Chapter 7
Mathematical Models of Functionally
Graded Beams in Temperature Field
7.1 Introduction
In this chapter, the analysis of nonlinear FGM (functionally graded material) straight
and curved beams behaviour based on the modified couple stress theory is carried
out. Defining the deflection curve in order to simplify the governing equations, we
investigate the influence of scale length parameter and the non-homogeneity coefficient on the dynamic characteristics and the scenario of transition from periodic to
chaotic beam vibrations. First, the properties of various FGM materials are revisited
(Sect. 7.2) with an emphasis on the dependence of material properties on temperature. Then, regular and chaotic vibrations of size-dependent Timoshenko beams
with functionally graded properties along their thickness are illustrated and analysed (Sect. 7.3). The following new properties regarding the research topic under
consideration can be derived.
(i) We have considered the dynamics of nonlinear FG Timoshenko beams on the
basis of the modified couple stress theory, using a novel concept of the bending
line.
(ii) The influence of the size-dependent coefficient and the grading parameter on the
load-deflection dependence is investigated for the static problem. In order to get
solutions for nonlinear static problem, the relaxation method was employed. It
has been shown that the minimum deflection is achieved by the beam when the
functional grading process is taken into account and the stiffer layer is located
on the upper side in both cases, i.e. with/without the size-dependent behaviour.
(iii) The functionally graded beam with the stiffer layer on the upper side is suitable
for application to carry dynamic loads for a given frequency and amplitude
of the harmonic excitation. This conclusion coincides with that formulated for
static problems. In the case of the homogeneous beam and the beam with the
stiffer layer located on the bottom side, the essential dependence of the obtained
results on the size-dependent coefficient is observed.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
J. Awrejcewicz et al., Mathematical Modelling and Numerical Analysis of Size-Dependent
Structural Members in Temperature Fields, Advanced Structured Materials 142,
https://doi.org/10.1007/978-3-030-55993-9_7
197
Mathematical Models of Functionally
Graded Beams in Temperature Field
7.1 Introduction
In this chapter, the analysis of nonlinear FGM (functionally graded material) straight
and curved beams behaviour based on the modified couple stress theory is carried
out. Defining the deflection curve in order to simplify the governing equations, we
investigate the influence of scale length parameter and the non-homogeneity coefficient on the dynamic characteristics and the scenario of transition from periodic to
chaotic beam vibrations. First, the properties of various FGM materials are revisited
(Sect. 7.2) with an emphasis on the dependence of material properties on temperature. Then, regular and chaotic vibrations of size-dependent Timoshenko beams
with functionally graded properties along their thickness are illustrated and analysed (Sect. 7.3). The following new properties regarding the research topic under
consideration can be derived.
(i) We have considered the dynamics of nonlinear FG Timoshenko beams on the
basis of the modified couple stress theory, using a novel concept of the bending
line.
(ii) The influence of the size-dependent coefficient and the grading parameter on the
load-deflection dependence is investigated for the static problem. In order to get
solutions for nonlinear static problem, the relaxation method was employed. It
has been shown that the minimum deflection is achieved by the beam when the
functional grading process is taken into account and the stiffer layer is located
on the upper side in both cases, i.e. with/without the size-dependent behaviour.
(iii) The functionally graded beam with the stiffer layer on the upper side is suitable
for application to carry dynamic loads for a given frequency and amplitude
of the harmonic excitation. This conclusion coincides with that formulated for
static problems. In the case of the homogeneous beam and the beam with the
stiffer layer located on the bottom side, the essential dependence of the obtained
results on the size-dependent coefficient is observed.
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer
Nature Switzerland AG 2021
J. Awrejcewicz et al., Mathematical Modelling and Numerical Analysis of Size-Dependent
Structural Members in Temperature Fields, Advanced Structured Materials 142,
https://doi.org/10.1007/978-3-030-55993-9_7
197