Linear to True Stress-Strain

Neuber Method

Neuber's Rule for Estimating Elastic-Plastic Stress and Strain at Notches

1. Introduction and Physical Basis

Neuber's rule, developed by Heinz Neuber in 1961, provides a method for estimating the true elastic-plastic stress and strain at a stress concentration (notch) from linear elastic finite element analysis (FEA) results. The rule is based on the principle of strain energy density conservation.

When a component with a notch or stress concentration is loaded beyond the elastic limit, the material at the notch root yields and undergoes plastic deformation. A linear elastic FEA cannot capture this plasticity, resulting in artificially high "pseudo-elastic" stresses. Neuber's rule corrects these elastic predictions to account for yielding.

Key Principle: Neuber's rule assumes that the total strain energy density at the notch root calculated from an elastic analysis equals the actual elastic-plastic strain energy density.

2. Stress Concentration Factors

Before deriving Neuber's rule, we must define three concentration factors:

Theoretical (Elastic) Stress Concentration Factor:
K_t = σ_elastic / σ_nom

Actual Stress Concentration Factor (with plasticity):
K_σ = σ_true / σ_nom

Actual Strain Concentration Factor (with plasticity):
K_ε = ε_true / ε_nom

Where:

• σ_nom = nominal (far-field) stress
• σ_elastic = elastic stress at notch from linear FEA = K_t · σ_nom
• σ_true = actual elastic-plastic stress at notch
• ε_nom = σ_nom / E = nominal strain
• ε_elastic = σ_elastic / E = elastic strain at notch from linear FEA
• ε_true = actual elastic-plastic strain at notch (includes plastic component)

3. Derivation of Neuber's Rule

3.1 Elastic Strain Energy Density

For a uniaxial stress state, the elastic strain energy density is:

W_elastic = (1/2) · σ · ε = σ² / (2E)

At the notch root, using the elastic FEA stress:

W_{elastic,notch} = σ_elastic² / (2E) = (K_t · σ_nom)² / (2E)

In the far field (nominal region):

W_elastic,nom = σ_nom² / (2E)

3.2 Neuber's Hypothesis

Neuber proposed that the product of stress and strain at the notch is preserved when transitioning from elastic to elastic-plastic behavior:

σ_elastic · ε_elastic = σ_true · ε_true

This can also be written in terms of concentration factors:

K_t² = K_σ · K_ε

3.3 Neuber's Rule in Practical Form

Substituting the definitions of the elastic stress and strain:

(K_t · σ_nom) · (K_t · σ_nom / E) = σ_true · ε_true

Simplifying:

K_t² · σ_nom² / E = σ_true · ε_true

Recognizing that σ_elastic = K_t · σ_nom:

Neuber's Rule: σ_true · ε_true = σ_elastic² / E

This is the fundamental form of Neuber's rule used in practice.

4. Solution Procedure with Ramberg-Osgood Material Model

To find the actual stress and strain at the notch, we must solve Neuber's rule simultaneously with the material's stress-strain relationship. The Ramberg-Osgood equation is commonly used:

Ramberg-Osgood Equation: ε = σ/E + (σ/K')^1/n'

Where:

• E = elastic modulus
• K' = strength coefficient (Ramberg-Osgood parameter)
• n' = strain hardening exponent (Ramberg-Osgood parameter)

4.1 System of Equations

We have two equations with two unknowns (σ_true and ε_true):

Equation 1 (Neuber): σ_true · ε_true = σ_elastic² / E

Equation 2 (Ramberg-Osgood): ε_true = σ_true/E + (σ_true/K')^1/n'

4.2 Combining into Single Nonlinear Equation

Substitute the Ramberg-Osgood equation into Neuber's rule:

σ_true · [σ_true/E + (σ_true/K')^1/n'] = σ_elastic² / E

Let RHS = σ_elastic² / E, then rearranging to find the root:

Function to solve (f(σ) = 0): f(σ) = σ · [σ/E + (σ/K')^1/n'] - RHS = 0

4.3 Iterative Solution (Newton's Method)

This nonlinear equation is solved iteratively using Newton's method:

σ_i+1 = σ_i - f(σ_i) / f'(σ_i)

The derivative is:

f'(σ) = 2σ/E + (1 + 1/n') · (σ/K')^1/n'

Once σ_true converges, the true strain is calculated directly from Ramberg-Osgood:

ε_true = σ_true/E + (σ_true/K')^1/n'

5. Graphical Interpretation

On a stress-strain diagram:

1. Neuber's rule defines a hyperbola with constant product σ · ε = RHS
2. The Ramberg-Osgood curve describes the material's actual stress-strain behavior
3. The intersection of these two curves gives the elastic-plastic solution

Note: The elastic point (σ_elastic, ε_elastic) lies on the elastic line ε = σ/E and has the same σ · ε product as the true elastic-plastic point.

6. Limitations and Accuracy

Important Limitations:

• Tends to OVERESTIMATE strains: Neuber's rule typically predicts higher strains than experimentally measured values
• Small-scale yielding: Most accurate when the plastic zone is small compared to the notch radius
• Uniaxial loading: Originally developed for uniaxial stress states; extensions exist for multiaxial loading
• Monotonic loading: Most accurate for monotonic (non-cyclic) loading; cyclic plasticity requires modifications

7. Comparison with Glinka's Method

Glinka's equivalent strain energy density (ESED) method is an alternative that typically provides more accurate results:

• Neuber: Conserves the product σ · ε (total strain energy density)
• Glinka: Conserves the integral of stress over strain (includes plastic work properly)
• Result: Glinka tends to give slightly lower (more accurate) strain predictions than Neuber

8. Applications

Neuber's rule is widely used in:

• Fatigue life prediction at notches and stress concentrations
• Low-cycle fatigue (LCF) analysis
• Post-processing of linear elastic FEA results
• Notch analysis when full nonlinear FEA is not practical
• Strain-life (ε-N) fatigue calculations

9. References

Key references for Neuber's rule:

• Neuber, H. (1961). "Theory of Stress Concentration for Shear-Strained Prismatical Bodies With Arbitrary Nonlinear Stress-Strain Law." Journal of Applied Mechanics, 28(4), 544-550.
• Dowling, N.E. (2013). Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, 4th Edition, Pearson.

Glinka Method

Glinka's Equivalent Strain Energy Density (ESED) Method

1. Introduction and Physical Basis

The equivalent strain energy density (ESED) method, developed by Molski and Glinka in 1981, provides an improved approach for estimating elastic-plastic stress and strain at stress concentrations compared to Neuber's rule. Like Neuber, it corrects linear elastic FEA results to account for plasticity, but it does so more accurately by properly accounting for plastic work.

The key innovation of Glinka's method is that it equates the total strain energy density (elastic plus plastic) rather than just the product of stress and strain.

Key Principle: Glinka's ESED method assumes that the total strain energy density (elastic energy plus plastic work) at the notch root from an elastic analysis equals the actual total strain energy density in the elastic-plastic case.

2. Strain Energy Density Components

For an elastic-plastic material, the total strain energy density has two components:

Elastic Strain Energy Density:
W_elastic = σ² / (2E)

Plastic Work (strain energy absorbed in plastic deformation):
W_plastic = ∫₀^ε_p σ dε_p

The total strain energy density is:

Total Strain Energy Density:
W_total = W_elastic + W_plastic = σ²/(2E) + ∫₀^ε_p σ dε_p

3. Derivation of Glinka's Method

3.1 Elastic Case (from linear FEA)

For the elastic case at the notch root:

W_{elastic,notch} = σ_elastic² / (2E)

Where σ_elastic is the pseudo-elastic stress from linear FEA.

3.2 Elastic-Plastic Case (actual behavior)

For the actual elastic-plastic state:

W_total,true = σ_true²/(2E) + ∫₀^ε_p,true σ dε_p

3.3 Glinka's Energy Balance

Glinka's hypothesis states that these energies are equal:

Energy Balance:
σ_elastic² / (2E) = σ_true²/(2E) + ∫₀^ε_p,true σ dε_p

Rearranging:

(σ_elastic² - σ_true²) / (2E) = ∫₀^ε_p,true σ dε_p

4. Integration for Ramberg-Osgood Material

For a Ramberg-Osgood material, the plastic strain is:

ε_p = (σ/K')^1/n'

Therefore:

dε_p = (1/n') · (1/K')^1/n' · σ^(1/n')-1 dσ

4.1 Evaluation of Plastic Work Integral

The plastic work integral becomes:

∫₀^ε_p σ dε_p = ∫₀^σ σ · (1/n') · (1/K')^1/n' · σ^(1/n')-1 dσ

Simplifying:

= (1/n') · (1/K')^1/n' ∫₀^σ σ^1/n' dσ

Integrating:

= (1/n') · (1/K')^1/n' · [σ^{1 + 1/n'} / (1 + 1/n')]

Simplifying the expression:

= σ^{1 + 1/n'} / [n'(1 + 1/n') · K'^1/n']

Factor out σ from the numerator:

= σ · σ^1/n' / [n'(1 + 1/n') · K'^1/n']

Recognize that σ^1/n' / K'^1/n' = (σ/K')^1/n' = ε_p:

= σ · (σ/K')^1/n' / [n'(1 + 1/n')]

Simplifying the denominator: n'(1 + 1/n') = n' + 1

Plastic Work:
W_plastic = σ · (σ/K')^1/n' / (1 + n')

5. Glinka's Rule in Practical Form

Substituting the plastic work expression into the energy balance:

σ_elastic² / (2E) = σ_true²/(2E) + σ_true · (σ_true/K')^1/n' / (1 + n')

Let RHS_G = σ_elastic² / (2E), then:

Glinka's Rule:
σ_true²/(2E) + [σ_true/(1+n')] · (σ_true/K')^1/n' = σ_elastic² / (2E)

This is the fundamental form of Glinka's ESED method.

6. Solution Procedure

6.1 System of Equations

We solve two equations simultaneously:

Equation 1 (Glinka): σ²/(2E) + [σ/(1+n')] · (σ/K')^1/n' = RHS_G

Equation 2 (Ramberg-Osgood): ε = σ/E + (σ/K')^1/n'

6.2 Nonlinear Equation to Solve

Rearranging Glinka's equation to find the root:

Function to solve (f(σ) = 0):
f(σ) = σ²/(2E) + [σ/(1+n')] · (σ/K')^1/n' - RHS_G = 0

6.3 Iterative Solution (Newton's Method)

Using Newton's method:

σ_i+1 = σ_i - f(σ_i) / f'(σ_i)

The derivative is:

f'(σ) = σ/E + [1/(1+n')] · (1 + 1/n') · (σ/K')^1/n'

Once σ_true converges, calculate ε_true from Ramberg-Osgood:

ε_true = σ_true/E + (σ_true/K')^1/n'

7. Comparison with Neuber's Rule

8. Physical Interpretation

Key Difference from Neuber:

• Neuber: Assumes all energy goes into elastic deformation (σ · ε), neglecting that plastic work dissipates energy differently
• Glinka: Properly accounts for the fact that plastic deformation involves irreversible work, which is correctly integrated as ∫σ dε_p
• Result: Glinka provides more physically realistic predictions, especially for materials with significant plastic deformation

9. Limitations and Applicability

Glinka's method shares some limitations with Neuber's rule:

• Small-scale yielding: Most accurate when plastic zone is small relative to notch size
• Monotonic loading: Originally formulated for monotonic loading paths
• Uniaxial stress: Extensions needed for multiaxial stress states
• Material model dependent: Requires accurate Ramberg-Osgood parameters

Advantage over Neuber: Glinka's method typically provides more accurate predictions than Neuber, particularly under plane strain conditions and for sharp notches (high K_t values).

10. Applications

Glinka's ESED method is used in:

• Accurate fatigue life predictions at notches
• Low-cycle fatigue (LCF) analysis requiring precision
• Sharp notches and crack-like defects (high stress concentrations)
• Situations where Neuber's overestimation is problematic
• Strain-based fracture mechanics
• Validation of nonlinear FEA results

11. References

Key references for Glinka's ESED method:

• Molski, K., and Glinka, G. (1981). "A Method of Elastic-Plastic Stress and Strain Calculation at a Notch Root." Materials Science and Engineering, 50(1), 93-100.
• Glinka, G. (1985). "Energy Density Approach to Calculation of Inelastic Strain-Stress Near Notches and Cracks." Engineering Fracture Mechanics, 22(3), 485-508.
• Dowling, N.E. (2013). Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, 4th Edition, Pearson.

Unified Estimate Method

Unified Estimate Method (Weighted Neuber-Glinka)

1. What the Calculator Computes

The Unified Estimate is a weighted average of the Neuber and Glinka solutions already calculated on this page. With the weighting parameter α entered in the form:

σ_U = (1 − α) · σ_N + α · σ_G
ε_U = (1 − α) · ε_N + α · ε_G

α = 0 returns the Neuber result, α = 1 returns the Glinka result, and α = 0.5 is the midpoint. The estimate is an interpolation between two bounds. It is not a derived notch rule and does not solve an additional energy balance; in particular it is not the unified expression of Ye, Hertel & Vormwald (2008), which is a separate method not implemented here.

2. Why the Two Methods Bracket the Answer

Neuber's rule conserves the product σ · ε and tends to over-predict notch strain, most noticeably under plane strain (thick sections, interior points, sharp notches). Glinka's equivalent strain energy density method accounts for plastic work and tends to under-predict notch strain, most noticeably under plane stress (thin sections, free surfaces). Experimental and finite element comparisons (for example Sharpe, Yang & Turner, 1992) place the true elastic-plastic response between the two. The weighting parameter lets the user move the estimate toward whichever bound better matches the constraint at the point of interest.

3. Selecting α

• α = 0: Neuber only - very thin sections, extreme plane stress, or when a conservative (high strain) estimate is wanted
• α = 0.2-0.4: Plane stress - thin sheets, skin panels, surface points
• α = 0.5: Balanced estimate when the constraint is uncertain
• α = 0.6-0.8: Plane strain - thick sections, interior points, forgings, sharp notches
• α = 1: Glinka only - extreme plane strain, very thick sections

Indicators of plane stress (use a lower α): section thickness less than about 3-5 times the plastic zone size, surface or edge locations, sheet metal structures, low stress triaxiality.

Indicators of plane strain (use a higher α): section thickness more than about 10 times the plastic zone size, interior points far from free surfaces, thick forgings or castings, high stress triaxiality.

4. Cautions

• The estimate has not been validated against elastic-plastic finite element results in this tool. Treat it as an engineering interpolation, not a calibrated method.
• For a conservative design value use the Neuber result directly.
• When the point of interest is critical, an elastic-plastic finite element submodel of the notch region with the same Ramberg-Osgood curve is the reference solution that all of these estimates approximate.
• All three results on this page are uniaxial. For biaxial notch-root states use the plane-strain modulus or a multiaxial extension (Hoffmann & Seeger, 1985).

5. References

• Neuber, H. (1961). "Theory of Stress Concentration for Shear-Strained Prismatical Bodies With Arbitrary Nonlinear Stress-Strain Law." Journal of Applied Mechanics, 28(4), 544-550.
• Molski, K., and Glinka, G. (1981). "A Method of Elastic-Plastic Stress and Strain Calculation at a Notch Root." Materials Science and Engineering, 50(1), 93-100.
• Sharpe, W. N., Yang, C. H., and Turner, R. L. (1992). "An Evaluation of the Neuber and Glinka Relations for Monotonic Loading." Journal of Applied Mechanics, 59(2S), S50-S56.
• Hoffmann, M., and Seeger, T. (1985). "A Generalized Method for Estimating Multiaxial Elastic-Plastic Notch Stresses and Strains, Part 1: Theory." Journal of Engineering Materials and Technology, 107(4), 250-254.
• Ye, D., Hertel, O., and Vormwald, M. (2008). "A Unified Expression of Elastic-Plastic Notch Stress-Strain Calculation in Bodies Subjected to Multiaxial Cyclic Loading." International Journal of Solids and Structures, 45(24), 6177-6189. [Source of the idea that Neuber and Glinka are limiting cases of a constraint-dependent family; its expression is not implemented here.]
• Dowling, N. E. (2013). Mechanical Behavior of Materials, 4th ed., Pearson. [General discussion of notch approximation methods]