Draft:Micro-cantilever fracture testing

Micro-cantilever testing is a valuable tool for measuring fracture toughness at the microscale. While it does have some limitations, its ability to provide detailed insights into fracture behaviour at the microscale makes it an essential technique in the field of fracture mechanics. As device miniaturisation continues to advance, the importance of microscale fracture toughness measurements is likely to increase.

Introduction

Fracture toughness is a critical material property that describes a material's ability to resist crack propagation. It is particularly important in the design and analysis of components that are subjected to stressful conditions. One of the methods used to measure fracture toughness at the microscale involves the use of micro-cantilevers.

Micro-cantilevers and fracture toughness

Micro-cantilevers are small, beam-like structures that can be subjected to bending or other forms of stress to induce and propagate cracks. These micro-cantilevers can be fabricated from the material of interest using techniques such as focused ion beam (FIB) milling. The cantilevers are then loaded until failure, and the resulting fracture surfaces are analysed to determine the material's fracture toughness.

The use of micro-cantilevers allows for the investigation of fracture behaviour at the microscale, which is essential due to the continuing miniaturisation of devices and the observed size effects in fracture behaviour. This method is particularly useful for studying how individual microstructural features contribute to a bulk material's crack resistance.

Geometries and equations

There are several equations used in the field of fracture mechanics to calculate fracture toughness. One common equation is the formula for the stress intensity factor (K), which is a function of the loading stress, the size of the existing or assumed crack, and the structural geometry. The formula is given by:

$K_{IC}=Y\sigma _{f}{\sqrt {\pi a}}$

where:

• $K_{IC}$ is the fracture toughness,
• $Y$ is a dimensionless geometric constant of the order of 1,
• $\sigma _{f}$ is the fracture stress, and
• $a$ is the edge crack length or half the length of an internal through crack.

Fracture occurs when the stress intensity factor reaches a critical value $K_{IC}$, known as the fracture toughness of the material.

Clamped beam bending

This section is an excerpt from Three-point flexural test § Fracture toughness testing.[edit]

Single-edge notch-bending specimen (also called three-point bending specimen) for fracture toughness testing.

The fracture toughness of a specimen can also be determined using a three-point flexural test. The stress intensity factor at the crack tip of a single edge notch bending specimen is

${\begin{aligned}K_{\rm {I}}&={\frac {4P}{B}}{\sqrt {\frac {\pi }{W}}}\left[1.6\left({\frac {a}{W}}\right)^{1/2}-2.6\left({\frac {a}{W}}\right)^{3/2}+12.3\left({\frac {a}{W}}\right)^{5/2}\right.\\&\qquad \left.-21.2\left({\frac {a}{W}}\right)^{7/2}+21.8\left({\frac {a}{W}}\right)^{9/2}\right]\end{aligned}}$

where $P$ is the applied load, $B$ is the thickness of the specimen, $a$ is the crack length, and $W$ is the width of the specimen. In a three-point bend test, a fatigue crack is created at the tip of the notch by cyclic loading. The length of the crack is measured. The specimen is then loaded monotonically. A plot of the load versus the crack opening displacement is used to determine the load at which the crack starts growing. This load is substituted into the above formula to find the fracture toughness $K_{Ic}$.

The ASTM D5045-14 and E1290-08 Standards suggests the relation

$K_{\rm {I}}={\cfrac {6P}{BW}}\,a^{1/2}\,Y$

where

$Y={\cfrac {1.99-a/W\,(1-a/W)(2.15-3.93a/W+2.7(a/W)^{2})}{(1+2a/W)(1-a/W)^{3/2}}}\,.$

The predicted values of $K_{\rm {I}}$ are nearly identical for the ASTM and Bower equations for crack lengths less than 0.6$W$.

Standards for measuring fracture toughness

Several standards guide the process of measuring fracture toughness. Some of the common ones include:

• ISO 12135:2021: This standard provides a unified method of testing for the determination of quasi-static fracture toughness in metallic materials.
• ASTM E1820-23b: This standard outlines the method for measuring the fracture toughness of metallic materials.
• ASTM E399: This standard is used for the determination of plane-strain fracture toughness of metallic materials.
• ASTM E740: Practice for Fracture Testing with Surface-Crack Tension Specimens
• ASTM E1823: Terminology Relating to Fatigue and Fracture Testing

These standards provide detailed procedures for preparing test specimens, conducting the tests, and analysing the results.

Advantages

Micro-cantilever testing offers several advantages. Firstly, it allows for the simultaneous recording of load-displacement information and crack initiation and propagation events, providing detailed insights into the fracture process. Secondly, it enables the testing of fracture properties of individual microstructural constituents, such as grain boundaries. Lastly, it allows for the investigation of size effects in fracture behaviour, which is crucial in the era of device miniaturisation.

Disadvantages

Despite its advantages, micro-cantilever testing also has some limitations. One of the main challenges is that size effects cannot be accounted for by direct extrapolation of known properties at the macroscale. Also, as the sample dimensions become smaller, maintaining plane strain and linear elastic fracture mechanics (LEFM) conditions becomes difficult. Furthermore, the presence of coarser grains in the material can impact the resultant fracture toughness and morphology.

History: Micro-cantilever fracture testing has its roots in the field of micro-/nano-mechanics of materials. The technique was developed to understand the fracture behaviour of materials at smaller scales. For instance, researchers have used micro-cantilever bending tests to study the semi-brittle fracture behaviour of single-crystalline tungsten.

Micro-cantilever fracture testing has been an active area of research for the past 30 years. However, it’s important to note that the exact date of the first use of this method might be difficult to pinpoint due to the gradual development and refinement of the technique over time. For instance, the use of the Continuous Stiffness Measurement (CSM) as a fast actuation technique for cyclically loading focused ion beam (FIB)-structured samples was pioneered by Merle and Höppel, and Lavenstein et al., as mentioned in a study published in 2020. This indicates that the method has been in use at least since the early 1990s.

see Measuring anisotropy in Young's modulus

Usage: Micro-cantilever fracture testing is used to measure the fracture toughness of low volume materials. It allows researchers to determine how individual microstructural features contribute to a material’s overall crack resistance. This is typically accomplished using micro-cantilever fracture tests by applying compression to notched cantilever beams, meticulously prepared using lithography or a Focused Ion Beam (FIB).

Prospects: The prospects of micro-cantilever fracture testing are promising. It has been used to understand the fracture behaviour of Mg/CNT composites synthesized by the sandwich technique. The technique has also been used to measure the strength of the interfacial bond between silica inclusions and iron. Moreover, it has been used to study the high-cycle fatigue behaviour of materials at the micrometre scale.

Future: The future of micro-cantilever fracture testing lies in its ability to probe ever smaller length scales. With the continuing miniaturization of devices, detecting and preventing fracture at small-length scales has become a priority. Furthermore, due to the gradient nature of a coating or irradiated material, depth profiling of fracture properties may be necessary. The development of new methods and enhancement of existing ones that allow accurate measurements of mechanical properties at the micron and submicron scales is expected.