Evaluation of the Damage Value of Steel Alloys Using a CDM Model

Abstract

Damage is a phenomenon experienced when a material is subjected to external factors such as load and temperature. Damage is quantified through the damage value of a material, and its value typically ranges from 0 to 1, with 1 indicating complete damage. The damage value in this context primarily refers to the crack initiation condition, indicating failure. The damage value corresponding to this condition is referred to as critical damage. However, most materials tend to fail at a critical damage value of less than one. Researchers have developed different models to evaluate damage, and some of the prominent models are Lemaitre, Rice & Tracy, Gurson, and Bhattacharya & Ellingwood. This study uses the Bhattacharya & Ellingwood model to evaluate the damage value of 113 selected steel materials that play crucial roles in aerospace, automobile, and other industrial applications. This model uses monotonic properties of the material as the input and estimates the critical damage value (D_c). The study revealed that, for steel materials, the D_c value generally ranges from 0.1 to 0.7. This study highlights the variation in damage with plastic strain under monotonic loading, and this helps to quickly select a specific material when the damage criterion is crack initiation.

1. Introduction

Damage mechanics, a branch of continuum mechanics, involves modeling material behavior and degradation and studying how materials fail (ductile, brittle, creep, and fatigue) under various loading and environmental conditions. The primary objective of damage mechanics is to forecast the onset and progression of damage, thereby enhancing the design and reliability of engineering components. By understanding how materials deteriorate under specific conditions, engineers can create safer and more reliable structures to reduce repairs in the future. To predict failure, researchers have developed damage-based models for various materials, such as steel and rubber [1,2]. Early detection and prevention of cracks will help to avoid severe failure. Failure prediction is important for improving component durability and extending its lifespan. By having a failure prediction mechanism, critical systems are safer in the aerospace, civil construction, and automotive industries. A good understanding of the behavior of materials under damage can be used to develop standards for numerical simulations. Several damage models have been developed in recent years. These models can be categorized as phenomenological models, porosity models, and Continuum Damage Mechanics (CDM) models. Phenomenological models use experimental evidence to describe material responses and failures without accounting for microstructure. Porosity model studies on the growth of voids and cavities in materials. The CDM integrates damage parameters in material equations to explain the degradation process. By using these approaches, engineers will be able to predict material behavior for enhanced structural integrity [3,4].

CDM is preferred for modeling material degradation because it effectively captures various damage processes, such as fatigue, creep, and ductile failure. By having damage variables in constitutive equations, the CDM helps to predict material behavior under different load and environmental conditions. This ability to predict cracks is important in various industries, such as construction, automotive, aerospace, railroad, manufacturing, and oil and gas. Techniques such as nondestructive testing and computational modeling are often used to improve the performance of materials in these important areas. These methods, along with CDM, provide a proactive approach for managing cracks to enable engineers to predict failures. Therefore, the application of the CDM model for crack prediction enhances the safety of structures through appropriate maintenance practices across various engineering fields. The CDM thus acts as a tool in engineering by providing a strong basis for addressing the challenges related to material degradation [5].

By using the concept of effective stress and thermodynamics, Tai and Young [6] developed an isotropic ductile plastic void damage mechanism. Triaxiality is derived from a modified damage criterion for ductile fracture. A comparison with the Rice–Tracey model and some experiments showed the results’ good agreement. Wang [7] developed a new parameter to address the elastic–plastic fracture toughness of materials. The relationship between the conventional elastic–plastic fracture toughness and crack tip constraint characterized by crack tip stress triaxiality is derived from a modified damage criterion for ductile fracture. Chandrakanth and Pandey [8] introduced an exponential ductile plastic damage framework based on void damage variables, which improved the understanding of damage mechanisms. The damage model shows a nonlinear variation with respect to plastic strain and is sensitive to stress triaxiality. The predictions via the model compare well with the experimental results for several aluminum alloys. Dhar et al. [9] proposed a damage mechanics model to study void growth and crack initiation. The damage variable is used to model the material behavior in the elastoplastic regime. It was concluded that the critical value of the damage variable can be taken as a crack initiation parameter. Bonora [10] developed a new nonlinear CDM plasticity damage model on the basis of experimental observations that the growth of microvoids results in nonlinear damage accumulation with plastic deformation. Damage evolution trends are addressed by a single damage model. Needleman and Tvergaard [11] conducted numerical studies of the ductile–brittle transition via an elastic–viscoplastic model for a porous plastic solid. Notably, Bhattacharya and Ellingwood’s model [12] uses the CDM to predict failure under monotonic, creep, and fatigue conditions, thus improving the assessment of structural integrity. Thakkar and Pandey [13] developed a CDM-based isotropic nonlinear plastic damage evolution model. The damage model is based on the concept of effective stress and the principle of strain equivalence. The model is studied by comparing it with experimental data available in the literature for materials such as aluminum, copper, and steel. The damage model is able to represent experimental damage evolution patterns with close agreement. Bouchard et al. [14] offered an enhanced Lemaitre model for complex multiaxial configurations by developing an anisotropic damage approach on the basis of a comparison between the grain flow orientation and principal loading directions. The application of the model in various cases demonstrates its robustness and accuracy. Kumar and Dixit [15] proposed a simple nonlinear ductile damage growth law with only two material constants for steel materials. The material constants of the damage growth law are determined from experimental measurements of void growth as well as triaxiality at various plastic strain levels in tension tests. This trend is consistent with the trends of damage growth reported in the literature. Gautam et al. [4] carried out a detailed review of CDM-based ductile models covering both isotropic and anisotropic damage models. They also highlighted different direct (qualitative) and indirect (quantitative) damage evaluation techniques. Among these various damage models, the Bhattacharya and Ellingwood model is a good framework because it handles various loading conditions by using only the available material data. Vishavbandhu et al. [16] conducted a damage analysis study of 32 aluminum alloys to help engineers select appropriate aluminum materials. Chow and Wei [17] stated that D_c is an intrinsic material property, i.e., its value could be used for critical conditions such as crack initiation in the case of high cycle fatigue. By knowing the D_c value, fatigue and creep analyses of materials can be carried out, as demonstrated by Bhattacharya and Ellingwood [12]. From an application point of view, the damage value at any number of cycles within the crack initiation limit refers to the quantification of damage under high cycle fatigue. Therefore, in this study, a detailed damage analysis of steel materials was performed via a CDM-based model.

2. Materials and Methodology

Steel alloys are classified into different series on the basis of their alloying elements, and each series is used for a specific application. The 1xxx series primarily includes pure iron-based alloys, which are ideal for general structural work because of their ductility and formability. The 4xxx series has silicon to improve fluidity during welding. The 5xxx series has magnesium for enhancing toughness, wear resistance, and corrosion resistance. This makes 5xxx alloys suitable for car framing and marine applications. The C and Ck series, considered as carbon steels, are good for mechanical parts and tools where durability is important. The St and StE series have high strength and are mainly used in structural construction and heavy machinery, where high load-bearing capacity is needed. Additionally, specific series such as B, Cr, Mn, Ni, V, and X have unique characteristics appropriate for special applications. Boron is added in the B series to improve hardenability, and the Cr series shows features similar to those of stainless steels, making it suitable for use in environments with high oxidation risk. The Mn series is known for its strong wear resistance, and the Ni series alloys provide good toughness at subzero temperatures. In contrast, the V series is known for its strength and heat resistance at high temperatures. The X series, which has a high alloy content, is suitable for aerospace and power plant applications, where the mechanical integrity and performance are critical. Therefore, it is required to choose the right materials suitable for different industrial applications from this large variety of steels. A total of 113 steel materials from various series whose monotonic properties are available are selected, and the details are listed in Table 1.

The present analysis concerns the growth of the plastic zone in materials as it moves toward failure and fracture. The Bhattacharya & Ellingwood model [12] calculates the damage value, which is dependent mainly on the plastic strain. This model is appropriate for the assessment of material damage. Using the Ramberg–Osgood relation [18], Bhattacharya & Ellingwood analyzed damage progression in a material under monotonic (uniaxial) loading and developed their model. This model works as an isotropic damage accumulation model where damage development is governed by plastic strain. Moreover, the critical damage value (D_c) is crucial in determining the failure of a material.

Critical damage is an important factor, as it helps to determine the transition point from damage confinement to failure. This study also emphasizes D_c as an intrinsic material parameter, as recommended by Chow and Wei [17]. This analysis considers the variation in the critical damage values of 113 different steel materials from pure brittle (D_c ≈ 0) to pure ductile (D_c ≈ 1) fracture [3]. The constitutive relationship of the Bhattacharya & Ellingwood model, which is based on the Ramberg–Osgood model [18], relates the effective stress to the actual strain. In this model, damage growth is described by the accumulation of plastic strain, and damage does not occur until the plastic strain reaches a critical value.

The mathematical representation of damage growth is expressed as

$$\mathrm{D}=1-{\displaystyle \frac{{\mathrm{C}}_2}{{\mathsf{\epsilon }}_{\mathrm{P}}^{1+\mathrm{n}}+{\mathrm{C}}_1}}$$

(1)

where

• ε_p denotes the plastic strain experienced by the material;
• n is the strain-hardening exponent, which characterizes the resistance of a material to deformation as the plastic strain increases;
• C₁ and C₂ are constants derived from the material’s stress–strain behavior.

The constants C₁ and C₂ are calculated via the following equations:

$${\mathrm{C}}_1={\displaystyle \frac{3}{4}}\left(1+\mathrm{n}\right){\displaystyle \frac{{\mathsf{\sigma }}_{\mathrm{f}}}{\mathrm{K}}}-{{\mathsf{\epsilon }}_0}^{(1+\mathrm{n})}$$

$${\mathrm{C}}_2={\mathrm{C}}_1+{{\mathsf{\epsilon }}_0}^{(1+\mathrm{n})}$$

(2)

where

• σ_f represents the fracture stress;
• K is the material’s strength coefficient;
• ε₀ is the threshold plastic strain, which can be considered negligible or zero if specific data are unavailable.

The D_c value is calculated via Equations (1) and (2) by substituting ε_{p =} ε_f. The value of the threshold plastic strain (ε₀) is close to zero for most engineering materials and is ε₀ ≤ 0.02 for the five alloys reported by Lemaitre and Desmorat [3]. Furthermore, in the absence of other information, it may conservatively be taken to be zero [12]. To assess the impact of the selected value for ε₀, a comparison is carried out for critical damage values with (assumed ε₀ = 0.02) and without (ε₀ = 0) threshold plastic strain. A comparison reveals that the deviation is less than 1% for 71 materials, 1–5% for 34 materials, 5–10% for 6 materials, and more than 10% for 2 materials. When the deviation is less than 5%, the materials have critical damage values less than 0.25. The maximum deviation was observed for 100Cr6 material, with critical damage values of 0.13 (with ε₀ = 0) and 0.11 (with ε₀ = 0.02). Therefore, in the absence of the nonavailability of a threshold plastic strain value in the literature, the assumption of approximating it to zero is valid, and accordingly, for all 113 steel materials, ε₀ is assumed to be zero in this study.

These equations show how critical damage is influenced primarily by plastic strain. Using the monotonic properties of the materials listed in Table 1, the CDM-based Bhattacharya and Ellingwood model estimates the damage value for all 113 steel materials. The analysis of these materials reveals that the critical damage value varies for ductile and brittle materials under uniaxial loading. Additionally, this model has the ability to estimate damage variation via plastic strain and material properties. This structured methodology forms the foundation for predictions of damage in materials and makes a substantial contribution to material science research in the field of damage mechanics. The critical damage value D_c is determined for 113 steel alloy materials, and the values are presented in Table 1. In the material selection, the materials having monotonic properties close to those of the other materials are not included in the current study. Several materials having the same designation but different monotonic properties were selected in the present study to emphasize the fact that D_c can vary for the same material. Separate material IDs (Examples: A1, A2, A3) were given to differentiate those materials having different monotonic properties but the same designation. There may be several reasons for the observed variations in the mechanical properties of these materials, such as testing conditions, heat treatment processes, and alterations in the manufacturing process.

Table 1. Tensile properties and critical damage values of steels [19,20,21,22,23,24,25,26,27,28,29,30].

Sl. No.	Material (Material ID *)	Ultimate Strength σu (MPa)	True Fracture Strength σf (MPa)	E (GPa)	K	n	True Fracture Ductility ϵf	Critical Damage Value Dc
1xxx series (22 materials)
1	1010	331	870	203	534	0.185	1.63	0.55
2	1020 (A1)	393	795	203	400	0.072	1.02	0.39
3	1020 (A2)	441	865	207	738	0.190	0.96	0.48
4	1020 (A3)	502	849	207	933	0.239	1.01	0.54
5	1022	604	1587	200	971	0.161	1.16	0.45
6	1025	547	1085	186	1142	0.281	0.98	0.52
7	1038 (B1)	610	956	207	511	0.071	0.59	0.27
8	1038 (B2)	582	898	201	1106	0.259	0.77	0.48
9	1038 (B3)	743	1292	218	1231	0.169	1.16	0.56
10	1045	747	1151	209	1334	0.199	1.00	0.56
11	1050(M) (C1)	829	1177	203	1313	0.163	0.42	0.32
12	1050(M) (C2)	821	1379	211	1819	0.274	0.68	0.46
13	1050(M) (C3)	821	1128	211	1819	0.274	0.70	0.52
14	1090 (D1)	1090	1254	203	1765	0.158	0.15	0.15
15	1090 (D2)	1090	966	219	1780	0.162	0.20	0.25
16	1090 (D3)	1124	840	214	1576	0.108	0.50	0.51
17	1141 (A1FG)	925	1405	227	1205	0.074	0.88	0.48
18	1141 (NbFG)	695	999	220	1287	0.217	0.76	0.50
19	1141 (VFG) (E1)	725	1087	214	1321	0.207	0.68	0.46
20	1141 (VFG) (E2)	797	1243	215	1244	0.141	0.88	0.50
21	1541 (F1)	906	1395	205	1924	0.204	0.54	0.42
22	1541 (F2)	783	1409	205	1576	0.235	0.80	0.48
4xxx series (6 materials)
23	4140 (G1)	1514	2071	201	1911	0.055	0.65	0.43
24	4140 (G2)	1043	1519	207	1303	0.059	1.00	0.52
25	4142 (H1)	1929	2719	200	2054	0.016	0.46	0.31
26	4142 (H2)	1551	2366	200	1765	0.032	0.64	0.38
27	4142 (H3)	1413	1827	207	1892	0.051	0.66	0.46
28	4620	998	1530	208	1448	0.109	0.90	0.50
5xxx series–9xxx series (9 materials)
29	5120	1008	1287	214	1277	0.074	0.90	0.52
30	5150	867	1382	210	1630	0.207	0.80	0.50
31	5160 (I1)	1584	2241	203	1941	0.046	0.51	0.35
32	5160 (I2)	1669	1931	193	2124	0.066	0.87	0.54
33	8620	991	1411	212	1624	0.140	0.78	0.50
34	8822 (J1)	1723	3387	208	2175	0.057	0.67	0.35
35	8822 (J2)	946	1170	212	1074	0.025	1.12	0.57
36	9262	1565	1855	200	1951	0.060	0.38	0.32
37	9310	1201	2172	195	1796	0.094	0.83	0.45
C & Ck series (11 materials)
38	C 10	566	1205	218	659	0.073	1.13	0.44
39	C 20	414	953	190	330	0.061	1.19	0.34
40	C 70	964	837	201	1315	0.090	0.20	0.25
41	Ck 15	434	848.7	205	394	0.067	1.13	0.40
42	Ck 25	464	982	210	924	0.276	1.05	0.51
43	Ck 35 (K1)	593	1169	210	1168	0.257	0.97	0.50
44	Ck 35 (K2)	656	1468	210	1196	0.207	1.35	0.56
45	Ck 45 (L1)	684	987	202	735	0.092	0.46	0.28
46	Ck 45 (L2)	790	1400	206	730	0.047	0.92	0.38
47	Ck 45 (L3)	844	1582	206	1208	0.108	1.02	0.48
48	Ck 45 (L4)	774	1559	207	1297	0.166	1.14	0.53
St series (9 materials)
49	St 37	435	835	210	829	0.275	1.02	0.52
50	St 42	457	923	206	906	0.252	1.02	0.52
51	St 52	549	1192	206	371	0.007	1.17	0.33
52	St 52-3 (M1)	540	1289	205	433	0.022	1.39	0.38
53	St 52-3 (M2)	597	1083	210	1061	0.225	0.98	0.51
54	St E460 (N1)	682	936	208	1026	0.157	0.39	0.30
55	St E460 (N2)	682	574	208	1026	0.157	0.66	0.56
56	St E690	872	1446	214	954	0.024	0.87	0.43
57	St E790	820	965	206	727	0.112	0.25	0.15
B series (5 materials)
58	15B27	847	1839	203	1230	0.075	1.17	0.50
59	51B60	1970	1968	200	2332	0.039	0.20	0.22
60	86B20 (O1)	1034	869	205	1213	0.037	1.01	0.65
61	86B20 (O2)	1502	968	206	2193	0.092	0.91	0.70
62	41B17M (PS19)	872	1304	213	1031	0.042	1.14	0.54
Cr series (14 materials)
63	41 Cr 4	904	1688	200	967	0.036	0.87	0.39
64	42 Cr 4 (P1)	952	1689	194	1288	0.086	0.97	0.47
65	42 Cr 4 (P2)	840	1617	193	1240	0.118	1.17	0.52
66	100 Cr 6	2016	2230	207	2281	0.031	0.12	0.13
67	30 CrMo 2	898	1692	221	1117	0.063	1.12	0.48
68	34 CrMo 4 (Q1)	1078	1818	197	1382	0.070	0.94	0.47
69	34 CrMo 4 (Q2)	881	1740	194	1299	0.116	1.24	0.53
70	40 CrMo 4 (R1)	1088	2085	249	989	0.011	0.92	0.36
71	40 CrMo 4 (R2)	940	1440	209	1300	0.094	1.04	0.53
72	42 CrMo 4	1111	1525	211	1469	0.069	0.50	0.36
73	50 CrMo 4 (S1)	1086	1609	205	1132	0.026	0.67	0.38
74	50 CrMo 4 (S2)	983	926	205	1042	0.018	0.16	0.18
75	30 CrMoNiV 5 11	773	1332	212	717	0.027	0.97	0.40
76	30 CrNiMo 8	910	1168	206	1128	0.079	0.71	0.45
Mn series (17 materials)
77	8 Mn 6	965	1579	198	1227	0.054	0.85	0.45
78	14 Mn 5	697	1222	206	858	0.067	1.15	0.50
79	20 Mn 3	960	1090	206	1190	0.060	0.56	0.43
80	23 Mn 4	1091	1616	207	1185	0.026	0.95	0.47
81	25 Mn 3	540	1173	200	992	0.236	1.10	0.51
82	25 Mn 5	1008	1284	207	1138	0.033	0.68	0.43
83	80 Mn 4	931	1060	188	1100	0.127	0.17	0.15
84	20 MnCr 5 (T1)	1337	2351	194	1816	0.085	0.74	0.41
85	20 MnCr 5 (T2)	1053	1991	194	1762	0.117	0.83	0.46
86	30 MnCr 5	950	1445	206	1250	0.097	1.07	0.53
87	28 MnCu 6	580	950	204	938	0.190	1.03	0.53
88	49 MnVS 3 (U1)	840	1152	210	1428	0.194	0.38	0.30
89	49 MnVS 3 (U2)	845	1318	206	1390	0.180	0.56	0.38
90	22 MnCrNi 3	1510	2034	198	2447	0.114	0.55	0.42
91	41 MnCr 3 4	930	1390	207	1350	0.112	0.96	0.53
92	17 MnCrMo 33 (W1)	830	1550	206	767	0.006	0.87	0.36
93	17 MnCrMo 33 (W2)	929	1446	214	1285	0.099	0.87	0.48
Ni series (8 materials)
94	23 NiCr 4	808	1215	209	762	0.007	1.08	0.47
95	43 NiCr 7 9	1174	1636	206	1360	0.036	0.83	0.47
96	4 NiCr Mn 4	623	1229	206	753	0.081	1.45	0.53
97	40 NiCrMo 7	829	1201	194	1175	0.098	0.57	0.39
98	16 NiCrMo 32	939	1491	209	963	0.011	0.99	0.46
99	11 NiMnCrMo 55	852	1327	210	1277	0.124	0.83	0.48
100	40 NiCrMo 6 (Y1)	1015	1808	190	1372	0.089	0.97	0.47
101	40 NiCrMo 6 (Y2)	884	1680	205	1378	0.142	1.11	0.52
V series (4 materials)
102	10V45 (Z1)	909	1197	216	1520	0.168	0.50	0.39
103	10V45 (Z2)	765	1131	213	1456	0.223	0.70	0.48
104	15V24	878	1363	207	1318	0.129	0.90	0.50
105	1151V	761	1319	206	1346	0.190	0.70	0.43
X series (8 materials)
106	X 3 CrNi 19 9 (V1)	745	1920	186	548	0.136	1.37	0.32
107	X 3 CrNi 19 9 (V2)	951	2037	172	1114	0.063	1.16	0.45
108	X 2 CrNi 18 9	601	971	192	455	0.097	0.62	0.25
109	X 10 CrNi 18 8	635	1908	204	1416	0.362	1.56	0.57
110	X 3 CrNiTi 18 10	569	1381	204	349	0.062	1.43	0.32
111	X 5 CrNiMo 18 10	650	1400	183	1210	0.193	1.73	0.65
112	X 6 CrNi 19 11	650	1400	183	1210	0.193	1.61	0.63
113	X 25 CrNiMn 25 20	642	1360	193	754	0.228	1.01	0.38

* Material ID is used to differentiate various values of material properties for the same material name.

Table 1. Tensile properties and critical damage values of steels [19,20,21,22,23,24,25,26,27,28,29,30].

Sl. No.	Material (Material ID *)	Ultimate Strength σu (MPa)	True Fracture Strength σf (MPa)	E (GPa)	K	n	True Fracture Ductility ϵf	Critical Damage Value Dc
1xxx series (22 materials)
1	1010	331	870	203	534	0.185	1.63	0.55
2	1020 (A1)	393	795	203	400	0.072	1.02	0.39
3	1020 (A2)	441	865	207	738	0.190	0.96	0.48
4	1020 (A3)	502	849	207	933	0.239	1.01	0.54
5	1022	604	1587	200	971	0.161	1.16	0.45
6	1025	547	1085	186	1142	0.281	0.98	0.52
7	1038 (B1)	610	956	207	511	0.071	0.59	0.27
8	1038 (B2)	582	898	201	1106	0.259	0.77	0.48
9	1038 (B3)	743	1292	218	1231	0.169	1.16	0.56
10	1045	747	1151	209	1334	0.199	1.00	0.56
11	1050(M) (C1)	829	1177	203	1313	0.163	0.42	0.32
12	1050(M) (C2)	821	1379	211	1819	0.274	0.68	0.46
13	1050(M) (C3)	821	1128	211	1819	0.274	0.70	0.52
14	1090 (D1)	1090	1254	203	1765	0.158	0.15	0.15
15	1090 (D2)	1090	966	219	1780	0.162	0.20	0.25
16	1090 (D3)	1124	840	214	1576	0.108	0.50	0.51
17	1141 (A1FG)	925	1405	227	1205	0.074	0.88	0.48
18	1141 (NbFG)	695	999	220	1287	0.217	0.76	0.50
19	1141 (VFG) (E1)	725	1087	214	1321	0.207	0.68	0.46
20	1141 (VFG) (E2)	797	1243	215	1244	0.141	0.88	0.50
21	1541 (F1)	906	1395	205	1924	0.204	0.54	0.42
22	1541 (F2)	783	1409	205	1576	0.235	0.80	0.48
4xxx series (6 materials)
23	4140 (G1)	1514	2071	201	1911	0.055	0.65	0.43
24	4140 (G2)	1043	1519	207	1303	0.059	1.00	0.52
25	4142 (H1)	1929	2719	200	2054	0.016	0.46	0.31
26	4142 (H2)	1551	2366	200	1765	0.032	0.64	0.38
27	4142 (H3)	1413	1827	207	1892	0.051	0.66	0.46
28	4620	998	1530	208	1448	0.109	0.90	0.50
5xxx series–9xxx series (9 materials)
29	5120	1008	1287	214	1277	0.074	0.90	0.52
30	5150	867	1382	210	1630	0.207	0.80	0.50
31	5160 (I1)	1584	2241	203	1941	0.046	0.51	0.35
32	5160 (I2)	1669	1931	193	2124	0.066	0.87	0.54
33	8620	991	1411	212	1624	0.140	0.78	0.50
34	8822 (J1)	1723	3387	208	2175	0.057	0.67	0.35
35	8822 (J2)	946	1170	212	1074	0.025	1.12	0.57
36	9262	1565	1855	200	1951	0.060	0.38	0.32
37	9310	1201	2172	195	1796	0.094	0.83	0.45
C & Ck series (11 materials)
38	C 10	566	1205	218	659	0.073	1.13	0.44
39	C 20	414	953	190	330	0.061	1.19	0.34
40	C 70	964	837	201	1315	0.090	0.20	0.25
41	Ck 15	434	848.7	205	394	0.067	1.13	0.40
42	Ck 25	464	982	210	924	0.276	1.05	0.51
43	Ck 35 (K1)	593	1169	210	1168	0.257	0.97	0.50
44	Ck 35 (K2)	656	1468	210	1196	0.207	1.35	0.56
45	Ck 45 (L1)	684	987	202	735	0.092	0.46	0.28
46	Ck 45 (L2)	790	1400	206	730	0.047	0.92	0.38
47	Ck 45 (L3)	844	1582	206	1208	0.108	1.02	0.48
48	Ck 45 (L4)	774	1559	207	1297	0.166	1.14	0.53
St series (9 materials)
49	St 37	435	835	210	829	0.275	1.02	0.52
50	St 42	457	923	206	906	0.252	1.02	0.52
51	St 52	549	1192	206	371	0.007	1.17	0.33
52	St 52-3 (M1)	540	1289	205	433	0.022	1.39	0.38
53	St 52-3 (M2)	597	1083	210	1061	0.225	0.98	0.51
54	St E460 (N1)	682	936	208	1026	0.157	0.39	0.30
55	St E460 (N2)	682	574	208	1026	0.157	0.66	0.56
56	St E690	872	1446	214	954	0.024	0.87	0.43
57	St E790	820	965	206	727	0.112	0.25	0.15
B series (5 materials)
58	15B27	847	1839	203	1230	0.075	1.17	0.50
59	51B60	1970	1968	200	2332	0.039	0.20	0.22
60	86B20 (O1)	1034	869	205	1213	0.037	1.01	0.65
61	86B20 (O2)	1502	968	206	2193	0.092	0.91	0.70
62	41B17M (PS19)	872	1304	213	1031	0.042	1.14	0.54
Cr series (14 materials)
63	41 Cr 4	904	1688	200	967	0.036	0.87	0.39
64	42 Cr 4 (P1)	952	1689	194	1288	0.086	0.97	0.47
65	42 Cr 4 (P2)	840	1617	193	1240	0.118	1.17	0.52
66	100 Cr 6	2016	2230	207	2281	0.031	0.12	0.13
67	30 CrMo 2	898	1692	221	1117	0.063	1.12	0.48
68	34 CrMo 4 (Q1)	1078	1818	197	1382	0.070	0.94	0.47
69	34 CrMo 4 (Q2)	881	1740	194	1299	0.116	1.24	0.53
70	40 CrMo 4 (R1)	1088	2085	249	989	0.011	0.92	0.36
71	40 CrMo 4 (R2)	940	1440	209	1300	0.094	1.04	0.53
72	42 CrMo 4	1111	1525	211	1469	0.069	0.50	0.36
73	50 CrMo 4 (S1)	1086	1609	205	1132	0.026	0.67	0.38
74	50 CrMo 4 (S2)	983	926	205	1042	0.018	0.16	0.18
75	30 CrMoNiV 5 11	773	1332	212	717	0.027	0.97	0.40
76	30 CrNiMo 8	910	1168	206	1128	0.079	0.71	0.45
Mn series (17 materials)
77	8 Mn 6	965	1579	198	1227	0.054	0.85	0.45
78	14 Mn 5	697	1222	206	858	0.067	1.15	0.50
79	20 Mn 3	960	1090	206	1190	0.060	0.56	0.43
80	23 Mn 4	1091	1616	207	1185	0.026	0.95	0.47
81	25 Mn 3	540	1173	200	992	0.236	1.10	0.51
82	25 Mn 5	1008	1284	207	1138	0.033	0.68	0.43
83	80 Mn 4	931	1060	188	1100	0.127	0.17	0.15
84	20 MnCr 5 (T1)	1337	2351	194	1816	0.085	0.74	0.41
85	20 MnCr 5 (T2)	1053	1991	194	1762	0.117	0.83	0.46
86	30 MnCr 5	950	1445	206	1250	0.097	1.07	0.53
87	28 MnCu 6	580	950	204	938	0.190	1.03	0.53
88	49 MnVS 3 (U1)	840	1152	210	1428	0.194	0.38	0.30
89	49 MnVS 3 (U2)	845	1318	206	1390	0.180	0.56	0.38
90	22 MnCrNi 3	1510	2034	198	2447	0.114	0.55	0.42
91	41 MnCr 3 4	930	1390	207	1350	0.112	0.96	0.53
92	17 MnCrMo 33 (W1)	830	1550	206	767	0.006	0.87	0.36
93	17 MnCrMo 33 (W2)	929	1446	214	1285	0.099	0.87	0.48
Ni series (8 materials)
94	23 NiCr 4	808	1215	209	762	0.007	1.08	0.47
95	43 NiCr 7 9	1174	1636	206	1360	0.036	0.83	0.47
96	4 NiCr Mn 4	623	1229	206	753	0.081	1.45	0.53
97	40 NiCrMo 7	829	1201	194	1175	0.098	0.57	0.39
98	16 NiCrMo 32	939	1491	209	963	0.011	0.99	0.46
99	11 NiMnCrMo 55	852	1327	210	1277	0.124	0.83	0.48
100	40 NiCrMo 6 (Y1)	1015	1808	190	1372	0.089	0.97	0.47
101	40 NiCrMo 6 (Y2)	884	1680	205	1378	0.142	1.11	0.52
V series (4 materials)
102	10V45 (Z1)	909	1197	216	1520	0.168	0.50	0.39
103	10V45 (Z2)	765	1131	213	1456	0.223	0.70	0.48
104	15V24	878	1363	207	1318	0.129	0.90	0.50
105	1151V	761	1319	206	1346	0.190	0.70	0.43
X series (8 materials)
106	X 3 CrNi 19 9 (V1)	745	1920	186	548	0.136	1.37	0.32
107	X 3 CrNi 19 9 (V2)	951	2037	172	1114	0.063	1.16	0.45
108	X 2 CrNi 18 9	601	971	192	455	0.097	0.62	0.25
109	X 10 CrNi 18 8	635	1908	204	1416	0.362	1.56	0.57
110	X 3 CrNiTi 18 10	569	1381	204	349	0.062	1.43	0.32
111	X 5 CrNiMo 18 10	650	1400	183	1210	0.193	1.73	0.65
112	X 6 CrNi 19 11	650	1400	183	1210	0.193	1.61	0.63
113	X 25 CrNiMn 25 20	642	1360	193	754	0.228	1.01	0.38

* Material ID is used to differentiate various values of material properties for the same material name.

3. Results and Discussion

The results obtained, i.e., critical damage values, indicate that materials with higher D_c values have higher true fracture ductility ε_f. However, D_c does not always remain linearly proportional to ε_f, as other factors, such as the true fracture strength, strength coefficient, and strain hardening exponent, also influence the D_c value. For example, materials such as 1010 and X5CrNiMo1810 have high ductility values of 1.63 and 1.73, and high D_c values of 0.55 and 0.65, respectively. On the other hand, materials with high fracture strength, for example, 5160 with a strength of 1931 MPa, do not always have the highest D_c value, which shows that the combined properties of the material influence the D_c value. The variation in D_c with respect to plastic strain increases with increasing plastic strain and becomes nonlinear beyond a particular value of plastic strain. The present analysis is useful in explaining the material response to these conditions and in the selection of materials for use in applications where both strength and ductility are desirable.

3.1. Parametric Study of the Effects of D_c Dependence on Material Parameters

A parametric study is carried out to provide an understanding of the D_C dependence on the material and parameters involved in Equations (1) and (2). The data in Table 1 related to D_C are plotted with respect to each variable in the equations, along with a quadratic regression line, as shown in Figure 1. The true fracture ductility (ε_f) has a relatively strong, direct, and positive correlation with critical damage, indicating a dominant influence within the Ellingwood and Bhattacharya damage model. Figure 1a shows the D_c–ε_f relationship; D_c increases with increasing ε_f, confirming that more ductile steels can sustain greater damage accumulation prior to failure. The changing behavior after ε_f = 1 indicates a diminishing rate of increase in D_c at higher ε_f values, suggesting a saturation effect. Figure 1b shows the D_c–n relationship; the regression curve indicates a positive influence on D_c and is linear. Owing to greater scatter in the data, the role of n in maximizing the D_C capacity cannot be confirmed. Figure 1c shows the D_c–σ_f relationship; scatter indicates a nonlinear dependence characterized by an initial mild increasing trend followed by a decreasing trend. This behavior reflects the trade-off between strength and ductility, i.e., a higher fracture stress enhances the load-carrying capability, but it is often accompanied by a reduced plastic deformation capacity, indicating that high-strength materials have a lower value of D_c. Figure 1d shows the D_c–K relationship; scatter indicates a mixed trend with noticeable scatter around the regression line. The relatively wide scatter observed in the subplots of Figure 1b–d indicates that each parameter alone is not a dominant controlling parameter for D_c.

3.2. Damage Analysis of All the Materials

The critical damage value (D_c) is determined for all 113 steel materials via the Bhattacharya and Ellingwood model, and the results are presented in Table 1 with the true fracture strength and true fracture ductility values of the materials. The D_c value of a material is influenced primarily by the true fracture strength (σ_f) and true fracture ductility (ϵ_f). For the selected steel materials, the D_c value varies in the range of 0.1–0.7, and the majority of the materials have D_c values in the range of 0.2–0.6 (Figure 2). The primary damage influencing parameter ϵ_f varies in the range of 0.12–1.73, and σ_f varies in the range of 574–3387 MPa, as shown in

Table 2. Range of D_c values for all series of materials.

Sl. No.	Material Series (No. of Materials)	σf Range	ϵf Range	Dc Range
1	1xxxs (22)	840–1587	0.15–1.63	0.15–0.56
2	4xxx (6)	1519–2719	0.46–1.00	0.31–0.52
3	5xxx–9xxx (9)	1170–3387	0.38–1.12	0.32–0.57
4	C & Ck series (11)	837–1582	0.20–1.24	0.25–0.56
5	St & StE series (9)	574–1446	0.25–1.39	0.16–0.56
6	B series (5)	847–1968	0.20–1.17	0.22–0.70
7	Cr series (14)	926–2230	0.12–1.24	0.13–0.53
8	Mn series (17)	950–2351	0.17–1.15	0.15–0.53
9	Ni series (8)	1201–1808	0.57–1.45	0.39–0.53
10	V series (4)	1131–1363	0.50–0.90	0.39–0.50
11	X series (8)	971–2037	0.62–1.73	0.25–0.65
	All Materials	574–3387	0.12–1.73	0.13–0.70

. Figure 1 shows the trend of the variation in D_c with respect to ϵ_f, and this variation is linear until approximately ϵ_f = 1; beyond this value, the relationship is nonlinear.

Figure 2 shows the frequency of occurrence of Dc values across the dataset, highlighting the spread and clustering. Clustering can be observed at Dc values of 0.38, 0.43, and 0.45–0.53. For clarity, only 3 materials (Figure 3) from the entire range of D_c values are considered. The plot (Figure 3) shows that 86B20 (O2) offers the highest resistance against damage at any value of plastic strain.

For materials with D_c values up to 0.3, ϵ_f varies in the limited range of 0.12–0.62, and only a few materials are in this range. Similarly, only a few materials have a D_c value above 0.6, and in such cases, ϵ_f is greater than 0.9. The majority of the materials have D_c values in the range of 0.3–0.6, and in such cases, ϵ_f varies in the wide range of 0.38–1.72.

3.3. Damage Analysis of Various Material Series

A detailed analysis of all series of materials is carried out to determine the range of damage and true fracture ductility for each series. Figure 4 shows the variation in the damage value with respect to the plastic strain for the 1xxx, 4xxx, Ck and C, and Cr series materials. As the material damage variation trend is almost the same, only four series are considered, and in each series, only a few materials have been identified. Among the selected series, the 4xxx series (high-strength materials with σ_u greater than 1000 MPa) has a maximum D_c value of 0.52, although the maximum value of ϵ_f is 1.

3.4. Damage Analysis of Materials with the Same D_c Value

An analysis of the D_c values presented in Table 1 reveals that some materials have the same D_c value even though their true fracture ductility is different. In total, 26 such cases of the same D_c values from 0.15 to 0.65 were observed with two materials in a few cases, and 10 materials had the same D_c value of 0.5. For further analysis, only 4 cases with the same D_c values of 0.15, 0.32, 0.48, and 0.65 were considered from 26 total such cases, as shown in Figure 5. Even in each case, only a few materials were considered as part of the detailed analysis. Although the D_c value is the same, the nature of the curve is different. This is due to the influence of other parameters, such as the true fracture strength, strength coefficient, and strain hardening exponent. The results also show that the damage variation trend is linear until the plastic strain reaches approximately 0.4, and beyond this value, the variation trend is nonlinear.

3.5. Summary

Analysis of the variation in the damage value for different steels reveals that the majority of the materials exhibit damage values in the range of 0.43–0.56. A good spread of damage values (0.22 to 0.7) is observed in the B series, although there are only five materials. Both the Ni and V series materials exhibited relatively high damage values (0.39–0.53). Nonlinear variation in D_c with respect to ϵ_p is observed above a D_c value of approximately 1.

4. Conclusions

The critical damage value (D_c) is an intrinsic material property and is independent of loading conditions such as monotonic, fatigue, and multiaxial loading. Therefore, the present study focused on the analysis of the D_c value of steel materials considering their wide application, and the following conclusions were drawn.

• The CDM-based Bhattacharya and Ellingwood model helps to determine the damage value at different strain values by considering the monotonic properties of steel alloys.
• There is no unique D_c value for one material because material processing conditions vary the monotonic properties. For example, three different types of 1090 materials presented a wide range of D_c values, i.e., 0.27 to 0.56, representing the maximum variation observed in the present study.
• The critical damage values of steel alloys vary over a wide range of 0.1–0.7, and this study can be used to quickly determine the specific material when the damage criterion is crack initiation.