frac{V}{V_0}\right)\left(\frac{\sigma}{\sigma_0}\right)^m\right],
where V_0 is a reference volume, \sigma_0 a characteristic strength, and m the Weibull modulus reflecting flaw variability (higher m indicates more consistent strength).[17] This "weakest-link" approach accounts for size effects, where larger volumes increase failure likelihood at lower stresses./06%3A_Yield_and_Fracture/6.03%3A_Statistics_of_Fracture) Environmental factors like corrosion and hydrogen embrittlement significantly degrade fracture strength by introducing surface degradation and internal stresses. Corrosion, through pitting or uniform attack, creates localized stress raisers that reduce effective cross-section and initiate cracks, often lowering strength by 20-50% depending on exposure duration and severity.[18] Hydrogen embrittlement, prevalent in high-strength steels, diffuses atomic hydrogen into the lattice, promoting brittle intergranular fracture and reducing ductility; it can diminish fracture strength by up to 50% or more in susceptible alloys by facilitating hydrogen-enhanced decohesion or localized plasticity.[19] The disparity between theoretical and actual fracture strength underscores the role of imperfections. In an ideal, flaw-free crystal, theoretical strength approximates E/10 (where E is the elastic modulus), derived from the stress needed to break atomic bonds uniformly.[15] However, real materials achieve only about E/1000 due to preexisting flaws, as explained by Griffith's criterion, which posits that cracks propagate when the stress intensity overcomes surface energy, yielding strengths orders of magnitude below the theoretical limit.[20]