# research

## Micro cantilever pre-stress

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Dear all,

Im a PhD student in Cambridge Uni, UK working in the field of MEMS, and as part of my work, of late Ive been looking at deriving materials properties of MEMS thin film materials by means of resonant testing. The basic outline of the experiment is first creating free standing rectangular cantilevers of the material under test, evaporate with gold to increase reflectivity (when needed), then (under reasonable vacuum) applying a base excitation using a chirp signal into a piezo actuator and logging the cantilever tip response using a laser/photodetector setup. The frequency response is then calculated and the modal frequencies noted.

To determine the materials' properties, both an analytical model (with bending/torsion modes) and finite element model (using 2D mindlin) are created with similar geometry as the sample, and by minimising the squared relative error between the measured modes and those from the models, the value of Young's modulus (known density) and poissons ratio may be determined iteratively. These yield fairly consistent results.

To take the work further I now feel I should also include the effects of residual stress in the cantilevers. The method Ive been looking at is by using finite element (via COMSOL) - the beam geometry is created and loaded with the stress model ('surface stress' as a force tangential to the top boundary, and gradient stress as a 'tangential' force that varies from +F to -F from top to bottom boundary of the cantilever). The model is solved statically, and the deformed shape is then saved as the linearisation point for the next model, which then computes the eigenfrequencies. Btw I can only do this in 3D FE, which makes computation times quite long hence using iteration to quantify this stress highly unlikely.

In any case, is there an analytical model I can use to model the effect of this stress on multiple modes of a cantilever. Id like to verify whether the FE is giving me anything close to ball park numbers before I work out a means to compare them with experimental results. I was thinking of using the Rayleigh method by representing the effect of prestress as an additional term in the potential energy. The original mode shapes, with some modification will be used to evaluate the two energy integrals. The potential energy due to stress is worked out by measuring the static deflected shape using a zygo inteferometer - some rough model is used, with the beam curvature and peak deflection as input to work out the amount of this energy. Not having much experience in mechanics (i was an electronics undergrad!), Im not sure how good an estimate this would be, if at all its a useable or even possible one. Will the extra energy factor in to the torsion modes just the same?

## Some numerical mechanics software

Recently, during one of my net searches, I came across this page of RPI, where I learnt about a couple of numerical mechanics software which might be of interest to some of you.

FMDB:

As for the effort toward the scalable engineering simulations on distributed environements, we addressed this challenge by developing a distributed mesh data management infrastructure that satisfies the needs of distributed domain of applications.

## 2. Is a mesh required in meshfree methods?

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In meshfree (this is more in vogue than the term meshless) methods, two key steps need to be mentioned: (A) construction of the trial and test approximations; and (B) numerical evaluation of the weak form (Galerkin or Rayleigh-Ritz procedure) integrals, which lead to a linear system of equations (Kd = f). In meshfree Galerkin methods, the main departure from FEM is in (A): meshfree approximation schemes (linear combination of basis functions) are constructed independent of an underlying mesh (union of elements).

However, since a Galerkin method is typically used in solid mechanics applications, (B) arises and the weak form integrals need to be evaluated. Three main directions have been pursued to evaluate these integrals: