Normal Tractions Research Articles

Elastic Wave Scattering off a Single and Double Array of Periodic Defects

The scattering problem of elastic waves impinging on periodic single and double arrays of parallel cylindrical defects is considered for isotropic materials. An analytic expression for the scattering matrix is obtained by means of the Lippmann–Schwinger formalism and analyzed in the long-wavelength approximation. It is proved that, for a specific curve in the space of physical and geometrical parameters, the scattering is dominated by resonances. The shear mode polarized parallel to the cylinders is decoupled from the other two polarization modes due to the translational symmetry along the cylinders. It is found that a relative mass density and relative Lamé coefficients of the scatterers give opposite contributions to the width of resonances in this mode. A relation between the Bloch phase and material parameters is found to obtain a global minimum of the width. The minimal width is shown to vanish in the leading order of the long wavelength limit for the single array. This new effect is not present in similar acoustic and photonic systems. The shear and compression modes in a plane perpendicular to the cylinders are coupled due to the normal traction boundary condition and have different group velocities. For the double array, it is proved that, under certain conditions on physical and geometrical parameters, there exist resonances with the vanishing width, known as Bound States in the Continuum (BSC). Necessary and sufficient conditions for the existence of BSC are found for any number of open diffraction channels. Analytic BSC solutions are obtained. Spectral parameters of BSC are given in terms of the Bloch phase and group velocities of the shear and compression modes.

First Failure Load of Rectangular Laminated and Sandwich Plates Using Isogeometric Analysis

This paper investigates the first failure load of simply supported and clamped laminate and sandwich plates loaded by a distributed normal traction on the top surface. A third-order shear and normal deformable plate theory, the isogeometric basis functions, and five failure criteria, namely, the maximum stress, the Tsai–Wu, the Tsai–Hill, the Hoffman, and the Hashin for laminated plates, and only the Tsai–Hill criterion for sandwich plates are used in this study. Of these, Hashin’s criteria distinguish between the fiber and the matrix failure. The in-plane stresses are found from the plate displacements and Hooke’s law, and the transverse stresses are recovered by integrating the equilibrium equations through the thickness. Effects of the plate aspect ratio, the fiber angle, the face sheet materials, and the core materials on the first failure load are identified. The computed results are found to agree well with those reported in the literature. Whereas the five failure criteria for laminated plates predict nearly the same value of the first failure load, they do not provide the same location of the failure initiation.

An Analytical Adhesion Model for Elastic Contact Electrification

Abstract Contact electrification is a universal phenomenon that commonly occurs in almost every solid–solid contact pair. The tribo-charges deposited on two surfaces by contact electrification can significantly affect adhesion; however, contact electrification is often overlooked in the study of adhesive contact. Here, we develop an analytical model to investigate electroadhesion during the contact phase between two initially uncharged dielectric surfaces, namely, an elastic parabolic surface and a rigid flat. A system of nonlinear equations is derived to describe the relationship between the indentation, normal load, radius of contact area, and radius of the charged zone using the Barthel–Maugis–Dugdale model (Barthel, 1999, “Modelling the Adhesion of Spheres: When the Form of the Interaction Is Complex, Colloids. Surf., A., 149, pp. 99105.). The analytical results show good agreement with the numerical results of the full self-consistent contact model. When contact electrification leads to a higher tribo-charge density and a larger charged zone, it has a greater impact on the normal traction, interfacial gap, force-approach curves, jump-out, and dissipated energy. The analytical model developed in this study serves as the foundation for advances in rough surface electroadhesive contact and electroadhesion testing, and it sheds light on the usage of adhesive joints in ultra-high vacuum environments and outer space, where contact electrification has a significant impact.

Surface effect on the partial-slip contact of a nano-sized flat indenter

The partial-slip contact at nano-scale is always influenced by a surface effect. A new model based on the Gurtin-Murdoch (G-M) surface elasticity theory is proposed in this paper to study the partial-slip contact behavior of a rigid flat nano-indenter on an elastic half-space. The surface effect in the partial-slip contact is characterized via a non-classical boundary condition involving a surface-induced normal traction related to the residual surface stress and a surface-induced tangential traction related to the surface elasticity, the influences of which on stress and displacement fields and the stick-slip state at the contact surface are investigated. It is found that, with the increase of residual surface stress, both the normal pressure and vertical and horizontal displacements in the contact zone decrease, the relative slip between indenter and substrate reduces and the stick region is enlarged. Such effects are basically attributed to the action of the surface-induced normal traction opposite to the externally applied compression. However, the increase of surface elastic constants is conductive to the relative slip and results in a shrinkage of the stick region, which is attributed to the action of the surface-induced tangential traction opposite to the frictional stress. An interesting phenomenon is further unveiled that, when the frictional coefficient increases, the dominant role in affecting the stick-slip state changes from the surface-induced tangential traction to the normal one, thus inspiring a feasible route to manipulate the surface effect by tuning the frictional coefficient of substrate. The present research enables one to better understand the partial-slip contact behavior of nano-indenters, which is of guiding value for anti-wear designs in nano-mechanical devices.

A Receding Contact Problem of Two Layers one of Functionally Graded, Loaded by Circular Rigid Block and Resting on a Pasternak Foundation

In this study, frictionless receding contact problem of two elastic layers which one is functionally graded material (FGM) resting on a Pasternak foundation is considered. The external load is applied to the homogeneous elastic layer by means of a circular rigid block and the functionally graded layer rests on a Pasternak foundation. The effect of gravity forces is neglected, and only compressive normal tractions can be transmitted through the interfaces. Displacement and stress expressions for the layers are obtained using the theory of elasticity and integral transformation technique. By applying the boundary conditions for the problem, reduced to two integral equations in which the contact stresses and contact lengths are unknown. The system of integral equations is numerically solved by making use of appropriate Gauss Chebyshev integration formulas. The equilibrium conditions are satisfied in the solution and the contact stresses and contact distances related to the problem are obtained for various dimensionless quantities.

Estimation of fatigue life of TiN coatings using cyclic micro-impact testing

We studied the behaviour of a thin titanium nitride (TiN) coating (1.5 µm thick) on a tool steel substrate under dynamic and cyclic impacts through an approach combining experimental testing and computational modelling. Dynamic impact testing was used to investigate the load-dependent dynamic hardness and assess the energy-dissipation capabilities of the coating system. In cyclic impact tests, the coating-substrate system experienced permanent plastic deformation in each cycle, ultimately leading to coating failure. Chemical analysis identified an interlayer between the coating and the substrate that influenced the coating response, while cross-sectional analysis revealed the extent of coating damage due to cycling and impact load. A three-dimensional map was constructed, connecting the acceleration load, sensed depth, and cycles to the coating failure, and an empirical equation was used to characterise the relationship between the depth and cycles to failure. The computational model examined the traction component distribution during loading and unloading, focusing on normal and shear tractions. These findings suggested the potential significance of normal traction in the interfacial fatigue failure due to impact.

An adhesion model for contact electrification

Contact electrification (CE) is a universal phenomenon that occurs at the contact interface where tribo-charges transfer from one surface to another. These CE-driven charges accumulate at the interface and can significantly affect the soft adhesive contact. So far, only a few analytical and numerical models have been proposed to partially solve the CE-induced electroadhesive contact problem. In this study, we have developed a full self-consistent contact model to study the CE-induced electroadhesion between a dielectric elastic axisymmetric parabolic surface and a dielectric rigid flat. Both surfaces are initially neutralized and undergo only one loading–unloading cycle. The surface interaction is determined by both the Lennard-Jones and the electrostatic interaction laws. The electrostatic problem is numerically solved to accurately calculate the electrostatic traction due to both the tribo-charges and bound charges. The normal traction, interfacial gap, and hysteresis loop of the force curve are thoroughly investigated. The effect of non-uniform tribo-charge density on the force curve and electrostatic traction is explored. The numerical model developed in the present work can serve as the foundation for more complex electroadhesive contact models. It also sheds light on the interfacial design of CE-related devices, e.g., contact-separation triboelectric nanogenerator.

Revisiting the Young’s model for ferrofluid droplets: Magnetowetting or magneto-dewetting?

Magnetic field-mediated wetting of ferrofluid droplets has emerged to be of fundamental importance because of its plethora of applications ranging from chemical and biomolecular analysis to soft robotics and medical imaging. However, while extensive studies continue to be reported on the underlying experimental facets, a fundamental theoretical depiction of the observed contact angle variation with the applied magnetic field remains grossly elusive, as attributable to a complex dependence of the induced magnetization on the droplet’s intrinsic magnetic properties. Circumventing these constraints, here we put forward a first-principle analytical model that delineates the variation in the contact angle as a function of an applied magnetic field on a sessile ferrofluid droplet, by appealing to energy minimization principles. One key feature of our analysis is the consideration of the local magnetization as a function of the magnetic field itself, resulting in a new contribution to the interfacial energy by virtue of normal traction at the droplet's contact with its non-magnetic surroundings. This theory is supported by experiments identified in prior research to elucidate the fundamental physics of relevance. Our results highlight the substantial influence of the saturation magnetization and the initial susceptibility of the ferrofluid on the resulting contact angle alterations, as well as possible non-trivial routes of highly selective and precise enhancement or attenuation in the extent of wetting. These findings are likely to be of fundamental importance towards establishing the design foundations of portable magnetic resonance imaging devices, micromixers, and microreactors, magnetically-manipulated automatic control of soft fluidic matters, among others.

High-resolution assessment of multidimensional cellular mechanics using label-free refractive-index traction force microscopy

A critical requirement for studying cell mechanics is three-dimensional assessment of cellular shapes and forces with high spatiotemporal resolution. Traction force microscopy with fluorescence imaging enables the measurement of cellular forces, but it is limited by photobleaching and a slow acquisition speed. Here, we present refractive-index traction force microscopy (RI-TFM), which simultaneously quantifies the volumetric morphology and traction force of cells using a high-speed illumination scheme with 0.5-Hz temporal resolution. Without labelling, our method enables quantitative analyses of dry-mass distributions and shear (in-plane) and normal (out-of-plane) tractions of single cells on the extracellular matrix. When combined with a constrained total variation-based deconvolution algorithm, it provides 0.55-Pa shear and 1.59-Pa normal traction sensitivity for a 1-kPa hydrogel substrate. We demonstrate its utility by assessing the effects of compromised intracellular stress and capturing the rapid dynamics of cellular junction formation in the spatiotemporal changes in non-planar traction components.

Analysis of adhesive contact of heterogeneous elastic materials

This paper reports a three-dimensional numerical model to solve adhesive elastic contact problem of heterogeneous materials with an engineered rough surface. This model integrates the Maugis-Dugdale (MD) adhesive contact model, the Boussinesq-Cerruti integral equations, the equivalent inclusion method (EIM), and the discrete convolution and fast Fourier transform (DC-FFT) algorithm, aiming at accurate determination of contact results for heterogeneous materials. An adhesive displacement-driven conjugate gradient method (CGM) was also developed to support rapid solutions of contact pressure and contact area. Four cases were analyzed to verify this model with satisfaction. This model was used to analyze the effects of inhomogeneity (location, size, and material properties) and adhesion on micro-adhesive-elastic contact, and the results lead to a curve-fitting equation for quick evaluation of the effects of inhomogeneity properties and adhesion characteristics on normal traction. It was also utilized to study the influence of the roughness level of a type of rough surface on the surface adhesion behavior.

Detachment forces during parallel-plate gap separation mediated by a simple yield-stress fluid.

In this work we have monitored the multiple stages of the normal traction force response of a yield-stress fluid confined between two circular parallel plates that are separated at constant velocity. At narrow initial gaps, the air-fluid interface suffers from the Saffman-Taylor instability, confirmed by visual inspection of fingering patterns imprinted on the fluid. At larger initial gaps, the fluid preserves the initially imposed circular symmetry of the confining plates, indicating the absence of instability. Due to the system characteristics and experimental environment, the multiple traction force contributions occurred in cascade, permitting us to isolate the adhesion responses associated with viscosity, capillarity, and yield stress. Employing a standard Herschel-Bulkley model, we assessed the scaling of the traction force in multiple regimes-specifically, evaluating the dependencies of the fingering to yield-stress transitions.

Elastic Stress Field beneath a Sticking Circular Contact under Tangential Load

Based on a potential theoretical approach, the subsurface stress field is calculated for an elastic half-space which is subject to normal and uniaxial tangential surface tractions that—in the case of elastic decoupling—correspond to rigid normal and tangential translations of a circular surface domain. The stress fields are obtained explicitly and in closed form as the imaginary parts of compact complex-valued expressions. The stress state in the surface and on the central axis are considered in detail. As, within specific approximations that have been discussed at length in the literature, any tangential contact problem with friction can be understood as a certain incremental series of such rigid translations, the solutions presented here can serve as the basis of very fast superposition algorithms for the analysis of subsurface stress fields in general tangential contact problems with friction. This idea is demonstrated by means of the frictional tangential contact between an elastic half-space and a rigid cylindrical flat punch with rounded corners.

Mode Ⅰ–Governed fracture energy and maximum normal traction of a CFRP rod

This study develops a double cantilever beam test by using a specific fixture for measuring the representative critical energy release rate of rigid composite rods. Carbon fiber reinforced epoxy composite rod was used for the evaluation. Finite element analysis applying Cohesive Zone Model was used to estimate the relationship between load and displacement. In addition, for verifying the effect of specimen width on the measurement, the representative critical energy release rate was estimated and compared with reference to the crack length measured from the center or outer tip of the crack. The numerical results showed close value to that of experiment. This suggests that an effective representative critical energy release rate can be measured by the proposed method.

Experimental determination of the interlayer fracture toughness of a composite material

Modeling the propagation of interlaminar defects in multilayer composite materials requires knowledge of the characteristics of the material under study associated with the fracture processes. The goal of the study is determination of the interlaminar fracture toughness in the case of normal traction (GIc). Finite element modeling of testing was carried out proceeding from the data obtained. The delamination process was modeled using the most common approaches within the finite element method: VCCT (virtual crack closure technique) and CZM (cohesive zone model), wherein the fracture toughness is used as the main parameter for the occurrence of delamination. The obtained data are compared with the experiment. To confirm the correctness of the obtained experimental data on the fracture toughness GIc, modeling of the process of compression testing of a strut-type specimen with a defect in the form of a through-the-width delamination was carried out. This problem includes a nonlinear static solution with buckling and subsequent post-buckling behavior, and, as a result, the spread of the delamination zone. The data obtained using finite element modeling are consistent with the available experimental data.

General Series Solutions for Stresses around an Arbitrarily Loaded Hole in the Infinite Plate

Based on the linear elasticity theory and the Michell-Fourier series technique, a general series solution for stresses around an arbitrarily loaded hole in the infinite plate is presented. An Airy stress function in the form of Michell-Fourier series is chosen. The stresses are expressed by polar coordinates and two sets of stress coefficients. Arbitrarily distributed normal and tangential tractions on the edge of the hole are expanded into Fourier series. Then the loads are expressed by two sets of loading coefficients. The stress boundary is utilized for conditions on the edge of the hole. The stress coefficients are related to the loading coefficients. The general series solution for stresses around an arbitrarily loaded hole in the infinite plate is obtained. The verification solution of stresses around a uniformly pressured hole, and the stresses around a hole loaded by uniform tangential force are presented. The stresses around a circular hole partially loaded by varying traction are also calculated. The results show that the presented analytical solution is in great consistence with that from ANSYS.

Plane and axisymmetric problems of an interfacial crack in a power-law graded material

Plane and axisymmetric problems of interfacial cracks in a power-law graded infinite medium are examined. It is shown that the models are governed by integral equations of Mellin’s convolution type whose kernels are expressed through the hypergeometric function. Exact solutions are derived by the method of orthogonal Jacobi polynomials in a series form and by the Wiener–Hopf method by quadratures. Mode-I stress intensity factors and the associated weight functions for power-law graded materials in the plane and axisymmetric cases are introduced and evaluated. It is shown that, although the displacement jump and the normal traction component have power singularities at the crack tip different from 1/2, the strain energy variation is proportional to the crack length change as in the case of homogeneous materials. A Griffith-type criterion of crack propagation in power-law graded materials is proposed and results of numerical tests are reported.

Is the out-of-plane shear strength an independent property? A micromechanical perspective about a macromechanical question

For unidirectional laminae, five elastic properties and six strengths are usually required. The goal of this paper is to evaluate if the out-of-plane shear strength is really an independent property or if it is somehow related to the transversal tensile and compressive strengths. The analytical VSPKc micromechanical model is used to derive closed-form relations. Fiber-matrix interface failure due to normal and shear interface tractions is evaluated considering a perfectly brittle behavior. The proposed model is compared with four analytical models from the literature and their estimations are verified against six experimental data. The relations proposed obtained the best results compared with the experimental data, with an average error equal to 8.84%, while the error using the second-ranked model is 17.40%, indicating with good reliability that the out-of-plane shear strength is not an independent property.

Experimental and numerical investigation on the mode I and mode II interlaminar fracture toughness of nitrogen-doped reduced graphene oxide reinforced composites

An experimentaland computational study is carried out to investigate the toughening effect of nitrogen-doped reduced graphene-oxide particles (ND-RGOP) in mode I and mode II delamination of carbon fiber/epoxy laminates. For mode I, both delamination initiation and propagation toughness are enhanced by 126.3% and 119.3% with 0.8 ND-RGOP addition. In addition to mode I toughness enhancement, ND-RGOP reinforcement also increases mode II toughness. The highest mode II delamination toughness value was obtained in 0.4ND-RGOP laminates with an increase of 45.6% compared to neat laminates. However, as seen in the bending tests, the maximum force of the end notch bending (ENF) specimens with 0.8 ND-RGOP added is not lower than that of neat laminates. The interlaminar fracture toughness is high due to the rough path of delamination with ND-RGOP addition, which is also observed by FESEM images of the delamination surfaces. In addition to the experiments, a cohesive zone model with parametric traction stress value was prepared in ABAQUS/CAE to obtain the critical traction stresses in the delamination of laminates. The critical normal traction stress of double cantilever beam (DCB) specimens is increased by over 20% with the addition of ND-RGOP. Besides, the critical interlaminar shear stress in mode II delamination for ENF samples is significantly improved.

Circular Nanoplate on Elastic Nanolayer under Axisymmetric Loading and Surface Effects

Influence of surface energy on an interaction problem between a flexible circular nanoplate and a nanolayer is examined by using a variational formulation and the GM surface theory. The nanoplate is resting in smooth contact on the supporting nanolayer, and subjected to axisymmetric vertical loadings. The normal traction at the plate–layer interface is written in terms of generalized coordinates obtained from the flexibility equations derived from Green’s function and Hankel integral transform technique. A numerical solution scheme is then implemented into a computer code, and the convergence and accuracy of the proposed solution are verified with existing solutions. A set of numerical solutions is illustrated to present an impact of the surface energy effects on this interaction problem. Both deflection and bending moment of the nanoplate show a considerable dependence on the relative plate stiffness and the surface material properties, and demonstrate the size-dependent behaviors.

New analytical model for thermomechanical responses of multi-layered structures with imperfect interfaces

A new analytical model is developed for thermomechanical responses of multi-layered structures with an arbitrary number of layers and subjected to general thermal and mechanical loading. The formulation is based on an extended Bernoulli–Euler beam theory and a slip-interface model. The former includes Poisson’s effect and covers both the plane stress and plane strain deformations, and the latter allows slipping between two adjacent layers but no jump in the normal displacement or traction. An analytical solution for a multi-layered structure under general thermomechanical loading is derived by using a new approach that first determines one interfacial shear stress and the curvature of the deformed structure. To illustrate the newly developed model, three example problems for two-, three- and five-layer structures respectively are analytically solved by directly applying the new model. In all three cases, the solutions are obtained in closed-form expressions by considering both temperature changes and mechanical loads including body forces, distributed normal and shear stresses on the top and bottom surfaces, and normal forces, transverse shear forces and bending moments at the two ends, unlike existing ones. It is shown that the current solution for two-layer structures recovers an existing solution without considering Poisson’s effect and mechanical loading and the classical solution of Timoshenko for perfectly bonded bi-metal thermostats as two special cases. The closed-form solution for five-layer structures with imperfect interfaces is derived here for the first time. In addition, numerical results are provided for five- and seven-layer transistor stacks to quantitatively demonstrate the new model. It is found that the current results for the five-layer transistor stack agree well with those obtained by others, thereby further validating the new model.