1 Introduction

Since the twenty-first century, the performance development of structural steel materials in civil engineering has primarily been focused on the increase of yielding and ultimate strengths to achieve desired resistance against natural disasters, such as strong earthquakes and winds. However, learned lessons from past catastrophic events such as the Hyogoken-Nanbu earthquake (1995), Chichi earthquake (1999) and the Wenchuan earthquake (2008), tens of thousands of civilian fatalities, and billions of direct economic losses are mainly caused by the destruction of building structures [1,2,3] and life-line infrastructure.

Steel plays an important role in infrastructure and overall economic development. Modern steel manufacturers actively participate in the development of new advanced steels by adjusting their composition or adopting new processing procedures to meet the growing demand for high-performance materials [4]. As one of the most essential alloy metals in engineering structures, steel with high strength (> 460 MPa) has been extensively used in buildings, bridges, and offshore structures [5]. High-strength steel is always in market demand for the special purpose use in the field of energy conservation, transportation and industrial infrastructure [6]. By using high-strength steel, we can reduce the weight of structures and vehicles, thereby reducing energy consumption and greenhouse gas emissions, which contributes to sustainable development.

Due to the significant deformation ability of high ductility steel without fracture, it is crucial in seismic zones and areas prone to natural disasters. This high-performance steel has excellent energy absorption capacity and is an ideal material for manufacturing earthquake resistant structures. Materials are always expected to achieve both high strength and high toughness. However, these two are often mutually exclusive [7]. Most metallurgical mechanisms for increasing strength would lead to ductility loss and fatigue failure [8] due to the strength-ductility tradeoff effect [7, 9]. Although high-strength Fe-based metals with high ductility have been developed, strength and stiffness will get loss inevitably as ductility increases due to the uncertainty of the scaling effect between micromaterials and macroscale components for mature production [10, 11]. Theoretically, damage and fracture mechanics provides solutions to predict fatigue life based on micro defects and materials fracture energy, Nguyen et al. [12] develop a new perspective of fracture mechanics inspired by a modified experiment with crack-parallel compression and bending. The above approach reveals the importance of developing damage and fracture mechanics for evaluating the full range of fatigue failure including crack initiation, propagation, opening, and closing, to promote new metallic materials such as high-strength steels with desirable damage tolerance [13].

The concept of damage mechanics describes the evolution of mechanically induced micro-defects of materials. A hypothesis of equivalence establishes the constitutive relation between the strain and effective stress that depends on an additional state variable (ω) to quantify the area of micro-defects (Ad) [14, 15]:

$$\overline\sigma=\frac P{A-A_d}=\frac\sigma{1-\omega}$$
(1)

where σ is the conventional Cauchy stress, and \(\tilde{\sigma }\) is the effective stress, which highly depends on the development of the damage variable. In the equivalence hypothesis of damaged micro-materials, the total strain is decomposed into three terms: an elastic strain εe that obeys the Hooke law, a plastic strain εp, and a damage strain εd, which introduces an additional flexibility term in the material. In other words, if the damage is equal to zero, then there is no additional flexibility, otherwise, the additional flexibility tends to infinity when the damage becomes one. For the alloys with plasticity such as steel materials, a yield function with kinematic hardening should be established using this strain equivalence hypothesis. In addition to the uniaxial damage concept for solid materials such as steel under compressive or tensile forces, flexural damages exist in nature and man-made structures such as beams and frames [16, 17]. For instance, the behavior of origami structures is primarily dominated by the flexural deformation capacity of folding ceases subjected to bending moment [18].

From the inspiration of the constitutive model in the framework of damage mechanics for micro-materials, this paper develops a new lumped damage concept for macro-meta structures with generalized forces and deformations. More critically, the rotational joints in these flexural members should be able to absorb plastic energy and prevent brittle failure concerning fatigue and extreme events that possibly occur in the service period.

2 Experimental design

2.1 Specimen preparation

To develop the mechanical properties of the HS-S and HD-S materials, a comprehensive test program was designed. For the design of specimen configuration and testing setup, we mainly cited the fatigue testing provisions in ISO 12106: 2003 code. The ductility and tensile strength were assessed through the tensile tests at room temperature. The fatigue resistance was assessed based on the fatigue tests, in which the controlled strain amplitudes vary from 0.9% to 1.5%.

The materials studied in this paper are two types of high-performance steels HS-S and HD-S, provided by Baosteel (Shanghai, China) and Yongtai Metal Materials (Dongguan, China) respectively. The chemical composition of the alloys is given in Table 1. The FeCrNi-based high-ductility steel used in this study is composed of (by weight) 8.94% carbon, 6.71% nickel, 16.86% chromium and balanced iron, which is developed to achieve larger ductility. And the high-strength steel is a type of industrially manufactured high-strength steel for engineering structures (Q960).

Table 1 Chemical composition of various steels in the percentage of weight
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A total of five round bars were manufactured for tensile tests. Figure 1 shows the dimensions of the tensile specimens. For each kind of material, two round bars were used to study the tensile property under cyclic tension stress and cyclic tension–compression stress. The remaining one round bar was used to execute the influence of cyclic cycles for the HD-S. The specimens for fatigue tests are 130.87 mm long and the dimensions are shown in Fig. 2. There are 4 round bars for the HD-S and 6 round bars for the steel HS-S to carry out traditional fatigue tests under different controlled strain amplitudes respectively. Besides, there are two additional round bars for both materials used to study the rain flow effect.

Fig. 1
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Tensile specimen: a dimensions of the specimen; b picture of the specimen

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Fig. 2
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Fatigue specimen: a dimensions of the specimen; b picture of the specimen

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Tables 2 and 3 show the measured dimensions of the steel HS-S and HD-S, respectively. The specimen labels, as shown in Tables 2 and 3, are designed in a way that the specimens with different critical information could be identified easily. In the labeling, the first letter Tension or Fatigue refers to the tensile test or fatigue test, respectively. The following notations can be used to distinguish the materials. The last notations indicate the controlled strain amplitude in the fatigue test which is expressed as a multiple of the yield strain. As for tensile tests, the last notation stress and strain present two different unloading controls that are stress unloaded to zero stress and strain unloaded to zero strain. It should be noted that all the tests are conducted at room temperature.

Table 2 Measured dimensions of tensile specimens
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Table 3 Measured dimensions of fatigue specimens
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2.2 Test example application

The tensile tests and fatigue tests were performed by a 250kN electro-hydraulic servo universal testing machine. The strain-controlling mode is adopted during all the tests. When the strain amplitude of each cycle is unloaded to zero strain, the stress of the specimen will transfer from tension to compression. In order to avoid buckling failure caused by out of plane deformation, the unloading rate was determined as a pure static protocol with a strain rate of 0.1%/s. The unloading strain rate is the minimum value of the actuator capacity. The gauge length of the extensometer is 20 mm for the tensile specimen and 10 mm for the fatigue specimen. To measure the local strain, two additional strain gauges are placed in the middle of the specimen longitudinally and horizontally shown in Fig. 3.

Fig. 3
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Test setup and extensometer placement

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For the tensile tests, three types of cyclic loading schemes are used. Table 4 describes the specified loading scheme for cyclic tensile specimens. It can be seen that the two different unloading patterns are compared for both materials. Specimens under unloading control of zero stress experience more strain amplitudes (0.5%, 1%, 2%, 3%, 4%, 6%, 8%, 10%, 12%, 16%, and 20%) compared with fewer strain amplitudes for specimens under unloading control of zero strain (0.5%, 1%, 2%, 3%, and 4%) to avoid bulking. The strain rate is 0.1%/s and the failure is defined when the specimen was completely fractured. The extensometer can be damaged when the specified range is exceeded. Therefore, displacement control is adopted when the strain is greater than 20%.

Table 4 Loading scheme for cyclic tensile specimens
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The low-cycle fatigue characters are studied in this paper, which limits the fatigue life to 1000 cycles. Therefore four different strain amplitudes (1.24εy, 1.22εy, 1.73εy, and 1.98εy) are employed for fatigue tests of high-strength carbon Ni steel, and six strain amplitudes (0.86εy, 1.04εy, 1.08εy, 1.15εy, 1.29εy, and 1.9εy) are adopted for HS-S steel and all the strain amplitudes are greater than the yield strain εy of each material. Furthermore, a supplementary test is designed to employ the dominating factor affecting fatigue life based on the rain flow rule. Alternating strain amplitudes are employed that are first applying half-cycles under 1.24εy and then applying a bigger stain amplitude of 1.73εy until fracture for the HD-S. The test is done again applying the bigger stain amplitude of 1.73εy at the beginning. As for the HS-S structural steel, the strain amplitude of 1.24εy and 1.73εy are replaced by 1.08εy and 1.29εy. The frequency for fatigue test is 0.05 Hz and failure is defined when one of the following happens: the specimen is completely fractured; the large tensile stress changes by 25% relative to the level determined by the test; the ratio of the elastic modulus of tension to the modulus of compression is changed. Normally, Et/Ec = 0. 5 is assessed as a failure; the maximum tensile stress varies by 25% relative to the maximum compressive stress.

3 Results and discussion

3.1 Strength and ductility

Monotonic and cyclic stress–strain properties of the high-strength steel (HS-S) are compared with that of the high-ductility steel (HD-S), as shown in Fig. 4. The steel HS-S behaved significantly higher yielding stress 1.011 GPa at the associated strain of 0.72%, comparative to 0.633 GPa for the HD-S. During the loading process, Young’s modulus is estimated approximately was 200 GPa by considering σy,0.1% as the end of linear elasticity. The maximum strength of HS-S material reaches 1.067 GPa with only elongated to 4% strain. In contrast, the associated stress at 4% strain of the HD-S as shown in Fig. 4 (a), is 7.93 GPa which remains to harden to 861 MPa until the strain reaches 42%. Compared with the hardening stress of the HS-S, a fourfold increase of elongation under ultimate strength is more remarkable for the HD-S. Additionally, the Bauschinger effect was found to play various roles in both materials, which demonstrates the modulus degradation during unloading that is 132 GPa for the HS-S and 100 GPa for the HD-S, respectively.

Fig. 4
figure 4

Monotonic and hysteretic behavior of two types of steel materials: a Monotonic engineering stress–strain curve for both specimens (b) The hysteretic response of stress and strain relation curves

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The hysteresis curves in Fig. 4 (b) indicate another unloading protocol to zero strain in each cycle. During the test, compressive stress is applied to recover the residual strain. Thus, the bulking is inevitable for the tensile specimens with a large slenderness ratio and the bulking is observed while loading to 4% strain. The ultimate stress of the HS-S in the failure cycle is 1031 MPa reduced 36 MPa compared with the cyclic tensile test, whilst the comparative value for the HD-S is 711 MPa at 4% strain. Besides, the degradation of modulus is more significant, which decreases by 40% from 200 to 119 GPa. It should be noted that the behavior during unloading deviates from Hook's law since the stress–strain curve during unloading does not follow a linear course. This curved progression indicates the so-called pseudo-elastic effect.

3.2 Failure mechanism to low-cycle fatigue

Figure 5 (a) shows the hysteretic stress–strain curves of the HD-S and the HS-S respectively, in which the evolution of stress–strain responses versus fatigue life can be compared. The ultimate stresses for the fatigue test bring in correspondence with that of the cyclic tensile skeleton curve, which is 953 MPa of the HS-S and 647 MPa of the HD-S. The plastic flow is inevitable as the accumulation of plastic strain. However, the strength degradation of the HD-S is not significantly contrasting with the HS-S. The failure can be identified as the strength reduction reaches 25% to the level determined by the fatigue test based on the specification (ISO 12106: 2003), there will be a distinct shrink for the fatigue life of the HS-S from 371 to 332 cycles, dropped by 11% while the degradation ratio is 1% for the HD-S. Furthermore, the unilateral asymmetric phenomenon is discovered for HD-S material when the crack propagates in tension, which indicates a significant effect due to crack closure.

Fig. 5
figure 5

Comparisons of behavior to failure under low cycle fatigue. a Hysteresis behavior during LCF history. b Hysteretic energy accumulation of plasticity. c S–N relation for fatigue life

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The dissipated energy per unit mass and accumulated energy during the whole fatigue life are displayed in Fig. 5 (b). Figure 5 (c) shows the data of fatigue lives of the two different high-performance steel materials under different standardized strain amplitude. A standardized strain amplitude that refers to the ratio of strain amplitude to yield strain is accepted here to compare the material fatigue performance. The low-cycle fatigue lives of both materials present high sensitivity to controlled strain amplitude that increases with the decrease of strain levels. The higher ductility of the HD-S displays better performance of fatigue resistance. Besides, the fitted regression equation can be determined as shown in Fig. 5 (c). Calculate the root mean square error(RMSE) of the fatigue life of HD-S and HS-S respectively by using Eq. 2, the calculated results are RHD-S = 0.054% and RHS-S = 0.045%.

$$RMSE=\sqrt{\frac1n}\sum_{i=1}^n\left(\phi_i-{\widehat\phi}_i\right)^2$$
(2)

For all the specimens, the life cycles decrease along with the logarithmic relation with the increase of strain amplitude. Contrarily, the dissipated energy per unit mass is closely related to the strain amplitude, rising with the increase of strain amplitude. Therefore, the different accumulation ratio of dissipated energy is easy to understand in the accumulated energy curves where the slopes represent the dissipated energy per unit mass under different strain amplitudes. When it comes to fracture, the arresting descent for dissipated energy can be seen for both materials although the distinct decrease modes are a longer-lasting degradation for the steel HS-S. Thus, significantly different fracture patterns are formed shown in Fig. 5 (a). The HS-S fractures along the direction of maximum shear stress while the HD-S cracks along with the maximum normal stress, therefore, developing two types of fracture. It is further noted that the fracture of the HS-S is much rougher and is prone to generate multiple cracks compared with the HD-S.

3.3 Fractography

To investigate the ductility properties of the two materials, the measurement of elongation and scanning electron microscope of fracture are employed, as shown in Figs. 6 and 7. From Fig. 6, it can be seen that the crack propagation zone of HD-S is larger than that of HS-S, and HS-S has obvious brittle crack traces. In addition, compared to HS-S, HD-S has smaller and more evenly distributed micro-voids. The HD-S exhibits better ductility than the HS-S and the elongation reaches up to 42%, three times the HS-S. The residual fracture surface is characterized by ductile deformation features (dimples), which this property can be estimated by the intensity and size of dimples. Therefore, the large size and uniform arrangement of dimples in the HD-S indicates more ductile performance than the HS-S with the same amplified factor.

Fig. 6
figure 6

SEM fractography of (a,c) HD-S and (b,d) HS-S steels under ultra-low cyclic tensile loading (3 cyclic per amplitude – unloading to zero stress), and (e) EDX mapping of the chemical composite around dimple area of HS-S steel

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Fig. 7
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SEM fractography of low cycle fatigue (a,c,e) HD-S specimen tested under 1.98εy strain amplitude and (b,d,f) HS-S specimen tested under 1.29εy strain amplitude. a,b Overall fatigue fracture surfaces, c,d crack propagation regions, e,f final rupture areas

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The fractography surfaces from cyclic tensile loading tests was examined macroscopically and microscopically to compare the fracture surfaces of two high-performance steel plates. Macroscopic images of both samples are seen in Fig. 6(a) and (b) consist of fibrous and shear-lip zones, while the HS-S sample has radial marks shown with the red arrows additionally (Fig. 6(b)). These patterns are indications of rapid and unstable crack propagation and are perpendicular to the crack front [19]. Higher magnification SEM fractography in Fig. 6(c) and (d) reveals that the specimens fractured in a ductile manner as the surfaces are predominately composed of large voids and dimples. It is known that particle size and spacing have an important role in the variation of dimple size among the fracture surfaces. Nevertheless, the HD-S consists of larger dimples compared to HS-S steel in general, pointing out that the former displays better ductile performance under the same loading conditions.

The fracture morphology of low-cycle fatigue specimens is presented in Fig. 7. While the left side of the figure belongs to the fracture surfaces of the HD-S specimen under 1.98εy strain amplitude which corresponds to about 1.09% strain, the right side depicts the HS-S steel specimen tested under 1.29εy strain amplitude that corresponds to 0.93% strain. These two specimens were chosen for fractography investigations since they were tested under close strain values. Compared with Fig. 7(a), the crack growth region of the HS-S sample is much rougher, as shown in Fig. 7(b), which can be the result of possessing a higher crack growth rate. Stage II crack propagation is trans-granular for both samples and the crack propagation regions seen in Fig. 7(c, d) are characterized by predominantly quasi-cleavage fractures with some secondary cracks (highlighted by red arrows). The density of secondary cracks in the HD-S specimen is higher compared to that of the HS-S specimen which can be another factor for deceleration of crack growth in the HD-S specimen due to the energy-consuming capacity of secondary cracks. Besides, micro-voids are only detected in the HD-S specimens as shown with blue arrows. Fatigue striations were not observed on the fracture surfaces that can be seen under special cases in high-strength steels [20]. The characteristics of final rupture areas under low cycle fatigue shown in Fig. 7(e, f) are similar to those of under cyclic loading. HD-S specimen consists of larger and deeper dimples compared to HS-S specimen as an indication of more ductile fracture. Moreover, Fig. 7(f) indicates that the failure criterion seems sensitive to material flaws on the microscopic scale.

4 Damage evolution law for fatigue crack propagation

The well-known Manson-Coffin law for low-cycle fatigue (LCF) establishes the exponential relation between the reversal number of cycles to fracture in a uni-axial test with constant plastic strain amplitude as follows [21]:

$$\mathrm{N}={\left(\frac{{\Delta \varepsilon }_{p}}{{\varepsilon }_{f}}\right)}^{\beta }$$
(3)

where Δεp is the plastic strain amplitude, and εf and β are material parameters. In particular, εf indicates the plastic strain amplitude that induces the fracture of the specimen in one cycle.

When the origami with plastic hinge is subjected to repeatedly LCF loading protocol, using high-performance steel materials tends to create material and geometric nonlinearity. The crease rotation of origami then consists of three general deformations, which are controlled by elasticity εe = [F(d)]·{σ}, plasticity εp, and LCF-induced damage variable εd [2, 3].

Based on the conceptual idea of the lumped damage mechanics, all plastic deformation is concentrated in the plastic hinge region. The plastic strain amplitude of fiber at a distance z from the neutral axis can be replaced by the rotation shown in Eq. (3), where Lp is the plastic hinge length. Therefore, the Manson-Coffin law can be represented by the generalized deformation of structural components without damage as shown in Eq. (4), in which ϕ = 2Lpεf/H is the generalized deformation corresponding to flexural force.

$${\Delta\varepsilon}_p\left(z\right)\cong z\frac{{\Delta\phi}_p}{L_p}$$
(4)
$$N\cong\left(\frac{{H\Delta\phi}_p}{{2L}_p\varepsilon_f}\right)^\beta=\left(\frac{{\Delta\phi}_p}\phi\right)^\beta$$
(5)

The damage factor can be introduced to the equation based on the intermediate quantity of the second moment of area I. The second moment of the area after crack (\(\overline{I }\)) has a close relationship with the damage factor shown in Eq. (5). Meanwhile, the loss of modulus after cracking can be estimated by the ratio of effective height of section in Eq. (6). By eliminating the second moment of area in the Eqs. (5) and (6), the relationship of damage factor d and the distance z is established shown in Eq. (7). Therefore, a new Manson-Coffin law in combination with damage can be derived in Eq. (8).

$$\frac{\overline{I}}{I }=1-{d}_{i}$$
(6)
$$\frac{\overline{I}}{I}\cong \frac{{z }^{3}}{{\left(H/2\right)}^{3}}$$
(7)
$$z\cong \frac{H}{2}{\left(1-{d}_{1}\right)}^{\alpha }$$
(8)
$$N\cong\left(\frac{\left({1-d}_i\right)^\alpha{\Delta\phi}_p}\phi\right)^\beta$$
(9)

where Δϕp and ϕ bring into correspondence with Δεp and εf, which can be calculated by Eq. (3) and Eq. (4). Δϕp is the plastic rotation of fiber layers at a distance z and the parameter ϕ is the amplitude of the plastic rotation that produces the fracture of the specimen in one cycle. The plastic hinge length can be estimated at 0.5H based on Eurocode 8. For the initial specimen without damage, the distance z is equal to the half-height of the Sect. (0.5H). Thus, the Δϕp can be considered equal to the Δεp based on Eq. (3). And the parameter of rotation ϕ can be determined by the regression equation to be 0.023 for ϕHD-S and 0.028 for ϕHS-S when the failure of life is one cycle.

Denote the plastic amplitudes of two tests as \({\Delta\phi}_p^a\) and \({\Delta\phi}_p^b\). Let Na and Nb be the corresponding numbers of cycles for failure. Then, based on Eq. (8):

$$N^a\cong\left(\frac{\left({1-d}_i\right)^a\Delta\phi_p^a}\phi\right)^\beta$$
(10)

The parameter β can be computed by taking the logarithm of Eq. (9). Then, Eq. (10) gives the determination method of the parameter β, in which the values of the parameter (Na, Nb, Δpa, and Δpb) can be obtained from the regression curve shown in Fig. 5(c). Here, the parameter βHD-S is -3.07, and βHS-S is -3.45.

$$\beta=\ln\left(\frac{N^b}{N^a}\right)/\ln\left(\frac{{\Delta\phi}_p^b}{{\Delta\phi}_p^a}\right)$$
(11)

The parameter α is the section shape coefficient that can be determined by the second moment of area, for the circular section of this test, to be estimated as 0.4. Thus, the cycles of crack initiation Ncr can be calculated while taking the value of zero for damage in Eq. (11). Besides, the damage evolution rule can be derived from the inverse of Eq. (8). Only up to the initiation cycles, there will be damage accumulation, and before that damage is considered to be zero.

$$N_{cr}=\left(\frac{\left({1-d}_i\right)^\alpha{\Delta\phi}_p}\phi\right)^\beta=\left(\frac{{\Delta\phi}_p}\phi\right)^\beta$$
(12 )
$$d_i=0\;\mathrm{if}\;N<N_{cr}$$
$$\begin{array}{cc}d_i\left(N\right)=\mathrm{Max}\left(1-\exp\left(\frac{\beta\ln\left(\phi_{cr}/{\Delta\phi}_i^p\right)+\ln\left(N\right)}{\beta\alpha}\right)\right)&\mathrm{if}\;N>N_{cr}\end{array}$$
(13)

where di indicates the damage variable in the phase of crack propagation, and ϕcr is the critical rotation corresponding to LCF-induced crack initiation. Figure 8 compares the differences between crack initiation and failure as well as the damage evolution during the two stages. The fatigue life corresponding to the initial crack is not linearly distributed as the failure cycles. With the transformation from low cycle fatigue to high cycle fatigue, the crack initiation cycles are close to the cycles of failure illustrating a sudden and rapid brittle fracture. Besides, the damage evolution curve shows damage develops rapidly in the early stage and materials with better toughness can withstand greater internal damage for HD-S materials.

Fig. 8
figure 8

Fatigue life and fracture toughness during crack propagation

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5 Conclusions

In conclusion, the experimental and analytical results of hysteretic behavior and low-cycle fatigue resistance on the two types of steel alloys highlight the potential of using macro-modeling methods for high-strength/ductility steels to provide an effective approach based on lumped damage mechanics, not simply to their stress–strain hysteresis responses and fatigue life but also their post-fatigue crack propagation and the quantification of remaining life. Provided the hysteresis curves for the HD-S and HS-S materials revealed that the balance between strength and ductility is more important for better hysteresis behavior with more energy dissipations. Moreover, the damage evolution laws of structural components sustaining large rotational deformations and the fatigue testing on associated materials under uniaxial loading are correctly agreed, both remain good ductility with a relatively high strength of 1 GPa and prevent the post-fatigue brittle failures. The proposed macro-fatigue lumped damage theory and approach demonstrate advancements to quantify the damaged extent and remaining life for flexural origami structures constructed with high-performance steel materials.