Multilevel Prediction of Mechanical Properties of Samples Additively Manufactured from Steel 308LSi

Abstract

This study employs a multilevel modeling approach to describe the deformation of specimens made from austenitic Wire Arc Additive Manufactured (WAAM) steel 308LSi. Two WAAM processing modes were investigated: (1) the Cold Metal Transfer (CMT) method and (2) Cold Metal Transfer combined with interlayer deformation strengthening (hammer peening/forging). Test specimens were cut from the deposited walls at 0° and 90° relative to the deposition direction. The grain and dendritic structures of the specimens were analyzed using optical stereomicroscopy. A statistical multilevel model has been developed, accounting for the features of the grain-dendritic and defect structures under various technological deposition modes. Parameter identification and model verification were conducted based on experimental data from uniaxial tensile tests of 308LSi steel specimens. The maximum deviation of the numerical results from the experimental data during the identification stage under uniaxial tensile loading did not exceed 3%, and during the verification stage it did not exceed 10%; the overall mean deviation did not exceed 1% for the identification stage and 2% for the verification stage. The model effectively captured the anisotropic mechanical behavior of WAAM-processed samples. The maximum calculated yield strength 360 MPa was obtained for specimens cut at an angle of 45°, while the minimum value 331 MPa was observed for vertically oriented specimens. Specimens subjected to interlayer forging (hammer peening) exhibited isotropic material properties. Explicit multilevel modeling, incorporating the presence of MnO oxide inclusions located within the austenite matrix, was performed. The results showed good correlation with experimental data and confirmed the localization of fatigue cracks at the phase boundary-matrix-oxide interface.

1. Introduction

Production of components from austenitic stainless steel 308LSi by arc wire deposition (WAAM (Wire Arc Additive Manufacturing)) represents one of the most promising and cost-effective methods for manufacturing large-scale and geometrically complex metallic structures for the aerospace industry [1,2,3]. Advancements of this technology require precise tuning of process parameters: input energy, scanning strategy, layer thickness, feed rate, and others. This is critical for obtaining products with the desired operational properties [4,5,6].

A key feature of additive manufacturing via arc welding is the formation of a unique material structure driven by directional crystallization and heat dissipation conditions during layer-by-layer fabrication [4,5,6]. This structure exhibits pronounced anisotropy, primarily due to the dominance of elongated γ-austenite columnar grains with dendritic substructure, oriented perpendicular to the layer deposition direction and possessing preferential <100> crystallographic orientation [7,8,9]. Such columnar grains can reach hundreds of micrometers in length and tens of micrometers in width [10], and are capable of penetrating through the deposited layers [11,12]. Within these grains, a dendrite-foam substructure forms with low-angle boundaries [8,9,13]. Mechanical testing of standard specimens cut at various angles to the deposition direction reveals significant variations in elastic, plastic, and strength properties [14,15]. The interlayer cold forging technology [8,16] is an effective method to reduce mechanical property anisotropy and enhance operational characteristics of WAAM products. This hybrid technology enables the refinement of the grain-dendritic structure and the dispersion of texture [17,18].

The relevance and growing adoption of WAAM technology have led to the development of a broad class of models used to describe the mechanical behavior of structures and components produced by this process. The most widely used are macrophenomenological models (see, for example, [19,20,21]). These models are computationally efficient but exhibit limited predictive capability, as they do not explicitly account for the state and evolution of the microstructure. It is well known that WAAM materials are characterized by elongated, oriented grain-dendritic structures and internal defects (lack of fusion, pores, and inclusions), which lead to anisotropy of mechanical properties. The absence of their explicit consideration reduces the predictive capability of such models.

Advancements of additive manufacturing technologies require efficient mathematical models capable of quantitatively and in-detail describing the relationship between microstructure and macro-scale material properties [4,5,6]. As noted above, the structure and properties of austenitic steel 308LSi significantly depend on the manufacturing process parameters, which determine the degree of anisotropy and substantially influence the operational characteristics of the products [22,23,24]. Precise models that can predict the behavior of products in various service conditions are essential, accounting for microstructure evolution, mechanisms of inelastic deformation, and failure [7,8].

The multilevel approach to mathematical modeling of mechanical properties effectively describes the state and evolution of the microstructure during thermomechanical loading, enabling quantitative prediction of the material’s response and its service life under different operational impacts [25,26,27]. Such models provide a physically justified account of property anisotropy, strengthening, and failure mechanisms in materials produced by additive manufacturing methods [28,29]. The advantage of the proposed approach lies in its capacity to comprehensively consider defects, morphology, and texture of grains (dendrites), allowing for the description of the macro-response of the material across a broad range of operating conditions with high reliability [9,10,13].

A wide range of multiscale models has been developed to describe specimens and structures manufactured by WAAM technology [30,31,32,33]. Among multiscale models, direct, self-consistent, and statistical approaches can be distinguished. Direct models reproduce the material microstructure with the highest fidelity and, consequently, provide the highest accuracy [30,31,32]. However, direct models require substantial computational resources and are currently not applied to full-scale boundary-value problems associated with manufacturing and in-service performance of structural components. Other widely used self-consistent models [33,34] describe the microstructure in an efficient, homogenized manner, resulting in increased computational efficiency. Nevertheless, such models do not account for local interactions between structural elements (for example, the influence of defects on the surrounding matrix), which reduces their accuracy.

A compromise solution is the development and application of statistical multiscale models. These models are computationally efficient and allow the main structural features (dendrites and grains), including those in 308LSi steel, to be taken into account with sufficient accuracy. In this work, a statistical multiscale model of a representative volume element is proposed that accounts for the state of the grain-dendritic structure. This model is an effective tool for predicting the anisotropic material response while considering the key microstructural constituents, i.e., grains and dendrites.

Finer-scale defect structures (e.g., secondary-phase particles), which form during WAAM, have a decisive influence on strength characteristics. Within the statistical framework, these defects were not explicitly modeled and were instead implicitly incorporated through the model parameters. Therefore, to describe the formation of potential fatigue crack initiation sites, a direct crystal plasticity finite element method (CPFEM) multiscale model is proposed. This class of direct models acts as a digital microscope and allows detailed investigation of other types of defects, such as porosity and lack of fusion [31,35]. This framework can be applied to individual critical regions of the material. In this way, a hierarchical modeling strategy is formed, in which each model type is used most effectively to solve its corresponding class of problems.

The aim of this work was to develop and apply a statistical multilevel material model to describe the deformation behavior of WAAM-processed steel 308LSi specimens, considering their microstructure obtained using additive and hybrid additive technologies. The developed model is intended for predicting the main anisotropic mechanical properties of the specimens under monotonic loading. The material model for the grain region was implemented within the CPFEM framework to describe the cyclic behavior of the 308LSi steel matrix containing MnO inclusions, with the aim of identifying potential regions of fatigue crack initiation.

2. Materials and Methods

2.1. Description of the Experimental Methods Used in the Study

The CMT layer-by-layer depositions of wire material and deformation treatment was implemented on one platform using a CNC machine with an integral SA7401H AIRPRO air hammer manufactured by Airpro Industry Corp. (New Taipei, Taiwan) [36]. The setup was equipped with Fronius welding (welding source, remote control setup, cooling setup, wire feeder, wire buffer, and welding torch (Fronius, Salzburg, Austria)), and forging systems (Airpro, Taichung, Taiwan). The Fronius Trans Puls Synergic 5000 welding source (Fronius, Salzburg, Austria) included an integrated feature set with digital control of CMT processing.

For the study, walls were fabricated using the CMT process and comparable walls were produced with additional interlayer deformation strengthening (hammer peening). Peening was performed using a SA7401H AIRPRO pneumatic hammer mounted on the system column and equipped with a spherical indenter with a tip radius of 15 mm. The standard operating pressure was 0.6 MPa, the indenter contact pressure was 0.2 MPa, and the feed rate was 3.3 mm/s, ensuring uniform treatment of the workpiece.

During deposition, walls with dimensions of 175 × 18 × 75 mm (length × width × height). The parameters of the CMT deposition mode are presented in

Table 1. Deposition parameters.

Process Parameter	Value
Arc current I, A	110
Filler wire feed rate vs, m/min	6
Travel speed vd, m/min	0.6
Shielding gas flow rate Q, L/min	30

.

Specimens were extracted from the WAAM-deposited walls using an electrical discharge machining (EDM) process at 0° and 90° relative to the deposition direction for uniaxial tensile testing (Figure 1a) and to evaluate the anisotropy of the mechanical properties of the fabricated deposits with respect to the deposited layers. The specimen geometry is shown in Figure 1b. The experiments were performed using an Instron 8801 servo-hydraulic testing system designed for static and dynamic mechanical testing.

Wire electrodes OK Autrod 308LSi ESAB (ESAB-SVEL Plant, Saint Petersburg, Russia) were used to produce the experimental deposits. The chemical composition of the wire is given in

Table 2. Chemical composition of wire and deposited specimens.

	Element Concentration, wt %
	C	Cr	Ni	Si	Mn	P	S	Fe
Wire308LSi
according to EN ISO 14343-2017 [37]	≥0.03	19.5–21.0	9.0–11.0	0.65–1.00	1.50–2.10	≥0.030	≥0.020	base
Deposited metal
Standard deviation (σ)	0.012	19.62	9.511	0.653	1.873	0.024	0.017	base
Coefficient of variation (υ, %)	0.00082	0.122	0.061	0.024	0.019	0.00082	0.00082	−
Standard deviation (σ)	6.833	0.622	0.641	3.675	1.014	3.417	4.824	−

. Microstructural investigations were performed on cross-sectional specimens and microsections were prepared by grinding and polishing the surface. To identify the microstructure etching was carried out using a Vasiliev’s reagent composed of 500 mL hydrochloric acid, 250 mL sulfuric acid, 100 g copper sulfate, and 500 mL water. Observations were performed using an Altami CM0745-T (Altami, Saint Petersburg, Russia) optical stereomicroscope and an Altami MET 1T (Altami, Saint Petersburg, Russia) inverted light microscope with up to 1000× magnification, using Altami Studio 3.5 software (Altami LLC., St. Petersburg, Russia).

The chemical composition of the deposited metal was determined by X-ray fluorescence (XRF) analysis, and metallographic studies were conducted on the specimens. To determine the chemical composition of the deposited samples, a high-resolution (3–10 nm) scanning electron microscope (HITACHI S-3400N, Hitachi Ltd., Tokyo, Japan) equipped with a Bruker EDS XFlash 4010 detector (Bruker Corp., Billerica, MA, USA) was used at an accelerating voltage of 5 kV. The results are summarized in Table 2. The observed structures of the experimental samples after deposition are shown in Figure 2 and Figure 3.

Metallographic analysis indicated that deposition without interlayer deformation produced a macro- and microstructure typical of austenitic steels, consisting of long columnar grains spanning multiple layers, with a dendritic microstructure of columnar dendrites. Deposition with interlayer deformation resulted in an equiaxed dendritic structure with a small fraction of columnar dendrites in certain layers, and a reduction in microstructural size was observed.

The obtained metallographic results (presented in the following paragraphs) are fully consistent with our previous studies [8] and those of other authors [9,10,13,38,39,40,41,42,43,44,45]. Therefore, generalized data from these works were used when developing the mathematical model.

The phase composition of AM-specimens is predominantly represented by γ-austenite with an FCC lattice, in combination with δ-ferrite (BCC lattice), whose volume fraction ranges from 3 to 8%, and, according to some data, up to 12% [8,10]. Additionally, finely dispersed carbides and oxides are observed, amounting to up to 2.5% [8,41]. Hence, the steel under consideration is characterized by a multiphase structure with pronounced heterogeneity.

Within the large columnar grains, a dendritic-cellular substructure of γ-austenite with low-angle boundaries has been identified [9,42]. Epitaxial growth of dendrites occurs across the deposited layers, forming elongated columnar elements oriented transversely to the build direction [8,13]. The material structure strongly depends on processing parameters. During layer-by-layer deposition without interpass deformation, large columnar grains are formed, consisting of bundles of first- and second-order γ-austenitic dendrites. In the interdendritic regions, δ-ferrite and carbides/oxides are observed, with respective volume fractions of 5–8% and up to 2.5%. Grain boundaries are predominantly high-angle, while dendritic boundaries are low-angle; the latter, enriched with secondary phases, hinder dislocation slip and thus strengthen the material [10,43]. The height of a single deposited layer is approximately 1.5 mm, while the crystallite height reaches 5–7 layers (7.5–10.5 mm). The interdendritic spacing varies between 30 and 40 μm, dendrite width between 60 and 80 μm, and crystallite width between 300 and 720 μm [10,44].

To improve the service performance of components, the application of interlayer cold forging technology is highly relevant [8,16]. Microstructural analysis of such specimens revealed the formation of predominantly fine-grained equiaxed grains with a dispersed dendritic substructure, as well as an increased fraction of carbide particles within the interdendritic regions [8,45]. Plastic deformation induced by cold forging increases internal stresses and dislocation density. Forging also alters grain growth directions during subsequent deposition and promotes the formation of new nucleation centers, resulting in a more crystallographically uniform and fine-grained structure [8,46].

During WAAM deposition (as in other multilayer welding processes), the resulting microstructure, in terms of phase composition and relative content, primarily depends on the chemical composition of the alloy [47]. Changes in the processing parameters of the heat source (total heat input, travel speed) mainly affect the size and dispersion of the microstructural constituents. For example, 308LSi steel crystallizes into a dendritic austenitic structure with small amounts of δ-ferrite and carbides in the interdendritic regions [48]. Increasing the cooling rate over a wide range can lead to the formation of dendritic-cellular and fully cellular structures, a reduction in δ-ferrite and carbide content, and smaller constituent sizes [49]. The range of heat input parameters typically used in WAAM mainly allows adjustment of the dendrite size, and only under extreme thermal conditions can a dendritic-cellular structure be achieved [49].

2.2. Description of Theoretical Methods Applied in the Study

Within multilevel mathematical modeling, the austenitic stainless steel 308LSi is considered as a hierarchical system in which the key structural elements are γ-austenite grains with a well-developed dendritic substructure.

Figure 4 schematically illustrates the structural states of specimens for two technological modes: (a) without forging, and (b) with interlayer forging. In the first case, elongated columnar grains with a pronounced texture are observed, while in the second, a more dispersed structure with equiaxed crystallites and a blurred texture character is formed [8,17]. This process is accompanied by an increase in carbide fraction [8,41]. After forging, the layer height is approximately 1.2 mm, the interdendritic spacing decreases to 10–30 μm, and the grain size ranges from 20 to 60 μm.

The primary mechanism of inelastic deformation in 308LSi steel is intragranular dislocation slip [50]. The strengthening of the material is governed by the interaction of dislocations with grain, dendrite boundaries and secondary-phase particles (δ-ferrite and carbides) [28,29,51]. The texture formed during additive manufacturing induces a pronounced anisotropy of plastic flow [28,29,51]. Interlayer forging results in an additional increase in yield strength according to the Hall-Petch relationship, due to grain refinement [45]. In some cases, rearrangement of the dislocation structure and possible phase transformations (TWIP and TRIP effects) may occur. However, their manifestation is limited by thermal conditions and is not always observed in WAAM-produced specimens [52,53,54].

The microstructure and phase composition have a direct impact on fracture mechanisms. Under monotonic and cyclic loading, cracks typically initiate near hard-phase particles (δ-ferrite, carbides, and oxides) due to the mismatch in elastoplastic properties between adjacent phases [55,56]. Stress localization in these regions leads to the formation of microcracks, which usually propagate along dislocation slip bands. Crack growth is predominantly transgranular. Although the trajectory often deviates at grain boundaries, slowing further propagation [56]. Larger austenitic grains facilitate stress relaxation and crack blunting, resulting in slower crack growth [57].

The construction of the mathematical model for inelastic deformation is based on a statistical two-level approach [27]. The schematic representation of the proposed multilevel model for the structure of 308LSi steel is shown in Figure 5. The complete mathematical formulation is detailed in [27]. At the macroscale, kinematic loading conditions, defined by the velocity gradient, are applied to the material’s RVE.

At the meso-scale (the level of individual grains), computations are performed to determine the mechanical response and internal structural variables. Effective quantities, such as stress and elastic and plastic properties, are determined by averaging over the macro-scale. This method accounts for both the state and the evolution of the microstructure under external loading.

The upper (macro-) level corresponds to the representative volume consisting of a statistically sufficient number of γ-austenitic grains, approximated as ellipsoids. For these grains, statistical laws of distribution are formulated for the lengths of principal semi-axes and their orientations, as well as for the crystallographic lattice orientations (texture).

At the mesoscale, individual grains are considered; each characterized by a specific ellipsoidal shape, crystallographic lattice orientation, and an effective internal substructure of first- and second-order dendrites (Figure 6). The structure of first-order dendrites (Figure 6a) is represented by an effective (average) ellipsoid (Figure 6b). Based on the characteristic sizes of first-order dendrites and grain dimensions, the number of dendrites within grain $n_{dn1}$ is determined as follows:

$$n_{dn1}={\displaystyle \frac{a_{gr}{b}_{gr}{c}_{gr}}{a_{dn1}{b}_{dn1}{c}_{dn1}}}$$

(1)

where $a_{gr},\hspace{0.33em}{b}_{gr},\hspace{0.33em}{c}_{gr}$ are the characteristic semi-axes of the grain ellipsoid, and $a_{dn1},\hspace{0.33em}{b}_{dn1},\hspace{0.33em}{c}_{dn1}$ are those of the effective first-order dendrite ellipsoid. Second-order dendrites are also described in an effective manner (Figure 6b). Their shape is assumed to be equivalent to a portion of a sphere, and, for a given dendrite surface area $s_{dn1}$ and surface number density $n_0$ the total number of second-order dendrites is calculated as follows:

$$n_{dn2}=n_0{s}_{dn1}{n}_{dn1},$$

(2)

The developed model explicitly incorporates the key physical mechanisms governing the material response: intragranular dislocation slip, interaction of dislocations with grain and dendrite boundaries as well as with secondary-phase particles, and lattice rotation within individual grains.

The formulation of the problem for describing the deformation of a representative volume element of a polycrystalline metal (alloy) using a two-level statistical model was stated as follows [27]. At each instant of time t within the considered loading interval, it is necessary to determine the macroscopic stresses K(t), the effective macroscopic elastic properties Π(t), and the mesoscopic variables that satisfy the following system of equations:

Macroscopic relations.

$$K=〈{\kappa }_{(i)}〉,i=1,\dots ,N,$$

(3)

$$\Pi =〈{\Pi }_{(i)}〉,i=1,\dots ,N$$

(4)

Mesoscopic relations.

Governing mesoscopic relations (for each crystallite $i=1,\dots ,N$, the crystallite number is omitted)

$${\kappa }^{cr}=\Pi :\left(\widehat{\nabla }{v}^{\mathrm{T}}-\omega -z^{in}\right),$$

(5)

relations between mesoscopic and macroscopic variables

$$l=\widehat{\nabla }{v}^{\mathrm{T}}=\widehat{\nabla }{V}^{\mathrm{T}}=L,$$

(6)

evolutionary and closure relations

$$z^{in}={\displaystyle \sum _{k=1}^K{\dot{\mathsf{\gamma }}}^{\left(k\right)}{b}^{\left(k\right)}{n}^{\left(k\right)}},$$

(7)

$${\mathbf{\tau }}^{\left(k\right)}={\mathbf{b}}^{\left(k\right)}{\mathbf{n}}^{\left(k\right)}:\kappa ,$$

(8)

$${\dot{\mathsf{\gamma }}}^{\left(j\right)}={\dot{\mathsf{\gamma }}}_0{\left({\displaystyle \frac{{\mathbf{\tau }}^{\left(k\right)}}{{\mathbf{\tau }}_c^{\left(k\right)}}}\right)}^{m}H\left({\mathbf{\tau }}^{\left(k\right)}-{\mathbf{\tau }}_c^{\left(k\right)}\right){\dot{\mathsf{\gamma }}}_0=\sqrt{L:L},$$

(9)

$${\mathit{\tau }}_c^{(k)}={\mathbf{\tau }}_b^{\left(k\right)}+{\mathbf{\tau }}_{dis}^{\left(k\right)},$$

(10)

$${\dot{\mathbf{\tau }}}_{dis}^{\left(k\right)}={\displaystyle \sum _{j=1}^K{h}^{(kj)}{\dot{\mathsf{\gamma }}}^{\left(j\right)}},$$

(11)

$$h^{(kl)}=\left[q_{lat}+(1-q_{lat}){\mathsf{\delta }}^{(kl)}\right]h^{(l)}, h^{(l)}=h_0{\left|1-{\mathbf{\tau }}_{dis}^{\left(l\right)}/{\mathbf{\tau }}_{sat}\right|}^a,$$

(12)

$$\omega =\dot{o}\cdot o^{\mathrm{T}}=I\times (k_3{k}_1{k}_2-k_2{k}_1{k}_3+k_1{k}_2{k}_3):l^e,$$

(13)

$$l^e=\widehat{\nabla }{v}^{\mathrm{T}}-z^{in}$$

(14)

initial conditions

$${\left.\kappa \right|}_{t=0}={\kappa }_0, {\left.o\right|}_{t=0}=o_0, {\left.{\mathbf{\tau }}_c^{\left(k\right)}\right|}_{t=0}={\mathbf{\tau }}_{dis0}^{\left(k\right)}+{\left.{\mathbf{\tau }}_b^{\left(k\right)}\right|}_{t=0},$$

(15)

$${\left.{\gamma }^{\left(k\right)}\right|}_{t=0}={\gamma }_0^{\left(k\right)}, {\left.d_{gr}\right|}_{t=0}=d_{gr0}, {\left.d_{dn1}\right|}_{t=0}=d_{dn10}, {\left.d_{dn2}\right|}_{t=0}=d_{dn20},$$

(16)

under prescribed (loading) conditions $\widehat{\nabla }{V}^{\mathrm{T}}\left(t\right)$.

In the formulation, uppercase letters are used to denote macroscopic variables, while corresponding lowercase letters indicate mesoscopic variables. The following notations are adopted in the equations:

• At the macroscopic level
  • K is weighted Kirchhoff stress tensor at the macroscopic level, Π is tensor of effective elastic properties, $L=\widehat{\nabla }{V}^{\mathrm{T}}$ is gradient of macroscopic displacement velocities ($\widehat{\nabla }$ is Hamilton operator in the current Lagrangian coordinate system, V is the macroscopic velocity vector).
• At the mesoscopic level
  • $\kappa ={\displaystyle \frac{\mathsf{\rho }}{\hat{\mathsf{\rho }}}}\sigma$ is the weighted Kirchhoff stress tensor at the mesoscopic level, ${\kappa }^{cr}=d\kappa /dt+\kappa \cdot \omega -\omega \cdot \kappa$ is its corotational derivative, $\mathsf{\rho },\hat{\mathsf{\rho }}$ is density in the reference and current configurations, σ is Cauchy stress tensor at the mesoscopic level, $l=\widehat{\nabla }{v}^{\mathrm{T}}$ is gradient of mesoscopic displacement velocities (v is mesoscopic velocity vector), $l^e$, $z^{in}$ is elastic and inelastic components of the transposed gradient of mesoscopic relative velocity, $\mathsf{\rho }\left(\hat{\mathsf{\rho }}\right)$ is material density in the reference (current) configuration, ${\dot{\mathsf{\gamma }}}^{\left(k\right)}$ is shear rate on the k-th slip system, ${\dot{\mathsf{\gamma }}}_0$ is shear rate when the resolved shear stress reaches the critical value, m is rate sensitivity exponent of the material, $b^{\left(k\right)},n^{\left(k\right)}$ unit vectors along the k-th slip direction and normal to the slip plane, H is Heaviside function, ${\mathbf{\tau }}^{\left(k\right)},{\mathbf{\tau }}_c^{\left(k\right)}$ is resolved and critical shear stresses for the k-th slip system, K is number of slip systems in the crystallite, $\Pi$ is elastic properties tensor of the crystallite, which components ${\Pi }^{ijmn}$ are determined and constant in the basis k_i of rigidly rotating local coordinate system (LCS), rotating with spin $\omega$, $o$ is orientation tensor of the LCS relative to the laboratory coordinate system (LCS), $\widehat{\nabla }{\mathbf{v}}^{\mathrm{T}}=\dot{f}\cdot f^{-1}$ is transposed gradient of displacement velocities, f is deformation gradient, $\widehat{\nabla }{v}^{\mathrm{T}}-\omega$ is objective measure of the strain rate, ${\mathbf{\tau }}_{dis}^{\left(k\right)}$ is resistance to dislocation motion due to the dislocation structure [58], ${\mathbf{\tau }}_b^{\left(k\right)}$ is resistance to dislocation motion from grain and dendrite boundaries [59], $h^{(kl)}$ is matrix describing crystallite hardening due to dislocation-dislocation interactions, q_lat is latent hardening parameter, ${\mathbf{\tau }}_{sat}$ is saturation stress for slip, h₀, a are parameters describing slip system hardening, ${\mathsf{\delta }}^{(kl)}$ is Kronecker delta, a dot over a variable indicates the time derivative t, superscript “T” indicates tensor transposition of the corresponding second-order tensor, superscript “−1” denotes the inverse operation of the corresponding tensor, $〈\hspace{0.33em}\cdot \hspace{0.33em}〉$ is volume averaging operator for the corresponding mesoscopic quantity, ${\kappa }_0$ is weighted Kirchhoff stress tensor for the grain at the initial time (zero tensor for natural initial configuration), $o_0$ is orthogonal tensor defining the initial grain orientation, ${\mathbf{\tau }}_{dis0}^{\left(k\right)}$ is initial resistance to dislocation motion, ${\left.{\mathbf{\tau }}_b^{\left(k\right)}\right|}_{t=0}$ is initial resistance from grain boundaries, ${\dot{\mathsf{\gamma }}}_0^{\left(k\right)}$ are initial slip on the slip systems (equal to zero for natural configuration).

At large plastic strains, the model also accounts for grain refinement resulting from substructural evolution. To describe the strengthening contribution of structural boundaries, a generalized Hall–Petch relationship is employed [60,61,62], where the resistance to dislocation motion depends not only on the grain size but also on the characteristic dimensions of dendritic substructures. The following modified relations, extending those formulated in [58,59], are proposed:

$${\left.{\mathbf{\tau }}_b^{\left(k\right)}\right|}_{t=0}=f^{gr}{k}_y^{gr}{\left({\left.d_{gr}^{(k)}\right|}_{t=0}\right)}^{-0.5}+f^{dn1}{k}_y^{dn}{\left({\left.d_{dn1}^{(k)}\right|}_{t=0}\right)}^{-0.5}+f^{dn2}{k}_y^{dn}{\left({\left.d_{dn2}\right|}_{t=0}\right)}^{-0.5}$$

(17)

$$f^{gr}={\displaystyle \frac{s_{gr}}{s_{gr}+n_{dn1}{s}_{dn1}+n_{dn2}{s}_{dn2}}}$$

(18)

$$f^{dn1}={\displaystyle \frac{n_{dn1}{s}_{dn1}}{s_{gr}+n_{dn1}{s}_{dn1}+n_{dn2}{s}_{dn2}}}$$

(19)

$$f^{dn2}={\displaystyle \frac{n_{dn2}{s}_{dn2}}{s_{gr}+n_{dn1}{s}_{dn1}+n_{dn2}{s}_{dn2}}}$$

(20)

where the following notations are used: $f^{gr}$ is volume fractions of grain boundaries; $f^{dn1}$ is volume fractions of primary dendrites; $f^{dn2}$ is volume fractions of secondary dendrites per grain; $n_{dn1}$ and $n_{dn2}$ are numbers of primary and secondary dendrites per grain, respectively; $s_{gr}$, $s_{dn1}$ and $s_{dn2}$ are surface areas of the grain and characteristic dendrites of first and second order; $k_y^{gr}$ coefficient defining the resistance intensity of the corresponding boundaries to dislocation motion; $k_y^{dn}$ is coefficient defining the resistance intensity of dendrite boundaries to dislocation motion; ${\left.d_{gr}^{(k)}\right|}_{t=0}$, ${\left.d_{dn1}^{(k)}\right|}_{t=0}$ and ${\left.d_{dn2}\right|}_{t=0}$ are initial dislocation mean free paths within the grain and dendritic structures, determined by the intersection of the Burgers vector direction with the ellipsoidal geometry of each structural element.

The evolution of grain structure under deformation is modeled in two stages: (1) a geometrical change in ellipsoidal grain shape according to the applied kinematic deformation, and (2) grain refinement, described by a modified phenomenological relationship linking grain size to accumulated plastic strain [63]. For anisotropic response, this relation is applied individually to each semi-axis of the ellipsoid. In the reference configuration, the first and third semi-axes of the ellipsoid were assumed to coincide.

The initial ellipsoidal shape of a grain is characterized by a symmetric positive-definite tensor A_g0. Its eigenvalues ${\lambda }_1\le {\lambda }_2\le {\lambda }_3$ define the semi-axis lengths, while the eigenvectors specify their orientations. Under a deformation gradient F(t) the ellipsoidal shape evolves, and in the current configuration it is described by the following tensor:

$$A_g\left(t\right)=F^{-\mathrm{T}}\left(t\right)\cdot A_{g0}\cdot F^{-1}\left(t\right),$$

(21)

where $\cdot$ denotes the scalar (inner) product.

Grain refinement is described using a phenomenological relation of the form [63]:

$$d=d_{UFG}{\left(1+\left({\left(d_{UFG}/d_0\right)}^2-1\right)\exp \left(-k{\mathsf{\epsilon }}^n\right)\right)}^{-0.5}$$

(22)

where d is the grain size, $d_{UFG}$ is the limiting grain size, $d_0$ is the initial grain size, k, n are parameters of the material identified experimentally. To account for anisotropy, this relationship is applied separately to each semi-axis of the ellipsoidal grain:

$$\begin{array}{l}a=d_{UFG}{\left(1+\left({\left(d_{UFG}/{\left.a\right|}_{t=0}\right)}^2-1\right)\exp \left(-k{\mathsf{\epsilon }}^n\right)\right)}^{-0.5}\\ b=d_{UFG}{\left(1+\left({\left(d_{UFG}/{\left.b\right|}_{t=0}\right)}^2-1\right)\exp \left(-k{\mathsf{\epsilon }}^n\right)\right)}^{-0.5}\\ c=d_{UFG}{\left(1+\left({\left(d_{UFG}/{\left.c\right|}_{t=0}\right)}^2-1\right)\exp \left(-k{\mathsf{\epsilon }}^n\right)\right)}^{-0.5}\end{array}$$

(23)

Thus, the developed multilevel mathematical model provides a physically consistent description of the mechanical response of 308LSi polycrystalline steel, incorporating the fundamental mechanisms of inelastic deformation and the influence of both grain and dendritic substructures.

3. Results

3.1. Identification and Verification of the Multilevel Model for Describing Deformation of the Representative Volume of 308LSi Steel Samples

The developed multilevel constitutive model, describing the mechanical behavior of 308LSi steel, was identified based on experimental data obtained in the study (Figure 7a). For model identification, results from mechanical tests of samples without interlayer forging (with large columnar grains and dendritic structure) and with forging (refined equiaxed structure) were used (see Figure 4). In numerical experiments, three types of specimens were considered: (1) vertical specimens cut along the built direction; (2) horizontal specimens cut perpendicular to the deposited layer; (3) specimens cut at a 45° angle to the deposition axis. The sizes of structural elements in both states were determined from metallographic analysis: columnar grain width in the non-forged state was 300–720 μm, the height was 7.5–10.5 mm; the first-order dendrites were 60–80 μm wide, the height was matching corresponding grains; the second-order dendrites were 15–20 μm. The obtained data agree with the literature [10,44]. In the forged state, grain sizes decreased to 20–60 μm, and the degree of anisotropy was significantly reduced [8,45].

Uniaxial tensile experiments at room temperature with a strain rate of 0.001 s⁻¹ for both non-forged and forged samples are shown in Figure 7a. The model parameters were identified by minimizing the squared discrepancy between calculated and experimental stress–strain curves. Optimization was performed separately for each sample type, accounting for the specific structure and mechanical properties. The mathematical formulation is based on the foundational two-level model [27], and extended with relations (1)–(23). All identified model parameters are listed in

Table 3. Identified parameters of the multilevel constitutive model.

Parameter	Value	Source
Density, $\mathsf{\rho }$, kg/m3	~7800	[66]
Anisotropic elastic moduli [27], ${\Pi }_{1111},\hspace{0.33em}{\Pi }_{1122},\hspace{0.33em}{\Pi }_{1212}$, GPa	$$\begin{gathered} {\Pi }_{1111}=81.84, \\ {\Pi }_{1122}=55.08, \\ {\Pi }_{1212}=50.48 \end{gathered}$$	Identification procedure (experimental research data—loading diagram Figure 7a)
The exponent in Hutchinson’s law for shear rates [27], m	83.3	[27]
Latent slip hardening parameter [27,58], qlat	1.4	[27]
Saturation stress of slip [27,58], ${\mathbf{\tau }}_{sat}$, MPa	500	Identification procedure (experimental research data—loading diagram Figure 7a)
Parameters describing hardening [27,58], h0, a	$\begin{array}{l}{h}_0=0.35,\\ a=1.97\end{array}$	Identification procedure (experimental research data—loading diagram Figure 7a)
Initial critical slip stresses due to dislocation structure [58], ${\mathbf{\tau }}_{dis0}$, MPa	100	Identification procedure (experimental research data—loading diagram Figure 7a)
Initial critical slip stresses due to grain and dendrite boundaries [58], ${\mathbf{\tau }}_{b0}$, MPa	24	Identification procedure (experimental research data—loading diagram Figure 7a)
Burgers vector modulus, $b$, Å	2.58	[67]
Initial grain and dendrite sizes, $a_{gr}, b_{gr}, c_{gr}$, (μm) *	$\begin{array}{l}{a}_{gr}\in [300;\hspace{0.33em}750]\\ b_{gr}\in [7500;\hspace{0.33em}\mathrm{10,500}]\\ c_{gr}\in [300;\hspace{0.33em}750]\end{array}$	Experimental data of the study
Initial dendrite sizes of 1 rank, $a_{dn1}, b_{dn1}, c_{dn1}$, (μm) *	$\begin{array}{l}{a}_{dn1}\in [60;\hspace{0.33em}80]\\ b_{dn1}\in [7500;\hspace{0.33em}\mathrm{10,500}]\\ c_{dn1}\in [60;\hspace{0.33em}80]\end{array}$	Experimental data of the study
Initial dendrite sizes of 2 rank $r_{dn2}$, μm *	$r_{dn2}\in [15;\hspace{0.33em}20]$	Experimental data of the study
Grain boundary resistance coefficient to dislocation motion [59], $k_y^{gr}$, MPa/m0.5	0.15	Identification procedure (experimental research data—loading diagram Figure 7a)
Grain boundary resistance coefficient to dislocation motion [59] $k_y^{dn}$, MPa/m0.5	0.23	Identification procedure (experimental research data—loading diagram Figure 7a)

* Uniform distribution law applied to interval values.

. In a previous study, the parameters of the baseline two-level model [27] were subjected to a comprehensive analysis to assess their sensitivity to parameter perturbations and variations in the input data [64,65].

Figure 7a presents the comparison of experimental and computed stress–strain curves for forged and non-forged samples. The maximum deviation does not exceed 3%, confirming the accuracy of identification and adequacy of the model. Series of parametric simulations were conducted for different initial grain sizes; results for forged samples are shown in Figure 7b and are consistent with the Hall–Petch relation (Equation (17)).

To reproduce the texture of the non-forged samples, experimental direct pole figures (DPFs) for planes (100), (110), (111) were used [7], and Monte Carlo simulations [68] generated a representative sample of grain lattice orientations (Figure 8).

During the verification stage, the experimental and numerical results for horizontal specimens under uniaxial loading were compared (Figure 9a), with the overall mean error not exceeding 2%. Uniaxial tensile tests indicate that horizontal specimens exhibit a yield strength 5–7% higher than vertical specimens. Maximum yield strength is observed for specimens cut at 45°, which is 6–8% higher than horizontal specimens [14,69,70]. This effect is explained by differences in dislocation mean free paths in elongated grains along different loading directions. The multilevel model reproduces this anisotropy accurately: calculated flow stress for horizontal non-forged specimens is 6% higher than vertical; 45° specimens show a 6% increase relative to horizontal (Figure 9a). Figure 9b shows the calculated pole figures for the horizontal specimen, exhibiting the most pronounced texture at the end of deformation, which qualitatively agree well with the experimental data [71,72,73].

The developed two-level model demonstrates high quantitative and qualitative correlation with uniaxial tensile experiments, accounting for the influence of grain and dendritic structures, accurately describing anisotropic mechanical behavior. Verification against experimental and the literature data (discrepancy ≤ 6%) confirms the model adequacy for engineering applications. Simulation results show sensitivity of 308LSi steel mechanical properties to grain size, dendrite size, and specimen orientation relative to the loading axis, validating the model for a wide range of operating conditions.

3.2. Modeling the Behavior of 308LSi Austenitic Steel with Explicit Topological Consideration of MnO Inclusions Under Cyclic Loading

The multilevel model allows analysis of the effects of structural factors (grain size, dendrite size, crystallographic orientations) on mechanical behavior. For predicting fatigue life of WAAM-produced components, it is necessary to explicitly consider lower-scale structural elements [74,75]. A major defect type includes non-metallic oxide inclusions, which interact with the matrix creating local stress concentrations and initiating fatigue cracks [75,76].

During additive manufacturing of 308LSi steel, interaction of alloying elements with ambient oxygen forms MnO inclusions takes place [69,77,78,79]. Thermal cycles influence inclusion formation; insufficient heat reduces melt homogenization time, promoting local oxygen enrichment [80]. Inclusion clusters degrade mechanical properties, accelerate crack growth, and reduce strength [69]. Fractography confirms presence of voids and dispersed inclusions under fatigue [79,81,82]. The primary mechanism of cyclic failure involves localized stress and strain near inclusions, promoting microcrack nucleation and evolution [83].

MnO inclusions are predominantly spherical, 0.5–2 μm in diameter, most commonly ~1 μm [78,81]. As a brittle phase, MnO has elastic-plastic properties significantly different from the austenitic matrix [84,85] forming local stiff zones and microcracks at the matrix-oxide interface. MnO inclusions are treated as the primary factor reducing mechanical performance and fracture toughness under cyclic loading.

Finite element simulations in Abaqus were performed on a representative grain region containing a 1 μm spherical MnO inclusion. The domain size was ten times the inclusion diameter [69,79,86], ensuring accurate reproduction of stress–strain states near the inclusion [69,87,88] (Figure 10a). The boundary conditions were defined based on the type of applied loading program. Tension along the unit vector $a$, illustrated in Figure 10b, corresponds to the following equations:

$$v\cdot a=\overline{v},\hspace{1em}n\cdot \sigma \cdot \left(I-aa\right)=0,\hspace{1em}{R}_0\in \partial {\mathcal{P}}_0^1,$$

(24)

$$n\cdot \sigma =0,\hspace{1em}{R}_0\in \partial {\mathcal{P}}_0^{\mathrm{f}}$$

(25)

where $\partial {\mathcal{P}}_0^1$ and $\partial {\mathcal{P}}_0^{\mathrm{f}}$ are, respectively, the loaded and free material surfaces. ($\partial {\mathcal{P}}_0=\partial {\mathcal{P}}_0^1\cup \partial {\mathcal{P}}_0^{\mathrm{f}}$, $\partial {\mathcal{P}}_0^1\cap \partial {\mathcal{P}}_0^{\mathrm{f}}=\varnothing$, where $\partial {\mathcal{P}}_0$ is boundary ${\mathcal{P}}_0$); $\overline{v}$ is prescribed tensile velocity along $a$; $n$ is unit normal vector to $\partial {\mathcal{P}}_0$; and $I$ is second-order identity tensor.

The matrix behavior employed the constitutive model described in Section 1, neglecting dendritic substructure terms. Contact between particle and matrix was ideal. Matrix properties corresponded to 308LSi austenite (Section 2), while the inclusion behaved elastically with brittle failure, ultimate strength ~3 GPa, and elastic constants consistent with BCC MnO ${\Pi }_{1111}=226.4\hspace{0.33em}\mathrm{GPa}$, ${\Pi }_{1122}=\hspace{0.33em}114.9\hspace{0.33em}\mathrm{GPa}$, ${\Pi }_{1212}=79.0\hspace{0.33em}\mathrm{GPa}$ [84,85]. Crystal coordinate systems of particle and matrix coincided.

Cyclic loading was applied at the top boundary as a sinusoidal stress, amplitude near matrix yield stress, simulating realistic multicycle conditions [89]. A total of 80 cycles were simulated, sufficient for capturing microcrack nucleation and evolution [74,76,90]. Equivalent von Mises stress and plastic strain distributions after 40 and 80 cycles are shown in Figure 11.

The analysis shows pronounced stress–strain heterogeneity near the stiff inclusion. Hysteresis loops averaged near the inclusion exhibit plastic strain offset, indicating energy dissipation [90], and micro-crack evolution [91] (Figure 12).

Fatigue life assessment considers local stress–strain parameters [92]. Stored energy density [74,89] is the most informative quantity for predicting fatigue crack nucleation sites. Most energy dissipates as heat; the remainder accumulates in dislocation structures [74]. The plastic strain energy per cycle is computed as follows [89]: $\mathrm{\Delta }G=\mathsf{\xi }{\displaystyle \int \sigma :d{\epsilon }^p}$ where $\sigma$ and $d{\epsilon }^p$ are plastic stress and strain tensors, respectively, and $\alpha$ is the fraction stored in defect structures, which was accepted to be 0.05 [74,89].

Figure 13 shows stored energy fields after 40 and 80 cycles, with maxima at matrix-inclusion interfaces along ~45° to the applied load, correlating with potential crack growth paths [74,93]. The obtained results comply well with works in [74,94].

4. Conclusions

For the analysis of the deformation of WAAM-fabricated 308LSi steel specimens, produced by the CMT process with and without interlayer forging, both proprietary experimental results and the relevant literature data were used. Metallographic analysis of the specimen microstructure enabled the development of a representative description, which was incorporated into a constitutive statistical material model. Based on the observed microstructure, crystallographic texture, and properties of individual structural elements, a multiscale statistical model of the material state was developed. This model describes the deformation of a representative volume (statistical ensemble of grains), accounting for the morphology and anisotropy of the grain-dendritic structure and its evolution under external loading. The model is used to determine macroscopic mechanical properties through statistical averaging. Building on the statistical model, a CPFEM model was developed and implemented for a selected region of the grain matrix, including MnO nonmetallic inclusions. The study yielded the following key findings:

(1)

The statistical model provided good agreement between the calculated and experimental uniaxial tensile curves, with an overall mean deviation during the verification stage not exceeding 3%. The model also captured the anisotropy of the yield strength: the maximum calculated value was obtained for specimens cut at 45° (360 MPa), while the minimum value was for vertical specimens (331 MPa).

(2)

The CPFEM model demonstrated that the presence of MnO inclusions leads to local stress concentrations and accumulation of plastic deformation in their vicinity, forming potentially critical zones for fatigue crack initiation.