Mathematical treatment of fracture

A. I NTRODUCTION

IV. FRACTOGRAPHY OF SILICON NITRIDE BASED CERAMICS TO GUIDE PROCESS IMPROVEMENTS 68

1. Mathematical treatment of fracture

In flexure, both compressive and tensile stresses develop in the specimen. In a typical flexure experiment, where the specimen is supported at its ends from below and load is being applied toward the center from above, the top surface will be subjected to maximum compression, and the bottom surface will experience maximum tension. These stresses vary linearly throughout the specimen from compression to tension, and as such,

stress in the central plane of the specimen is zero. This plane is known as the neutral axis, as represented by the beam cross section in Figure 43.

Figure 43. Schematic representation of the stresses developed throughout the cross section of a bar in flexure. Maximum compressive stress is located on top, maximum tensile stress on the bottom, and zero stress develops in the center, or neutral axis.

The stresses developed in the specimen can be described by Equation 18, where σ is the stress in the flexure sample at a given load (MPa), F is the load (N), L is the support span length (mm), y is the distance from the neutral axis, b is the width of the sample (mm), and h is the height of the sample (mm). Equation 19 is found by substituting the maximum possible distance from the neutral axis, ½ h, for y. Equation 20 describes the maximum strain developed in the outermost surface of the specimen in bending, where ε is the axial strain induced in the sample at a given crosshead displacement, and w is the measured crosshead displacement. The derivations for these expressions can be found in Appendix B.

𝜎 =^{3𝐹𝐿𝑦}

2𝑏ℎ³ (18)

𝜎_𝑚𝑎𝑥 = ^3𝐹𝐿

4𝑏ℎ² (19)

𝜖_𝑚𝑎𝑥 =^432𝑤ℎ

77𝐿² (20)

Note that Equation 19 may be used to calculate fracture stress if and only if the critical flaw is at the surface of the specimen. If it is in the interior of the specimen, it will be subjected to lower stresses than the outermost surface, or “extreme fiber”, of the specimen, so Equation 18 must be applied. It is therefore important to measure the location of internal origins so that an accurate value for the stress experienced at the critical flaw may be calculated.

The critical stress intensity factor, KIC, is a measure of the stress state around a stress concentrator required to induce crack extension. It is a rather elusive parameter for which many experimental techniques and mathematical treatments have been developed and critically reviewed. Quinn⁷ presents a simple relation using fractographic measurements for the calculation of KIC, detailed in Equation 21

𝐾_𝐼𝐶 = 𝑌𝜎_𝑓√𝑐 (21)

where Y is a geometric factor accounting for the shape of the critical flaw, (1 < Y < 2), σf

is the fracture stress at the origin, and c is the size of the flaw parallel to the neutral axis.

This relation was first used by Leighton Orr of the Pittsburgh Plate Glass (PPG) Company as early as 1945.⁷ Another parameter accessible through fractographic analysis is the fracture mirror constant. This parameter, A, is defined in Equation 22:

𝐴 = 𝜎_𝑓√𝑟 (22)

where r is the fracture mirror radius. This constant is usually given as a range since it is quite difficult, particularly in polycrystalline ceramics, to measure the radius of the fracture mirror with sufficient precision to confidently yield a single value for A. Regardless, Equation 22 illustrates that if A is constant, then σf is proportional to r^-1/2. That is to say that the stronger the material is, the smaller the mirror can be expected to be. Equation 22 is useful because in a laboratory experiment, the stress is known, the mirror radius can be measured, and A can be calculated. However, the fracture stresses for components in use may not be readily known. Fractographic investigation may be undertaken to determine the

mirror radius for a fractured component, and if a database of mirror constants (A) exists for the material, the stress at failure may be estimated. This is one example of the utility of a tabulated collection of calculated material parameters from laboratory experiments.

It is common for four point flexure experiments to yield lower fracture stress values than three-point experiments. This phenomenon has to do with the volume of specimen being subjected to the maximum stress at a given load. In three-point geometry, the maximum stress developed in a specimen is concentrated at a point directly below the loading point. However, in four point geometry, maximum stress is generated throughout the entire region between the two loading points. The results of an example finite element analysis found in Figure 44 illustrate this difference. The consequence of different volumes of the specimen being subjected to the maximum stress is rooted in weakest link theory. A greater volume of specimen under maximum stress yields a greater probability of activating the most severe flaw. Therefore, a four point flexure experiment is likelier to subject this critical flaw to the maximum stress developed in the specimen than a three-point experiment. Similarly, a uniaxial tensile experiment typically yields the lowest values for tensile strength as the entire specimen volume is being interrogated by a uniform maximum stresses field.

Figure 44. Finite element analysis of a bar in a) three-point bending geometry and b) four point bending geometry. In three-point bending, maximum stress is concentrated at a single point directly below the loading point. In four point bending, a uniform maximum stress region is evolved between the loading points. FEA performed in MOOSE software (Idaho National Laboratory, USA), results visualized in ParaView (Kitware, USA).