Crystal plasticity

Last updated December 12, 2023

Crystal plasticity is a mesoscale computational technique that takes into account crystallographic anisotropy in modelling the mechanical behaviour of polycrystalline materials. The technique has typically been used to study deformation through the process of slip, however, there are some flavors of crystal plasticity that can incorporate other deformation mechanisms like twinning and phase transformations.^[1] Crystal plasticity is used to obtain the relationship between stress and strain that also captures the underlying physics at the crystal level. Hence, it can be used to predict not just the stress-strain response of a material, but also the texture evolution, micromechanical field distributions, and regions of strain localisation.^[2] The two widely used formulations of crystal plasticity are the one based on the finite element method known as Crystal Plasticity Finite Element Method (CPFEM),^[3] which is developed based on the finite strain formulation for the mechanics, and a spectral formulation which is more computationally efficient due to the fast Fourier transform, but is based on the small strain formulation for the mechanics.^[4]^[5]

Basic concepts

Crystal plasticity assumes that any deformation that is applied to a material is accommodated by the process of slip, where dislocation motion occurs on a slip system. Further, Schmid's law is assumed to be a valid, where a given slip system is said to be active when the resolved shear stress along the slip system exceeds the critical resolved shear stress of the slip system. Since the applied deformation occurs in the macroscopic sample reference frame and slip occurs in the single crystal reference frame, in order to consistently apply the constitutive relations, an orientation map (e.g. using Bunge Euler angles) is required for each grain in the polycrystal. This orientation information can be used to transform the relevant tensors between the crystal frame of reference and the sample frame of reference. The slip systems are described by the Schmid tensor, which is tensor product of the Burgers vector and the slip plane normal, and the Schmid tensor is used to obtain the resolved shear stress in each slip system. Each slip system can undergo different amounts of shearing, and obtaining these shear rates lies at the crux of crystal plasticity. Further, by keeping track of the accumulated strain, the critical resolved shear stress is updated according to various hardening models (e.g. Voce hardening law), and this recovers the observed macroscopic stress-strain response for the material. The texture evolution is captured by updating the crystallographic orientation of the grains based on how much each grain deforms.^[2]^[5]

Related Research Articles

In continuum mechanics, stress is a physical quantity that describes forces present during deformation. For example, an object being pulled apart, such as a stretched elastic band, is subject to tensile stress and may undergo elongation. An object being pushed together, such as a crumpled sponge, is subject to compressive stress and may undergo shortening. The greater the force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Stress has dimension of force per area, with SI units of newtons per square meter (N/m²) or pascal (Pa).

In physics and materials science, plasticity is the ability of a solid material to undergo permanent deformation, a non-reversible change of shape in response to applied forces. For example, a solid piece of metal being bent or pounded into a new shape displays plasticity as permanent changes occur within the material itself. In engineering, the transition from elastic behavior to plastic behavior is known as yielding.

In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to plasticity, in which the object fails to do so and instead remains in its deformed state.

A crystallite is a small or even microscopic crystal which forms, for example, during the cooling of many materials. Crystallites are also referred to as grains.

Solid mechanics is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and other external or internal agents.

<span class="mw-page-title-main">Texture (chemistry)</span>

In physical chemistry and materials science, texture is the distribution of crystallographic orientations of a polycrystalline sample. A sample in which these orientations are fully random is said to have no distinct texture. If the crystallographic orientations are not random, but have some preferred orientation, then the sample has a weak, moderate or strong texture. The degree is dependent on the percentage of crystals having the preferred orientation.

<span class="mw-page-title-main">Crystal twinning</span> Two separate crystals sharing some of the same crystal lattice points in a symmetrical manner

Crystal twinning occurs when two or more adjacent crystals of the same mineral are oriented so that they share some of the same crystal lattice points in a symmetrical manner. The result is an intergrowth of two separate crystals that are tightly bonded to each other. The surface along which the lattice points are shared in twinned crystals is called a composition surface or twin plane.

Zirconium alloys are solid solutions of zirconium or other metals, a common subgroup having the trade mark Zircaloy. Zirconium has very low absorption cross-section of thermal neutrons, high hardness, ductility and corrosion resistance. One of the main uses of zirconium alloys is in nuclear technology, as cladding of fuel rods in nuclear reactors, especially water reactors. A typical composition of nuclear-grade zirconium alloys is more than 95 weight percent zirconium and less than 2% of tin, niobium, iron, chromium, nickel and other metals, which are added to improve mechanical properties and corrosion resistance.

In materials science, critical resolved shear stress (CRSS) is the component of shear stress, resolved in the direction of slip, necessary to initiate slip in a grain. Resolved shear stress (RSS) is the shear component of an applied tensile or compressive stress resolved along a slip plane that is other than perpendicular or parallel to the stress axis. The RSS is related to the applied stress by a geometrical factor, $m$ , typically the Schmid factor:

In materials science, slip is the large displacement of one part of a crystal relative to another part along crystallographic planes and directions. Slip occurs by the passage of dislocations on close/packed planes, which are planes containing the greatest number of atoms per area and in close-packed directions. Close-packed planes are known as slip or glide planes. A slip system describes the set of symmetrically identical slip planes and associated family of slip directions for which dislocation motion can easily occur and lead to plastic deformation. The magnitude and direction of slip are represented by the Burgers vector, $b$ .

Micromechanics is the analysis of composite or heterogeneous materials on the level of the individual constituents that constitute these materials.

A shear band is a narrow zone of intense shearing strain, usually of plastic nature, developing during severe deformation of ductile materials. As an example, a soil specimen is shown in Fig. 1, after an axialsymmetric compression test. Initially the sample was cylindrical in shape and, since symmetry was tried to be preserved during the test, the cylindrical shape was maintained for a while during the test and the deformation was homogeneous, but at extreme loading two X-shaped shear bands had formed and the subsequent deformation was strongly localized.

In geology, a deformation mechanism is a process occurring at a microscopic scale that is responsible for changes in a material's internal structure, shape and volume. The process involves planar discontinuity and/or displacement of atoms from their original position within a crystal lattice structure. These small changes are preserved in various microstructures of materials such as rocks, metals and plastics, and can be studied in depth using optical or digital microscopy.

The microplane model, conceived in 1984, is a material constitutive model for progressive softening damage. Its advantage over the classical tensorial constitutive models is that it can capture the oriented nature of damage such as tensile cracking, slip, friction, and compression splitting, as well as the orientation of fiber reinforcement. Another advantage is that the anisotropy of materials such as gas shale or fiber composites can be effectively represented. To prevent unstable strain localization, this model must be used in combination with some nonlocal continuum formulation. Prior to 2000, these advantages were outweighed by greater computational demands of the material subroutine, but thanks to huge increase of computer power, the microplane model is now routinely used in computer programs, even with tens of millions of finite elements.

Paleostress inversion refers to the determination of paleostress history from evidence found in rocks, based on the principle that past tectonic stress should have left traces in the rocks. Such relationships have been discovered from field studies for years: qualitative and quantitative analyses of deformation structures are useful for understanding the distribution and transformation of paleostress fields controlled by sequential tectonic events. Deformation ranges from microscopic to regional scale, and from brittle to ductile behaviour, depending on the rheology of the rock, orientation and magnitude of the stress, etc. Therefore, detailed observations in outcrops, as well as in thin sections, are important in reconstructing the paleostress trajectories.

Anthony Rollett is a British materials scientist and engineer currently at Carnegie Mellon University and a Fellow of the Institute of Physics. His research interests are within computational materials science, specifically mesoscale methods and microstructure evolution.

Geometrically necessary dislocations are like-signed dislocations needed to accommodate for plastic bending in a crystalline material. They are present when a material's plastic deformation is accompanied by internal plastic strain gradients. They are in contrast to statistically stored dislocations, with statistics of equal positive and negative signs, which arise during plastic flow from multiplication processes like the Frank-Read source.

André Zaoui is a French physicist in material mechanics, born on 8 June 1941. He is a corresponding member of the French Academy of sciences and a member of the French Academy of Technologies.

Ulrich Fred Kocks known as Fred Kocks, is a physicist cum materials scientist.

Slip bands or stretcher-strain marks are localized bands of plastic deformation in metals experiencing stresses. Formation of slip bands indicates a concentrated unidirectional slip on certain planes causing a stress concentration. Typically, slip bands induce surface steps and a stress concentration which can be a crack nucleation site. Slip bands extend until impinged by a boundary, and the generated stress from dislocations pile-up against that boundary will either stop or transmit the operating slip depending on its (mis)orientation.

References

↑ Courtney, Thomas H. (2000). Mechanical behavior of materials (2nd ed.). Boston: McGraw Hill. ISBN 978-1577664253.
1 2 Pokharel, Reeju; Lind, Jonathan; Kanjarla, Anand K.; Lebensohn, Ricardo A.; Li, Shiu Fai; Kenesei, Peter; Suter, Robert M.; Rollett, Anthony D. (March 2014). "Polycrystal Plasticity: Comparison Between Grain - Scale Observations of Deformation and Simulations". Annual Review of Condensed Matter Physics. 5 (1): 317–346. Bibcode:2014ARCMP...5..317P. doi: 10.1146/annurev-conmatphys-031113-133846 . OSTI 1763197.
↑ Roters, F.; Eisenlohr, P.; Hantcherli, L.; Tjahjanto, D.D.; Bieler, T.R.; Raabe, D. (February 2010). "Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications". Acta Materialia. 58 (4): 1152–1211. Bibcode:2010AcMat..58.1152R. doi:10.1016/j.actamat.2009.10.058.
↑ Moulinec, H.; Suquet, P. (April 1998). "A numerical method for computing the overall response of nonlinear composites with complex microstructure". Computer Methods in Applied Mechanics and Engineering. 157 (1–2): 69–94. arXiv: 2012.08962 . Bibcode:1998CMAME.157...69M. doi:10.1016/S0045-7825(97)00218-1. S2CID 120640232.
1 2 Lebensohn, Ricardo A.; Rollett, Anthony D. (February 2020). "Spectral methods for full-field micromechanical modelling of polycrystalline materials". Computational Materials Science. 173: 109336. doi: 10.1016/j.commatsci.2019.109336 .

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[1] Courtney, Thomas H. (2000). Mechanical behavior of materials (2nd ed.). Boston: McGraw Hill. ISBN 978-1577664253.

[AnnualReviews-2] 1 2 Pokharel, Reeju; Lind, Jonathan; Kanjarla, Anand K.; Lebensohn, Ricardo A.; Li, Shiu Fai; Kenesei, Peter; Suter, Robert M.; Rollett, Anthony D. (March 2014). "Polycrystal Plasticity: Comparison Between Grain - Scale Observations of Deformation and Simulations". Annual Review of Condensed Matter Physics. 5 (1): 317–346. Bibcode:2014ARCMP...5..317P. doi: 10.1146/annurev-conmatphys-031113-133846 . OSTI 1763197.

[Acta-3] Roters, F.; Eisenlohr, P.; Hantcherli, L.; Tjahjanto, D.D.; Bieler, T.R.; Raabe, D. (February 2010). "Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications". Acta Materialia. 58 (4): 1152–1211. Bibcode:2010AcMat..58.1152R. doi:10.1016/j.actamat.2009.10.058.

[4] Moulinec, H.; Suquet, P. (April 1998). "A numerical method for computing the overall response of nonlinear composites with complex microstructure". Computer Methods in Applied Mechanics and Engineering. 157 (1–2): 69–94. arXiv: 2012.08962 . Bibcode:1998CMAME.157...69M. doi:10.1016/S0045-7825(97)00218-1. S2CID 120640232.

[CMS-5] 1 2 Lebensohn, Ricardo A.; Rollett, Anthony D. (February 2020). "Spectral methods for full-field micromechanical modelling of polycrystalline materials". Computational Materials Science. 173: 109336. doi: 10.1016/j.commatsci.2019.109336 .

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