A high cycle fatigue numerical framework for component-level virtual fatigue testing: application to a light-duty vehicle lower control arm

Engineering Structures 311 (2024) 118198 Available online 21 May 2024 0141-0296/© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/). Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct A high cycle fatigue numerical framework for component-level virtual fatigue testing: Application to a light-duty vehicle lower control arm L.A. Gonçalves Junior a,b,∗, S. Jiménez a,b, A. Cornejo a,b, M.M. Tedescoc, L.G. Barbu a,b aCentre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Campus Norte UPC, 08034 Barcelona, Spain bUniversitat Politècnica de Catalunya (UPC), Campus Norte UPC, 08034 Barcelona, Spain cStellantis, Metals and Anticorrosion Department, Corso Settembrini 40, 10135 Turin, Italy ARTICLE INFO Keywords: Virtual fatigue test High cycle fatigue simulation Isotropic damage Continuum damage mechanics Finite element method Advance in time strategy ABSTRACT The conception of lightweight structures constitutes a fundamental design principle of the contemporary automotive industry. In this sense, the chassis is one of the areas with major potential for the implementation of solutions for mass reduction, as it accounts for around 25% of the total weight of a regular road vehicle. When it concerns the design process, the fatigue strength of the components that integrate this system represents a factor of utmost importance because these parts are usually subjected to a high number of load cycles throughout the vehicle service life. In this work, a numerical framework to perform fatigue highfidelity simulations of metallic engineering structures such as chassis components is presented. This framework builds upon an isotropic damage-based high cycle fatigue constitutive model coupled to an advance in time strategy algorithm used to improve the computational efficiency of the simulations. A novel methodology to calibrate the model material parameters consistent with the local nature of the fatigue phenomenon is proposed. Two different durability tests carried out with the steel lower control arm of a light-duty vehicle are virtually reproduced to assess the ability of this numerical framework to replicate the main features of real experiments. The performed simulations show the capability of the present numerical scheme to accurately capture the location of the fatigue cracks while providing a physically sound representation of their morphology. In addition, they also provide an excellent estimation for the number of cycles quantified in both experiments for the propagation of the cracks. The obtained results evidence the predictive capabilities of this numerical framework in performing fatigue high-fidelity simulations of metallic structures at engineeringrelevant scales, which potentially allows the reduction of the number of experiments required to support engineering decision-making processes. 1. Introduction The modern era of the automotive industry is essentially characterized by a high market competitiveness in which increasingly tighter environmental regulations have to be fulfilled by carmakers. Due to this challenging conjuncture, the segment of road vehicles has been mainly driven in the past few decades toward the conceptualization of lightweight solutions that enable the design of ever more efficient cars at optimum production costs. Accordingly, a considerable part of these solutions have been traditionally concentrated in the chassis, as it accounts for around 25% of the vehicle total weight and, therefore, represents one of the regions with major potential for mass reduction [1]. Down to the component-level conception, among all the aspects to be considered in the design process of chassis parts, an special attention should be attained to their durability, since they integrate subsystems ∗Corresponding author at: Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Campus Norte UPC, 08034 Barcelona, Spain. E-mail address: [email protected] (L.A. Gonçalves Junior). (e.g. suspension, wheels, half shafts, etc.) that are exposed to a high number of load cycles during the vehicle service life [2,3]. It is a common practice in the automotive industry the utilization of the stress-life method as a benchmark approach to estimate the durability of components subjected to a high number of cycles [4– 8]. In general lines, this methodology relies on the assessment of the part critical stress points, usually by means of finite element analysis, followed by the fatigue life estimation with the aid of S–N curves of the component material. In some cases, the influence of aspects such as surface roughness, part size and stress concentration are explicitly included in the component life assessment through the reduction of the material nominal fatigue limit by the so-called Marin semi-empirical factors [4]. The effect of the non-zero mean stress, in turn, is commonly taken into account in the calculations by different fatigue failure criteria, e.g., Soderberg, Goodman, Gerber, etc. [5–7]. Furthermore, https://doi.org/10.1016/j.engstruct.2024.118198 Received 25 January 2024; Received in revised form 29 April 2024; Accepted 12 May 2024 Engineering Structures 311 (2024) 118198 2 L.A. Gonçalves Junior et al. Nomenclature 𝛼𝑡(𝑅),𝐵0(𝑅, 𝜎𝑚𝑎𝑥)Coefficient of the HCF constitutive model 𝝈,𝝈0,𝝈𝑖Stress, effective stress and inelastic stress tensors 𝜺Strain tensor 𝜇 Damage consistency factor  𝛯Dissipation potential 𝜂,𝑡𝑜𝑙 Relative error and convergence tolerance of the advance in time strategy algorithm C0,C𝑠,C𝑡Constitutive, secant constitutive and tangent constitutive tensors ,0Damage and effective damage thresholds 𝜈Poisson’s ratio F(𝑓(𝝈0),0, 𝑓𝑟𝑒𝑑 (𝜸))Threshold damage function 𝜓Free energy per unit mass 𝜎0Uniaxial stress at the proportional limit 𝜎𝑎Stress amplitude 𝜎𝑚𝑎𝑥,𝜎𝑚𝑖𝑛 Maximum and minimum equivalent stresses at each load cycle 𝐶,𝑘Basquin model coefficients 𝑑Damage internal variable 𝐸Material Young’s modulus 𝑓(𝝈0),𝑓𝐻𝐶𝐹 (𝝈0, 𝑓𝑟𝑒𝑑 (𝜸))Stress norm function and stress norm function modified by fatigue state function 𝑓𝑟𝑒𝑑 (𝜸)Fatigue state function 𝐺(⋅),𝐺𝐻𝐶𝐹 (𝑓(𝝈0), 𝑓𝑟𝑒𝑑 (𝜸))Monotonically increasing scalar function and monotonically increasing scalar function modified by the fatigue state function 𝐺𝑓,𝑔𝑓Fracture energy and volumetric fracture energy 𝐽2Second invariant of the deviatoric stress 𝐾𝑡Stress concentration factor 𝑙𝑐Characteristic length 𝑚0Material density 𝑁𝑐,𝑁𝑑,𝑁𝑓Number of cycles, number of cycles to the damage initiation point and number of cycles at the intersection between the normalized S-N-R surface and the fatigue state function 𝑃𝐴,𝑠𝑁,𝑇𝑁Failure probability, standard deviation and scatter band associated with the S–N curves 𝑅Stress ratio 𝑆(𝑅, 𝑁𝑐),𝑆𝑡ℎ (𝑅)Fatigue strength and fatigue limit functions 𝑆𝑒,𝑆𝑢Material fatigue limit at 𝑅= −1 and ultimate tensile strength 𝑆𝑡ℎ,𝑅1,AUX R1, 𝛼𝑓, 𝛽𝑓, 𝑆𝑡ℎ,𝑅2,AUX R2 Fatigue material parameters when variable amplitude loads are involved in the analysis, the rainflow technique and Palmgren–Miner rule are often employed as cycle counting method and linear damage accumulation law [3]. As an alternative to the stress-life method, the strain-life approach may also find some real applications in the automotive sector [9], despite the limited amount of fatigue data available due to the practical difficulties in performing a material characterization in terms of strains, i.e., its elastic and plastic parts. In spite of their practical applicability, both methods often produce poor correlation with the experimental measurements for the component life prediction [4,9]. In addition, the deterioration of the material properties (i.e., stiffness and strength) intrinsic to the fatigue phenomenon is not taken into account by them. In this sense, the damage parameter commonly computed in these calculations acts as a merely remaining life estimator so that it does not produce any change in the material load-bearing capacity. For this reason, the fatigue damage evolution and the consequent crack propagation process are not captured by these approaches. Thus, although useful for design purposes, the results provided by these methods should be only interpreted as a general guide for the prediction of fatigue failures, and therefore, should be accompanied by extensive testing campaigns. As a complex engineering system, a road vehicle undergoes numerous testing campaigns from the beginning of its design process until the moment that the final product hits the market. These tests range from coupon-level experiments, as for the characterization of material mechanical properties, up to system-level campaigns in which the full vehicle is employed [10]. As a middle ground, component-level experiments are, whenever possible, employed in place of subsystem and system-level tests due to their lower cost and greater flexibility for setup customization. In this work, the durability of the metallic lower control arm (LCA) of Fig. 1 used in the suspension of light-duty vehicles (LDV) was assessed. For this purpose, fatigue tests using two different bench setups were performed. The first aimed at assessing the overall component durability, whereas the second arrangement focused on the fatigue response of critical design areas close to welding seams. Following these tests, high-fidelity numerical models were created to reproduce the experimental findings, serving as virtual fatigue testing tools that are able to support engineers in decision-making processes. These models were built upon the isotropic damage-based high cycle fatigue (HCF) constitutive law first proposed by Oller et al. [11] and further developed by Barbu et al. [12], Jiménez et al. [13], Granados et al. [14] and Gonçalves Junior et al. [15]. As such, they are able to capture the onset and propagation of fatigue cracks up to the complete component failure, while providing its life prediction. A novel methodology to calibrate the material parameters required by the models was applied. In this approach, the average stress data of S–N curves obtained from standardized fatigue test (e.g., based on the staircase method [16]) is converted into material point information that is used as a reference to perform the calibration. Finally, the cycle-jump algorithm presented in [17] was employed as advance in time strategy to improve the computational efficiency of the simulations. The remainder of this paper is organized as follows. In Section 2, the HCF constitutive formulation is devised. In Section 4, the model calibration methodology is presented. Section 3describes both experimental setups and their virtual counterparts. In Section 4, the obtained numerical results are validated against the experimental observations. Finally, concluding remarks are provided in Section 5. 2. High cycle fatigue constitutive framework The HCF model presented in this section consists of a particular case of the thermo-mechanical constitutive law proposed by Oller et al. [11] in which the thermal effects are neglected. This formulation relies on the notion that, from the phenomenological standpoint, HCF is mainly driven by two dominant mechanisms: material strength reduction under sub-critical stresses (well below the material ultimate strength) and degradation of its stiffness. Grounded in the framework of the continuum damage mechanics (CDM), this model incorporates both effects by means of an isotropic damage formulation modified by a Engineering Structures 311 (2024) 118198 3 L.A. Gonçalves Junior et al. Fig. 1. Metallic lower control arm used in the suspension of light-duty vehicles. fatigue state function that accounts for the deterioration of the material strength due to the application of cyclic loads. A brief description of this model is provided in the following. For a more detailed presentation, the reader is referred to Gonçalves Junior et al. [15]. 2.1. Thermodynamic principles and constitutive equation The constitutive framework presented in this section is restricted to purely mechanical processes, i.e., with no thermal effects involved, under the regime of infinitesimal strains. Thus, to introduce this model, the assumption of a specific free energy (per unit mass) 𝜓(𝜺, 𝑑)=1 2𝑚0(1 − 𝑑)𝜺∶C0∶𝜺,(1) defined as a function of the strain tensor, 𝜺, and the damage internal variable, 𝑑, may be taken as a starting point for its development. Here, 𝑑is a scalar-valued variable that accounts for the degradation of the material stiffness. It ranges from 1 in the undamaged state to 0 in the fully damaged condition. Additionally, 𝑚0refers to the material density and C0to the constitutive tensor in the undamaged state. Similarly, all other variables denoted with the subscript 0 in the following are related to the undamaged space (i.e., effective space), whereas the rest refers to the real one. As a second ingredient, the second law of thermodynamics in the form of the Clausius–Planck inequality  𝛯=(𝝈−𝑚0𝜕𝜓 𝜕𝜺) ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ constitutive relation ∶ 𝜺−𝑚0𝜕𝜓 𝜕𝑑 ⋅ 𝑑 ⏟⏞⏞⏟⏞⏞⏟ damage dissipation ⩾0,(2) is employed to account for irreversible phenomena in the constitutive formulation. The inequality (2) must hold for any mechanical process (i.e., ∀ 𝜺 admissible). Thus, substituting the derivative of Eq. (1) with respect to 𝜺into the resulting constitutive relation, 𝜎=𝑚0𝜕𝜓 𝜕𝜺, one can write the Cauchy stress tensor as follows 𝝈=𝑚0𝜕𝜓 𝜕𝜺 =(1 − 𝑑)⋅C0∶𝜺=C0∶𝜺 ⏟⏟⏟ effective stress, 𝜎0 −𝑑⋅C0∶𝜺 ⏟⏞⏞⏞⏟⏞⏞⏞⏟ inelastic stress, 𝜎𝑖 .(3) Accordingly, taking the derivative of Eq. (3) with respect to 𝜺, the secant constitutive tensor, that relates strain and stress when damage is not evolving, i.e.,  𝑑= 0, is obtained C𝑠=𝜕𝝈 𝜕𝜺 =𝑚0𝜕2𝜓 𝜕𝜺2=(1 − 𝑑)⋅C0.(4) Finally, the remaining part of inequality (2) leads to the energy dissipation rate which is given by 𝑚0𝜕𝜓 𝜕𝑑 ⋅ 𝑑⩾0.(5) 2.2. Threshold damage function, evolution law and tangent constitutive tensor To identify whether a given material point is in an elastic or inelastic state, a threshold damage function, F, of the form F(𝑓(𝝈0),0, 𝑓𝑟𝑒𝑑 (𝜸))=𝐺(𝑓(𝝈0)) 𝑓𝑟𝑒𝑑 (𝜸)−𝐺(0) =𝐺𝐻𝐶𝐹 (𝑓(𝝈0), 𝑓𝑟𝑒𝑑 (𝜸))−𝐺(0)≤0, (6) is here introduced. Where 𝐺(⋅)is a suitable monotonically increasing scalar function, 𝑓(𝝈0)a stress norm function, 0the effective damage threshold, 𝑓𝑟𝑒𝑑 (𝜸)a fatigue state function defined in terms of a set of arguments, 𝜸, and 𝐺𝐻𝐶𝐹 (𝑓(𝝈0), 𝑓𝑟𝑒𝑑 (𝜸))the ratio between 𝐺(𝑓(𝝈0)) and 𝑓𝑟𝑒𝑑 (𝜸). Following the same line, all the functions with the subscript 𝐻𝐶𝐹 are analogously modified by the 𝑓𝑟𝑒𝑑 (𝜸)in the rest of this text. From a geometrical perspective, the inequality (6) may be interpreted as the region limited by a bounding surface, the so-called damage surface, in the undamaged stress space, i.e., the space of effective stresses [18]. The definition of such a surface depends on the choice of a stress norm function that suits the response of the material to be modeled. In this work, the investigated component is constituted by metals alloys, which are pressure-insensitive materials, therefore, the von Mises equivalent stress 𝑓(𝝈0)=√3𝐽2,(7) is used as the stress norm function here, with 𝐽2referring to the second invariant of the deviatoric stress tensor. In addition to that, 𝑓𝑟𝑒𝑑 (𝜸)in inequality (6) consists of an irreversible function ranging from 1 to nearly 0 that, ultimately, acts amplifying the effect of the stress norm function at a given material point due to the application of cyclic loads. In the general form, 𝜸={𝑁𝑐, 𝑅, 𝜎𝑚𝑎𝑥}, so that the 𝑓𝑟𝑒𝑑 function is defined according to the number of cycles, 𝑁𝑐, the stress ratio, 𝑅= 𝜎𝑚𝑖𝑛 𝜎𝑚𝑎𝑥 , and maximum equivalent stress, 𝜎𝑚𝑎𝑥, resulting from the applied load. A more detailed description of this function is provided in the next subsection. The combination of the stress norm concept with a fatigue state function that is sensitive 𝑁𝑐,𝑅and 𝜎𝑚𝑎𝑥 into a single threshold damage function enables the present constitutive model to naturally handle Engineering Structures 311 (2024) 118198 4 L.A. Gonçalves Junior et al. Fig. 2. Uniaxial stress–strain curve for damage [15]. multi-axial stress states while also accounting for the mean stress effect on the material response under cyclic loads. Apart from a threshold function, the full characterization of a dissipative constitutive model requires the definition of the evolution laws of its internal variables. In this particular case, the evolution law for the damage variable, 𝑑, follows the format proposed by Simo and Ju [18]  𝑑=𝜇 ⋅ 𝜕F(𝑓(𝝈0),0, 𝑓𝑟𝑒𝑑 (𝜸)) 𝜕𝑓 (𝝈0)=𝜇 ⋅ 𝜕𝐺𝐻𝐶𝐹 (𝑓(𝝈0), 𝑓𝑟𝑒𝑑 (𝜸)) 𝜕𝑓 (𝝈0);  0=𝜇, (8) where 𝜇 is a non-negative scalar-valued parameter called damage consistency factor which allows the definition of loading/unloading conditions according to the Kuhn–Tucker relations 𝜇 ≥0; F(𝑓(𝝈0),0, 𝑓𝑟𝑒𝑑 (𝜸))≤0; 𝜇 ⋅F(𝑓(𝝈0),0, 𝑓𝑟𝑒𝑑 (𝜸))= 0.(9) This consistency factor may be easily obtained by combining the threshold damage function (6), the evolution law of Eq. (8) and the Kuhn–Tucker conditions (9). Thus, it is given by 𝜇 = 𝑓𝐻𝐶𝐹 (𝝈0, 𝑓𝑟𝑒𝑑 (𝜸))= 0.(10) Eq. (10) reveals that the damage effective threshold may be interpreted as a historical variable that stores the maximum modified stress norm reached at a given material point, i.e. 0||𝑡=𝑚𝑎𝑥 {0, 𝑚𝑎𝑥 {𝑓𝐻𝐶𝐹 (𝝈0, 𝑓𝑟𝑒𝑑 (𝜸))|||𝑠}} 0≤𝑠≤𝑡. (11) where 0defines the initial damage threshold. Following this, the integration with respect to time of Eq. (8)1 combined with Eq. (10) gives rise to the damage internal variable, 𝑑, as follows 𝑑|𝑡=∫𝑡  𝐺𝐻𝐶𝐹 (𝑓(𝝈0), 𝑓𝑟𝑒𝑑 (𝜸))𝑑𝑡. (12) Eq. (12) may be explicitly integrated for a suitable monotonically increasing 𝐺𝐻𝐶𝐹 (𝑓(𝝈0), 𝑓𝑟𝑒𝑑 (𝜸))function. Thus, following the approach proposed by Gonçalves Junior et al. [15], 𝑑is explicitly obtained here so that it fulfills the hardening-exponential softening material response illustrated by the stress–strain curve of Fig. 2. Accordingly, 𝑑is computed, in Region II, through the interpolation of linear piecewise segments as follows 𝑑𝑗 II =𝑑𝑖−1 ⋅ 𝑖−1 0 𝑓𝑗 𝐻𝐶𝐹 (𝜎0, 𝑓𝑟𝑒𝑑 (𝜸)) ⋅ 𝑖 0−𝑓𝑗 𝐻𝐶𝐹 (𝜎0, 𝑓𝑟𝑒𝑑 (𝜸)) 𝑖 0−𝑖−1 0 +𝑑𝑖⋅ 𝑖 0 𝑓𝑗 𝐻𝐶𝐹 (𝜎0, 𝑓𝑟𝑒𝑑 (𝜸)) ⋅ 𝑓𝑗 𝐻𝐶𝐹 (𝜎0, 𝑓𝑟𝑒𝑑 (𝜸)) − 𝑖−1 0 𝑖 0−𝑖−1 0 , (13) where 𝑑𝑗 II, such that 𝑑𝑖−1 ≤𝑑𝑗 II ≤𝑑𝑖, is the damage corresponding to the modified stress norm 𝑓𝑗 𝐻𝐶𝐹 (𝜎0, 𝑓𝑟𝑒𝑑 (𝜸)). Here, 𝑖= 1 ∶ 𝑛+ 1 are the reference points at which the damages 𝑑𝑖−1 and the effective thresholds 𝑖−1 0are computed. In Region III, 𝑑is given by the following exponentially decaying function 𝑑III =𝐺(𝑓(𝝈0))III = 1 − 𝑛 𝑓𝐻𝐶𝐹 (𝜎0, 𝑓𝑟𝑒𝑑 (𝜸)) ⋅exp ⎧ ⎪ ⎨ ⎪ ⎩ 𝑛 𝐸 ⋅[𝑓𝐻𝐶𝐹 (𝜎0, 𝑓𝑟𝑒𝑑 (𝜸)) − 𝑛 0] 𝑛𝑛 0 2𝐸−𝑔𝑓III ⎫ ⎪ ⎬ ⎪ ⎭ , (14) where 𝑛and 𝑛 0are the damage thresholds in the real and effective spaces at the 𝑛th point, 𝐸the material Young’s modulus and 𝑔𝑓III the volumetric fracture energy corresponding to Region III, computed as follows 𝑔𝑓III =𝑔𝑓−{1 2 (𝜎0)2 𝐸+ 𝑛 ∑ 𝑖=2 [(𝑖+𝑖−1)⋅(𝑖 0−𝑖−1 0) 2𝐸]−𝑛𝑛 0 2𝐸}. (15) In Eq. (15),𝜎0is the uniaxial stress at the material proportional limit which defines the initial damage threshold, 0. In addition, 𝑔𝑓 represents the total volumetric fracture energy resulting from the ratio of the material fracture energy, 𝐺𝑓, and the characteristic length, 𝑙𝑐. This ensures the consistency between regularized and non-regularized models in terms of energy dissipated in the crack [19]. Finally, under damage evolution, i.e.,  𝑑 > 0, the tangent constitutive tensor relates the strain and stress rates. Thus, for the rate constitute equation expressed by  𝝈=C𝑡∶ 𝜺,(16) one may write the tangent operator as follows C𝑡=C𝑠−𝜕𝐺𝐻𝐶𝐹 (𝑓(𝝈0), 𝑓𝑟𝑒𝑑 (𝜸)) 𝜕𝑓 (𝝈0)⋅(C0∶𝜺)⊗𝜕𝑓 (C0∶𝜺) 𝜕𝜺,(17) where C𝑠is the secant constitutive tensor devised in Eq. (4). 2.3. Fatigue state function Following the rationale proposed by Oller et al. [11], the fatigue state function, 𝑓𝑟𝑒𝑑 (𝑁𝑐, 𝑅, 𝜎𝑚𝑎𝑥), is formulated here so that it benefits Engineering Structures 311 (2024) 118198 5 L.A. Gonçalves Junior et al. Fig. 3. Normalized Wöhler curve and 𝑓𝑟𝑒𝑑 as function of the number of cycles, 𝑁𝑐, for a given stress ratio 𝑅and maximum stress 𝜎𝑚𝑎𝑥. Source: Adapted from [13]. from the wide amount of fatigue data available in terms of Wöhler (S-N) curves. For this purpose, a S-N-R surface is first postulated through the analytical expression 𝑆(𝑅, 𝑁𝑐)=𝑆𝑡ℎ (𝑅)+[𝑆𝑢−𝑆𝑡ℎ (𝑅)]⋅exp [−𝛼𝑡(𝑅)⋅(log10 𝑁𝑐)𝛽𝑓],(18) where 𝑆𝑡ℎ (𝑅)is the fatigue limit function 𝑆𝑡ℎ (𝑅)∶⎧ ⎪ ⎨ ⎪ ⎩|𝑅|≤1⇒𝑆𝑡ℎ (𝑅)=𝑆𝑒+(𝑆𝑢−𝑆𝑒)⋅(1 + 𝑅 2)𝑆𝑡ℎ,𝑅1 |𝑅|>1⇒𝑆𝑡ℎ (𝑅)=𝑆𝑒+(𝑆𝑢−𝑆𝑒)⋅(1 + 𝑅 2𝑅)𝑆𝑡ℎ,𝑅2, (19) that establishes the minimum stress level above which cyclic loads induce fatigue degradation on a specific material point and 𝛼𝑡(𝑅)is a model parameter given by 𝛼𝑡(𝑅)∶⎧ ⎪ ⎨ ⎪ ⎩|𝑅|≤1⇒𝛼𝑡(𝑅)=𝛼𝑓+(1 + 𝑅 2)⋅AUX R1 |𝑅|>1⇒𝛼𝑡(𝑅)=𝛼𝑓−(1 + 𝑅 2𝑅)⋅AUX R2. (20) Additionally, 𝑆𝑢and 𝑆𝑒represent the material ultimate strength and fatigue limit for 𝑅= −1, respectively, whereas 𝑆𝑡ℎ,𝑅1,𝑆𝑡ℎ,𝑅2,𝛼𝑓,𝛽𝑓, AUX R1 and AUX R2 are material parameters to be calibrated from S–N experimental data. After the definition of the S-N-R surface, the expression of 𝑓𝑟𝑒𝑑 (𝑁𝑐, 𝑅, 𝜎𝑚𝑎𝑥)is then postulated according to the following exponential function 𝑓𝑟𝑒𝑑 (𝑁𝑐, 𝑅, 𝜎𝑚𝑎𝑥)= exp [−𝐵0(𝑅, 𝜎𝑚𝑎𝑥)⋅(log10 𝑁𝑐)𝛽2 𝑓],(21) such that, it intercepts the S-N-R surface normalized by the ultimate strength, i.e., 𝑆(𝑅, 𝑁𝑐)∕𝑆𝑢, at the number of cycles, 𝑁𝑓, related to the normalized maximum stress, 𝜎𝑚𝑎𝑥∕𝑆𝑢, as schematically shown in Fig. 3 for a given 𝑅, i.e., in a plane representation. The term 𝐵0in Eq. (21) is a model coefficient obtained by the substitution of the intersection point, (𝑁𝑓,𝑆(𝑅,𝑁𝑐) 𝑆𝑢), into this equation. Thus, it is given by 𝐵0(𝑅, 𝜎𝑚𝑎𝑥)= − ln (𝜎𝑚𝑎𝑥∕𝑆𝑢) (log10 𝑁𝑓)𝛽2 𝑓 ,(22) in which 𝑁𝑓can be determined imposing the 𝜎𝑚𝑎𝑥 value in Eq. (18), as follows 𝑁𝑓(𝑅, 𝜎𝑚𝑎𝑥)= 10 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ −1 𝛼𝑡(𝑅) ⋅ln[𝜎𝑚𝑎𝑥 −𝑆𝑡ℎ (𝑅) 𝑆𝑢−𝑆𝑡ℎ (𝑅)]1 𝛽𝑓⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭.(23) In a finite element framework, the whole formalism devised in this section at the material point level is translated into Gauss point (GP) calculations. 2.4. Advance in time strategy The constitutive framework presented in this section is destined for the modeling of fatigue cases in the high cycle domain, i.e., those in which the material is expected to withstand at least 104load cycles before its failure. Conceptually, this model is conceived to represent the progressive degradation of the material in a cycle-by-cycle manner. In practical terms, the application of this approach requires its implementation into a numerical framework, e.g., based on the finite element method. However, the realization of cycle-by-cycle HCF simulations is compromised by the high computational cost associated with even small numerical cases. Thus, the advancing in time strategy (AITS) algorithm first proposed by Oller et al. [11], further developed by Barbu et al. [17] and extended here is often used in conjunction with this constitutive model to enable the execution of these simulations at affordable analysis times. This algorithm consists of a load-tracking phase and a large increment one, as briefly summarized in the following. 2.4.1. Load-tracking phase In this phase, each load cycle is applied to the model according to a user-defined number of load increments, while the equivalent stress at each GP of the model is monitored. When, the respective maximum, 𝜎𝑚𝑎𝑥, and minimum, 𝜎𝑚𝑖𝑛, stresses are detected, the stress ratio, 𝑅is computed. Following that, the convergence of 𝑅at each GP is checked comparing its current value with the one calculated in the previous cycle. If they converge to constant values at each GP of the finite element mesh, then, the model reaches a stable condition, which occurs when the following inequality 𝜂=∑ 𝐺𝑃 ‖‖‖‖‖‖ 𝑅𝑗+1 𝐺𝑃 −𝑅𝑗 𝐺𝑃 𝑅𝑗+1 𝐺𝑃 ‖‖‖‖‖‖ ≤𝑡𝑜𝑙, (24) is fulfilled. Accordingly, 𝑡𝑜𝑙 is a user-defined convergence tolerance and 𝑅𝑗 𝐺𝑃 =𝜎𝑚𝑖𝑛 𝜎𝑚𝑎𝑥 |||| 𝑗 𝐺𝑃 the ‘‘𝑗-𝑡ℎ’’ stress ratio computed at the ‘‘GP-𝑡ℎ’’ Gauss point. Once this condition is reached, the large increment phase is activated. Engineering Structures 311 (2024) 118198 6 L.A. Gonçalves Junior et al. 2.4.2. Large increments phase The condition of stability given by the inequality (24) leads to constant values of 𝜎𝑚𝑎𝑥 and 𝑅GP point-wise so long no stress redistribution takes place in the model due to a damage evolution process. As a consequence of that, the number of cycles, 𝑁𝑐, becomes the only variable of the fatigue state function in this phase, i.e., 𝑓𝑟𝑒𝑑 (𝑁𝑐). Taking this into consideration, an advance in the analysis time may be performed accounting for the evolution of 𝑓𝑟𝑒𝑑 (𝑁𝑐)according to the corresponding jump in the number of cycles. After this, the condition of stability is checked again and, if the model remains stable, another jump is executed, otherwise the load-tracking phase is activated. Following this procedure, the AITS algorithm is able to detect any change in the loads and boundary conditions applied to the model adjusting the cycle-jumps accordingly. In addition, different types of cycle-jumps are applied by default depending on the model state. Prior to the damage initiation, one single jump takes the model to the number of cycles, 𝑁𝑑, defined by the mesh GP at which damage first starts. From this point on, user-defined jumps, 𝑁𝑢𝑠𝑒𝑟 𝑐, are applied whenever the condition of stability is reached. The expression for the calculation of the 𝑁𝑑value is devised in the following. 2.4.3. Cycle-jump increment to the damage initiation point According to stress–strain curve of Fig. 2, the initial damage threshold, 0=𝜎0, defines the beginning of the material non-linear behavior. Considering this, so long the modified stress norm function, 𝑓𝐻𝐶𝐹 (𝝈0, 𝑓𝑟𝑒𝑑 (𝜸)), at a given GP does not reach this threshold for the first time, the material remains in the elastic regime where the 𝑓𝑟𝑒𝑑 (𝑁𝑐)evolution depends only on the number of cycles, 𝑁𝑐. Consequently, damage only starts in this GP after the cycle 𝑁𝑑||𝐺𝑃 at which the 𝑓𝑟𝑒𝑑 (𝑁𝑑||𝐺𝑃 )function assumes the value, 𝜎𝑚𝑎𝑥 𝑓𝐻𝐶𝐹 (𝝈0,𝑓𝑟𝑒𝑑 (𝜸)). Thus, imposing this condition on Eqs. (21) and (22) combined and solving the resulting expression for 𝑁𝑑||𝐺𝑃 , one obtains 𝑁𝑑(𝑅, 𝜎𝑚𝑎𝑥)|||𝐺𝑃 =𝑁⎡⎢⎢⎣ 𝑙𝑛 (𝜎𝑚𝑎𝑥∕𝜎0) 𝑙𝑛 (𝜎𝑚𝑎𝑥∕𝑆𝑢)⎤⎥⎥⎦ 𝑓.(25) Finally, if all of the model GPs are yet to enter the material nonlinear regime for the first time, the AITS computes the cycle-jump to be applied as the minimum 𝑁𝑑||𝐺𝑃 value across all GPs, i.e., 𝑁𝑑= 𝑚𝑖𝑛 {𝑁𝑑||𝐺𝑃 }. Since the exponent of Eq. (25) pertains to the interval [0,1), then 𝑁𝑑< 𝑁𝑓, when the type of stress–strain curve for damage of Fig. 2 is considered. 3. Material calibration The material parameters required by the model presented in Section 2are calibrated through experimental data obtained from monotonic tensile and fatigue tests on coupon specimens. As described in the following, the calibration was performed here for the highstrength steel MPH780Y980T which was used as the base material in the manufacturing of the investigated lower control arm. 3.1. Monotonic tensile test The hardening–softening stress–strain curve for damage shown in Fig. 2 was calibrated through a standard unaxial tensile test [20]. For this purpose, the uniform gauge specimen schematically represented in Fig. 4 was employed. According to the performed test, 𝜎0= 778 MPa, was obtained as the stress limit for the material linear elastic response. Following this, the inelastic behavior of Region II was calibrated through curve fitting. For that, a 6th-order polynomial beginning at 𝜎0was fitted to the experimental curve. Then, this polynomial was evaluated for equally-spaced strain values. The stress–strain pairs obtained from this procedure were used to define the piecewise linear segments of Region II of the stress–strain curve for damage. Furthermore, the Young’s Fig. 4. Uniform gauge specimen with a thickness of 3.4 mm for the monotonic tensile testing. Dimensions are expressed in mm. Fig. 5. Comparison between numerical and experimental uniaxial stress–strain curves for damage. Fig. 6. Hourglass specimen with a thickness of 3.4 mm for the uniaxial fatigue testing. Dimensions are expressed in mm. modulus, 𝐸= 201 GPa, and Poisson’s ratio, 𝜈= 0.263, provided by the material supplier were adopted as remaining properties. Finally, the actual test was numerically reproduced to check the performed calibration. For this purpose, the constitutive law presented in Section 2was implemented in the open-source code KratosMultiphysics [21–23]. In addition, the universal preand post-processor GiD [24] was used to prepare the numerical model and visualize the results. Benefiting from symmetry, only one-eighth of the test specimen was modeled. The geometry was discretized with 21033 8-node hexahedral elements. A monotonically increasing load was applied as prescribed displacement to the specimen free end. Appropriate boundary conditions were set at the symmetry faces. Fig. 5 shows the stress–strain curves obtained numerically and through the actual test. As in the experiment, the average stress, numerically computed through the reaction force on symmetry plane 1, was considered in the numerical result. The excellent correlation between them both ensures a proper calibration of the material model for the condition of monotonic loading. Engineering Structures 311 (2024) 118198 7 L.A. Gonçalves Junior et al. Fig. 7. Finite element simulation to determine the stress concentration factor 𝐾𝑡of fatigue test specimen: (a) model mesh and (b) axial stress contour. 3.2. Fatigue test S–N curves resulting from uniaxial fatigue testing are traditionally represented in terms of either the amplitude, 𝜎𝑎=(1−𝑅) 2𝜎𝑚𝑎𝑥, or the maximum, 𝜎𝑚𝑎𝑥, nominal stresses [16]. It implies that, in any of these cases, the stresses are average quantities obtained as a result of the actuation force applied to the test specimen divided by its initial cross-sectional area. However, it is well established that the fatigue phenomenon is characterized by a permanent, progressive and localized degradation of the material properties [25], in which the real critical stresses occurring locally play a crucial role not only in the fatigue crack nucleation but also in its propagation. In line with that, the constitutive model presented in Section 2naturally accounts for this localization effect as the maximum and minimum equivalent stresses driving fatigue are computed at the material point level. Nevertheless, to be consistent with this constitutive framework, in particular, with the calculation of the number of cycles 𝑁𝑓given by Eq. (23), instead of using S–N points based on average stresses, the corresponding local critical values should be employed to calibrate the material parameters defining the fatigue S-N-R surface described by Eq. (18). However, measuring locally the critical stresses occurring in the specimen during the test execution is rather complicated. Alternatively, it is proposed here an approach to transform the average stresses coming from experiments into the corresponding critical values at the material point level. For this purpose, the stress concentration factor, 𝐾𝑡, that relates the average and critical stresses in the hourglass test specimen shown in Fig. 6 was first obtained by means of a simple simulation in which it is elastically pulled by a constant load. Analogous to the monotonic case, only one-eighth of the specimen geometry was considered in a model discretized with 10740 8-node hexahedral elements, as shown in Fig. 7(a). Load and boundary conditions also followed the same arrangement of the monotonic simulation, whereas the Young’s modulus, 𝐸= 201 GPa and Poisson’s ratio, 𝜈= 0.263 were provided as material properties. Accordingly, Fig. 7(b) illustrates the results for both, the stress evaluated at the most stressed model GP and the average stress computed from the reaction force on the symmetry plane 1 divided by the respective cross-sectional area. The ratio between the local and average values leads to a 𝐾𝑡of about 1.032. Following this, the stresses from the experimental S–N points are magnified by the obtained concentration factor, 𝐾𝑡, resulting in a set of S–N pairs defined at the material point level. Similarly, the fatigue limit, 𝑆𝑒, is modified by, 𝐾𝑡, whereas the ultimate strength, 𝑆𝑢, as a property related to the material monotonic response, is kept unaltered. Then, the S–N points with stress values above the limit of proportionality, 𝜎0, are neglected as they lie in the material inelastic regime. Finally, the calibration of the model parameters, 𝑆𝑡ℎ,𝑅1,𝑆𝑡ℎ,𝑅2, 𝛼𝑓,𝛽𝑓, AUX R1 and AUX R2, is performed through a curve fitting of the remaining S–N pairs. Fig. 8 summarizes schematically this procedure in which a set of experimental S–N points expressed in terms of average stresses are converted into the corresponding pairs at the material point level. The respective calibrated S–N curves are also shown in the same illustration. Fig. 8. Schematic representation of the experimental S–N points transformation into the corresponding pairs at the material point level for the calibration of the model parameters 𝑆𝑡ℎ,𝑅1,𝑆𝑡ℎ,𝑅2,𝛼𝑓,𝛽𝑓, AUX R1 and AUX R2. The calibration of the material parameters defining the S-N-R surface of Eq. (18) is performed using S–N pairs of at least two different stress ratios, 𝑅. S–N points coming from 𝑅= −1 and 𝑅= 0.1are employed for this purpose here. Since the calibration was carried out for |𝑅|≤1 (non-negative mean stress), only four out of the six fatigue material parameters were determined, namely: 𝛼𝑓,𝛽𝑓,𝑆𝑡ℎ,𝑅1and AUX R1. The material ultimate strength, 𝑆𝑢= 912 MPa, obtained from the monotonic test and the fatigue limit, 𝑆𝑒= 368, MPa were used as material properties. Before performing the curve fitting, both the average stresses from the S–N points and the fatigue limit were converted into the corresponding material point quantities by the stress concentration factor, 𝐾𝑡. Finally, a Matlab [26] code based on a built-in fitting function was implemented to perform the parametric adjustment. The flowchart of Fig. 9 summarizes this calibration procedure. Fig. 10 shows the S-N-R surface resulting from the performed calibration and also the material point S–N pairs used as reference for the curve fitting. Furthermore, this illustration also evidences the ability of this approach to naturally handle the experimental scatter intrinsic to the fatigue phenomenon. With all the material parameters set, fatigue simulations were carried out to check the ability of the calibrated model to reproduce the tendency exhibited by actual S–N curves obtained through the Basquin equation [27] expressed in the logarithmic form 𝑙𝑜𝑔(𝑁𝑐) = 𝑙𝑜𝑔(𝐶) − 𝑘⋅𝑙𝑜𝑔(𝑆),(26) for 𝑅= −1 and 𝑅= 0.1. In Eq. (21),𝑘and 𝐶are model coefficients fitted through the least squares method. The finite element model of Fig. 7(a) was also employed to perform these simulations, with the Engineering Structures 311 (2024) 118198 8 L.A. Gonçalves Junior et al. Fig. 9. Flowchart of the calibration procedure of the fatigue material parameters 𝛼𝑓,𝛽𝑓,𝑆𝑡ℎ,𝑅1and AUX R1 defining the S-N-R surface. Fig. 10. S-N-R surface resulting from the material parameters 𝛼𝑓,𝛽𝑓,𝑆𝑡ℎ,𝑅1and AUX R1 calibration. Parametric adjustment performed with converted S–N pairs of stress ratios R = −1 and R =0.1. Runout points at 2 × 106cycles are illustrated with arrows. loads applied as sinusoidal prescribed displacements of constant amplitude. Fig. 11 shows that the S–N points obtained numerically provide an accurate prediction of the curves originated from experimental ones, evidencing that the calibration executed at the material point level is able to adequately describe the behavior of actual S–N diagrams in which the stresses are typically computed as average quantities. Finally, Table 1 summarizes quantitatively the main features of the curve fitting performed for both cases, 𝑅= −1 and 𝑅= 0.1, as well as the corresponding statistical study. 4. Lower control arm fatigue assessment The LCA is a stamped component that, due to its complex shape, requires several steps to be manufactured, generally ranging from six to eight. The first step is performed through blanking operations in which the steel coil is cut in a transfer press, giving rise to the preshape of Fig. 12(a). Following this, it is formed with different dies Table 1 Main quantitative features of the designed S–N curves. Stress ratio (𝑅), Basquin coefficients (𝑘and 𝐶), fatigue limit (𝑆𝑒), standard deviation and scatter band (𝑠𝑁and 𝑇𝑁) related to the number of cycles 𝑁𝑐and failure probability (𝑃𝐴). 𝑅 𝑚 𝐶 𝑆𝑒[𝑀𝑃 𝑎]𝑠𝑛𝑇𝑁 𝑃𝐴= 50% 𝑃𝐴= 5% 𝑃𝐴= 95% 𝑃𝐴= 50% 𝑃𝐴= 5% 𝑃𝐴= 95% −1 8.988 1714 1552 1893 368 334 407 0.24 4.02 0.1 15.854 1559 1522 1596 652 637 668 0.10 1.80 to reach its final geometric shape including all the holes and flanges, as shown in Fig. 12(b). Then, the resulting component is washed to remove the stamping lubricants and visually inspected to ensure its geometrical conformity. Subsequently, the sleeve, schematically illustrated in Fig. 12(c), is welded to the LCA main body. After the welding operation, a painting layer is deposited on the whole component to protect it against environmental corrosion during its lifespan. Finally, Engineering Structures 311 (2024) 118198 9 L.A. Gonçalves Junior et al. Fig. 11. S–N curves obtained through the Basquin equation together with the numerical prediction of the S–N pairs for (a) 𝑅= −1 and (b) 𝑅= 0.1. The curves are represented in a log–log scale, the test runout S–N points at 2 × 106cycles are illustrated with arrows and 𝑃𝐴indicates the failure probability of each corresponding S–N curve. Fig. 12. LCA manufacturing operations: (a) blanking, (b) forming, (c) sleeve welding and (d) bushing and ball joint assembly. the bushings and ball joint of Fig. 12(d) are fixed to it before the final assembly in the vehicle suspension. The LCA is directly connected to the wheels and, as such, it plays a crucial role in the vehicle stability and steering. For this reason, it is classified as a safety component and, since it is submitted to several load cycles during the vehicle service life, its fatigue performance needs to be carefully assessed. Owing to that, the durability tests described in the following were carried out together with their respective computational simulations. As the numerical analyses presented in Section 3, these simulations were also performed using the open-source code Kratos-Multiphysics, whereas GiD was employed as preand postprocessor. Finally, the steel MPH780Y980T properties, as calibrated in that section, were adopted as material properties. They are summarized in Table 2. 4.1. General case The general assessment of the LCA durability focused on the investigation of the fatigue damage effects produced by operational cyclic loads, i.e., those carried by the component in the longitudinal direction. To perform the tests, a tailored experimental setup was built, as shown in the following together with the corresponding finite element model. 4.1.1. Experimental setup During operation, the LCA is subjected to longitudinal loads as a result of the vehicle acceleration and deceleration. To simulate this condition, the test bench illustrated in Fig. 13(a) was built. Accordingly, the LCA was fully fixed at points from 2 to 4 and in the out-of-plane direction at point 5. The anti-slippage washers shown in Fig. 13(b) were employed to avoid slipping at the fixture points. Finally, a cyclic load of ±7500 N was applied at point 1 by the corresponding hydraulic piston as a sinusoidal signal of 6 Hz. 4.1.2. Finite element model The finite element model shown in Fig. 14 was built to carry out the virtual counterpart of the fatigue general test described in the previous subsection. It was discretized with 282411 10-node tetrahedral Engineering Structures 311 (2024) 118198 16 L.A. Gonçalves Junior et al. [4] Topaç MM, Ercan S, Kuralay NS. Fatigue life prediction of a heavy vehicle steel wheel under radial loads by using finite element analysis. Eng Fail Anal 2012;20:67–79. http://dx.doi.org/10.1016/j.engfailanal.2011.10.007. [5] Abebe M, Koo B. Fatigue life uncertainty quantification of front suspension lower control arm design. Vehicles 2023;5(3):859–75. http://dx.doi.org/10. 3390/vehicles5030047. [6] Topaç MM, Günal H, Kuralay NS. Fatigue failure prediction of a rear axle housing prototype by using finite element analysis. Eng Fail Anal 2009;16(5):1474–82. http://dx.doi.org/10.1016/j.engfailanal.2008.09.016. [7] Sadiq M, Pachapuri A, Lingannavar Ravi G, Kelageri Nagaraj K, Phadate Kritesh K. Design and analysis of lower control arm of suspension system. Mater Today: Proc 2021;47:2949–56. http://dx.doi.org/10.1016/j.matpr.2021. 05.035. [8] Zanchini M, Longhi D, Mantovani S, Puglisi F, Giacalone M. Fatigue and failure analysis of aluminium and composite automotive wheel rims: Experimental and numerical investigation. Eng Fail Anal 2023;146:107064. http://dx.doi.org/10. 1016/j.engfailanal.2023.107064. [9] Ballo F, Frizzi R, Mastinu G, Mastroberti D, Previati G, Sorlini C. Lightweight design and construction of aluminum wheels. SAE Technical Papers, 2016-01-1575, 2016, http://dx.doi.org/10.4271/2016-01-1575. [10] Moustafa K, Hu Z, Mourelatos ZP, Baseski I, Majcher M. System reliability analysis using component-level and system-level accelerated life testing. Reliab Eng Syst Saf 2021;214:107755. http://dx.doi.org/10.1016/j.ress.2021.107755. [11] Oller S, Salomón O, Oñate E. A continuum mechanics model for mechanical fatigue analysis. Comput Mater Sci 2005;32(2):175–95. http://dx.doi.org/10. 1016/j.commatsci.2004.08.001. [12] Barbu LG, Oller S, Martinez X, Barbat AH. High-cycle fatigue constitutive model and a load-advance strategy for the analysis of unidirectional fiber reinforced composites subjected to longitudinal loads. Compos Struct 2019;220:622–41. http://dx.doi.org/10.1016/j.compstruct.2019.04.015. [13] Jiménez S, Barbu LG, Oller S, Cornejo A. On the numerical study of fatigue process in rail heads by means of an isotropic damage based high-cycle fatigue constitutive law. Eng Fail Anal 2022;131:105915. http://dx.doi.org/10.1016/j. engfailanal.2021.105915. [14] Granados JJ, XM, Barbu LG, Di Capua D. Fatigue prediction of composite materials, based on serial/parallel mixing theory. Compos Struct 2022;291:115516. http://dx.doi.org/10.1016/j.compstruct.2022.1155165. [15] Gonçalves LA, Jiménez S, Cornejo A, Barbu LG, Parareda S, Casellas D. Numerical simulation of a rapid fatigue test of high Mn-TWIP steel via a high cycle fatigue constitutive law. Int J Fatigue 2023;168:107444. http://dx.doi.org/10.1016/j. ijfatigue.2022.107444. [16] ASTM-E466-15. Standard practice for conducting force controlled constant amplitude axial fatigue tests of metallic materials. Technical report, West Conshohocken, PA: ASTM International; 2021, URL https://www.astm.org. [17] Barbu LG, Oller S, Martinez X, Barbat A. High cycle fatigue simulation: A new stepwise load-advancing strategy. Eng Struct 2015;97:118–29. http://dx.doi.org/ 10.1016/j.engstruct.2015.04.012. [18] Simo JC, Ju JW. Strainand stress-based continuum damage models – I. formulation. Int J Solids Struct 1987;23:821–40. http://dx.doi.org/10.1016/00207683(87)90083-7. [19] Barbat GB, Cervera M, Chiumenti M, Espinoza E. Structural size effect: Experimental, theoretical and accurate computational assessment. Eng Struct 2020;213:110555. http://dx.doi.org/10.1016/j.engstruct.2020.110555. [20] ISO-6892-1. Metallic materials – tensile testing. Part 1: Method of test at room temperature. Technical report, Vernier, Geneva: International Organization for Standardization; 2016, URL https://www.iso.org. [21] Dadvand P, Rossi R, Oñate E. An object-oriented environment for developing finite element codes for multi-disciplinary applications. Arch Comput Methods Eng 2010;17:253–97. http://dx.doi.org/10.1007/s11831-010-9045-2. [22] Dadvand P, Rossi R, Gil M, Martorell X, Cotela J, Juanpere E, et al. Migration of a generic multi-physics framework to HPC environments. Comput Fluid 2013;80:301–9. http://dx.doi.org/10.1016/j.compfluid.2012.02.004. [23] Ferrándiz VM, Bucher P, Zorrilla R, Rossi R, Cotela J, Cornejo A, et al. KratosMultiphysics/Kratos: release 9.2. 2022, http://dx.doi.org/10.5281/zenodo. 7084882. [24] Ribó R, Pasenau M, Escolano E, Ronda JS, Coll A. GiD user manual. CIMNE; 2007. [25] ASTM-E1823. Standard terminology relating to fatigue and fracture testing. Technical report, West Conshohocken, PA: ASTM International; 2013, URL https: //www.astm.org. [26] MATLAB. version 9.7.0 (R2019b). Natick, Massachusetts: The MathWorks Inc.; 2019. [27] Basquin OH. The exponential law of endurance tests. In: American society for testing and materials proceedings. Vol. 10, 1910, p. 625–30.