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# Finite Elements Analysis and Materials Science

Very often, our goal as materials scientists is to develop materials with “better properties”. While trying to get to some property you might have to compromise others. For example, it is common knowledge that it is possible to increase the tensile strength in polymers by adding inorganic additives, such as silica or carbon black. However, while doing so, you also increase the elastic modulus of the material. So, how do you know that the material is strong enough and not too stiff for your part. Obviously, you can fabricate the detail and examine it. But, it is time and money consuming process. Another option is to estimate by using analytical models. It might work in simple or symmetrical geometries. But, it is impossible to use this approach for complicate case (and life is complicate). This is exactly where finite elements analysis gets in.

The finite element analysis is a numerical method that allows to compute an approximate solution for field problems (or boundary value problems) involving differential equations. In essence, there is a domain with well-defined boundaries over which some variables, known as field variables need to be determined. These field variables must satisfy the differential equations and boundary conditions of the domain. Most real world problems don't have an analytical solution; instead a numerical approximation is sought. The basic idea behind the finite element method is to divide the structure ((A)) into a large number of finite elements ((B)). The discretization process is termed "meshing". Each subdomain is called "element" and the connections points between them are called "nodes". The values of the field variables are calculated explicitly at the nodes of the element, while inside the element they are interpolated.

Although for the sake of discussion we used mechanical properties, finite elements analysis is a very useful technique for wide range of problems. For example, in SciMAP, we have experience solving problems for materials with linear mechanical behavior, hyperelastic problems, rubber failure analysis, light-matter interaction (plasmons) and charge carriers function distribution in colloidal nanoparticles.

(A) Sphere cutout

(B) Same sphere discretized into many triangular finite elements. Prepared with COMSOL ®