Thin-Film Mechanics

Introduction N/MEMS devices involve thin films, with thicknesses from submicrons to few microns, which are expected to carry loads in many cases. Quantitative analysis of stress in thin films is not available: In thin films atomic bonding forces become important in determining film strength. All film deposition methods result in residual and intrinsic stresses in the film. These are due to thermal expansion mis-match or are due to the reaggregation of crystalline grains of thin-film materials during the deposition process. Distribution of material in thin films is not likely to be uniform. The use of continuum mechanics theory using average material behavior in assessing the stresses in the film. Total stress in a thin film, sT, is sT = sth(thermal stress)+sm(stress from applied mechanical load)+sint(intrinsic stress)

Overview of Finite Element (FE) Stress Analysis MEMS have complex 3-dimensional geometries. They are fabricated under severe thermal and mechanical conditions and subjected to harsh loading during operation. They also involve different films of dissimilar materials bonded together. Reliable stress analysis of MEMS can not be achieved using conventional methods. FE method (FEM) is the only viable way. FE Principle : Discretization: divide the whole solid structure into a finite number of of subdivisions of special shapes (elements) interconnected at the corners or specific points on the edges of the element (called the nodes). Engineering analysis is performed in these elements instead of the whole structure. Solutions obtained at the element level are assembled to get the corresponding solutions for the whole structure.

Credible results from FE analysis (FEA) are attainable with intelligent discretization. Two rules need to be followed : 1- Place denser and smaller elements in the parts where there are abrupt changes in geometry where high stress or strain concentrations are expected. 2- Avoid using elements with high aspect ratio. Keep the aspect ratio less than 10.

Input Information to FEA General information:(a) structure geometry, and (b) establish coordinates. Develop FE mesh : desired element and node densities in specific regions, element and nodal coordinates, nodal conditions (constraints, applied forces), element description. Material properties : E, n, G, a etc. Boundary and loading conditions : nodes with constrained displacements, concentrated forces at specific nodes etc.

Output from FEA Nodal and element information. Displacements at nodes. Stresses and strains in each element. Normal stress and strain components : e. g., sxx, exx. Shear stress and strain components : e. g., sxy, exy. Maximum and minimum principal stresses. The von Mises stress defined as and used as the representative stress in a multi-axial stress situation : it is compared with the yield strength, sY, for plastic yielding, and with ultimate strength, su, for the prediction of the rupture of the structure.

Graphical Output of FEA Popular commercial FEA codes, such as the ANSYS code, offer the following graphical output: A solid model of the structure. User input discretized solid structure with FE mesh. Deformed or undeformed model of the structure. Color-coded zones to indicate distribution of stresses, strains, and displacements over the selected plane of the structure. Color-coded zones to indicate other required output parameters such as temperature or pressure in the structure. Animated movements or deformation of the structure under specified operating conditions.

Fracture Mechanics

Introduction Many MEMS devices involve the binding of thin films of distinct materials. These materials are bonded using different techniques, such as chemical and physical deposition, thermal diffusion, soldering, adhesion, ionic bonding etc. Interfaces of bonded structures are likely places for structural failure : fracture mechanics is a way to assess the structural integrity of these interfaces. Fracture of bonded interfaces can occur due to excessive forces normal to the interface and/or shear forces acting on the plane of the interfaces. Linear elastic fracture mechanics (LEFM) principles can be applied to the design analysis in order to mitigate possible delamination of interfaces in MEMS. LEFM is an engineering discipline on its own. It is not possible to cover it here. However, we will try to learn the basic concepts and the formulation of LEFM that may be used in MEMS design.

A crack in a solid subjected to a stress field

Three Modes of Fracture Mode I : opening mode Is associated with local displacements in which the crack surfaces tend to move apart in a direction perpendicular to these surfaces. Mode II : edge-sliding mode Displacements in which the crack surfaces slide over one another and remain perpendicular to the leading edge of the crack. Mode III : tearing mode is defined by the crack surfaces sliding with respect to one another parallel to the leading edge of the crack.

Stress Intensity Factor, K KI, KII, and KIII are the stress intensity factors for modes I, II, and III. The functions fij and gi depend on the crack’s length and solid’s geometry : they are available in handbooks of fracture mechanics. Note that In reality this does not happen, however, sij is very large at r = 0 : this is called stress singularity near the crack tip.

Fracture Toughness, KC KC is used as a criterion for assessing the stability of the crack in a structure. KIC, KIIC, and KIIIC are for the modes I, II, and III, respectively. The measured critical load, Pcr, is that which causes a structure with a crack to fail. Common geometries of specimens for KIC include the compact tension (CT) specimens and three-point bending beam specimens.

Compact Tension The specimen is subjected to tension to open the crack. A notch is maintained to facilitate fracture. The CT specimen needs to satisfy The fracture toughness is c is the crack length and Pcr is the applied load necessary to cause fracture. F(c/b) is found in handbook, and as an approximation F=1 for a uniform tensile load.

Three-Point Bending The expression for the CT specimen for KIC is again used here. Except F(c/b) takes the form

Interfacial Fracture Mechanics Many MEMS are made of bonded dissimilar materials. A major complication in fracture analysis is that these interfaces are subjected to simultaneous mode I and mode II (and mode III in some cases) loading. The two bonded materials have distinct strength properties (E and n), and the interface is subjected to loading in both the normal and lateral shearing directions, with respective stress fields of syy and sxy.

ro is a short range from the crack tip, in which the stresses vary linearly, whereas re is the closest distance from the crack tip with which stress values are available from numerical analysis.

Fracture Criterion for Interfaces