Non-linear fracture of Single Cantilever Beam (SCB) is studied theoretically using J-integral. It is assumed that the beam is made of unidirectional fiber reinforced polymer composite which obeys the stress-strain relation of an elastic-perfectly plastic material. The lower crack arm is loaded by an external moment while the upper crack arm is stress free. Closed form analytical solutions of J-integral are found for different magnitudes of the external load corresponding to the different distribution of stresses and strains in the beam cracked and un-cracked portions. For this purpose, a Mechanics of materials based model is used. The validity of the solution is proved by comparison with formula for strain energy release rate in the linear-elastic stage of the work of the material. A numerical example is presented to illustrate the influence of the material non-linearity on J-integral. It is found that the material non-linearity leads to the increase of J-integral values. This is attributed to the strain energy dissipation due to the non-linear deformation.

By using the generalized Almansi's theorem and the Schmidt method, the solution of two 3D rectangular cracks in an orthotropic elastic media is investigated. The problems are solved through 2D Fourier transform as three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. The effects of the geometric shape of the rectangular crack and the distance between two rectangular cracks on the stress intensity factors (SIFs) in an orthotropic elastic media are concluded.

We consider three Sturm–Liouville boundary value problems (two of them are coercive and the other one is not) in a bounded Lipschitz domain for a perturbed Lamé operator with boundary conditions of Robin type. We prove that the problems are Fredholm in suitable weighted Sobolev type spaces. Conditions, providing the completeness of root functions related to the boundary value problems, are described.

In this paper we study the regularity of solutions to the Stokes and the Navier-Stokes system in polyhedral domains contained in ℝ^{3}. We consider the scale B^{s}_{τ}(L_{τ}), 1/τ = s/3 + 1/2 of Besov spaces which determines the approximation order of adaptive numerical wavelet schemes and other nonlinear approximation methods. We show that the regularity in this scale is large enough to justify the use of adaptive methods. The proofs of the main results are performed by combining regularity results in weighted Sobolev spaces with characterizations of Besov spaces by wavelet expansions.

In this note we show that the relaxed linear micromorphic model recently proposed by the authors can be suitably used to describe the presence of band-gaps in metamaterials with microstructures in which strong contrasts of the mechanical properties are present (e.g. phononic crystals and lattice structures). This relaxed micromorphic model only has 6 constitutive parameters instead of 18 parameters needed in Mindlin- and Eringen-type classical micromorphic models. We show that the onset of band-gaps is related to a unique constitutive parameter, the *Cosserat couple modulus* μ_{c} which starts to account for band-gaps when reaching a suitable threshold value. The limited number of parameters of our model, as well as the specific effect of some of them on wave propagation can be seen as an important step towards indirect measurement campaigns.

The paper deals with the elastic and cohesive interface behavior of pre-cracked bi-material ceramic-metal structures under dynamic time harmonic load. The shear lag model as well as the Fourier method is applied to find the dynamic response of the considered bi-material structure, assuming the cohesive interface behaviour, accompanied before of the elastic-brittle one. In both cases, the growth of debond length is not considered, e.g. at a given loading condition the only corresponding debond length is found. The inertia forces of the already elastic debond parts of the bi-material structure are neglected. Appropriate contact conditions are proposed in order to fit together both elastic and cohesive solutions. The numerical predictions for the cohesive debond length of the bi-material structures is calculated by the aid of the corresponding value of the elastic debond length at the same loading condition. The influence of loading characteristics i.e. frequencies and amplitude fluctuations on the debond length and the interface shear stress distribution is discussed. The parametric analysis of the results obtained is illustrated by examples of the modern ceramic-metal composites on metal substrates and is depicted in figures.

An overview of one specific design of continuously variable transmission (CVT) is presented. An approach to the modeling of contact interactions is described. Several CVT models of different complexity are developed and used in numerical simulations. The results of simulations can be used to estimate local behavior of CVT parts, as well as its global characteristics.

We derive a scaling-relation, for the infimum of the energy

- \[ J_{\varepsilon,\delta}(u,\gamma)=\int\limits_\Omega\frac{1}{\varepsilon}\operatorname{dist}^q\left(\nabla u\left(\mathds{1}-\gamma\vec{e}_1\otimes \vec{e}_2\right),SO(2)\right)+|\gamma|^p d\lambda^2(x,y)+\delta V_y \left(\chi_{\{\gamma=0\}},\Omega\right), \]

for small ε,δ > 0, where p, q ≥ 1, u: Ω ℝ^{2} is a deformation with suitable affine boundary conditions and γ: Ω ℝ is a suitable slip variable. This model is motivated by a two-dimensional single-slip model in finite crystal plasticity. We show, that the infimum of the energy J_{ε,δ} scales as

. This scaling-relation is attained by an asymptotically self-similar branching construction.

Fractal fault systems are analyzed mechanically by means of the fractional calculus. Small elastic deviations from equilibrium are captured by vectorial wave equations which imply elastic energy and conservation of momentum with spatio-temporal isofractality. Laplace and Fourier transformations lead to an eigenvalue problem which enables a diagonalization for the stable range with convex elastic energy. A degenerate fractional wave equation is proposed for a collapse at the verge of stability. The divergence at collapse is limited to small ranges and times. Substituting such a jerk by a stress jump, its propagation into a stable near-field is analyzed with a commuted isofractional wave equation. Novel solutions are presented which capture some features of earthquakes. These findings can be extended with less symmetry than first assumed for the ease of presentation. The outlook comprehends anelastic effects, coupling with pore water and multi-fractality.

A comparison of two different probabilistic techniques in the context of a homogenization of the polymer filled with rubber particles is demonstrated in this work. Homogenization approach is based on deformation energy of the Representative Volume Element containing a single spherical particle, which must be equal for the heterogeneous and equivalent homogenized bodies. Both probabilistic methods are based on a series of the Finite Element Method experiments that leads to determination of effective characteristics as the polynomial forms of the input random variables and it is done by using the optimized Weighted Least Squares Method. These functions are next integrated in the semi-analytical probabilistic approach and embedded into the Taylor series expansions as the derivatives in the framework of the alternative stochastic perturbation-based approach. These two strategies are employed to study a polymeric matrix with rubber particle as the filler because such a composite has enormously large contrast in-between expectations of the two random Young's moduli of its constituents. Numerical optimization concerns the order of the approximating polynomial that needs to maximize correlation with the set of trial points and minimize at the same time both computational error with the least squares standard deviation.

In this paper, based on the framework of the Flügge's shell theory, the transfer matrix approach and the Romberg integration method, the vibration behavior of an elastic oval cylindrical shell with parabolically varying thickness along of its circumference resting on the Winkler-Pasternak foundations is investigated. The theoretical analysis of the governing equations of the shell is formulated to overcome the mathematical difficulties of mode coupling of variable curvature and thickness of shell. Using the transfer matrix of the shell, the vibration equations based on the Winkler-Pasternak foundations are written in a matrix differential equation of first order in the circumferential coordinate and solved numerically. The proposed model is applied to get the vibration frequencies and the corresponding mode shapes of the symmetrical and antisymmetrical vibration modes. The sensitivity of the vibration characteristics and bending deformations to the Winkler-Pasternak foundations moduli, thickness variation, ovality and orthotropy of the shell is studied for different type-modes of vibration.

This research work considers a more efficient vibro-impacting piezoelectric energy harvester (VIPEH) structure, which is intended both to prevent the device from excessive displacements as well as to increase its operational bandwidth in actual excitation conditions. Multi-physics finite element model of the VIPEH was developed in Comsol with the objective to analyze influence of stopper location on the mechanical and electrical characteristics of the piezoelectric transducer, followed by the experimental study of two device configurations. Numerical and experimental results revealed that stopper location influences the magnitude of generated voltage since, at certain stopper location points, higher vibration modes are excited during the impact.

We consider a steady-state heat conduction problem in 2D unbounded doubly periodic composite materials with temperature independent conductivities of their components. Imperfect contact conditions are assumed on the boundaries between the matrix and inclusions. By introducing complex potentials, the corresponding boundary value problem for the Laplace equation is transformed into a special R-linear boundary value problem for doubly periodic analytic functions. The method of functional equations is used for obtaining a solution. Thus, the R-linear boundary value problem is transformed into a system of functional equations which is analysed afterwards. A new improved algorithm for solving this system is proposed. It allows to compute the average property and reconstruct the temperature and the flux at an arbitrary point of the composite. Computational examples are presented.

A transient torsional response of an elastic solid is discussed. Introducing a semi-infinite needle-like torque source, exact closed form expressions for torsional displacement and stress are obtained. Due to the needle-like shape of the source, two waves are produced. One is a cylindrical wave centered at its symmetry axis and the other is a spherical one centered at the edge of the needle source. The cylindrical wave shows the inverse square root singularity at its front, but the spherical wave does a finite jump at the front. Some additional discussions for the static response to the torque source are also carried out. The proposed needle-like torque source might be used as a primary model for the bore drilling technology.

The present paper contributes to the modeling of contacts with viscoelastic materials. An indenter is pressed into an elastomer and slides on its surface with a constant velocity. Contact regions typically experience periodical loading due to surface roughness with characteristic length scales. Loading within a medium frequency range is studied, in which the viscous properties of the elastomer dominate. After passing a transition zone a stationary state is reached. An one-dimensional model is used to determine estimations for the indentation depth and the coefficient of friction. Results for two simply shaped indenters are presented and compared to boundary element simulations.

This contribution presents a method to improve the energy efficiency of walking bipedal robots over 50% in a range of speed from 0.3 to 2.3 m/s by the use of constant elastic couplings. The method consists of modeling the robot as underactuated system – so that it is making use of its natural dynamics instead of fighting against it – controlling its joint-angle trajectories with input-output feedback linearization and optimizing the joint-angle trajectories as well as the elastic couplings numerically. The mechanism of minimizing energy expenditure consists of reducing impact losses by choosing smaller steps, which gets favorable by a higher natural frequency due to elastic couplings. The method is applied to a planar robot with upper body, two stiff legs, two actuators in the hip joints and one simple rotational spring between the legs as elastic coupling. The mechanism of energy expenditure is investigated for the robot with and without elastic coupling between legs in detail.

An energy harvesting concept has been proposed comprising a piezoelectric patch on a vertical cantilever beam with a tip mass. The cantilever beam is excited in the transverse direction at its base. This device is highly nonlinear with two potential wells for large tip masses, when the beam is buckled. For the pre-buckled case considered here, the stiffness is low and hence the displacement response is large, leading to multiple solutions to harmonic excitation that are exploited in the harvesting device. To maximise the energy harvested in systems with multiple solutions the higher amplitude response should be preferred. This paper investigates the amplitude of random noise excitation where the harvester is unable to sustain the high amplitude solution, and at some point will jump to the low amplitude solution. The investigation is performed on a validated model of the harvester and the effect is demonstrated experimentally.

In this research, the axial buckling and postbuckling configurations of single-walled carbon nanotubes (SWCNTs) under different types of end conditions are investigated based on an efficient numerical approach. The effects of transverse shear deformation and rotary inertia are taken into account using the Timoshenko beam theory. The nonlinear governing equations and associated boundary conditions are derived by the virtual displacements principle and then discretized via the generalized differential quadrature method. The small scale effect is incorporated into the model through Eringen's nonlocal elasticity. To obtain the critical buckling loads, the set of linear discretized equations are solved as an eigenvalue problem. Also, to address the postbuckling problem, the pseudo arc-length continuation method is applied to the set of nonlinear parameterized equations. The effects of nonlocal parameter, boundary conditions, aspect ratio and buckling mode on the critical buckling load and postbuckling behavior are studied. Moreover, a comparison is made between the results of Timoshenko beam model and those of its Euler-Bernoulli counterpart for various magnitudes of nonlocal parameter.

In this work, the interaction between a moving anti-plane crack and a screw dislocation in the magneto-electro-elastic material is studied. The exact solutions are derived by the Muskhelishvili theory and the mechanical-electric-magnetic fields are presented. Comparing with the recent researching status, the present work has three prominent features. Firstly, both the remote loads with eight possible combinations and the coupling of the mechanical-electric-magnetic fields are considered. Secondly, the crack face is assumed to have the general boundary condition which is not the single permeable or impermeable case. Finally, the general solution is derived with the consideration of the crack moving velocity and the screw dislocation. Results presented in this paper would have potential applications on the analysis and design of magneto-electro-elastic structures, especially for the influence of microdefects on the initiation and propagation of cracks.

Based on the Mindlin plate theory the eigenfunctions for a through-thickness crack in a bending and twisting plate have been derived. The results are given as power series in terms of deflection, rotation and stresses. By introducing two auxiliary functions in the Mindlin plate theory we obtained two decoupled partial differential equations of the fourth and second order. This system of partial differential equations allows for each crack face to describe all three types of static boundary conditions. The first eigenfunction of the stress state shows the same singular near-tip field at the crack tip known from two- and three-dimensional crack analyses as well as from Reissner's plate theory. The second eigenfunction similarly characterizes the constant stress parallel to the crack as the T-stress in plane elastic problems.

Volume deformation of solid/ice after 5 hours by a one-dimensional freezing process of water and ice, see in this issue, **94** 7–8, (2014) page 601

The gradient scheme family, which includes the conforming and mixed finite elements as well as the mimetic mixed hybrid family, is used for the approximation of Richards equation and the two-phase flow problem in heterogeneous porous media. We prove the convergence of the approximate saturation and of the approximate pressures and approximate pressure gradients thanks to monotony and compactness arguments under an assumption of non-degeneracy of the phase relative permeabilities. Strong convergence results stem from the convergence of the norms of the gradients of pressures, which demand handling the nonlinear time term. Numerical results show the efficiency on these problems of a particular gradient scheme, called the Vertex Approximate Gradient scheme.

The gradient scheme family, which includes the conforming and mixed finite elements as well as the mimetic mixed hybrid family, is used for the approximation of Richards equation and the two-phase flow problem in heterogeneous porous media. The authors prove the convergence of the approximate saturation and of the approximate pressures and approximate pressure gradients thanks to monotonicity and compactness arguments under an assumption of non-degeneracy of the phase relative permeabilities. Strong convergence results stem from the convergence of the norms of the gradients of pressures, which demand handling the nonlinear time term. Numerical results show the efficiency on these problems of a particular gradient scheme, called the Vertex Approximate Gradient scheme.

In civil engineering, the frost durability of partly liquid saturated porous media under freezing and thawing conditions is a point of great discussion. Ice formation in porous media results from coupled heat and mass transport and is accompanied by ice expansion. The volume increase in space and time corresponds to the moving freezing front inside the porous solid. In this paper, a macroscopic model based on the Theory of Porous Media (TPM) is presented which describes energetic effects of freezing and thawing processes. For simplification a ternary model consisting of the phases solid, ice and liquid is used. Attention is paid to the description of the temperature development, the determination of energy, enthalpy and mass supply as well as volume deformations due to ice formation during a freezing and thawing cycle. For the detection of energetic effects regarding the characterization and control of phase transition of water and ice, a physically motivated evolution equation for the mass exchange between ice and liquid is presented. Comparing experimental data with numerical examples shows that the simplified model is indeed capable of simulating the temperature development and energetic effects during phase change.

In civil engineering, the frost durability of partly liquid saturated porous media under freezing/thawing conditions is a point of great discussion. Ice formation in porous media results from coupled heat and mass transport and is accompanied by ice expansion. The volume increase corresponds to the moving freezing front inside the porous solid. A macroscopic model based on the Theory of Porous Media (TPM) is presented which describes energetic effects of freezing/thawing processes. To simplify a ternary model consisting of the phases solid, ice and liquid is used which describes the temperature development, the determination of energy, enthalpy and mass supply as well as volume deformations due to ice formation during a freezing and thawing cycle. Energetic effects regarding the phase transition are modelled by a physically motivated evolution equation for the mass exchange between ice and liquid. Comparing experimental data with numerical examples shows that the simplified model is capable of simulating the temperature development and energetic effects during phase change.

This study focuses on a formulation within the theory of porus media for continuum multicomponent modeling of bacterial driven methane oxidation in a porous landfill cover layer which consists of a porous solid matrix (soil and bacteria) saturated by a liquid (water) and gas phase. The solid, liquid, and gas phases are considered as immiscible constituents occupying spatially their individual volume fraction. However, the gas phase is composed of three components, namely methane (CH_{4}), oxygen (O_{2}), and carbon dioxide (CO_{2}). A thermodynamically consistent constitutive framework is derived by evaluating the entropy inequality on the basis of Coleman and Noll [8], which results in constitutive relations for the constituent stress and pressure states, interaction forces, and mass exchanges. For the final set of process variables of the derived finite element calculation concept we consider the displacement of the solid matrix, the partial hydrostatic gas pressure and osmotic concentration pressures. For simplicity, we assume a constant water pressure and isothermal conditions. The theoretical formulations are implemented in the finite element code FEAP by Taylor [29]. A new set of experimental batch tests has been created that considers the model parameter dependencies on the process variables; these tests are used to evaluate the nonlinear model parameter set. After presenting the framework developed for the finite element calculation concept, including the representation of the governing weak formulations, we examine representative numerical examples.

This study focusses on a formulation within the Theory of Porous Media for continuum multicomponent modelling of bacterial-driven methane oxidation in a porous landfill cover layer which consists of a porous solid matrix (soil and bacteria) saturated by a liquid (water) and gas phase. The solid, liquid, and gas phases are considered as immiscible constituents occupying spatially their individual volume fraction. However, the gas phase is composed of three components, namely methane, oxygen, and carbon dioxide. A thermodynamically consistent constitutive framework is derived by evaluating the entropy inequality, which results in constitutive relations for the stress and pressure states, interaction forces, and mass exchanges. The final set of variables for the derived finite element calculations consists of the displacement of the solid matrix, the partial hydrostatic gas pressure and osmotic concentration pressures. The model is implemented in the finite element code FEAP [29]. A new set of experimental batch tests has been created that considers the model parameter dependencies on the process variables; these tests are used to evaluate the nonlinear model parameter set by representative numerical examples.

We consider a mathematical model for reactive flow in a channel having a rough (periodically oscillating) boundary with both period and amplitude ε. The ions are being transported by the convection and diffusion processes. These ions can react at the rough boundaries and get attached to form the crystal (precipitation) and become immobile. The reverse process of dissolution is also possible. The model involves non-linear and multi-valued rates and is posed in a fixed geometry with rough boundaries. We provide a rigorous justification for the upscaling process in which we define an upscaled problem defined in a simpler domain with flat boundaries. To this aim, we use periodic unfolding techniques combined with translation estimates. Numerical experiments confirm the theoretical predictions and illustrate a practical application of this upscaling process.

The authors consider a mathematical model for reactive flow in a channel having a rough (periodically oscillating) boundary with both period and amplitude. The ions are being transported by the convection and diffusion processes. These ions can react at the rough boundaries and get attached to form the crystal (precipitation) and become immobile. The reverse process of dissolution is also possible. The model involves non-linear and multi-valued rates. They provide a rigorous justification for the upscaling process in which they define an upscaled problem defined in a simpler domain with flat boundaries. To this aim, they use periodic unfolding techniques combined with translation estimates. Numerical experiments confirm the theoretical predictions and illustrate a practical application of this upscaling process.

We study flow problems in unsaturated porous media. Our main interest is the gravity driven penetration of a dry material, a situation in which fingering effects can be observed experimentally and numerically. The flow is described by either a Richards or a two-phase model. The important modelling aspect regards the capillary pressure relation which can include static hysteresis and dynamic corrections. We report on analytical existence and instability results for the corresponding models and present numerical calculations. We show that fingering effects can be observed in various models and discuss the importance of the static hysteresis term.

This contribution deals with flow problems in unsaturated porous media. The main interest is the gravity driven penetration of a dry material, a situation in which fingering effects can be observed experimentally and numerically. The flow is described by either a Richards or a two-phase model. The important modelling aspect regards the capillary pressure relation which can include static hysteresis and dynamic corrections. Analytical existence and instability results for the corresponding models are given. By numerical calculations it is shown that fingering effects can be observed in various models, the importance of the static hysteresis term is discussed.

Neglecting capillary pressure effects in two-phase flow models for porous media may lead to non-physical solutions: indeed, the physical solution is obtained as limit of the parabolic model with small but non-zero capillarity. In this paper, we propose several numerical strategies designed specifically for approximating physically relevant solutions of the hyperbolic model with neglected capillarity, in the multi-dimensional case. It has been shown in [Andreianov & Cancès, Comput. Geosci., DOI: 10.1007/s10596-012-9329-8, 2013] that in the case of the one-dimensional Buckley-Leverett equation with distinct capillary pressure properties of adjacent rocks, the interface may impose an upper bound on the transmitted flux. This transmission condition may reflect the oil trapping phenomenon. We recall the theoretical results for the one-dimensional case which are used to motivate the construction of multi-dimensional finite volume schemes. We describe and discuss a coupled scheme resulting as the limit of the scheme constructed in [Brenner & Cancès & Hilhorst, HAL preprint no.00675681, 2012] and two IMplicit Pressure – Explicit Saturation (IMPES) schemes with different level of coupling. We finally provide numerical evidences of the good behavior of the fully decoupled version of the IMPES scheme.

Neglecting capillary pressure effects in two-phase flow models for porous media may lead to nonphysical solutions. The authors propose several numerical strategies designed specifically to approximate physically relevant solutions of the hyperbolic model with neglected capillarity. It has been shown in [Andreianov & Cancèes, Comput.Geosci., DOI: 10.1007/s10596-012-9329-8, 2013] that in the case of the one-dimensional Buckley-Leverett equation with distinct capillary pressure properties of adjacent rocks, the interface may impose an upper bound on the transmitted flux. This transmission condition may reflect the oil trapping phenomenon. Theoretical results for the one- dimensional case which are recalled. A coupled scheme resulting as the limit of the scheme constructed in [Brenner & Cancès & Hilhorst, HAL preprint no.00675681, 2012] and two Implicit Pressure – Explicit Saturation (IMPES) schemes with different levels of coupling are constructed for the multidimensional case.

Liquefaction phenomena are encountered in many engineering applications, especially, in geomechanics and earthquake engineering. Drawing our attention to fluid-saturated granular materials with heterogeneous microstructures, the modelling is carried out within a continuum-mechanical framework by exploiting the macroscopic Theory of Porous Media (TPM) together with thermodynamically consistent constitutive equations. In this regard, the solid skeleton of the water-saturated soil is described as an elasto-plastic material with isotropic hardening and a stress-dependent failure surface. The underlying equations are discretised and implemented into the coupled porous-media finite-element solver PANDAS and linked to the commercial finite-element package Abaqus via a general interface. This coupling allows the definition of complex intial-boundary-value problems through Abaqus, thereby using the sophisticated material models of PANDAS. To reveal the capabilities of this approach, two types of simulations have been carried out. At first, in order to get a detailed understanding of the porous-media soil model under transient loading conditions, a cyclic torsion benchmark is computed. In a second step, specific liquefaction phenomena are addressed, where the underlying initial-boundary-value problems are inspired by practically relevant scenarios.

Liquefaction phenomena are encountered in many engineering applications, especially, in geomechanics and earthquake engineering. Drawing our attention to fluid-saturated granular materials with heterogeneous microstructures, the modelling is carried out within a continuum-mechanical framework by exploiting the macroscopic Theory of Porous Media (TPM) together with thermodynamically consistent constitutive equations. In this regard, the solid skeleton of the water-saturated soil is described as an elasto-plastic material with isotropic hardening and a stress-dependent failure surface. The underlying equations are discretised and implemented into the coupled porous-media finite-element solver PANDAS and coupled to the commercial finite-element package Abaqus. To reveal the capabilities of this approach, two types of simulations have been carried out. At first, in order to get a detailed understanding of the porous-media soil model under transient loading conditions, a cyclic torsion benchmark is computed. In a second step, specific liquefaction phenomena are addressed.

We consider conservation laws with spatially discontinuous flux that are perturbed by diffusion and dispersion terms. These equations arise in a theory of two-phase flow in porous media that includes rate-dependent (dynamic) capillary pressure and spatial heterogeneities. We investigate the singular limit as the diffusion and dispersion parameters tend to zero, showing strong convergence towards a weak solution of the limit conservation law.