Efficient Crystal Plasticity Modeling of Lath Martensite Considering Block Boundary Effects

Abstract

An efficient extension of conventional rate-dependent crystal plasticity is formulated to include the influence of block boundaries in lath martensite. It is shown that this can be achieved by minor modifications to a conventional crystal plasticity model, providing an efficient modeling approach with a negligible additional computational cost. With this modification, the parameter related to lattice friction is made dependent on a Hall–Petch-type block size dependency, resulting in block strengthening. The boundaries are modeled as obstacles which restrict the motion of dislocations, resulting in lower slip activity and promoting dislocation accumulation close to the boundaries. The model is calibrated and validated using experimental data from the literature, and its capabilities are demonstrated in a series of simulation examples.

1. Introduction

Martensite can form in austenitic steel either through quenching or deformation of the material. Whereas common cooling sequences usually produce a body-centered tetragonal (bct) martensite phase, deformation of austenitic steel tends to render a martensitic phase which is either body-centered cubic (bcc) or hexagonal close-packed (hcp) [1]. Deformation-induced martensite is classically categorized as either stress-assisted or strain-induced, where the stress-assisted transformation typically occurs at stress levels below the yield stress of the austenite phase. At higher stresses, phase transformation takes place concurrently with plastic slip and gives rise to strain-induced phase transformation [2]. In [3], it is shown that martensite formed via stress-assisted transformation involves spontaneous nucleation of martensitic platelets, which results in a morphology similar to that obtained when martensite forms via quenching. Strain-induced martensite usually has a lath morphology where the laths form at defects such as deformation twins, stacking faults and isolated regions of hcp martensite in the parent grain [1,4]. In [3], it was found that the formation of lath-like martensite is associated with plastic slip activity in the parent phase. The effects of deformation mode on the martensite transformation were studied in [5]. It was shown that uniaxial tension promotes the growth of martensite with a lath structure, while biaxial tension results in the formation of martensite characterized by an irregular and blocky morphology.

As discussed in [6,7,8], the transformation of low-carbon austenitic steel into lath martensite is characterized by a subdivision of the austenite grain into packets, as illustrated schematically in Figure 1. A packet is defined as a region of parallel laths with a common habit plane. Since there are four closely packed $\left\{111\right\}$ planes in a face-centered cubic (fcc) austenitic grain, four different crystallographic packets can potentially exist in any given austenitic grain. Each closely packed plane in the austenitic grain has three different slip directions, which implies that each packet can be divided into three different and parallel blocks that correspond to the slip directions. Each block may comprise thin laths of two different martensite variants with a low relative misorientation of about ${10}^{\circ }$ and a sub-block between the two variants. This results in a total of 24 variants of lath martensite. The orientation relationship between martensite laths and the parent austenite grain is usually described according to the Kurdjumov–Sachs (KS) scheme [7].

The martensite lath morphology depends on the chemical composition of the alloy and on the size of the parent austenitic grain. It was, for example, shown in [9,10] that for low-carbon lath martensite, refinement of the parent austenite grain size results decreased packet and block sizes, making the packet size proportional to the parent austenite grain size. For smaller austenitic grains, namely those less than 10 $\mathsf{\mu }$m in size, not all blocks are formed, and proportionality is lost for such small parent austenite grain sizes. In [9], iron-nickel lath martensite was compared to low-carbon martensite and it was shown that the lath morphology was quite similar. The size of the martensite blocks can vary significantly. In [11], it was found that the block sizes for low-carbon lath martensite were between 2 and $8\mathsf{\mu }$m, depending on the size of the parent austenite grain. In [12], lath martensite was studied at different deformation temperatures, and block sizes in the range from 2 to $5\mathsf{\mu }$m were found.

In high-strength steels such as dual-phase steels and transformation-induced plasticity (TRIP) steels, the presence of lath martensite is primarily responsible for enhancing the material’s strength. The lath microstructure and its misalignment with the parent austenitic grain result in the high strength, toughness and fracture resistance of the material. The block morphology of lath martensite has a significant influence on the strength and toughness of the material, as discussed in [13]. A large body of research on the morphology and crystallography of lath martensite and its effect on the behavior of dual-phase materials at different scales exists. Numerical models, and in particular crystal plasticity models, are central to this research. In [14], a crystal plasticity model that accounts for mechanically induced martensitic transformation was considered. Each martensitic variant was, based on the KS relations, given a specific nucleation probability, and it was shown that during uniaxial tensile loading, martensite variants having the (001) axis parallel to the loading direction were less likely to form. For uniaxial compression loading, however, a larger amount of martensite was transformed as variants having the (001) axis parallel to the loading direction, as shown in [14,15]. A coupling approach of cellular automata with the crystal plasticity finite element model was used in [16] to simulate shear band formation in strain-induced martensitic transformation.

In [17,18], a finite-strain elastic-viscoplastic constitutive model was used to study the influence of the microstructure on the macroscopic stress–strain response in a ferritic-martensitic dual-phase steel. It was, for example, shown that the morphology of the martensite phase in dual-phase steel influences the overall ductility of the steel. Cyclic crystal plasticity finite element simulations of lath martensite were found in [19], where representative volume elements were used together with two power law-based flow rules to predict local stress and strain distributions within the martensitic microstructure. In [20,21], crystal plasticity simulations were used to investigate the influence of the retained interlath austenite on the mechanical behavior of lath martensite subgrains. It was found that the presence of austenite facilitated local deformation of the martensite. Dislocation density-based crystal plasticity models were used in [13,22,23] to investigate the dislocation density evolution ahead of a crack front in martensitic steel containing retained austenite. The effect of the martensitic variant distributions and orientations as well as the effect of block and packet boundaries on crack growth were studied. It was shown that by controlling the distribution of variant orientations of martensite and retained austenite packets, the fracture toughness can be improved.

The most important strengthening mechanism related to lath martensite is the fracture resistance provided by its block structure. In [24], uniaxial micro-tensile tests on specimens consisting of either a single packet or a single block were performed using a range of different block and sub-block boundaries. It was concluded that sub-block boundary strengthening was less effective than strengthening of the block boundary. Block boundaries are stronger obstacles to dislocation motion and hence contribute more to the hardening of the material. In [25], a micro-bending test of lath martensite was performed, and it was concluded that a block boundary contributes more than a sub-block boundary to the strength of lath martensite.

Extensive research has been carried out to show the relationship between the hardening of lath martensite and its sub-structure. In [26], tensile tests on martensitic steel quenched at different temperatures were performed. Hall–Petch relationships were found between the yield stress and the parent austenitic grain size, packet size and block width. In [11], two low-carbon lath martensites were studied, and a Hall–Petch relationship was found to be representative of the block size influence on the yield stress. In [12], where the morphology and strength of lath martensite was studied, it was shown that a Hall–Petch relationship exists between the hardening of lath martensite and the block size, suggesting that the block is the main microstructural unit that determines the strength of lath martensite.

The focus of the present study is to provide a compact and efficient means for incorporating the influence of block boundaries on the mechanical response of lath martensite without having to resort to more involved formulations or causing a severe reduction in the computational efficiency. A thermomechanically consistent crystal plasticity model that was previously established in [27] is used as a basis for the investigations. The model in [27] is enhanced by a Hall–Petch-type block size dependence, and it is shown that this enhancement only requires minor modifications of a standard crystal plasticity model.

This paper is structured such that the central aspects of the crystal plasticity model are provided first in Section 2. Next, Section 3 details the proposed model modification, whereby block strengthening is added and some initial examples of the effects are given. Section 4 and Section 5 provide two simulation cases—bending of micro-cantilever beam and a micro-tensile test—to give a more complete view of the capabilities of the proposed formulation. In the latter case, block boundaries are parallel and perpendicular to the tensile direction. Concluding remarks close the paper in Section 6.

2. Constitutive Model

The basis for the present work is the crystal plasticity model outlined in [27]. This section briefly outlines the key properties of the model before proceeding to add the martensite block strengthening effect that is at the core of the present work.

A multiplicative split of the deformation gradient $\mathit{F}$ into an elastic and plastic part is assumed:

$$\mathit{F}={\mathit{F}}^e{\mathit{F}}^p.$$

(1)

The spatial velocity gradient $\mathit{L}=\dot{\mathit{F}}{\mathit{F}}^{-1}$ is additively split into an elastic and a plastic part as follows:

$${\mathit{L}}^e={\dot{\mathit{F}}}^e{\mathit{F}}^{e-1},{\mathit{l}}^p={\mathit{F}}^e{\mathit{L}}^p{\mathit{F}}^{e-1}$$

(2)

where ${\mathit{L}}^p={\dot{\mathit{F}}}^p{\mathit{F}}^{p-1}$.

The elastic part of the Helmholtz free energy depends on the volume change $J=\det \left(\mathit{F}\right)$, and the isochoric part of the elastic right Cauchy deformation tensor is defined as ${\mathit{C}}_i^e=J^{-2/3}{\mathit{F}}^{eT}{\mathit{F}}^e$. In [27], the elastic part of the Helmholtz free energy was chosen to be of a Neo-Hookean type, providing

$${\psi }^e({\mathit{C}}_i^e,J)={\displaystyle \frac{K}{2}}\left[{\displaystyle \frac{1}{2}}\left(J^2-1\right)-\ln \left(J\right)\right]+{\displaystyle \frac{G}{2}}\left[\mathrm{tr}\left({\mathit{C}}_i^e\right)-3\right]$$

(3)

where K and G are the bulk and shear moduli in the limit of small strains and where $\mathrm{tr}(·)$ defines the trace of a tensorial quantity.

The macroscopic plastic deformation is on the microscale, manifested through slip on individual slip systems. For an austenitic fcc crystal, slip can occur on four closely packed slip planes in three different slip directions. Thus, slip in an fcc crystal is defined by the 12 $\left\{111\right\}\langle 110\rangle$ slip systems. For the martensitic bcc crystal, slip occurs on six closely packed planes in two different slip directions, providing 12 $\left\{110\right\}\langle 111\rangle$ slip systems. The plastic part of the Helmholtz free energy is assumed to be a function of the internal variables $g^{\alpha }$ and associated with hardening on each slip system $\alpha$, which is chosen as follows:

$${\psi }^p\left(g^{\alpha }\right)={\displaystyle \frac{1}{2}}Q\sum _{\alpha =1}^n\sum _{\beta =1}^n{h}_{\alpha \beta }{g}^{\alpha }{g}^{\beta },$$

(4)

where $n=12$ is the number of slip systems, Q is a material parameter and the hardening matrix $h_{\alpha \beta }={\delta }_{\alpha \beta }+q(1-{\delta }_{\alpha \beta })$ models both self-hardening and latent hardening. The ratio between the hardening of the current slip system and the hardening due to interaction with neighboring slip systems is governed by q. Thermodynamic force $G^{\alpha }$, associated with the hardening $g^{\alpha }$, is found from Equation (4) as follows:

$$G^{\alpha }={\displaystyle \frac{\partial {\psi }^p}{\partial g^{\alpha }}}=Q\sum _{\beta =1}^n{h}_{\alpha \beta }{g}^{\beta }.$$

(5)

The Cauchy stress is defined as follows:

$$\mathit{\sigma }={\displaystyle \frac{1}{J}}{\displaystyle \frac{\partial \psi }{\partial \mathit{F}}}{\mathit{F}}^T.$$

(6)

Following [27], the evolution of the plastic part of the spatial velocity gradient ${\mathit{L}}^p$ is given by

$${\mathit{L}}^p=\sum _{\alpha =1}^n{\dot{\gamma }}^{\alpha }{\mathit{M}}^{\alpha }\otimes {\mathit{N}}^{\alpha },$$

(7)

where $n=12$ is the number of slip systems, ${\dot{\gamma }}^{\alpha }$ is the slip rate and ${\mathit{M}}^{\alpha }$ and ${\mathit{N}}^{\alpha }$ indicate the slip direction and the normal to the slip plane for a slip system $\alpha$, respectively. The slip direction and the normal to the slip plane are defined in the isoclinic intermediate configuration.

The slip rate ${\dot{\gamma }}^{\alpha }$ is governed by the power law:

$${\dot{\gamma }}^{\alpha }={\dot{\gamma }}_0{\left({\displaystyle \frac{|{\tau }^{\alpha }|}{G_0+G^{\alpha }}}\right)}^m\mathrm{sgn}\left({\tau }^{\alpha }\right),$$

(8)

where ${\dot{\gamma }}_0$ is the reference slip rate and m is the rate sensitivity parameter. With this choice of power law, slip is active at all times, and for large values of m, the slip rate evolution law becomes nearly rate-independent. In Equation (8), $G_0$ is introduced as a material parameter that governs the lattice friction. The resolved shear stress on the slip system $\alpha$, denoted by ${\tau }^{\alpha }$, is defined as follows:

$${\tau }^{\alpha }={\mathit{M}}^{\alpha }\mathsf{\Sigma }{\mathit{N}}^{\alpha },$$

(9)

where the Mandel stress tensor $\mathsf{\Sigma }$ is derived from the elastic part of the Helmholtz free energy in Equation (3):

$$\mathsf{\Sigma }=2{\mathit{C}}^e{\displaystyle \frac{\partial {\psi }^e}{\partial {\mathit{C}}^e}}.$$

(10)

The evolution of the hardening variables $g^{\alpha }$ is defined by

$${\dot{g}}^{\alpha }=\left(1-B{g}^{\alpha }\right){\displaystyle \frac{|{\tau }^{\alpha }|}{G_0+G^{\alpha }}}\left|{\dot{\gamma }}^{\alpha }\right|,$$

(11)

where the parameter B controls the saturation of the hardening. For further details and background on the consitutive model, see [27].

3. Block Size Strengthening Model

The block size strengthening can be explained by the fact that the block boundary acts as an obstacle to dislocation motion, causing dislocation pile-ups at the block boundary (see Figure 2). The lattice friction $G_0$ (i.e., the slip resistance) should be sensitive to the dislocation motion resistance posed by block boundaries. As the block boundary retards slip activity, it is assumed that the presence of block boundaries contributes to dislocation pile-ups and local slip accumulation. This retarding effect is modeled here by letting $G_0$ depend on the distance to the block boundaries. This mechanism suppresses slip activity ${\gamma }^{\alpha }$ and accelerates slip hardening $g^{\alpha }$ close to the block boundary, as illustrated in Figure 2.

As observed in experiments (e.g., [9,11,24,26]) where lath martensite samples of different alloy compositions are examined, the block size strengthening mechanism was found to be similar to a Hall–Petch type of block size dependence. Thus, the parameter $G_0$ in Equations (8) and (11), which governs the lattice friction, is proposed to depend on the presence of block boundaries according to

$$G_0\left(d\right)={\widehat{G}}_0\left(1+{\displaystyle \frac{k}{\sqrt{d}}}\right).$$

(12)

The material parameters ${\widehat{G}}_0$ and k control the influence of the block size strengthening, and d is the distance to the block boundary, indicated in Figure 2.

The effect of the block size dependence introduced in Equation (12) is schematically illustrated in Figure 3 for an idealized slip system $\alpha$ carrying a constant resolved shear stress ${\tau }^{\alpha }$. The hardening variable $g^{\alpha }$ and slip variable ${\gamma }^{\alpha }$ are then integrated using Equations (8) and (11) for different values of the distance d to the block boundary. The idealized hardening and slip variables are found by numerical integration using the Matlab ode45 solver with the material parameters in

Table 1. Model parameters needed to solve the idealized hardening and slip variable (Equations (8) and (11)).

Parameter	Value	Description
$\tau$	48 MPa	Resolved shear stress
Q	300 MPa	Hardening parameter
${\widehat{G}}_0$	40 MPa	Lattice friction parameter
${\dot{\gamma }}_0$	${10}^{-3}$ 1/s	Initial slip rate
m	30	Rate sensitivity parameter
$B_0$	150	Initial hardening saturation
$g_0^{\alpha }$	$7·{10}^{-3}$	Initial hardening

. The idealized hardening and slip variables are plotted as functions of the distance d in Figure 3. The hardening variable g takes on higher values closer to the block boundary, which reduce the slip rate $\dot{\gamma }$ in the vicinity of the block boundary. The slip $\gamma$ can be seen to increase with an increasing distance to the block boundary. Also, the influence of the parameter k is illustrated in Figure 3. It can be noted that for higher values of k, the influence originating from the block boundary persists over extended distances from the boundary. In summary, Equation (12) models the effect of dislocation accumulation (i.e., pile-up (increasing g)) and the decrease in slip rate ($\dot{\gamma }$) near the block boundary. In [9], different low-carbon lath martensite alloys were studied, and it was concluded that different ranges of the Hall–Petch parameter exist and must be identified for each specific alloy.

To further illustrate the behavior of the proposed model, the effect of block size strengthening between two block boundaries separated by $4\mathsf{\mu }$m was studied. Using the material parameters in Table 1 and $k=0.5\mathsf{\mu }$ ${\mathrm{m}}^{-0.5}$, the variation in the idealized hardening and slip variables between the block boundaries are illustrated in Figure 4. It is shown that the hardening variable g had its maximum at the block boundaries (i.e., at $x=\pm 2\mathsf{\mu }$m), and the slip variable $\gamma$ had its minimum at the block boundaries at $x=\pm 2\mathsf{\mu }$m. It can also be noted that both the slip and hardening variables were symmetric around the block boundaries, and they intersected at $x=0$.

4. Micro-Bending Test of Lath Martensite

In [25,28], micro-bending experiments on Fe–$23.00$ mass% Ni alloy were performed to study the influence of block boundaries and sub-block boundaries on the mechanical response of lath martensite. The tested micro-cantilever beam had a $10\times 10$$\mathsf{\mu }$m² cross-section and length of $50\mathsf{\mu }$m, and it was subject to a line load that was applied 10 $\mathsf{\mu }$m from the free end of the beam (see Figure 5). In this study, the crystal plasticity model, detailed in Section 2, together with the proposed block size strengthening model in Section 3 are used to simulate the micro-bending test. Following [28], four specimens were studied: one containing a sub-block boundary (SP1) and three containing different block boundaries (SP2–SP4) (see Figure 6). In

Table 2. The six considered Kurdjumov–Sachs variants in a given packet, with $\gamma$ and ${\alpha }^{\prime }$ denoting austenite and martensite, respectively.

Variant No.	Plane Parallel ($\mathit{\gamma }$)//(${\mathit{\alpha }}^{\mathbf{\prime }}$)	Direction Parallel ($\mathit{\gamma }$)//(${\mathit{\alpha }}^{\mathbf{\prime }}$)
v1	$\left(111\right)//\left(011\right)$	$\left[\overline{1}01\right]//\left[\overline{1}\overline{1}1\right]$
v2		$\left[\overline{1}01\right]//\left[\overline{1}1\overline{1}\right]$
v3		$\left[01\overline{1}\right]//\left[\overline{1}\overline{1}1\right]$
v4		$\left[01\overline{1}\right]//\left[\overline{1}1\overline{1}\right]$
v5		$\left[1\overline{1}0\right]//\left[\overline{1}\overline{1}1\right]$
v6		$\left[1\overline{1}0\right]//\left[\overline{1}1\overline{1}\right]$

, the KS variants v1–v6, which form a packet, are specified. The cantilever beam in SP1 consisted of lath martensite of variant v1 above the sub-block boundary and variant v4 below the sub-block boundary. In samples SP2–SP4, the cantilever beams consisted of variant v1 and variant v2, as shown in Figure 6. It was assumed that the parent austenitic grain had an initial orientation defined by the Euler–Bunge angle set $({\phi }_1,\mathsf{\Phi },{\phi }_2)=(0,0,0)$.

The simulations were performed in 3D, and the specimens were discretized using 6-node linear triangular prism elements with 2 integration points. The elements triangular faces were oriented parallel to the $xy$ plane. The crystal plasticity model was implemented as a user subroutine, UMAT, in Abaqus Standard. A total of 24,216 elements and 16,860 nodes were used in the simulation. The linear triangular prism elements were computationally efficient and facilitated a smoother transition from 2D to 3D simulations. Following [25], a prescribed displacement with a displacement rate ${\dot{u}}_y=0.1\mathsf{\mu }$m/s was applied as shown in Figure 5. The material parameters are presented in

Table 3. Material and model parameters obtained from the calibration of the constitutive model against the micro-bending experimental data adapted from Ref. [28].

Parameter	Value	Description
G	70 GPa	Shear modulus
K	180 GPa	Bulk modulus
Q	400 MPa	Hardening parameter
${\widehat{G}}_0$	370 MPa	Lattice friction parameter
q	$1.4$	Ratio between self-hardening and latent hardening
${\dot{\gamma }}_0$	$0.05$ 1/s	Initial slip rate
m	30	Rate sensitivity parameter
$B_0$	150	Initial hardening saturation
$g_0^{\alpha }$	${10}^{-5}$	Initial hardening
k	$0.5\mathsf{\mu }$ ${\mathrm{m}}^{-0.5}$	Block size dependence parameter (block boundary)
k	0	Block size dependence parameter (sub-block boundary)

. For the specimen with a sub-block boundary (SP1), where $k=0\mathsf{\mu }$ ${\mathrm{m}}^{-0.5}$, and for the specimens with block boundaries (SP2–SP4), the parameter value $k=0.5\mathsf{\mu }$ ${\mathrm{m}}^{-0.5}$ was used.

The model response is plotted in Figure 7 along with experimental data from [25,28]. It can be concluded that the load level for the specimens with the block boundaries (SP2–SP4) was higher than the load level for the specimen with the sub-block boundary (SP1). This is in accordance with the findings in [28], where it was observed that a block boundary contributes more to the strength than a sub-block boundary. The load drop seen in the experimental results for (SP2–SP4) was explained in [25,28] as being caused by dislocation pile-ups and further dislocation propagation across the block boundary, which was not included in the model proposed in this study.

Figure 8a shows the Cauchy stress component ${\sigma }_{xx}$ distribution obtained at the displacement $u_y=5\mathsf{\mu }$m. It can be seen that the compressive stress magnitude at the lower surface was higher for the specimens with the block boundary (SP2), where ${\sigma }_{xx}^{min}=-2000$ MPa, compared with the specimen with the sub-block boundary (SP1), where it was ${\sigma }_{xx}^{min}=-1700$ MPa. Similarly, the maximum tensile stresses at the upper surface were also higher in the specimen with the block boundary (SP2), where ${\sigma }_{xx}^{max}=1000$ MPa, than in the specimen with the sub-block boundary (SP1), in which ${\sigma }_{xx}^{max}=900$ MPa. This hardening effect was due to the presence of the block boundary.

In Figure 8b,c, the distributions of ${\gamma }^{sum}$ and $g^{sum}$ are defined by

$$\begin{array}{ccc}\hfill {\gamma }^{sum}& =& \sqrt{\sum _{\alpha }{\left({\gamma }^{\alpha }\right)}^2}\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill g^{sum}& =& \sqrt{\sum _{\alpha }{\left(g^{\alpha }\right)}^2}\hfill \end{array}$$

(14)

and are shown for SP1 and SP2 at $u_y=5\mathsf{\mu }$m, respectively. These quantities represent the overall slip and hardening across the specimen, thus simplifying the interpretation rather than analyzing slip and hardening for an individual slip system. Figure 8b shows that the distribution of the accumulated slip ${\gamma }^{sum}$ was higher in the specimen containing a sub-block boundary (SP1) compared with the specimen with the block boundary (SP2). This was a result of the block boundary acting as an obstacle for dislocation movement, and this is in line with the observations made in [25]. Due to the block boundary being an obstacle to dislocation movement, the sum of slip hardening $g^{sum}$, defined in Equation (14), was higher for the specimen with the block boundary (SP2) than in the specimen with the sub-block boundary (SP1) (see Figure 8c).

5. Micro-Tensile Test on Lath Martensite

In [29], a micro-tensile test was performed on multi-layered steel to characterize the stress–strain behavior of a lath martensite structure. This experiment is simulated in this section to complement the micro-cantilever study in the previous section.

5.1. Block Boundaries Aligned with the Tensile Load

The micro-tensile specimen had the gauge dimensions $20\times 50\mathsf{\mu }$ ${\mathrm{m}}^2$ with a thickness of $20\mathsf{\mu }$m. It was discretized using 6-node linear triangular prism elements with 2 integration points. A total of 15,464 elements and 10,175 nodes were being used in the simulations. Following [29], the specimen contained lath martensite of variant v1 and variant v3, as illustrated in Figure 9, with the material parameters in

Table 4. Material and model parameters obtained from calibration of the constitutive model against the micro-tensile experimental data adapted from Ref. [29].

Parameter	Value	Description
G	300 GPa	Shear modulus
K	600 GPa	Bulk modulus
Q	700 MPa	Hardening parameter
${\widehat{G}}_0$	200 MPa	Lattice friction parameter
q	$1.4$	Ratio between self-hardening and latent hardening
${\dot{\gamma }}_0$	$5·{10}^{-4}$ 1/s	Initial slip rate
m	30	Rate sensitivity parameter
$B_0$	100	Initial hardening saturation
$g_0^{\alpha }$	$0.02$	Initial hardening

. It was assumed that the parent austenitic grain had an initial orientation defined by the Euler–Bunge angle set $({\phi }_1,\mathsf{\Phi },{\phi }_2)=(0,0,0)$. A prescribed displacement rate of $({\dot{u}}_x,{\dot{u}}_y)=(0.1\mathsf{\mu }$m/s$,0)$ was applied on the right-hand side of the specimen. The left hand side of the specimen $(x=0)$ was fully clamped.

In the simulated stress–strain response shown in Figure 10, the true stress was calculated as $F/A$, where F is the total force applied on the right-hand side of the specimen and A is the deformed minimum cross-sectional area of the gauge section of the specimen. The logarithmic strain is calculated as follows:

$$\epsilon =ln\left({\displaystyle \frac{l}{L}}\right)$$

where l is the deformed length of the gauge section and L its original length. For $k=0$, the yield stress was significantly lower than when the block boundary effects were being considered (i.e., for $k=0.5\mathsf{\mu }$ ${\mathrm{m}}^{-0.5}$), and the yield stress coincided with the experimental results in [29].

The stress distribution at $\epsilon =0.05$ in Figure 11 shows that stress concentrations were present along the block boundaries. In Figure 12a, the distribution of the accumulated slip ${\gamma }^{sum}$ at $\epsilon =0.05$ shows high slip activity at the center of the specimen. Interestingly, the stress and slip concentration regions seen in Figure 11 and Figure 12a coincide with the location of the ductile fracture of the specimen found in [29]. In [24], it was concluded that the fracture propagates from both specimen sides and intercepts at the block boundaries. This is a direct result of block boundaries restricting dislocation motion and slip activity being interrupted at the block boundaries.

The distribution in Figure 12b shows that the hardening variable $g^{sum}$ at $\epsilon =0.05$ attained its maximum at the block boundaries and decreased further away from the block boundary. This stems from of the proposed block boundary-strengthening formulation introduced in Equation (12).

5.2. Block Boundaries Perpendicular to the Tensile Load

To examine the effect of the block size strengthening model on the yield stress, several micro-tensile tests were simulated with different numbers of block boundaries. In Figure 13, an illustration of the tensile tests is shown, with l denoting the variable block width. The specimen dimensions, material parameters and boundary conditions were the same as in the Section 5.1 example.

Block widths of $l=3,5,10,15$ and $46\mathsf{\mu }$m, corresponding to $18,11,6,4$ and 2 block boundaries being present in the gauge section, respectively, were considered. Figure 14 shows that the yield stress depended linearly on the inverse square root of the block width. This means that a Hall–Petch-type relationship, as frequently observed in experiments (e.g., [9,11,24,26]), was retrieved from the model. The slope of the linear relationship between the yield stress and the inverse square root of the block width l was found for $k=0.1\mathsf{\mu }$ ${\mathrm{m}}^{-0.5}$ to be $0.2$ MPa ${\mathrm{m}}^{0.5}$ and for $k=0.8\mathsf{\mu }$ ${\mathrm{m}}^{-0.5}$ to be $1.4$ MPa ${\mathrm{m}}^{0.5}$. These values are in accordance with the results in [9,11,12,26], where different lath martensite alloys were examined and the slope was found to vary between $0.1$ and $1.8$ MPa ${\mathrm{m}}^{0.5}$.

For the specimen with four block boundaries and a block width $l=15\mathsf{\mu }$m, the distribution of the x component of the Cauchy stress ${\sigma }_{xx}$ at $\epsilon =0.05$ is shown in Figure 15. It was observed that the stress concentration was at its maximum in the center of the gauge section between the block boundaries. The accumulated slip variable ${\gamma }^{sum}$ and hardening variable $g^{sum}$ distributions at $\epsilon =0.05$ are shown in Figure 16. The accumulated slip was observed to be concentrated at the center of the gauge section, with the block boundaries acting as barriers to slip activity. The hardening reached its maximum between the block boundaries, while the entire gauge section exhibited generally high values for the hardening variable. These findings suggest that block boundaries, oriented perpendicular to the loading direction, weaken the gauge section by inducing high stress concentrations and slip accumulation at its center while simultaneously restricting dislocation movement. As a result, slip activity was hindered throughout the gauge section. Additionally, the central region of the gauge section experienced greater deformation between the block boundaries, further emphasizing their role as obstacles to dislocation motion.

6. Conclusions

A compact and efficient rate-dependent crystal plasticity model that incorporates block boundary effects in lath martensite was presented. The block boundaries are modeled as obstacles that restrict dislocation motion. This results in lower slip activity in the vicinity of a block boundary, which is accompanied by increased local hardening through dislocation accumulation.

The model was calibrated and validated using micro-bending experiments from the literature. It was shown that bending a specimen containing a block boundary requires higher load levels compared with a specimen containing a sub-block boundary, which is a direct result of the block boundary contributing to the macroscopic strength of the specimen. It was also shown that less slip activity occurred in the specimen with a block boundary compared with the specimen with a sub-block boundary. These simulation results agree with experimental data found in the literature.

The stress distribution in a micro-tensile test specimen of lath martensite was also studied using the proposed model, and it was observed that stress and slip concentration lines were inclined in the loading direction and provided the precursor to ductile fracture. These results are in accordance with the experimental results found in the literature, where the fracture surface of lath martensite was found to have a chisel edge shape and provided additional validations of the proposed model formulations.

In conclusion, the proposed crystal plasticity model provided simulation results which agree well with findings from experimental studies while also providing a Hall–Petch-type relationship of the yield stress as found in the literature. The proposed model comprises only a minor modification to standard crystal plasticity and offers an efficient avenue for simulation of the effects due to block size strengthening of lath martensite with the minimum added computational expense. The model can be applied for studying lath martensite, particularly to explore how the orientation of block boundaries relative to the applied load direction affects the material’s strength. Additionally, the model can be used to investigate the role of block boundaries in crack propagation and how blocks affect the fracture toughness.