Since any stress inside a material can be represented in the Cartesian coordinate system
formed by the three principal stresses, any stress can also be expressed in the cylindrical coordinate system
formed by the equivalent stress, stress triaxiality, and Lode angle parameters. Bao and Wierzbicki [
19] extended the application range of the Mohr–Coulomb criterion by transforming Equation (1) into a function of
and proposed the MMC failure criterion for predicting ductile fracture in metallic materials. Its original expression is shown in Equation (2):
where
K and
n are the hardening coefficients and hardening exponents, respectively, in the hardening criteria, which can be obtained by fitting the true stress–strain curve from uniaxial tension tests.
and
are material parameters in the original Mohr–Coulomb criterion, which can be calibrated through a combination of experimental tests and numerical simulations using optimization algorithms to determine the parameters of the three-dimensional fracture surface,
and
. Parameter
,
is associated with the asymmetry of the fracture surface. When the material follows the von Mises yield criterion (i.e., isotropic hardening),
and
, Equations (2) and (3) can be simplified as follows:
In the plane stress state, there is a one-to-one correspondence between the Lode angle parameter and stress triaxiality, as shown in Equation (5). By substituting Equation (5) into Equation (4), which represents the fracture surface of the MMC failure criterion in
three-dimensional (3D) space, the fracture failure trajectory in
two-dimensional (2D) space can be obtained and is displayed in
Figure 1.
2.1. MMC Failure Criterion Parameter Fitting
In the parameter calibration process of the MMC failure criterion, Bai and Wierzbicki [
11] adopted the basic procedure proposed for determining the failure criterion parameters in the
two-dimensional space, and further applied it to the calibration of parameters for the three-dimensional fracture surface in the
space. The specific steps are as follows:
Firstly, material testing is performed on specimens under different stress states to obtain their load–displacement curves. According to the setup of the experimental conditions, corresponding finite element simulation simulations are carried out. Critical data, such as the equivalent strain, stress triaxiality, and Lode angle parameters, are extracted from the finite element model, and their evolution curves are plotted with respect to the equivalent strain changes over the course of stress triaxiality and Lode angle parameters. Based on these curves, the average stress triaxiality and average Lode angle or average Lode angle parameters (where this study’s MMC fitting selects the average Lode angle parameter) are calculated using Equation (6). The fracture strain, average stress triaxiality, and average Lode angle parameters obtained from these calculations are then used to calibrate the undetermined parameters within the fracture criterion.
Yu [
20] conducted experimental and simulation studies on the plastic and fracture characteristics of marine high-strength steel under different stress states. Their tensile tests primarily included tensile and shear stress states. The literature provides the average stress triaxiality and average Lode angle parameters of the critical element at the failure location of different specimens, along with the corresponding failure strain (see
Table 1). This paper, based on the data from the literature, employs MATLAB 2018b optimization tools to fit the MMC failure criterion.
Generally speaking, the structural steel of a ship’s hull can be considered an isotropic material. The MMC failure criterion can be simplified from Equations (2)–(4) based on the von Mises criterion’s premises, where
and
reference the parameters of the Swift model in the literature [
20], which are
= 909.29 MPa,
= 0.1992. Equation (7) is obtained.
At this point, the MMC failure criterion only involves two undetermined parameters,
and
. Utilizing the data from
Table 1 and the optimization tool lsqcurvefit in MATLAB, the parameters
and
in Equation (7) are optimized to obtain the best-fitting 3D fracture surface. The lsqcurvefit function in MATLAB requires an initial set of solutions
and
. By substituting the data from specimens 4 and 5 into the aforementioned equation, initial values of
= −0.0207 and
= 278.23 MPa can be determined. Taking these values as the initial solutions and applying the least squares error optimization, the final MMC failure criterion parameters can be acquired. The MMC failure criterion parameters are then listed in
Table 2.
The final derived MMC failure criterion is presented as Equation (8):
Using the cftool utility in MATLAB, the aforementioned equation is plotted in the
space as shown in
Figure 2, where the red line represents the fracture trajectory under the plane stress state.
In ship collision simulation computations, where the finite element type employed is a plane stress shell element, the relationship between the stress triaxiality and the Lode angle parameter under plane stress conditions, as described in Equation (5), is substituted into Equation (8). This yields the fracture trajectory for the MMC failure criterion under plane stress conditions. The curve within the
space is depicted in
Figure 3.
It should be noted that the parameters for the MMC failure criterion obtained here are based on solid elements with a size of 0.1 mm acquired through experimental–simulation hybrid analysis. These parameters are not directly applicable to ship collision simulations that employ shell elements. Consequently, a modification is necessary to adapt them for use in the simulation analysis of ship structural collisions.
2.2. Mesh Size Sensitivity Correction of the MMC Failure Criterion
The MMC failure criterion is established based on solid elements [
11]. When this criterion is applied to shell elements, which operate under the assumption of plane stress and thus neglect variations in stress across the thickness, it results in a heightened sensitivity to the size of the shell element mesh. As the grid size of the shell elements is substantially greater than the plate thickness, in conducting finite element failure simulations with ductile metal plate materials using shell elements, scaling of the failure strain becomes necessary when the dimensions of the shell elements surpass the necking length. This scaling ensures that the necking phenomenon can be evenly distributed along the length of the element, thus enabling a precise determination of structural failure. Due to the necking, the failure strain of shell elements is significantly affected by the size of the element. Consequently, for shell elements with disparate grid sizes, it is imperative to adjust the failure strain accordingly to maintain the fidelity of the simulation.
Adjustments to failure criteria are commonly grounded on the results of uniaxial tension tests. For instance, the critical equivalent plastic strain criterion as well as the RTCL failure criterion can be scaled using the relation presented in Equation (9):
Herein, represents the input value for fracture strain at the occurrence of , while is the strain hardening exponent, indicative of the strain at the onset of necking during uniaxial tension. The grid size sensitivity of the failure strain is attributed to the necking phenomenon; given that the degree of necking varies under different stress states, it can be surmised that any adjustments for grid size sensitivity of the failure strain should also take into account the influence of the stress state (stress triaxiality).
This article adopts a correction method that considers both grid size sensitivity and stress state. The crux lies in altering the fracture strain
, in Equation (9), and the diffuse necking condition
n to become functions of the stress triaxiality
. Utilizing
to substitute
in Equation (9), the following formula can be obtained:
In this context,
and
represent the characteristic element size and the plate thickness, respectively, used when calibrating the failure criterion parameters. For the MMC failure criterion discussed in this paper, a solid element size of 0.1 mm is adopted for calibration, and the plate specimen thickness used for parameter calibration is 2 mm. Thus, it follows that
mm and
mm. By incorporating the stress-triaxiality-dependent failure fracture trajectory, denoted by
, and the Swift local necking curve, denoted by
, into Equation (10), one can obtain the failure trajectory
for
shell elements, as depicted in
Figure 4.
Subsequently, through the inversion of Equation (10), the failure strain function that corresponds to any characteristic element length, denoted as
, can be derived, as demonstrated in Equation (11). With Equation (11), a series of the MMC failure criterion curves associated with different
values can be obtained through interpolation, as illustrated in
Figure 5.
It should be mentioned that the Swift necking curve typically exists in the
space. The expression of the load stress ratio, represented by the symbol
, can be described as follows, assuming that it stays constant:
In order to convert Equation (12), which exists within the space, to the curve within the space as reflected in Equation (10) to Equation (11), the following transformations, represented by Equations (13)–(15), need to be applied to Equation (12).
Under the condition of plane stress, the equivalent plastic strain can be expressed as follows:
In the equation,
represents the strain ratio, defined by
. Based on the von Mises criterion, there is a one-to-one correspondence between stress and strain under plane stress conditions. Consequently, the strain ratio
is related to the stress ratio
as follows:
Therefore, the stress triaxiality can also be obtained by calculating through the strain ratio.
Based on Equatios (13)–(15), the Swift instability curve in the
space can be transformed into a curve
in the
space. Since the Swift curve corresponds to a stress range between uniaxial tension and equi-biaxial tension
, to facilitate scaling the MMC failure criterion throughout the entire range of plane stress states,
is selected within the
range, resulting in Equation (16). This is plotted in the
space as shown in
Figure 5.