Uncoupled Fracture Indicator Dependent on the Third Invariant and the Level of Ductility: Applications in Predominant Shear Loading

Abstract

This contribution presents an improved ductile fracture indicator that is dependent on the normalized third invariant and the plastic strain at fracture, which is in turn dependent on the stress state of the material. Lemaitre’s damage model is used to numerically evaluate the location and level of displacement required to initiate ductile fracture of the alloy. Along with this analysis, the constitutive model based on Gao’s criterion was implemented, to which the fracture indicator was coupled. The proposed fracture indicator is considered a damage denominator function dependent on the normalized third invariant and the plastic strain at the fracture. This damage denominator function seeks to regulate the rate of evolution of the degradation of the material according to its state of stress and level of ductility, which makes the indicator, coupled with the post-processing step of Gao’s model, satisfactorily predict the correct level of force, the appropriate location, and the displacement necessary for the fracture of the material. The predictions made with the help of the indicator were compared to the performance of Lemaitre’s model. In addition, specimens that describe the low triaxiality ratio were evaluated using AISI 4340 alloy at different ductility levels. The new indicator proved to be more advantageous in predicting the mechanical behavior in the fracture of the material for different types of loads, allowing a good estimate of the displacement in the fracture and the levels of force.

1. Introduction

The correct description of the mechanical behavior in the fracture of materials allows the optimized and safe development of structures and mechanical components in different industry sectors. In this sense, investigations concerning ductile fracture are still a challenge for researchers and engineers worldwide. Determining the start and location of the fracture is of great importance for the design and dimensioning of these structures and components.

The main limitation of current models is the excessive dependence on calibration for specific stress states. Studies such as those by [1] show that, in low-stress triaxiality states, traditional indicators fail to estimate displacement at fracture. This work proposes a new indicator that incorporates the dependence of the normalized third invariant and plastic strain at fracture, aiming to improve predictability under predominant shear loads.

It is possible to verify that the stress triaxiality defines the state of stress generated in the material, via experimental data. Thus, the ranges of low-stress triaxiality are established, in which pure shear effects and combined loading states prevail and the region of high-stress triaxiality are where purely tractive effects predominate. The work developed by [1] presented an important advance in determining these effects in ductile materials, correlating the level of plastic strain in the fracture with the value of the stress triaxiality. In the literature, one can find several proposals for works that generate experimental data for the most diverse states of stress, such as tensile, compression and shear, or biaxial, where one may quantify combinations of tensile and shear, or even triaxial or multiaxial stress states, such as shear–tensile–compression. The work of [2,3] made fundamental contributions to understanding the effect of stress triaxiality and the third invariant on the behavior of ductile fracture. However, its application remains limited in cases of predominantly shear loads, justifying the need for a new indicator capable of improving predictions in these cases.

According to the study presented by [4], the classical theory of plasticity, applied to metallic materials, i.e., the approach based on the second deviatoric stress invariant tensor, assumes that the hydrostatic stress and the third deviatoric stress invariant tensor do not influence the plastic flow of the material. However, when it comes to ductile materials, these assumptions must be carefully evaluated, as these parameters define the stress state and can affect the plastic flow of the material. For example, it is known that the third invariant is related to the shape of the yield surface.

Cao et al. in 2014 [5] studied ductile fracture for low-stress triaxiality ratios. In this region, shear loads prevail and a modification of the Lemaitre damage model is proposed, to which the third invariant of the deviator tensor is incorporated through a parabolic function to quantify the dependence of the model. Application results were presented for two materials: one with a high elastic limit, and the other a zirconium alloy. The research results showed a comparison between the experimental observations and the numerical predictions, the values being very close for both materials evaluated, but different models were used to perform the numerical simulations because the materials studied present very different levels of plasticity.

The description of the plastic behavior and the prediction of the fracture site for ductile materials were also addressed by [6]. In this work, a model based on the second and third invariant was presented, with damage decoupled from exponential evolution. Smooth cylindrical and tubular specimens were analyzed, in which experimental tests were carried out in tensile, torsion, and biaxial (tensile/torsion) configurations. The model was tested on steel alloys used in gas pipelines, oil pipelines, and automotive applications. The results indicated a satisfactory description of the plastic behavior of the materials and an improvement in failure prediction through the evolution of the exponential decoupled damage. However, to consolidate this approach, it is essential to investigate further the effects of specific loading conditions, the model’s sensitivity to variations in the microstructure of the materials, and possible limitations in the calibration of the parameters in complex stress states. These aspects may impact the accuracy of fracture prediction and the applicability of the model in broader industrial scenarios.

In the proposals by [7,8,9], experiments were also carried out with rectangular specimens, reproducing biaxial states. Another work of great relevance, developed for materials with very complex behavior, was carried out by [10], in which experimental tests were carried out for DIN 1623 St12 steel. Although significant advances have been made in ductile fracture modeling, recent studies by [11,12] show that traditional models fail to predict fractures in combined stress states. Furthermore, experimental results obtained for the AISI 4340 alloy indicate divergences in the estimation of displacement and force in existing models. Thus, we propose a new indicator that incorporates the dependence of the third normalized invariant and plastic strain on fracture.

Refs. [11,13] performed experiments concerning damage and fracture in the negative region of the triaxiality ratio, that is, the effect of compression as a state of stress was considered in the evaluation of the behavior of the material. Furthermore, proposals for experimental tests under non-proportional load, and under triaxial conditions, were also addressed in the literature, as in the work presented by [14]. A study published by [15] presented a numerical model for ductile materials and proposed a formulation that incorporates damage into the flow function. In this way, plastic strains can be quantified, as well as the nucleation and propagation of micro-defects through the internal variable of damage. The model was tested on a microscale and presents itself as a great alternative, since the damage detection, as well as the fracture mechanisms at the micro-level, are too complex to be identified through experimental techniques.

When checking the literature, it can be seen that, despite the significant advance in research and the numerous contributions related to the topic, there is still a need to continue investigating the behavior of materials to better predict the point and location of the fracture, as the models proposed are still imprecise, are highly dependent on the calibration point, and only provide answers for specific materials. This work presents a new ductile fracture indicator based on a damage denominator function dependent on the normalized third invariant and on the ductility of the material. The indicator was associated with Gao’s model and the simulations were performed for the rectangular specimens, considering only the region of low-stress triaxiality.

This article presents an improved ductile fracture indicator for better prediction of material failure under predominant shear loading. A modification of the damage denominator function is performed, running the coupling of the indicator to the post-processing step of Gao’s model. The results were validated through experimental tests and numerical simulations using AISI 4340 alloy (ref. [16]), a material widely used in the offshore industry, especially in components for deep water oil and gas extraction. The choice of the normalized alloy and the annealed alloy was based on the differences in the deformation levels, evaluated through the accumulated plastic deformation and the true deformation at fracture, allowing for a more detailed analysis of the fracture indicator in different stress states. To characterize the material elastically and plastically, rectangular specimens under low-stress triaxiality conditions were considered. The academic numerical simulation tool Hyplas, based on the finite element method, was used. Hyplas considers large deformations, implicit integration of the constitutive model, nonlinear isotropic hardening, associative plasticity, and the Gao yield criterion. This demonstrated that the proposed ductile fracture indicator, coupled with the post-processing step of the Gao model, achieved high efficiency in predicting fracture displacement and estimating force levels [4].

2. Theoretical Review

In this section, the theoretical aspects that serve as a basis for the formulation of the proposed damage indicator are presented. Gao’s model and the equations associated with its formulation are described.

In the description of the ductile fracture of materials, there are important parameters that allow the study of this phenomenon and some fundamental definitions are presented in the context of the research into the behavior of materials. The stress triaxiality and the normalized third invariant of the deviator tensor are defined (refs. [4,17]).

In problems that address situations regarding the elastoplastic behavior of materials, it is necessary to describe parameters that are widely used in the definition of the stress state of a material point. Several works have presented contributions in the sense of proving the influence of parameters that describe the mechanical aspects of materials.

Gao-Based Model

In this study, we propose the coupling of the new ductile fracture indicator in the post-process stage of the model by [18]. It is important to note that the Lemaitre model, which is based on the von Mises flow criterion, while being extremely conservative in predicting ductile fracture in shear conditions, also overestimates the reaction forces. Gao’s model in its complete form is represented mathematically as follows:

$$\varphi =c{\left[a{I_1}^6+27{J_2}^3+b{J}_3\right]}^{\frac{1}{6}}-{\sigma }_{y0}-H\left({\overline{\epsilon }}^p\right){\overline{\epsilon }}^p,$$

(1)

where $I_1$ represents the first invariant of the Cauchy tensor and is responsible for coupling the effect of hydrostatic stress on the yield criterion. $J_2$ and $J_3$ are, respectively, the second and third invariants of the deviator tensor $,\left({\overline{\epsilon }}^p\right)$ corresponds to the isotropic hardening modulus, and the $c$ term is a material parameter calculated through the following equation:

$$c={\left[a+\frac{4}{728}b+1\right]}^{-\frac{1}{6}},$$

(2)

in which $a$ and $b$ are adjustment parameters of the Gao criterion. For this work, the term $a=0$ is considered because, under shear conditions, the volumetric term can be neglected. Thus, Gao’s criterion is now written in the following form:

$$\varphi =c{\left[27{J_2}^3+b{J}_3\right]}^{\frac{1}{6}}-{\sigma }_{y0}-H\left({\overline{\epsilon }}^p\right){\overline{\epsilon }}^p.$$

(3)

The parameter $c$ is described by the following expression:

$$c={\left[{\displaystyle \frac{4}{728}}b+1\right]}^{-\frac{1}{6}}.$$

(4)

The evolution of the equivalent plastic strain can be determined by means of the equivalence of the plastic work:

$${\dot{\overline{\epsilon }}}^p=\dot{\mathsf{\gamma }}{\displaystyle \frac{\mathit{\sigma }:\mathit{N}}{{\sigma }_y}} .$$

(5)

In this context, the law of plastic flow can be defined as follows:

$${\dot{\mathit{\epsilon }}}^p\equiv \dot{\gamma }\mathit{N}=\dot{\gamma }c\left[\frac{1}{6}{\left(\kappa \right)}^{-\frac{5}{6}} {\displaystyle \frac{\partial \kappa }{\partial \mathit{\sigma }}}\right] ,$$

(6)

In Equation (6), ${\dot{\mathit{\epsilon }}}^p$ is the rate of evolution of the plastic strain tensor and corresponds to the plastic multiplier.

Considering associative plasticity, the plastic flow vector $\left(\mathit{N}\right)$, as shown by [19], is written as follows:

$$\mathit{N}\equiv {\displaystyle \frac{\partial \varphi }{\partial \mathit{\sigma }}}=\mathrm{c}\frac{1}{6}{\left(\kappa \right)}^{-\frac{5}{6}} {\displaystyle \frac{\partial \kappa }{\partial \mathit{\sigma }}} .$$

(7)

The coefficient $\kappa$ is a parameter defined as follows:

$$\kappa =27{J_2}^3+\mathrm{b}{J}_3 .$$

(8)

The derivative of the parameter related to the Cauchy stress tensor can be seen in the expression below:

$${\displaystyle \frac{\partial \kappa }{\partial \mathit{\sigma }}}=81{J_2}^2{\displaystyle \frac{\partial J_2}{\partial \mathit{\sigma }}}+\mathrm{b}{\displaystyle \frac{\partial J_3}{\partial \mathit{\sigma }}} ,$$

(9)

with

$$\begin{gathered} {\displaystyle \frac{\partial J_2}{\partial \mathit{\sigma }}}=\mathit{S}, \\ {\displaystyle \frac{\partial J_3}{\partial \mathit{\sigma }}}=\det \mathit{S}\left({\mathit{S}}^{-T}:{\mathbb{I}}^d\right) \end{gathered}$$

(10)

where ${\mathbb{I}}^d={\mathbb{I}}^4-\frac{1}{3}\mathit{I}\otimes \mathit{I}$ is the fourth-order identity tensor, with $\mathit{I}$ as the second-order identity tensor.

Box 1 presents the elements for determining Gao’s mathematical model, considering isotropic hardening.

Box 1. Mathematical model of Gao modified with isotropic hardening.

(i)

Additive decomposition of total strain:

$$\mathit{\epsilon }={\mathit{\epsilon }}^e+{\mathit{\epsilon }}^p$$

(ii)

Elastic law:

$$\mathit{\sigma }={\mathbb{D}}^e:{\mathit{\epsilon }}^e$$

(iii)

Yield function:

$$\varphi =c{\left(27{J_2}^3+b{J}_3\right)}^{\frac{1}{6}}-{\sigma }_{y0}-H\left({\overline{\epsilon }}^p\right){\overline{\epsilon }}^p$$

$$c={\left[\frac{4}{728}b+1\right]}^{-\frac{1}{6}}$$

$$\kappa =27{J_2}^3+b{J}_3$$

(iv)

Plastic flow rule:

$${\dot{\mathit{\epsilon }}}^p\equiv \dot{\gamma }\mathit{N}=\dot{\gamma }\mathit{N}=\dot{\gamma }\left[c\frac{1}{6}{\left(\kappa \right)}^{-\frac{5}{6}} {\displaystyle \frac{\partial \kappa }{\partial \mathit{\sigma }}}\right]$$

$${\displaystyle \frac{\partial \kappa }{\partial \mathit{\sigma }}}=81{J_2}^2\mathit{S}+\mathrm{b} \det \mathit{S}\left({\mathit{S}}^{-T}:{\mathbb{I}}^d\right)$$

(v)

Evolution of the accumulated plastic strain:

$${\dot{\overline{\epsilon }}}^p=\dot{\gamma }{\displaystyle \frac{\mathit{\sigma }:\mathit{N}}{{\sigma }_y}}$$

(vi)

Loading/unloading rule:

$\dot{\gamma }\ge 0, \varphi \le 0, \dot{\gamma }\varphi =0$

3. Proposal Fracture Indicator and Numerical Strategy

3.1. Uncoupled Fracture Indicator

The contribution of [20] included the first proposals in the literature that presented a modification of Lemaitre’s model, regarding the construction of a damage denominator function dependent on the stress state, so the law of Lemaitre’s damage evolution is represented as follows:

$$\dot{D}=\dot{\gamma }{\displaystyle \frac{1}{\left(1-D\right)}}{\left({\displaystyle \frac{-Y}{S\left(\eta ,\xi \right)}}\right)}^s.$$

(11)

where $\dot{D}$ is the damage evolution rate, $-Y$ represents the energy released due to damage, $s$ is the damage exponent, and $S\left(\eta ,\xi \right)$ is the damage denominator function, in this case dependent on the stress triaxiality and the normalized third invariant. (The definitions of these parameters, as well as studies of their influence on the mechanical behavior of materials, can be found in the works of [21,22]). Ref. [23] demonstrated that the proposed damage denominator function was flawed for certain conditions and materials because the damage denominator function proposed by [20] does not distinguish the level of ductility of the materials, as can be seen in Equation (4):

$$S={\displaystyle \frac{S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{3\left|\eta \right|+\frac{S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{S_0}\left(1-{\xi }^2\right)}},$$

(12)

where $S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$ represents the damage denominator calibrated through an experimental tensile test on a smooth cylindrical specimen and $S_0$ is the damage denominator calibrated through a pure shear test.

The results presented by [1], demonstrate that the description of the mechanical behavior of ductile materials presents a sensitivity to the effects of the normalized third invariant. Refs. [23,24] show that, for materials with different levels of ductility, the models’ traditional methods do not allow for adequate estimates to be made either for the levels of reaction forces or for the fracture site. To improve the description of materials, it is necessary to consider the characteristics of the stress state (ref. [25]). This work presents a new ductile fracture indicator, which incorporates the accumulated plastic strain and the plastic strain at fracture. The contribution made by [26], in which a ductile fracture indicator was proposed, can be mathematically represented as follows:

$$I={\int }_0^{{\overline{\epsilon }}_f^p}{\left[{\displaystyle \frac{1}{S}}\left({\displaystyle \frac{q^2}{6G}}+{\displaystyle \frac{p^2}{2K}}\right)\right]}^{s}d{\overline{\epsilon }}^p,$$

(13)

where $G$ and $K$ represent the shear modulus of elasticity and the volumetric modulus of elasticity, respectively. The term $q$ is the equivalent stress, where $p$ is the hydrostatic stress and $S$ corresponds to the damage denominator.

The indicator, which is derived from the proposal by [19], also considers a constant damage denominator and proposes the integration of the total damage work. As with the Lemaitre model, the [26] indicator presents a prediction of ductile fracture in disagreement with the experimental observations for stress states that are different from the stress state used to calibrate the indicator. Therefore, following the proposal by [20], we propose replacing the constant value of denominator by a function dependent on the normalized third invariant ($\xi )$ and on the ductility, represented here by the plastic strain at fracture (${\overline{\epsilon }}_f^p$). In this sense, the [26] indicator can be rewritten as follows:

$$I={\int }_0^{{\overline{\epsilon }}_f^p}{\left[{\displaystyle \frac{1}{f(\xi ,{\overline{\epsilon }}^p)}}\left({\displaystyle \frac{q^2}{6G}}+{\displaystyle \frac{p^2}{2K}}\right)\right]}^{s}d{\overline{\epsilon }}^p ,$$

(14)

where $f(\xi ,{\overline{\epsilon }}^p)$ is the damage denominator function. It is important to highlight that, when there is a plane stress state, a condition observed for all rectangular specimens manufactured here, only the stress triaxiality or the normalized third invariant can characterize the stress state of the material. Furthermore, for Lemaitre, the damage exponent for most ductile materials can be assumed as equal to 1. Thus, one can rewrite the [26] indicator as follows:

$$I={\int }_0^{{\overline{\epsilon }}_f^p}{\displaystyle \frac{-Y}{f(\xi ,{\overline{\epsilon }}^p)}}d{\overline{\epsilon }}^p,$$

(15)

where the energy released due to damage, $-Y$, is determined as follows:

$$-Y={\displaystyle \frac{q^2}{6G}}+{\displaystyle \frac{p^2}{2K}} ,$$

(16)

To define the damage denominator function, we initially seek to build a function that recovers the behavior of the material at the two defined calibration points: (a) tensile in a smooth cylindrical specimen, and (b) shear in a rectangular specimen. Thus, it is possible to establish a relationship as follows:

$$f\left(\xi \right)=S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\xi }^2+S_0\left(1-{\xi }^2\right),$$

(17)

where $f\left(\xi \right)$ is the damage denominator function, $S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$ the denominator of calibration damage in pure traction, and $S_0$ the denominator of damage calibrated in pure shear. Analyzing the behavior of the function,

(a)

Pure traction: in this case, $\xi =1$ and the function returns $f\left(\xi \right)=S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.};$

(b)

Pure shear: in this case, $\xi =0$ and the function returns $f\left(\xi \right)=S_0;$

(c)

Tensile/shear combinations: in this case, $0<\xi <1$ and the function returns values in the range of $S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}<f\left(\xi \right)<S_0.$

To consider the ductility of the material, we propose adding a term dependent on the accumulated plastic strain and the plastic strain at fracture. In the expression, it is necessary to include a term that is dependent on the accumulated plastic strain, so the function can be written as follows:

$$f\left({\overline{\epsilon }}^p\right)=\mathrm{T}{\displaystyle \frac{{\overline{\epsilon }}^p}{{\overline{\epsilon }}_f^p}},$$

(18)

where $\mathrm{T}$ is a constant to be calibrated, ${\overline{\epsilon }}^p$ is the accumulated plastic strain, and ${\overline{\epsilon }}_f^p$ is the plastic strain at fracture that is obtained as a consequence of calibrating the isotropic hardening curve of the material. Although calibration can be performed for any stress state, in this research the combined tensile and shear state was considered as $(\mathrm{T}=S_0-S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.})$, thereby limiting the number of calibration parameters. The complete analysis and sensitivity evaluation for these stress states are presented in [24]. When adding the two effects through Equations (17) and (18), we have the following:

$$f\left(\xi ,{\overline{\epsilon }}^p\right)=S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\xi }^2+S_0\left(1-{\xi }^2\right)+\left(S_0-S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right){\displaystyle \frac{{\overline{\epsilon }}^p}{{\overline{\epsilon }}_f^p}}\left({1-\xi }^2\right).$$

(19)

Note that the term ${(1-\xi }^2)$ is multiplied by the ductility term because, at this stage of the work, only the region of low-stress triaxiality is studied. Therefore, the damage denominator function has the following behavior:

(a)

Pure traction: in this case, $\xi =1$ and the function returns $f\left(\xi \right)=S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}.$

(b)

Pure shear: in this case, $\xi =0$ and the function returns $f\left(\xi \right)\approx 2S_0-S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}.$

It is important to note that the fact that the damage denominator function does not return the value of $S_0$, when $\xi =0,$ does not necessarily make the indicator inaccurate under shear conditions. In fact, the value of $\xi$ for the rectangular specimen in pure shear changes dramatically throughout the deformation process, so the additional ductility term ends up being compensated by the variation of $\xi$. Under these conditions, the damage denominator function starts from a value equal to $S_0$ and evolves to values close to $2S_0-S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$.

(c)

Tensile/shear combinations: in this case, $0<\xi <1$ and the function returns values in the range of $S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}<f\left(\xi \right)<\left({2S}_0-S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right).$

After analyzing the behavior of the normalized third invariant in the different states of stress, an expression is obtained from the substitution of Equation (19) for Equation (15), and the mathematical representation of the new proposal for the ductile fracture indicator is shown below:

$$I={\int }_0^{{\overline{\epsilon }}_f^p}{\displaystyle \frac{-Y}{\left[S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\xi }^2+S_0\left(1-{\xi }^2\right)+\left(S_0-S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)\frac{{\overline{\epsilon }}^p}{{\overline{\epsilon }}_f^p}\left({1-\xi }^2\right)\right]}}d{\overline{\epsilon }}^p.$$

(20)

It is important to emphasize that the definition of the constant T as the difference between the two damage denominators was chosen to ensure that the evolution of the fracture indicator correctly fits the characteristics of the material’s stress state. This approach allows the model to adequately recover the damage behaviors observed in pure shear (ξ = 0) and pure tensile (ξ = 1) conditions, ensuring that the damage denominator evolves coherently with the loading regime. For situations with high triaxiality and predominant tensile loading, the constant T could be refined by introducing a triaxiality-dependent adjustment factor or an additional term based on hydrostatic stress. This approach would improve the accuracy of fracture predictions in these regimes, avoiding overestimations or underestimations of displacements and force levels.

3.2. Integration Strategy for Gao-Based Model and Damage Indicator

The integration methodologies used for the model by [18], as well as the proposed damage indicator, are described below. Corresponding systems of nonlinear equations are presented as well as linearization and solution using the Newton–Raphson method.

For the integration of Gao’s model, the following system of nonlinear equations needs to be solved according to the Newton–Raphson method:

$$\left\{\begin{array}{c}{R}_{{\mathit{S}}_{n+1}}={\mathit{S}}_{n+1}-{\mathit{S}}_{n+1}^{trial}+2G\mathsf{\Delta }\gamma {\mathit{N}}_{n+1}\\ R_{\mathsf{\Delta }\gamma }=c{\left(27{J_{2 n+1}}^3+b{J}_{3 n+1}\right)}^{\frac{1}{6}}-{\sigma }_{y0}-H\left({\overline{\epsilon }}^p\right) {{\overline{\epsilon }}^p}_{n+1}\\ R_{{{\overline{\epsilon }}^p}_{n+1}}={{\overline{\epsilon }}^p}_{n+1}-{{\overline{\epsilon }}^p}_n-\mathsf{\Delta }\gamma {\displaystyle \frac{{\mathit{S}}_{n+1}:{\mathit{N}}_{n+1}}{{\sigma }_y}}\end{array}\right.$$

(21)

The system of equations is solved iteratively using the Newton–Raphson method. A summary of the iterative process is shown in Box 2.

Box 2. Algorithm for solving the nonlinear system (Newton–Raphson).

(i)

Given the trial stress state as initial step, $k:=0$, initial parameters: ${\mathit{S}}_{n+1}^{\left(0\right)}={\mathit{S}}_{n+1}^{trial}, {∆\gamma }^{\left(0\right)}=0, {{{\overline{\epsilon }}^p}_{n+1}}^{\left(0\right)}={{\overline{\epsilon }}^p}_n$ and residual equations:

(ii)

$\left[\begin{array}{c}\begin{array}{c}{R}_{{\mathit{S}}_{n+1}}\left({\mathit{S}}_{n+1}, \mathsf{\Delta }\gamma ,{{\overline{\epsilon }}^p}_{n+1}\right)\\ R_{\mathsf{\Delta }\gamma }\left({\mathit{S}}_{n+1}, \mathsf{\Delta }\gamma ,{{\overline{\epsilon }}^p}_{n+1}\right)\end{array}\\ R_{{{\overline{\epsilon }}^p}_{n+1}}\left({\mathit{S}}_{n+1}, \mathsf{\Delta }\gamma ,{{\overline{\epsilon }}^p}_{n+1}\right)\end{array}\right]=\left[\begin{array}{c}\begin{array}{c}{\mathit{S}}_{n+1}-{\mathit{S}}_{n+1}^{trial}+2G\mathsf{\Delta }\gamma {\mathit{N}}_{n+1}\\ c{\left(27{J_{2 n+1}}^3+b{J}_{3 n+1}\right)}^{\frac{1}{6}}-{\sigma }_{y0}-H\left({\overline{\epsilon }}^p\right){{\overline{\epsilon }}^p}_{n+1}\end{array}\\ {{\overline{\epsilon }}^p}_{n+1}-{{\overline{\epsilon }}^p}_n-\mathsf{\Delta }\gamma {\displaystyle \frac{{\mathit{S}}_{n+1}:{\mathit{N}}_{n+1}}{{\sigma }_y}}\end{array}\right]$

(iii)

Solve the system of equations:

$${\left[\begin{array}{cc}{\displaystyle \frac{\partial R_{{\mathit{S}}_{n+1}}}{\partial {\mathit{S}}_{n+1}}}& \begin{array}{cc}{\displaystyle \frac{\partial R_{{\mathit{S}}_{n+1}}}{\partial \mathsf{\Delta }\gamma }}& {\displaystyle \frac{\partial R_{{\mathit{S}}_{n+1}}}{\partial {{\overline{\epsilon }}^p}_{n+1}}}\end{array}\\ \begin{array}{c}{\displaystyle \frac{\partial R_{\mathsf{\Delta }\gamma }}{\partial {\mathit{S}}_{n+1}}}\\ {\displaystyle \frac{\partial R_{{{\overline{\epsilon }}^p}_{n+1}}}{\partial {\mathit{S}}_{n+1}}}\end{array}& \begin{array}{c}\begin{array}{cc}{\displaystyle \frac{\partial R_{\mathsf{\Delta }\gamma }}{\partial \mathsf{\Delta }\gamma }}& {\displaystyle \frac{\partial R_{\mathsf{\Delta }\gamma }}{\partial {{\overline{\epsilon }}^p}_{n+1}}}\end{array}\\ \begin{array}{cc}{\displaystyle \frac{\partial R_{{{\overline{\epsilon }}^p}_{n+1}}}{\partial \mathsf{\Delta }\gamma }}& {\displaystyle \frac{\partial R_{{{\overline{\epsilon }}^p}_{n+1}}}{\partial {{\overline{\epsilon }}^p}_{n+1}}}\end{array}\end{array}\end{array}\right]}^k.{\left[\begin{array}{c}\begin{array}{c}\delta {\mathit{S}}_{n+1}\\ \delta \mathsf{\Delta }\gamma \end{array}\\ \delta {{\overline{\epsilon }}^p}_{n+1}\end{array}\right]}^{k+1}=-{\left[\begin{array}{c}\begin{array}{c}{R}_{{\mathit{S}}_{n+1}}\left({\mathit{S}}_{n+1},\mathsf{\Delta }\gamma ,{{\overline{\epsilon }}^p}_{n+1}\right)\\ R_{\mathsf{\Delta }\gamma }\left({\mathit{S}}_{n+1},\mathsf{\Delta }\gamma ,{{\overline{\epsilon }}^p}_{n+1}\right)\end{array}\\ R_{{{\overline{\epsilon }}^p}_{n+1}}\left({\mathit{S}}_{n+1},\mathsf{\Delta }\gamma ,{{\overline{\epsilon }}^p}_{n+1}\right)\end{array}\right]}^k$$

(iv)

New guess for ${\mathit{S}}_{n+1}, ∆\gamma \mathrm{and} {{\overline{\epsilon }}^p}_{n+1}$:

$\begin{array}{c}{{\mathit{S}}_{n+1}}^{\left(k+1\right)}={{\mathit{S}}_{n+1}}^{\left(k\right)}+{\delta {\mathit{S}}_{n+1}}^{(k+1)} ; { ∆\gamma }^{\left(k+1\right)}={∆\gamma }^{\left(k\right)}+{\delta \mathsf{\Delta }\gamma }^{(k+1)} ;\\ {{{\overline{\epsilon }}^p}_{n+1}}^{\left(k+1\right)}={{{\overline{\epsilon }}^p}_{n+1}}^{\left(k\right)}+{\delta {{\overline{\epsilon }}^p}_{n+1}}^{(k+1)} ;\end{array}$

(v)

Update variables:

$\begin{array}{c}{\mathit{\epsilon }}_{n+1}^e={\left[{\mathbf{D}}^e\right]}^{-1}:{\mathit{\sigma }}_{n+1} ; {\mathit{\sigma }}_{n+1}={\mathit{S}}_{n+1}+p_{n+1}\mathit{I}\end{array}$

(vi)

Verify the convergence: $\tilde{\mathsf{\Phi }}=c{\left(27{J_{2 n+1}}^3+b{J}_{3 n+1}\right)}^{\frac{1}{6}}-{\sigma }_{y0}-H\left({\overline{\epsilon }}^p\right){{\overline{\epsilon }}^p}_{n+1, }$

if $\left|\tilde{\mathsf{\Phi }}\right|\le tolerance ⟹$ End, otherwise return to step (ii)

End

The numerical strategy for integrating the damage indicator proposed in the post-processing stage of Gao’s model is presented in Box 3.

Box 3. Integration of the damage indicator.

(i)

Run Box 2 for Gao’s model to determine: ${\mathit{S}}_{\mathit{n}+1}, ∆\mathit{\gamma } \mathrm{and} {{\overline{\mathit{\epsilon }}}^{\mathit{p}}}_{\mathit{n}+1}$

(ii)

Calculate ${\mathit{\xi }}_{\mathit{n}+1}$, according to Equation (18) to determine $\mathit{f}\left({\mathit{\xi }}_{\mathit{n}+1},{{\overline{\mathit{\epsilon }}}^{\mathit{p}}}_{\mathit{n}+1}\right)$:

$$\mathit{f}\left({\mathit{\xi }}_{\mathit{n}+1},{{\overline{\mathit{\epsilon }}}^{\mathit{p}}}_{\mathit{n}+1}\right)={\mathit{S}}_0-\left({\mathit{S}}_0-{\mathit{S}}_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right){{\mathit{\xi }}_{\mathit{n}+1}}^2+\left({\mathit{S}}_0-{\mathit{S}}_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)\left({\displaystyle \frac{{{\overline{\mathit{\epsilon }}}^{\mathit{p}}}_{\mathit{n}+1}}{{{\overline{\mathit{\epsilon }}}^{\mathit{p}}}_{\mathit{f}}}}\right)\left(1-{{\mathit{\xi }}_{\mathit{n}+1}}^2\right)$$

(iii)

To determine the damage in pseudo-time ${\mathit{t}}_{\mathit{n}+1}$

$${\mathit{I}}_{(\mathit{n}+1)}={\mathit{I}}_{\left(\mathit{n}\right)}+\left[{\displaystyle \frac{1}{\mathit{f}\left({\mathit{\xi }}_{\mathit{n}+1},{{\overline{\mathit{\epsilon }}}^{\mathit{p}}}_{\mathit{n}+1}\right)}}\left({\displaystyle \frac{{{\mathit{q}}_{\mathit{n}+1}}^2}{6\mathit{G}}}+{\displaystyle \frac{{{\mathit{p}}_{\mathit{n}+1}}^2}{2\mathit{K}}}\right)\right]\left({{\overline{\mathit{\epsilon }}}^{\mathit{p}}}_{\mathit{n}+1}-{{\overline{\mathit{\epsilon }}}^{\mathit{p}}}_{\mathit{n}}\right)$$

(iv)

End

4. Calibration Strategy and Numerical Results

The results obtained through numerical simulations are presented below. The details and values obtained for the calibration of the material are described as well as the parameters associated with the Lemaitre damage model. Finally, fracture prediction values for normalized and annealed AISI 4340 are also shown.

4.1. Calibration of Material Parameters

From the experimental data of the smooth specimens, the calibration parameters were obtained for the material in the two heat treatment conditions. Initially, the nonlinear hardening curve proposed by [27] and the calibration parameters obtained through the parametric identification search method are described. Subsequently, the results for the calibration of the damage indicator and the parameters of Gao’s model are presented.

4.1.1. Elastoplastic Material Parameters and Damage for Lemaitre Model

Adopting the experimental results for the smooth cylindrical specimen, a parametric identification strategy and the equation of [27] were used to define the isotropic hardening curve. The following parameters define the normalized and annealed AISI 4340 alloy: for the determination of the isotropic hardening parameters of the materials, the inverse method of parametric identification was used (ref. [28]). This method considers the reaction curve obtained experimentally for each calibration condition using the finite element technique to obtain the optimized curve with the gradient-based multivariable search method. The equation of [27] is used since it considers the description of the isotropic hardening curve of the material and accounts for this through four parameters, whose dependence is shown in the mathematical expression of Equation (22):

$${\sigma }_y={\sigma }_{y0}+\varsigma {\overline{\epsilon }}^p+\left({\sigma }_{\infty }-{\sigma }_{y0}\right)\left(1-e^{-\delta {\overline{\epsilon }}^p}\right),$$

(22)

where ${\sigma }_y$ represents the yield point of the material, ${\sigma }_{y0}$ corresponds to the initial yield point, ${\overline{\epsilon }}^p$ is the equivalent plastic strain, and the set represents the curve fitting parameters.

Table 1. Material parameters—AISI 4340 normalized and annealed alloy.

Description	Symbol	Values
		Normalized	Annealed
		Tensile	Tensile
Young’s modulus	E [MPa]	199,960	206,880
Poisson’s ratio	$\upsilon$	0.30	0.30
Initial yield stress	${\sigma }_{y0}$ [MPa]	657.00	449.00
First parameter calibration	$\varsigma$ [MPa]	819.10	568.20
Second parameter calibration	${\sigma }_{\infty }$ [MPa]	1022.00	747.00
Calibration exponent	$\delta$	51.90	28.90
Denominator damage $\left(S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)$	$S$ [MPa]	16.65	25.02
Damage exponent	$s$	1	1
Critical damage	$D_c$	0.21	0.21

presents the elastic and plastic parameters for the material used in the tensile and shear calibrations. It is important to note that the Poisson ratio was assumed from the literature and is equal to 0.30.

4.1.2. Set of Parameters for Fracture Indicator and Gao-Based Model

The isotropic hardening curve presented in Equation (22) was used, whose values are presented in Table 1, considering that the constitutive model based on Gao’s criterion recovers the behavior of the von Mises model at the point where, for the behavior of the smooth cylindrical specimen, this is subjected to pure tensile. Subsequently, to calibrate the parameter of Gao’s model, the reaction curve obtained experimentally for the rectangular specimen, subjected to pure shear, was considered. In this condition, the same parametric identification strategy was used, using the gradient-based method as the search procedure (ref. [28]).

In the calibration of the damage parameters for the ductile fracture indicator, implemented in Box 3 in the post-processing stage of Gao’s model, a multi-objective strategy was used, having both the critical damage and the displacement in the fracture for the bodies’ smooth cylindrical test specimens subjected to tensile, and rectangular ones subjected to pure shear, as an objective function. In this case, the optimized set of parameters was determined when the numerically calculated displacement became equal to the experimentally observed displacement and the maximum damage calculated reaches the critical damage level, considering the stress states in tensile and shear when associated. Equation (23) mathematically represents the multi-objective function used.

Table 2. Parameters for the damage indicators of Vaz Jr. and Owen and the new proposal—AISI 4340 alloy.

Description	Symbol	Vaz Jr. and Owen		Proposed
		Normalized	Annealed	Normalized	Annealed
Gao’s parameter	$b$	−100	−130 and −140	−100	−130 and −140
Denominator damage–tensile	$S_{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$ [MPa]	16.65	25.02	16.65	25.02
Denominator damage–shear	$S_0$ [MPa]	-	-	14.40	62.04
Critical damage	$D_c$	0.17	0.16	0.17	0.16
Plastic strain at fracture	${{\overline{\epsilon }}^p}_f$	-	-	0.67	1.20

presents all the parameters necessary for the execution of the model.

$$\begin{array}{c} g={\left[\sqrt{{\left({\displaystyle \frac{D_c-D_{max}}{D_c}}\right)}^2+{\left({\displaystyle \frac{u_{f exp}-u_{f num}}{u_{f exp}}}\right)}^2}\right]}_{tensile}\\ +{\left[\sqrt{{\left({\displaystyle \frac{D_c-D_{max}}{D_c}}\right)}^2+{\left({\displaystyle \frac{u_{f exp}-u_{f num}}{u_{f exp}}}\right)}^2}\right]}_{shear} .\end{array}$$

(23)

The proposed model incorporates critical damage ($D_c$) and plastic strain at fracture (${{\overline{\epsilon }}^p}_f$) to more accurately predict material failure under different stress states. While $D_c$ defines the irreversible degradation limit, ${{\overline{\epsilon }}^p}_f$ captures the material’s ability to withstand deformation before rupture. This approach, based on [18,19], improves the model’s sensitivity to material ductility, as highlighted by [2,21]. Furthermore, works such as [23] reinforce that the inclusion of these parameters enables more precise adjustments in fracture prediction under varying loading conditions.

Two values were used for the Gao model coefficient, both for the Vaz Jr. and Owen damage indicators and for the newly proposed indicator. For both materials, simulations were performed based on the parametric identification process [27,28]. Thus, the parameters used were found for the model calibration condition (values presented in Table 2). Therefore, it was possible to evaluate the model’s accuracy in estimating fracture displacement and predicting force level. This approach allowed comparing the model’s performance under different loading and calibration conditions [18].

Next, comparisons of the experimental and numerical results are presented, referring to the evolution of damage and equivalent plastic strain. The estimates obtained in the simulations carried out using Gao’s model, in its simplified formulation, were compared to the experimental results when considering the reaction forces, as well as the evolution of the plastic strain in the fracture and the damage.

Ref. [1] presented advances in the numerical and experimental description of an aluminum alloy, addressing the effects of the stress triaxiality and the third invariant but, in the research presented by [24], an experimental and numerical study is presented, which evaluates the alloy AISI 4340 in the ratios of high and low triaxiality. In this study, the effects of the normalized third invariant of the deviator tensor are considered as well as the plastic strain in the fracture. Furthermore, in the context of the fracture of ductile materials, recent studies have deepened the understanding of damage and failure mechanisms under different stress states. Works such as those by [29,30] explore the resistance to fracture propagation under predominant shear conditions, while [31] investigates the influence of stress triaxiality and the third invariant on damage nucleation. These contributions are fundamental for improving fracture prediction models, allowing for more accurate calibration of parameters and enhancing the applicability of the proposed model.

4.2. Results for AISI 4340 Annealed Alloy

For the experimental tests, cylindrical and rectangular specimens were manufactured. In accordance with current technical standards, dimensional control and surface finish verification were performed during fabrication of the geometries. In the tests, a 25 mm long extensometer was used for the cylindrical specimens. Due to the extensometer’s length limitation, a force-control configuration was used. All tests were performed at room temperature with a mechanical actuator at a speed of 2 mm/min, and the load was applied axially to the specimen until rupture, following the guidelines of the ASTM E8/E8M [32] standard, using an MTS-810 universal testing machine with a maximum capacity of 100 kN. The metal alloy 4340 used in the study was acquired in normalized and annealed heat treatment conditions, with the appropriate certifications.

Initially, the isotropic hardening parameters were calibrated through tensile tests on cylindrical specimens, ensuring that the yield curve adequately captured the evolution of plastic deformation. Subsequently, shear tests were conducted on rectangular specimens to determine the damage denominator, allowing the model to accurately capture the effects of the stress state on fracture nucleation and propagation.

The model was validated by comparing the experimentally obtained fracture displacements with the numerical predictions, ensuring that the simulated damage mechanisms corresponded to those observed empirically. Studies such as those by [21] demonstrate that stress triaxiality and the third invariant significantly influence the mechanical behavior of ductile materials, justifying the adopted approach. The works of [23] reinforce the importance of calibration based on multiple stress states to improve the accuracy of fracture prediction, ensuring that the model is applicable to a wide range of loading conditions.

Next, comparisons of the experimental and numerical results are presented, which refer to the evolution of damage and equivalent plastic strain. The estimates obtained in the simulations carried out using Gao’s model, in its simplified formulation, were compared to the experimental results considering the reaction forces and the evolution of the plastic strain in the fracture and the damage. For the AISI 4340 annealed material, the simulations of the specimens were considered: smooth, pure shear, and the mixed (15°, 45°, 60°). For the normalized material, the specimens were simulated: smooth, pure shear, and mixed 30°. In the graphical representations, the results of the numerical simulations, obtained by Gao’s model using the damage indicator proposed by [26], were identified as Gao/Vaz. The results obtained by Gao’s model, using the new damage indicator, were represented as Gao/Morales.

For the beginning of the study, Figure 1 shows the graphic representations of the smooth cylindrical specimen of the AISI 4340 annealed alloy. The reaction force as displacement function is shown in Figure 1a, in which the models of Gao and Lemaitre are compared, and it is possible to verify the agreement between the experimental results, both for the reaction forces and the displacements. The result was satisfactory because it is the calibration point of the models. When looking at Figure 1b, it is possible to see that the damage evolution for Gao’s model showed good coincidence in the two simulations. The evolution of the equivalent plastic strain for the Gao and Lemaitre models is shown in Figure 1c.

Figure 2a shows the experimental failure. The damage contour for the models used in this research is shown in Figure 2b–d. One can see that the damage distribution is very close to a critical point, with the critical value reaching a value of around 0.21 for Lemaitre’s model and approximately 0.16 for both the damage indicators of Vaz Jr. and Owen regarding the new proposal. For all specimens, simulations were performed up to the point at which the damage variable reached this critical value.

The analysis of the new damage indicator was performed for specimens in a stress state within the low-stress triaxiality ratio. Numerical simulations were performed using the parameters of Gao’s model, b = −130 and b = −140, and compared with Lemaitre’s model. Figure 3a shows and compares the reaction force with the values obtained experimentally for the rectangular specimen in pure shear. It is possible to notice that Lemaitre’s model makes a premature prediction of the failure. Using the indicators proposed by [26] and the new damage indicator, Gao’s model significantly improves the fault displacement prediction and reaction force estimation. On the other hand, when the damage evolution is observed, ref. [26] presents a fracture displacement below that obtained experimentally, since the proposed damage indicator greatly improves the prediction of the failure point, as shown in Figure 3b. The evolution of the equivalent plastic strain appears to be practically coincident in the two conditions of Gao’s model, as seen in Figure 3c.

The Lemaitre model predicts higher strength levels and elevated damage rates because it considers continuous degradation of the material without directly incorporating void nucleation and growth mechanisms. This leads to a faster damage evolution, especially under high triaxiality, resulting in optimistic predictions for fracture toughness. Furthermore, the absence of a porosity softening effect maintains a high load capacity until fracture. Works such as [2,21] indicate that adjustments in the calibration can enhance this prediction.

Figure 4a shows the region where the specimen fails after the experimental test. The contours of the damage distribution, in all situations, are distributed on the face of the specimen, very close to the critical point, as can be seen in Figure 4b–d,f.

When the reaction force versus displacement was evaluated for the specimen in a tensile state with force applied at 15°, it could be verified that the estimate of displacement in the fracture performed by Lemaitre’s model was underestimated when compared to the experimental result, and Gao’s model makes a more realistic estimate. The levels of forces, as well as the displacement in the fracture, were closer to the experimental example when using the damage estimator of [26] and the new damage indicator (Figure 5a). These results are corroborated when the damage evolution and the equivalent plastic strain are observed, as shown in Figure 5a,b.

The experimental result for the mixed specimen 15° is shown in Figure 6a. The contour of the damage distribution is spread out on the face and with higher values within the critical region of the specimen, both for Lemaitre’s model and for Gao’s model, a characteristic that can be seen in Figure 6b–d,f.

The mixed loading condition at 45° is analyzed in Figure 7a–c. The prediction of fracture displacement for Lemaitre’s model remains below what was experimentally predicted since the level of reaction force was overestimated by this model. The results for Gao’s model (b = −130 and b = −140) provide an improved estimate when compared to Lemaitre’s model for both reaction forces and displacement at fracture (see Figure 7b). The evolution of the damage estimated by the new indicator presents better results when compared to the prediction made by the damage indicator of [26]. The behavior of the equivalent plastic strain was practically identical for Gao’s model (b = −130 and b = −140), as can be seen in Figure 7c.

For the 45° mixed specimen, the evidence of fracture is shown in Figure 8a. The damage tends to the interior of the specimen face, both for Lemaitre’s model and for Gao’s model, as can be seen in Figure 8b–d,f.

The results for the 60° mixed specimen of the annealed AISI 4340 material are shown in Figure 9a–c. For Lemaitre’s model, the results follow the trend shown for the 45° mixed loading condition. Gao’s model, in both simulated conditions, presents itself as an excellent alternative for obtaining the level of reaction force, but it loses precision when observing the displacement in the fracture, due to the stress triaxiality dependence, the lack of an explicit term for the third invariant (J₃) and the limitations on damage evolution. This can lead to inconsistent predictions in combined stress states, such as shear. Studies such as [18] suggest adjustments in the calibration to improve accuracy (See Figure 9a). Even so, the improvement in the estimation of the fracture site is evident. This trend is accompanied by the evolution of the damage for both the new indicator and the estimator from Vaz Jr. and Owen (Figure 9b). The evolution of plastic strain at fracture can be seen in Figure 9c.

Figure 10 allows evaluation of the experimental failure (Figure 10a), as well as the damage distribution on the surface of the critical region for the mixed specimen 60°. The damage contour also tends to the interior of the specimen face, both for Lemaitre’s model and for Gao’s model, as can be seen in Figure 10b–d,f.

4.3. Results for AISI 4340 Normalized Alloy

The verification of the behavior of the new damage denominator function and its comparison with the indicator from [26], using Lemaitre’s and Gao’s models, were also performed for the normalized AISI 4340 material. Because this material has characteristics of greater ductility compared to the annealed specimen, other parameters were used in the models. For Lemaitre’s model, the critical damage level was kept equal to 0.21 and, in the case of Gao’s model, the critical value of 0.17 was used. Figure 11 shows the results for the smooth specimen, which was used to calibrate the models. One can see that both models perfectly describe the levels of force and displacements in the fracture, when compared with the experimental results (Figure 11a).

Figure 12 shows the appearance of the fracture, after the experimental test (Figure 12a, and the way in which the damage is distributed in the critical region of the smooth cylindrical normalized specimens, for Lemaitre’s and Gao’s models. In Figure 12b–d the damage is concentrated very close to the critical point.

For the case of the normalized AISI 4340 material, Gao’s model was tested using the coefficient b = −100 and results were generated for the damage indicator of [26] and the new damage indicator for these values. The results for the pure shear condition are shown in Figure 13. Lemaitre’s model estimates the reaction force above the values obtained experimentally, and the same happens when observing the fracture displacement prediction. The results for Gao’s model (b = −100) also estimate the failure after the experimentally observed failure (Figure 13a). The damage evolution is shown in Figure 13b, in which it is observed that the growth proposed by [26] and the new damage function present similar and coherent results with the reaction forces.

The experimental fracture and the way in which the damage is distributed are shown in Figure 14. The specimen subjected to pure shear loading presents its failure in the center of the critical region, as can be seen in Figure 14a. The phenomenon can be seen by observing the numerical results of Figure 14b–d, in which the damage concentration, for the cases of Lemaitre and Gao, show very similar distributions.

The other condition studied, for the normalized AISI 4340 material, was the 30° combined state, whose results are shown in Figure 15. In the same way as the pure shear condition, for this material, Lemaitre’s and Gao’s models provided an estimate above the values observed in the experimental tests (Figure 15a). The damage gauge of [26] and the new damage indicator show similar evolution, as can be seen in Figure 15b. The growth of the equivalent plastic strain obtained by Gao’s model is slower, which justifies the failure in prediction for displacements different from those calculated using Lemaitre’s model (Figure 15c).

Figure 16 initially shows the experimental failure (Figure 16a) and then the outline of the damage parameter. Observing the images, it is possible to see that the concentration is located on the side wall and grows into the specimen (Figure 16b–d). This phenomenon is corroborated in the visualization of the experimental failure.

5. Conclusions

A new damage indicator for the fracture of ductile materials was proposed. To reach the objective, the models of [18,19] were used, both with isotropic hardening and damage. The behavior of the damage evolution for the new indicator was evaluated and compared with the damage indicator proposed by [26].

The new ductile fracture damage indicator and the prediction of the fracture site of materials, with different levels of ductility, as well as the levels of reaction forces, were obtained; this damage indicator is coupled to the flow function of Gao’s model in the post-processing stage. Considering the results, it was possible to infer a significant improvement in the prediction of the exact point at which the failure occurs. This means that it is possible to approximate the displacement at the fracture, both for loading conditions at the calibration point and for situations far from calibration. On the other hand, the reaction force was also estimated much more accurately when comparing the predictions of Gao’s and Lemaitre’s models.

The new damage indicator proposed in this research incorporates the normalized third invariant and the plastic strain at the fracture, which significantly reduces the sensitivity concerning the calibration point. This is demonstrated when predictions of fracture displacement and reaction force in pure and combined shear stress states are observed and allow good predictions, both for normalized and annealed material. Therefore, it presents itself as a good alternative for studying materials with different levels of ductility. In short, the incorporation of the third normalized invariant and the plastic strain had the expected effects since the proposed damage indicator is a function dependent on the stress state of the material.

When observing the results related to the cylindrical specimens, it was possible to perceive that Lemaitre’s and Gao’s models can perfectly recover the behavior of the normalized and annealed material, both for the levels of reaction forces and the levels of displacement in the fracture. This is an indication that both models can make a very accurately predict the onset of failure under load conditions that are close to the calibration point.

When evaluated at the calibration point, the proposed damage indicator presented the expected behavior since, at this point, the third normalized invariant remains constant; thus, and the damage evolution curve coincided with the indicator proposed by [26]. This behavior causes the delay perceived in the prediction, performed by Lemaitre’s model, to significantly improve the failure detection, verified both for the normalized and annealed material.

The third normalized invariant is null when evaluating the models in pure shear conditions. However, the prediction of the fracture point is excellent as is the estimation of forces for the two materials (AISI 4340 normalized and annealed). The proposed damage indicator, in both cases, showed excellent results, both for Gao’s coefficients and for those used for the annealed AISI 4340 material, whereas, for the normalized AISI 4340 material, Gao’s coefficient was used. The simulations were carried out until the specimens reached the critical damage for both materials.

Under the combined stress state conditions, the predictions in which the third normalized invariant range from 0 to 1 for both materials are satisfactory. Reaction force predictions are close to experimentally observed values. In general, Gao’s model improves the prediction regarding fracture displacements and, as the combined stress state approaches the tensile limit, close to the 1/3 stress triaxiality, the estimate loses precision. However, the damage evolution rate shows a deceleration and, thus, significantly improves the predictions made by Gao’s model.

The ability to predict the fracture point of the materials considered in this research through the proposed damage indicator is confirmed by observing the contours of the damage variable. For smooth cylindrical specimens, the fracture starts on the side of the critical section and is concentrated at the center of the specimen. For rectangular specimens under all load conditions, the fracture starts at the face of the critical section and propagates to the center of the specimen.

Figure 17 presents the percentage error of fracture displacements for the models tested. The percentage differences between the numerical estimates and the values obtained experimentally were represented. Observing the distribution of errors, it is possible to perceive a significant improvement in the prediction of the fracture site when the estimate is performed by the proposed damage indicator in all the situations studied.