Critical Flaw for Fracture (Linganna T).pptx

critical flaws for fracture

1. Critical Flaw for Fracture LINGANNA T 120CR1044

2. CONTENTS What is fracture? Failure ..? Crack propagation. Weakest Link Theory. Failure and weakest Link Theory. Weibull Distribution. Some Limitations of Weibull Statistics. Summary.

3. What is fracture? Fracture is the separation of a body into two or more pieces in response to an imposed stress. The applied stress may be tensile or compressive or shear or torsional. Any fracture process involves two steps crack formation and propagationin response to imposed stress. Fracture in ceramics can occur due to various reasons such as mechanical forces, defects in the materials, or sudden changes in temperatures .

4. Failure…? Failure refers to the point at which a structure can no longer carry the loads it was designed to support . Failure occurs imminently without any prior warning- i.e.no visible deformation nor do seemingly identical brittle components appear to break at the same applied stress or at the same point. This is due to the predominance of stronger ionic and covalent bonds in the ceramics.

5. Failure…? Understanding the failure modes of structural ceramics is crucial for designing reliable components. Engineers must consider flaws ,environmental conditions and stress concentrations to prevent failures. Mode I  – Opening mode (a  tensile stress  normal to the plane of the crack), Mode II  – in plane shear mode (a  shear stress  acting parallel to the plane of the crack and perpendicular to the crack front), and Mode III  – out of plane shear mode (a shear stress acting parallel to the plane of the crack and parallel to the crack front).

6. Crack propagation In brittle materials lack of plastic deformation or yielding at the crack front,causes sufficient concentrations to cause the crack to propagate without very little expended energy.

7. When does a crack propagate? Crack propagates if the applied stress is above critical stress s c . i.e., s m > s c Or K t > K c where , E = modulus of elasticity  s = specific surface energy a = one half length of internal crack K c = s c / s The rapid propagation of the crack from the crack tip, especially in a monolithic material results in low fracture toughness or resistance to crack propagation

8. Weakest Link Theory failure of a ceramic material follows the “Weakest Link” theory. The physical properties of the flaw- such as size, shape, orientation and thermal expansion coefficient also play a role . Modern fracture mechanics relates the applied fracture stress at the fracture origin, σf , to a flaw size, c Where, Y is the stress intensity shape factor, is a dimensionless, material independent constant, related to the flaw shape, stress configuration, location. The idea of failure being associated with a largest flaw, or “weakest link theory”, is not a recent theory . Leonardo Da Vinci is reputed to have conducted tests involving baskets suspended by different lengths of wire of nominally identical diameter

9. Weakest Link Theory Da Vinci gradually filled the baskets with sand and noted the baskets suspended by shorter wires could hold more sand, an outcome that is expected if it is assumed that there is a lower probability of encountering a large flaw in a shorter wire. Da Vinci did not know exactly where a particular wire would break, recommended multiple tests be done for each wire length. There was a statistical variation in the strengths and failure locations of wires of a the same length If the flaw size distribution of a large number of samples of the same material is plotted, the distribution of flaws may have normal, or Gaussian distribution as illustrated in the Fig

10. Failure and Weakest Link Theory There are three commonly recognized families of extreme value distributions where G(x) is the probability distribution function for an outcome being less than x for a sample set of n independent measurements . Type I . Gumbel G(x) = exp(- e * x * p ^ (- (x - mu) / sigma))for all x Type II . Fréchet: G(x) { = exp(-((x – μ)/σ)) for x >= mu = 0 otherwise. Type III . Weibull: G(x) { = exp(-(( μ – x)/ σ)) for x <= mu = 1 otherwise. where μ, sigma (>0) and xi (>0) are the location, scale and shape parameters, respectively.

11. Weibull Distribution Type III, (Weibull distribution), is usually considered the best choice because it is bounded (the lowest possible fracture strength is zero). The parameters can provide reasonably accurate failure forecasts with small numbers of test specimens and it provides a simple and useful graphical plot In Weibull fracture strength analysis, the cumulative probability function (the probability of failure, Pf ), increases with the fracture stress variable, σ Weibull three parameter strength distribution σu -the threshold stress parameter (minimum stress below which a test specimen will not break), σθ -characteristic strength -dependent on the stress configuration and test specimen size. The distribution shape parameter, m, =Weibull modulus

12. Some Limitations of Weibull Statistics Bends or kinks in a Weibull distribution function are often indicative of fracture resulting from multiple flaw types. Large number of test specimens required to characterize an entire strength distribution (different from estimating a mean strength value) The optimal number of test specimens depends on many variables- material and testing costs, the accuracy of the required distribution parameters and the scrutiny for an intended application.

13. Limitations and Considerations Material Heterogeneity: Ceramics have a varying microstructure, making it difficult to define a single critical flaw size applicable everywhere. Multi-size Flaw Distribution: Various flaw sizes exist within a ceramic. The critical flaw concept focuses on the largest one, neglecting the potential contribution of smaller interacting flaws. Complex Crack Propagation: Crack paths aren't always straight. Microstructure, stresses, and environment can influence the path, making prediction more complex.

14. Summary The concept of critical flaws plays a crucial role in understanding and predicting fracture in ceramics. A critical flaw is the smallest pre-existing crack or imperfection within the material that can propagate under applied stress and lead to complete failure. The weakest link theory relates to the Weibull distribution . Larger specimens are more likely to contain larger flaws. The probability of finding a critical flaw increases with specimen size. Smaller critical flaw size implies that the material is more resistant to fracture, as smaller flaws are less likely to act stress concentrators that lead to failure. Therefore ,reducing critical flaw size in a material can improve its fracture resistance and overall mechanical strength.

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