Strain-Life (E-N) Approach

Neuber Correction

Neuber correction is the most popular practice to correct elastic analysis results into elastic-plastic results.

In order to derive the local stress from the nominal stress that is easier to obtain, the concentration factors are introduced such as the local stress concentration factor $K_{σ}$ , and the local strain concentration factor $K_{ε}$ .

K_{σ} = σ / S

K_{ε} = ε / e

Where, $σ$ is the local stress, $ε$ is the local strain, $S$ is the nominal stress, and $e$ is the nominal strain. If nominal stress and local stress are both elastic, the local stress concentration factor is equal to the local strain concentration factor. However, if the plastic strain is present, the relationship between $K_{σ}$ and $K_{ε}$ no long holds. Thereafter, focusing on this situation, Neuber introduced a theoretically elastic stress concentration factor $K_{t}$ defined as:

K_{t}^{2} = K_{σ} K_{ε}

Substitute Equation 1 and Equation 2 into Equation 3, the theoretical stress concentration factor $K_{t}$ can be rewritten as:

K_{t}^{2} = (\frac{σ}{S}) (\frac{ε}{e})

Through linear static FEA, the local stress instead of nominal stress is provided, which implies the effect of the geometry in Equation 4 is removed, thus you can set $K_{t}$ as 1 and rewrite Equation 4 as:

σ ε = σ_{e} ε_{e}

Where, $σ_{e}$ , $ε_{e}$ is locally elastic stress and locally elastic strain obtained from elastic analysis, $σ$ , $ε$ the stress and strain at the presence of plastic strain. Both $σ$ and $ε$ can be calculated from Equation 5 together with the equations for the cyclic stress-strain curve and hysteresis loop.

Monotonic Stress-Strain Behavior

Relative to the current configuration, the true stress and strain relationship can be defined as:

σ = P / A

ε = \int_{l}^{l} \frac{d l}{l} = \ln (1 + \frac{l - l_{0}}{l_{0}})

Where, $A$ is the current cross-section area, $l$ is the current objects length, $l_{0}$ is the initial objects length, and $σ$ and $ε$ are the true stress and strain, respectively, Figure 1 shows the monotonic stress-strain curve in true stress-strain space. In the whole process, the stress continues increasing to a large value until the object fails at C.

The curve in Figure 1 is comprised of two typical segments, namely the elastic segment OA and plastic segment AC. The segment OA keeps the linear relationship between stress and elastic strain following Hooke Law:

σ = E ε_{e}

Where, $E$ is elastic modulus and $ε_{e}$ is elastic strain. The formula can also be rewritten as:

ε_{e} = σ / E

by expressing elastic strain in terms of stress. For most of materials, the relationship between the plastic strain and the stress can be represented by a simple power law of the form:

σ = K {(ε_{p})}^{n}

Where, $ε_{p}$ is plastic strain, $K$ is strength coefficient, and $n$ is work hardening coefficient. Similarly, the plastic strain can be expressed in terms of stress as:

ε_{p} = {(\frac{σ}{K})}^{1 / n}

The total strain induced by loading the object up to point B or D is the sum of plastic strain and elastic strain:

ε = ε_{e} + ε_{p} = \frac{σ}{E} + {(\frac{σ}{K})}^{1 / n}

Cyclic Stress-Strain Curve

Material exhibits different behavior under cyclic load compared with that of monotonic load. Generally, there are four kinds of response.

Stable state
Cyclically hardening
Cyclically softening
Softening or hardening depending on strain range

Which response will occur depends on its nature and initial condition of heat treatment. Figure 2 illustrates the effect of cyclic hardening and cyclic softening where the first two hysteresis loops of two different materials are plotted. In both cases, the strain is constrained to change in fixed range, while the stress is allowed to change arbitrarily. If the stress range increases relative to the former cycle under fixed strain range, as shown in the upper portion of Figure 2, it is called cyclic hardening; otherwise, it is called cyclic softening, as shown in the lower portion of Figure 2. Cyclic response of material can also be described by specifying the stress range and leaving strain unconstrained. If the strain range increases relative to the former cycle under fixed stress range, it is called cyclic softening; otherwise, it is called cyclic hardening. In fact, the cyclic behavior of material will reach a steady-state after a short time which generally occupies less than 10 percent of the material total life. Through specifying different strain ranges, a series of hysteresis loops at steady-state can be obtained. By placing these hysteresis loops in one coordinate system, as shown in Figure 3, the line connecting all the vertices of these hysteresis loops determine cyclic stress-strain curve which can be expressed in the similar form with monotonic stress-strain curve as:

Figure 3. Definition of Stable Stress-Strain Curve

ε = ε_{e} + ε_{p} = \frac{σ}{E} + {(\frac{σ}{K^{'}})}^{1 / n^{'}}

Where,

$K^{'}$: Cyclic strength coefficient
$n^{'}$: Strain cyclic hardening exponent

Hysteresis Loop Shape

Bauschinger observed that after the initial load had caused plastic strain, load reversal caused materials to exhibit anisotropic behavior. Based on experiment evidence, Massing put forward the hypothesis that a stress-strain hysteresis loop is geometrically similar to the cyclic stress strain curve, but with twice the magnitude. This implies that when the quantity ( $Δ ε, Δ σ$ ) is two times of ( $ε, σ$ ), the stress-strain cycle will lie on the hysteresis loop. This can be expressed with formulas:

Δ σ = 2 σ

Δ ε = 2 ε

Expressing $σ$ in terms of Δσ, $ε$ in terms of Δε, and substituting it into Equation 13, the hysteresis loop formula can be calculated as:

Δ ε = \frac{Δ σ}{E} + 2 {(\frac{Δ σ}{2 K'})}^{1 / n'}

Almost a century ago, Basquin observed the linear relationship between stress and fatigue life in log scale when the stress is limited. He put forward the following fatigue formula controlled by stress:

σ_{a} = σ'_{f} {(2 N_{f})}^{b}

Where, $σ_{a}$ is the stress amplitude, $σ_{f}^{'}$ is the fatigue strength coefficient, and $b$ is the fatigue strength exponent. Later in the 1950s, Coffin and Manson independently proposed that plastic strain may also be related with fatigue life by a simple power law:

ε_{a}^{p} = ε'_{f} {(2 N_{f})}^{c}

Where, $ε_{a}^{p}$ is the plastic strain amplitude, $ε'_{f}$ is the fatigue ductility coefficient, and $c$ is the fatigue ductility exponent. Morrow combined the work of Basquin, Coffin and Manson to consider both elastic strain and plastic strain contribution to the fatigue life. He found out that the total strain has more direct correlation with fatigue life. By applying Hooke Law, Basquin rule can be rewritten as:

ε_{a}^{e} = \frac{σ_{a}}{E} = \frac{σ'_{f}}{E} {(2 N_{f})}^{b}

Where, $ε_{a}^{e}$ is elastic strain amplitude. Total strain amplitude, which is the sum of the elastic strain and plastic stain, therefore, can be described by applying Basquin formula and Coffin-Manson formula:

ε_{a} = ε_{a}^{e} + ε_{a}^{p} = \frac{σ'_{f}}{E} {(2 N_{f})}^{b} + ε'_{f} {(2 N_{f})}^{c}

Where,

ε_{a}

is the total strain amplitude, the other variable is the same with above.

Figure 4. Strain-Life Curve in Log Scale

Mean Stress Correction

The fatigue experiments carried out in the laboratory are always fully reversed, whereas in practice, the mean stress is inevitable, thus the fatigue law established by the fully reversed experiments must be corrected before applied to engineering problems.

Morrow:

Morrow is the first to consider the effect of mean stress through introducing the mean stress $σ_{0}$ in fatigue strength coefficient by:

ε_{a}^{e} = \frac{(σ'_{f} - σ_{0})}{E} {(2 N_{f})}^{b}

Thus, the entire fatigue life formula becomes:

ε_{a}^{} = \frac{(σ'_{f} - σ_{0})}{E} {(2 N_{f})}^{b} + ε_{f}^{'} {(2 N_{f})}^{c}

Morrow's equation is consistent with the observation that mean stress effects are significant at low value of plastic strain and of little effect at high plastic strain.

MORROW2:

Improves the MORROW method by ignoring the effect of negative mean stress.

Smith, Watson and Topper:

Smith, Watson and Topper proposed a different method to account for the effect of mean stress by considering the maximum stress during one cycle (for convenience, this method is called SWT in the following). In this case, the damage parameter is modified as the product of the maximum stress and strain amplitude in one cycle.

ε_{a}^{S W T} σ_{\max} = ε_{a} σ_{a} = σ_{a} (\frac{σ'_{f}}{E} {(2 N_{f})}^{b} + ε'_{f} {(2 N_{f})}^{c})

The SWT method will predict that no damage will occur when the maximum stress is zero or negative, which is not consistent with the reality.

When comparing the two methods, the SWT method predicted conservative life for loads predominantly tensile, whereas, the Morrow approach provides more realistic results when the load is predominantly compressive.

Damage Accumulation Model

In the E-N approach, use the same damage accumulation model as the S-N approach, which is Palmgren-Miner's linear damage summation rule.

Bergmann

Bergmann method is an extension to SWT introducing a parameter ‘a’ to account the mean stress sensitivity of the material.

ε_{a} σ_{\max} = ε_{a} σ_{a} = σ_{a} (\frac{σ_{f}^{'}}{E} {(2 N_{f})}^{b} + ε_{t}^{'} {(2 N_{f})}^{c})

where

σ_{\max} = σ_{a} + a

and a is the Bergmann constant.