Doctoral Thesis

Constitutive Modelling of Composites with Elastomer Matrix and Fibres with Significant Bending Stiffness

Final Thesis 3.44 MB Summary of Thesis 1.45 MB

Author of thesis: Svitlana Fedorova, Ph.D.

Acad. year: 2018/2019

Supervisor: prof. Ing. Jiří Burša, Ph.D.

Reviewer's: prof. RNDr. Michal Kotoul, DrSc., prof. Andreas Menzel


Constitutive modelling of fibre reinforced solids is the focus of this work. To account for the resulting anisotropy of material, the corresponding strain energy function contains additional terms. Thus, tensile stiffness in the fibre direction is characterised by additional strain invariant and respective material constant. In this way deformation in the fibre direction is penalised.
Following this logic, the model investigated in this work includes the term that penalises change in curvature in the fibre direction. The model is based on the large strain anisotropic formulation involving couple stresses, also referred to as “polar elasticity for fibre reinforced solids”. The need of such formulation arises when the size effect becomes significant.
         Mechanical tests are carried out to confirm the limits of applicability of the classical elasticity for constitutive description of composites with thick fibres. Classical unimaterial models fail to take into account the size affect of fibres and their bending stiffness contribution.
The specific simplified model is chosen, which involves new kinematic quantities related to fibre curvature and the corresponding material stiffness parameters. In particular, additional constant k3 (associated with the fibre bending stiffness) is considered. Within the small strains framework, k3 is analytically linked to the geometric and material properties of the composite and can serve as a parameter augmenting the integral stiffness of the whole plate. The numerical tests using the updated finite element code for couple stress theory confirm the relevance of this approach.         
An analytical study is also carried out, extending the existing solution by Farhat and Soldatos for the fibre-reinforced plate, by including additional extra moduli into constitutive description.
Solution for a pure bending problem is extended analytically for couple stress theory. Size effect of fibres is observed analytically.
Verification of the new constitutive model and the updated code is carried out using new exact solution for the anisotropic couple stress continuum with the incompressibility constraint. Perfect agreement is achieved for small strain case. Large strain problem is considered by finite element method only qualitatively.
Three cases of kinematic constraints on transversely isotropic material are considered in the last section: incompressibility, inextensibility and the double constraint case. They are compared with a general material formulation in which the independent elastic constants are manipulated in order to converge the solution to the “constraint” formulation solution. The problem of a thick plate under sinusoidal load is used as a test problem. The inclusion of couple stresses and additional bending stiffness constant is considered as well. The scheme of determination of the additional constant d31 is suggested by using mechanical tests combined with the analytical procedure.


Fibre-reinforced materials, fibre bending stiffness,  hyperelasticity,  constitutive modelling,  polar elasticity

Date of defence


Result of the defence

Defended (thesis was successfully defended)


Process of defence

Práce přinesla podstatné rozšíření poznatků v oboru v oblasti modelování a simulací. Konkrétně modifikovala předchozí model tak, aby jeho konstanty měly jasnější fyzikální interpretaci a mohly být určeny experimentálně. Také realizovala řadu verifikačních výpočtů, porovnávajících výsledky MKP s analytickým řešením. Nebyl plně využit publikační potenciál práce.

Language of thesis




Study programme

Applied Sciences in Engineering (D4A-K)

Study branch

Applied Mechanics

Composition of Committee

prof. RNDr. Petr Dub, CSc. (předseda)
prof. RNDr. Michal Kotoul, DrSc. (člen)
prof. Dr. Ing. Eduard Rohan, DSc. (člen)
prof. Ing. Jindřich Petruška, CSc. (člen)
doc. Ing. Lukáš Horný, Ph.D. (člen)

Supervisor’s report
prof. Ing. Jiří Burša, Ph.D.

viz. posudek v pdf.
File inserted by supervisor Size
stanovisko školitele.pdf 39.63 kB

viz. posudek v pdf.
File inserted by the reviewer Size
prof. Kotoul.pdf 136.01 kB

Reviewer’s report
prof. Andreas Menzel

viz. posudek v pdf.
File inserted by the reviewer Size
review-Fedorova_Menzel.pdf 120.72 kB