3.3 Mechanical Strength
3.3.3 Fracture Strength
Any reverse engineered part shall never operate beyond its fracture strength.
The complexity of fracture mechanism prevents engineers from developing
FIgurE 3.7
Micro slip steps in an aluminum alloy.
a universal fracture strength theory for all materials. However, the fol- lowing theories discussed in this section will provide engineers with the fundamental knowledge on failure analysis for reverse engineering applica- tions. The maximum shear strength theory estimates the maximum shear strength, τmax, of a perfect crystal by assuming that shearing results from the displacement of one whole layer of atoms over another. It approximately equals G2π, as mathematically expressed by Equation 3.13, where G is the shear modulus. However, it is 100 to 1,000 times larger than the measured value due to the line defect of dislocation. Dislocations are a linear atomic misalignment in crystalline materials. There are two basic types of disloca- tions: edge and screw dislocations. They also sometimes combine together to form a mixed dislocation. The required stress to move the dislocation line, one atomic distance at a time, only needs to break the atomic bond between the upper and lower atoms involved at any time. This is much smaller than the yield stress otherwise required in a perfect crystal to break all the bonds between all the atoms crossing the slip plane simultaneously. The existence of dislocations in a crystalline material has made yielding significantly easier. This explains the above-mentioned discrepancy between theoretical and nominal fracture strengths.
τmax= Gπ
2 (3.13)
Cohesive tensile strength is based on the theory that estimates fracture strength in tension. The atoms of crystalline metals are bound together by
N τcrss
λ φ A
P P
directionSlip
Slip plan
FIgurE 3.8
Critical resolved shear stress.
an attractive force and simultaneously repelled apart by a repulsive force between them. These two forces balance each other to keep the atoms at equilibrium. If the crystal is subject to a tensile load, initially the repulsive force decreases more rapidly with increased atomic spacing than the attrac- tive force. A net attractive force is therefore formed. Though the externally applied tensile load is first resisted by the net attractive force, eventually it reaches the peak point where the net attractive force starts to decrease due to increased separation between atoms. This corresponds to the maximum cohesive strength of the crystal. Beyond it an unstable state is reached. The required stress to further separate the atoms decreases, and the atoms con- tinuously move apart at the applied stress until fracture occurs. Equation 3.14 presents a good approximation for the theoretical cohesive tensile strength, σmax , where E is the Young’s modulus. The theoretical cohesive strength in tension can only be observed in tiny, defect-free metallic whiskers and very fine diameter silica fibers. The measured fracture strength for most engi- neering alloys is only 1/100 to 1/1,000 of the theoretical strength. This leads to the conclusion that existing flaws or cracks are responsible for the nominal fracture stress of engineering alloys.
σmax= Eπ
2 (3.14)
Fracture mechanics was first introduced to the engineering community in the 1930s by A. A. Griffith, an English aeronautical engineer, to explain the discrepancy between the actual and theoretical strengths of brittle materi- als. Later it was further developed by G. R. Irwin at the U.S. Naval Research Laboratory (NRL) in the 1940s. It explains that a part failure is dependent on not only a material’s inherent strength, but also the preexisting cracks in the subject part. Fracture mechanics is a theory that analyzes the mechan- ical strength with the acknowledgment of existing cracks. This is in con- trast to the classic mechanics that calculates the mechanical strength with the assumption that the part is defect-free. When fracture occurs in a brittle solid, all the work consumed goes to the creation of two new surfaces. This theory leads to the fracture strength described by Equation 3.15, where γs is surface energy and ao is the atomic distance:
σ γ
max=
E a
s o
12
(3.15) Consider an infinitely wide plate subject to an average tensile stress σ, with a thin elliptical crack of length 2c and a radius of curvature at its tip of ρt ,
as depicted in Figure 3.9. The maximum stress at the tip of the crack due to stress concentration is given by Equation 3.16 (Inglis, 1913):
σ σ
ρ σ
max= + ρ
≈
1 2 2
12 12
c c
t t
(3.16)
Assume that the theoretical cohesive strength can be reached at the crack tip, while the average tensile stress represents the nominal fracture strength, σf . Set Equations 3.15 and 3.16 equal to each other; then the nominal fracture strength can be calculated by Equation 3.17:
σ σ γ ρ
f s t
o
E
≈ = a c
4
12
(3.17) The sharpest possible crack has a radius of curvature at the tip equal to the atomic distance, ρt = ao , and the fracture strength can be approximated by Equation 3.18:
σ γ
f E s
≈ c 4
12
(3.18) If a crack of length 2c = 10 μm exists in a brittle material having E = 100 GPa, γs = 2 Jm2, the nominal fracture strength can be numerically calculated by Equation 3.19:
σmax σmax
σ
2c
σ FIgurE 3.9
Elliptical crack in an infinitely wide plate.
σ γ
f E s
= c
= × ×
× ×
= 4 −
100 10 2
4 5 10 10
12 9
6 12
( 166 12) =0 1. GPa (3.19) This example demonstrates that the existence of a very small crack can sig- nificantly reduce the fracture strength from the theoretical cohesive strength by a factor of 100. In some cases, it can even be reduced by a factor of 1,000.
The level of preexisting cracks in a part is primarily determined by the man- ufacturing process and the quality control system. From a reverse engineer- ing perspective, the reproduced part should be manufactured under such a quality control system that only introduces the same level of or less preexist- ing cracks than what the original part is allowed.