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Neff P.,University of Duisburg - Essen | Ghiba I.D.,University of Duisburg - Essen | Ghiba I.D.,Al. I. Cuza University | Ghiba I.D.,Octav Mayer Institute of Mathematics of the Romanian Academy | And 4 more authors.
Quarterly Journal of Mechanics and Applied Mathematics | Year: 2015

We consider the equilibrium problem in the relaxed linear model of micromorphic elastic materials. The basic kinematical fields of this extended continuum model are the displacement u ε R3 and the non-symmetric micro-distortion density tensor P ε R3×3. In this relaxed theory, a symmetric force-stress tensor arises despite the presence of microstructure and the curvature contribution depends solely on the micro-dislocation tensor Curl P. However, the relaxed model is able to fully describe rotations of the microstructure and to predict non-polar size-effects. In contrast to classical linear micromorphic models, we allow the usual elasticity tensors to become positive-semidefinite. We prove that, nevertheless, the equilibrium problem has a unique weak solution in a suitable Hilbert space. The mathematical framework also settles the question of which boundary conditions to take for the micro-distortion. Similarities and differences between linear micromorphic elasticity and dislocation gauge theory are discussed and pointed out. © The Author, 2015.

Lu Y.,Warsaw University of Technology | Lekszycki T.,Warsaw University of Technology | Lekszycki T.,Medical University of Warsaw | Lekszycki T.,International Center Mand Mathematics and Mechanics of Complex Systems
Continuum Mechanics and Thermodynamics | Year: 2014

In case of the secondary bone fracture healing, four characteristic steps are often distinguished. Thefirst stage, hematoma and clot formation, which is an object of our study, is important because it preparesthe environment for the following stages. In this work, a new mathematical model describing basic effectspresent short after the injury is proposed. The main idea is based on the assumption that blood leaking fromthe ruptured blood vessels propagates into a poroelastic saturated tissue close to the fracture and mixes withthe interstitial liquid present in pores. After certain time period from the first contact with surrounding tissue,the solidification of blood in the fluid mixture starts. This results in clot formation. By assuming the timenecessary to initiate solidification and critical saturation of blood in the mixture, the shape and the structure ofblood clot could be determined. In numerical example, proposed mathematical formulas were used to studythe size of the gap between fractured parts and its effect in blood clot formation. © 2014 The Author(s).

Madeo A.,INSA Lyon | Madeo A.,International Center Mand Mathematics and Mechanics of Complex Systems | Ghiba I.-D.,University of Duisburg - Essen | Ghiba I.-D.,Al. I. Cuza University | And 4 more authors.
European Journal of Mechanics, A/Solids | Year: 2016

In this paper we consider the Grioli-Koiter-Mindlin-Toupin linear isotropic indeterminate couple stress model. Our main aim is to show that, up to now, the boundary conditions have not been completely understood for this model. As it turns out, and to our own surprise, restricting the well known boundary conditions stemming from the strain gradient or second gradient models to the particular case of the indeterminate couple stress model, does not always reduce to the Grioli-Koiter-Mindlin-Toupin set of accepted boundary conditions. We present, therefore, a proof of the fact that when specific "mixed" kinematical and traction boundary conditions are assigned on the boundary, no "a priori" equivalence can be established between Mindlin's and our approach. © 2016 Elsevier Masson SAS. All rights reserved.

Neff P.,University of Duisburg - Essen | Munch I.,Karlsruhe Institute of Technology | Ghiba I.-D.,University of Duisburg - Essen | Ghiba I.-D.,Al. I. Cuza University | And 3 more authors.
International Journal of Solids and Structures | Year: 2016

In a series of papers which are either published [Hadjesfandiari, A., Dargush, G. F., 2011a. Couple stress theory for solids. Int. J. Solids Struct. 48 (18), 2496-2510; Hadjesfandiari, A., Dargush, G. F., 2013. Fundamental solutions for isotropic size-dependent couple stress elasticity. Int. J. Solids Struct. 50 (9), 1253-1265.] or available as preprints [Hadjesfandiari, A., Dargush, G. F., 2010. Polar continuum mechanics. Preprint arXiv:1009.3252; Hadjesfandiari, A. R., Dargush, G. F., 2011b. Couple stress theory for solids. Int. J. Solids Struct. 48, 2496-2510; Hadjesfandiari, A. R., 2013. On the skew-symmetric character of the couple-stress tensor. Preprint arXiv:1303.3569; Hadjesfandiari, A. R., Dargush, G. F., 2015a. Evolution of generalized couple-stress continuum theories: a critical analysis. Preprint arXiv:1501.03112; Hadjesfandiari, A. R., Dargush, G. F., 2015b. Foundations of consistent couple stress theory. Preprint arXiv:1509.06299] Hadjesfandiari and Dargush have reconsidered the linear indeterminate couple stress model. They are postulating a certain physically plausible split in the virtual work principle. Based on this postulate they claim that the second-order couple stress tensor must always be skew-symmetric. Since they do not consider that the set of boundary conditions intervening in the virtual work principle is not unique, their statement is not tenable and leads to some misunderstandings in the indeterminate couple stress model. This is shown by specifying their development to the isotropic case. However, their choice of constitutive parameters is mathematically possible and we show that it still yields a well-posed boundary value problem. © 2015 Elsevier Ltd. All rights reserved.

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