Towards a theory of flow stress in multimodal polycrystalline aggregates. Effects of dispersion hardening

A.A. Reshetnyak; D. Cevizovic; Yu.P. Sharkeev

REVIEW 3 major objections 6 minor 13 references

Towards a theory of flow stress in multimodal polycrystalline aggregates. Effects of dispersion hardening

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Adding grain-boundary and dispersion phases to a polycrystalline aggregate shifts and lowers the yield-strength maximum; temperature moves the peak across the nanoscale.

desk verdict A narrow, honest extension of a two-phase model to dispersion hardening, but the phase-average scheme is stretched near the nanocrystalline maxima and the key constant is fitted to experiment. read the letter →

arxiv 1908.09338 v1 pith:VKZV7G7Z submitted 2019-08-25 cond-mat.mtrl-sci cond-mat.soft

D. Cevizovic , A.A. Reshetnyak , Yu.P. Sharkeev This is my paper · ORCID

classification cond-mat.mtrl-scicond-mat.soft

keywords flowstressHall-Petchlawmultimodalpolycrystallineaggregatesdispersionhardeninggrain-boundaryphasetemperature-dimensionaleffectdisloconnanocrystallinematerials

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends a quantized statistical theory of flow stress from single-mode polycrystalline metals to multimodal aggregates containing grain-boundary and dispersion phases. The authors claim that a volume-weighted mixture rule for crystalline, grain-boundary, pore, and third-phase stresses yields an analytic generalized Hall–Petch law that works across the whole grain-size range, from nanocrystalline to coarse-grained, and across temperatures from 150 K to 350 K. Their central quantitative prediction is that adding second- and third-phase contributions shifts the yield-strength maximum to smaller grain sizes and lowers its height, with the size of the shift set by grain-boundary angle, pore size, particle size, and temperature. If correct, this gives materials designers a closed-form stress–strain relation for dispersion-hardened and bimodal polycrystalline alloys and a temperature-dependent optimal grain size.

What carries the argument

The load-bearing object is the "dislocon", a quasiparticle carrying the unit dislocation energy $\tfrac{1}{2}Gb^3$, obtained from a quantized energy spectrum in each crystallite with $N=[d/b]$ levels. The flow stress follows from Boltzmann probabilities for defect generation and Taylor hardening, and the multi-phase extension is carried by the coherent equal-strain mixture rule of Eqs. (5)–(6): the integral stress is $(1-U_{\mathrm{dis}})$ times the weighted sum of crystalline, grain-boundary, and pore phases plus $U_{\mathrm{dis}}$ times the dispersion phase, with weights $f_1=1-nb/d$, $f_2=(n-m)b/d$, $f_3=mb/d$. This rule turns the one-phase formula into two- and three-phase generalized Hall–Petch relations, with a single parameter $m_0$ fixed by the experimental Hall–Petch coefficient $k(\varepsilon)$ in the coarse-grained limit and then applied at all grain sizes.

What would settle it

A direct test is to measure yield strength versus grain size for aluminum containing about 5% Cu particles of diameter $d_{\mathrm{dis}}=1.5\,d_0(\mathrm{Cu},T)$ at 150, 200, 250, 300, and 350 K with large-angle grain boundaries. The model predicts the yield-strength maximum moves from about 22.5 nm at 150 K to 8.7 nm at 350 K while its height falls; a measurement showing a fixed maximum grain size, a reversed temperature shift, or no reduction in maximum height would falsify the central claim. The constant-pore version predicts instead a maximum stabilized near 70 nm, so the two cases are distinguishable in the same experiment.

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Extended reading notes

Core claim

The central claim is that Eqs. (5)–(6), which combine the crystalline-phase stress with grain-boundary, pore, and dispersion-particle phase stresses weighted by volume fractions that scale linearly with $b/d$, produce an analytic stress–strain dependence for multimodal polycrystalline aggregates "in the entire range of grain sizes, values of temperature and accumulated strain, correctly reflecting the experimental data." On this basis the paper derives generalized Hall–Petch curves for $\alpha$-Fe, Cu, Al, Ni, $\alpha$-Ti, and Zr at $T=300$ K and finds that the extreme grain size $d_{\Sigma\mathrm{dis}0}$ and the maximum $\Delta\sigma_{\Sigma\mathrm{dis}}^m$ of yield strength are both reduced relative to the one-phase values ($d_{\Sigma\mathrm{dis}0}<d_0$; $\Delta\sigma_{\Sigma\mathrm{dis}}^m<\Delta\sigma_m$). It further predicts that second- and third-phase contributions shift the maximum: for small-angle grain boundaries the maximum remains higher than for large-angle boundaries, constant pores of size $d_P\approx d_0$ move the maximum into the experimentally accessible 80–150 nm range, and lowering temperature shifts the maximum toward larger grains while raising it. For Al with Cu particles, the model gives $d_{\Sigma\mathrm{dis}0}$ from about 8.7 nm at 350 K to 22.5 nm at 150 K in the large-angle case, and a temperature-stabilized value near 70 nm when pore size is held constant.

Load-bearing premise

The load-bearing premise is that all phases deform with the same strain and the integral stress is simply the volume-weighted sum of phase stresses, with one constant fitted at coarse grain sizes applied down to nanocrystalline sizes; the paper's own Footnote 1 flags this equal-strain, coherent-mixture treatment as an idealization because plastic deformation in each phase is generally heterogeneous.

Editorial extensions

If this is right

Engineers get a closed-form $\sigma(\varepsilon,d,T)$ for two- and three-phase polycrystalline alloys, so the optimum grain size for yield strength can be computed rather than scanned experimentally.
Adding a dispersion phase with the right size and stiffness hardens coarse- and ultrafine-grained regions but can weaken the nanocrystalline region, making particle size relative to $d_0$ a design lever.
The temperature-dimensional effect survives in multimodal aggregates: lowering temperature moves the yield-strength peak to larger grains, and increasing accumulated strain does the same, giving a predictable operating window.
With constant pores at $d_P\approx d_0$, the yield-strength maximum shifts into the sub-microcrystalline range (about 80–150 nm) and its height drops below 1 GPa, identifying a grain-size regime worth testing.
The same mixture rule predicts how grain-boundary angle changes the peak: small-angle boundaries keep a higher maximum than large-angle boundaries at fixed composition.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: the same volume-weighted mixture machinery could be inverted to design bimodal grain-size distributions, choosing $n$, $m$, $U_{\mathrm{dis}}$, and $d_{\mathrm{dis}}$ to place the yield-strength maximum at a target grain size; this inversion is not tested in the paper.
Editorial extension: if the equal-strain assumption fails, the most visible breakdown should be in the nanocrystalline region where the grain-boundary phase carries a large fraction of strain; in-situ diffraction during loading could measure phase strains and test the mixture rule directly.
Editorial extension: the pore phase enters with a negative sign, suggesting the model could be extended to damage and ductile failure by letting pore fraction grow with strain, a path the paper does not develop.
Editorial extension: the temperature scaling in Eq. (7) could be checked against existing published yield-strength data for Al and Cu without new experiments, since the paper presents theory only.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the paper is a narrow but honest extension of a two-phase model to three-phase (dispersion-hardened) polycrystals. The new items are Eq (6), the extremal-grain-size entries in Table 1, and the Al temperature scans in Fig 2. They are not in the authors' earlier papers [1-4]. Credit where due: the model is laid out explicitly, and the Summary openly calls for experimental verification of the temperature shift. The paper is not pretending to have data it does not have.

The soft spots are structural. The constant m0 is fitted to the experimental Hall-Petch coefficient k(ε), so the coarse-grained slope is imported from experiment rather than predicted. The base distribution (1) and flow stress (2) are quoted from self-cited preprints, not derived, and 9 of 13 references are to the authors' own program, which makes the paper hard to evaluate in isolation. The dispersion term in Eq (6) uses hand-chosen Udis and ddis and is never benchmarked against Orowan/Ashby or similar.

The more serious issue is the phase averaging. The weights in (5)-(6) are asserted to be volume fractions, but f1=1-nb/d goes negative for d < n b. The authors are aware of the boundary: they introduce d_LB and state that σΣ<0 below it. That already contradicts the Summary's claim of an analytic description for the 'entire range of grain sizes'. The stress-test note says the predicted maxima sit in the unphysical region; strictly, the Table 1 maxima d0 are just above d_LB (e.g., Al 13.6 nm vs 12 nm; Cu 14.4 vs 12 nm), so f1 is positive there—but only 0.12-0.25. The averaging scheme is therefore stretched to the point of near-invalidity exactly where the main predictions are made, and for any grain size below d_LB the model produces negative crystalline volume fraction and negative stress. That is a real limitation, not a cosmetic one.

The paper is for readers inside the authors' 'dislocon' program. It is a natural incremental step and shows clear thinking, but it is not experimentally anchored and the new predictions are untested. I would not cite it in my own work, and I would not put it on a reading group agenda. For peer review: I would send it to a referee rather than desk-reject, because the model is explicitly stated and the failure modes are identifiable. A competent referee should flag the m0 fit and the d_LB/negative-fraction problem.

Referee Report

3 major / 6 minor

Summary. This manuscript extends a previously proposed statistical, quantized-dislocation model of flow stress to multimodal polycrystalline aggregates by adding a second (grain-boundary) phase and a third (dispersion-hardening) phase. The central equations are Eq. (5) for the two-phase integral stress and Eq. (6) for the three-phase stress with dispersion particles, where the total stress is a volume-weighted sum of crystalline, grain-boundary, pore, and dispersion contributions. The authors report generalized Hall-Petch curves for BCC (α-Fe), FCC (Cu, Al, Ni), and HCP (α-Ti, Zr) materials at 300 K, and they study the temperature dependence of the yield strength and the extremal grain size for Al with Cu particles over 150-350 K. The headline claims are that the model provides an analytic description of the stress-strain dependence 'in the entire range of grain sizes, values of temperature and accumulated strain' and that the extreme grain size dΣdis0 and the maximum yield-strength difference ΔσΣdism are shifted by the second and third phases (dΣdis0 < d0; ΔσΣdism < Δσm).

Significance. If the model were valid, it would offer a closed-form, analytically tractable description of flow stress in multiphase polycrystalline materials, with falsifiable predictions for the location and height of the yield-strength maximum and its temperature shift. The paper has some strengths in this direction: it provides explicit equations, tabulated parameter values for six metals, concrete numerical predictions, and a clear statement of the equal-strain idealization in Footnote 1. However, the central quantitative predictions are undermined by an internal inconsistency in the mixture rule, described below, and by the fact that the coarse-grained Hall-Petch branch is fitted rather than predicted. As a result, the claimed generality and predictive power are not currently established.

major comments (3)

[Eq. (5), Table 1, Summary] The phase weights in Eq. (5) are f1 = 1 - n b/d, f2 = (n-m)b/d, and f3 = m b/d, and the manuscript repeatedly calls them volume fractions. With n ~ 100-10^2 and b ~ 0.25-0.3 nm, n b is tens of nanometers, whereas the predicted extremal grain sizes d0 and dΣdis0 in Table 1 lie in the range 12-28 nm. For Cu, taking n = 100 and b = 0.256 nm gives n b = 25.6 nm, so at the predicted d0 = 14.4 nm the crystalline-phase weight is f1 ≈ 1 - 25.6/14.4 = -0.78, which is negative. The Summary itself concedes that σΣ < 0 in the nanocrystalline region and introduces a lower bound d_LB for sample existence, but the predicted maxima are located precisely at or near that boundary. Therefore the defining equations become unphysical in exactly the regime where the paper's main predictions are made, and the claim of an analytic description 'in the entire range of grain sizes' is internally inconsistent. This is a load-bearing error, not a presentation issue.
[Introduction, text near Eq. (4)] The model constant m0 is determined from the experimental Hall-Petch coefficient k(ε) through the relation m0 = (π/(6√2)) · k²(ε)/((α_m G)² ε b) · M(ε)/M0. Consequently, the coarse-grained branch of every predicted σ(d) curve is enforced by the fit to k(ε), so the statement that the theory 'correctly reflects the experimental data' in the coarse-grained region is not an independent validation. More importantly, the same fitted m0 is then used to extrapolate into the nanocrystalline region where the phase weights become negative; the extrapolation is therefore uncontrolled with respect to the very parameter that sets the CG slope.
[Footnote 1 and Eqs. (5)-(6)] Footnote 1 explicitly admits that the equal-strain, coherent-mixture rule is an idealization because in general plastic deformation at each phase is heterogeneous. The quantitative predictions dΣdis0 < d0 and ΔσΣdism < Δσm are direct consequences of the volume-weighted sum in Eqs. (5)-(6) with equal strains. If the phases deform heterogeneously, the computed extremal sizes and their temperature shifts do not follow. The manuscript does not provide a test of this assumption or a bound on the resulting error, so the central predictions are conditional on an unvalidated mixture rule.

minor comments (6)

[Abstract] The abstract contains a duplicated and garbled phrase: 'FCC (Cu, Al, Ni) and HCP (Cu, Al, Ni) and HCP (α-Ti, Zr)' should list the lattice types once and correctly.
[General formatting] Several equations and inline formulas appear as corrupted or fragmentary text, for example the expression for m0 following Eq. (4) and the polynomial fit in Eq. (4). The authors should ensure that all mathematical notation is cleanly typeset and legible.
[Table 1] Table 1 is overloaded, with several rows (notably dΣdis0 and ΔσΣdism) combining multiple model variants in a way that is very hard to read. A clearer layout with separate columns or panels for small-angle GB, large-angle GB, and constant-pore cases would substantially improve accessibility.
[Introduction] The quasiparticle 'dislocon' is introduced with only a provisional name and is not used later in the derivation; if it is not needed, it should be removed or defined more precisely.
[References] Reference [12] appears in the reference list but is not cited anywhere in the body of the manuscript.
[Eqs. (5)-(6) and Table 2] The dispersion weight Udis is set to 0.01 in the main Hall-Petch section and to 0.05 in the temperature-dependence study for Al; the reason for this change and the range of admissible Udis values should be stated explicitly.

Circularity Check

1 steps flagged · score 6.0 of 10

The coarse-grained Hall-Petch slope and the maximum yield-strength increments are fitted inputs: m0 is defined from experimental k, so the 'theoretically reproduced' sigma(d) curves and Delta sigma_m values inherit that fit; the extremal-grain-size shift retains some independent content.

fitted input called prediction [HALL-PETCH LAW section, paragraph after Eq. (4) (m0 calibration); used in Table 1 and Summary]
"The value of 𝑚0 in (2) is determined for CG materials from the limit of the normal HP law (d>>b) at 𝜀=0.002, using a relation with an experimental value of the HP coefficient 𝑘(𝜀), 𝑚0 = 𝜋/(6√2) 𝑘²(𝜀)/((𝛼𝑚 𝐺)² 𝜀𝑏) ⋅ 𝑀(𝜀)/𝑀0, within the one-phase model approximation."

m0 is the only material-dependent coefficient of the grain-size term in Eq. (2), and the paper solves for it from the experimental Hall-Petch coefficient k. Because k fixes the d^{-1/2} slope, the 'theoretical' sigma(0.002,d) curves and the numerical maxima Delta_sigma_m and Delta_sigma_Sigma_dism in Table 1 are not independent predictions: they reproduce the input slope and inherit the fitted magnitude. The paper's 'good coincidence' with experimental sigma(0.002,d) is therefore partly guaranteed by construction. The extremal size d0 in Eq. (3) does not depend on m0, so the size-shift predictions retain some non-circular content; hence partial rather than total circularity.

full rationale

The clearest reduction is in the definition of m0: the paper calibrates its sole strength constant to the experimental Hall-Petch coefficient k and then presents the resulting Hall-Petch curves and maximum yield-strength increments as theoretical output. That is a fitted input called a prediction, and it directly affects the quantitative claims in Table 1 and the Summary. I did not score the author's reliance on [1-4] as circular by itself: this is a continuation paper that explicitly extends a previously introduced model, and the cited prior work is not invoked as a forbidden 'uniqueness theorem'. The temperature shift in Eq. (7) and the sign of the d_Sigma_dis0 changes are not forced by m0 alone, so the paper has independent content. A separate internal-consistency problem is noted: with f1 = 1 - n b/d, the weights in Eq. (5) become negative for d below n b (~25-58 nm), while the predicted one-phase extrema d0 in Table 1 lie at 13.6-28 nm; the Summary itself concedes sigma_Sigma < 0 near the NC boundary. This makes the 'entire range of grain sizes' claim physically invalid in the very regime of the predictions, but this is a correctness issue outside the circularity classification, so it is mentioned here rather than scored as a circular step.

Assumptions & free parameters 6 free parameters · 7 assumptions · 1 invented entities

The central claims rest on three unpaid layers: (1) the quantized dislocation-energy statistical theory of Refs [1-4], quoted without derivation; (2) the equal-strain coherent mixture rule of Eqs (5)-(6), which Footnote 1 flags as an idealization; and (3) hand-chosen or literature-fitted constants (m0 fitted to k, n, Udis, ddis, phase size ratios). The dislocon is an interpretative quasiparticle with no independent falsifiable handle in this paper.

free parameters (6)

m0 (per material) = α-Fe 3.66-5.11; Cu 2.57; Al 2.28; Ni 1.11; α-Ti 5.83-7.47; Zr 3.69
Defined from the experimental HP coefficient k(ε) via m0 = (π/(6√2))k²/((αmG)²εb)·M(ε)/M0; it sets the stress scale of every computed curve.
n = 100-10² (order only)
Constant controlling the second-phase weight (n b/d); stated to depend strongly on grain-boundary preparation, not measured.
Udis = 0.01 (Fig 1); 0.05 (Fig 2)
Weight of the dispersion (third) phase, chosen by hand; the Summary notes results change if Udis is increased.
ddis(Cu) = 1.5·d0(Cu,300K) = 21.6 nm
Average Cu-particle diameter chosen by hand; also used as a T-dependent input 1.5·d0(Cu,T) in Fig 2b.
Phase size ratios and pore sizes = (dGBs,dPs)=(0.025,0.025)d; (dGBg,dPg)=(0.114,0.114)d; dGB=0.15d; dP=d0
Second-phase grain and pore sizes are set as fixed fractions of the first-phase grain size d; dP = d0(Cu or Al) for the constant-pore runs. These choices drive the predicted extremal shifts.
α (dislocation interaction constant) = 0.38 (α-Fe), 0.35 (Cu), 0.97 (Al); ranges for Ti, Zr
Taken from literature [6,7] with wide quoted ranges; enters Eq (2) as the multiplier of Gb√ρ and therefore directly scales all predicted stresses and extrema.

assumptions (7)

ad hoc to paper Each crystallite has a quantized, equal-spaced energy spectrum of dislocation states with energy quantum ½Gb³, and defect-emergence probabilities are Boltzmann-distributed, P_n ~ exp(-nGbε³/(2NkT)).
Restated in the Introduction from the authors' prior works [1-3]; no independent experimental evidence is given for the quantization or for the 'dislocon'.
ad hoc to paper The dislocation density formula (1), ρ = M(0)(6√2/π)(m0/d²)ε(e^{M(ε)b/d}-1)^{-1}, holds as stated.
Quoted without derivation in this paper; its derivation resides in self-cited [1].
domain assumption Taylor strain hardening holds: τ = τf + αGb√ρ, with orientation factor m = 3.05.
Standard Taylor hardening framework from the plasticity literature, invoked in the Introduction.
domain assumption G(T) and bε(T) vary linearly with temperature with handbook coefficients αG, αd, via Eq (7).
Linear temperature approximation used for the temperature-dimensional effect; coefficients from [10,11].
ad hoc to paper All phases deform coherently as non-interacting objects with equal strain, so the integral stress is the volume-weighted sum in Eqs (5)-(6).
Footnote 1 explicitly calls this an idealization and says that in general plastic deformation in each phase is heterogeneous.
domain assumption The experimental Hall-Petch coefficients k(0.002) for the six metals (Table 1, from [2,6,7,10]) are correct input data.
The model constant m0 is defined through these literature k-values.
ad hoc to paper The constant n ~ 100-10² and the geometric phase ratios (0.025d, 0.114d, 0.15d) represent grain-boundary phase geometry.
n is said to depend strongly on the preparation of grain-boundary states; neither n nor the ratios are measured or derived.

invented entities (1)

dislocon
purpose: Quasiparticle interpretation of the unit-dislocation energy quantum ½Gb³, invoked to describe localized deformation bands and the Portevin-Le Chatelier effect.
Introduced in the authors' prior work [2]; this paper offers no falsifiable prediction or direct observation that would test the entity's existence independently.

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Pith. "Pith review of Towards a theory of flow stress in multimodal polycrystalline aggregates. Effects of dispersion hardening." pith.science (2026). https://pith.science/paper/VKZV7G7Z

@misc{pith2026190809338,
  author       = {Pith},
  title        = {Pith review of: Towards a theory of flow stress in multimodal polycrystalline aggregates. Effects of dispersion hardening},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VKZV7G7Z}},
  note         = {Machine review of arXiv:1908.09338}
}

abstract

We elaborate the recently introduced theory of flow stress, including yield strength, in polycrystalline materials under quasi-static plastic deformations, thereby extending the case of single-mode aggregates to multimodal ones in the framework of a two-phase model which is characterized by the presence of crystalline and grain-boundary phases. Both analytic and graphic forms of the generalized Hall-Petch relations are obtained for multimodal samples with BCC ($\alpha$-phase Fe), FCC (Cu, Al, Ni) and HCP (Cu, Al, Ni) and HCP ($\alpha$-Ti, Zr) crystalline lattices at $T=300K$ with different values of the grain-boundary (second) phase. The case of dispersion hardening due to a natural incorporation into the model of a third phase including additional particles of doping materials is considered. The maximum of yield strength and the respective extremal grain size of samples are shifted by changing both the input from different grain modes and the values at the second and third phases. We study the influence of multimodality and dispersion hardening on the temperature-dimensional effect for yield strength within the range of $150-350K$.

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Reference graph

Works this paper leans on

13 extracted references · 6 canonical work pages

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А.А.Reshetnyak, Peculiriaties of temperature dependence for generalized Hall-Petch Law and two-phase model for de- formable polycrystalline materials, submitted to Russ. Phys. Journal (2018). Arxiv:1805.08623[cond-mat.mtr-sci]

arXiv 2018
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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

А.А.Reshetnyak, Statistical approach to flow stress and generalized Hall-Petch law for polycrystalline materials under plastic deformations, submitted in Russ. Phys. Journal (2018), Arxiv:1803.08247[cond-mat.mtr-sci]

arXiv 2018

[2] [2]

А.А.Reshetnyak, Peculiriaties of temperature dependence for generalized Hall-Petch Law and two-phase model for de- formable polycrystalline materials, submitted to Russ. Phys. Journal (2018). Arxiv:1805.08623[cond-mat.mtr-sci]

arXiv 2018

[3] [3]

Reshetnyak, On statistical quantized approach to flow stress and generalized Hall -Petch law for deformable polycrystalline materials

A.A. Reshetnyak, On statistical quantized approach to flow stress and generalized Hall -Petch law for deformable polycrystalline materials. Temperature-dimensional effect. (submitted to Phys.Rev.B)

[4] [4]

Reshetnyak, Sharkeev Yu.P., Two -phase model of the polycrystalline aggregate with account for grain-boundary states under quasi -static deformation // AIP Conference Proceedi ngs

A.A. Reshetnyak, Sharkeev Yu.P., Two -phase model of the polycrystalline aggregate with account for grain-boundary states under quasi -static deformation // AIP Conference Proceedi ngs. – 2018. V.2051- p.020251 – e-print - arXiv:1809.03628[cond-mat.mes-hall]

arXiv 2018

[5] [5]

E.O. Hall. Proc. Roy. Soc. B 64, 474 (1951); N.J. Petch. J. Iron Steel Inst. 174, 25 (1953)

1951

[6] [6]

Glezer, E.V

A.M. Glezer, E.V. Kozlov, N.A. Koneva et al. Plastic Deformation of Nanostructured Ma terials – CRC Press, 2017, 334p

work page 2017

[7] [7]

Firstov, Yu.F

S.A. Firstov, Yu.F. Lugovskoi and B.A. Movchan, Structure, strength and fatigue resistance of micro - crystalline and micro-layer materials. – Kiev: Naukova Dumka, 2016, 171p. (in Russian)

2016

[8] [8]

Karlsson and G

B. Karlsson and G. Linden, Mat. Sci Eng., 17, 209 (1975)

work page 1975