Multiscale Model Synthesis to Clarify the Relationship between Microstructures of Steel and Macroscopic Brittle Crack Arrest Behavior - Part I: Model Presentation

Abstract

A new multiscale model is proposed by a “model synthesis” approach, as the first attempt to clarify the relationship between microstructures of steel and macroscopic brittle crack propagation and arrest behavior. The first part of the present paper shows the model presentation. The multiscale model consists of two models: (1) a microscopic model to simulate cleavage fracture in the grain scale and (2) a macroscopic model to simulate brittle crack propagation and arrest behavior in the steel plate scale. In both the models, we utilize the same framework, where a simple two-dimensional domain discretization is employed but a three-dimensional crack propagation can be effectively modeled. The discretized unit cells in the microscopic model correspond to the grain size. On the other hand, the discretized unit cells in the macroscopic model correspond to the entire domain of the microscopic model. The microscopic model proposed by Aihara and Tanaka is basically employed except the integration with the macroscopic model. The effective surface energy, which is used for the integration between microscopic and macroscopic models, is assumed as the plastic work to form tear-ridge. The proposed model synthesis for multiscale model as an integrated macroscopic model is performed by systematically incorporating (1) the preparatory macroscopic finite element analysis and (2) the Monte Carlo simulation of microscopic analysis into (3) the macroscopic analysis for brittle crack propagation and arrest in steel plate. The integration procedure is implemented by the assignment of physical quantities based on interpolation methods as a one-way coupling algorithm for simplification.

1. Introduction

Prevention of brittle crack propagation as well as crack initiation is essential as a “double integrity” for large steel structures. Although it is generally the most critical to prevent brittle crack initiation by controlling welding defects and fatigue cracks by repeated load during service, it is actually difficult to remove welding defects completely. It is thus significant for a realistic integrity approach how to arrest the brittle crack without serious damages.

Recently, increasing strength and thickness of steel plate are promoted due to the requirement of the significant enlargement in of scale for many ships and offshore plants in the heavy industries. Because of the fact that the increase of the thickness in steel makes the risk of brittle fracture higher, Nippon Kaiji Kyokai and International Association of Classification Societies (IACS) published a guideline on brittle crack arrest design.^1,2) The guideline recommends that the arrestability of steel is evaluated as a crack arrest toughness, which is obtained by the crack arrest test prescribed in the standard published by The Japan Welding Engineering Society.³⁾

The application of steels with high arrestability to the structures is directly effective to ensure the integrity. Therefore, there have been a lot of efforts on researches and developments of the steels with higher arrestability. In particular, it has been recognized that there is a strong correlation between microstructures and arrest toughness as an empirical knowledge supported by many experiments.^4,5,6,7,8,9)

One of the early investigations on the relationship between microstructures and arrest toughness was performed by Ohmori et al.⁴⁾ Their work focused on the ferrite-pearlite steels, which is the most widely used as structural steels. The results evaluated by the tapered double cantilever beam tests showed that the finer pearlite colony makes the higher arrest toughness. Such grain refining has been one of the major approach of the development of steels with high arrestability even in the recent years.^6,9) However, the detail of the theoretical mechanism has been scarcely clarified.

As mentioned above, in the recent years, the high strength steels have been widely spread for actual use. They are generally made by low temperature rolling, so that the high strength steel shows not only stronger texture but also more nonhomogeneous distributions of grain size and orientation in the thickness direction than the conventional steels. It was reported that the well-controlled texture has a potential to enhance the arrestability of steel.^7,8) The mechanism is qualitatively explained, i.e., some kinds of texture make many micro-branching of crack, which not only reduce the fracture driving force but also increase the effective surface energy, on the fracture surface during crack propagation. It was also reported that the nonhomogeneous distributions of grain size and orientation in the thickness direction has a possibility to macroscopically enhance the arrestability of steel. In particular, a steel plate with higher arrestability at the mid-thickness position rather than at the quarter-thickness position shows a characteristic fracture surface morphology, called as “split-nail”, and also shows higher brittle crack arrest performance.¹⁰⁾ However, there are not any theories which have quantitatively explained the above phenomena in the past investigations.

According to the above mentioned facts, a reasonable theory to explain the quantitative and universal relationship between microscopic cleavage fracture and macroscopic brittle crack propagation/arrest behavior has never been established. For the macroscopic brittle crack propagation in steel plate, the past studies has been restricted in the field of the continuum mechanics where the crystallographic factors such as grain size and orientation were not considered.^11,12,13) For the microscopic cleavage fracture in the past studies, the details of the dominant factors have not been sufficiently clarified. Although some models to simulate microscopic cleavage crack propagation were proposed in the past,^{14,15,16,17,18,19)} their achievements have been limited in the qualitative evaluation such as a trend of fracture surface morphology. Consequently, there has never been a reasonable model to bridge the relationship between the microscopic cleavage fracture and the macroscopic brittle fracture. One of the most major reasons is caused by a large “scale gap” on the dominant factors between macroscopic and microscopic phenomena. Brittle fracture of the steel plate is generally on the scale of 10⁰ m. On the other hand, the microstructures as polycrystalline are generally on the scale of 10⁻⁶–10⁻⁴ m, and moreover, the cleavage fracture condition on a crystal is on the scale of 10⁻⁹ m. In addition, the brittle fracture in steel occurs in the significantly fast process with extremely strong material nonlinearity, which is difficult to be experimentally measured in detail.

In the present paper, we propose a multiscale model to simulate the complex behavior of brittle crack propagation and arrest. In the proposal of multiscale model, we make an attempt to solve the problem of the scale gap between macroscopic and microscopic phenomena, which is described above, by a new approach of “model synthesis”, which integrates the multiple models and analyses to systematically evaluate complicated macroscopic and microscopic phenomena, based on the information of microstructures of the steel.

2. Concept of the Multiscale Model

In the present paper, we show the first attempt to clarify the relationship between microstructures of steel and macroscopic brittle fracture by a new proposal of multiscale model by a “model synthesis” approach. Figure 1 shows an outline of the proposed multiscale model. The multiscale model consists of two models: (1) a microscopic model and (2) a macroscopic model. The microscopic model simulates cleavage fracture in the grain scale. The macroscopic model simulates brittle fracture in the steel plate scale. A framework for domain discretization and criterion of crack propagation in both the models is based on the same manner as the studies of Aihara et al.^14,17) although their works are restricted only to simulate microscopic cleavage fracture. In particular, a three-dimensional crack propagation has been effectively modeled by two-dimensional domain discretization. The entire domain for microscopic model is defined as a square whose size is 1 mm by 1 mm in the width and thickness directions of a plate, respectively. The entire domain is divided into square unit cells with the same size which is equal to mean grain size. That is, each unit cell simulates a grain of steel in the microscopic model. On the other hand, the entire domain for the macroscopic model is defined as an actual plate size. The entire domain is divided into the square unit cells. Each unit cell corresponds to the entire domain of the microscopic model, so that the size of the unit cell in the macroscopic model is 1 mm by 1 mm in the width and thickness directions.

Fig. 1.

Outline of the proposed multiscale model incorporated by the model synthesis between microscopic and macroscopic models. (Online version in color.)

The newly proposed model synthesis for multiscale model as an integrated macroscopic model is performed by systematically integrating (1) a preparatory macroscopic finite element analysis and (2) a microscopic analysis for cleavage fracture into (3) a macroscopic analysis for brittle crack propagation and arrest in steel plate. In particular, it is expected that there is a large scatter in the results of the microscopic analysis due to the variation of distributions of microstructures such as grain size and orientation, so that the Monte Carlo simulation is employed and integrated into the multiscale model as a process of the model synthesis. The integration procedure is implemented by a one-way coupling algorithm for simplification.

The details of the proposed multiscale model are described in the following three sections. The domain discretization and criterion of crack propagation, which are the common procedures between microscopic model and macroscopic model, are shown in Section 3. The detail of the microscopic model to simulate cleavage fracture in the grain scale is then shown in Section 4. The detail of the integrated macroscopic model by model synthesis approach to simulate brittle fracture in the steel plate scale is shown in Section 5.

3. Domain Discretization and Criterion of Crack Propagation

We adopt the framework for domain discretization and criterion of crack propagation in both the microscopic and macroscopic models. The framework is developed based on the works by McClintock¹⁴⁾ and Aihara et al.¹⁷⁾ to simulate microscopic cleavage fracture.

The brittle crack propagation and arrest behavior in steel is an exceedingly complicated phenomenon. In particular, the crack path shows a complicated three-dimensional morphology. In addition, the brittle crack propagation occurs as a significantly fast dynamic process with extremely strong material nonlinearity. In the present study, the three-dimensional crack propagation is effectively simplified by adopting the following four assumptions;

(1) Overlap of crack surfaces in the normal direction of the macroscopic fracture surface is not taken into account. It makes the domain discretization performed in the two dimensional plane.

(2) The shapes of an entire domain and a unit cell are both modeled as rectangles. In addition, each unit cell is defined as the same size.

(3) Dynamic effect is not strictly taken into account during crack propagation. The crack propagation is modeled by a step-by-step calculation based on the linear fracture mechanics theory.

(4) Calculation of the stress intensity factor, which is employed as the crack driving force, is performed by a superposition of the approximate solutions for crack shape^20,21) and crack closure effect by tear-ridge¹⁷⁾ (see Fig. 4).

Fig. 4.

Fracture condition and calculation of stress intensity factor. (Online version in color.)

Figure 2 shows the schematic of the domain discretization and crack propagation modeling by the step-by step calculation. The above assumptions can make a significant merit for simplification of calculation.

Fig. 2.

Domain discretization and crack propagation modeling by step-by step calculation. (Online version in color.)

The crack propagation is evaluated by the criterion comparing between driving force and resistance of crack propagation in terms of stress intensity factor, as   

$$k\ge k_{\text{f}}$$(1)

where k is the equivalent stress intensity factor as the driving force of crack propagation, and k_f is the fracture toughness as the resistance of crack propagation.

In the microscopic model, k_f (=k_fm) is defined as a material constant. On the other hand, in the macroscopic model, k_f (=k_fM) is calculated from the result of the microscopic model. The details of this topic are found in the following Section 3 and Section 4, respectively.

The equivalent stress intensity factor k is calculated by a vector of the stress intensity factors k_i (i=1,2,3), where i means the fracture mode, and normal unit vector of next fracture surface n, as   

$$k=\sum _{i=1}^3{\mathbf{n}}^{\text{T}}\cdot k_i{\mathbf{f}}_i\left[{\theta }_{\text{s}}\right]\cdot \mathbf{n}$$(2)

A graphical description of the relationship between planes of global fracture surface and next fracture surface with the normal unit vector n is shown in Fig. 3. θ_s is an incline angle between planes of global fracture surface and next fracture surface, which is calculated as   

$${\theta }_{\text{s}}={tan}^{-1}\left(\frac{n_{\xi }}{n_{\eta }}\right)$$(3)

where n_ξ and n_η are components of n in the local coordinate system (ξ,η,ζ) (see Fig. 3). f_i[θ] is coefficient tensor functions of θ corresponding to the fracture mode i, used in the expression of asymptotic solution of stress tensor σ near the crack tip,²²⁾ as   

$$\sigma \left[r,\theta \right]=\frac{1}{\sqrt{2\pi r}}{k}_i{\mathbf{f}}_i\left[\theta \right]$$(4)

and   

$$\begin{array}{l}{\mathbf{f}}_1\left[\theta \right]=\\ \left[\begin{array}{ccc}cos\frac{\theta }{2}\left(1-sin\frac{\theta }{2}sin\frac{3\theta }{2}\right)& sin\frac{\theta }{2}cos\frac{\theta }{2}cos\frac{3\theta }{2}& 0\\ sin\frac{\theta }{2}cos\frac{\theta }{2}cos\frac{3\theta }{2}& cos\frac{\theta }{2}\left(1+sin\frac{\theta }{2}sin\frac{3\theta }{2}\right)& 0\\ 0& 0& 2\nu cos\frac{\theta }{2}\end{array}\right]\end{array}$$(5)

  

$$\begin{array}{l}{\mathbf{f}}_2\left[\theta \right]=\\ \left[\begin{array}{ccc}-sin\frac{\theta }{2}\left(2+cos\frac{\theta }{2}cos\frac{3\theta }{2}\right)& cos\frac{\theta }{2}\left(1-sin\frac{\theta }{2}sin\frac{3\theta }{2}\right)& 0\\ cos\frac{\theta }{2}\left(1-sin\frac{\theta }{2}sin\frac{3\theta }{2}\right)& sin\frac{\theta }{2}cos\frac{\theta }{2}cos\frac{3\theta }{2}& 0\\ 0& 0& 2\nu sin\frac{\theta }{2}\end{array}\right]\end{array}$$(6)

  

$${\mathbf{f}}_3\left[\theta \right]=\left[\begin{array}{ccc}0& 0& sin\frac{\theta }{2}\\ 0& 0& cos\frac{\theta }{2}0\\ sin\frac{\theta }{2}& cos\frac{\theta }{2}& 0\end{array}\right]$$(7)

where (r,θ) is a polar coordinate with origin at the crack-tip and θ=0 for crack propagation direction. ν is a Poisson’s ratio.

Fig. 3.

Relationship between the planes of global fracture surface and next fracture surface. (Online version in color.)

The vector of the stress intensity factors k_i (i=1,2,3) in Eq. (2) is calculated by a superposition of the approximate solutions for various crack shapes. In this calculation, the stress intensity factor for a crack having flat surface and straight crack front perpendicular to the crack propagation direction in the infinite body, $k_1^{\infty }$ , is adopted as a basis, as²²⁾   

$$k_1^{\infty }={\sigma }_{yy}\left[r_{\text{c}}\right]\sqrt{2\pi r_{\text{c}}}$$(8)

where r_c is the characteristic length, assumed as r_c=0.2 mm. σ_yy is tensile stress in front of the crack tip, which is obtained by the dynamic elasto-plastic finite element analysis performed in the macroscopic model.

In the superposition of approximate solutions, three effects are taken into account (1) Non-straightness of the crack front;²⁰⁾ (2) Non-planar of the crack surface;²¹⁾ (3) Crack closure effect by the tear-ridge.¹⁷⁾ The calculation procedure of the superposition of approximate solutions is the same as the work by Aihara and Tanaka,¹⁷⁾ so that the detail can be found in their work.

The schematic of fracture condition and approximate calculation of stress intensity factor is shown in Fig. 4. The approximate calculation is elaborated in the sections of microscopic model.

In the following sections, the subscripts m and M are introduced to distinguish the parameters for microscopic model and macroscopic model, respectively. As the example of the subscript m, k_m and k_fm mean the equivalent stress intensity factor and the fracture toughness, respectively, only for microscopic model.

4. Microscopic Model

The cleavage fracture on grain scale is simulated in the microscopic model. The model proposed by Aihara and Tanaka is basically utilized except the integration of the microscopic model into the integrated macroscopic model. A qualitative comparison between an observation result by 3D-SEM and a simulation result for cleavage fracture surface is shown in Fig. 5. The detail can be found in their work.¹⁷⁾

Fig. 5.

A qualitative comparison between an observation result by 3D-SEM and a simulation result for cleavage fracture surface. (Online version in color.)

The aim of calculation by the microscopic model is the evaluation of (1) effective surface energy, and (2) direction of fracture surface, which are used as the input data for the integrated macroscopic model. This integration procedure is shown in the following Section 5.2.

The required input data for microscopic model is listed as (1) average grain size d, (2) distribution of grain orientation, (3) remote applied stress tensor σ and (4) yield stress σ_Y. (1) Average grain size d and (2) distribution of grain orientation are the characteristic parameters of steel plate. These parameters are easily measured by microscopic observations such as EBSD.²³⁾ (3) Remote applied stress tensor σ is acting on the entire domain of the microscopic model. (4) Yield stress σ_Y considers the dependence of temperature and strain rate. σ and σ_Y are obtained by the finite element analysis which is incorporated in the macroscopic model and assumed as constants because the calculation domain is sufficiently small. The detail of the finite element analysis is shown in Section 5. In addition, the fracture toughness for cleavage fracture k_fm, and the critical shear strain to form tear-ridge ε_fm is required to be defined to perform the microscopic model simulation. In the present study, these parameters are assumed as constants, as k_fm= $1\text{MPa}\sqrt{\text{m}}$ and ε_fm=0.7 in reference to the past study.^16,19)

Cleavage fracture surfaces generally form on {100} planes in a BCC polycrystal including ferrite.^15,24,25,26) According to the fact, the equivalent of the stress intensity factor k as expressed in Eq. (2) is replaced by the following expression in the microscopic model, as   

$$k_{\text{m}}=\underset{j=1,2,3}{max}\sum _{i=1}^3{\left({\mathbf{n}}_j\right)}^{\text{T}}\cdot k_i{\mathbf{f}}_i\left[{\theta }_{\text{s}}^j\right]\cdot {\mathbf{n}}_j$$(9)

where n_j is a unit normal vector of the j-th {100} plane and ${\theta }_{\text{s}}^j$ is an incline angle between the planes of global fracture surface and j-th {100} plane.

For microscopic model analysis, the size of entire domain is defined as a square whose size is 1 mm by 1 mm in the width and thickness directions, which corresponds to a unit cell in the macroscopic model. The entire domain is discretized into square unit cells of the average grain size d. That is, each unit cell corresponds to a grain of steel. The grain orientation is assigned to each unit cell to define {100} planes of the grain according to the distribution of the grain orientation.

The incorporating into the integrated macroscopic model is performed by using two physical quantities: (1) effective surface energy γ, and (2) direction of fracture surface n_M. These physical quantities γ and n_M can be calculated from the cleavage fracture surface obtained as a result of microscopic analysis.

The energy absorbing mechanism during brittle crack propagation has not been sufficiently clarified. In the present study, we assume that the plastic work to form tear-ridge is dominant in the total absolute energy to form the macroscopic fracture surface. The tear-ridge is formed as a result of the ductile fracture of uncracked ligament along grain boundary as shown in Fig. 6(a). The effective surface energy as the plastic work to form tear-ridge per unit area can be calculated by a line integral along grain boundaries s. Assuming the Tresca yielding condition without strain hardening, the plastic work to form tear-ridge per unit volume is expressed as the product of shear strength τ_Y (=0.5σ_Y) and critical shear strain ε_fm. The width of uncracked ligament, where the plastic deformation is produced, is assumed as 10% of the height of uncracked ligament based on the preliminary microscope observation. A schematic of the above assumed energy absorbing mechanism is shown in Fig. 6(b). As a summary of the above assumptions, the surface energy γ is approximately evaluated as   

$$\gamma =\frac{1}{2A}{\int }_{s}c{h}^2{\tau }_{\text{Y}}{\epsilon }_{\text{fm}}\text{d}s$$(10)

where A is an area of the entire domain, c is a ratio of width and height of uncracked ligament, i.e., c=0.1 as described above.

Fig. 6.

Tear-ridge formation in brittle fracture surface and assumed energy absorbing mechanism. (Online version in color.)

For actual calculation in the macroscopic model, the surface energy γ is introduced as the arrest toughness k_fM in the same dimension with the stress intensity factor, as shown in Eq. (1). The relationship between γ and k_fM are expressed based on the linear elastic fracture mechanics theory, as   

$$k_{\text{fM}}=\sqrt{2\gamma E}$$(11)

where E is a Young’s modulus.

The direction of fracture surface n_M is obtained as the normal vector of approximated plane of the cleavage fracture surface by the least square method, as is shown in Fig. 7.

Fig. 7.

Calculation of direction of fracture surface n_M. (Online version in color.)

5. Integrated Macroscopic Model

For the macroscopic model, a brittle fracture in the actual steel plate, which is generally 20–100 mm in thickness and 200–3000 mm in length, is simulated. That is, the macroscopic brittle fracture is on the scale of 10⁰. On the other hand, the microstructures, which is modeled in the microscopic model, is on the scale of 10⁻⁶–10⁻⁴ m, and moreover, the cleavage fracture condition on a crystal is on the scale of 10⁻⁹ m. To solve this problem, the reasonable integration to bridge the large gap in scale between microscopic and macroscopic models is the most important for the establishment of the multiscale model. In the present paper, we therefore propose as a new multiscale model that the above large gap in scale between microscopic and macroscopic models is bridged by only two simple parameters, i.e., the fracture toughness k_fM and the direction of fracture surface n_M, on each unit cell in the macroscopic model.

To bridge a large gap in scale, an integrated macroscopic model is formed as the multiscale model by a new approach of “model synthesis” by systematically incorporating (1) the preparatory macroscopic finite element analysis and (2) the Monte Carlo simulation for the microscopic analysis into (3) the macroscopic analysis for crack propagation and arrest. That is, the integrated macroscopic model is composed of the three-staged analyses. Figure 8 shows the procedures of the respective analyses in the integrated macroscopic model.

Fig. 8.

Procedure of the integrated macroscopic model by model synthesis in the multiscale model. (Online version in color.)

5.1. Preparatory Macroscopic Finite Element Analysis

The preparatory macroscopic finite element analysis as the first stage of the integrated macroscopic model is performed as a dynamic elasto-plastic analysis. The aim of the finite element analysis is to obtain (1) stress tensor σ and (2) yield stress σ_Y, at the characteristic distance, i.e., r_c=0.2 mm, from crack front. It is noted that the yield stress σ_Y is obtained as the numerical result of the finite element analysis because it depends on the strain rate $\dot{\epsilon }$ . Both the parameters of (1) stress tensor σ and (2) yield stress σ_Y are used as the input data of the following microscopic and macroscopic analyses, as show in Fig. 8. An example of the detail of the finite element analysis is shown in the second part of the present paper.²⁷⁾

5.2. Monte Carlo Simulation for Microscopic Analysis

The Monte Carlo simulation for microscopic model analysis as the second stage of the integrated macroscopic model is performed at the discrete evaluation points of the plate for efficiency of the whole analysis. A schematic of the microscopic analysis is shown in Fig. 9. The input data of the microscopic model are (1) average grain size d, (2) distribution of grain orientation, (3) remote applied stress tensor σ and (4) yield stress σ_Y, at each evaluation point. Sufficient number of trials of the Monte Carlo simulation is required to make reasonable distributions of (1) the fracture toughness k_fM and (2) the direction of fracture surface n_M. The number of evaluation points can generally be reduced due to the symmetry.

Fig. 9.

Schematic of the Monte Carlo simulation of the microscopic analysis. (Online version in color.)

5.3. Macroscopic Analysis by Model Synthesis

An integrated macroscopic model as the multiscale model bridging the large gap in scale is formed using a new approach of “model synthesis” by systematically incorporating (1) the preparatory macroscopic finite element analysis and (2) the Monte Carlo simulation for the microscopic analysis into (3) the macroscopic analysis for crack propagation and arrest. The procedure in the integrated macroscopic model is composed of the two parts, i.e., in the part 1: assignment of the results (σ and σ_Y) obtained by the above (1) the preparatory macroscopic finite element analysis and (k_fM and n_M) obtained by the above (2) the Monte Carlo simulation of the microscopic analysis in each unit cell, and in the part 2: simulation of crack propagation and arrest by the above (3) the macroscopic analysis.

More detailed procedures of the proposed model synthesis applied to the multiscale model are described below for the integrated macroscopic analysis.

The stress tensor σ and the yield stress σ_Y obtained as the results of the finite element analysis are directly assigned to each unit cell in the macroscopic model. The distribution of the fracture toughness k_fM obtained by the Monte Carlo simulation of the microscopic analysis is assigned to each unit cell by the interpolation expressed as   

$$F\left[k_{\text{fM}},\mathbf{x}\right]=\sum _i{N}_i\left[\mathbf{x}\right]F_i\left[k_{\text{fM}}\right]$$(12)

where F[k_fM,x] is a cumulative probability distribution (CPD) function of k_fM at the unit cell whose coordinate is x (=(x,z)). F_i[k_fM] is a CPD function of k_fM at the i-th evaluation point. N_i[x] is a bi-linear interpolation function corresponding with the i-th evaluation point, which is the same as the shape function of the finite element method, as shown in Fig. 10. According to the calculated F[k_fM,x], the value of k_fM at the unit cell is specified at random. The same procedure for the k_fM cannot be adopted for the distribution of the direction of fracture surface n_M because it is not a scalar. Therefore, n_M at each unit cell is selected at random from the data sets of the respective evaluation points with the weighted probability of N_i[x]. A schematic of the calculation process of k_fM and n_M at each unit cell is shown in Fig. 11.

Fig. 10.

Interpolation function N_i[x]. (Online version in color.)

Fig. 11.

Calculation process of k_fM and n_M at each unit cell. (Online version in color.)

After the assignment processes of (1) stress tensor σ, (2) yield stress σ_Y, (3) fracture toughness k_fM, and (4) direction of fracture surface n_M, a macroscopic crack propagation analysis is performed. It is noted that the yield stress σ_Y is used for the calculation of crack closure effect by tear-ridge in the same manner as the microscopic model. For domain discretization and criterion of crack propagation in the macroscopic analysis, the same framework as that in the microscopic model is used as shown in Section 3.

6. Conclusion

The present paper proposed a new multiscale model by a “model synthesis” approach, as the first attempt to clarify the relationship between microstructures of steel and macroscopic brittle crack propagation and arrest behavior.

The multiscale model consists of two models: (1) a microscopic model to simulate cleavage fracture in the grain scale and (2) a macroscopic model to simulate brittle crack propagation and arrest behavior in the steel plate scale.

The same framework for the domain discretization and the criterion of crack propagation were employed in both the models, based on the same manner as the studies of Aihara and Tanaka.¹⁷⁾ In the framework, a simple two-dimensional domain discretization was performed but a three-dimensional crack propagation can be effectively modeled. The discretized unit cells in the microscopic model correspond to the respective grains. On the other hand, the discretized unit cells in the macroscopic model correspond to the entire domain of the microscopic model.

The microscopic model proposed by Aihara and Tanaka¹⁷⁾ is basically employed except the integration with the macroscopic model. The integration procedure between microscopic and macroscopic models is performed by using two physical quantities: (1) effective surface energy and (2) direction of fracture surface. In particular, for the calculation of the effective surface energy, it is assumed that the plastic work to form tear-ridge is dominant in the total absolute energy to form the macroscopic fracture surface. In addition, the Monte Carlo simulation is employed because it is expected that there is a large scatter in the results of the microscopic analysis due to the variation of distributions of microstructures such as grain size and orientation.

The integration between microscopic and macroscopic models to simulate the complex behavior of brittle crack propagation and arrest is required to bridge a large scale gap between 10⁰ m in macroscopic scale and 10⁻⁹–10⁻⁴ m in microscopic scale. Therefore, the proposed model synthesis for multiscale model as an integrated macroscopic model is performed by systematically incorporating (1) the preparatory macroscopic finite element analysis and (2) the Monte Carlo simulation of microscopic analysis into (3) the macroscopic analysis for brittle crack propagation and arrest in steel plate. The integration procedure is implemented by the assignment of physical quantities into each unit cell based on the interpolation methods as a one-way coupling algorithm for simplification.

Validation of the proposed model by comparing with the experiments and the detailed discussions of simulation results will be found in the second part of the present study.²⁷⁾

Acknowledgements

A part of this work was supported by Fundamental Research Developing Association for Shipbuilding and Offshore (REDAS). The authors would like to thank them.

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