Line and point defects in nonlinear anisotropic solids

Golgoon, Ashkan; Yavari, Arash

doi:10.1007/s00033-018-0973-2

Cited by 16 publications

(8 citation statements)

References 82 publications

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“…Contact mechanics is one of the most common problems in mechanical engineering and tribology, with a variety of applications in collision mechanics [1][2][3], joint structures [4,5], electrical contact [6], thermal contact [7], solid mechanics [8][9][10][11], seals and bearings [12], biomechanical systems [13,14], turbines [15], and additive manufacturing [16,17]. The studies in contact mechanics can be categorized into: single asperity spherical, elliptical, cylindrical and flat contacts, with single asperity spherical contact being the most employed one [18][19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

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Strain Hardening From Elastic–Perfectly Plastic to Perfectly Elastic Flattening Single Asperity Contact

Ghaednia

Brake

Berryhill

et al. 2018

Journal of Tribology

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For elastic contact, an exact analytical solution for the stresses and strains within two contacting bodies has been known since the 1880s. Despite this, there is no similar solution for elastic–plastic contact due to the integral nature of plastic deformations, and the few models that do exist develop approximate solutions for the elastic–perfectly plastic material model. In this work, the full transition from elastic–perfectly plastic to elastic materials in contact is studied using a bilinear material model in a finite element environment for a frictionless dry flattening contact. Even though the contact is considered flattening, elastic deformations are allowed to happen on the flat. The real contact radius is found to converge to the elastic contact limit at a tangent modulus of elasticity around 20%. For the contact force, the results show a different trend in which there is a continual variation in forces across the entire range of material models studied. A new formulation has been developed based on the finite element results to predict the deformations, real contact area, and contact force. A second approach has been introduced to calculate the contact force based on the approximation of the Hertzian solution for the elastic deformations on the flat. The proposed formulation is verified for five different materials sets.

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Section: Introductionmentioning

confidence: 99%

“…579 The reason for this trend can be the effect of E t /E s on the accuracy 580 of Eq. (11). To find the large deformation limit Eq.…”

mentioning

confidence: 99%

Strain Hardening From Elastic–Perfectly Plastic to Perfectly Elastic Flattening Single Asperity Contact

Ghaednia

Brake

Berryhill

et al. 2018

Journal of Tribology

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“…This means that Q 2 G X if and only if Q = Q a b e a (X ) # b (X ) for some ½Q a b 2 G. Hence, the material symmetry group G X for T X B depends on the natural moving frame, as already observed in Wang and Bloom [12]. For a discussion on material symmetry and defects in solids see Golgoon and Yavari [25].…”

Section: Energy Functionsmentioning

confidence: 73%

“…The deformation gradient must satisfy local compatibility equations, which are written as ∂ A F a B = ∂ B F a A . 5 Compatibility equations can also be written in terms of the two-point derivatives of equation (25) as…”

Section: The Piola Transformation In the Two Structuresmentioning

confidence: 99%

“…Finally, as is well-known, the balance of linear momentum can be written in the ambient space in terms of the Cauchy stress s as r b s ab + .b a = .a a . Note that, using the two-point derivative notation (equation (25)), for a spatial tensor the three symbols r,r, and r are equivalent. Thus, there is only one form for the balance of linear momentum when it is written in terms of the Cauchy stress.…”

Section: The Balance Of Linear Momentum In Terms Of the Two Structuresmentioning

confidence: 99%

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Riemannian and Euclidean material structures in anelasticity

Sozio

Yavari

2020

Mathematics and Mechanics of Solids

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In this paper, we discuss the mechanics of anelastic bodies with respect to a Riemannian and a Euclidean geometric structure on the material manifold. These two structures provide two equivalent sets of governing equations that correspond to the geometrical and classical approaches to non-linear anelasticity. This paper provides a parallelism between the two approaches and explains how to go from one to the other. We work in the setting of the multiplicative decomposition of deformation gradient seen as a non-holonomic change of frame in the material manifold. This allows one to define, in addition to the two geometric structures, a Weitzenböck connection on the material manifold. We use this connection to express natural uniformity in a geometrically meaningful way. The concept of uniformity is then extended to the Riemannian and Euclidean structures. Finally, we discuss the role of non-uniformity in the form of material forces that appear in the configurational form of the balance of linear momentum with respect to the two structures.

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Controllable deformations in compressible isotropic implicit elasticity

Yavari,

Goriely

2024

Z. Angew. Math. Phys.

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For a given material, controllable deformations are those deformations that can be maintained in the absence of body forces and by applying only boundary tractions. For a given class of materials, universal deformations are those deformations that are controllable for any material within the class. In this paper, we characterize the universal deformations in compressible isotropic implicit elasticity defined by solids whose constitutive equations, in terms of the Cauchy stress $$\varvec{\sigma }$$ σ and the left Cauchy-Green strain $$\textbf{b}$$ b , have the implicit form $$\varvec{\textsf{f}}(\varvec{\sigma },\textbf{b})=\textbf{0}$$ f ( σ , b ) = 0 . We prove that universal deformations are homogeneous. However, an important observation is that, unlike Cauchy (and Green) elasticity, not every homogeneous deformation is constitutively admissible for a given implicit-elastic solid. In other words, the set of universal deformations is material-dependent, yet it remains a subset of homogeneous deformations.

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Line and point defects in nonlinear anisotropic solids

Cited by 16 publications

References 82 publications

Strain Hardening From Elastic–Perfectly Plastic to Perfectly Elastic Flattening Single Asperity Contact

Strain Hardening From Elastic–Perfectly Plastic to Perfectly Elastic Flattening Single Asperity Contact

Riemannian and Euclidean material structures in anelasticity

Controllable deformations in compressible isotropic implicit elasticity

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