Fully analytic approach to study bifurcation phenomenon for a shape memory alloy beam under temperature variation and lateral white-noise excitation

Abstract

The main aim of this article is to obtain the probability density function of the response of an SMA beam under lateral white-noise excitation and study the bifurcation phenomenon through a fully analytic approach. Firstly, an efficient constitutive stress–strain relation for a shape memory alloy material supporting both martensite and austenite phases is considered. Secondly, the governing equation of motion for a typical SMA simply supported beam is derived using some dimensionless parameters. Thirdly, the corresponding probability density function of the response is computed analytically using the powerful Fokker–Planck–Kolmogorov equation, and finally, the bifurcation phenomenon for parameter variations of the beam is investigated. The results show how geometric (beam aspect ratio), force (the mean value of the white-noise excitation), and environmental (working temperature) factors can affect the nonlinear behavior and response bifurcation of the beam. A numerical validation is also done that guarantees the correctness of the method and results.

Keywords

shape memory alloys white noise probability density function Fokker–Planck–Kolmogorov method bifurcation phenomenon structural stability

1. Introduction

One of the main challenges of designing engineering structures is to have robust dynamic stability under different working conditions. In the last two decades, with the development of the use of smart materials, significant progress has been made in optimizing the behavior of structures using these materials. Advanced or smart materials can change their mechanical and physical properties under environmental and loading variations to reach a desired response. This capability makes the design of structures more adaptive, providing desired behaviors under different environmental and loading conditions. This feature leads to an efficient and robust passive control algorithm to overcome dynamic instabilities in various conditions for high-tech structures. The range of these variations may be wide and different such as applied stress, electric current, light, temperature, humidity, and electromagnetic field (Maheswari and Anusuri Lavanya, 2022). The most well-known smart materials and advanced structures are shape memory alloys (SMAs), electro/magneto-strictive materials, piezoelectrics, electro/magneto-rheological fluids, optical fibers, etc. (Maheswari and Anusuri Lavanya, 2022; Tzou et al., 2004). There are several and varied applications for these types of materials in general engineering, mechatronics, robotics, medical and biomedical applications, aeronautical structures, and structronic systems (Mrinalini and Prasanthkumar, 2019; QADER et al., 2019).

Among the above intelligent materials, SMAs have received a lot of attention recently due to their unique properties such as high strength/weight ratio, large recovery of strain, super elasticity, and fatigue resistance (Chang and Araki, 2016; Johnson et al., 2001). The main feature of these materials is the ability to return to the original shape after the process of mechanical deformations. They can resist stress fields or temperature variations and recover thermomechanical deformations. For example, an SMA crystalline structure reveals the martensite phase that can be transformed into the austenite phase when heated. In the martensitic phase, more compliant (lower elastic modulus) behavior is seen; while in the austenitic phase, more rigidity is observed. This temperature variation may vary from −200°C to +200°C depending on the alloys (QADER et al., 2019; Tzou et al., 2004). So, SMAs present different thermomechanical behaviors. The main phenomena associated with these materials due to temperature variation are the shape memory effect (one-way (SME) or two-way (TWSME)), phase transformation, and pseudoelasticity (Paiva and Savi, 2006).

Among many applications of SMA materials, the control of vibrations and unstable behaviors is of particular importance. In fact, due to the presence of a hysteresis loop in the constitutive stress–strain relation of these materials, on one hand, the damping capacity increases, and on the other hand, with the increase of the vibration amplitude, the amount of damping capacity also increases. Therefore, in the presence of dynamic and vibratory forces, especially if there are large temperature changes (like many aerospace applications), the use of SMA materials is recommended (Costanza and Tata, 2020; Paiva and Savi, 2006; Sohn et al., 2023; Tabrizikahou et al., 2022; Wang et al., 2023; Wanhill and Ashok, 2017).

Since mechanical, civil, and aerospace structures are composed of fundamental elements such as plates, panels, and beams, investigating the vibration behavior of these components is of particular importance. There are many articles in the literature about the numerical investigation of the behavior and nonlinear response of beams and other elementary structures; nevertheless, few of them have used fully analytical methods (see, e.g., Asnafi, 2020; Asnafi, 2022b; Hasan and Jabbar, 2019). The number of these studies decreases when SMA materials are used in the main elements or layers of the structure. Zheng et al. studied the time and frequency response of nonlinear vibration suppression of a laminated beam reinforced by SMA wire ropes (Zheng et al., 2021). Hao et al. studied the nonlinear vibration of axially moving laminated SMA beams with internal resonance (Hao et al., 2021). Subharmonic and superharmonic resonance and bifurcation phenomenon of a laminated SMA beam was studied by Zhang et al. (Zhang et al., 2021). Rezaie et al. considered nonlinear free vibration of a fixed–fixed composite beam containing SMA wires (Rezaei Da et al., 2012). Razavillar et al. used a semi-analytic approach to analyze the free and forced vibration of a pseudoelastic SMA beam (Razavilar et al., 2018).

In the case of analytic and semi-analytic approaches to studying nonlinear vibration of structures with SMA layers or contents, the research area has become narrower but there are some articles in recent years. A semi-analytic method was employed to study the damping performance of SMA beams (Danesh et al., 2023). Asadi et al. considered an SMA fiber-reinforced hybrid composite beam and studied the thermal stability and nonlinear vibration of the system using an analytic approach (Asadi et al., 2013). Using a combination of analytic modeling and the finite element method, the response of corrugated beams reinforced by SMA plates was studied by Koudzari et al. in (Koudzari et al., 2019). Asnafi investigated the chaos phenomenon in an SMA beam under simultaneous transverse and in-plane harmonic excitation using a fully analytic approach (Asnafi, 2022a).

By considering random forces or parameters for a structure with SMA contents, the number of researches becomes smaller. Recently, Hao et al. used Monte Carlo simulation as one of the most well-known numerical methods to obtain probability density functions (PDFs) of stochastic systems, to study parametric random vibration analysis of an axially moving laminated SMA beam (Hao et al., 2022). Zhang et al. studied the nonlinear behavior of an SMA beam under ore-strained conditions using the finite element method (Zhang et al., 2023). If randomness and the use of analytical methods, which increase the validity of the results, are considered together, the number of research will decrease again. Some articles use analytic methods to analyze local threshold values of chaos for an SMA beam under random loads (Costa et al., 2019; Ge and Xu, 2014). Bu et al. studied the nonlinear behavior of an SMA oscillator under simultaneous periodic and colored Gaussian noise excitations using the semi-analytic stochastic averaging method (Bu et al., 2023). Also, a manuscript by the author of this article in the same field to find the boundary of chaotic behavior in SMA webs under non-Gaussian noises has been lately accepted for publication and now is in “published online” situation. (Asnafi, A., 2023, The effect of bandwidth on the noise-induced chaos of shape memory alloy webs under mixed transverse and parametric excitations: an analytic approach. Acta Mechanica, 1-15.)

In this article, using the Fokker–Planck–Kolmogorov (FPK) equation, which is one of the most efficient analytical methods for evaluating the PDF of the response of nonlinear oscillators, the effect of temperature variation (material-phase changes) on the bifurcation of the response of an SMA beam under lateral white-noise excitation is investigated. As far as the author has considered the literature, this subject has been less studied and received less attention.

2. Materials and methods

2.1. Governing equation of motion of the system under study

As shown in schematic Figure 1, consider a simply supported SMA beam with a rectangular cross-section area b × h under a lateral white-noise excitation $ξ (t)$ fluctuating about a mean value ${\bar{F}}_{0}$ and whose intensity is $\bar{k}$ .

For a longitudinal element of the beam under this forcing function, the governing equation of lateral vibration obeys the following partial differential equation (Asnafi, 2021; Asnafi, 2022a; Ge and Xu, 2014):

ρ b h \frac{\partial^{2} W (x, t)}{{\partial t}^{2}} = - \bar{c} \frac{\partial W (x, t)}{\partial t} - \frac{\partial^{2} M}{{\partial x}^{2}} + \bar{F_{0}} + \bar{k} ξ (t),

(1)

where

W (x, t)

is the lateral deflection,

\bar{c}

is the linear damping coefficient,

ρ

is the beam density, and

M

is the internal bending moment which is a function of longitudinal normal stress at each section. Based on the material phases due to temperature variation in an SMA material, the following constitutive relation has been recommended in the literature (Falk and Konopka, 1990; Paiva and Savi, 2006):

σ = α (T - T_{M}) ε - β ε^{3} + \frac{β^{2}}{4 α (T_{A} - T_{M})} ε^{5},

(2)

where

α

and

β

are positive constant material coefficients,

T_{M}

is the temperature below which the martensite phase is stable, and

T_{A}

is the one above which the austenite phase is stable. Equation (2) is an almost simple model that also shows both the martensite detwinning process and pseudoelasticity in a consistent qualitative way. Noting also that the longitudinal strain has the following relation with lateral deformation:

ε = - y \frac{\partial^{2} W (x, t)}{{\partial x}^{2}}, y \in [- \begin{array}{l} \frac{h}{2}, & \frac{h}{2} \end{array}]

(3)

To obtain the minimum requirement of dynamic stability, the first or fundamental mode of lateral vibration of the beam whose instability is sufficient to destabilize the whole dynamics is considered (see Abedi and Asnafi, 2015; Abedi and Asnafi, 2017; Asnafi and Abedi, 2015). The fundamental mode of a simply supported beam is obtained from the free vibration analysis as

W (x, t) = \bar{w} (t) \sin (\frac{π x}{l})

(4)

where

l

is the beam length. Now, to compute the sectional bending moment, one can simply insert equations (3) and (4) into equation (2) and then integrate over the cross-section area to reach the following relation:

M = \int_{- \frac{h}{2}}^{\frac{h}{2}} {- y σ}_{x} d A = b \int_{- \frac{h}{2}}^{\frac{h}{2}} - y (α (T - T_{M}) ε - β ε^{3} + \frac{β^{2}}{4 α (T_{A} - T_{M})} ε^{5}) d y = - α (T - T_{M}) \frac{b h^{3}}{12} {(\frac{π}{l})}^{2} \bar{w} \sin (\frac{π x}{l}) + β \frac{b h^{5}}{80} {(\frac{π}{l})}^{6} {\bar{w}}^{3} \sin^{3} (\frac{π x}{l}) - {\frac{β^{2}}{4 α (T_{A} - T_{M})} \frac{b h^{7}}{448} (\frac{π}{l})}^{10} {\bar{w}}^{5} \sin^{5} (\frac{π x}{l})

(5)

Using one of the discretizing methods of partial differential equation of continuous media such as the Galerkin method (read more about this method in Abedi and Asnafi, 2016; Abedi et al., 2014), the following equation of nonlinear oscillation is obtained by substituting equations (4) and (5) into (1) and then integrating over the whole domain as

\ddot{\bar{w}} + \frac{\bar{c}}{ρ A} \dot{\bar{w}} + \frac{α (T - T_{M})}{ρ A} {(\frac{π}{l})}^{4} \frac{b h^{3}}{12} \bar{w} - {\frac{3 β}{4 ρ A} (\frac{π}{l})}^{8} \frac{b h^{5}}{80} {\bar{w}}^{3} {+ \frac{5 β^{2}}{4 ρ A 4 α (T_{A} - T_{M})} (\frac{π}{l})}^{12} \frac{b h^{7}}{448} {\bar{w}}^{5} = \frac{4}{ρ A π} (\bar{F_{0}} + \bar{k} ξ (t))

(6)

A set of non-dimensional coefficient is defined in this manuscript, to have a more general and comprehensive study, so equation (6) becomes

w^{″} + c w^{'} + k_{1} w + k_{3} w^{3} + k_{5} w^{5} = F_{0} + k ξ (t)

(7)

where

w = \frac{\bar{w}}{l}, τ = \hat{ω} t, {(*)}^{'} = \frac{\partial}{\partial τ}, {\hat{ω}}^{2} = \frac{b h^{3} π^{4}}{12 ρ A α T_{0} l^{4}}, T_{0} = 300 K, c = \frac{\bar{c}}{ρ A \hat{ω}}, k_{1} = \frac{(T - T_{M})}{T_{0}}, k_{3} = - \frac{9 π^{4}}{80} \frac{β}{α T_{0}} {(\frac{h}{l})}^{2}, k_{5} = \frac{5 π^{8}}{1344} \frac{β^{2}}{4 α^{2} (T_{A} - T_{M}) T_{0}} {(\frac{h}{l})}^{4}, F_{0} = \frac{4}{ρ A π {\hat{ω}}^{2}} {\bar{F}}_{0}, k = \frac{4}{ρ A π {\hat{ω}}^{2}} \bar{k}

(8)

Equations (7) and (8) let everyone study the variation of these dimensionless coefficients on the dynamic stability and bifurcation phenomenon of the whole system.

2.2. Probability density function

Here, there is a stochastic system that cannot be analyzed via conventional deterministic approaches. To study the behavior of a stochastic system, the statistical moments of the response rather than the response itself are used. Since all statistical moments and dependent variables such as mean and variance can be extracted from the PDF, the focus is on the calculation of this function (Abedi and Asnafi, 2016; Asnafi and Abedi, 2015). There are analytic/numeric/semi-analytic methods in the literature to compute the PDF of a random variable but one of the best is the Fokker–Planck–Kolmogorov (FPK) equation (Asnafi, 2017; Asnafi and Mahzoon, 2005). This partial differential equation, at each time step, determines the density of the random variable in the spatial domain. Although the analytical solution of this equation does not exist in its general form, it provides the PDF of the response in stationary cases for a wide range of linear and nonlinear oscillators (Asnafi and Mahzoon, 2005). Without any loss of generality, consider the following nonlinear stochastic oscillator:

w^{″} + c w^{'} + g (w) = k ξ (t)

(9)

where

g

is the nonlinear elasticity function and

ξ (t)

is a white-noise excitation whose intensity is

k

. The stationary joint probability density function (JPDF) of the response can be computed using the FPK equation (see more about the method in Abedi and Asnafi, 2015; Abedi and Asnafi, 2017):

P (w, w^{'}) = κ \exp (- \frac{2 c}{k} (\frac{{w^{'}}^{2}}{2} + \int_{0}^{w} g (ζ) d ζ)),

(10)

where

κ

is the normalization factor. A simple comparison between equations (7) and (9) shows that the JPDF of the beam response in phase space can be presented as

P (w, w^{'}) = κ \exp (- \frac{2 c}{k} (\frac{{w^{'}}^{2}}{2} + k_{1} \frac{w^{2}}{2} + k_{3} \frac{w^{4}}{4} + k_{5} \frac{w^{6}}{6} - F_{0} w)), κ = \frac{1}{\int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} P (w, w^{'}) d w d w^{'}}

(11)

Since the PDF for a general random variable X must satisfy the following conditions (Asnafi, 2017; Asnafi and Abedi, 2015)

\begin{array}{l} P \geq 0, \\ \int_{- \infty}^{+ \infty} P (X) d X = 1, \\ A t b o u n d a r i e s (X \to \infty) : \frac{\partial P}{\partial X} \to 0, P (X) \to 0, \end{array}

(12)

the number and type of equilibrium points and their stability conditions and potential bifurcations can be investigated by this function (see also Asnafi, 2017; Asnafi and Abedi, 2015 for more details). Here, the relative minima/maxima of the PDF plays a crucial role in locating equilibrium points and studying the stability conditions. Equation (11) which is derived through a complete analytic manner is used in this article to obtain the results and study the effect of parameter variations on the nonlinear response and bifurcation of the SMA beam in following sections.

3. Results and discussion

According to equations (7) and (8), coefficients

k_{3}

and

k_{5}

relate to material properties and aspect ratio of the beam, and

(h / l)

with non-varying values but coefficient

k_{1}

depends completely on the working temperature and its variation. Other coefficients such as

c

F_{0}

, and

k

depend on more factors and may have more diverse values in applied problems. The parameters of a material presented in Paiva and Savi (2006) are listed in Table 1 and used in this article to study the instability phenomenon.

Table 1.

The parameters of a material presented in Paiva and Savi (2006).

Parameter	$α$ (Mpa/⁰K)	$β$ (Mpa)	$T_{A}$ (K)	$T_{M}$ (K)
Value	1 × 10³	40 × 10⁶	313	287

The values or variations of other parameters in studying the bifurcation phenomenon are listed in Table 2. As mentioned earlier, the main goal of this article is to investigate the effect of temperature variation on the dynamic stability and bifurcation response of the beam. Therefore, the values of working temperature are considered variable in the range of 250 to 450 K. Also, the values of aspect ratio

(h / l)

are assumed to be variable between 1:5 to 1:20. Three positive, zero, and negative mean values for the lateral noise are considered for investigation. Since the variations of the linear damping coefficient and noise intensity values do not affect the behavior qualitatively, to avoid prolixity, the effect of changing these last two variables on behavior has not been investigated.

Table 2.

The values or variations of some parameters of the SMA beam.

Parameter	$c$	$k$	$T$ (K)	$F_{0}$	$h / l$
Value/range	0.2	0.02	[250, 450]	[−0.02, 0.02]	[1:20, 1:5]

3.1. The effect of temperature variation on the instability and bifurcation phenomenon

According to equation (8), a variation of working temperature leads to a change in coefficient $k_{1}$ , which acts as the linear elasticity term. The values of parameters $k_{3}$ , $k_{5}$ , $c$ , and $k$ are also presented in Tables 1 and 2. In this section and without any loss of generality, the values of $F_{0}$ and the aspect ratio $(h / l)$ are considered zero and 1:10, respectively. Figure 2(a) shows the evolution of the PDF (evaluated from equation (12)) when the working temperature varies from 250 to 450 K. Three different zones at low, intermediate, and high temperatures can be detected concerning this figure. In Figure 2(b), the PDFs for three specific temperatures are drawn. Note also that these diagrams are obtained via a complete analytic approach.

The following outcomes can be concluded from Figure (2):

As it is known, at low temperatures, the most likely presence of the response (deflection of the beam) is around two peaks of the PDF. These two equilibrium areas are the stable equilibrium points of the system.

At intermediate temperatures and especially after the temperature of 287 K, the probability density peaks increase to 3, which indicates a new equilibrium domain. A bifurcation occurs once the temperature passes through 287 K.

At high temperatures and specifically over $T = 375 K$ , three peaks converge to only one stable peak at the origin which means that another bifurcation occurs and the response oscillates around only one equilibrium point.

To have a better investigation, the joint PDFs of the response for three cases of Figure 2(b) are plotted and drawn in Figures 3 and4. These figures show the behavior in the phase space and make proper analogies with the numerical results as will be done also in the next sections of this article.

3.2. The effect of noises with non-zero mean values on the PDF

In the previous section, the results were obtained for a zero mean white-noise excitation. Here, and in this section, the effect of the non-zero mean value on the PDF of the response is investigated. In Figure 5, the graphs of Figure 2(b) are examined for two situations where the mean value of white noise has a positive or negative value.

From the graphs of Figure 5, it is clear that:

For non-zero mean values, the symmetry of the equilibrium points (peaks of the PDF) is disturbed and the dominant peak (with higher height) is formed in line with the sign of the mean value.

In terms of bifurcation location, no difference was observed—as expected—but the weight of equilibrium domains changed.

3.3. The effect of changing in aspect ratio on the PDF

In this section, the effect of aspect ratio $(h / l)$ variation on the PDF is examined. Figures 6 and 7 clearly show the modifications of PDFs of the case studies in Figure 2(b) when the aspect ratio increases from 1:20 to 1:5.

From Figures 6 and 7, it is concluded that:

As the aspect ratio becomes larger, the PDF peaks are formed at farther distances from the origin. This means that the random variable has more spread in the phase plane, which physically means that more fluctuation can be observed.

In Sections 3.1 to 3.3, the instability and bifurcation phenomenon were investigated for the response of the system under the mentioned conditions. Due to the use of dimensionless parameters, these results show the relations between environmental conditions (temperature), loading conditions (average and intensity of random load), and geometric conditions (aspect ratio) with the dynamic stability of the beam. The results of this research can be used to estimate the dynamic stability of structures under wide-bandwidth loads and temperature variations. The main contributions are fuselages and instruments of aerospace structures, and also the sensors and actuators of drilling deep wells. Noting that also, due to the analytical nature of the method, the results of this research have proper validity.

3.4. Numerical confirmation

Since the results presented in the previous sections were obtained through a completely analytical process with equations whose correctness has already been verified with mathematical theorems, it is meaningful to validate them with some corresponding ones obtained via numerical approaches. Instead, it is shown that analytic results for some specific case studies can be confirmed by the corresponding numerical ones. Using data from Section 3.1, in Figure 8(a) the maxima/minima locus of the corresponding PDF concerning temperature variation is drawn. The black lines represent the peaks (relative maxima) of the PDF function, which represent stable manifolds. Likewise, the red lines represent valleys (relative minima) of the same function that represent unstable manifolds.

Figure 8(a) also shows how the temperature variation can affect the stable and unstable manifolds quantitatively and qualitatively. In Figure 8(b), using the 4th-order Runge–Kutta numeric solver, the bifurcation diagram of the response concerning temperature variation has been calculated directly from equation (7) and drawn. In fact, in each step of the temperature domain, the equation was solved with several initial conditions, and the steady-state solutions were reported. In other words, at a temperature lower than 287 K, there are two stable equilibrium points and one unstable saddle point. It is expected to have two symmetrical equilibrium domains around the origin at lower temperatures of 287 K in numerical solution, which is seen in Figure 8(b). At intermediate temperatures (between 287 and 375 K), there are three stable equilibrium points and two unstable saddle points in Figures 2–4 and 8(a). The bifurcation diagram of Figure 8(b) confirms also the same issue. At high temperatures (above 375 K), there is only one stable equilibrium point, which is confirmed in Figure 8(b).

4. Conclusion

Using the Fokker–Planck–Kolmogorov equation, the PDF of the response for an SMA beam under white-noise excitation was derived analytically in this article. The function was then used to study the nonlinear behavior and bifurcation response of the beam when one of the following parameters varied: working temperature, beam aspect ratio, and the mean value of the white-noise excitation. Three different behaviors were observed in the behavior of the beam according to the working temperature; at a temperature lower than 287 K, which is mainly the dominant martensite phase, the beam oscillates between two equilibrium domains around the origin. At a higher temperature of 375 K where the austenite phase is dominant, there is only one equilibrium domain at the origin of the phase plane. Between these two temperatures, three equilibrium domains are formed, one at the origin and two around it. It was shown that once the temperature passes through the mentioned critical temperatures, a bifurcation occurs in the PDF and consequently in the response. It was shown that for non-zero mean values of the white-noise excitation, an asymmetric PDF was constructed locating the dominant and greater equilibrium point; however, it did not affect the occurrence of bifurcation. It was shown that for the larger aspect ratio of the beam, the probability of the presence of the response is more scattered in the phase plane which means larger fluctuation about the origin. Finally, the analytic results were validated by corresponding numeric ones.

Considering the response of the system under other random loads such as narrow band and wide frequency band—for which, of course, there may not be an analytical solution—can be the next step of this research. Also, using other constitutive stress–strain models to study and compare the accuracy is the next step to improve the quality of the research. Studying some practical examples such as high-tech sensors and actuators that are exposed to temperature changes can also be assumed as the further steps.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Alireza Asnafi

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