Mechanics of indentation for piezoelectric thin films on elastic substrate

Frictionless normal indentation problem of rigid flat-ended cylindrical, conical and spherical indenters on piezoelectric film, which is either in frictionless contact with or perfectly bonded to an elastic half-space (substrate), is investigated. Both conducting and insulating indenters are considered. With Hankel transform, the general solutions of the homogeneous governing equations for the piezoelectric layer and the elastic half-space are presented. Using the boundary conditions for a vertical point force or a point electric charge, and the boundary conditions on the film/substrate interface, the Green’s functions can be obtained by solving sets of simultaneous linear algebraic equations. The solution of the indentation problem is obtained by integrating these Green’s functions over the contact area with unknown surface tractions or electric charge distribution, which will be determined from the boundary conditions on the contact surface between the indenter and the film. The solution is expressed in terms of dual integral equations that are converted to a Fredholm integral equation of the second kind and solved numerically. Numerical examples are also presented. The comparison between two film/substrate bonding conditions is made. It shows that the indentation rigidity of the film/substrate system is lower when the film is in frictionless contact with the substrate. The effects of the Young’s modulus and Poisson’s ratio of the elastic substrate, indenter electrical condition and indenter prescribed electric potential on the indentation responses are presented.     

A numerical analysis of spherical indentation response of thin hard films on soft substrates

In this paper, finite element simulations of spherical indentation of a thin hard film deposited on a soft substrate are carried out. The primary objective of this work is to understand the operative mode of deformation of the film corresponding to various stages of indentation. The transition from contact dominant behaviour to that governed by flexure of the film on the plastically yielding substrate is investigated from analysis of the load versus displacement curve as well as the stress distribution in the film. It is found that onset of bending deformation in the film occurs when the contact radius is about 0.2–0.3 of the film thickness. Further, distinct membrane stresses arise in the film for indentation depth greater than half the film thickness. The implications of these results on indentation fracture of the film are briefly discussed. Finally, the effects of substrate yield strength and presence of residual stresses on the indentation response are examined.     

Nonlinear analysis of compressed elastic thin films on elastic substrates: From wrinkling to buckle-delamination

Nonlinear buckling of elastic thin films on compliant substrates is studied by modeling and simulations to reveal the roles of pre-strain, elastic modulus ratio, and interfacial properties in morphological transition from wrinkles to buckle-delamination blisters. The model integrates an interfacial cohesive zone model with the Föppl–von Kármán plate theory and Green function method within the general framework of energy minimization. A kinetics approach is developed for numerical simulations. Subject to a uniaxial pre-strain, the numerical simulations confirm the analytically predicted critical conditions for onset of wrinkling and wrinkle-induced delamination, with which a phase diagram is constructed. It is found that, with increasing pre-strain, the equilibrium configuration evolves from flat to wrinkles, to concomitant wrinkles and buckle-delamination, and to an array of parallel straight blisters. The height and width of the buckle-delamination blisters can be approximately described by a set of scaling laws with respect to the pre-strain and interfacial toughness. Subject to an equi-biaxial pre-strain, the critical conditions are determined numerically to construct a similar phase diagram for the buckling modes. Moreover, by varying the pre-strain, modulus ratio, and interfacial toughness, a rich variety of equilibrium configurations are simulated, including straight blisters, and network blisters with or without wrinkles. These results provide considerable insight into diverse surface patterns in layered material systems as a result of the mechanical interactions between the film and the substrate through their interface, which suggests potential control parameters for designing specific surface patterns.     

Use of thin polyparaxylene films to study the plastic deformation bismuth single crystals

The effect of thin polyparaxylene films on the mechanical twinning of bismuth single crystals with the (111) surface subjected to local deformation. It is found that the number of twins formed near the stress concentrator increases in the presence of the film. Possible mechanisms are proposed to explain an increase in the mobility of twin dislocations in a deformable crystal whose surface is coated with a polyparaxylene film. Spalling of bismuth is found in the regions deformed by the indenter. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 162–166, July–August, 2006.     

Deformability of thin metal films on elastomer substrates

Recent applications in flexible electronics require that thin metal films grown on elastomer substrates be deformable. However, how such laminates deform is poorly understood. While a freestanding metal film subject to tension will rupture at a small strain by undergoing a necking instability, we anticipate that a substrate will retard this instability to an extent that depends on the relative stiffness and thickness of the film and the substrate. Using a combination of a bifurcation analysis and finite element simulations, we identify three modes of tensile deformation. On a compliant elastomer, a metal film forms a neck and ruptures at a small strain close to that of a freestanding film. On a stiff elastomer, the metal film deforms uniformly to large strains. On an elastomer of intermediate compliance, the metal film forms multiple necks, deforms much beyond the initial bifurcation, and ruptures at a large strain. Our theoretical predictions call for new experiments.     

Buckling-induced dislocation emission in thin films on substrates

Atomistic simulations of the evolution of a strained thin film on a substrate has been reported and the formation of dislocations has been observed in the film/substrate interface after the film has buckled. In the framework of the linear elasticity theory, an analytical model has been developed to explain the buckle effect on the formation of the dislocations. A stability diagram with respect to the buckling and dislocation emission phenomena is finally presented for the film as a function of the uniaxial strain and the Burgers vector.     

Controlled wrinkling analysis of thin films on gradient substrates

The paper investigates continuously changing wrinkle patterns of thin films bonded to a gradient substrate. Three types of gradient substrates including exponential,power-law, and symmetry models are considered. The Galerkin method is used to discretize the governing equation of film bonded to gradient substrates. The wavelength and the normalized amplitude of the wrinkles for substrates of various material gradients are obtained. The numerical simulation based on the finite element method(FEM) is used to evolve the wrinkle patterns. The result agrees well with that of the analytical model.It is concluded that localization of wrinkle patterns strongly depends on the material gradient. The critical membrane force depends on both the minimum value of wrinkle stiffness and the gradient of wrinkle stiffness when the wrinkle stiffness is at its minimum.This work provides a better understanding for local wrinkle formation caused by gradient substrates.     

Wedge indentation of thin films modelled by strain gradient plasticity

A plane strain study of wedge indentation of a thin film on a substrate is performed. The film is modelled with the strain gradient plasticity theory by Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52, 1379–1406] and analysed using finite element simulations. Several trends that have been experimentally observed elsewhere are captured in the predictions of the mechanical behaviour of the thin film. Such trends include increased hardness at shallow depths due to gradient effects as well as increased hardness at larger depths due to the influence of the substrate. In between, a plateau is found which is observed to scale linearly with the material length scale parameter. It is shown that the degree of hardening of the material has a strong influence on the substrate effect, where a high hardening modulus gives a larger impact on this effect. Furthermore, pile-up deformation dominated by plasticity at small values of the internal length scale parameter is turned into sink-in deformation where plasticity is suppressed for larger values of the length scale parameter. Finally, it is demonstrated that the effect of substrate compliance has a significant effect on the hardness predictions if the effective stiffness of the substrate is of the same order as the stiffness of the film.     

Micromechanical deformation analysis of beta alloy titanium in elastic and elastic/plastic tension

An experimental analysis was conducted to document micromechanical behavior of a large grain beta alloy titanium in elastic and elastic/plastic tension. The deformation associated with individual grains was measured by microscopic moiré interferometry. After subtracting uniform strains, the multiplied fringe patterns revealed strong anomalies. The anomalous deformation was documented even at the early stage of loading. Slip along grain boundaries was not observed, but strong shear strains developed within the grains near grain boundaries. Paper was presented at the 1994 SEM spring conference on Experimental Mechanics held in Baltimore, MD, on June 6–8.     

Spherical indentation of freestanding circular thin films in the membrane regime

We present theoretical and experimental results to describe the mechanics of indentation of a clamped circular membrane with a frictionless spherical indenter. Analytical expressions and numerical simulations are presented for the relationships between contact radius, finite indentation strains (and stresses), pre-stretch, loads and deflection. These closed-form solutions are contrasted with point-load models that neglect the contact size (i.e. classical Schwerin-type solutions), and lead to important differences in the indentation strain and load-deflection response. The accuracy of these closed form expressions is illustrated by comparisons with detailed numerical results and experiments on thin elastomer films. We show that the closed-form solutions can be used to extract mechanical properties from indentation testing of freestanding films, with important implications for developing new tests on nanoscale films and/or compliant materials such as polymers and biological substances.     

The activation volume associated with the plastic deformation of ice

Stress relaxation experiments are carried out on single, bi, tri and quadri crystals of ice to obtain information on the rate controlling process. The results are analyzed with the rate theory developed previously. The study indicated that the energy barrier is asymmetrical and that the rate controlling process is associated with the Peierls barrier, with the dislocation intersection mechanism, or with the non-conservative motion of jogs. It seems unlikely that climb would control the dislocation motion in ice.Formerly Research Officer, Geotechnical Section, Division of Building Research, National Research Council of Canada, Ottawa.     

Linearized equations of nonlinear elastic deformation of thin plates

A linearized system of equations governing elastic deformation of a thin plate with arbitrary boundary conditions at its faces in an arbitrary curvilinear coordinate system is proposed. This system of equations is the first approximation of a oneparameter sequence of equations of twodimensional problems obtained from the initial threedimensional problem by approximating unknown functions by truncated series in Legendre polynomials. The stability problem of an infinite plate compressed uniaxially is solved. The results obtained are compared with the existing solutions.     

Spinodal surface instability of soft elastic thin films

When the thicknesses of thin films reduce to microns or even nanometers, surface energy and surface interaction often play a significant role in their deformation behavior and surface morphology. The spinodal surface instability induced by the van der Waals force in a soft elastic thin film perfectly bonded to a rigid substrate is investigated theoretically using the bifurcation theory of elastic structures. The analytical solution is derived for the critical condition of spinodal surface morphology instability by accounting for the competition of the van der Waals interaction energy, elastic strain energy and surface energy. Detailed examinations on the effect of surface energy, thickness and elastic properties of the film show that the characteristic wavelength of the deformation bifurcation mode depends on the film thickness via an exponential relation, with the power index in the range from 0.749 to 1.0. The theoretical solution has a good agreement with relevant experiment results.     

A simplified analysis on buckling of stressed and pressurized thin films on substrates

This paper is concerned with effect of mismatched pressure on buckling of stressed thin films on a semi-infinite rigid substrate. Analytical approximate solutions are established in explicit form by using total potential energy of the system and the Rayleigh–Ritz’s method, and their stabilities have also been determined. The dependences of the critical stress and film-center deflection on the mismatched pressure have been formulated. Results are compared with exact or numerical shooting solutions, and excellent agreements are observed for a large range of film-center deflection. These expressions are brief and can easily be used to derive the effects of various parameters on mechanical behavior of films.     

Multiple bifurcations in wrinkling analysis of thin films on compliant substrates

Wrinkling phenomena of stiff thin films on compliant substrates are investigated based on a non-linear finite element model. The resulting non-linear equations are then solved by the Asymptotic Numerical Method (ANM) that gives interactive access to semi-analytical equilibrium branches, which offers considerable advantage of reliability compared with classical iterative algorithms. Bifurcation points are detected through computing bifurcation indicators well adapted to the ANM. The effect of boundary conditions and material properties of the substrate on the bifurcation portrait is carefully studied. The evolution of wrinkling patterns and post-bifurcation modes including period-doubling has been observed beyond the onset of the primary sinusoidal wrinkling mode in the post-buckling range.     

On quasi-non-destructive strength and toughness testing of elastic–plastic materials

Non-destructive evaluation of mechanical material properties, like strength and fracture toughness, is impossible for principal reasons. However, there are possibilities of quasi-non-destructive estimation methods, which can be quite useful in practice. Instrumented indentation tests are often suitable to get information about the elastic–plastic behaviour, where the indentation depth is measured as a function of indentation force. By approximate analytical methods, key parameters like ultimate tensile strength, work-hardening exponent or even yield stress can be derived from these measurements. A mobile indenter is presented here and its use in ambulant testing is described. To obtain the uniaxial stress–strain curve more directly and more exactly, the same instrument can be used for a miniature compression test, where a small pin is machined out from the surface of the material. Furthermore, to get information about the toughness of materials, a carving instrument has been developed, which allows the energy required to introduce a defined furrow to be measured and correlated with toughness parameters.     

Effect of elastic modulus variation during plastic deformation on uniaxial and multiaxial ratchetting simulations

In order to identify different variables that affect ratchetting simulations, variation of elastic modulus during loading and unloading is considered and discussed based on the experimental observations which pointed out by Morestin and Boivin, 1996, Ishikawa, 1997, Cleveland and Ghosh, 2002, Zhou et al. (2005) and recently by Khan et al., 2009a, Khan et al., 2009b, Khan et al., 2009c. Then the effect of such variation on simulations is scrutinized from the theoretical point of view by considering simulations of ratchetting experiments conducted on stainless steel 304L by Hassan et al. (2008) using the well-known Armstrong–Frederick model. It is shown that, using two different values for the elastic modulus during loading and unloading could have a significant effect on simulations of uniaxial ratchetting. On the other hand, such significant effect hardly occurs in the case of simulations of biaxial ratchetting experiments under consideration. The importance of such findings is that the excessive ratchetting over-prediction resulting from any specific kinematic hardening rule is expected to decrease significantly by taking into consideration this effect. In this case, modeling of kinematic hardening rules could necessitate more attention and reconsideration.