# Mathematical and numerical modelling of thin nonlinear curvilinear interphases in composite materials

• Daria Andreeva

Traethawd ymchwil myfyriwr: Traethawd Ymchwil DoethurolDoethur mewn Athroniaeth

### Crynodeb

My research has been predominantly dedicated to obtaining, by means of asymptotic analysis, transmission conditions that would accurately represent thin curvilinear interphases in composite materials. The precision of the models is particularly crucial now that such materials are extensively used in safety-conscious areas of industry and construction. Since the classical approach (finite elements method) often brings about inaccuracy and instability of the solution, there is a necessity of a more reliable and realistic way to model a composite with fine structural elements (thin inclusions). Replacing the original layer of small yet non-zero thickness with an imperfect interface of zero thickness with the carefully derived transmission conditions that incorporate the physical behaviour of the interphase provides an opportunity to reach the desired accuracy. Such transmission conditions were first of all obtained in the context of heat transfer for a low conductive interphase, and verified for cases of different shapes of the inclusion, with a particular focus on the impact of geometrical properties (curvature of the boundaries) on the accuracy of the conditions. Next, transmission conditions for a highly conductive interphase were evaluated and verified for interphases of varying shapes, from circular to close to circular but with greater curvature of the boundaries. This way, the effect of the geometry on the precision of the model was again analysed. For all the described numerical examples, a benchmark case of a perfectly circular geometry was considered in the first place. This allowed to reduce the problem to a one-dimensional one, solve it analytically and use this exact solution to estimate the accuracy of both the analytical solution to the problem with the transmission conditions and of the numerical solution obtained via finite element method. Seeing the satisfactory accuracy of the latter in the circular case, the numerical solution was then used to test the accuracy of the transmission conditions in the more complicated cases, for which the exact solutions could not be evaluated. In addition, two problems related to the two main cases were considered. In connection with the low conductivity, an example of a thin circular interphase with the conductivity proportional to those of the external layers was considered, and it was shown that the transmission conditions derived for the highly resistant layer are valid also in this case. On the other hand, the case of highly transmissive layer has been continued in the context of elastic deformation and displacement with similar problems that can be adapted to,
again, use the derived transmission conditions. However, a profound and detailed analysis of this latter example is yet to be conducted, as is highlighted in the outline of future work.