skip to main content

Title: Fifth-degree elastic energy for predictive continuum stress–strain relations and elastic instabilities under large strain and complex loading in silicon

Materials under complex loading develop large strains and often phase transformation via an elastic instability, as observed in both simple and complex systems. Here, we represent a material (exemplified for Si I) under large Lagrangian strains within a continuum description by a 5th-order elastic energy found by minimizing error relative to density functional theory (DFT) results. The Cauchy stress—Lagrangian strain curves for arbitrary complex loadings are in excellent correspondence with DFT results, including the elastic instability driving the Si I → II phase transformation (PT) and the shear instabilities. PT conditions for Si I → II under action of cubic axial stresses are linear in Cauchy stresses in agreement with DFT predictions. Such continuum elastic energy permits study of elastic instabilities and orientational dependence leading to different PTs, slip, twinning, or fracture, providing a fundamental basis for continuum physics simulations of crystal behavior under extreme loading.

more » « less
Award ID(s):
1943710 1904830
Author(s) / Creator(s):
; ; ; ;
Publisher / Repository:
Nature Publishing Group
Date Published:
Journal Name:
npj Computational Materials
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Scale-free phase-field approach (PFA) at large strains and corresponding finite element method (FEM) simulations for multivariant martensitic phase transformation (PT) from cubic Si I to tetragonal Si II in a polycrystalline aggregate are presented. Important features of the model are large and very anisotropic transformation strain tensor εt = {0.1753; 0.1753; −0.447} and stress-tensor dependent athermal dissipative threshold for PT, which produce essential challenges for computations. 3D polycrystals with 55 and 910 stochastically oriented grains are subjected to uniaxial strain- and stress-controlled loadings under periodic boundary conditions and zero averaged lateral strains. Coupled evolution of discrete martensitic microstructure, volume fractions of martensitic variants and Si II, stress and transformation strain tensors, and texture are presented and analyzed. Macroscopic variables effectively representing multivariant transformational behavior are introduced. Macroscopic stress-strain and transformational behavior for 55 and 910 grains are close (less than 10% difference). This allows the determination of macroscopic constitutive equations by treating aggregate with a small number of grains. Large transformation strains and grain boundaries lead to huge internal stresses of tens GPa, which affect microstructure evolution and macroscopic behavior. In contrast to a single crystal, the local mechanical instabilities due to PT and negative local tangent modulus are stabilized at the macroscale by arresting/slowing the growth of Si II regions by the grain boundaries and generating the internal back stresses. This leads to increasing stress during PT. The developed methodology can be used for studying similar PTs with large transformation strains and for further development by including plastic strain and strain-induced PTs. 
    more » « less
  2. Abstract

    Instability-induced pattern transformations of the architectured multi-phase soft metamaterial under bi-axial compression were explored. The soft metamaterial is composed of two phases: a soft matrix and a reinforcing hexagonal network embedded in the matrix. Equi-biaxial loading is found to induce both micro- and macro- instabilities in the networked architecture. Two types of instability patterns were observed, dependent upon the architecture geometry and the material combination. The critical strain for triggering instability and the two resulting types of patterns was derived, and a theoretical criterion for the transition between the two patterns was determined. Type I patterns retain the original periodicity of the architecture but wrinkles the network walls whereas Type II patterns transform the overall periodicity of the architecture while bending the network walls. Elastic wave propagation analysis was performed for the two distinct patterns under both stressed and stress-free conditions: a change in band gaps is found for both instability-induced pattern transformations, but differs for each type due to their dramatic difference in structure transformation (i.e. Type I wall wrinkling vs. Type II periodicity switching). The distinguished mechanical behavior and the rich properties of this category of multi-phase soft metamaterial can be used to design new smart materials with switchable functionalities controllable by deformation.

    more » « less
  3. null (Ed.)
    Recent theoretical and computational progress has led to unprecedented understanding of symmetry-breaking instabilities in 2D dynamic fracture. At the heart of this progress resides the identification of two intrinsic, near crack tip length scales — a nonlinear elastic length scale ℓ and a dissipation length scale ξ — that do not exist in Linear Elastic Fracture Mechanics (LEFM), the classical theory of cracks. In particular, it has been shown that at a propagation velocity v of about 90% of the shear wave-speed, cracks in 2D brittle materials undergo an oscillatory instability whose wavelength varies linearly with ℓ, and at larger loading levels (corresponding to yet higher propagation velocities), a tip-splitting instability emerges, both in agreements with experiments. In this paper, using phase-field models of brittle fracture, we demonstrate the following properties of the oscillatory instability: (i) It exists also in the absence of near-tip elastic nonlinearity, i.e. in the limit ℓ→0, with a wavelength determined by the dissipation length scale ξ. This result shows that the instability crucially depends on the existence of an intrinsic length scale associated with the breakdown of linear elasticity near crack tips, independently of whether the latter is related to nonlinear elasticity or to dissipation. (ii) It is a supercritical Hopf bifurcation, featuring a vanishing oscillations amplitude at onset. (iii) It is largely independent of the phenomenological forms of the degradation functions assumed in the phase-field framework to describe the cohesive zone, and of the velocity-dependence of the fracture energy Γ(v) that is controlled by the dissipation time scale in the Ginzburg-Landau-type evolution equation for the phase-field. These results substantiate the universal nature of the oscillatory instability in 2D. In addition, we provide evidence indicating that the tip-splitting instability is controlled by the limiting rate of elastic energy transport inside the crack tip region. The latter is sensitive to the wave-speed inside the dissipation zone, which can be systematically varied within the phase-field approach. Finally, we describe in detail the numerical implementation scheme of the employed phase-field fracture approach, allowing its application in a broad range of materials failure problems. 
    more » « less
  4. Abstract

    The atomic‐scale cracking mechanism in clay is vital in discovering the cracking mechanism of clay at the continuum scale in that clay is a nanomaterial. In this article, we investigate mechanisms of modes I and II crack propagations in pyrophyllite and Ca‐montmorillonite with water adsorption through reactive molecular dynamics (MD) with a bond‐order force field. Clay water adsorption is considered by adding water molecules to the clay surface. During the equilibration stage, water adsorption could cause bending deformation of the predefined edge crack region. The relatively small orientating angle of water molecules indicates the formation of hydrogen bonds in the crack propagation process. The peak number density of adsorbed water decreases with the increasing strains. The atomistic structure evolution of the crack tip under loading is analyzed to interpret the nanoscale crack propagation mechanism. The numerical results show that the crack tip first gets blunted with a significant increase in the radius of the curvature of the crack tip and a slight change in crack length. The crack tip blunting is studied by tracking the crack tip opening distance and O–Si–O angle in the tetrahedral Si–O cell in modes I and II cracks. We compare bond‐breaking behaviors between Al–O and Si–O. It is found that Si–O bond breaking is primarily responsible for crack propagation. The critical stress intensity factor and critical energy release rate are determined from MD simulation results.

    more » « less
  5. Abstract

    Crystallographic theory based on energy minimization suggests austenite-twinned martensite interfaces with specific orientation, which are confirmed experimentally for various materials. Pressure-induced phase transformation (PT) from semiconducting Si-I to metallic Si-II, due to very large and anisotropic transformation strain, may challenge this theory. Here, unexpected nanostructure evolution during Si-I → Si-II PT is revealed by combining molecular dynamics (MD), crystallographic theory, generalized for strained crystals, and in situ real-time Laue X-ray diffraction (XRD). Twinned Si-II, consisting of two martensitic variants, and unexpected nanobands, consisting of alternating strongly deformed and rotated residual Si-I and third variant of Si-II, form$$\{111\}$${111}interface with Si-I and produce almost self-accommodated nanostructure despite the large transformation volumetric strain of$$-0.237$$0.237. The interfacial bands arrest the$$\{111\}$${111}interfaces, leading to repeating nucleation-growth-arrest process and to growth by propagating$$\{110\}$${110}interface, which (as well as$$\{111\}$${111}interface) do not appear in traditional crystallographic theory.

    more » « less