Finite Element Calculation of the Linear Elasticity Problem for Biomaterials with Fractal Structure

Aims: The aim of this study was to develop the mathematical models of the linear elasticity theory of biomaterials by taking into account their fractal structure. This study further aimed to construct a variational formulation of the problem, obtain the main relationships of the finite element method to calculate the rheological characteristics of a biomaterial with a fractal structure, and develop application software for calculating the components of the stress-strain state of biomaterials while considering their fractal structure. The obtained results were analyzed. Background: The development of adequate mathematical models of the linear elasticity theory for biomaterials with a fractal structure is an urgent scientific task. Finding its solution will make it possible to analyze the rheological behavior of biomaterials exposed to external loads by taking into account the existing effects of memory, spatial non-locality, self-organization, and deterministic chaos in the material. Objective: The objective of this study was the deformation process of biomaterials with a fractal structure under external load. Methods: The equations of the linear elasticity theory for the construction of the mathematical models of the deformation process of biomaterials under external load were used. Mathematical apparatus of integro-differentiation of fractional order to take into account the fractal structure of the biomaterial was used. A variational formulation of the linear elasticity problem while taking into account the fractal structure of the biomaterial was formulated. The finite element method with a piecewise linear basis for finding an approximate solution to the problem was used. Results: The main relations of the linear elasticity problem, which takes into account the fractal structure of the biomaterial, were obtained. A variational formulation of the problem was constructed. The main relations of the finite-element calculation of the linear elasticity problem of a biomaterial with a fractal structure using a piecewise-linear basis are found. The main components of the stress-strain state of the biomaterial exposed to external loads are found. Conclusion: Using the mathematical apparatus of integro-differentiation of fractional order in the construction of the mathematical models of the deformation process of biomaterials with a fractal structure makes it possible to take into account the existing effects of memory, spatial non-locality, self-organization, and deterministic chaos in the material. Also, this approach makes it possible to determine the residual stresses in the biomaterial, which play an important role in the appearance of stresses during repeated loads.

Structural model of heterogeneous material (microsphere foam) straining and failure under hydrostatic loading

Object and purpose of research. The paper investigates polymeric composite material of syntactic foams type being by nature a heterogeneous medium and consisting of polymeric matrix, filled with spherical inclusions: microspheres. The main purpose of this this paper is to develop a structural model of straining and failure for this type of materials under hydrostatic pressure and software and mathematical apparatus for model implementation. Materials and methods. The input data for this research were composition and structure of syntactic foam material as well as the performance of its components (polymeric matrix and glass microspheres). Structural model was developed on the basis of solutions to linear elasticity theory problems using Lubachevsky – Stillinger algorithm for the formation of structure, homonization methods, etc. A calculation algorithm implemented in code in the С++ language was developed on the basis of the designed mathematical apparatus. Verification of calculation results was carried out by comparison with failure test results of samples of one of the grades of syntactic foam under short-term hydrostatic pressure loading. Main results. Structural model of syntactic foam type material straining and failure under hydrostatic pressure was developed. A calculation algorithm implemented in program code written in the С++ language which is relatively highly efficient for analysis of real structures with a large number of microspheres of the order of 105. Correlation with experimental results showed compatibility of modelling results in terms of both quantitative and qualitative estimates. Conclusion. The developed structural model allows with a high degree of confidence to describe the processes of damage and failure accumulation in syntactic foam under hydrostatic pressure. For practical purposes the model can be used applied for prediction of syntactic foam performance (strength, bulk strain and buoyancy), based on the properties of the initial components – microspheres and polymeric matrix.

Mathematical models of supersonic and intersonic crack propagation in linear elastodynamics

This paper presents mathematical models of supersonic and intersonic crack propagation exhibiting Mach type of shock wave patterns that closely resemble the growing body of experimental and computational evidence reported in recent years. The models are developed in the form of weak discontinuous solutions of the equations of motion for isotropic linear elasticity in two dimensions. Instead of the classical second order elastodynamics equations in terms of the displacement field, equivalent first order equations in terms of the evolution of velocity and displacement gradient fields are used together with their associated jump conditions across solution discontinuities. The paper postulates supersonic and intersonic steady-state crack propagation solutions consisting of regions of constant deformation and velocity separated by pressure and shear shock waves converging at the crack tip and obtains the necessary requirements for their existence. It shows that such mathematical solutions exist for significant ranges of material properties both in plane stress and plane strain. Both mode I and mode II fracture configurations are considered. In line with the linear elasticity theory used, the solutions obtained satisfy exact energy conservation, which implies that strain energy in the unfractured material is converted in its entirety into kinetic energy as the crack propagates. This neglects dissipation phenomena both in the material and in the creation of the new crack surface. This leads to the conclusion that fast crack propagation beyond the classical limit of the Rayleigh wave speed is a phenomenon dominated by the transfer of strain energy into kinetic energy rather than by the transfer into surface energy, which is the basis of Griffiths theory.

The lattice dislocation trapping mechanism at the ferrite/cementite interface in the Isaichev orientation relationship

We analyze the lattice dislocation trapping mechanism at the ferrite/cementite interface of the Isaichev orientation relationship by atomistic simulations combined with the anisotropic linear elasticity theory and disregistry analysis. We find that the lattice dislocation trapping ability is varied by initial position of the lattice dislocation. The lattice dislocation near the interface is attracted to the interface by the image force generated by the interface shear, while the lattice dislocation located far is either attracted to or repelled from the interface, or even oscillates around the introduced position, depending on the combination of the stress field induced by the misfit dislocation array and the image stress field induced by the lattice dislocation.

Effect of Seepage Force on the Wellbore Breakdown of a Vertical Wellbore

As a fluid flows through a porous media, a drag force, called seepage force in the paper, will be formed on the matrix of the media in the fluid flowing direction. However, the seepage force is normally ignored in the analysis of wellbore fracturing during hydraulic fracturing operation. In this paper, an analytical model for seepage force around a vertical wellbore is presented based on linear elasticity theory, and the effect of the seepage force on wellbore breakdown has been analyzed. Also studied are the effects of the two horizontal principal stresses and the reservoir permeability on the action of seepage force. The paper proves that seepage force lowers formation breakdown pressure of a vertical wellbores; the deeper a formation is, the greater action of the seepage force; seepage force contributes more to breakdown formation with small difference of the two horizontal stresses such as unconventional reservoirs; seepage force increases as rock permeability decreases, and it should not be ignored in hydraulic fracturing analysis, especially for low-permeability formation.

The Lattice Dislocation Trapping Mechanism at the Ferrite/Cementite Interface: The Isaichev Orientation Relationship

We analyze the lattice dislocation trapping mechanism at the ferrite/cementite interface (FCI) of the Isaichev orientation relationship (OR) by atomistic simulations combined with the anisotropic linear elasticity theory and disregistry analysis. We find that the lattice dislocation trapping ability is varied by initial position of the lattice dislocation. The lattice dislocation near the interface is attracted to the interface by the image force generated by the interface shear, while the lattice dislocation located far is either attracted to or repelled from the interface, or even oscillates around the introduced position, depending on the combination of the stress field induced by the misfit dislocation array and the image stress field induced by the lattice dislocation.

Asymptotic Behavior of Stable Structures Made of Beams

In this paper, we study the asymptotic behavior of an $\varepsilon $ ε -periodic 3D stable structure made of beams of circular cross-section of radius $r$ r when the periodicity parameter $\varepsilon $ ε and the ratio ${r/\varepsilon }$ r / ε simultaneously tend to 0. The analysis is performed within the frame of linear elasticity theory and it is based on the known decomposition of the beam displacements into a beam centerline displacement, a small rotation of the cross-sections and a warping (the deformation of the cross-sections). This decomposition allows to obtain Korn type inequalities. We introduce two unfolding operators, one for the homogenization of the set of beam centerlines and another for the dimension reduction of the beams. The limit homogenized problem is still a linear elastic, second order PDE.

Boundary value problems for the Lamé-Navier system in fractal domains

<abstract><p>The aim of this paper is to establish a representation formula for the solutions of the Lamé-Navier system in linear elasticity theory. We also study boundary value problems for such a system in a bounded domain $ \Omega\subset {\mathbb R}^3 $, allowing a very general geometric behavior of its boundary. Our method exploits the connections between this system and some classes of second order partial differential equations arising in Clifford analysis.</p></abstract>