Displacement Boundary Conditions Research Articles

Experimental Evaluation of Fluid Connectivity in Two‐Phase Flow in Porous Media During Drainage

AbstractThis study aims to experimentally investigate the possibility of combining two extended continuum theories for two‐phase flow. One of these theories considers interfacial area as a separate state variable, and the other explicitly discriminates between connected and disconnected phases. This combination enhances our potential to effectively model the apparent hysteresis, which generally dominates two‐phase flow. Using optical microscopy, we perform microfluidic experiments in quasi‐2D artificial porous media for various cyclic displacement processes and boundary conditions. Specifically for a number of sequential drainage processes, with detailed image (post‐)processing, pore‐scale parameters such as the interfacial area between the phases (wetting, non‐wetting, and solid), and local capillary pressure, as well as macroscopic parameters like saturation, are estimated. We show that discriminating between connected and disconnected clusters and the concept of the interfacial area as a separate state variable can be an appropriate way of modeling hysteresis in a two‐phase flow scheme. The drainage datasets of capillary pressure, saturation, and specific interfacial area, are plotted as a surface, given by f (Pc, sw, awn) = 0. These surfaces accommodate all data points within a reasonable experimental error, irrespective of the boundary conditions, as long as the corresponding liquid is connected to its inlet. However, this concept also shows signs of reduced efficiency as a modeling approach in datasets gathered through combining experiments with higher volumetric fluxes. We attribute this observation to the effect of the porous medium geometry on the phase distribution. This yields further elaboration, in which this speculation is thoroughly studied and analyzed.

On the adhesive nanocontact of a graded coating

This paper analyzes the adhesive nanocontact properties between a rigid cylinder and an exponentially graded coating-substrate structure. While adhesion between the indenter and the graded coating is accommodated by Johnson–Kendall–Roberts adhesive model, surface effects are addressed by the full version of Steigmann–Ogden surface mechanical theory. In the presence of surface effects alone, the fundamental Flamant solution of the graded coating-substrate structure is determined first, with the aid of Fourier integral transforms. The displacement boundary condition under the rigid cylinder and the static equilibrium equation lead to dual Fredholm integral equations of the first kind. The adhesion is subsequently introduced in terms of the principle of minimum potential energy, by which both the adhesive nanocontact length and pressure are uniquely determined. In order to clarify the effects of adhesion and surface elasticity, parametric studies are carried out with respect to adhesive energy density, surface material constants, shear modulus gradation of the coating and indenter size. Surface effects tend to reduce contact length, maximum contact pressure and subsidence displacement. On the other hand, adhesion increases these three properties. Nonetheless, both effects result in higher pull-off forces. In general, the relation between adhesion and surface elasticity can be qualified as competitive. The current study systematically reveals the feature of adhesive nanocontact of a graded coating-substrate system in the presence of both adhesion and surface effects. It helps the community to extend adhesive nanocontact mechanics from homogeneous to graded materials and structures.

A Novel Photo Elasto-Thermodiffusion Waves with Electron-Holes in Semiconductor Materials with Hyperbolic Two Temperature

In this paper, a novel mathematical—physical model of the generalized elasto-thermodiffusion (hole/electron interaction) waves in semiconductor materials is studied when the hyperbolic two-temperature theory in the two-dimensional (2D) deformation is taken into account. Shear (purely transverse) waves are dissociated from the remainder of the motion and remain unaffected by external fields. The coupled system of partial differential equations of the main interacting fields has been solved. Using the Laplace transform method, the governing equations of motion and heat conduction can be formulated in 2D. The hole charge carrier, displacement, thermal, and plasma boundary conditions are applied on the interface adjacent to the vacuum to obtain the basic physical quantities in the Laplace domain. The inversion of the Laplace transform with the numerical method is applied to obtain the complete solutions in the time domain for the main physical fields under investigation. The effects of thermoelastic, the phase-lag of the temperature gradient and the phase-lag of the heat flux, the hyperbolic two-temperature parameter, and comparing between silicon and germanium materials on the displacement component, carrier density, hole charge carrier, and temperature distribution have been discussed and obtained graphically.

Numerical investigation on tensile behavior of three-dimensional six-directional braided composites

Based on the mesostructure, a parametric finite element (FE) model of the three-dimensional six-directional (3D6d) braided composites was developed considering the real cross-sectional shape of the yarn and its contact relationship. The 3D Hashin criterion and maximum stress criterion were used for failure determination and periodic displacement boundary conditions were imposed on the model to numerically simulate the initiation, propagation and final failure of yarns and matrix damage. The FE model accurately predicts the longitudinal tensile strength of 3D6d braided composites and simulates the progressive damage process associated with the stress-strain behavior. The longitudinal mechanical properties of the 3D6d braided composites decrease with increasing braiding angle, while increasing the fiber volume fraction leads to enhancement of these properties. Furthermore, the change rate of longitudinal mechanical properties with braiding angle is relatively stable, and the influence of fiber volume fraction on longitudinal properties decreases with the increase of braiding angle. For the damage mode, the increase of braiding angle makes the damage of braiding and sixth-directional yarns gradually change from single direction to multiple directions, and the decrease of fiber volume fraction makes the damage of braiding and fifth-directional yarns at the peak stress more serious.

A FEM-based direct method for identification of Young’s modulus and boundary conditions in three-dimensional linear elasticity from local observation

Simultaneously identifying unknown Young’s modulus and boundary conditions of a linear elastic material with the measurements only on a part of its boundary is an inverse problem, which is usually solved by iterative methods in most studies. In this paper, we present a direct inverse method based on the finite element method (FEM) to recover these unknown parameters without iteration. A novel energy-like regularization term is developed to ensure the convergence of the estimated parameters with noisy input. In order to increase the accuracy of estimation, a novel regularization coefficient and its related preprocessing steps are designed based on the Morozov’s discrepancy principle to obtain the optimal regularization parameter, which is independent to the observation noise. In numerical experiments, several three-dimensional objects with different shapes and distribution of observed and unknown regions were used to verify the proposed method under different observation noises. Physical experiments were also conducted on a silicone phantom. The results of both numerical and physical experiments successfully recovered the Young’s modulus and displacement boundary conditions with expected accuracies close to their corresponding observation noise levels.

Synthesis of patient-specific multipoint 4D flow MRI data of turbulent aortic flow downstream of stenotic valves.

We propose to synthesize patient-specific 4D flow MRI datasets of turbulent flow paired with ground truth flow data to support training of inference methods. Turbulent blood flow is computed based on the Navier–Stokes equations with moving domains using realistic boundary conditions for aortic shapes, wall displacements and inlet velocities obtained from patient data. From the simulated flow, synthetic multipoint 4D flow MRI data is generated with user-defined spatiotemporal resolutions and reconstructed with a Bayesian approach to compute time-varying velocity and turbulence maps. For MRI data synthesis, a fixed hypothetical scan time budget is assumed and accordingly, changes to spatial resolution and time averaging result in corresponding scaling of signal-to-noise ratios (SNR). In this work, we focused on aortic stenotic flow and quantification of turbulent kinetic energy (TKE). Our results show that for spatial resolutions of 1.5 and 2.5 mm and time averaging of 5 ms as encountered in 4D flow MRI in practice, peak total turbulent kinetic energy downstream of a 50, 75 and 90% stenosis is overestimated by as much as 23, 15 and 14% (1.5 mm) and 38, 24 and 23% (2.5 mm), demonstrating the importance of paired ground truth and 4D flow MRI data for assessing accuracy and precision of turbulent flow inference using 4D flow MRI exams.

Equilibrium problem for a thermoelastic Kirchhoff-Love plate with a delaminated flat rigid inclusion.

A new mathematical model describing an equilibrium of a thermoelastic heterogeneous Kirchhoff-Love plate is considered. A corresponding nonlinear variational problem is formulated with respect toa two-dimensional domain with a cut. This cut corresponds to an interfacial crack located on a given part of the boundary of a flat rigid inclusion. The flat inclusion is described by a cylindrical surface. Due tothe presence of the flat rigid inclusion in the plate, restrictions of the functions describing displacements to the corresponding curves satisfy special constraints having a linear form. Displacement boundary conditions of an inequality type are set on the crack faces that ensure a mutual non-penetration of opposite crack faces. Solvability of the problem is proved. Under the assumption that the solution of the variational problem is smooth enough, an equivalent differential formulation is found. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Effect of boundary conditions on values of homogenized fabrics’s elastic parameters and inelastic response

The effect of displacement, periodic and surface traction boundary conditions (BCs) on predicted elastic parameters and inelastic response of a homogenized material equivalent to either a unidirectional or a randomly distributed short fibers reinforced composite has not been extensively studied. A few such studies have focused on their elastic analyses. Here, we analyze the effect of the three types of BCs on both elastic and inelastic deformations of weaves and braids with continuum damage mechanics used in the inelastic analyses. We employ simple unit cells, Hill’s theorem, and the displacement, periodic and uniform traction BCs to deduce from the finite element stiffness matrix K the matrix C of homogenized moduli. For both elastic and inelastic problems, it is seen that predictions using the displacement and the periodic BCs are close to those reported in the literature while those from the traction BCs are not. For unidirectional loading, it is found that the nonlinearity in the axial stress - axial strain curve, the localized deformations and the resin damage occurring in regions unsupported by fibers are significantly more for the traction BCs than that for the other two BCs. Also, there is more damage for the periodic BCs than that for the affine displacement BCs.

The consistent coupling boundary condition for the classical micromorphic model: existence, uniqueness and interpretation of parameters

We consider the classical Mindlin–Eringen linear micromorphic model with a new strictly weaker set of displacement boundary conditions. The new consistent coupling condition aims at minimizing spurious influences from arbitrary boundary prescription for the additional microdistortion field \(\varvec{P}\). In effect, \(\varvec{P}\) is now only required to match the tangential derivative of the classical displacement \(\varvec{u}\) which is known at the Dirichlet part of the boundary. We derive the full boundary condition, in adding the missing Neumann condition on the Dirichlet part. We show existence and uniqueness of the static problem for this weaker boundary condition. These results are based on new coercive inequalities for incompatible tensor fields with prescribed tangential part. Finally, we show that compared to classical Dirichlet conditions on \(\varvec{u}\) and \(\varvec{P}\), the new boundary condition modifies the interpretation of the constitutive parameters.

Mechanically-grown morphogenesis of Voronoi-type materials: Computer design, 3D-printing and experiments

The present work introduces a novel and versatile computer-design and experimental strategy to obtain random Voronoi-type geometries, called M-Voronoi (from mechanically grown), with smooth void shapes and variable intervoid ligament sizes that can reach very low relative densities. This is achieved via a numerical, large strain, nonlinear elastic, void growth mechanical process. Originally small circular voids embedded in a cell of arbitrary shape (triangle, circle, rectangle, trapezoid) grow when subjected to displacement (Dirichlet) boundary conditions. The deformed voids evolve into smooth Voronoi-type geometrical shapes leading to macroscopic isotropy or anisotropy depending on the prescribed boundary conditions. The void growth process is a direct consequence of mass conservation and the incompressibility of the surrounding nonlinear elastic matrix phase and the final achieved relative density may be analytically estimated in terms of the determinant of the applied deformation gradient. In order to study the mechanical properties of the M-Voronoi materials, we focus on two-dimensional porous polymer square representative isotropic and anisotropic geometries in terms of void size and realization, which are 3D-printed and experimentally tested under uniaxial compression. For comparison, we also test random polydisperse porous materials with circular voids, standard eroded Voronoi geometries and hexagonal honeycombs. The first two are also isotropic while the latter are only isotropic in the linear elastic regime. We show that the randomness of the M-Voronoi geometry and their non-uniform intervoid ligament size leads to enhanced mechanical properties at large compressive strains with no apparent peak-stress and strong hardening well before densification. By comparing them with the hexagonal geometries, which tend to exhibit a peak-stress and a plateau-type response, we show that the hardening response of the M-Voronoi is mainly due to their geometrical characteristics and less due to the polymer hardening response. Anisotropic M-Voronoi are also produced and tested indicating that anisotropy only enhances the initial stiffness along the longitudinal direction but instead leads to lower buckling loads and hardening rates than the corresponding isotropic M-Voronoi in the nonlinear regime.

A multi-grid sampling multi-scale method for crack initiation and propagation

In this study, a multi-grid sampling multi-scale (MGSMS) method is proposed by coupling with finite element (FEM), extended finite element (XFEM) and molecular dynamics (MD) methods. Crack is studied comprehensively from microscopic initiations to macroscopic propagation by MGSMS method. In order to establish the coupling relationship between macroscopic and microscopic model, the multi-grid FEM is used to transmit the macroscopic displacement boundary conditions to the atomic model and the multi-grid XFEM is used to feedback the microscopic crack initiations to the macroscopic model. Moreover, an image recognition based crack extracting method is proposed to extract the crack coordinate from the MD result files efficiently and the Latin hypercube sampling method is used to reduce the computational cost of MD. Numerical results show that MGSMS method can be used to calculate micro-crack initiations and transmit it to the macro-crack model. The crack initiation and propagation simulation of plate under mode I loading is completed.

Structural dynamic properties of stiffened composite plates with embedded multi-layered viscoelastic damping membranes

Aiming at optimizing stiffened composite structures with embedded multi-layered viscoelastic damping membranes, we developed a dynamic analytical model based on the first-order shear deformation theory, in which the vibration equations were derived from both the modulus theory and the composite material mechanics theory. The theoretical solution that satisfied the displacement boundary conditions was obtained by employing the Navier method. Results were simulated by the finite element method to verify correctness of the dynamic analytical model. In addition, the effects of structural parameters on dynamic performance (including first-order natural frequency and modal loss factor) were studied using the developed model.

Why coal permeability changes under unconstrained displacement boundary conditions: Considering damage effects

Permeability is an important indicator of coalbed methane production. Previous classical theoretical permeability models under unconstrained boundary conditions have difficulty matching experimental results. Consideration of the actual cleat geometry can explain the non-equilibrium evolution of permeability to some extent, but the matrix swelling strain ignores the impact of damage effects. In this study, cleat damage is considered and the damage ratio is calculated. Damage as a function of strain can be solved iteratively by post-processing calculations using COMSOL Livelink with MATLAB software for correcting the permeability equation. Unlike conventional permeability models, this modified permeability model incorporates damage effects. Damage effects and matrix adsorption swelling are interactive and coupled processes. The main parameters controlling the matrix swelling strain are Young's modulus, Heterogeneous coefficient, gas injection pressure, Langmuir strain constant, and Poisson's ratio. Based on these parameters, eight working conditions are designed to compare the models without and with damage effects, and the numerical solutions of the two models are also compared with experimental data. The results show that the cleat damage model is valid better in coal permeability prediction than conventional models without damage effect. This is because the cleat damage inhibits fracture aperture rebound, which has a crucial influence on the evolution of permeability. Through this study, it is shown that the experimentally tested permeability results can indeed be explained by the impact of damage deformation on the cleat aperture.

Bond- and state-based peridynamic analysis in a commercial finite element framework with native elements

This study presents a framework to perform bond-based (BB), ordinary state-based (OSB) and non-ordinary state-based (NOSB) peridynamic (PD) analysis in ANSYS, a commercial software, with native MATRIX27 elements. The PD equilibrium equations as well as the PD form of the traction components are constructed by using these elements and solved within the ANSYS framework through implicit methods. The domain is divided into three regions in order to satisfy the equilibrium equations and to directly impose the displacement and traction boundary conditions without a fictitious layer. The results are free of displacement kinks; thus, removing the unphysical stress concentrations. Also, the displacement predictions maintain the smoothness throughout the domain. Its accuracy is demonstrated by considering isotropic elastic plates subjected to various types of boundary conditions under quasi-static loading conditions. For all combinations of boundary conditions, the displacement predictions agree well with FE results. Failure is introduced gradually through the KILL option in ANSYS.

The shear stress reverse phenomenon in FRP sheet–concrete interface

Negative shear stress values often appear in the falling section of a bond-slip experimental curve of a fiber-reinforced polymer (FRP) sheet–concrete interface; the reason for this phenomenon is unclear. Defining this phenomenon as the “shear stress reverse phenomenon,” we attempt to explore its mechanism and influence on the bond-slip behavior and bond-slip model through theoretical and numerical analyses, and experimental verification. The theoretical analysis revealed that an “interface mismatch” often occurs during the shear failure of an FRP sheet–concrete interface, resulting in the formation of discontinuous microcracks at the interface, which cause fluctuations and make the interface slip field discontinuous. Owing to the quadratic derivative relationship between the slip and interfacial shear stress, the fluctuations in the slip curve lead to the shear stress reverse phenomenon. The validity of this theoretical concept was verified by analyzing the experimental interfacial bond-slip curves obtained by scholars in previous studies, and by performing double shear experiments on three fully-bonded FRP sheet–concrete interface specimens. The experimental results confirmed the existence of fluctuations in the interfacial slip field and the shear stress reverse phenomenon in the bond-slip curve. Finally, the failure process of the FRP sheet–concrete interface was analyzed through the finite element method, and the fluctuation of the slip and existence of the shear stress reverse phenomenon at the interface were observed in the calculation results. The results revealed that the shear stress reverse phenomenon is caused by special displacement boundary conditions, rather than the material properties. Therefore, while establishing a bond-slip model by fitting experimental curves, bond-slip curves that exhibit the shear stress reverse phenomenon should be deleted in their entirety, instead of only deleting the negative shear stress data in such curves.

An experimental study of morphological formation in bilayered tubular structures driven by swelling/growth

Circumferential wrinkling in soft tubular tissues is vital in supporting normal physiological functions. Most existing literature was dedicated to theoretical modeling and finite element simulations based on a specific growth model. This paper presents an experimental investigation on pattern formation and evolution in bilayered tubular organs using swelling deformation of polydimethylsiloxane (PDMS) and aims at supplying a thorough comparison with theoretical and finite element results. To create a twin model in modeling and simulation, the shear modulus in the incompressible neo-Hookean material is estimated via uni-axial tensile and pure shear tests. Five bilayered tubes with different material or geometrical parameters are fabricated. Swelling experiments are carried out for these samples in an individual experimental setup where a plane-strain deformation is guaranteed, and several surface patterns and the associated mode transformations are observed, namely, creases, wrinkles, period-doubling profiles, wrinkle-to-crease transition, and wrinkle-to-period-doubling transition. In particular, an interfacial wrinkling pattern is also observed. To make comparisons, a buckling analysis is conducted within the framework of finite elasticity by means of the Stroh formulation and a refined surface impedance matrix method. In addition, a finite element analysis (FEA) is performed to trace the evolution of surface instabilities. It turns out that the experimental findings agree well with the theoretical predictions as well as the finite element results. From our experiments, it is found that creasing mode may appear instead of wrinkling mode when both layers share a similar mechanical property. It is expected that the current work could provide novel experimental insight into pattern formation in tubular structures. In particular, the traditional impedance matrix method has been adapted, which enables us to resolve eigenvalue problems with displacement boundary conditions, and the good agreement among experimental, theoretical, and simulation consequences supplies strong evidence that a phenomenological growth model is satisfactory to reveal mechanisms behind intricate surface morphology in tubular tissues.

An extended fictitious node method for surface effect correction of bond-based peridynamics

The surface effect of peridynamics seriously affects the accuracy of numerical results near the boundary. The fictitious node method can modify the surface effect of peridynamics, but the previous method (Le and Bobaru, 2018) is only applicable to problems with only stress boundary conditions. This paper proposes an extended fictitious node method for deriving the fictitious node displacement constraint equations that apply to the displacement boundary conditions, based on the corresponding equivalence of stress components along with the boundary normal and tangential directions on the left and right sides of the boundary node. Consequently, the extended fictitious node method can handle problems with mixed boundary conditions of stress and displacement constraints. Then, this method is verified through numerical examples with mixed boundary conditions. The results after the surface correction show good agreements with those of the finite element method or with analytical solutions.

Stress average rule derived through the principle of virtual power

AbstractStress and strain average rules are the key conceptual pillars of the wide field of continuum micromechanics of materials. The aforementioned rules express that the spatial average of (micro‐)stress and (micro‐)strain fields throughout a microscopically finite representative volume element (RVE) are equal to the (macro‐)stress and (macro‐)strain values associated with the corresponding macroscopically infinitesimal volume element (macroscopic material point). According to the famous contribution of Hashin, stress and strain average rules are derived from equilibrium and compatibility conditions, together with (micro‐)displacement and (micro‐)traction boundary conditions associated with homogeneous (macro‐)strains and (macro‐)stresses, respectively. However, as, strictly speaking, only displacements or tractions can be described at the boundary, the remaining average rule turns out as a mere definition. We here suggest a way to do without such a definition, by resorting to the principle of virtual power (PVP) as a means to guarantee mechanical equilibrium: at the boundary of the RVE, we prescribe virtual (micro‐)velocities, which are linked to arbitrary, but homogeneous virtual (macro‐)velocities and (macro‐)strain rates, while the latter are also linked, in a multilinear fashion, with the microscopic virtual strain rate fields inside the RVE. Considering, under these conditions, equivalence of the macroscopic and the microscopic expressions for the virtual power densities of the internal and the external forces yields the well‐known stress average rule and, in case of microscopically uniform force fields, a volume force average rule. The same strategy applied to an RVE hosting single forces between atomistic mass points, readily yields the macroscopic “internal virial stress tensor.”

Analytical Solution for Elastic Analysis around an Ellipse with Displacement-Controlled Boundary

This paper presents novel analytical solutions for the analysis of an elliptical cavity within an infinite plane under plane strain conditions, considering typical displacement-controlled boundaries at the inner cavity and biaxial stresses at infinity. The problem is investigated by the plane theory of elasticity using Muskhelishvili’s complex variable method. The complex displacement boundary conditions are represented using the conformal mapping technique and Fourier series, and stress functions are evaluated using Cauchy’s integral formula. The proposed solutions are validated at first by comparing them with other existing solutions and then used to show the influences of displacement vectors on the distributions of induced stresses and displacements. The new solutions may provide useful analytical tools for stress and displacement analysis of an elliptical hole/opening in linear elastic materials which are common in many engineering problems.

A road modulus test method based on rigid spherical indentation

Currently, the modulus test method such as falling weight deflectometer has an inevitable problem of stress concentration, which may lead to a negative impact on the results. This paper attempted to apply the spherical indentation test equipment into road detection to solve this problem. Firstly, by following the displacement boundary conditions, a general expression of modulus, force, and displacement under the semi-infinite system was established. Compared with the Sneddon’s formula (a classic formula in materials industry), this expression was consistent with it under the small deformation situations, which verified the rationality of this method. Moreover, this expression can also be utilized in the larger deformation situations, which was more in line with the pavement environment. Then, by further improving the method, this paper established the method of calculating the structure modulus of new pavement layers. Finally, a specific test method for modulus detection based on the spherical indentation was proposed.