Handbook of Continuum Mechanics: General Concepts, Thermoelasticity

Salençon, Jean; Bechtel, Stephen E.

doi:10.1115/1.1470668

Cited by 52 publications

(77 citation statements)

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“…Besides the vector β associated to the angular velocity can also be discontinuous, whereas only its normal component can be discontinuous in the thin plate case due to the Love-Kirchhoff condition β = ∇ w and Hadamard's compatibility condition [20].…”

Section: Upper Bound Kinematic Approachmentioning

confidence: 99%

Locking‐free discontinuous finite elements for the upper bound yield design of thick plates

Bleyer

Buhan

2015

Numerical Meth Engineering

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This work investigates the formulation of finite elements dedicated to the upper bound kinematic approach of yield design or limit analysis of Reissner-Mindlin thick plates in shear-bending interaction. The main novelty of this paper is to take full advantage of the fundamental difference between limit analysis and elasticity problems as regards the class of admissible virtual velocity fields. In particular, it has been demonstrated for 2D plane stress, plane strain or 3D limit analysis problems that the use of discontinuous velocity fields presents a lot of advantages when seeking for accurate upper bound estimates. For this reason, discontinuous interpolations of the transverse velocity and the rotation fields for Reissner-Mindlin plates are proposed. The subsequent discrete minimization problem is formulated as a second-order cone programming (SOCP) problem and is solved using the industrial software package Mosek. A comprehensive study of the shear-locking phenomenon is also conducted and it is shown that discontinuous elements avoid such a phenomenon quite naturally, whereas continuous elements cannot without any specific treatment. This particular aspect is confirmed through numerical examples on classical benchmark problems and the so-obtained upper bound estimates are confronted to recently developed lower bound equilibrium finite elements for thick plates.

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Section: Upper Bound Kinematic Approachmentioning

confidence: 99%

Locking‐free discontinuous finite elements for the upper bound yield design of thick plates

Bleyer

Buhan

2015

Numerical Meth Engineering

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“…π is a symmetric tensor known as the second PiolaKirchhoff stress tensor [104] (also called thermodynamic tension by some authors [105,106]). Tensor π can be used to express the power P/Ω 0 of internal forces, per unit volume in the reference (undeformed) configuration.…”

Section: Appendix C: Stress-controlled Molecular Dynamicsmentioning

confidence: 99%

Internal states of model isotropic granular packings. I. Assembling process, geometry, and contact networks

2007

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This is the first paper of a series of three, in which we report on numerical simulation studies of geometric and mechanical properties of static assemblies of spherical beads under an isotropic pressure. The influence of various assembling processes on packing microstructures is investigated. It is accurately checked that frictionless systems assemble in the unique random close packing (RCP) state in the low pressure limit if the compression process is fast enough, higher solid fractions corresponding to more ordered configurations with traces of crystallization. Specific properties directly related to isostaticity of the force-carrying structure in the rigid limit are discussed. With frictional grains, different preparation procedures result in quite different inner structures that cannot be classified by the sole density. If partly or completely lubricated they will assemble like frictionless ones, approaching the RCP solid fraction ΦRCP ≃ 0.639 with a high coordination number: z * ≃ 6 on the force-carrying backbone. If compressed with a realistic coefficient of friction µ = 0.3 packings stabilize in a loose state with Φ ≃ 0.593 and z * ≃ 4.5. And, more surprisingly, an idealized "vibration" procedure, which maintains an agitated, collisional régime up to high densities results in equally small values of z * while Φ is close to the maximum value ΦRCP. Low coordination packings have a large proportion (>10%) of rattlers -grains carrying no force -the effect of which should be accounted for on studying position correlations, and also contain a small proportion of localized "floppy modes" associated with divalent grains. Low pressure states of frictional packings retain a finite level of force indeterminacy even when assembled with the slowest compression rates simulated, except in the case when the friction coefficient tends to infinity. Different microstructures are characterized in terms of near neighbor correlations on various scales, and some comparisons with available laboratory data are reported, although values of contact coordination numbers apparently remain experimentally inaccessible.

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“…Following standard tensor calculus (Salencon 2001), the tensor components of P poly disc (w, 4) = S esh disc (w, 4) : C −1 poly , being related to differently oriented inclusions, are transformed into one, single base frame (e 1 , e 2 , e 3 ), in order to evaluate the integrals in equations (3.1) and (3.2).…”

Section: (A) Stiffness Estimatementioning

confidence: 99%

The role of disc-type crystal shape for micromechanical predictions of elasticity and strength of hydroxyapatite biomaterials

Fritsch

Hellmich

Dormieux

2010

Phil. Trans. R. Soc. A.

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The successful design of ceramic bone biomaterials is challenged by two competing requirements: on the one hand, such materials need to be stiff and strong, which would suggest a low porosity (of pore sizes in the 10-100 mm range) to be targeted; on the other hand, bone biomaterials need to be bioactive (in particular vascularized), which suggests a high porosity of such materials. Conclusively, reliable information on how porosity drives the stiffness and strength properties of ceramic bone biomaterials (tissue engineering scaffolds) is of great interest. In this context, mathematical models are increasingly being introduced into the field. Recently, self-consistent continuum micromechanics formulations have turned out as expressedly efficient and reliable tools to predict hydroxyapatite biomaterials' stiffness and strength, as a function of the biomaterialspecific porosity, and of the 'universal' properties of the individual hydroxyapatite crystals: their stiffness, strength and shape. However, the precise crystal shape can be suitably approximated by specific ellipsoidal shapes: while it was shown earlier that spherical shapes do not lead to satisfactory results, and that acicular shapes are an appropriate choice, we here concentrate on disc-type crystal shape as, besides needles, plates are often reported in micrographs of hydroxyapatite biomaterials. Discbased model predictions of a substantial set of experimental data on stiffness and strength of hydroxyapatite biomaterials almost attain the quality of the very satisfactory needle-based models. This suggests that, as long as the crystal shape is clearly nonspherical, its precise shape is of secondary importance if stiffness and strength of hydroxyapatite biomaterials are predicted on the basis of continuum micromechanics, from their micromorphology and porosity.

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Handbook of Continuum Mechanics: General Concepts, Thermoelasticity

Cited by 52 publications

References 0 publications

Locking‐free discontinuous finite elements for the upper bound yield design of thick plates

Locking‐free discontinuous finite elements for the upper bound yield design of thick plates

Internal states of model isotropic granular packings. I. Assembling process, geometry, and contact networks

The role of disc-type crystal shape for micromechanical predictions of elasticity and strength of hydroxyapatite biomaterials

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