Normal Stress Boundary Condition Research Articles

Physics-informed UNets for discovering hidden elasticity in heterogeneous materials

Soft biological tissues often have complex mechanical properties due to variation in structural components. In this paper, we develop a novel UNet-based neural network model for inversion in elasticity (El-UNet) to infer the spatial distributions of mechanical parameters from strain maps as input images, normal stress boundary conditions, and domain physics information. We show superior performance – both in terms of accuracy and computational cost – by El-UNet compared to fully-connected physics-informed neural networks in estimating unknown parameters and stress distributions for isotropic linear elasticity. We characterize different variations of El-UNet and propose a self-adaptive spatial loss weighting approach. To validate our inversion models, we performed various finite-element simulations of isotropic domains with heterogenous distributions of material parameters to generate synthetic data. El-UNet is faster and more accurate than the fully-connected physics-informed implementation in resolving the distribution of unknown fields. Among the tested models, the self-adaptive spatially weighted models had the most accurate reconstructions in equal computation times. The learned spatial weighting distribution visibly corresponded to regions that the unweighted models were resolving inaccurately. Our work demonstrates a computationally efficient inversion algorithm for elasticity imaging using convolutional neural networks and presents a potential fast framework for three-dimensional inverse elasticity problems that have proven unachievable through previously proposed methods.

On flow of power-law fluids between adjacent surfaces: Why is it possible to derive a Reynolds-type equation for pressure-driven flow, but not for shear-driven flow?

Flows of incompressible Navier–Stokes (Newtonian) fluids between adjacent surfaces are encountered in numerous practical applications, such as seal leakage and bearing lubrication. In seals, the flow is primarily pressure-driven, whereas, in bearings, the dominating driving force is due to shear. The governing Navier–Stokes system of equations can be significantly simplified due to the small distance between the surfaces compared to their size. From the simplified system, it is possible to derive a single lower-dimensional equation, known as the Reynolds equation, which describes the pressure field. Once the pressure field is computed, it can be used to determine the velocity field. This computational algorithm is much simpler to implement than a direct numerical solution of the Navier–Stokes equations and is therefore widely employed by engineers. The primary objective of this article is to investigate the possibility of deriving a type of Reynolds equation also for non-Newtonian fluids, using the balance of linear momentum. By considering power-law fluids we demonstrate that it is not possible for shear-driven flows, whereas it is feasible for pressure-driven flows. Additionally, we demonstrate that in the full 3D model, a normal stress boundary condition at the inlet/outlet implies a Dirichlet condition for the pressure in the Reynolds equation associated with pressure-driven flow. Furthermore, we establish that a Dirichlet condition for the velocity at the inlet/outlet in the 3D model results in a Neumann condition for the pressure in the Reynolds equation.

Elasticity imaging using physics-informed neural networks: Spatial discovery of elastic modulus and Poisson's ratio.

Elasticity imaging is a technique that discovers the spatial distribution of mechanical properties of tissue using deformation and force measurements under various loading conditions. Given the complexity of this discovery, most existing methods approximate only one material parameter while assuming homogeneous distributions for the others. We employ physics-informed neural networks (PINN) in linear elasticity problems to discover the space-dependent distribution of both elastic modulus (E) and Poisson's ratio (ν) simultaneously, using strain data, normal stress boundary conditions, and the governing physics. We validated our model on three examples. First, we experimentally loaded hydrogel samples with embedded stiff inclusions, representing tumorous tissue, and compared the approximations against ground truth determined through tensile tests. Next, using data from finite element simulation of a rectangular domain containing a stiff circular inclusion, the PINN model accurately localized the inclusion and estimated both E and ν. We observed that in a heterogeneous domain, assuming a homogeneous ν distribution increases estimation error for stiffness as well as the area of the stiff inclusion, which could have clinical importance when determining size and stiffness of tumorous tissue. Finally, our model accurately captured spatial distribution of mechanical properties and the tissue interfaces on data from another computational model, simulating uniaxial loading of a rectangular hydrogel sample containing a human brain slice with distinct gray matter and white matter regions and complex geometrical features. This elasticity imaging implementation has the potential to be used in clinical imaging scenarios to reliably discover the spatial distribution of mechanical parameters and identify material interfaces such as tumors. STATEMENT OF SIGNIFICANCE: Our work is the first implementation of physics-informed neural networks to reconstruct both material parameters - Young's modulus and Poisson's ratio - and stress distributions for isotropic linear elastic materials by having deformation and force measurements. We comprehensively validate our model using experimental measurements and synthetic data generated using finite element modeling. Our method can be implemented in clinical elasticity imaging scenarios to improve diagnosis of tumors and for mechanical characterization of biomaterials and biological tissues in a minimally invasive manner.

Discrete Element Simulation on Macro-Meso Mechanical Characteristics of Natural Gas Hydrate-Bearing Sediments under Shearing

In order to study the macro-meso shear mechanical characteristics of natural gas hydrate-bearing sediments, the direct shear simulations of natural gas hydrate-bearing sediment specimens with different saturations under different normal stress boundary conditions were carried out using the discrete element simulation program of particle flow, and the macro-meso shear mechanical characteristics of the specimens and their evolution laws were obtained, and their shear damage mechanisms were revealed. The results show that the peak intensity of natural gas hydrate-bearing sediments increases with the increase in normal stress and hydrate saturation. Hydrate particles and sand particles jointly participate in the formation and evolution of the force chain, and sand particles account for the majority of the force chain particles and take the main shear resistance role. The number of cracks produced by shear increases with hydrate saturation and normal stress. The average porosity in the shear zone shows an evolutionary pattern of decreasing and then increasing during the shear process.

Experimental Study on the Influence of Hydromechanical Boundary Conditions on Shear-Flow Coupling Characteristics of Granite Joints

The instability of jointed rock mass is usually the shear process of the rock mass along discontinuities under the influence of groundwater flow. By conducting laboratory tests and numerical experiments on the shear-flow coupling of rock joints under constant normal stiffness (CNS) and constant normal stress (CNL) boundary conditions, the influence of normal boundary conditions and seepage pressure on the shear mechanical and flow characteristics of joints were investigated. The test results were as follows: The joint shear stiffness, peak, and residual shear strength under the CNS boundary condition were predominantly larger than those under the CNL boundary condition. Overall, these parameters were positively correlated with the initial normal stress σ n 0 . When σ n 0 > 2 MPa, the postpeak shear stress of the CNS boundary condition showed a sharp decrease, whereas that of the CNL boundary condition changed from a slowly decreasing type ( σ n 0 = 4 MPa, 6 MPa) to a sharply decreasing type at σ n 0 = 8 MPa. The peak dilation rate under the CNS boundary condition at all levels of normal stress was lower than that of CNL, and the strain softening in postpeak of the latter was more remarkable. In the process of joint shear, the hydraulic aperture displayed a four-stage variation law of “steady-sudden increase-slow increase-basically stable.” Moreover, the hydraulic aperture under the CNS boundary condition was always lower than that under the CNL boundary condition. The seepage pressure increased from 0.5 MPa to 1.5 MPa, and the average hydraulic aperture in the stable stage under normal stress at all levels increased from 0.146 mm to 0.187 mm. In addition, the average peak shear stress and average shear stiffness decreased by 0.9 MPa and 0.83 GPa/m, respectively. We also established a numerical model of a real rough three-dimensional joint, compiled a calculation program for the shear-flow process of a joint under CNS boundary conditions, and visualized the flow channel inside the joint. The seepage flow bypassed the area where the joints contacted each other, forming obvious flow channels. The flow rate increased at the intersection of the flow channels.

On flow simplification occurring in viscous three-dimensional water flows with constant non-vanishing vorticity

We show that a viscous steady water flow of constant non-vanishing vorticity situated below a free surface (assumed to have the shape of a two-dimensional time dependent graph) and above a flat bottom has to be two-dimensional; that is, while satisfying the viscous three-dimensional water wave equations, it turns out that the free surface, the velocity field and the pressure present no variation in one of the horizontal directions. Moreover, the vorticity must have only one non-zero component which points in the horizontal direction orthogonal to the direction of the surface wave propagation. Compared with previous works, our study here has the advantage that it considers the Navier–Stokes equations and also takes into account the normal and tangential stress boundary conditions.

Photogrammetry based characterization of hydro-mechanical properties of a rock fracture

Hydro-mechanical properties of a single fracture are governed by several parameters such as contact area, roughness, tortuosity, aperture, channeling, matedness, sample sizes, normal stress, flow regime, and flow boundary conditions. In this study, photogrammetry with numerical modeling and laboratory measurements were used to investigate the influences of roughness, normal stress, aperture, water pressure, and different flow boundary conditions on fluid flow in an artificial granite fracture. A Finnish Kuru grey granite block was mechanically split, and a 250 mm × 250 mm × 100 mm slab pair sample with a tensile crack in the middle was extracted. A photogrammetry-based method was used to reconstruct a 3D model of the fracture geometry. The resulting model was numerically simulated with COMSOL using the Navier-Stokes equations. Fluid flow experiments were computed with the Forchheimer equation. Comparison between the numerical modeling results and the analytical solution confirms that the 3D roughness geometry has a crucial role in defining the transmissivity, especially for nonlinear flow. On the other hand, normal stress increases fracture closure and increases contact areas, decreasing the hydraulic aperture and changing the flow paths. The presented method can be used in the contactless estimation of fluid flow properties of rock joints.

A Theoretical Model of Wind-Wave Growth Over an Ice-Covered Sea

A wind-wave generation model over an ice-covered sea is proposed. The wind velocity over the ice upper surface is decomposed into the mean velocity profile of the boundary-layer flow and small perturbations, while the ice cover is modelled as a viscoelastic layer, with the water part modelled as an inviscid fluid. The present model is based on two-dimensional linear flow-instability theory, with no-slip boundary conditions at the air–ice interface, and both normal and shear stress boundary conditions matched on the air–ice interface. It is shown that the model converges to the field and experimental data for open-water cases. The ice elasticity is found to be the critical factor for generating wind waves, and the generation of flexural-gravity waves and elastic waves in ice is analyzed.

Electrified cone formation in perfectly conducting viscous liquids: Self-similar growth irrespective of Reynolds number

Above a critical field strength, the free surface of an electrified, perfectly conducting viscous liquid, such as a liquid metal, is known to develop an accelerating protrusion resembling a cusp with a conic tip. Field self-enhancement from tip sharpening is reported to generate divergent power law growth in finite time of the forces acting in that region. Previous studies have established that tip sharpening proceeds via a self-similar process in two distinguished limits—the Stokes regime and the inviscid regime. Using finite element simulations to track the shape and forces acting at the tip of an electrified protrusion in a perfectly conducting Newtonian liquid, we demonstrate that the conic tip always undergoes self-similar growth irrespective of the Reynolds number. The blowup exponents at the conic apex for all terms in the Navier-Stokes equation and the normal stress boundary condition at the moving interface reveal the dominant forces at play as the Reynolds number increases. Rescaling of the tip shape by the power law representing the divergence in capillary stress at the apex yields an excellent collapse onto a universal cone shape with an interior half-angle dependent on the Maxwell stress. The rapid acceleration of the liquid interface also generates a thin interfacial boundary layer characterized by a significant rate of strain. Additional details of the modeled flow, applicable to cone growth in systems such as liquid metal ion sources, help dispel prevailing misconceptions that dynamic cones resemble conventional Taylor cones or that viscous stresses at a finite Reynolds number can be neglected.

Temporal stability of free liquid threads with surface viscoelasticity

We analyse the effect of surface viscoelasticity on the temporal stability of a free cylindrical liquid jet coated with insoluble surfactant, extending the results of Timmermans & Lister (J. Fluid Mech., vol. 459, 2002, pp. 289–306). Our development requires, in particular, deriving the correct expressions for the normal and tangential stress boundary conditions at a general axisymmetric interface when surface viscosity is modelled with the Boussinesq–Scriven constitutive equation. These stress conditions are applied to obtain a new dispersion relation for the liquid thread, which is solved to describe its temporal stability as a function of four governing parameters, namely the capillary Reynolds number, the elasticity parameter, and the shear and dilatational Boussinesq numbers. It is shown that both surface viscosities have a stabilising influence for all values of the capillary Reynolds number and elasticity parameter, the effect being more pronounced at low capillary Reynolds numbers. The wavenumber of maximum amplification depends non-monotonically on the Boussinesq numbers, especially for very viscous threads at low values of the elasticity parameter. Finally, two different lubrication approximations of the equations of motion are derived. While the validity of the leading-order model is limited to small enough values of the elasticity parameter and of the Boussinesq numbers, a higher-order parabolic model is able to accurately capture the linearised behaviour for the whole range of values of the four control parameters.

Influence of viscosity and surface tension of fluid on the motion of bubbles

Boundary integral simulation has been conducted to study the motion and deformation of bubbles with weak viscous and surface tension effects in fluid. Both normal and tangential stress boundary conditions are satisfied and the weak viscous effects are confined to the thin boundary layers around bubble surfaces, which is also known as boundary layer theory of bubble. By using this method, the influence of viscosity and surface tension of fluid on the motion of bubbles has been studied. Both axisymmetric and three-dimensional numerical results are compared with analytical results of Rayleigh-Plesset equation. Good agreement between them is achieved, which validates the numerical model. On this basis, interaction model between two vertically placed bubbles is established, by taking the surface tension, gravity, and viscous effects into consideration. Variations of physical quantities including bubble deformation, jet velocity, and energy of fluid are studied. Last but not least, the influence of viscosity and surface tension on the motion of a spherical bubble is investigated. It is found that viscous effects of fluid depress the pulsation of bubble and part of fluid energy is transformed into viscous dissipation energy. As a result, the development of bubble jet, the radius of the bubble, and the jet velocity are reduced gradually. On the other hand, the surface tension of fluid does not change the range of the bubble pulsation but reduces the period of the bubble pulsation and enhances the potential energy of the bubble. This model and numerical results aim to provide some references for bubble dynamics in bioengineering, chemical engineering, naval architecture, and ocean engineering, etc.

Accurate buckling analysis of thin rectangular plates under locally distributed compressive edge stresses

The buckling analysis of thin rectangular plates under locally distributed compressive edge stresses is a challenging problem if the point discrete methods are to be used. To obtain accurate buckling stress, one of the important factors is that the in-plane stress distributions within the plate prior to buckling should be accurate enough. Although it is possible to get analytical solutions for the in-plane stress distributions, but the expressions are very complicated since a stress-diffusion phenomenon exists. The differential quadrature method (DQM), being a point discrete method, has been successfully used in a variety of fields including the buckling analysis of thin rectangular plates under nonlinearly distributed edge compressions. However, it is rare to employ the DQM directly to solve problems of rectangular plates under locally distributed or point loads. To solve the challenging problem by using the DQM, novel formulations are presented in this paper. The locally distributed stress is first work-equivalently to point loads at all inner grid points on the loaded edge, then the normal stress boundary condition is numerically integrated before being discretized in terms of the differential quadrature. In this way accurate in-plane stress distributions can be obtained by the DQM without any difficulties. Buckling analysis of rectangular plates under either uniaxial or biaxial locally distributed compressive stresses is successfully performed. The accuracy of the DQ data is validated by comparing them with existing analytical solutions and finite element data. It is demonstrated that the compactness and computational efficiency of the DQM are retained. Accurate buckling loads are presented for rectangular plates with nine combinations of boundary conditions, various aspect ratios and load ratios. Some new results are also provided for references.

Buckling analysis of thin rectangular plates under uniaxial or biaxial compressive point loads by the differential quadrature method

To obtain accurate buckling load for rectangular plates under compressive point loads, one of the important factor is that the in-plane stress distributions within the plate prior to buckling should be accurate enough. Although the differential quadrature method (DQM) has been successfully used in a variety of fields including the buckling analysis of thin rectangular plates under non-uniformly distributed edge compressions, however, the DQM has some difficulty in dealing with singular functions such as the Dirac-delta function appeared in the stress boundary condition. In this paper, novel formulations are presented to overcome the difficulty encountered in dealing with the Dirac-delta functions by using the DQM. The normal stress boundary condition is numerically integrated before being discretized in terms of the differential quadrature. Detailed formulations are given. Buckling of rectangular plates under either uniaxial or biaxial compressive point loads is successfully analyzed. It is demonstrated that accurate buckling loads can be obtained by the DQM for rectangular plates with nine combinations of boundary conditions and various aspect ratios. The compactness and computational efficiency of the DQM are retained in solving the partial differential equations with boundary conditions involving Dirac-delta functions. The accuracy of the differential quadrature (DQ) results is verified by comparing them with existing analytical solutions and finite element data. New results are tabulated which can be a reference for other researchers to develop new numerical methods.

A second order virtual node algorithm for Navier–Stokes flow problems with interfacial forces and discontinuous material properties

We present a numerical method for the solution of the Navier–Stokes equations in three dimensions that handles interfacial discontinuities due to singular forces and discontinuous fluid properties such as viscosity and density. We show that this also allows for the enforcement of normal stress and velocity boundary conditions on irregular domains. The method improves on results in [1] (which solved the Stokes equations in two dimensions) by providing treatment of fluid inertia as well as a new discretization of jump and boundary conditions that accurately resolves null modes in both two and three dimensions. We discretize the equations using an embedded approach on a uniform MAC grid to yield discretely divergence-free velocities that are second order accurate. We maintain our interface using the level set method or, when more appropriate, the particle level set method. We show how to implement Dirichlet (known velocity), Neumann (known normal stress), and slip velocity boundary conditions as special cases of our interface representation. The method leads to a discrete, symmetric KKT system for velocities, pressures, and Lagrange multipliers. We also present a novel simplification to the standard combination of the second order semi-Lagrangian and BDF schemes for discretizing the inertial terms. Numerical results indicate second order spatial accuracy for the velocities (L∞ and L2) and first order for the pressure (in L∞, second order in L2). Our temporal discretization is also second order accurate.

Three-dimensional boundary integral simulations of motion and deformation of bubbles with viscous effects

Based on the velocity potential theory, a three-dimensional bubble model with the viscous effects is explored by using boundary element method. The velocity is irrotational, and the pressure is given by Bernoulli’s equation. The viscous component of the normal stress is evaluated and the viscous correction pressure is introduced to compensate for the non-zero shear stress. In this way, both normal stress and tangential stress boundary conditions are satisfied, and the weak viscous effects confined to the thin boundary layer around bubble surface are included. A spherical bubble moving in a viscous fluid is chosen as the case, and the 3d numerical results are compared with axi-symmetric ones and the analytical results. The good agreements between them validate the 3d model erected in this paper. On this basis, the non-spherical deformation of a single bubble under buoyancy force and the interaction of two bubbles are studied. The results of the bubbles with and without viscous effects are compared, and it is found that the jet velocity is slowed due to the viscous dissipation. The effects of Reynolds number and Froude number on the bubble dynamics are also investigated.

DEM Simulation of Direct Shear: 1. Rupture Under Constant Normal Stress Boundary Conditions

A particle-based distinct element method and its grain-based method are used to generate and simulate a synthetic specimen calibrated to the rupture characteristics of an intact (non-jointed) low-porosity brittle rock deformed in direct shear. The simulations are compared to the laboratory-generated ruptures and used to investigate rupture at various normal stress magnitudes. The fracturing processes leading to shear rupture zone creation and the rupture mechanism are found to be normal stress dependent (progressing from tensile splitting to shear rupture) and show partial confirmation of rupture zone creation in nature and in experiments from other materials. The normal stress dependent change is found to be due to the orientation of the major principal stress and local stress concentrations internal to the synthetic specimens being deformed. The normal stress dependent rupture creation process results in a change to the rupture zone’s geometry, shear stress versus horizontal displacement response, and thus ultimate strength.

The effect of clamping stress on the fatigue strength of bonded high-strength steel interfaces

It is well known that the fatigue strength of a welded joint does not increase proportionally with steel strength. For this reason, alternative joining techniques for high-strength steels are of great interest. Some studies have shown that hybrid joints, which combine mechanical fasteners and adhesives, have improved fatigue resistance as compared to welded joints in high-strength steel structures. For reliable design, an in-depth understanding of the clamped and bonded interface under cyclic loading is needed. In this study, the napkin ring specimen geometry has been chosen for a series of constant-amplitude fatigue tests. The test setup provides a bonded interface with well-defined normal and shear stress boundary conditions. Test conditions included four different static normal pre-stresses for the grit-blasted surface finish and one pre-stress level for the fine ground. Fatigue strengths of all the pre-stress/surface type conditions were obtained by a statistical method. For grit-blasted surfaces, the bonded specimens with high static normal pre-stresses behaved identically to the non-bonded specimens with the same pre-stress, i.e. the fatigue strength was not improved by the addition of adhesive even though the static strength had previously been shown to improve significantly for the same conditions.

Electric-field-induced interfacial instabilities of a soft elastic membrane confined between viscous layers

We explore the electric-field-induced interfacial instabilities of a trilayer composed of a thin elastic film confined between two viscous layers. A linear stability analysis (LSA) is performed to uncover the growth rate and length scale of the different unstable modes. Application of a normal external electric field on such a configuration can deform the two coupled elastic-viscous interfaces either by an in-phase bending or an antiphase squeezing mode. The bending mode has a long-wave nature, and is present even at a vanishingly small destabilizing field. In contrast, the squeezing mode has finite wave-number characteristics and originates only beyond a threshold strength of the electric field. This is in contrast to the instabilities of the viscous films with multiple interfaces where both modes are found to possess long-wave characteristics. The elastic film is unstable by bending mode when the stabilizing forces due to the in-plane curvature and the elastic stiffness are strong and the destabilizing electric field is relatively weak. In comparison, as the electric field increases, a subdominant squeezing mode can also appear beyond a threshold destabilizing field. A dominant squeezing mode is observed when the destabilizing field is significantly strong and the elastic films are relatively softer with lower elastic modulus. In the absence of liquid layers, a free elastic film is also found to be unstable by long-wave bending and finite wave-number squeezing modes. The LSA asymptotically recovers the results obtained by the previous formulations where the membrane bending elasticity is approximately incorporated as a correction term in the normal stress boundary condition. Interestingly, the presence of a very weak stabilizing influence due to a smaller interfacial tension at the elastic-viscous interfaces opens up the possibility of fabricating submicron patterns exploiting the instabilities of a trilayer.

A BOUNDARY-LAYER APPROACH TO STRESS ANALYSIS IN THE SIMPLE SHEARING OF RUBBER BLOCKS

Abstract Rubbers are usually modeled as being perfectly incompressible. In the simple shear of rubber blocks, the normal stress components, therefore, contain an arbitrary constant pressure term to be determined from the boundary conditions. There is therefore a fundamental ambiguity in the determination of this pressure because the normal stresses are expected to be identically zero on different faces of the sheared block. It is proposed here that the stress distribution near a face should be determined by the normal stress boundary condition at that face and that this distribution is valid only within a short distance from the face, giving rise to boundary layers at the faces of the sheared block. At the intersection of these boundary layers it will be assumed that the stress is additive. It is further assumed that the stresses within the bulk of the material should be determined by treating the perfectly incompressible material as equivalent to a slightly compressible material. The form of slight compressibility adopted here is that usually assumed in the finite element simulation of rubbers. These reasonable assumptions give rise to a complex stress pattern within the block. The results are qualitatively similar to results obtained by other authors on using a finite element approach for a neo-Hookean material. The possible occurrence of cavitation at the corners of the block is also examined.

The Amplitude Equation for the Rosensweig Instability in Magnetic Fluids and Gels

The Rosensweig instability has a special character among the frequently discussed instabilities. One distinct property is the necessary presence of a deformable surface, and another very important fact is, that the driving force acts purely via the surface and shows no bulk effect. These properties make it rather difficult to give a correct weakly nonlinear analysis. In this paper we give a detailed derivation of the appropriate amplitude equation based on the hydrodynamic equations emphasizing the conceptually new procedures necessary to deal with the distinct properties mentioned above. First the deformable surface requires a fully dynamic treatment of the instability and the observed stationary case can be interpreted as the limiting case of a frozen-in characteristic mode. Second, the fact that the driving force is manifest in the boundary conditions, only, requires a considerable change in the formalism of weakly nonlinear bifurcation theory. To obtain the amplitude equations a combination of solubility conditions and (normal stress) boundary conditions has to be invoked in all orders of the expansions.