A theoretical and numerical study of a thin clamped circular film under an external load in the presence of a tensile residual stress

Abstract

Tensile residual stress in a plate or membrane clamped at the perimeter can be measured by either applying a uniform hydrostatic pressure or a central load via a cylindrical punch (with several different loading configurations). Analytical constitutive relations are derived here based on an average membrane stress approximation and are compared to finite element analysis results. The thickness and flexural rigidity of the film are not confined to a small range but will span a wide spectrum. The elastic responses of the blistering films are shown to be linear when the film is thick, relatively rigid, or subjected to a large residual stress, and cubic when the film is thin, flexible, or under a small residual stress. The linear-to-cubic transition is formulated.

Introduction

Thin film technology is indispensable in the microelectronics industry. Mechanical integrity (e.g. thin film adhesion) of coated parts has a significant impact on reliability, lifespan, and structural stability. Residual stresses can be introduced by volume changes associated with crosslinking or crystallization, by temperature changes, and by variations in moisture content or other diluents that may occur during processing, curing, or service life. Residual stresses within a film can lead to (i) delamination and premature failures due to the significant interfacial stresses that can occur near free edges and other flaws, (ii) damage such as environmental stress cracking in service conditions, and (iii) dimensional instability. Mechanical characterization of residual stress is essential for understanding and predicting the performance of thin film coated devices.

Many techniques have been developed to measure tensile residual stress such as lattice strain measurement using X-ray and neutron diffraction methods [1], [2], surface curvature measurement [3], [4], [5] and vibrational holographic methods [6], [7]. One classical way is to measure the elastic response of a suspended, circular film. A hole is drilled or chemically etched in the substrate until it reaches the film–substrate interface. Upon application of an external pressure or load, the drumhead will deform to form a blister with the elastic deflection depending on the magnitude of the tensile residual stress. In a residual stress-free film, the membrane stress increases from zero to a value determined by the external load. In a pre-stressed film, the total membrane stress is a superimposition of the residual stress and a concomitant stress due to change of blister profile. A diminished deflection is expected in the pre-stressed film compared to a stress-free film.

Application of a hydrostatic pressure to load a thin blistering film is well documented. The blister height is measured and related to the residual stress based on linear elasticity. Bennett [8] studied the elastic response of a thin stress-free plate and formulated analytical elastic solution as well as finite element analysis (FEA); Cotterell [9] studied blister delamination of thin films without residual stress; Williams [10] considered blistering film with a tensile residual stress but only for a thin flexible membrane under pure stretching; Jensen [11] investigated the presence of a compressive residual stress and its effects on thin film delamination; Allen [12], Lin [13] and Sizemore [14] verified experimentally their elastic model for thin flexible film and considered a perturbation of bending moment for thick films; Voorthuyzen [15] solved numerically the von Karman equation for a pressurized blistering film with residual stress; Sheplak [16] considered the transition from a rigid plate to a flexible membrane without residual stress; and Wan [17] presented an analytical constitutive equation for films spanning the entire range of film thickness and flexural rigidity but with zero residual stress. An alternative way of loading a blistering film is by a mechanical force applied via a rigid shaft. Williams [18], Malyshev [19] and Wan [20] studied the elastic response of a thin plate without residual stress; Wan investigated thin flexible stress-free films [21], [22] and films of intermediate thickness and flexural rigidity [23]. Jennings et al. [24] derived the most comprehensive elastic solution for thin films in the presence of large residual stress and introduced perturbation terms to account for bending moment in thick and rigid films. A detailed comparison between Jennings's solution and the one proposed herein will be discussed in Section 3.

In this paper, we will reconstruct the elastic models used in the pressurized blister test and three different shaft-loaded blister geometries based on an average equi-biaxial stress approximation, i.e. radial and tangential membrane stresses are identical and are independent of radial distance from the film center. This assumption allows us to derive an analytical solution for the elastic response rather than a numerical solution from the exact von Karman equations. The validity of the assumption will be discussed in Section 3. We will extend the previous work to include (i) films spanning the entire range of flexural rigidity and thickness, (ii) film deformed under a mixed mode of plate bending and membrane stretching and (iii) the full range of tensile residual stresses coupled with the concomitant membrane stress. Nonlinear FEA will be carried out to compare with and verify the closed-form solutions. Simple equations will be derived for gauging residual stress experimentally.