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Fig. 1 Ericksen–Landau strain energy landscape for the s – h phase transformation: ( a ) plot of the GL-invariant s – h potential σ d in Eq. (5) for β = 1.2 and μ = 1 , displayed over a portion of the Dedekind tessellation on the hyperbolic plane ... More about this image found in Ericksen–Landau strain energy landscape for the s – h phase trans...

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$Snapshot illustrating the bursty evolution of the strain field over a portion of H during a β-driven h→s transformation test (β decreasing from 1.4 to 0.5). Supplemental Video available in the Supplemental Materials for the entire simulation. Panel (a) overlays the strain-field cloud directly on the energy landscape defined on the Dedekind tessellation of Fig. 1(b). For each β, the unimodular strain tensor of each body cell is represented by means of the bijection (Eq. (3)) as a dot on the energy surface, given by the GL-energy density σd in Eq. (5). The snapshot shows the case for β=0.74, with the s-wells (e.g., i, i+1, ζ,…) deeper than the h-wells (e.g., ρ, ρ−1). During avalanches, the strain-cloud predominantly follows paths along the β-evolving energy valley-floors (indicated by dashed red lines on the surface), see Refs. [26,27]. Avalanching cells (i.e., cells jumping energy basin during the last imposed β-decrement) are marked as green dots on the surface, the rest are shown in black. A large avalanche (in green) is highlighted in this snapshot. Beneath the energy surface, a corresponding 2D histogram depicts the clustering of strain values evolving with β on the Dedekind tessellation of H. The fraction of cells involved in the last strain avalanche is shown in green at the top of each bar; (b) vertical log-scale representation of the same 2D histogram as in (a), highlighting the fraction of cell-strains elastically stabilized on the non-convex regions of the energy surface.$

Snapshot illustrating the bursty evolution of the strain field over a porti...

in Reconstructive Martensitic Phase Transitions: Intermittency, Anti-Transformation, Plasticity, Irreversibility > Journal of Applied Mechanics

Published Online: May 9, 2025

Fig. 2 Snapshot illustrating the bursty evolution of the strain field over a portion of H during a β -driven h → s transformation test ( β decreasing from 1.4 to 0.5). Supplemental Video available in the Supplemental Materials for the entire simulation. Panel ( a ) ... More about this image found in Snapshot illustrating the bursty evolution of the strain field over a porti...

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$Transformation intermittency due to strain avalanching under thermal driving: (a)–(d) numerical simulation; (e)–(h) empirical observation. Panel (a) shows the computed discontinuous β-dependent hexagonal phase fraction in the h→s transforming body for decreasing β (Fig. 2). Snapshots (b) and (c) show examples of the transformation avalanches computed in the crystal, i.e., bursts of microstructural change in the simulation. The arrows indicate the corresponding phase-fraction jumps. The color coding in (b) and (c) distinguishes the h→s transformation avalanches, in blue, versus the smaller attending local anti-transformation s→h events, in yellow. Panel (d) gives the associated log-log plots of the heavy-tailed probability density distributions P(S) for the avalanche size S (see text), separately for the transformation (blue) and anti-transformation (yellow) events in the simulation (the dashed lines, with the indicated slopes computed from least-squares fitting, are drawn to guide the eye). Panel (e) shows the empirical phase fraction plot obtained from the analysis of the optically-recorded microstructures in a temperature-driven phase-transforming crystal from Ref. [18]. Snapshots in (f)–(g) show examples of transformation avalanches in the body during the thermally induced austenite-to-martensite phase change, with the same color coding as in (b)–(c). Panel (h) gives, as in (d), the associated heavy-tailed avalanche-size distributions, separately for the observed transformation and anti-transformation events.$

Transformation intermittency due to strain avalanching under thermal drivin...

in Reconstructive Martensitic Phase Transitions: Intermittency, Anti-Transformation, Plasticity, Irreversibility > Journal of Applied Mechanics

Published Online: May 9, 2025

Fig. 3 Transformation intermittency due to strain avalanching under thermal driving: ( a )–( d ) numerical simulation; ( e )–( h ) empirical observation. Panel ( a ) shows the computed discontinuous β -dependent hexagonal phase fraction in the h → s transforming bo... More about this image found in Transformation intermittency due to strain avalanching under thermal drivin...

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$Bursty evolution of mixed-phase s–h microstructure and associated strain-field cloud on H. The images refer to the thermally driven phase-transforming crystal, with gradually decreasing β from β=1.4 as described in Sec. 3. Panels (a)–(c) show three snapshots of the evolution of the β-dependent strain clustering during the test, as a 2D heat-map histogram on the Dedekind tessellation of H in Fig. 1(b). See also the 2D strain histogram on the “floor” in Fig. 2; the Supplemental Video available in the Supplemental Materials shows the strain-cloud’s evolution along the entire simulation. We see in these panels that, as in Refs. [26,27], under the slow thermal driving the strain-cloud path on H follows the valley floors in the energy landscape (red dashed curves), crossing EPN-boundaries and visiting ever larger portions of strain space as the RMT progresses. Panels (d)–(f) show the associated body deformation, characterized by the bursty evolution of the LIS-driven phase microstructure, with domains occupied by different variants of both the h and s lattice phases (color coding for the cell strains indicates the energy basin, as in panel (a)). For reference, panel (g) marks the position of the snapshots along the hexagonal phase fraction plot for decreasing β (blue line), the same as in Fig. 3(a). The discontinuity of this plot tracks a global aspect of deformation’s intermittency, indicated also by the orange spikes in the same panel, which show the bursty fraction of cell strains jumping energy basin at each β. Each burst is generated by a number of spatially separated strain avalanches in the body, as exemplified in Figs. 3(b)–3(c). The corresponding evolution of the s–h phase mixtures is accompanied by the creation and movement of lattice defects, as the dislocation in the detail inset to panel (e); see also Fig. 5(a). The inset to panel (f) shows one of the grain-like homogeneous-lattice domains produced by LIS activity, such as LIS-layering, as discussed in Sec. 3.2.$

Bursty evolution of mixed-phase s – h microstructure and associat...

in Reconstructive Martensitic Phase Transitions: Intermittency, Anti-Transformation, Plasticity, Irreversibility > Journal of Applied Mechanics

Published Online: May 9, 2025

Fig. 4 Bursty evolution of mixed-phase s – h microstructure and associated strain-field cloud on H . The images refer to the thermally driven phase-transforming crystal, with gradually decreasing β from β = 1.4 as described in Sec. 3 . Panels ( a )–( c ) s... More about this image found in Bursty evolution of mixed-phase s – h microstructure and associat...

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https://doi.org/10.1115/1.4068470

Published Online: May 8, 2025

View Articletitled, A Thermomechanical Framework for the Critical Temperature in Nanotube Structural Transitions

Topics: Entropy, Nanotubes, Temperature, Molecular dynamics simulation, Thermomechanics, Stiffness

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