A New Model for the Shear of Atomic Planes

Liubomyr Kozak ^* 

1. Ivano-Frankivsk National Technical Oil and Gas University, 15 Carpathian St., Ivano-Frankivsk, 76019, Ukraine

* Correspondence: Liubomyr Kozak 

Academic Editor: Changming Fang

Special Issue: New Advances in the Analysis and Applications of Crystalline Materials

Received: December 21, 2025 | Accepted: February 26, 2026 | Published: March 18, 2026

Recent Progress in Materials 2026, Volume 8, Issue 1, doi:10.21926/rpm.2601002

Recommended citation: Kozak L. A New Model for the Shear of Atomic Planes. Recent Progress in Materials 2026; 8(1): 002; doi:10.21926/rpm.2601002.

© 2026 by the authors. This is an open access article distributed under the conditions of the Creative Commons by Attribution License, which permits unrestricted use, distribution, and reproduction in any medium or format, provided the original work is correctly cited.

1. Introduction

Plastic deformation is a common phenomenon in crystalline materials and plays a key role in determining their mechanical behavior under external loading. In general, plastic deformation is an irreversible change in the shape of a solid caused by applied forces. At the microscopic level, it involves the shear of atoms over relatively large distances and leads to changes in the material’s structure and properties without macroscopic loss of integrity.

Early studies of plastic deformation in metals associated this process with the relative shear of atomic planes. This concept was illustrated by the classical Boas-Schmid model (Figure 1), in which the motion of atomic planes was compared to the slip of a stack of coins [1].

Figure 1 Boas-Schmid model.

In this model, shear was assumed to occur simultaneously across an entire atomic plane. Estimates based on this assumption led to theoretical values of the critical shear stress that exceeded experimentally measured values for technical metals by three to four orders of magnitude. This discrepancy indicated that a more detailed microscopic description of atomic-plane shear in crystalline materials was required.

By the 1930s, this problem motivated the development of the dislocation concept. In 1934, Taylor [2], Orowan [3], and Polanyi [4], independently and building on the theory of defects introduced by Volterra, proposed that plastic deformation does not involve the collective motion of all atoms in a shearing plane but is instead mediated by localized regions of atomic rearrangement associated with dislocations. This idea provided a physically consistent explanation for the relatively low stresses observed in experiments and became the foundation of the modern theory of plastic deformation in crystalline solids.

According to the current understanding, plastic deformation of crystalline materials is predominantly described in terms of dislocation motion, which produces relative shear between different parts of the crystal [1,5,6,7]. At the same time, the atomic-scale origin of low shear stresses remains closely related to the form of the interatomic potential and, ultimately, to the electronic structure of the material [8,9,10,11,12].

The dislocation model has played an important role in the development of solid-state physics and materials science, enabling the description of many experimentally observed features of plastic deformation. However, the dislocation approach does not provide an exhaustive explanation of the physical origins of plasticity.

In particular, its limitations become apparent when addressing the origin of direction-dependent low shear resistance and the magnitude of the Peierls barrier in metals.

In particular, the theoretical shear strength of an ideal crystal remains significantly higher than experimentally observed values [6]. The theory does not yet offer a fundamental explanation for the origin of mobile dislocations with low activation barriers, does not account for changes in the electronic structure in dislocation cores, and struggles to describe rotational deformation modes and collective plastic flow. At high dislocation densities, the assumption of a stable crystal lattice serving as a background for isolated defects becomes invalid.

An important manifestation of these limitations is the Peierls relief [7], which reflects a periodic variation in the potential energy during the displacement of atoms within a crystal plane. In dislocation theory, the discrepancy between the ideal and experimental shear strength is attributed to the localized motion of dislocations. However, this explanation presupposes the existence of mobile dislocations with low activation barriers in metals, whereas in covalent crystals, such barriers are substantially higher.

Moreover, the dislocation approach does not explain the selective nature of slip: in some crystallographic directions, dislocations move under low stresses, while in others their motion is strongly suppressed. Referring solely to energy barriers does not explain why the Peierls distortion is low in metals only along certain directions and high in covalent crystals, despite similar crystal symmetries. In general, the physical origin of these differences lies beyond the scope of the dislocation model.

The electronic structure of solids plays the key role in resolving this problem.

In this work, a model of atomic-plane shear in metals is developed that explicitly accounts for the role of electronic structure in shaping the interatomic potential. The proposed approach provides an analytical description of shear processes in crystalline materials and offers additional insight into the conditions under which low shear stresses can arise.

2. Model of Shear of Atomic Planes According to an Alternative Concept

In materials science, solids are represented as three-dimensional crystal lattices, with nodes corresponding to the centers of atomic oscillations. A characteristic feature of such a model is that the lattice nodes are located at the minima of the potential energy. Such crystal lattices are stable with respect to shear deformations. Therefore, the classical theory explaining the mechanical properties of solids—strength and plasticity—is based on the concept of a stable crystal lattice. Within this framework, solids are considered strong, while their plasticity is attributed to the presence and motion of special defects—dislocations. In perfect crystals, plastic deformation is therefore generally associated with the nucleation and motion of dislocations.

In contrast to the classical concept, an alternative concept of plasticity is proposed to explain the plastic behavior of materials, primarily metals.

According to the proposed concept, the mechanical behavior of solids is governed by the spatial distribution of the valence-electron density, which may be isotropic or anisotropic (Figure 2). This distribution determines the geometry of the interatomic interaction potential and, consequently, the stability of the crystal lattice.

Figure 2 Electronic structure and mechanical properties of solids [11].

Two limiting cases of electron-density distribution can be distinguished. In the isotropic case, a spherically symmetric (or nearly symmetric) interatomic potential is formed, for which the interaction force depends only on the interatomic distance, F = f(r), and the bonds are nondirectional. This type of interaction is characteristic of metallic bonding.

In the opposite limiting case, where the valence-electron density is anisotropically distributed, the interatomic force depends on both distance and spatial orientation, F = f(r, θ, ϕ).

Such a potential gives rise to directional (covalent) bonding, which is typically associated with high lattice stability, high strength, and limited plasticity, as observed in semiconductors and dielectrics.

For solids with a spherically symmetric interatomic potential—primarily metals—the crystal lattice may exist in a compressed state [11,12,13,14,15,16] such that atoms occupy positions that do not correspond to the minimum of their potential energy. As a result, the lattice becomes unstable with respect to shear, which provides the physical basis for plastic deformation.

The instability of the crystal lattice originates from self-compression caused by the spherically symmetric interatomic interaction potential typical of many solids, particularly metals. In metals, the crystal lattice can become compressed due to long-range interatomic interactions, which increase the potential energy of atoms and contribute to lattice instability. In this context, the term “instability” refers to a low energy barrier for the relative translation of adjacent atomic planes. Self-compression is accompanied by an increase in the potential energy of atoms, as a result of which atoms occupy positions that do not correspond to the minima of the interatomic potential. The instability of the crystal lattice thus determines its ability to undergo plastic deformation.

The assumption that plastic deformation of metals is caused by self-compression of the crystal lattice is supported by the well-known fact that non-plastic solids, when subjected to high hydrostatic pressure, become plastic and in some cases electrically conductive, thereby acquiring metallic properties.

3. Theoretical Studies of Shear of Atomic Planes in Two-Dimensional Models

The assumption of compression of the crystal lattice in metals was supported by theoretical studies of two-dimensional models using molecular mechanics simulations [17]. These studies showed that, for a spherically symmetric long-range interatomic potential, the crystal lattice becomes compressed [16]. The change in the potential energy of atoms during simultaneous shear of the entire atomic plane in different crystallographic directions was also analyzed for an infinite crystal lattice. When an atom is displaced together with the atomic plane (Figure 3a) along the [100] direction, which is the most densely populated by atoms, its potential energy decreases (Figure 3b, curve 1). The same behavior applies to all atoms in that plane. This result implies lattice instability with respect to small shear strains in the [100] direction. Consequently, the shear of atomic planes can occur with minimal energy input.

Figure 3 (a) Scheme for calculating the energy expenditure dE during the simultaneous shear of all atoms; (b) Potential energy change of an internal atom by a distance z in a two-dimensional crystal lattice along the [100] (curve 1) and [110] (curve 2) directions.

In contrast, shear along the [110] direction is accompanied by an increase in potential energy (Figure 3b, curve 2), which indicates that the lattice is stable in this direction. Therefore, the sliding of atomic planes during the deformation of real crystals occurs only along certain crystallographic directions.

While Figure 3 presents results for a simplified two-dimensional lattice under homogeneous shear, this study is intended as a conceptual proof of principle rather than a full description of three-dimensional, dislocation-mediated plasticity. Simplified models allow the essential physics to be illustrated more clearly, much as the operation of a gearbox can be more easily understood on a kinematic diagram than on a fully assembled mechanical drawing. The directional differences in shear energetics revealed in 2D provide qualitative insight into how low plane-to-plane translation barriers could facilitate plastic deformation along specific crystallographic directions in 3D crystals, a topic for future work.

The energy required for the shear of an atomic plane in a two-dimensional crystal was also calculated (Figure 4). The crystal was considered to have a two-phase structure consisting of the internal regions of an unstable lattice and a surface layer whose structure and properties differ from those of the internal regions. Such a crystal is in an unstable equilibrium (metastable state). Its stability was found to depend on size. This dependence is due to the variation in the energy required to shear atomic planes with crystal size. At small sizes, crystal stability is high since the energy required for the shear is large. As the crystal size increases, the required energy decreases, and at a certain size, such crystals become unstable, because the energy required for shear becomes negative.

Figure 4 (a) Calculation scheme; (b) The energy required for the shear of an atomic plane by a distance z = 0.1 Å in the [100] direction as a function of the number of atoms N in the plane (crystal cross-section).

The transformation of crystals from stable to unstable with increasing size occurs because energy is released when the internal atoms of the lattice shear - they move to positions of lower potential energy (Figure 3b).

However, the shear of surface atoms requires energy input. At small crystal sizes, the energy required for the displacement of surface atoms exceeds the energy released by the shear of internal atoms. As the number of atoms in the cross-section increases, the released energy also increases. At a certain number (about 500), it exceeds the energy required to displace surface atoms.

4. Comparison of Plastic Deformation Processes in Classical and Alternative Concepts

Plastic deformation of crystalline materials can be described within both the classical dislocation-based framework and alternative atomistic approaches. In both cases, plastic deformation proceeds through two principal mechanisms: slip and twinning. During slip, one atomic plane is displaced relative to another by several interatomic distances. In contrast, in twinning, a coordinated displacement of many atomic planes occurs over distances smaller than the interatomic spacing. While these deformation mechanisms are formally identical in both frameworks, their microscopic realization and physical interpretation differ.

Figure 5 schematically compares the processes of plastic deformation as described by the classical dislocation model and by the alternative atomic-plane shear concept proposed in this work.

Figure 5 (a) Shear of atomic planes by single shear of atoms; (b) Simultaneous shear of all atoms.

4.1 Microscopic Realization of Atomic-Plane Shear

According to the classical dislocation model, the shear of an atomic plane is described as a sequence of localized atomic rearrangements associated with the motion of dislocations (Figure 5a). In contrast, within the alternative concept, atomic-plane shear is treated as a cooperative displacement of atoms along the entire plane (Figure 5b). This distinction applies to both slip and twinning. In this sense, the alternative description corresponds to the original Boas-Schmid picture of collective plane shear (Figure 1), reformulated here in terms of electronic-structure-controlled interatomic interactions.

4.2 Surface Effects and Lattice Restoration

Within the dislocation framework, it is commonly assumed that when an edge dislocation reaches the free surface, the crystal lattice is locally restored. However, experimental observations such as Lüders bands—regions characterized by locally disturbed lattice structure along shear planes—indicate that lattice restoration at the surface may not always be complete. These observations are consistent with a cooperative shear scenario, in which extended regions of the lattice participate simultaneously in deformation (Figure 5b), rather than being confined to isolated defect cores.

4.3 Development of Plastic Deformation

In the classical description, dislocation motion accounts for the elementary act of plastic deformation, corresponding to the shear of an atomic plane by approximately one interatomic spacing. Continued deformation is then described through the nucleation and interaction of additional dislocations, for which various mechanisms have been proposed, including Frank-Read sources, dislocation climb, and the formation of different dislocation configurations (e.g., partial, extended, prismatic, or helical dislocations).

Within the alternative concept, plastic deformation is described as a cooperative shear process that can occur at low applied stresses without invoking pre-existing defects. Depending on the magnitude of atomic displacement, this shear may correspond to twinning (sub-interatomic displacements) or slip (displacements exceeding the interatomic spacing), as illustrated schematically in Figure 5b.

4.4 Interpretation of Strengthening

The two approaches also differ in their interpretation of strengthening during deformation. In the classical framework, strengthening is attributed to the accumulation and interaction of dislocations, which progressively increase the resistance to further shear. In the alternative concept, strengthening is associated with a structural transformation of the crystal lattice during deformation: atoms move toward configurations corresponding to lower potential energy, leading to a transition from an energetically unstable to a more stable lattice state.

4.5 Conceptual Comparison and Historical Context

Despite their differences, the two approaches share several common features. In both cases:

• plastic deformation is associated with lattice instability—localized and defect-related in the classical model, and cooperative and extended in the alternative one;
• deformation proceeds through slip and twinning mechanisms;
• atomic-plane shear occurs under relatively low applied stresses.

From the perspective of the alternative concept, plastic deformation can, within an idealized framework, occur in a crystal lattice without invoking pre-existing defects such as dislocations. In this view, the ability of crystalline solids to deform plastically may be regarded as an intrinsic property related to lattice instability in a defect-free crystal.

Similar ideas were explored in earlier studies that sought to construct physical theories of crystal plasticity based primarily on the properties of a regular crystal lattice rather than on defects [18,19,20]. These approaches emphasized that the defining feature of crystals is their periodic atomic arrangement, while plastic deformability is a fundamental material property. However, at the time, no atomic-scale model was available that could account for low-stress atomic-plane shear with predictive power comparable to that of the dislocation concept.

M. Born also articulated the idea of lattice instability as a driving factor for plastic deformation in the 1930s. Born demonstrated theoretically that certain crystal lattices typical of metals are unstable with respect to small shear deformations in specific crystallographic directions and suggested that this instability could play a key role in the physical theory of plasticity [20,21].

The alternative concept proposed in the present work is consistent with Born’s conclusions but extends them by identifying a physical origin of this instability. Specifically, the isotropic distribution of valence electrons in metals gives rise to a spherically symmetric interatomic potential, which leads to self-compression of the lattice and reduces its resistance to shear in selected directions. In this way, the proposed model provides a microscopic physical basis for lattice instability and offers a complementary interpretation of plastic deformation phenomena alongside classical dislocation-based descriptions.

5. Experimental Confirmation of Theoretical Results

From the theoretical results (Figure 6a)¹, it follows that the energy required to shear atomic planes in two-dimensional crystals depends on their size. The same relationship is expected for three-dimensional crystals. At small diameters, strength is expected to be maximized and to decrease with increasing diameter, following the predicted curve in Figure 6a.

Figure 6 (a) Predicted; (b) experimental – dependences of the “whiskers” elasticity limit from their diameter [22].

Experimental results supporting this prediction are presented in Figure 6b and Figure 7. Thin filamentary crystals, commonly referred to as whiskers, are known to exhibit extraordinarily high strength [22,23,24,25,26,27,28,29,30].

Figure 7 Dependence of the yield strength of metal whiskers on their diameter [26].

As whisker cross-sectional area increases, whisker strength decreases, reflecting the reduced influence of surface-related effects. At characteristic sizes exceeding approximately 50 μm, the strength approaches that of conventional technical metals.

A similar trend is observed in polycrystalline metals [28], where the yield strength decreases with increasing grain size (Figure 8), following the well-known Hall-Petch relationship.

Figure 8 Tensile curves of Al 99.3% with different grain sizes [28].

Within the framework of classical dislocation theory, whiskers' high strength is commonly attributed to their nearly ideal crystal structure. It is often regarded as experimental confirmation of the conventional dislocation-based concept of plasticity.

At the same time, these classical observations form the experimental basis of what is now more broadly recognized as size-dependent strength in crystalline materials, a phenomenon that has been discussed extensively in later reviews and modern interpretations [29,30,31].

Therefore, the force required to shear atomic planes in crystals is determined by the size of the crystals. In large, high-purity single crystals with an almost perfect structure, this force becomes extremely small—approaching zero. Such single crystals are intrinsically unstable, and thus metals in nature typically occur as polycrystalline aggregates of small crystallites. Nevertheless, bulk single crystals can be grown artificially in the laboratory.

It has long been recognized [1,18,32,33,34,35,36] that plastic deformation in crystalline materials may begin at extremely small stresses. As noted in the classical work of Erich Schmid and Walter Boas, “plastic flow in metal single crystals may occur at arbitrarily small stresses; thus, the true elastic (yield) limit of such crystals is essentially zero” ([1], p. 23). This conclusion is consistent with later theoretical considerations by Yakov Ilyich Frenkel, who noted that along the most densely packed crystallographic planes and directions, the hardness of an undeformed single crystal may become extremely small and, in the limiting case of an ideal crystal, may approach zero [19].

The low strength of bulk metallic single crystals is one of the main pieces of experimental evidence that the ideal crystal lattice of plastic materials is unstable with respect to small shear deformations and that such deformations can occur at very low shear stresses. Moreover, the more perfect these crystals are, the lower the required stresses become: “Experimental observations show that the more perfect the crystal structure, the greater its plasticity. For example, the slower a single crystal grows from a melt, the more perfect its structure becomes and the lower its yield strength. Annealing and relaxation significantly reduce the yield strength, while purer crystals with fewer impurities exhibit higher plasticity. Conversely, distortions such as grain boundaries in polycrystals or mosaic block boundaries in single crystals hinder plastic shear. These findings contradict the classical premise that plasticity is solely the result of dislocations” ([18], p. 143).

6. Strength of Thin Filamentary Crystals Containing Defects

As discussed earlier in Section 5, within the alternative concept, the strength of a defect-free crystal is primarily determined by its surface layer. At the same time, the internal regions do not initially carry the applied load. However, during plastic deformation, the internal layers of the crystal undergo structural rearrangement as atoms shift toward lower potential-energy positions. As a result, these internal regions become progressively strengthened.

Consequently, if relatively thick whiskers are subjected to controlled plastic deformation, their strength is expected to increase substantially. This expectation is confirmed experimentally. Figure 9 presents tensile data for copper single-crystal whiskers [25]. During plastic deformation of a thick whisker (curve 2), pronounced strengthening was observed, with the stress reaching approximately 1200 MPa, which exceeds the elastic limit of a thin whisker (curve 1). For comparison, curve 3 corresponds to a bulk copper single crystal.

Figure 9 Tensile stress-strain curves of copper single crystals: (1) thin and (2) thick whiskers; (3) bulk single crystal [25].

This behavior is not an isolated case. In [26], additional experimental results are reported for filamentary crystals and structurally similar formations that exhibit high strength despite a large number of defects.

These observations challenge the classical view that whiskers' high strength is solely a consequence of their nearly ideal, defect-free crystal structure. Instead, they indicate that the presence of defects does not necessarily reduce strength and, under certain conditions, may even contribute to strengthening, which is naturally explained within the framework of the alternative concept.

7. Experimental Confirmation of Simultaneous Atomic Shear During Slip and Twinning

Experimental evidence for the simultaneous displacement of atoms during plastic deformation is provided by the Portevin-Le Chatelier (PLC) effect [37,38,39,40,41]. At the macroscopic level, plastic deformation, as recorded by highly sensitive instrumentation, appears as small stress serrations (Figure 10) that are not always visible in conventional stress-strain diagrams.

Figure 10 The stress-strain and temperature-strain curves of 310S stainless steel at a deformation rate of 3.3·10^-4 s^-1 [40].

The PLC effect represents discontinuous yielding, or jerky flow, and corresponds to unstable plastic deformation. During loading, it manifests as the periodic nucleation and propagation of localized deformation bands, producing single or multiple short-term strain bursts.

The serrated stress-strain curve indicates the aperiodic occurrence of shear events. Within the classical dislocation model, shear of an atomic plane proceeds via successive individual atomic displacements (Figure 5a). In contrast, the proposed model assumes that shear occurs through the simultaneous displacement of all atoms with an atomic plane (Figure 5b). Thus, the PLC effect provides experimental evidence for simultaneous atomic shear.

7.1 Heat Release During Shear of Atomic Planes

Shear of atomic planes is accompanied by significant heat release, evidenced by a sharp temperature increase following each stress drop (Figure 10). This heat release is associated with atomic displacement into lower-potential-energy positions.

Direct measurements of the true specimen temperature during discontinuous slip were reported in [40]. Experiments were conducted on samples immersed in liquid helium at 4.2 K, using 310S stainless steel, which is stable against martensitic transformation (Figure 10). Each stress drop was accompanied by a temperature rise, followed by a relatively slow relaxation to the coolant temperature. The magnitude of the temperature jumps increased with strain, ranging from 4.2 to 10 K up to 50 K.

Temperature effects were also studied during compression of high-purity tantalum single crystals at 4.2 K [41]. As shown in Figure 11, a single stress drop during deformation at a rate of 3.3·10^-4 s^-1 was accompanied by a corresponding increase in temperature. Larger stress drops produced higher temperature rises.

Figure 11 (1) Stress drop during a single deformation burst; (2) temperature rise during deformation of tantalum [42].

The release of latent heat results from atomic displacements into lower-energy positions, analogous to the heat released during crystallization. Intense heat release and its fluctuations significantly affect interatomic bond strength. This process promotes crack nucleation in surface layers, reduces whisker yield stress, and influences the amplitude of elastic serrations.

Overall, heat release provides important experimental confirmation that, during slip and twinning, atoms are simultaneously displaced into positions of lower potential energy.

It is also noteworthy that the step-like temperature increases are accompanied by bursts of acoustic emission [42], which provides additional evidence that shear occurs via the simultaneous displacement of all atoms within an atomic plane.

The above experimental results provide qualitative support for the proposed model of atomic-plane shear in crystals. Additional experimental evidence supporting this model and demonstrating its advantages is presented in Refs. [11,12,13,14,15,16].

8. Conclusions

1. In solids with an isotropic distribution of valence electrons, the interatomic interaction is characterized by a spherically symmetric potential. As a result, the crystal lattice becomes compressed, potentially leading to instability.
2. Because the crystal lattice is unstable, the shear of atomic planes can occur with a low energy barrier, with atoms moving toward positions of lower potential energy.
3. The stability and strength of real crystals are ensured by surface layers and internal structural defects, including dislocations.

The experimental data discussed in this paper suggest that the spatial distribution of valence electrons in solids largely determines the resistance of atomic planes to shear and thus governs the stability of the crystal lattice. At the same time, dislocation-based interpretations of plastic deformation may be viewed as a particular manifestation of this more general mechanism.

Acknowledgments

The author thanks Alexander Andreikiv, Alexander Cyrulnyk and Yaroslav Saliy for constructive comments on earlier drafts of the paper.

Author Contributions

L.K.: conceptualization, formal analysis, investigation, methodology, validation, writing—original draft, writing—review and editing.

Competing Interests

The author has declared that no competing interests exist.

Data Availability Statement

This article has no additional data.

AI-Assisted Technologies Statement

AI-assisted tools (ChatGPT) were used only for language editing. All scientific content was developed by the author, who reviewed and verified all AI-edited text and accepts full responsibility for the manuscript.