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1

On geometrical interpretation of the fractional strain concept

Sumelka W.

Journal of Theoretical and Applied Mechanics

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2016

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Vol. 54 nr 2

671--674

EN

In this paper, for the first time, the geometrical interpretation of fractional strain tensor components is presented. In this sense, previous considerations by this author are shown in a new light. The fractional material and spatial line elements concept play a crucial role in the interpretation.

2

Non-local Kirchhoff–Love plates in terms of fractional calculus

Sumelka W.

Archives of Civil and Mechanical Engineering

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2015

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Vol. 15, no. 1

231--242

EN

Modern continuum mechanics needs new mathematical techniques to describe the complexity of real physical processes. Recently fractional calculus, a branch of mathematical analysis that studies differential operators of an arbitrary (real or complex) order, emerged as a powerful tool for modelling complex systems. It is due to the fact that fractional differential operators introduce non-locality to the description considered in a natural way. In this sense they generalize classical (local) formulations and make the description more realistic. This paper deals with the generalisation of the Kirchhoff–Love plates theory using fractional calculus. This new formulation in non-local, thus all common fields like e.g. internal forces or displacements at a specific point contain somehow information from its finite surroundings, which is in agreement with experimental observations.

3

Fractional continua for linear elasticity

Sumelka W., Blaszczyk T.

Archives of Mechanics

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2014

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Vol. 66, nr 3

147--172

EN

Fractional continua is a generalisation of the classical continuum body. This new concept shows the application of fractional calculus in continuum mechanics. The advantage is that the obtained description is non-local. This natural non-locality is inherently a consequence of fractional derivative definition which is based on the interval, thus variates from the classical approach where the definition is given in a point. In the paper, the application of fractional continua to one-dimensional problem of linear elasticity under small deformation assumption is presented.

4

A comparison of meshless and finite element approaches to ductile damage in forming processes

Cesar de Sa J., Zheng C.

Computer Methods in Materials Science

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2007

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Vol. 7, No. 2

262-268

EN

In metal forming processes the damage associated with large deformations is a phenomenon that should be minimized or simply avoided as it usually leads to flawed parts. The initiation of plasticity and damage is caused by movement and accumulation of dislocations in metals but their nature and evolution is different. Ductile damage evolution in metals is usually associated with the initiation and growth of micro cracks and cavities, resulting in a progressive material softening. Damage growing influences indirectly the plastic behaviour by locally reducing the elementary area of resistance and therefore plasticity and damage should be coupled at the constitutive level. In the theory of Continuum Damage Mechanics the damage is represented by internal variables (of scalar, vectorial or tensor type) which give a measure of the deteriorated state at each representative volume of the material. This variable may then be used to define the effective stress state. Another important aspect is related with the fact that in ductile damage localization is similar to that associated with plastic strain. These physical phenomena are characterised by the accumulation of damage and large deformations within narrow bands. In experiments these localization zones display a finite width which may be related to the micro structure of the material. Classical theories of plasticity and damage mechanics, based on internal variable approaches, are local theories and do not include size effects associated to a characteristic dimension of the material. Their implementation in a finite element setting shows a pathologic effect of spatial mesh dependence because the constitutive models are unable to capture the limitation of the localization upon mesh refinement. In fact, the original hypothesis of homogeneous continuous models does not take into account large changes in the internal variables, like plastic strain and damage, in the localization zone. The aforementioned effect can be adequately explained by micro mechanical theories but their numerical implementation is still rather expensive. Non-local models have been proposed to bridge the gap between classical continuum theories and the micromechanical ones. In these models the evolution of some internal variables describing strain and damage in a specific point is also determined by the history of the surrounding material by including in the formulation averages or gradients of part or all of them. Some of theses models have proved to be effective when implemented in a finite element framework. Some claims have been made that the new class of computational methods, i.e. meshless methods, could be more effective when dealing with localization problems. Typically these new methods use a set of points and local support functions to represent the problem domain with no need of an additional mesh. This local support functions could then be broadened for the evaluation of the evolution of the internal variables, giving a non-local character to the solution. Therefore, in this work, an incursion was made into the application of these methods to this particular type of problems in order to investigate how meshless methods deal with ductile damage phenomena, if the unacceptable discretization dependence is also present and to assess how effectively the non-local and gradient models work in these settings. The chosen meshless method was the Reproducing Kernel Particle Method (RKPM). The material model was extended in order to include ductile damage effects by coupling the elastoplastic constitutive law with the damage evolution equations. Non-local and related gradient (explicit and implicit) models were also implemented using the RKPM. A set of numerical examples showed that the meshless solution scheme on ductile damage, exhibits the same type of dependence of solutions upon refinement of the geometrical discretization. Both implicit and explicit gradient and non-local models can alleviate this pathological behaviour. Nevertheless the explicit gradient model still presents a local behaviour by concentrating the damage on a narrower zone.

PL

Możliwość wykorzystania metody bez siatkowej Reproducing Kernel Particle Method (RKPM) do symulacji plastycznego pękania w procesach przeróbki plastycznej jest tematem niniejszej pracy. Zalety metody RKPM są porównane z konwencjonalnymi modelami MES, szczególnie pod względem problemów z dyskretyzacją badanego obszaru. Zastosowany model pękania bazuje na podejściu Lemaitre z uwzględnieniem rozgraniczenia pękania dla lokalnych obszarów rozciąganych i spęczanych. Zaimplementowane lokalne i globalne modele w formie jawnej i niejawnej są porównane i omówione w niniejszej pracy.