3.3 Mechanical Strength
3.3.5 Notch Effects
warranted for the reverse engineered part that is a critical structure element serving in an environment across the ductile–brittle transition temperature.
In many reverse engineering projects, the determination of fracture tough- ness is yet to become a mandatory requirement despite its being a critical parameter that determines if a part will fail. However, when more and more reverse engineered parts are life-limited or critical parts, and the original part was designed based on fracture toughness, the fracture toughness test should be conducted to demonstrate the equivalency whenever feasible.
specimens at the tip of the notch. If each tensile specimen is free to deform, a lateral strain εx , resulting from contraction, will be produced due to Poisson’s ratio, ν. To maintain the material continuity, a tensile stress σx must exist across the tensile specimen interface. At the free surface of the notch (x = 0) the tensile specimen can be laterally contracted freely without any restric- tion, and σx = 0. The σx rises steeply near the tip, and then falls slowly as the σy distribution flattens out, σx = Eεx = –E(νεy ) = –E[ν(σy E)]. In the plane stress condition of a thin plate, the stress in the thickness direction z is negligibly small, that is, σz = τxz = τyz = 0, and can be ignored.
As the thickness, B, increases, it becomes a plane strain condition. The strain in the thickness direction is approximately zero, that is, εz = 0. It is assumed that all deformation occurs in one plane, and the stress in the z direction, σz = υ(σx + σy), becomes more significant and cannot be deemed as zero anymore. The principal stresses and strains of plane stress and plane strain conditions are summarized in Table 3.1. The stress distributions for a thick-notched plate loaded uniaxially in the y direction are illustrated in Figure 3.13a, showing a high degree of triaxial stress configuration with stress components in all three, x, y, and z, directions. The value of σz falls to zero at the notch root where x = 0 on both surfaces of the plate (z = ±B/2), but rises rapidly with distance from the free surfaces. The distribution of σz with z at the notch root is shown in Figure 3.13b. As the thickness B decreases, the values of σx and σy only fall by less than 10%; however, the value of σz
decreases to 0 as the thickness approaches zero to assume a plane stress con- figuration (Dieter, 1986).
For a ductile metal, as the applied stress increases to yield strength, it starts yielding plastically, and a plastic zone will be established at the notch tip. According to the maximum shear stress yielding theory, the existence of transverse stresses σx and σz will raise the yielding stress in the longitudinal y direction, where the external stress is applied. The maximum shear stress yielding theory predicts yielding when the maximum shear stress reaches the value of the shear strength in the uniaxial-tension test. This criterion is mathematically expressed as σyield = σ1 – σ3 , where σyield is the yield strength, and σ1 and σ3 are the algebraically largest and smallest principal stresses, respectively. In an unnotched tension specimen subject to a uniaxial stress, the material yields at σyield = σ1 – 0, and therefore σyield = σ1 = σy . In a thick plane strain plate, σ1 = σy and σ3 = σx , as illustrated in Figure 3.13a. The yielding first starts at the notch root, which requires the smallest stress for yielding because σ3 = σx = 0 at the free root surface. The required externally
TablE 3.1
Principal Stresses and Strains for Plane Stress and Plane Strain Conditions
Condition Stress Strain
Plane stress (thin plate) σx ≠ 0, σy ≠ 0, σz = 0 εx ≠ 0, εy ≠ 0, εz ≠ 0 Plane strain (thick plate) σx ≠ 0, σy ≠ 0, σz = ν(σx + σy) ≠ 0 εx ≠ 0, εy ≠ 0, εz = 0
applied stress in the longitudinal y direction (σy ) to yield increases with the distance from the notch root following the criterion σyield = σy – σx , because
σyield is a constant material parameter, while σx increases with the distance
from the notch root near the tip.
As explained above, the existence of a sharp notch can strengthen the ductile metal due to the triaxiality of stress. The ratio of notched-to-unnotched yield stress is referred to as the plastic constraint factor, q. In contrast to the elastic stress concentration factor that can reach values in excess of 10, the value of q does not exceed 2.57 (Orowan, 1945). However, brittle metals could prema- turely fail due to stress increase at the notch before plastic yielding occurs.
When plastic deformation occurs at the notch root, σy drops from its high elastic value to σyield . Once the first imaginary tensile element at the notch root starts yielding, it deforms plastically at a constant volume that requires Poisson’s value to be ν = 0.5 instead of about 0.3 during elastic deforma- tion. Therefore, a higher transverse stress, σx = Eεx = –E(νεy ) = –E[ν(σy E)] , will be developed to maintain the material continuity. The stress σx will also increase with the distance from the notch root more quickly than in the
B
σx σz
Y
X σnorm
σy
(a)
B
Z Z
Y σz
(b)
FIgurE 3.13
(a) Elastic stress profiles at a sharp notch in a thick plate. (b) Stress distribution of σz at the notch tip x = 0.
elastic case. Within the plastic zone, the stresses σy and σz increase accord- ing to σy = σyield + σx and σz = 0.5(σy + σx ) until they reach the plastic-elastic boundary. The three principal stress, σx , σy , and σz , profiles at the notch with a plastic zone are illustrated in Figure 3.14, where rx is the length of the plas- tic zone in the x direction (Dieter, 1986).
The stress profiles for elastic and plastic deformation in front of a notch, illustrated in Figures 3.12 to 3.14, have demonstrated the complex notch effects that could drastically affect a comparative analysis of mechanical strength in reverse engineering. Many questions related to notch effects frequently come up in reverse engineering when a component test is not conducted. Can the tensile strength obtained from a smooth test specimen be used to estimate the strength of a notched specimen? How much debit should be factored in if an analysis is based on smooth tensile strength against a part with a notch?
Why and when should a test with a notched specimen be mandated? The answers to these questions are often based on the specific part configura- tions and criticality.