Abstract
The current study covers thermal buckling analysis and stacking sequence
optimization of shape memory alloy hybrid composite (SMAHC) conical shells. To
formulate the theoretical model, the First Order Shear Deformation Theory (FSDT)
is adopted. Linear membrane pre-buckling analysis, followed by the adjacent
equilibrium criterion, is utilized to establish the stability equations at the onset of
buckling. The recovery stress of SMA fibers through the heating stage is computed
using the one-dimensional constitutive law of Brinson. The semi-analytical solution
to the problem is accomplished via employing Generalized Differential Quadrature
(GDQ) method in the longitudinal direction and exact exponential functions in the
circumferential direction. The effects of different edge supports, semi-vertex angles,
length to thickness ratios, small radius to thickness ratios, angle-ply lamination
sequences, and SMA volume fractions are investigated as comprehensive parametric
studies. Finally, stacking sequence optimization is performed via implementing the
Genetic Algorithm (GA) to maximize the buckling temperature of SMAHC conical
shells. The results demonstrate that embedding SMA fibers in the longitudinal
direction is most effective on postponing the thermal buckling phenomenon,
whereas embedding SMA fibers in the circumferential direction, escalates the axial
pre-buckling thermal force and subsequently, the critical buckling temperature
decreases. The optimization results indicate that even considerable variations in the
geometrical parameters of SMAHC conical shells, such as semi-vertex angle and
length, have insignificant effects on the optimal stacking sequence with respect to
thermal buckling.
Key Words:
Thermal buckling, composite conical shells, shape memory alloys, Brinson model,
GDQ method, optimization
1. Introduction
Numerous engineering and bio-engineering applications of smart SMAHC have
made this field an extremely intriguing branch of study in the last decades. Due to
the unique characteristics of composite materials such as great stiffness to weight
ratio, low thermal coefficient and high usability in different industries, great
consideration have been devoted to this subject [1]. Additionally, augmenting
composite structures with SMA fibers, which represent exceptional behaviors under
thermal load, has been of great use in controlling and enhancing the behavior of
various engineering structures.
Constraining the edges of a structure subjected to thermal load could result in
developing the thermal forces and therefore, the possibility of the buckling
phenomenon at a particular temperature is increased. The primary objectives of the
current study are to determine such critical temperatures, study the impact of
embedding SMA fibers on the thermal behavior of SMAHC conical shells, and
furthermore, optimize the stacking sequence in order to maximize the buckling
temperature.
Based on a survey of the relevant literature, first studies on buckling analysis of
SMA hybrid structures were conducted by Birman [2] in 1997 by analyzing the
instability of rectangular plates with embedded SMA fibers. In this study, only
simply-supported plates under uniaxial loads are considered. The results show that
non-uniform distribution of SMA fibers in the transverse direction of plates has
higher impact on the plate stability than uniform distribution. Birman [3] studied the
effect of the composite reinforcements and SMA fibers on the stability of cylindrical
shells and rectangular plates under compressive loads. In this research, the governing
equations of cylindrical shells are established based on the Love’s firstapproximation theory. Ostachowicz et al. [4] developed a finite element formulation
to investigate the effect of embedding SMA fibers on the free vibrational and thermal
buckling behavior of SMAHC plates.
Thompson and Loughlan [5] and Loughlan et al. [6] performed several experimental
studies regarding the influence of SMA fibers on the stability characteristics of
SMAHC plates. Results of their studies corroborate the importance of SMA fibers
in postponing buckling and reducing the post-buckling deformations of SMAHC
plates. Tawfik et al. [7] performed stability analysis of panels with embedded SMA
fibers. They developed the finite element formulation considering the non-linear
strains of Von-Karman type. In their study, the unique capability of SMA fibers in
recovering large strains at temperatures higher than austenite finish temperature is
shown as an effective parameter on the stability behavior of SMA hybrid panels.
Roh et al. [8] employed the finite element method to study the thermal snapping
phenomenon in cylindrical panels reinforced with SMA fibers. Their results show
that embedding SMA fibers in panels could significantly postpone this phenomenon.
Park et al. [9] performed a comprehensive study regarding the buckling, postbuckling and low-amplitude vibration analysis of post-buckled SMAHC plates
based on the finite element formulation. Kuo et al. [10] implemented the finite
element method to examine the buckling of SMAHC plates. Their results
demonstrate that embedding SMA fibers in the middle layers of a laminate could
lead to higher buckling stability.
Thermal post-buckling analysis of circular plates with embedded SMA fibers was
performed by Li et al. [11] via the shooting method. Ibrahim et al. [12] utilized the
finite element method to investigate the thermal buckling and post-buckling of
initially imperfect SMAHC plates. Thermal buckling and post-buckling of SMAHC
plates was conducted by Kumar and Singh [13] via adopting the layer-wise theory.
Panda and Singh [14] conducted a research on the thermal post-buckling behavior
of SMAHC panels under uniform temperature distribution. The governing
equations, incorporating the geometrical nonlinearities of Green-Lagrange type, are
derived based on the FSDT and solved using the finite element method.
Asadi et al. [15-17] performed several studies on evaluating the efficiency of SMA
fibers in enhancing forced and free vibrational behavior of SMAHC beams. Asadi
et al. [18, 19] presented an analytical closed-form solution for the thermal postbuckling behavior of geometrically imperfect SMAHC beams. Buckling and postbuckling deformation analysis of SMAHC plates with initial imperfections was
conducted by Asadi et al. [20]. Abdollahi et al. [21] utilized GDQ method to analyze
the non-linear thermal stability of SMAHC Timoshenko beams resting on a nonlinear hardening elastic foundation. Asadi et al. [1] studied the thermal buckling of
bifurcation type in SMAHC cylindrical shells. In this research, the governing
equations are derived based on the FSDT and a semi-analytic solution including the
harmonic differential quadrature method is employed to solve the governing
equations. Furthermore, the one-dimensional constitutive law of Brinson is
employed to predict the SMA behavior in the heating stage and linear-membrane
analysis is implemented to obtain the pre-buckling forces.
Hasanali and Samali [22] studied the buckling of curved SMA hybrid panels under
axial, lateral and combined loads. They investigated the effects of SMA fibers
distribution on the critical buckling load and revealed that buckling load is
immensely dependent on the non-uniform distribution coefficient of SMA fibers.
Shao et al. [23] analyzed the flutter and thermal buckling behavior of SMAHC
panels based on the classical theory. Their study shows that embedding SMA fibers
in the outer layers has more significant effect on flutter characteristics. Kamarian
and Shakeri [24] investigated the thermal buckling behavior of SMAHC skewed
plates and implemented the Firefly Algorithm to optimize the stacking sequence
with respect to the buckling temperature. Bayat and EkhteraeiToussi [25] presented
a closed-form solution to the non-linear thermal buckling and post-buckling
behavior of SMAHC beams based on the layer-wise theory.
A thorough search of the relevant literature revealed that, to the best of authors’
knowledge, there is no published article on thermal buckling analysis and stacking
sequence optimization of SMAHC conical shells. Accordingly, the present study can
be considered as a novel research. This paper aims to provide a semi-analytical
solution to the linear thermal buckling of SMAHC conical shells under uniform
thermal load, employing 1-D GDQ method in the longitudinal direction and
exponential displacement functions in the circumferential direction. The first aim of
this paper embraces the study of the thermal buckling behavior of SMAHC conical
shells with various geometrical parameters, angle-ply lamination sequences, and
SMA volume fractions. To formulate the problem, the displacement field is
supposed to be based on the FSDT and linear pre-buckling analysis, accompanied
by the adjacent equilibrium criterion, is utilized to establish the stability equations
at the onset of buckling. To calculate the recovery stress of SMA fibers during
heating stage, the one-dimensional constitutive law of Brinson is employed. The
second aim involves the stacking sequence optimization of SMAHC conical shells
in order to obtain the maximum buckling temperature. To this end, the well-known
GA is implemented to attain the best solutions.
2. SMA Constitutive Equation
SMAs, after being deformed, have the ability to return to their pre-deformed size
and shape upon being subjected to a thermal load. In other words, the material
“remembers” its previous size and shape [26]. The exceptional behavior of SMA is
due to the phase transformation between two crystallographic structures named
Austenite (A) and Martensite (M). Four characteristic temperatures associated with
the phase transformation can be defined as austenite start, austenite finish, martensite
start and martensite finish, respectively indicated by 𝐴𝑠 , 𝐴𝑓 , 𝑀𝑠 , and 𝑀𝑓 .
Consider SMA at first in austenite phase. Starting from the parent phase shown by
point A (Fig. 1) upon cooling without applied mechanical load, austenite phase
transforms to heavily twinned martensite. Subjecting twinned martensite to a
sufficient stress level denoted by 𝜎𝑠 initiates the reorientation process and
subsequently, twinned martensite transforms to detwinned martensite.
Transformation to detwinned martensite completes at the stress level 𝜎𝑓 . The
necessary stress level for reorientation is considerably lower than the permanent
plastic yield stress of martensite [27]. Detwinned martensite phase is stable at low
temperatures. For this reason, upon unloading, SMA remains at detwinned
martensite and a significant residual strain is left in SMA. Upon heating in the
absence of stress, transformation to austenite is initiated as the temperature
reaches 𝐴𝑠 and is completed at 𝐴𝑓 . At this temperature, the original shape of SMA
is regained.