Geometrical Foundations of Continuum Mechanics

Steinmann, Paul

doi:10.1007/978-3-662-46460-1

Cited by 57 publications

(29 citation statements)

References 123 publications

(221 reference statements)

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“…The GND methodology adopted originates from Nye (1953), and is consistent with Cermelli and Gurtin (2001) when elastic lattice strains are small (Clayton, 2010;Gurtin, 2002;Steinmann, 2015). More details of the formation adopted in the current study can be found in Arsenlis and Parks (1999) and Busso et al (2000).…”

Section: The Gnd and Ssd Contributionmentioning

confidence: 78%

Quantitative investigation of micro slip and localization in polycrystalline materials under uniaxial tension

Zhang

Lunt

Abdolvand

et al. 2018

International Journal of Plasticity

106

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Micro slip activation and localization in Ti-6Al-4V deformed in tension have been examined quantitatively using high-resolution (HR) digital image correlation (DIC), HR-electron backscatter diffraction (EBSD) and crystal plasticity finite element modelling. The measured polycrystal slip, strain, lattice rotation and geometrically necessary dislocation (GND) density distributions are generally well captured by the a priori crystal plasticity model based on the rate-sensitive properties of α-titanium. An overall slip trace analysis showed over 80% agreement between HR-DIC and crystal plasticity modelling of the primary slip activation. The texture beneath the characterised free-surface has been found to affect the local slip, stress distribution, lattice curvature and GND density and three texture variations have been considered. Grain-level slip trace analysis shows that the crystal plasticity modelling can capture single (straight) slip, multiple slip activation and complex wavy slip. The latter has been found to result from the interaction of independently activated basal and prismatic slip systems with common slip direction. Initial inter-granular misorientations greater than about 5 o have been shown to influence the subsequent micromechanical grain behaviour including slip, lattice rotation and GND density. This work contributes to the understanding of slip localization and load shedding in dwell fatigue in polycrystalline hexagonal materials.

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Section: The Gnd and Ssd Contributionmentioning

confidence: 78%

Quantitative investigation of micro slip and localization in polycrystalline materials under uniaxial tension

Zhang

Lunt

Abdolvand

et al. 2018

International Journal of Plasticity

106

View full text Add to dashboard Cite

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“…Generalized pseudo-Finsler geometry Cartan's tensor components and coefficients of Levi-Civita, Chern-Rund, and Cartan connections are still defined via Eqs. (5), (4), and (6); these are referred to by the same names regardless of whether or not the space is Finsler, pseudo-Finsler, or more general. Various modifications of classical or strict Finsler geometry not requiring homogeneity of the metric tensor were earlier analyzed in [54], along with generalized Finsler connections defined independently of a fundamental scalar or even a metric.…”

Section: -6mentioning

confidence: 99%

“…Many early works [1,2] focused on crystals with dislocations, the primary carriers of plastic deformation in these kinds of solids [3,4]. Differential geometry continues to be prominently featured in modern treatments of continuum physics of solids with dislocations [5,6] as well as those with other kinds of lattice defects such as disclinations [4,7], point defects [8][9][10], fractures [11], and deformation twins [3,12]. Sophisticated geometric methods have also been forwarded for analysis of thermomechanics coupled with electromagnetism in oriented solids [13,14].…”

Section: Introductionmentioning

confidence: 99%

Generalized pseudo-Finsler geometry applied to the nonlinear mechanics of torsion of crystalline solids

Clayton

2018

Int. J. Geom. Methods Mod. Phys.

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A continuum theory of the mechanical behavior of solid materials is presented wherein fundamental geometric quantities such as the metric tensor and connection coefficients can depend on one or more director vectors, also called internal state vectors. This theory, referred to as generalized pseudo-Finsler geometric continuum mechanics, enables depiction of a very broad class of physical phenomena in deformable solid bodies. The general nonlinear theory is reported first, primarily summarizing prior work by the author. Next, a new application of the theory to torsional deformation of solids is presented, whereby a cylindrical sample of material may be simultaneously subjected to twisting, extension or compression along its axis, as well as possible radial confinement or contraction. The internal state variable, when normalized by a regularization length, is identified with an order parameter associated with inelastic deformation that may include slippage, fracture, and/or structural collapse. Evolution of the internal state follows a generalized Ginzburg–Landau type of kinetic equation. For axially homogeneous fields, a coupled system of nonlinear partial differential equations is obtained that can be integrated numerically. Results are first documented for generic solids representative of many crystals that exist in nature. Then solutions corresponding to realistic properties of crystals of boron carbide ceramic and ice are reported. Results for boron carbide predict a dominant effect of shearing over compression on the structural transformation process, in agreement with observations from atomistic simulations. Results for ice demonstrate periods of steady plastic flow under constant applied average shear stress as well as torsional rigidity varying with sample size. Existence of both phenomena agrees with experimental observations.

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“…Though further details are outside the scope of the present paper, a lack of integrable kinematics may be associated with the presence of structural defects in a (crystalline) material [Lazar and Maugin, 2007;Clayton et al, 2005Clayton et al, , 2006Clayton, 2011;Steinmann, 2015]. A multiplicative split of the director gradient function ϑ ϑ ϑ of (2.30) is newly introduced:…”

Section: Multiplicative Kinematicsmentioning

confidence: 99%

Finsler-geometric continuum mechanics and the micromechanics of fracture in crystals

Clayton

2016

J. Micromech. Mol. Phys.

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A continuum theory for the mechanical response of solid bodies subjected to potentially finite deformation is further developed and applied to solve several new problems in the context of the micromechanics of crystalline solids. The theory invokes concepts from Finsler differential geometry, and it provides a diffuse interface description of fracture surfaces. The director or internal state vector is associated with an order parameter describing degradation of the solid. Here, the deformation gradient between pseudo-Finsler reference and spatial configuration spaces is decomposed into a product of two terms, neither necessarily integrable to a vector field. The first is the recoverable elastic deformation, the second is the residual deformation attributed to changes in free volume in failure zones. The latter is restricted to spherical or isotropic symmetry; resulting Euler–Lagrange equations for mechanical and state variable equilibrium are derived. Metric tensors and volume elements depend on the internal state via a conformal transformation, i.e., Weyl scaling. This version of the theory is first applied to tensile fracture of magnesium. Analytical solutions demonstrate the model's capability to predict ductile versus brittle fracture depending on incorporation of Weyl scaling, with results aligned with molecular dynamics (MD) simulations. The second application is shear fracture in boron carbide: solutions depict weakening and tensile pressure in conjunction with structural collapse in shear transformation zones, as suggested by experiments, quantum mechanics, and/or MD simulations.

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Geometrical Foundations of Continuum Mechanics

Cited by 57 publications

References 123 publications

Quantitative investigation of micro slip and localization in polycrystalline materials under uniaxial tension

Quantitative investigation of micro slip and localization in polycrystalline materials under uniaxial tension

Generalized pseudo-Finsler geometry applied to the nonlinear mechanics of torsion of crystalline solids

Finsler-geometric continuum mechanics and the micromechanics of fracture in crystals

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