Indentation Problem Research Articles

Spherical Cavity Expansion Approach for the Study of Rigid-Penetrator’s Impact Problems

In recent years, Spherical Cavity Expansion (SCE) theory has been extensively utilized to model dynamic deformation processes related to indentation and penetration problems in many fields. In this review, the SCE theory is introduced by explaining the different mathematical features of this theory, its solution, and a potential application to model the penetration of a rigid penetrator into a deformable target. First, a chronologically literature review of the most common models used to study this kind of penetration problems is introduced, focusing on the SCE theory. Then, the engineering model of penetration is presented using the SCE approach. The model is then compared and validated with some FE numerical simulations and with previous penetration results. It is concluded that this engineering model based on the SCE theory can be utilized to predict the projectile deceleration and penetration depth into the semi-infinite and finite targets impacted by rigid penetrators.

Measurement of nanoindentation properties of polymers considering adhesion effects between AFM sharp indenter and material

Nanomechanical properties of polymer samples were calculated using an adhesive contact model appropriate for AFM indentation problems. A series of Polydimethylsiloxane (PDMS) samples were indented by the sharp indenter in the air by using an AFM, and dozens of the force–displacement curves of each sample were obtained. An adhesive contact model suitable for sharp indentation with adhesion was established based on the same assumptions of the JKR model which is only suitable for spherical indentation at small penetration depth. Differences between sharp indentation problems with and without adhesion were discussed, and the limitations of the traditional adhesion model were given. The elastic modulus was obtained by fitting experimental force–displacement curves with theoretical ones, and results were compared to those macroscopic values in literature. The adhesion energy between the indenter and the sample surface was accurately calculated using the adhesion model based on the calculated elastic modulus. The influence of the indenter tip angle on the calculation results of the elastic modulus was also discussed theoretically. In this study, the mechanical properties of polymer samples were calculated at the nanoscale considering the adhesion effect.

A simple method for spherical indentation of an elastic thin layer with surface tension bonded to a rigid substrate

An alternative method is proposed to solve the spherical indentation problem of an elastic thin layer with surface tension bonded to a rigid substrate. Based on the Kerr model, we establish a simple modified governing equation incorporating the surface tension effects for describing the relationship between the pressure and downward deflection of the impressed surface of the layer. This modified governing equation holds both inside and outside the contact zone, making it possible to analyze the whole layer by a unified differential equation. Numerical results are presented for the contact pressure inside the contact zone, the surface deflection of the elastic layer and the load-contact zone width relation to illustrate the present method. The validity and accuracy of the present method are demonstrated by comparing our results with those available in the existing literature.

Expanding Cavity Model Combined With Johnson–Cook Constitutive Equation for the Dynamic Indentation Problem

Abstract The indentation formed on a metallic component by the high-velocity impingement of a small object can fracture the component, and this is known as foreign object damage. In this type of dynamic indentation, it is necessary to consider the effects of work hardening, strain rate hardening, and thermal softening in the impinged material. In this study, in order to consider these effects, the expanding cavity model based on a spherical formulation is modified via the Johnson–Cook constitutive equation for the dynamic indentation problem. Additionally, an equation is developed based on energy conservation and the modified expanding cavity model to predict the size of the indentation formed by an impingement of a solid sphere (EPIS). The distributions of equivalent plastic strain, equivalent plastic strain rate, temperature, and equivalent von Mises stress obtained via the expanding cavity model were in good agreement with the data obtained from the finite element analysis (FEA). Furthermore, it was demonstrated that EPIS accurately predicted the indentation size formed on various metallic materials at several impingement velocities in the range of 50–300 m/s. Consequently, EPIS can be effectively applied to an impingement problem of a hard sphere onto a sufficiently thick ductile material within 300 m/s without any help of FEA.

A micromorphic computational homogenization framework for auxetic tetra-chiral structures

Auxetic chiral structures exhibit many unique and enhanced mechanical properties, which emerge from the coupling of stretching and rotational mechanisms within the underlying unit cell. The standard first-order Computational Homogenization (CH) approach is inadequate in capturing this coupling effect. Moreover, a size effect arising from the interactions between micro and macro length scales cannot be accounted for. In this contribution, the micromorphic CH framework developed in Biswas and Poh (2017), Biswas et al. (2019), is reformulated for 2D tetra-chiral solids to address these limitations. Together with the standard macro-distortion tensor, an additional kinematic variable is introduced to characterize the deformation and rotation of the central ring. These two kinematic fields, each characterizing a particular aspect of the unit cell deformation, enable the micromorphic framework to adequately characterize the underlying dominant deformation modes. The predictive capability of the micromorphic CH framework is demonstrated through three benchmark problems in plane strain condition. Considering an axial loading problem, it is shown that the proposed micromorphic framework accurately captures the coupling between internal ring rotation and stretching of tangential ligaments. The predictive capability of the micromorphic approach is next demonstrated with a flat punch indentation problem in a regime without a clear separation of length scales. The homogenized micromorphic model predicts the correct global force-displacement response, as well as the underlying deformation mechanisms. Finally, the capability of the micromorphic framework in capturing both distortional and rotational modes of the central ring is demonstrated through the axial loading of a tapering plate.

Strain localization analysis of Hill’s orthotropic elastoplasticity: analytical results and numerical verification

In this work the strain localization analysis of Hill’s orthotropic plasticity is addressed. In particular, the localization condition derived from the boundedness of stress rates together with Maxwell’s kinematics is employed. Similarly to isotropic plasticity considered in our previous work, the plastic flow components on the discontinuity surface vanish upon strain localization. The resulting localization angles in orthotropic plastic materials are independent from the elastic constants, but rather, they depend on the material parameters involved in the plastic flow in the material axes. Application of the above localization condition to Hill’s orthotropic plasticity in 2-D plane stress and plane strain conditions yields closed-form solutions of the localization angles. It is found that the two discontinuity lines in plane strain conditions are always perpendicular to each other, and for the states of no shear stresses, the localization angle depends only on the tilt angle of the material axes with respect to the global ones. The analytical results are then validated by independent numerical simulations. The B-bar finite element is employed to deal with the incompressibility due to the purely isochoric plastic flow. For a strip under vertical stretching in plane stress and plane strain as well as Prandtl’s problem of indentation by a flat rigid die in plane strain, numerical results are presented for both isotropic and orthotropic plasticity models with or without tilt angle. The influence of various parameters is studied. In all cases, the critical angles predicted from the localization condition coincide with the numerical results, giving compelling supports to the analytical prognoses.

Axisymmetric indentation problem of a transversely isotropic elastic medium with surface stresses

The frictionless contact problem between an axisymmetric rigid indenter and a layered transversely isotropic medium with surface stresses is considered. The contact pressure is represented as a product of two series based on the solutions of the bulk material and the elastic surface. By using Hankel transforms, the coefficients in the product-series representation are determined by the normal displacement condition inside the contact area and the finite-pressure condition at the contact edge. Taking the spherical indentation as a specific example, the effectiveness of the solution procedure is verified for various contact scenarios. Comparing with the Green’s function method, this solution procedure is not only computationally efficient but also may give the contact pressure in its analytical form. For some specific problems, the effects of the material anisotropy and the layer thickness on the contact process with surface stresses are investigated.

New Axisymmetric Slip-Line Theory for Metal and Its Application in Indentation Problem

AbstractExisting axisymmetric slip-line theories for metal are established based on certain hypotheses of circumferential stress and thus lack strictness. This article presents a technique for deri...

Contact Problem for a Rigid Punch and an Elastic Half Space as an Inverse Problem

We solve a contact problem of indentation of a punch into an elastic half space with regard for the friction and in the presence of the zones of adhesion, sliding, and separation. The applied approach is based on the statement of the problem in the form of the inverse problem in which the Coulomb law of friction is used as an additional condition in the regions with friction. In the formulation of the inverse problem, we take into account the presence of the zones of adhesion whose sizes are unknown. The correctness of the solution of the inverse problem is analyzed. The proposed approach, in combination with the procedure of discretization, enables us to determine the zones of microsliding alternating with the zones of adhesion and separation.

Modeling and analysis of the nonlinear indentation problems of functionally graded elastic layered solids

In tribological and thermo-electro-mechanical applications with an appropriate gradation of properties, functionally graded materials provide superior performance for damage and wear resistance of contact systems and thermal and mechanical responses for other mechanical devices. This paper presents a nonlinear mixed variational-based computational model for the analysis of nonlinear plane indentation problems of elastic functionally graded layered elastic solids. The functionally graded elastic layer is perfectly bonded to the elastic substrate and is normally loaded by a cylindrical rigid punch. In addition to nonlinearity inherent to the contact problem, the developed model accounts for the geometric nonlinearity in the sense of larger displacements and rotations, but smaller strains. The modulus of elasticity as well as Poisson's ratio of the graded layer are varying along the thickness of the layer according to both exponential and power laws. On the contrary, in the conventional homogeneous finite element formulation the isoparametric graded finite element formulation is adopted on the level of Gaussian integration points to realize the gradation in material properties. The friction at the contact interface is modeled using Coulomb's friction law. Moreover, the inequality contact constraints are exactly satisfied throughout the contact interface by employing the Lagrange multiplier method, where the indentation force and the displacement fields are treated as independent variables. The obtained results of contact pressure, tangential contact stress, stress distribution, and indentation force show a significant influence of the material distribution, gradation law, gradient index, and thickness of the functionally graded layer.

Experimental determination of the parameters of the nonlinearly elastic Hencky model

This work presents a variant of physically nonlinear constitutive elasticity relations derived using the Hencky strain tensor. A test plant for determining the model’s constants is designed. These constants for weakly compressible materials are derived by compression tests of prismatic elastomeric specimens. The results of solving the problem of indentation of a rigid spherical stamp into a cylindrical specimen are provided. The suggested constitutive relations were used with the constants found by the compression tests. In addition, calculations were performed according to the relations known for the Hencky material and to the hypoelasticity relations with the neutral derivative of the stress tensor. To stabilize the calculation procedures, the Hencky tensor and its derivative were expanded by the monotonous parameter in a series by the Cauchy strain tensor degrees without any restrictions for allowable strain values. A numerical algorithm was built for tracking the variable area of contact. The calculation procedures of discretizing the stated problem by the finite elements method and step-by-step loading were implemented as an applied software suite. One of the integral characteristics of the process considered was the indentation curve defined as the dependence of the stamp immersion effort on the stamp immersion depth. The curves derived by computation are compared with the data of the indentation test conducted on the designed plant. It has been shown that the calculations according to the Hencky model and hypoelasticity relations provide a fair description of the curve to strains of around 30%. The suggested physical nonlinear model allows describing the continued strain growth to a sufficient degree of accuracy.

The influence of strain hardening on steel erosion wear studied by analytical modelling of impeller testing

ABSTRACTAbrasive wear of steel is a common problem in the mining industry. It was tackled by many researches using analytical and numerical approaches. Despite the advantages of the analytical models to provide quick and elegant solutions, they were derived using several assumptions limiting their applicability, such as e.g. rigid-plastic flow. However, the consideration of strain hardening during impact is crucial to depict a real behaviour of tool material, which changes its mechanical properties during collision. In this research, a new analytical model is invented describing the impact of steel plate with a solid rock, while the material of the steel plate is hardening during penetration and scratching. The model provides a frame of analytical equations based on the second Newton law and equations of motion in vertical and horizontal directions. The motion in vertical direction is considered as an indentation problem and the motion in the horizontal direction as a scratch problem. This model incorporates the most general Holloman representation of strain-hardening law to capture the relationship between microstructural characteristics of steels and wear resistance. It was shown that the indentation depth, the pile-up height, length of the scratch and erosion ratio directly dependant on the strain-hardening parameters in Holloman equation. The model predicts the maximum indentation depth and scratch length as a function of strength coefficient and strain hardening exponent, but also of impact angle, mass of the rock, position of impact at the surface of steel plate and coefficient of friction during metal movement. The model was tested qualitatively by comparison with impeller-tumbler experiments using different steel plates. The solution obtained from this model could be used for quick and easy evaluation of the steel for mining tools in an industrial environment.

Corrections to the stiffness relationship in 3-sided and conical indentation problems

One key relationship in the depth-sensing indentation technique is the proportionality between the contact stiffness and the contact size, as can be proved from the Sneddon's solution of axisymmetric frictionless contact. However, Sneddon's solution is only accurate when the indenter approaches a half-space (e.g., for conical indenter, the half-apex angle approaches 90º) and the interface is frictionless. As Hay et al. (J. Mater. Res., 1999) pointed out, sharp indenters lead to a radial inward displacement on the sample surface, thus leading to extra indentation force needed to push the surface back to conform with the conical indenter. In this paper, we argue that the physical origin arises from the incorrect use of reference and deformed coordinates in the boundary conditions that define Sneddon's problem. This yields two correction factors for both load and depth solutions, which are needed for sharp pyramidal indenters and frictional contact. Approximate solutions are derived which compare favorably well with the finite element simulations. We also find that the stiffness correction factor of three-sided indenter is about 11∼15% times higher than that of conical indenter, and this multiplicative factor is only a weak function of the indenter angle but does not depend on the friction coefficient and Poisson's ratio.

Finite strain analysis of size effects in wedge indentation into a Face-Centered Cubic (FCC) single crystal

The size effect in wedge indentation of an FCC single crystal is investigated. Conventional plasticity fails to describe the mechanical response of crystalline materials at the micron level due to the accumulation of Geometrically Necessary Dislocations (GNDs) which experiments have shown to introduce a size-dependent increase in the apparent yield stress and subsequent hardening. GND-densities scale with the gradient of plastic deformation and their effect on the mechanical response is here modelled by adopting a dissipative strain gradient single-crystal plasticity theory. Numerical solutions to the classical wedge indentation problem are obtained from a purpose-built Finite Element model accounting for finite strains. A special 2D plane strain set-up, with three effective in-plane slip systems, is adopted to comply with state-of-the-art experimental results. The indentation process is modelled for a nearly flat wedge as well as a wedge with an included angle of 90∘. The distribution of slip and the GND-densities are investigated and compared to conventional plasticity predictions.

An experimental study on the contact problem of spherical indentation on the anisotropic elastic half-space based on imaging

The contact problem of spherical indentation on the anisotropic elastic half-space is considered based on an experimental method in this work. The mechanical anisotropy is simulated by uniaxially stretching isotropic material made of silica gel. A special optical system is adopted to capture the contact images during indentation tests. The experimental results further prove the reasonability of the assumption that the contact area of spherical indentation on the anisotropic half-space is elliptical. And the results also show that the direction of strengthening by the uniaxial tension is consistent with the minor axis of the elliptical contact area. Such results validate the associated theory. Besides, the error analysis is made to examine the reliability of the fitting results, where the relative errors are less than 0.8% under the influence of given drawbacks. The indentation modulus can be determined through experimental data. Results show that the indentation modulus monotonically increases with the increase of the prestrain value. In the case of isotropy, the initial elastic modulus is determined under the assumption of incompressibility. The result by indentation test is in a good agreement with that by uniaxial tensile test.

Оценка механических характеристик кристаллов соляных пород на основе математической модели процесса наноиндентирования на зондовом силовом микроскопе Dimension Icon

Зондовый силовой микроскоп (ЗСМ) Dimension Icon применяется как для оценки рельефа поверхности исследуемого образца, так и для получения силового отклика образца при взаимодействии кантилевера (индентора специальной формы, расположенного на конце упругой консоли) с поверхностью образца. При этом, в отличие от других приборов, например NanoTest-600, где в результате индентирования исследователь получает величину твердости и эффективного модуля упругости с помощью программного обеспечения, «зашитого» в данный прибор, ЗСМ позволяет получить только кривую отклонения индентора в зависимости от перемещения основания кантилевера. Зная коэффициент жесткости на изгиб упругой консоли, исследователь может самостоятельно определить кривую «усилие-перемещение» индентора на этапе нагрузки и разгрузки. Далее возникает задача интерпретации полученной кривой: каким образом можно оценить механические характеристики материала на ее основе? Ответ на этот вопрос зависит, в частности, от характера механического поведения материала. Для диапазона глубин индентирования, превышающих головную сферическую часть зонда кантилевера, при предположении упруго-идеально-пластической модели материала рассмотрена двумерная осесимметричная задача индентирования образца на этапе нагрузки и разгрузки. Численное моделирование осуществлено в пакете ANSYS в рамках контактной задачи в предположении абсолютно жесткого наконечника кантилевера. Путем обработки результатов вычислительного эксперимента предложена методика оценки предела текучести и модуля упругости поверхностных слоев образца. На основе данных ранее проведенных экспериментов получены конкретные значения механических характеристик для кристаллов соляных пород.

Numerical Simulation of Unsteady Channel Flow with a Moving Indentation

The FSI problem - unsteady channel flow with a moving indentation problem, which represents flow features of oscillating stenosis of a blood vessel, is numerically simulated. The flow inside the channel with moving boundary results in transient and complex flow phenomena mainly due to the interaction between the moving boundary and the flowing fluid. In this paper, an accurate Harten Lax and van Leer with contact for artificial compressibility Riemann solver have been used for flow computation. The Riemann solver is modified to incorporate Arbitrarily Lagrangian-Eulerian (ALE) formulation in order to take care of mesh movement in the computation, where radial basis function is used for dynamically moving the mesh. Higher order accuracy over unstructured meshes is achieved using quadratic solution reconstruction based on solution dependent weighted least squares (SDWLS). The present numerical scheme is validated here and the numerical results are found to agree with experimental results reported in literature.

Indentation of a surface-stiffened elastic substrate

The paper deals with the problem of the indentation of a substrate in the form of an isotropic elastic halfspace that is reinforced at the surface by a bonded layer, which has flexibility characteristic of a Poisson-Kirchhoff-Germain thin plate. The presence of the reinforcing layer changes the characteristics of the indentation problem, in that the techniques that are applicable for the solution of the direct indentation of the halfspace cannot be applied to develop a load-displacement relationship for the indenter. The integro-differential equation governing the indentation of the surface reinforced halfspace is solved using a discretization approach and numerical results are presented to demonstrate the influence of the stiffness of the reinforcing layer on the Boussinesq indention problem.

Elastic response of coating materials in thermoelasticity: Indented by a hot punch

The indentation problem of coating materials in thermoelasticity is investigated. Based on the Papkovitch–Neuber potential functions, general displacement and stress solutions of the coating layer and the substrate body are derived in thermoelasticity. Fourier transform and Bessel function are used to obtain the explicit elastic response. The influences of the layer thickness on the elastic response are studied. Also, three heated indenters are employed to study the indentation behavior involving heat flow. Besides, the influences of the temperature on the elastic response are distinguished. The results involving heat flow show significant effects on the contact behaviors.

Computational modeling of the large deformation and flow of viscoelastic polymers

Deformation of soft polymeric materials often involves complex nonlinear or transient mechanical behaviors. This is due to the dynamic behaviors of polymer chains at the molecular level within the polymer network. In this paper, we present a computational formulation to describe the transient behavior (e.g., viscoelasticity) of soft polymer networks with dynamic bonds undergoing large to extreme deformation. This formulation is based on an Eulerian description of kinematics and a theoretical framework that directly connects the molecular-level kinetics of dynamic bonds to the macroscopic mechanical behavior of the material. An extended finite element method is used to discretize the field variables and the governing equations in an axisymmetric domain. In addition to validating the framework, this model is used to study how the chain dynamics affect the macroscopic response of material as they undergo a combination of flow and elasticity. The problems of cavitation rheology and polymer indentation under extreme deformation are investigated in this context.