Effect of damage parameter in assessment of low cycle fatigue

Abstract

A low-cycle fatigue (LCF) analysis is one of the main design stages for highly loaded structural elements used in various applications. For this analysis, it is necessary to determine the values of local stresses and deformations, taking into account both elastic and plastic regions in the zones of stress concentration. This study presents and assesses the engineering methods used for prediction of low-cycle fatigue in structural elements. For zones of stress (strain) concentration, the Neuber-Makhutov method for LCF, taking into account the type of material stress-strain diagrams, is employed. The concept of distributed damage, based on the main ideas of the continuum damage mechanics of Kachanov-Rabotnov, was used. An approach employing the damage parameter for assessment of damage accumulation in LCF in highly loaded areas of structural elements is presented.

Keywords

Damage damage model effective stress stress concentration low-cycle fatigue failure criterion

Introduction

Assessment of critical and highly loaded structural elements at the stages of design and in-service, in addition to static strength, is generally, carried out also for fatigue. However, with the increasing intensity of loading of modern equipment with a simultaneous light weighting, the low-cycle fatigue (LCF) an analysis was also introduced into engineering practice. LCF effects can be found in the most loaded areas of components and structures. In such zones (generally, they are zones of stress concentration), operational damage appears in the form of LCF cracks.

The LCF analysis is a complex problem. It includes assessment of nominal and local stresses, as well as elastic and elastic-plastic deformations in the highly loaded zones for various combinations of operational mechanical and thermal loads. Such loads define the most damaging cycles in the low-cycle loading history with the number of cycles in the range from 10²–10⁵. A detailed analysis of LCF problems was described in the works of Serensen, Makhutov and Gusenkov (Gusenkov, 1979; Gusenkov and Kotov, 1988; Liu et al., 2022; Liu and Liu, 2023; Makhutov, 1981; 2005; Makhutov et al., 1983; Nourian-Avval and Khonsari, 2021; Serensen, 1985a, 1985b; Serensen et al., 1975; Yang and Sun, 2022). The complexity of the calculation and experimental assessment of the structural element’s strength under low-cycle loading conditions is due to wide variation of the main design, technological and in-service factors, e.g., forces, temperatures and levels of stress (strain) concentration. This, relatively new, calculation method is mandatory for highly loaded structures in the aerospace and power engineering, shipbuilding and other industries.

The reliability of the life assessment for structural elements at the design stage and residual life at the operation stage, is mainly determined by the level of accuracy of the constitutive equations describing the effects of deformation taking into account the history of thermal force loading. For now, there are several approaches to describe the cyclic deformation of metallic materials: a generalized Mazing model (Gokhfeld and Sadakov, 1984; Gusenkov and Moskvitin, 1973; Moskvitin, 1965; Troschenko, 1978; Troschenko et al., 2000), a Morrow cycle diagram (Gusenkov, 1979; Makhutov, 1981; Morrow, 1965; Troschenko et al., 2000) and a model based on the concept of generalized cyclic diagram, which is currently the most widely used (Gusenkov, 1979; Gusenkov and Moskvitin, 1973; Makhutov, 1981; Makhutov et al., 1981; Serensen, 1985b; Shnejderovich, 1968; Troschenko, 1978; Troschenko et al., 2000). However, the last concept does not take into account the change (degradation) of the elastic modulus during unloading, that depends on the number of loading half-cycles and the level of elastic-plastic deformation. At the same time, in continuum damage mechanics (CDM), the degradation of the material’s elastic modulus is one of the main approaches for assessing the state of its damage (Bonora et al., 2004; Bonora and Newaz 1998; Hambli, 2002; Hansen and Schreyer, 1994; Lemaitre, 1985; Marcilio et al., 2000; Mashayekhi and Ziaei-Rad, 2006; Tang et al., 2002; Timoshenko et al., 2011; Voyiadjis et al., 2013; Wei et al., 2022). One of the promising approaches in this direction is the solution of the problem of assessment of combined deformation and fracture processes in structural materials at micro and mesolevels. Such a scheme makes it possible to determine the behavior, as well as the time and locations of occur range of the first macroscopic cracks in the loaded component.

Structural damage also affects the shape of the stress-strain diagram; this becomes crucial in the elastoplastic deformation region.

The aim of this work is to expand the scope of CDM application to improve the calculation accuracy for LCF of structural elements using the Neuber-Makhutov engineering approach, which is the basis for prediction of the service life under low-cycle loading. The improved method is based on the use of an effective deformation diagram, which takes into account the damage parameter.

The study considers the issues of parameter quantification for damage accumulation in metallic materials, establishing the lower stress threshold for the onset of damage, and demonstrates the extent of damage effect on the deformation diagram in elastic and elastoplastic regions.

Improvement of the Neiber-Makhutov engineering approach for introduction of effective stress- and strain-concentration factors for analysis of low-cycle and static loading conditions, using effective material deformation diagrams, is also discussed.

Study also provides the description of the energy method for assessment of LCF, taking into account material damage, and comparison of the obtained results with the existing standard method. The effectiveness of the proposed method for calculation of LCF of materials and components was demonstrated.

Damage in structural metallic materials

In-service loading of structural elements is accompanied by the accumulation of distributed damage (DD) at micro- and meso-levels causing the onset of first macrocrack in the most loaded area. The features of DD accumulation in metallic materials, including those under cyclic loading, are currently estimated with various methods, which are described in detail in several works (Castagne et al., 2003; Huo et al., 2022; Jiang et al., 2021; Lacy et al., 1997; Lebedev et al., 1996, 2002; Lemaitre, 1985, 1990; Lemaitre and Desmorat, 2005; Li et al., 2020; Liu and Ma, 2023; Ochsner et al., 2001; Tang et al., 2002; Wei et al., 2022; Yuan et al., 2015; Zhou et al., 2020). Generally, they introduce the changes of the physical and mechanical parameters of materials (modulus of elasticity, electrical resistance, density, etc.) depending on the level and nature of loading. According to some studies (Bobyr et al., 2004, 2006; Grabovskij et al., 2002; Timoshenko et al., 2011) an effective method for evaluation of SD accumulation in metals is the measurement of the specific value of electrical resistivity in test samples. Unlike other methods, it makes a continuous evaluation of the material state in the measurement zone (working zone of the samples) possible. The change of the material’s electrical resistivity reflects the processes of initiation and evolution of microvoids, microcracks and other discontinuities that may occur in it during elastic-plastic deformation.

According to the CDM fundamentals, the damage in an isotropic material can be described with a number of parameters, including scalars, vectors or second-rank tensors (Brunig et al., 2023; Ganczarski and Barwacz, 2004; Lemaitre, 1990; Lemaitre and Desmorat, 2005; Namestnikova and Shesterikov, 1985; Tang et al., 2002; Wri, 1988). Basically, as a first approximation for engineering calculations, a scalar damage parameter and the concept of effective stress are commonly used.

In the proposed approach, the current value of the scalar damage parameter of the sample under tension is determined as follows:

D_{i} = D_{i - 1} + Δ D

(1)

where

D_{i - 1}

is the value of the damage in accumulated the previous measurement period,

Δ D

is the damage increase over the current measurement period:

D_{i} = \frac{ρ_{i} - ρ_{0}}{ρ_{0}}; Δ D = \frac{|Δ ρ|}{ρ_{0}}; ρ_{i} = \frac{R_{i} A_{i}}{l_{i}}

(2)

where

Δ ρ = ρ_{i} - ρ_{i - 1}

is change in the value of electrical resistivity the test sample measurement interval;

ρ_{0}

ρ_{i - 1}

and

ρ_{i}

are the initial and current values of electrical resistivity at the beginning and at the end of the measurement interval, respectively;

R_{i}

A_{i}

and

l_{i}

are the current values of electrical resistance, cross-sectional area and working length of the test sample, respectively.

The research methodology and experimental equipment used in this study are described in detail in (Grabovskij et al., 2003; Timoshenko et al., 2011). Here, a brief description is given.

An experimental test machine for determining the damage parameter using the proposed method is presented in Figure 1. A cylindrical specimen 1 with a diameter of 5 mm and a working gage area of 25 mm is fixed in specially designed electrically insulated grips 2. One of the grips is connected directly to a load cell 3, which provides the force measurements in the range of up to 10 tons with an accuracy of 1%. The second one is mounted on a movable crossbeam 4, the movement of which is provided by an electric motor 5 with a computer remote control. The specimen’s longitudinal deformation is measured using two optoelectronic extensometers 6 Megatron MS30-1-TTL installed in parallel. The displacement measurement accuracy is ±0.001 mm. The magnitude of longitudinal deformation is calculated as the arithmetic mean of the readings of two extensometers. The measurement of the current value of the specimen’s diameter during the experiment is carried out using a lever system. The main measuring element of this scheme is also a Megatron MS30-1-TTL extensometer 7. To measure the current value of the electrical resistance of the specimen, an HIOKI 3541 Resistance HiTester 8 with a lower measurement limit of 0.1 µOhm is used. The resistance tester has two main channels: one connected directly to the grips and used to supply the electric current to the specimen, the second used to directly measure the electrical resistance of the specimen’s area. The schematic diagram of the electrical-resistance measurement system is shown in Figure 1. The general control of the specimen’s loading process and data acquisition is performed with a PC. The damage parameter was assessed in accordance with equations (1) and (2), based on the results of measurements taken from the standard tensile test of the specimen.

The experiments were carried out on a wide range of metallic materials. Their parameters and chemical composition are given in Tables 1 and 2, were $σ_{p r}$ is the proportionality limit, $σ_{y}$ $σ_{y}$ is the yield stress, $σ_{u}$ is the ultimate stress, $δ$ and $ψ$ the limit values of relative elongation (residual plastic strain after rupture) and transverse contraction of the cross-section of the specimen, respectively.

Table 1.

Parameters of studied materials.

Material	Analogue	Mechanical properties					Material parameters
Material	Analogue	$σ_{p r}$ , MPa	$σ_{y}$ , MPa	$σ_{u}$ , MPa	$δ$ , %	$ψ$ , %	$D_{T}$	$D_{B}$	$D_{R}$	$T_{1}$	$T_{2}$
D16Т	Alloy 2024	355	398	532	11.1	12.2	0.0075	0.04	0.12	2.0	4.0
VТ22	Ti6-4	900	1236	1414	5.1	9.9	0.011	0.048	0.065	4.8	3.8
18H2N4VA	17CrNiMo6	752	960	1343	16.6	66.8	0.043	0.08	0.575	2.1	1.6
15HSND	13Mn6	303	349	515	34.6	61.2	0.012	0.051	0.5	27.0	1.1
12H18N10Т	X10CrNiTi18-10	139	248	627	68.2	72.2	0.0059	0.045	0.52	10.6	13.5
07H16N6	X10CrNi18-8	340	710	951	45	45	0.004	0.19	0.295	5.0	2.2

Table 2.

Chemical composition of studied materials.

Material	C	Si	Mn	Ni	S	P	Cr	N	Cu	As	Fe	Ti	Va	Mo	W	V	Zr	Al	Mg	Zn
D16Т		0.33	0.64				0.02		4.10		0.47	0.03						base	1.60	0.18
VT22	<0.1	<0.15					0.5–1.5	<0.05			0.5–1.5	79.4–86.3		4.0–5.5		4.0–5.5	<0.3	4.4–5.7
18H2N4VA	0.16	0.30	0.43	4.10	0.01	0.014	1.47		0.13			0.01	0.01	0.06	0.92
15HSND	0.12–0.18	0.4–0.7	0.4–0.7	0.3–0.6	<0.04	<0.035	0.6–0.9	<0.008	0.2–0.4	<0.08	97
12H18N10Т	<0.12	<0.8	<2.0	9.0–11.0	<0.02	<0.035	17.0–19.0		<0.3		67	0.4–1.0
07H16N6	0.05–0.09	<0.8	<0.8	5.0–8.0	<0.02	<0.035	15.5–17.5				75

The process of damage accumulation in each material depended on its ductility (Figure 2).

Each damage evolution curve could be characterized, generally, by three regions. In the first region, which could be defined in range from $ε = 0$ to $ε = ε_{y}$ , an almost instant increase in the damage level was observed. Strain $ε_{y}$ corresponds to the proportional limit (yield stress) of the material and is generally in the elastic domain. The second region could be characterized by the strain range from $ε_{y}$ to $ε_{u}$ which corresponds to the ultimate stress $σ_{u}$ . It is the longest part on the damage accumulation curve for each material. The third part of the damage curve corresponds to the range of strains from $ε_{u}$ to macroscopic fracture of the sample at strain $ε_{R} .$ It is characterized by the accelerated growth of damage in the local necking part of the specimen.

So far, in the literature the main attention has been paid to analysis of the features of DD accumulation in the second region of the damage diagram, although damage in many cases is present in the loaded material below the proportionality limit. This is sufficient to solve the technological problems related to the theory of plasticity. However, for prediction of the in-service life of critical structural elements operating under cyclic (low- and high-cycle) loading, the first zone of this diagram has a strong effect on the accuracy of describing the processes of degradation of physical and mechanical properties of metallic materials and, accordingly, the formulation of the constitutive equations and fracture criteria.

Consider in more details the first region of the DD accumulation curves for the materials under study at the level of the proportionality limit $σ_{p r}$ , which are shown on Figure 3.

It can be concluded that for brittle materials, such as 2024 and Ti6-4, the first region of the damage diagram has an almost linear character. At the same time, for ductile materials, such as steels X10CrNiTi18-10, 17CrNiMo6, etc., this region has a curvilinear character with a decrease in the accumulation rate. The obtained results showed that DD appeared and developed in metals at the stage of elastic deformation, starting from the stress level, corresponding to the endurance limit of the material for symmetric load cycles $σ_{- 1}$ . At these stresses, local microplastic deformations occur in the material, together with microdamage in the form of micropores and microcracks. These processes were also confirmed by numerous experimental studies of fatigue, summarized in (Troschenko, 1981), where it was demonstrated that at the level of stresses corresponding to local microplastic deformations were ${(ε_{H})}_{σ_{- 1}} = 1 \times 1 0^{- 5} - 20 \times 1 0^{- 5}$ . The value ${(ε_{H})}_{σ_{- 1}}$ plays a determining role in the criteria used to assess the service life at the design stage and the residual in-service life of structural elements. The value of this strain can be taken as threshold $ε_{i D}^{p}$ , at which the process of damage accumulation in low- multicycle fatigue begins ( $ε_{i D}^{p} \geq {(ε_{H})}_{σ_{- 1}}$ ).

The phenomenological model of DD presented in (Khalimon, 2006) is a generalization of the model by Lemaitre (Lemaitre, 1990, 1992). It describes the evolution of DD in the stress range $σ_{y} \leq σ \leq σ_{u}$ :

\frac{d D}{d t} = {\{\frac{σ_{i}^{2}}{2 E T_{1}} [\frac{2}{3} (1 + ν) {[χ + (1 - χ) \frac{σ_{1}}{σ_{i}}]}^{2} + 3 (1 - 2 ν) K_{σ}^{2}]\}}^{T_{2}} \frac{d ε_{i}^{p}}{d t}

(3)

where

T_{1}

and

T_{2}

are the material parameters;

ν

is the Poisson's ratio;

K = \frac{σ_{0}}{σ_{i}}

is the loading stiffness parameter;

σ_{0} = \frac{1}{3} δ_{i j} σ_{i j} σ_{i j}

is the mean stress;

σ_{i} = \sqrt{\frac{3}{2} S_{i j} S_{i j}}

is the stress intensity,

0 \leq χ \leq 1

is the Pisarenko-Lebedev parameter. The last parameter takes into account the level of plasticity of the material and is defined for static loading as the ratio of the yield strength of the material in compression

σ_{T}^{(-)}

to that in tension

σ_{T}^{(+)}

, namely,

χ = \frac{σ_{T}^{(-)}}{σ_{T}^{(+)}}

The equivalent stress by Pisarenko-Lebedev, (Khalimon, 2006) was used in this damage model:

σ_{e q} = χ σ_{i} + (1 - χ) σ_{1}

(4)

where

σ_{1}

is the major principal stress. For ductile materials, when

= 1

, equation (3) reduces, as a special case, to the Lemaitre model (Lemaitre, 1985, 1990). The limiting value

χ = 0

characterizes the brittle fracture of the material. The use of equation (4) in model (3) makes it possible to take into account two main types of fracture: opening and shearing.

According to equation (3), the function of the stress state type, taking into account equation (4), can be shown as:

R_{σ} = \frac{2}{3} (1 + ν) {[χ + (1 - χ) \frac{σ_{1}}{σ_{i}}]}^{2} + 3 (1 - 2 ν) K_{σ}^{2}

(5)

Depending on the loading regime, equation (5) takes the form: for the case of active uniaxial tension, $R_{σ} = 1$ ; for compression, $R_{σ} = \frac{2}{3} χ^{2} (1 + ν) + \frac{1}{3} (1 - 2 ν)$ ; for torsion, $R_{σ} = \frac{2}{9} (1 + ν) {(1 + \sqrt{3} χ - χ)}^{2}$ .

Equation (3) was integrated for each individual type of loading. Parameters $T_{1}$ and $T_{2}$ were found from a simple tensile experiment on specimens by assessing the damage parameters at several levels of plastic deformation. Let us take the equation of the deformation diagram in the elastic-plastic region according to the Ramberg-Osgood law as $σ_{i} = b {(ε_{i}^{p})}^{m}$ , where $b$ and $m$ are the approximation parameters of the deformation curve. For uniaxial tension:

D = {[\frac{b^{2}}{2 E T_{1}}]}^{T_{2}} \frac{1}{T_{3}} [{(ε^{p})}^{T_{3}} - {(ε_{D}^{p})}^{T_{3}}]

(6)

where

ε^{p}

and

ε_{D}^{p}

are the current and threshold values of plastic deformation, respectively.

ε_{D}^{p}

corresponds to the limit of proportionality (elasticity) of the material;

T_{3} = 2 m T_{2} + 1

. Equation (6) does not take into account the presence of damage (

D_{p r}

) at the level of

ε_{D}^{p}

, i.e., the first section of the damage accumulation diagram in the material. Then, the total damage of the material, taking into account (6) takes the following form:

D_{Σ} = D_{T} + {[\frac{b^{2}}{2 E T_{1}}]}^{T_{2}} \frac{{(ε^{p})}^{T_{3}}}{T_{3}}

(7)

where

D_{T}

is the value of the damage parameter at the limit of proportionality (yield limit) of the material.

The values of damage parameters corresponding to the proportionality limit ( $D_{T}$ ), ultimate stress ( $D_{B}$ ) and fracture ( $D_{R}$ ) are shown in Table 1. They are obtained using the method of measuring the specific electrical resistance of the material.

The analysis of these data demonstrated the necessity to take into account the damage parameters $D_{T}$ in predictions of durability of components for different materials, it makes from 2% to 54% of $D_{B}$ . The level of damage at proportionality limit for the considered materials was 80–90% of that at yield strength.

The dependence of the scalar damage parameter $D_{T}$ at the yield strength level $σ_{y}$ and $D_{R}$ at the fracture level $σ_{R}$ on plasticity characteristics of the material (Bobyr and Koval, 2011) is shown in Figure 4.

These dependencies for metallic materials can be described by the following equations:

\begin{matrix} D_{T} = C_{1} {(\frac{ψ_{R}}{δ})}^{2} + C_{2} \frac{ψ_{R}}{δ} \\ D_{R} = K_{3} ψ_{R} \end{matrix}

(8)

where

C_{1} = 19 \cdot 1 0^{- 4}

;

C_{2} = 31 \cdot 1 0^{- 4}

;

C_{3} = 8 \cdot 1 0^{- 3}

are constants.

Obtaining the damage parameter directly from the experiment or using equations (7) and (8) makes it possible to calculate the effective stresses according to the Kachanov-Rabotnov hypothesis (Kachanov, 1986; Rabotnov, 1966). This hypothesis of invariance of deformations that occur in the damaged and undamaged states of the material is accepted. The effective stress that takes into account the damage of the material for a uniaxial tensile condition is determined as

σ_{e f} = \frac{σ_{r e}}{1 - D (ε)}

(9)

where

σ_{r e}

is real stress, taking into account specimen’s cross-section reduction.

The stress-strain diagram for a material with hardening can be described as follows:

\bar{σ} = \{\begin{array}{l} {\bar{ε}}^{\bar{m}} for σ \geq σ_{y} \\ \bar{ε} for σ < σ_{y} \end{array}

(10)

where

\bar{σ} = \frac{σ}{σ_{y}}

\bar{ε} = \frac{ε}{ε_{y}}

σ_{y}

and

ε_{y}

are the stresses and strains at proportionality limit,

\bar{m}

is the hardening factor. Figure 5 shows the comparison of experimental deformation diagrams in the elastic-plastic region for uniaxial tension.

Conventional stress-strain curve does not take into account specimen’s cross-section reduction, real – takes into account cross-section reduction and effective – can be build based on the effective stress approach. Apparently, the use of conventional diagrams in calculations can lead to errors exceeding the safety margin. The use of effective deformation diagrams allows a reasonable increase in the allowable loads in design of critical structural elements. For this, it is necessary to employ the Kachanov-Rabotnov approach (Kachanov, 1986; Rabotnov, 1966) and the basic hypotheses of continuum damage mechanics.

Consider the constitutive equation in the form (10). For the actual $σ_{r e}$ and effective $σ_{e f}$ stresses the diagrams are described as

{\bar{σ}}_{e f} = {\bar{ε}}^{{\bar{m}}_{e f}}; {\bar{σ}}_{r e} = {\bar{ε}}^{{\bar{m}}_{r e}}

(11)

For equations (9) and (11), after transformations

{\bar{σ}}_{e f} = \frac{σ_{e f}}{{(σ_{e f})}_{y}} = \frac{σ_{r e}}{1 - D} \cdot \frac{1 - D_{T}}{{(σ_{r e})}_{y}}; {\bar{ε}}^{{\bar{m}}_{e f}} = {\bar{ε}}^{{\bar{m}}_{r e}} \frac{1 - D_{T}}{1 - D}

(12)

where

{(σ_{r e})}_{y}

is the real yield stress;

{(σ_{e f})}_{y}

is the effective yield stress.

After taking logarithms:

{\bar{m}}_{e f} = {\bar{m}}_{r e} + \log_{\bar{ε}} (\frac{1 - D_{T}}{1 - D})

(13)

Equation (13) makes it possible to obtain the effective stress-strain diagram of the material from the known real tensile curve of the specimen and the damage accumulation. In the absence of the data, $D_{T} \approx 0$ .

When using a conventional tensile diagram in the following form:

\bar{σ} = {\bar{ε}}^{\bar{m}}; \bar{σ} = {\bar{σ}}_{r e} [1 - ψ (ε)]

(14)

after appropriate transformations, the dependence of the hardening parameter for the effective diagram

m_{e f}

is obtained:

{\bar{m}}_{e f} = {\bar{m}}_{у} + \log_{\bar{ε}} (\frac{1 - D_{T}}{1 - D}) + \log_{\bar{ε}} (\frac{1 - ψ_{y}}{1 - ψ (ε)})

(15)

where

ψ (ε)

and

ψ_{y}

are the relative values of contraction of the sample’s cross-section for the current state of elastic-plastic deformation and the yield strength of the material, respectively. For approximate calculations, as well as for brittle materials,

D_{T} = 0

and

ψ_{y} = 0

are taken. Comparison of the calculated results and experimental data for

m_{e f}

according to equations (13) and (15) for studied metallic materials showed a discrepancy within 2%.

Determination of concentration factors

Distributions of stresses and strains in the zones of increased loading, which are areas of stress/strain concentrators, have a complex character. Therefore, to assess the load-bearing capacity of structural elements in current engineering practice Neuber-Makhutov approach (Makhutov, 1981) can be in the following form:

\frac{K_{σ} \cdot K_{ε}}{α_{σ}^{2}} = F (α_{σ}, {\bar{σ}}_{n}, f ({\bar{σ}}_{n}, {\bar{ε}}_{n}))

(16)

where

K_{σ}

and

K_{ε}

are the effective stress and strain concentration factors, respectively;

α_{σ}

is the theoretical stress concentration factor;

{\bar{σ}}_{n}

and

{\bar{ε}}_{n}

are the relative values of current levels of stresses and strains, respectively.

They are defined as

K_{σ} = \frac{σ_{\max}}{σ_{n}}; K_{ε} = \frac{ε_{\max}}{ε_{n}}; α_{σ} = \frac{σ_{\max}^{e}}{σ_{n}}

(17)

where

σ_{\max}^{e}

and

σ_{\max}

are the maximum stresses in the concentration zone in case of elastic and elastic-plastic calculation approach, respectively;

ε_{\max}

and

ε_{n}

are the maximum elastic-plastic and nominal values of strain, respectively;

σ_{n}

is the nominal stress.

The function $F$ in equation (16) is determined on the basis of a conventional experimental deformation diagram of a structural material. The account of the damage parameter and the use of the effective deformation diagram in the form of $σ_{e f} = f (ε_{e f})$ causes a significant refinement compared to the Neuber-Makhutov definition:

F [α_{σ}, {\bar{σ}}_{n}, f ({\bar{σ}}_{n}, {\bar{ε}}_{n})] = \frac{1}{[α_{σ} {({\bar{σ}}_{e f})}_{n}] n (1 - {\bar{m}}_{e f}) \{1 - [{({\bar{σ}}_{e f})}_{n} - \frac{1}{α_{σ}}]\}}

(18)

where

{\bar{m}}_{e f}

is the hardening factor of the effective strain diagram, according to equation (10);

n

is the parameter (in general case

0 \leq n \leq 1

), in most cases n = 0.5 (the suitability of this magnitude for the function

F [α_{σ}, {\bar{σ}}_{n}, f ({\bar{σ}}_{n}, {\bar{ε}}_{n})]

was discussed in detail in works by Makhutov, 1981, Makhutov et al., 1983, Makhutov N et al., 1981);

{({\bar{σ}}_{e f})}_{n} = \frac{{(σ_{e f})}_{n}}{{(e f)}_{y}}

;

{(σ_{e f})}_{n}

;

{(σ_{e f})}_{y}

are the effective values of nominal stresses in the zone of stress concentration and stresses at the level of the proportional (yield) limit of the material, respectively..

The effective strain and stress concentration factors taking into account the conventional and effective strain diagrams differ only by the values of hardening coefficients $\bar{m}$ . For the effective stress-strain diagram the following expressions are obtained:

For ${({\bar{σ}}_{e f})}_{n} \leq 1$ :

\{\begin{matrix} {(K_{e f})}_{ε} = \frac{α_{σ}^{\frac{2}{1 + {\bar{m}}_{e f}}} {({\bar{σ}}_{e f})}_{n}^{\frac{1 - {\bar{m}}_{e f}}{1 + {\bar{m}}_{e f}}}}{{[α_{σ} {({\bar{σ}}_{e f})}_{n}]}^{\frac{n (1 - {\bar{m}}_{e f})}{1 + {\bar{m}}_{e f}} \{1 - [({\bar{σ}}_{e f}) - \frac{1}{α_{σ}}]\}}} \\ {(K_{e f})}_{σ} = \frac{α_{σ}^{\frac{2 {\bar{m}}_{e f}}{1 + {\bar{m}}_{e f}}}}{{({\bar{σ}}_{e f})}_{n}^{\frac{1 - {\bar{m}}_{e f}}{1 + {\bar{m}}_{e f}}} {[α_{σ} {({\bar{σ}}_{e f})}_{n}]}^{\frac{{\bar{m}}_{e f} \cdot n (1 - {\bar{m}}_{e f}) \{1 - [({\bar{σ}}_{e f}) - \frac{1}{α_{σ}}]\}}{1 + {\bar{m}}_{e f}}}} \end{matrix}

(19)

For ${({\bar{σ}}_{e f})}_{n} > 1$ :

\{\begin{matrix} {(K_{e f})}_{ε} = \frac{α_{σ}^{\frac{2 {\bar{m}}_{e f}}{1 + {\bar{m}}_{e f}}}}{{[α_{σ} {({\bar{σ}}_{e f})}_{n}]}^{\frac{n (1 - {\bar{m}}_{e f}) \{1 - [({\bar{σ}}_{e f}) - \frac{1}{α_{σ}}]\}}{1 + {\bar{m}}_{e f}}}} \\ {(K_{e f})}_{σ} = \frac{α_{σ}^{\frac{2 {\bar{m}}_{e f}}{1 + {\bar{m}}_{e f}}}}{{[α_{σ} {({\bar{σ}}_{e f})}_{n}]}^{\frac{{\bar{m}}_{e f} \cdot n (1 - {\bar{m}}_{e f}) \{1 - [({\bar{σ}}_{e f}) - \frac{1}{α_{σ}}]\}}{1 + {\bar{m}}_{e f}}}} \end{matrix}

(20)

The results of calculation of the concentration coefficients $K_{σ}$ and $K_{ε}$ for two types of metallic materials are presented in Figures 6 and 7 using conventional, real and effective deformation diagrams.

Apparently, the account for damage in calculations results in an increase in the value of $K_{σ}$ and a decrease in $K_{ε}$ . Thus, for the nominal relative stress at the level of elastic limit (proportionality limit) ${\bar{σ}}_{n} = 1$ , using the effective diagram, the value of the effective coefficient ${(K_{e f})}_{σ}$ became larger by 10%–12%, and for ${(K_{e f})}_{ε}$ - smaller by 9% in compared to those in the conventional strain diagram of the studied material.

The effect of theoretical stress concentration factor $α_{σ}$ on the values of effective concentration factors ${(K_{e f})}_{σ}$ and ${(K_{e f})}_{ε}$ for 17CrNiMo6 steel and 13Mn6 steel is shown in Figures 8 and 9.

For the stress concentration level with $α_{σ} = 3$ the effective stress concentration factor exceeded that of the conventional stresses by 15%, and the effective strain factor decreased by 12%.

Energy based approach for LCF assessment

The energy-based justification of the process of elastic-plastic deformation and the process of distributed damage led to the consideration of the absorbed mechanical energy. In this case, the thermal component was neglected. From the deformation diagrams (Figure 10), the specific energy spent on the process of accumulation of DD $W_{D}$ , is defined as the difference of the areas under the effective and actual deformation diagrams:

W_{D} = W_{e f} - W_{d} = \frac{1}{2} Δ σ_{y} \cdot ε_{y} + \int_{ε_{y}}^{ε_{R}} Δ σ (ε) d ε = W_{e} + W_{p}

(21)

where

W_{e}

and

W_{p}

are the specific components of strain energy in elastic and elasto-plastic deformations, respectively.

For symmetric cyclic loading (Figure 11), the specific energy that corresponds to the accumulation of distributed damage in a cycle can be conventionally divided into two components: the energy in the positive and negative half-cycles of loading:

\begin{matrix} W_{N}^{+} = \frac{Δ σ_{y}^{+} \cdot ε_{y}^{+}}{2} + \int_{ε_{y}^{+}}^{Δ ε_{N}^{+}} Δ σ (ε) d ε - \frac{Δ σ_{N}^{+} \cdot ε_{N}^{+}}{2} \\ W_{N}^{-} = \frac{Δ σ_{y}^{-} \cdot ε_{y}^{-}}{2} + \int_{ε_{y}^{-}}^{Δ ε_{N}^{-}} Δ σ (ε) d ε - \frac{Δ σ_{N}^{-} \cdot ε_{N}^{-}}{2} \end{matrix}

(22)

Since the magnitude of plastic deformation in the cycle is insignificant, the change in the damage parameter $D$ in a half cycle can be neglected and $D$ can be taken as a constant value. Then, the effective stress in a half cycle is defined as $σ_{e f}^{(k)} = \frac{σ_{r e}^{(k)}}{1 - D^{(k)}}$ , where $D^{(k)}$ is the damage parameter in the half cycle $k$ , determined experimentally; $σ_{r e}^{(k)}$ is the maximum real stress in the half cycle $k$ . $D^{(k)}$ value can be determined as (Lemaitre, 1985, 1990)

D^{(k)} = \{\begin{matrix} D^{+} = 1 - \frac{E_{k}}{E_{0}} for positive half cycle \\ D^{-} = h \cdot D^{+} for negative half cycle \end{matrix}

(23)

where

E_{i}

and

E_{0}

are the elastic modulus for the half cycle

k

and initial half cycle respectively;

h

is the experimentally determined healing parameter which, for one cycle, could be calculated following (Lemaitre and Desmorat, 2005)

h = \frac{E_{0} - E^{+}}{E_{0} - E^{-}}

(24)

were

E^{+}

and

E^{-}

are the elastic moduli for the tension and compression half cycles respectively. In this work, the healing parameter was determined as the arithmetic mean based on the results of three cyclic tests carried out for different levels of loading amplitudes.

For studied materials $h$ values are shown in Table 3.

Table 3.

Healing parameter.

Material	Alloy 2024	17CrNiMo6	13Mn6	Ti6-4
h	0,32	0,51	0,70	0,46

Thus, using equation (23) effective stress cycle can be obtained and energy values from equation (22) can be found.

In a case of stress concentration, effective stress cycle can be defined by using Makhutov (Makhutov, 1981) approach for cyclic loading combined with Kachanov-Rabotnov effective stress approach. Maximum stress ${({\bar{σ}}_{e f}^{c y})}_{\max}^{(k)}$ and strain ${({\bar{ε}}_{e f}^{c y})}_{\max}^{(k)}$ for effective cyclic diagram can be written as

{({\bar{S}}_{e f})}_{\max}^{(k)} = {({\bar{S}}_{e f})}_{n} \cdot {(K_{e f}^{c y})}_{σ}^{(k)}, {({\bar{ε}}_{e f}^{c y})}_{\max}^{(k)} = {\bar{ε}}_{n}^{c y} \cdot {(K_{e f}^{c y})}_{ε}^{(k)}

(25)

${({\bar{S}}_{e f})}_{n} = \frac{S_{n}}{{(S_{y})}_{e f}}$ and ${\bar{ε}}_{n}^{c y} = \frac{ε_{n}^{c y}}{ε_{y}^{c y}}$ , where ${({\bar{S}}_{e f})}_{n}$ and $ε_{n}^{c y}$ are the nominal cyclic stress and strain respectively, ${(S_{y})}_{e f}$ and $ε_{y}^{c y}$ are the effective cyclic yield stress and deformation; ${(K_{e f}^{c y})}_{σ}^{(k)}$ and ${(K_{e f}^{c y})}_{ε}^{(k)}$ are the effective stress and strain concentration factors for half cycle $k$ . Concentration factors ${(K_{e f}^{c y})}_{σ}^{(k)}$ and ${(K_{e f}^{c y})}_{ε}^{(k)}$ can be found for every half cycle substituting ${\bar{σ}}_{e f}$ with ${\bar{S}}_{e f}$ and ${\bar{m}}_{e f}$ with ${\bar{m}}_{e f} (k)$ in equations (19) and (20). The effective cyclic diagram hardening factor ${\bar{m}}_{e f} (k)$ defined as

\{\begin{matrix} {\bar{m}}_{e f} (k) = \frac{{\lg ({\bar{ε}}_{n} \cdot {(K_{ε}^{(0)})}_{e f})}^{\bar{m}}}{\lg [{({\bar{ε}}_{n} \cdot {(K_{ε}^{(0)})}_{e f})}^{\bar{m}} + ({\bar{ε}}_{n} \cdot {(K_{ε}^{(0)})}_{e f} - 1) \cdot F (k)]}, for {\bar{σ}}_{n} \leq 1 \\ {\bar{m}}_{e f} (k) = \frac{{\lg ({\bar{ε}}_{n})}^{\bar{m}}}{\lg [{({\bar{ε}}_{n})}^{\bar{m}} + ({\bar{ε}}_{n} - 1) \cdot F (k)]}, for {\bar{σ}}_{n} > 1 \end{matrix}

(26)

where A is the material parameter (from experiments it was found that for Alloy 2024 A = 0.77, for 13Mn6 A = 1.04, for 17CrNiMo6 A = 0.44);

{(K_{ε}^{(0)})}_{e f}

is the effective strain concentration factor for

k = 0

according to (19);

F (k)

is the material cyclic state function. Real (conditional) stress cycle can be defined by classical Makhutov (Makhutov, 1981) approach.

Function $F (k)$ can be found experimentally from dependence of plastic deformation in cycle on half cycles: $F (k) = \frac{δ^{(k)}}{δ^{(1)}}$ (Makhutov, 1981). This data for initial cycles (for three different cycle amplitude each) for materials Alloy 2024, 13Mn6 and 17CrNiMo6 is shown in Figure 12.

It is obvious from Figure 12, that 13Mn6 steel can be considered as material in cyclically-stabilized state, alloy 2024 – in cyclically hardening state and 17CrNiMo6 – in cyclically softening state. Therefore, cyclic state function

\{\begin{matrix} F (k) = \frac{5 - \ln (ε^{(0)})}{6} for 13 M n 6 steel \\ F (k) = \frac{1}{k^{α}} = \frac{1}{k^{3.21 ({\bar{σ}}^{(0)} - 1)}} for alloy 2024 \\ F (k) = e^{β (k - 1)} = e^{0.0047 ({\bar{ε}}^{(0)} - 1) (k - 1)} for 17 C rNiMo 6 steel \end{matrix}

(27)

The total energy of the conditionally stabilized cycle (performed at the level $0, 1 N_{R}$ ) under cyclic loading can be written in the form:

W_{N} = W_{N}^{+} + W_{N}^{-}

(28)

According to equations (21) and (28), it is possible to determine the number of cycles $N_{R}$ before the first macrocrack appears. For this purpose, the part of specific energy under static loading $W_{D}$ , released in fracture, is taken as a basic one:

N_{R} = \frac{W_{D}}{q (r_{σ}; f (σ, ε); R_{σ}) W_{N}}

(29)

where

q (r_{σ}, f (σ, ε), R_{σ})

is the function that depends on the material and the type of stress state. Experimental studies showed that for symmetric loading case of unnotched specimen it was constant. For a uniform stress state, this function becomes a constant and its values for symmetric loading cycle (

r_{σ} = - 1

) are shown in Table 4. For an non-uniform stress state with the theoretical concentration factor

α_{σ} > 1

q (r_{σ}, f (σ, ε), R_{σ}) = M_{1} \cdot N_{R}^{M_{2}}

, where

M_{1}

and

M_{2}

are the constants. Therefore, for the conditions

α_{σ} = 1

.8 (a concentrator in the form of an annular notch) and symmetric loading cycle (

r_{σ} = - 1

) the values of parameters for

q (r_{σ}; f (σ, ε); R_{σ})

are also shown in Table 4.

Table 4.

Material parameters.

		Alloy 2024	17CrNiMo6	13Mn6	Ti6-4
$α_{σ} = 1$	$q (r_{σ}, f (σ, ε), R_{σ})$	1.0	2.8	0.2	0.7
$α_{σ} = 1, 8$	$M_{1}$	1.65	3.32	0.32	–
$α_{σ} = 1, 8$	$M_{2}$	−0.52	−0.36	−0.22	–

Comparison of the durability values calculated according to equation (29) and the experiment is presented in Figure 13.

The proposed assessment method for low-cycle fatigue was compared with the standard one according to (Makhutov et al., 1987):

σ_{a}^{*} = \frac{A \cdot E \cdot α_{σ}^{2}}{N^{m_{1}}} \cdot \frac{1 - r}{2} \ln \frac{100}{100 - ψ_{R}} + \frac{σ_{- 1}}{{(4 N)}^{m_{e}} + \frac{1 + r}{1 - r}}

(30)

where

σ_{a}^{*} = E \cdot ε_{a}

is the maximum elastic stress,

r

is the cycle asymmetry factor,

ψ_{R}

is the relative decrease in the cross-sectional area of the specimen after rupture,

А = 0.16 [1 + {(1 - \frac{σ_{y}}{σ_{u}})}^{- 1}]

is the material constant,

m_{1} = \frac{1 - r}{2} (1.2 \frac{σ_{y}}{σ_{u}} - 0.35)

and

m_{e} = 0.132 \lg \frac{σ_{u}}{σ_{- 1}}

are the material parameters,

σ_{y}

and

σ_{u}

are the yield and ultimate stress, respectively.

A comparison of the calculated results and experimental data for low-cycle fatigue for the studied metallic materials is shown in Figure 14 with the corresponding confidence interval. It is obvious that criterion (29) corresponds more accurately to the experimental data. The maximum discrepancy between the experimental data and the calculated results for $α_{σ} = 1.8$ is 18%.

Conclusions

For metallic materials, it was established that the process of elasto-plastic deformation is continuously accompanied by the accumulation of scattered damage from the very start of the loading process. It is shown that damage occurs at the proportional limit level. This effect must be taken into account when new structural elements is designed. For metal forming process it is sufficient to take the yield strength of the material as the threshold corresponding to the beginning of the damage accumulation process.

It was found that the character of the damage accumulation curve did not depend on the plasticity of the metallic material or its cyclic type: cyclically hardening, softening or stabilizing. The damage diagram consists of three characteristic parts (by analogy with the creep curve): the first is characterized by a decrease in the damage accumulation rate, the second - by stabilization of this process, and the third demonstrates an increase in the rate of damage accumulation until the formation of a macrocrack. The second section is the longest (Figure 2).

The equations that describe the dependence of damage parameters $D_{T}$ and $D_{R}$ on the materials plasticity parameters for tested materials were obtained.

It was shown that the introduction of the scalar damage parameter made it possible to specify the effective stress and strain concentration coefficients.

It was established, that with a negative value of the first invariant of the stress tensor (in compression half-cycle) a damage accumulation process occurs with a factor $h < 1$ .

On the basis of the energy analysis of the low-cycle loading process, a method of durability assessment was developed and compared with the standard one, as well as with the experiment. For a structural element with the theoretical coefficient $α_{σ} = 1, 8$ , the maximum discrepancy between the calculation and the experiment was about 18%.

The developed method can be used to calculate L-N curves more accurately, thus enhancing the design and optimization of components exposed to low-cycle loading conditions.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article

ORCID iDs

Mykola Bobyr

Vadim Silberschmidt

Viktor Koval

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