Keywords

1 Introduction

Engineering structures and components when subjected to loading fail in the region of high stress concentration; therefore, it is necessary to know the inelastic stresses and strains. To assess the crack initiation and fatigue lives, notch stress/strain analysis and low-cycle fatigue concepts are often used. For such analysis, it is useful and essential to estimate the local inelastic notch stresses and strains [1,2,3]. Different methods, viz experimental methods [4, 5], inelastic FEA, and the NSSC rules, are available to estimate notch root stresses and strains. Experimental method is tedious and consumes lot of time. Among these, Neuber method [6] and equivalent strain energy method [7] are most popular and commonly used methods by designers. The objective of this paper is to review and assess the local stresses and strains in the vicinity of notches using NSSC rules, viz Neuber’s rule and ESED method. Both the methods are programmed using Microsoft Excel to find the local plastic stresses and strains.

2 Johnson–Cook Elastic Plastic Material Model

Johnson–Cook elastic plastic material model [8] is used to define the material nonlinearity. In this paper, Aluminum 6063T7 material is used, even though in the present methodology any material can be used.

At normal strain rate and room temperature, the Johnson–Cook model is given by

$$\sigma = \left( {a + b \in_{p}^{n} } \right)$$
(1)

a is yield stress at low strains. b and n are strain-hardening constants.

Plastic strain, \(\in_{p} \,=\, \in - \in_{e}\) and \(\in_{e} \,=\, \frac{\sigma }{E}\)

σ and \(\in\) are true stresses and strains. \(E\) is Young’s modulus of the material.

For Aluminum 6063T7, Johnson–Cook parameters are

a = 90.26 MPa, b = 223.13 MPa, n = 0.374, and ultimate stress (σm) = 175 MPa.

The stress–strain curve is generated using Eq. (1) as shown in Fig. 1.

Fig. 1
figure 1

Stress-strain response using Eq. (1) [9, 10]

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3 Notch Stress–Strain Conversion Rules

3.1 Neuber’s Rule

The relation between local stress \(\left( {\sigma_{ {\max} } } \right)\), local strain \(\left( { \in_{{\max} } } \right)\), theoretical or linear stress concentration factor (\(k_{t}\)), and nominal stress (\(\sigma_{n }\)) is given by Neuber’s rule

$$\sigma_{{\max} } \in_{{\max} }\, = \, k_{t}^{2} \frac{{\sigma_{n}^{2} }}{E}$$
(2)

The actual or local stresses and strains may be obtained by solving Eqs. (1) and (2) as shown in Fig. 2.

Fig. 2
figure 2

Assessment of local stresses and strains using Neuber’s rule

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3.2 Neuber’s Rule to Calculate Local Stresses in Plastic Region

An Excel-based program is prepared to calculate the local stresses for given nominal stress (\(\sigma_{n }\)). Johnson–Cook material model for Aluminum 6063T7 and Neuber’s hyperbola is plotted in Excel sheet. For given values of nominal stress (\(\sigma_{n }\)), local stress \(\left( {\sigma_{ {\max} } } \right)\) can be calculated graphically using the Excel-based program.

3.2.1 Input to the Program

In the worksheet, only yellow colored cells as given in Table 1 are to be given as input to find local stress and strain.

Table 1 Input to the program
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3.2.2 Excel-Based Program

To find local stress and strain by Neuber’s rule, two equations need to be solved, stress-strain equation (Eq. 1) and Neuber’s hyperbola (Eq. 2). The screen shot of Excel sheet shown in Fig. 3 represents stress and corresponding strain values using Johnson–Cook material model for Aluminum 6063T7 with yield stress (a) = 90.26 MPa, hardening parameter (b) = 223.13 MPa, hardening exponent (n) = 0.374618, and Young’s modulus (E) = 60,400 MPa. The Excel sheet shown in Fig. 4 presents data for Neuber’s hyperbola. Figure 4 also shows screen shot image of Excel sheet to generate stress–strain curve for Neuber’s hyperbola.

Fig. 3
figure 3

Screen shot image of stress and strain values for Al 6063T7

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Fig. 4
figure 4

Screen shot image of stress-strain values for Neuber’s hyperbola

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3.2.3 Output of the Program

The data in the two sheets (Figs. 3 and 4) is used to plot a graph of stress-strain curve and Neuber’s hyperbola. As explained in Sect. 3.1, the intersection of these two curves gives local stress and strain at the notch and is shown in Fig. 5.

Fig. 5
figure 5

Determination of notch stress using Neuber

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3.3 Equivalent Strain Energy Density (ESED) Method

One more popular method to estimate local stress is ESED approach given by Molski and Glinka [7]. The strain energy per unit volume is called as strain energy density, and mathematically, it is expressed as W = \(\int\nolimits_{0}^{ \in } {\sigma\left( \in \right)\,.\,d \in }\), and graphically, it is considered as area under stress-strain curve (Fig. 6). From stress-strain constitutive relation, within an elastic limit σ = E . \(\in\). From the above equation, \(w = \frac{{\sigma^{2} }}{2E}\).

Fig. 6
figure 6

Energy density due to nominal and notch stress [7]

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Let, \(w_{s}\) = Strain energy density due to nominal stress and

\(w_{\sigma }\) = Strain energy density due to notch stress

$$w_{s} = \frac{{\sigma_{n}^{2} }}{2E}$$
(3)
$$w_{\sigma } = \frac{{\sigma_{{\max} }^{2} }}{2E}$$
(4)
$$k_{t} = \frac{{\sigma_{{\max} } }}{{\sigma_{n} }}$$
(5)

From Eqs. (3), (4), and (5), the relation between \(k_{t}\), \(w_{s}\), and \(w_{\sigma }\) is

$$w_{\sigma } = k_{t}^{2} w_{s}$$
(6)

Graphical interpretation of \(w_{\sigma }\) and \(w_{s}\) is shown in Fig. 6. As per Glinka’s approach [7], \(k_{t}\), \(w_{s}\), and \(w_{\sigma }\), can be correlated as \(k_{t}^{2} = \frac{{W_{\sigma } }}{{W_{s} }}\).

For finding maximum stress \(\sigma_{{\max} }\) in the plastic region, ESED method [7] states that “it is reasonable to assume that the energy distribution does not change significantly if localized plasticity is surrounded by predominantly elastic material,” i.e., the energy ratio given in Eq. (6) should be almost the same for elastic region and local yielding due to the effect of relatively large volume of the elastically strained material. The volume of the material that is strained elastically is more near the material that has undergone plastic deformation. Therefore, the assumption made in the above statement is reasonable. The relation can also be used if the material undergoes plastic deformation at the notch root. Equation (6) is used to find local stress in plastic region.

The comparison between ESED method and Neuber’s method is shown in Fig. 7. The strain energy \(w_{\sigma }\) is represented by the triangle as shown in Fig. 7 by Neuber’s rule which can be expressed as follows.

Fig. 7
figure 7

Comparison between Glinka’s and Neuber’s method [7]

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From Eqs. (2) and (6)

$$\frac{1}{2}\sigma_{{\max} } \in_{{\max} } = w_{\sigma }$$
(7)

Left hand side of Eq. (7) is area of a triangle under stress-strain curve as shown in Fig. 7, and RHS of the Eqn. is strain energy density at notch root. According to ESED method [7], the same energy \(w_{\sigma }\) is represented by area as shown in Fig. 7. From Fig. 7, it can be seen that the stresses and strains calculated on the basis of Neuber’s rule are always more than or equal to those obtained using ESED method, i.e., Neuber’s rule overestimates the results.

3.3.1 ESED (Molski and Glinka) Method to Calculate Local Stresses in Plastic Region

To find local stress and strains in plastic region knowing E, \(\sigma_{n }\) and \(k_{t}\), first \(W_{s} = \frac{{\sigma_{n}^{2} }}{2E}\) is determined, then \(W_{\sigma } = k_{t}^{2} W_{s}\), and finally, \(\sigma_{{\max} }\) is located on stress-strain curve such that area under the curve is \(W_{\sigma }\). Excel sheet is prepared to calculate \(W_{s}\) and \(W_{\sigma }\) for known values of E, \(\sigma_{n } ,\) and \(k_{t}\) as shown in Table 2. Locating \(\sigma_{{\max} }\) for given \(\sigma_{n }\) is done using hypergraph [11]. Stress-strain curve is imported to hypergraph. Hypergraph has facility to calculate area under curve between two limits. First point is set to (0, 0), and second point (\(\sigma_{{\max} }\), \(\varepsilon_{{\max} }\)) is manipulated using curser in order to get area under curve equal to \(W_{\sigma }\) as shown in Fig. 8. The procedure is repeated for all values of \(\sigma_{n }\) and (\(\sigma_{{\max} }\), \(\varepsilon_{{\max} }\)) values are tabulated in Table 2.

Table 2 Data for ESED (Molski and Glinka’s) method
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Fig. 8
figure 8

Calculation of local stress and strains by area under stress-strain curve using ESED method (Tool used: Hypergraph-2D)

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4 Conclusion

From the reported work on nonlinear analysis of assessment of local stresses and strains, it was observed that the past researchers used ESED method proposed by Glinka and Neuber for comparing their methodology. Therefore in this paper, the procedure to evaluate local stresses and strains using NSSC rules (Neuber and ESED) is explained. An Excel-based program was developed to assess inelastic stresses and strains by using Neuer’s rule. The systematic approach was also explained to assess inelastic stresses and strains by area under stress-strain curve using hypergraph-2D for ESED method. The method used in this paper would help design engineer to assess notch stresses and strains by NSSC rules without tedious numerical calculations.