Quantitative investigation of internal friction components in NiTiCu shape memory alloy using dynamic mechanical analyser and iterative method

Abstract

This study primarily investigates the internal friction (IF) behaviour of Ti₅₀Ni₄₈Cu₂ (TNC2) shape memory alloy (SMA) through temperature, frequency, isothermal, and strain sweep analysis using a Dynamic mechanical analyser (DMA). The total internal friction (IF_Total) value of TNC2 alloy is 0.0686 at 323.4 K. The frequency sweep analysis reveals that increasing the frequency reduces the IF_Total by 15.97% due to restricted martensitic interface movement. Isothermal analysis shows a significant drop in IF_Total during the B2-B19′ transformation peak, with reductions of 79.02%, stabilising as intrinsic damping dominates. Strain sweep tests demonstrate that higher strain at isothermal temperature (323.4K) increases inherent internal friction (IF_PT + IF_Int)_B2–B19′, with a frequency-dependent trend. The secondary focus is decomposing the IF_Total into its intrinsic (IF_Int) and inherent internal friction (IF_PT + IF_Int)_B2–B19′, components using an iterative method. The iterative method effectively separates IF_Total, allowing precise calculation of the transformed volume fraction n(T). After eight iterations, convergence is achieved with a minimal error rate. The predicted IF spectrum closely matches experimental data, highlighting the method’s reliability for decomposing IF_Total into its contributions.

Keywords

shape memory alloy NiTiCu DSC DMA isothermal iterative analysis

Introduction

Thermoelastic martensitic transformation in shape memory alloys (SMAs) results in distinctive characteristics, including shape memory effect and superelasticity (Jani et al., 2014). Multiple investigations have demonstrated that SMA possess strong damping characteristics during martensitic transformation, making them suitable for energy dissipation purposes (Frenzel et al., 2015; Nespoli et al., 2021; Wang et al., 2018; Yuan et al., 2011). Recent review articles have extensively examined the damping properties of SMAs, highlighting the influence of various factors on their performance. These studies explore the potential of SMAs in vibration control and noise reduction across multiple applications, including space structures, where they serve as passive isolators for spaceborne cryocoolers, superelastic tyres, and micro dampers for MEMS devices (Saedi et al., 2023). Additionally, SMAs are investigated for their role in aerospace engineering, particularly in structural vibration control applications such as variable geometric chevrons for noise reduction and adaptive structures (Ndukwe, 2024; Trehern et al., 2023).

Damping property, Internal friction (IF), and Tan δ are closely related concepts. The influence of IF on the damping characteristic of a shape memory material can be measured using Tan δ with a dynamic mechanical analyser (DMA), a spectroscopic method (Radhamani and Balakrishnan, 2023b). So, IF and Tan δ are both terms that are repeatedly used in this manuscript. IF is observed when a material converts mechanical energy into heat during deformation or when subjected to mechanical stress. In SMAs, two distinct types of IF peaks emerge during heating and cooling, each serving unique applications. The sharp martensitic transformation peak arises from the movement of the austenite/martensite interface during phase transformation, making it useful for active damping, actuation, and precise phase transformation monitoring. In contrast, the broad relaxation-type peak results from the movement and interaction of martensitic variant boundaries under stress, contributing to passive damping, structural health monitoring, and impact absorption (Liu et al., 2024). One of the most promising and widely used NiTi-based alloys is NiTiCu. The inclusion of Cu in place of Ni in binary equiatomic NiTi has a considerable impact on the system’s martensitic transformation and thermomechanical behaviour. Cu addition shifts the thermoelastic transformation temperature to the lower sides and improves the damping properties of the SMAs. Therefore, NiTiCu can be used at various working temperatures different from those of the NiTi binary alloy (Radhamani and Balakrishnan, 2023a, 2024; Rajeshkannan et al., 2022; Sampath et al., 2020).

Few studies have explored the damping properties of Ti₅₀Ni_50-xCu_x SMAs. Initial research investigated the total internal friction [IF(T)] of these alloys using the inverted torsional pendulum method at low frequencies ranging from 1 to 3 Hz, revealing that the Ti₅₀Ni₃₀Cu₂₀ alloy exhibits an exceptionally high IF value of 0.2 (Yoshida et al., 2003). Subsequent studies examined the impact of heat treatment on the IF of NiTiCu SMAs, finding that solution treatment at 1173 K is preferable over 1373 K due to the higher volume of Ti₂NiCu precipitates (Yoshida et al., 2004). Over the years, researchers have employed DMA to evaluate IF across various compositions, which provides the opportunity to quantitatively analyse the IF under different conditions, such as temperature, frequency, strain sweep, and isothermal conditions (Chang and Hsiao, 2014; Chang and Wu, 2007; Chien et al., 2014; Fabregat-Sanjuan et al., 2015; Nespoli et al., 2013, 2016).

Research indicates that Cu additions influence the damping properties of NiTiCu alloys by stabilising the B19 phase and modifying IF across a broad temperature range. Among NiTiCu compositions, high Cu content (20 at.%) enhances martensitic phase damping, while Ni₄₀Ti₅₀Cu₁₀ demonstrates superior damping at both room temperature and higher temperatures (Nespoli et al., 2021; Villa et al., 2021a). The strain glass transition (SGT) and reentrant strain glass transition (RSGT) have been studied in NiTi-based SMAs, including NiTi, NiTiCu, and NiTiHf. SGT in NiTi and NiTiHf occurs when local strain heterogeneities, induced by atomic disorder or rapid cooling, suppress the long-range martensitic transformation, leading to a frozen strain state with only short-range strain ordering (Li et al., 2025; Liang et al., 2024). RSGT, observed in NiTiCu SMAs, involves a strain-ordered martensitic phase transforming into a strain glass state upon cooling. Wang et al. reported RSGT in Ti₅₀Ni₃₄Cu₁₆, where this transition challenges conventional thermodynamic stability and provides valuable insights into the functional behaviour of NiTiCu alloys. Additionally, the RSTG Ti₅₀Ni₃₄Cu₁₆ alloy exhibits the Elinvar effect, featuring an ultra-low modulus of 24 GPa and high damping capability (tan δ > 0.075) over a wide temperature range, making it highly promising for damping applications (Wang et al., 2022). Similarly, other researchers also reported a comparative study on other SMAs, such as Mn-doped Co-V-Ga, Cu-Al-Ni, Cu-Al-Fe, Ni-Ti-Hf-Nb alloys, further illustrate the role of microstructural modifications, precipitation effects and elemental addition and doping in altering IF behaviour (Araújo et al., 2024; Eftifeeva et al., 2024; Liu et al., 2024; Santosh et al., 2022a). Recent studies have explored the integration of SMA wires into 3D-printed polymer (e.g. PLA, PETG) and pressureless infiltration of a Mg melt into 3D-printed NiTi to enhance mechanical and damping properties (Santosh et al., 2022b; Sun et al., 2022; Zhang et al., 2020). These findings emphasise the significance of DMA in optimising SMA compositions and processing conditions for enhanced damping performance. Despite extensive studies on the damping properties of NiTiCu SMA’s, there is a lack of research on alloys with lower copper additions.

The total internal friction [IF(T)] during a martensitic phase transformation is widely recognised to consist of three distinct contributions, as shown in Figure 1(a) and (b).

IF (T) = I F_{Tr} + I F_{PT} + I F_{Int}

(1)

IF (T) or I F_{Total} - Total IF

(1)

I F_{Tr} - Transient IF

(1)

I F_{PT} - Phase transition IF

(1)

I F_{Int} - Intrinsic IF

(1)

The $I F_{Int}$ is the combination of the inherent IF contributions of martensite and austenite. The $I F_{Tr}$ is a kinetic term that arises exclusively during heating or cooling and is intricately linked to the transformed volume fraction n(T), which represents the fraction of a material that has undergone a phase transformation at a given temperature T. The $I F_{PT}$ represents the movement of the interfaces between martensite and austenite over significant distances caused by the transformation’s kinetics. Understanding these individual IF components is crucial for optimising damping applications of SMAs in adaptive structures, aerospace, and biomedical devices (Pérez-Sáez et al., 1998; Radhamani and Balakrishnan, 2023a).

Few groundbreaking studies have focussed on the quantitative and accurate analysis of IF for Cu and NiTi-based SMA (Li et al., 2007; Pérez-Sáez et al., 1998, 2000). The quantitative separation of different contributions from the IF_Total spectra of SMA can be achieved in several ways. The isothermal method provides stable damping but requires time for temperature stabilisation. The temperature rate approach involves measuring multiple IF spectra at different rates. In contrast, the iterative approach offers a simpler alternative by using a single IF spectrum at a consistent temperature change rate (Nespoli et al., 2016; Pérez-Sáez et al., 1998). However, there is a dearth of research on the quantitative evaluation of IF in NiTiCu SMA.

NiTi SMAs are extensively studied for their exceptional damping characteristics, which are vital in applications that demand efficient energy dissipation, including aerospace, biomedical devices, and structural engineering. NiTiCu SMAs have gained attention due to their tunable transformation temperatures and improved damping characteristics. However, the damping response of Ti₅₀Ni₄₈Cu₂ remains underexplored despite its potential advantages in adaptive damping systems. This study addresses this gap by systematically investigating the IF behaviour of Ti₅₀Ni₄₈Cu₂ under various conditions, including temperature sweep, frequency sweep, isothermal analysis at critical transformation temperatures (martensite and austenite), and strain sweep. A key aspect of this research is applying an iterative method to quantitatively decompose IF into its fundamental components: transient IF, intrinsic IF, and transformed volume fraction n(T). This decomposition allows a more precise understanding of their contributions to the total IF signal. By isolating these components, this study provides a more comprehensive framework for understanding damping mechanisms in NiTiCu SMAs, particularly in engineering applications where temperature stabilisation renders transient damping effects negligible. The findings contribute to optimising NiTiCu SMAs for high-performance structural applications, where controlled damping is essential for longevity and functionality.

Materials and methods

Alloy preparation

Ni, Ti, and Cu, with high purity levels of 99.9%, 99.8%, and 99.5%, purchased from Alpa Aesar, were used to prepare billet weighing 30 g. The billet was composed of Ti₅₀Ni₄₈Cu₂ (in atomic percent) labelled TNC2. The alloy was prepared using a vacuum arc remelting furnace (M/S Vacutech System), which is widely recognised for producing high-purity alloys. Prior to material loading, the furnace chamber, the water-cooled copper mould (anode) and the tungsten electrode (cathode) were thoroughly cleaned using a fine grit sheet and acetone to ensure no residual material or contamination from previous melts, and the water-cooling system was inspected for proper operation. The pure raw materials were accurately weighed using an electronic balance and loaded into a rectangular-shaped water-cooled copper crucible along with a Titanium getter. The furnace chamber was then sealed, and a high-vacuum pressure of 10⁻⁵ mbar was achieved in stages using rotary and diffusion pumps, effectively removing residual gases and volatile elements. A GE-402 TIG power supply was used to control the melting process. An initial arc was struck between the tungsten electrode and the sample material, initiating the melting process. The titanium getter was initially melted to capture any residual impurities, after which the loaded raw materials were melted to create a unified NiTiCu alloy. After the first melting cycle, the solidified billet was flipped over, again the chamber was evacuated to 10⁻⁵ mbar, and the sample was remelted. This process of melting, flipping, and remelting was repeated for four cycles to ensure uniform composition and minimal contamination.

Quartz tube sealing

The as-cast billet was ground using a belt grinder to achieve a uniform surface finish. After grinding, the billet was roughly polished to enhance its surface quality and remove imperfections for further processing. The quartz tube with an outer diameter of 22 mm is filled with the sample and Titanium getter, and a constriction is created at the neck to seal it. The tube is subsequently connected to a high-vacuum system, which includes a rotary and diffusion pump. Initially, the rotary pump is activated to achieve a vacuum level of around 10⁻² mbar. After the initial evacuation, the diffusion pump is engaged further to accomplish the vacuum level to 10⁻⁵ mbar. After obtaining this high vacuum, the tube is backfilled with argon gas. The process is repeated three times to achieve a high vacuum. Finally, the quartz tube is sealed at the constricted neck with an LPG-oxygen torch.

Heat treatment

To examine the alloy’s intrinsic properties in its as-treated condition, the sealed ampoule is homogenised in a box furnace at 1173 K for 24 h at a heating rate of 5 K/min, followed by rapid quenching in water to prevent the formation of precipitates.

Metallographic sample preparation

The heat-treated samples were meticulously sectioned with a low-speed diamond saw to reduce deformation and heat generation, thereby preserving structural integrity. The sectioned samples were subsequently mounted with an automatic mounting press to ensure stability during polishing. This was followed by a sequential polishing process, which involved grinding with silicon carbide abrasive papers ranging from 200 to 2000 grit and fine polishing using diamond paste on a disc polisher to achieve a mirror finish. Due to the complexity of morphology, extensive trials and optimisation led to the final etching solution of 10% HF (Hydrofluoric acid), 60% HNO₃ (Nitric acid), and 30% H₂₀O (Deionised water). The mirror-polished samples were immersed in this modified Kroll’s reagent inside a fume hood for approximately 10 s to reveal the microstructural features effectively.

Material characterisation

The elemental composition and impurity analysis of the TNC2 SMA were conducted using ICP-OES (PerkinElmer Avio 200) and CHNS/O (PerkinElmer-2400). Approximately 0.5 g of the sample was dissolved via microwave acid digestion for ICP-OES, ensuring accuracy through calibration with standard solutions. Impurity elements, mainly carbon and oxygen, were analysed using CHNS and oxygen determination methods. For CHNS analysis, ∼2 mg of the sample was combusted in a high-temperature furnace with oxygen, while oxygen determination involved pyrolysis under a helium atmosphere. Calibration was performed using certified reference materials (CRMs) to ensure precision.

The alloy’s crystal structure was meticulously analysed using a Malvern Panalytical X-ray diffractometer. The measurements were taken over a range of 2θ values from 20° to 90°, with a precise step rate of 7°/min. CuKα radiation with a specific wavelength of 1.5406 Å was used for the analysis. The crystalline phase was analysed using X′Pert High Score Plus software, where the obtained diffraction patterns were matched with JCPDS data and past literature to identify the phases present in the TNC2 alloy accurately. Origin 2023b software was used to plot the XRD patterns for clear visualisation and analysis.

The microstructure of etched samples was examined using an Olympus BX53M Optical Microscope and a Field Emission Scanning Electron Microscope (FESEM) using a Thermo Fisher FEI QUANTA 250 FEG instrument. Additionally, Energy-Dispersive X-ray Spectroscopy (EDS) analysis was conducted to validate the chemical composition of the TNC2 alloy.

The alloy’s phase transformation temperatures were determined using a NETZSCH DSC 214 calorimeter following ASTM F2004-21. A small specimen weighing 25–35 mg was carefully cut using a slow-speed diamond cutter and subjected to heating and cooling cycles from 175 to 425 K at a rate of 10 K/min. The phase transformation temperatures (As, Af, Ms, and M_f) were accurately determined.

Dynamic mechanical analysis

The billet was cut using wire electrical discharge machining (WEDM) into dimensions of 40x5x2 mm² for DMA to maintain an L/T ratio > 10, ensuring reliable DMA measurements (Radhamani and Balakrishnan, 2023b). The analysis was performed using a TA Q800 instrument in single cantilever mode under two conditions. The instrument sensitivity for internal friction (Tan δ) was 0.0001. The temperature-dependent behaviour of the IF was investigated by conducting measurements using a constant cooling rate of 5 K/min within the temperature range of 400–200 K. The measurements were performed at a frequency of 1 Hz and an amplitude of 5 µm ( $1.25 \times 10^{- 4} strain$ ). Using identical settings (clamp, cooling rate, and strain amplitude), frequency sweep analysis was conducted by varying the frequencies to 0.5, 1, 5, and 10 Hz. Isothermal analysis was carried out at the temperature corresponding to the peak of martensite and at the temperatures of both martensite and austenite to examine the inherent IF of the TNC2 sample, as these alloys exhibit a one-step phase transformation. The experimental procedure was conducted as follows: The sample was subjected to a steady cooling rate of 5 K/min, starting from an initial temperature of 400 K. After reaching the desired temperature, it was maintained at that temperature for a minimum of 30 min.

Iterative analysis

The iterative procedure decomposes the IF_Total into IF_Int, IF_Tr, and n(T) as shown in Figure 2. The process begins with the experimental IF spectrum, recorded as a function of temperature [IF_Total(T)], spanning from an initial temperature T_s to a final temperature T_f. The temperature range is selected to be sufficiently broad to capture the background regions on both sides of the IF peak. During the cooling phase, the temperatures follow the sequence: T_s > M_s > M_f > T_f, where M_s and M_f represent the martensitic start and finish temperatures, respectively (Pérez-Sáez et al., 1998).

Step 1: The procedure started with an assumption of arbitrary initial function for the intrinsic term $[I F_{Int} (T)]_{Initial}$

This function is subject only to the condition, $I F_{m} \geq [I F_{Int} (T)]_{Initial} \geq I F_{a}$

$I F_{m} & I F_{a}$ are the internal friction values of Austenite and Martensite.

Step 2: From the chosen $[I F_{Int} (T)]_{Initial}$ function, $I F_{T} - [I F_{Int} (T)]_{Initial}$ is estimated.

Integral $[J (T)]_{Initial}$ , is calculated, which is directly related to $n (T)$

J {(T)}_{Initial} = \int_{Ts}^{T} (I F_{T} - {[I F_{Int} (T)]}_{Initial}) f^{l} dT = n (T)

(2)

f –The frequency at which experimental analysis is performed,

T – Temperature against each IF value,

l – Coefficient chosen to 1.

Since most models consider the l coefficient equal to one, the actual l value could be computed empirically by measuring it as a function of the frequency. Many proofs have been conducted using various methods, and only minor variations have been noted (Nespoli et al., 2016; Pérez-Sáez et al., 2000).

Step 3: The obtained $n (T)$ is introduced in the below equation to calculate the $[I F_{Int} (T)]_{1}$

[I F_{Int} (T)]_{1} = n (T) I F_{m} + [(1 - n (T)] I F_{a}

(3)

Step 4: Using this new $[I F_{Int} (T)]_{1}$ an iteration is performed.

J (T) = \int_{Ts}^{T} (IF - {[I F_{Int} (T)]}_{1}) f^{l} dT

(4)

Step 5: The iteration cycle is repeated until the difference between the input function $[I F_{Int} (T)]_{j - 1}$ and the output function $[I F_{Int} (T)]_{j}$ become a smaller value than the prefixed value. The process converges rapidly. The square root of the sum of the squared differences between successive iterations is used as a metric to evaluate the convergence rate. This parameter helps quantify how quickly the iterative cycles approach stability.

Step 6: Once the Final $[I F_{Int} (T)]$ is reached, final $[I F_{Int} (T)]$ versus T is plotted.

Step 7: After getting the Final $[I F_{Int} (T)]$ ;

n (T) = \frac{[I F_{Int} (T)] - I F_{a}}{I F_{m} - I F_{a}} is calculated

(5)

Finally, n(T) is plotted against T.

The iterative method was executed using Python in Jupyter Notebook 6.5.4 via Anaconda 3 Navigator.

Results and discussion

Chemical analysis

The ICP-OES findings are displayed in Table 1, along with impurity analysis. The composition variations are due to the powder loss during melting and spattering. Despite certain variations from the standard compositions, the alloys will be identified by their previously assigned label (TNC2) for simplicity in the subsequent discussion. Similar variations in the nominal compositions have been reported in earlier studies (Yoshida et al., 2004; Zhang et al., 2006.). However, the alloy’s total (Ni + Cu) content remains below 50 at.% (49.4 at.%), which is significant because compositions exceeding this threshold, even marginally by 0.1 at%, are reported in the literature to cause abrupt and nonlinear changes in phase transformation temperatures (Frenzel et al., 2015).

Table 1.

ICP-OES & CHNSO analyser results.

Nominal Composition	ICP-OES analysis	Impurity analysis, wt. %
Nominal Composition	ICP-OES analysis	C	H	N	S	O
Ti₅₀Ni₄₈Cu₂	Ti_50.5Ni_46.6Cu_2.8	0.0156	0.00017	0.001182	0.0000486	0.00316

The presence of impurity elements, mainly carbon and oxygen, substantially influences the phase transformation temperature and microstructure. Undesirable compounds such as TiC and Ti₄Ni₂O_x (Zhang et al., 2006) are prone to develop, leading to a rise in nickel concentration in the matrix. Hence, it is imperative to meticulously monitor and reduce the use of oxygen and carbon while achieving complete eradication, which is unattainable. Therefore, it is essential to continuously monitor the assimilation of carbon and oxygen by the liquefied substance. Consequently, the carbon, hydrogen, nitrogen, oxygen, and Sulphur quantities were assessed and documented. According to industry norms and ASTM F2063, their concentration was much below 500 ppm, achieved by utilising high-quality raw materials and employing a highly efficient vacuum system during processing.

XRD analysis

The XRD analysis of sample TNC2 at room temperature reveals distinct phases, as shown in Figure 3. The alloy exhibits cubic (B2) austenite and monoclinic (B19′) martensite phases, undergoing a one-step phase transformation from B2 to B19′ when the Cu content is below 7.5 at.% (Yang et al., 2015). The cubic B2 phase is observed at 41.7°, 62.08°, and 78.1°, corresponding to the (110), (200), and (211) lattice planes, respectively, matching JCPDS card no. 044-1288, while the monoclinic B19′ phase appears at 39.63°, 42.7°, 45.3°, and 60.1° with planes (002), (111), (012), and (112), corresponding to JCPDS card no. 035-1281 (Bricknell et al., 1979; Callisti et al., 2013; Tatar et al., 2021).

Microstructure of NiTiCu SMA

Figure 4(a) shows the optical microstructure of TNC2, revealing tiny needle-shaped martensite plates dispersed within the austenitic matrix. The needles are oriented in various directions throughout the matrix, creating distinct microstructural features. Micrographs also show regions of retained austenite, indicating the coexistence of austenitic and martensitic phases. The microstructure aligns with Santosh and Sampath (2020). This needle-like morphology supports the shape recovery mechanism by facilitating self-accommodation, a critical factor in the alloy’s shape memory behaviour. The FESEM image in Figure 4(b) further confirms the presence of needle-shaped martensite. The martensite length ranges from 0.3 to 14.8 μm, with an average length of 9.64 μm. The figure also highlights the presence of tiny precipitates, with a small circle marking their location. The chemical composition of the matrix and precipitate was studied through EDS analysis, as shown in Figure 4(c) and (d). The EDS analysis confirms the minor variation in chemical composition, as discussed in Section 3.1, and highlights the Ti-rich precipitates, likely Ti₂NiCu (Cai et al., 2024; Tatar et al., 2020, 2021). The presence of Ti-rich precipitates has a notable influence on the damping properties, particularly through their interaction with the microstructure, such as interfaces and twin boundaries (Acar, 2015; Fabregat-Sanjuan et al., 2018). Ti₂NiCu precipitates primarily influence the IF_Int by acting as obstacles to the movement of dislocations and twin boundaries. This pinning effect leads to increased energy dissipation and higher IF_Int values. However, they can reduce the IF_Tr and IF_PT components by hindering the motion of phase boundaries, thereby slowing down transformation kinetics (Li et al., 1994; Villa et al., 2021b).

DSC analysis

DSC provides a fundamental reference for determining the temperature range in which DMA measurements should be performed. DSC identifies the key phase transformation temperatures (Ms, Mf, As, and Af) by directly measuring latent heat changes, which define the expected transformation window. Table 2 highlights the phase transformation temperature of TNC2. Figure 5(a) shows the corresponding DSC plots, which confirms the alloys have phase transformation during heating and cooling. Heating the alloy produced an endothermal peak, which soaked up energy, and cooling off made an exothermal peak, which liberated energy in the form of heat due to the variation in total heat (enthalpy) during the heating and cooling cycle. The area under the peak curve represents the enthalpy of transformation. The alloy exhibits a one-step phase transformation sequence from B2 to B19′. The elastic modulus, affected by temperature and alloy chemistry, is crucial in determining phase transformation temperatures in SMAs. Transition metals, with valence electrons acting as an adhesive to the nucleus, establish a bulk and shear modulus reduction as their atomic number increases, weakening this adhesive effect. SMAs experience pre-martensitic softening during cooling as their elastic modulus decreases. A lower bulk modulus results in higher martensite start (Ms) temperatures, while a higher bulk modulus lowers these temperatures (Vedamanickam et al., 2023; Zarinejad and Liu, 2008).

Table 2.

DSC analysis.

Composition	A_s (K)	A_f (K)	M_s (K)	M_f (K)	ΔH^M→A (J/g)	ΔH^A→M (J/g)
TNC2	349.3	375.6	339.8	313.1	20.4	−20.1

Dynamic mechanical analysis

Temperature sweep

The DMA experiment for the TNC2 SMA shown in Figure 5(b) provides vital insights into phase change behaviour during cooling from 400 to 200 K via internal friction (IF), represented by Tan δ. The alloy exhibits two IF peaks on Tan δ versus temperature curves, that is, a broad plateau on the low-temperature side and a sharp peak on the high-temperature side. A sharp, distinct peak (IF_Total) is observed during the cooling process, corresponding to the martensitic phase transformation from the B2 to B19′, which is frequency-dependent. The occurrence of this peak is a result of the combined effects of intrinsic internal friction (IF_Int), transient internal friction (IF_Tr), and phase transformation internal friction (IF_PT), and the observed surge in IF_Total during the martensitic phase transformation is primarily attributed to the IF_Tr (Chang et al., 2016; Chang and Wu, 2019). The IF_Total value of TNC2 is 0.0686, at a peak temperature of 323.4 K, comparatively higher than equiatomic NiTi SMA (Cai et al., 2005). This phenomenon may be attributed to the enhanced mobility of martensitic interfaces and twin boundaries in NiTiCu alloys, resulting in the increased material capacity to dissipate mechanical energy (Chien et al., 2014). Conversely, the broad and smooth IF plateaus on the low-temperature side span a wide range and exhibit relatively low Tan δ values. These plateaus are the relaxation-type IF peaks stemming from the martensite twin boundaries and the absence of twin variants in B19. In practical applications of damping, the relaxation-type IF peaks stand out due to their extensive operating temperature range and their capacity to sustain damping effectiveness over time, making them particularly advantageous (Liu et al., 2024). The DSC and DMA results show a shift in the peak of the phase transformation temperature and peak of IF, likely due to stress effects, differences in sample size, frequency effects, microstructural inhomogeneities and cooling rate (Wu et al., 2023).

The IF of the B19′ phase is significantly higher than the B2 phase due to numerous twin boundaries, which can reorient under stress to accommodate strain and dissipate energy (Helbert et al., 2021; Wu and Lin, 2003). Conversely, the B2 phase has fewer twin boundaries, resulting in lower IF values (Fan et al., 2006). The storage modulus (E′) decreases during the martensitic phase transformation, reflecting the softening of the material as it transforms from the stiff austenite phase to the more compliant martensite phase. The reduction in storage modulus indicates that martensite accommodates more strain, further supporting its higher damping capacity than austenite (Chang and Wu, 2006).

Table 3 provides a comparative analysis of the Peak Internal Friction (Tan δ) and IF Peak Temperature of Ti₅₀Ni₃₀Cu₂₀ alloys alongside other well-studied NiTiCu compositions from the literature. The data highlights key trends in damping behaviour and transformation temperatures based on composition. The increase in peak internal friction with higher Cu content suggests an improvement in damping capacity. For example, Ti₅₀Ni₄₈Cu₂ shows a Tan δ of 0.0686, while Ti₅₀Ni₃₀Cu₂₀ achieves 0.172, indicating that adding Cu leads to more significant lattice distortion and internal stress interactions, improving damping properties. Nonetheless, inconsistencies are evident in the reported values for identical compositions (e.g. Ti₅₀Ni₃₀Cu₂₀ exhibits Tan δ values of 0.172 and 0.11 across various studies), underscoring the impact of differences in experimental conditions, processing methods, and measurement techniques. On the other hand, the IF peak temperature exhibits a non-linear relationship with Cu content, showing variations that are mainly affected by the processing history, experimental conditions, measurement techniques and frequency. Furthermore, variations in compositional inhomogeneity and elemental distribution across various studies may lead to discrepancies in IF peak temperature and damping behaviour.

Table 3.

Peak internal friction (Tan δ) and IF peak temperature of Ti₅₀Ni₄₈Cu₂ alloys and other NiTiCu compositions reported in literature.

Composition	Experimentalconditions	Frequency(Hz)	Peak IF value(Tan δ)	Temperature range of IF peak (K)	Ref
Ti₅₀Ni₄₈Cu₂	Single cantilever	1	0.0686	323.4	This work
Ti₅₀Ni₄₅Cu₅	Single cantilever	1	0.082	290.13	Chang and Hsiao (2014)
Ti₅₀Ni₄₀Cu₁₀	Single cantilever	1	0.116	290.73
Ti₅₀Ni₃₀Cu₂₀	Single cantilever	1	0.172	308.43
Ti₅₀Ni₃₀Cu₂₀	Single cantilever	1	0.11	335.13	Villa et al. (2021a)
Ti₅₀Ni₄₅Cu₅	Single cantilever	1	0.1	291.23	Chien et al. (2015)
Ti₅₀Ni_37.5Cu_12.5	Single cantilever	1	0.17	302.8	Chien et al. (2015)
Ti_44.6Ni_50.4Cu₅	3-point Bending	0.2	0.1374	309.384	Fabregat-Sanjuan et al. (2015)
Ti₅₀Ni₂₇Cu₂₃	Dual cantilever	1	0.16	364.13	Li et al. (2007)

Frequency sweep analysis

The frequency sweep analysis examines the time-dependent behaviour of the SMA. Figure 6(a) shows the IF_Total of sample TNC2 against the temperature in K., with frequencies varying from 0.5, 1, 5, and 10 Hz. The IF_Total values against frequency under a constant amplitude of 5 µm and cooling rate of 5 K/min are displayed in Figure 6(b). The low-frequency domain is highly relevant for several real-world applications, including launch vibration damping in small satellites, vibration suppression in spacecraft solar arrays, and seismic or structural vibration control in civil engineering systems (Jani et al., 2014; Saedi et al., 2023). The increase in frequency decreases the IF_Total by 15.97%, which is mainly ascribed to the restricted movement of martensitic interfaces and twin boundaries (Blanter et al., 2007). The IF_Total variation between 0.5 and 1 Hz is slightly higher, about 12.14%, than the variation between 1 and 10 Hz, around 4.46% for TNC2. The movement of martensitic interfaces and twin boundaries is time-dependent. At lower frequencies, they move freely, but at higher frequencies, their motion is restricted, resulting in more elastic behaviour, reduced strain accommodation, and decreased energy dissipation, leading to lower IF (Otsuka and Wayman, 1999).

Isothermal analysis

The IF_Total obtained during the martensitic phase transformation through temperature sweep at a constant cooling rate, fixed frequency and amplitude is primarily attributed to the IF_Tr. Removing the IF_Tr from the IF_Total is necessary to provide steady damping solutions for continuous vibration at set temperatures. IF_Tr should progressively decrease when the specimen is maintained isothermally at a constant temperature. Figure 7 presents the isothermal IF analysis of TNC2 at three critical temperatures: the peak temperature corresponding to the martensitic transformation (B2–B19′) at 323 K, the martensite phase (B19′) at 243 K, and the austenite phase (B2) at 368 K. The isothermal IF analysis was performed at a fixed frequency of 1 Hz and an amplitude of 5 microns for 30 min. This duration was carefully chosen based on prior studies and preliminary tests to ensure thermal equilibration and complete stabilisation of phase transformation phenomena. Furthermore, the DMA chamber used includes a temperature feedback control system, which minimises the risk of thermal gradients. The observed drop in internal friction under isothermal conditions is therefore attributed to the stabilisation of phase interfaces and reduction in interface mobility over time, rather than experimental artefacts or thermal gradients.

During the isothermal analysis conducted at the B2–B19′ transformation peak, it was observed that the IF_Tr vanishes over time, leaving the combination of IF_PT and IF_int. After 30 min, these two components stabilise, representing the inherent IF of the material. For TNC2, the IF_Total dropped from 0.06864 to 0.0144 over the 30 min, showing a decrease of about 79.02% to reach (IF_PT + IF_Int)_B2–B19′. This reduction in IF_Total indicates that once the transient effects of the phase transformation diminish, the material stabilises, and its inherent damping properties (IF_PT + IF_int) become dominant. The attenuation of IF_Total is attributed to two main factors: First, no new interphase forms when the temperature is kept constant in the B2–B19′ transformation region, and the irreversible movement of interfaces contributes to the IF_Tr stops. Second, a stabilisation phenomenon occurs during the martensitic transformation, driven by the rearrangement of lattice defects. Under external stress, these defects accumulate at the interface, strengthening the pinning effect. As a result, the mobility of the interface is gradually weakened until an equilibrium state is achieved (Helbert et al., 2021). Consequently, the IF_Total, which is influenced by the movement of these defects, diminishes under constant conditions (Zeng et al., 2012). Figure 7 shows that the IF values of the austenite (B2) phase remained nearly consistent over the whole isothermal period. The final and steady IF value of 0.0024485 after a 30-min isothermal interval was interpreted as the (IF_Int)_B2. The IF values of the martensite (B19′) phase slightly decreased with an increase in the isothermal interval of approximately 23.48% and reached a steady value after 10 min. The final and steady IF value of 0.01445 after a 30-min isothermal interval was interpreted as the (IF_Int)_B19′. The (IF_Int)_B19′ is higher than (IF_Int)_B2 owing to the prevalence of numerous twin boundaries in the martensite, which can be readily displaced by external stress to accommodate the imposed strain. On the other hand, (IF_Int)_B2 arises only due to the dynamic/static hysteresis of elastic defects leading to low IF (Chang and Hsiao, 2014). These findings have practical implications for the design and application of NiTiCu SMA, enhancing the research’s relevance and applicability.

Strain sweep analysis

The strain sweep test evaluates the combined internal friction values of phase transformation and intrinsic friction (IF_PT+IF_Int)_B2–B19′ under isothermal conditions with different frequencies. Figure 8 shows the Tan δ (IF_PT + IF_Int)_B2–B19′ values against strain analysed at martensite peak temperatures of 323 K. The (IF_PT + IF_Int)_B2–B19′ increases when the applied strain increases, irrespective of the frequency. The increase in (IF_PT + IF_Int)_B2–B19′ at different frequencies (0.5, 1, 5, and 10 Hz) exhibits distinct trends. They vary modestly at lower frequencies (0.5 and 1 Hz), showing 46.7% and 45.8%, respectively. However, as the frequency rises to 5 and 10 Hz, the increase diminishes slightly, from 46.4% to 43.9%. The increase in (IF_PT+IF_Int)_B2–B19′ concerning applied strain is due to the enhanced twin boundary motion and phase transformations. At lower frequencies, the TNC2 exhibits high (IF_PT+IF_Int)_B2–B19′ due to more free movement of martensitic interfaces and twin boundaries. At higher frequencies, the movement of these interfaces is restricted, reducing (IF_PT+IF_Int)_B2–B19′ (Chien et al., 2015; Radhamani and Balakrishnan, 2023a).

Iterative analysis of internal friction spectrum

The non-isothermal experimental analysis provides quantitative information on IF_Total. As discussed in the introduction, The IF_Total is composed of three different terms, namely IF_Tr, IF_PT, and IF_Int. Separating these IF_Total spectra into their respective components is required to obtain quantitative information like the transformed volume fraction n(T) of changed material. This is possible through isothermal, temperature rate, and iterative methods. Isothermal IF analysis is often challenging and time-consuming, as stabilising the temperature at each data point requires significant effort and precision, making it difficult to gather accurate data across a broad range of temperatures. Similarly, the temperature rate method introduces complexities due to rate-dependent shifts in transformation peaks, and the need to control and maintain consistent temperature rates can obscure critical details of the phase transitions (Pérez-Sáez et al., 1998; San Juan in Schaller et al., 2001). The iterative method is preferred over the isothermal and temperature rate methods because it offers a more efficient and flexible approach to decomposing the IF_Total spectrum. However, it also requires an initial assumption of IF_Int. The choice of the initial IF function strongly affects convergence, as a poor assumption may lead to slow convergence or failure. This can be minimised by making an initial guess based on previous experimental trends or models found in the literature and by defining a piecewise initial IF function for different temperature ranges to ensure smooth behaviour. A poorly chosen temperature range can lead to inaccurate intrinsic friction estimation, while noise in experimental data may propagate through iterations, reducing accuracy. It is essential to ensure adequate background coverage by choosing a temperature range that encompasses baseline IF behaviour, expanding the dataset if necessary, and implementing noise reduction methods and outlier detection to minimise artefacts and enhance stability. The initial IF_Int function used in this process is selected based on one of the three models proposed by Pérez-Sáez et al., each employing the same temperature range (T_s to T_f) to ensure adequate coverage of the background regions (Pérez-Sáez et al., 1998). The process begins with the experimental IF spectrum measured over this range. To minimise the influence of experimental noise on the iterative decomposition results, Savitzky-Golay smoothing was applied to the raw DMA data using OriginPro software. This method was chosen for its ability to preserve key features such as peak shape and inflexion points while effectively reducing high-frequency noise (Liu et al., 2016). The smoothed dataset was then used as the input for the iterative analysis to ensure reliable and stable convergence. The iterative decomposition procedure is clearly outlined in Section 2.7 of the manuscript.

Table 4 summarises the convergence rates of TNC2. After eight iteration cycles, the convergence parameter falls below the assumed experimental error of the IF data, set at 1.E−5. To demonstrate the mathematical convergence more clearly, the table lists values until the convergence rate drops below 1.E−10. Figure 9(a) and (b) illustrates the carefully analysed IF_Int contribution, transformed volume fraction n(T), and IF_Tr(T)+ IF_PT(T) contributions at an oscillation frequency of 1 Hz, utilising the iterative technique. The transformation temperatures can be easily derived from the n(T) curve, and in many technological applications, these temperatures are determined at specific transformation percentages. The transformation temperatures displayed in Figure 9(a) are identified at 0.05 and 0.95 of n(T) to avoid uncertainties that might happen at the beginning or end of the transformation process.

Table 4.

Convergence rate of TNC2 SMA at 1 Hz frequency.

No of iteration	Convergence rate
No of iteration	TNC 2
1	0.0032
2	0.00173
3	6.61837E−4
4	3.12076E−4
5	1.18332E−4
6	4.19E−5
7	1.31E−5
8	3.91E−6
9	1.13E−6
10	3.29E−7
11	9.73E−8
12	2.92E−8
13	8.77E−9
14	2.63E−9
15	7.86E−10
16	2.35E−10
17	0

The relevant transient plus phase transformation term (IF_Tr(T) + IF_PT(T)) can be readily computed by deducting the intrinsic internal friction (IF_Int) term from the original IF (IF_Total) spectrum. Figure 9(b) shows the IF_Tr(T) + IF_PT(T) contributions with a peak value of 0.05284. Since the phase transition term is negligible, a reliable approximate transient IF (IF_Tr) term can be determined (Pérez-Sáez et al., 1998). The relative error of peak values between the experimental and predicted IF spectra at 1 Hz frequency is 2.58% for TNC2. The anticipated IF spectrum closely corresponds with the experimental findings, indicating that this predictive approach for the IF spectrum is feasible. Extending the iterative analysis to other frequencies (0.5, 5, and 10 Hz) yields similar results. Table 5 summarises the convergence rates of TNC2 at 0.5, 5 and 10 Hz frequencies. Figure 10(a)–(c) illustrates the carefully analysed IF_Int contribution, corresponding to the transformed volume fraction n(T), and transient plus phase transformation IF contributions (IF_Tr(T) + IF_PT(T)) of TNC2 SMA at an oscillation frequency of 0.5, 5, and 10 Hz, utilising the iterative technique. At 0.5 Hz, the convergence rate stabilised after six iteration cycles, with a relative error of 1.19%. For 5 Hz, convergence was achieved after eight cycles, with a relative error of 2.98%. At 10 Hz, the convergence also occurred after eight cycles, showing the highest accuracy with a relative error of 3%. These findings demonstrate consistent convergence behaviour across all frequencies, with the model’s predictive accuracy remaining reliable and within acceptable limits.

Table 5.

Convergence rate for TNC2 SMA at 0.5, 5, and 10 Hz frequency.

Iteration	Convergence rate
Iteration	0.5 Hz	5 Hz	10 Hz
1	0.003046291	0.003086536	0.002932543
2	0.001357118	0.001619382	0.001491629
3	0.000406906	0.000608692	0.000548539
4	0.000142666	0.000291812	0.000252088
5	3.84E−05	0.000118598	9.78E−05
6	9.91E−06	4.86E−05	3.80E−05
7	2.36E−06	1.94E−05	1.44E−05
8	5.73E−07	7.89E−06	5.53E−06
9	1.43E−07	3.25E−06	2.16E−06
10	3.64E−08	1.35E−06	8.45E−07
11	9.33E−09	5.62E−07	3.32E−07
12	2.38E−09	2.33E−07	1.30E−07
13	6.01E–10	9.69E−08	5.10E−08
14	1.52E−10	4.02E−08	2.00E−08
15	0	1.67E−08	7.83E−09
16		6.92E−09	3.07E−09
17		2.87E−09	1.20E−09
18		1.19E−09	4.71E−10
19		4.95E−10	1.85E−10
20		0	0

This decomposition enhances the understanding of their distinct contributions to the total IF signal. By isolating these components, this study provided a comprehensive framework for analysing damping mechanisms in NiTiCu SMAs, particularly in engineering applications where temperature stabilisation minimises transient damping effects. These findings contribute to optimising NiTiCu SMAs for high-performance structural applications, where controlled damping is crucial for longevity and functionality. A more detailed study on the evolution of IF under cyclic loading is necessary to assess the long-term stability and functional fatigue of NiTiCu SMAs. Investigating how different frequencies and strain amplitudes affect IF degradation over multiple thermal/mechanical cycles would provide valuable insights for real-world applications.

Conclusion

This study provides a comprehensive analysis of the IF behaviour of Ti₅₀Ni₄₈Cu₂ SMA through various tests, and the results highlight several key findings:

Dynamic mechanical analysis of the TNC2 alloy, conducted at a frequency of 1 Hz, amplitude of 5 microns, and a cooling rate of 5 K/min, revealed that the IF_Total reached a value of 0.0686 at a peak temperature of 323.4 K.

The IF_Total decreases significantly with increasing frequency, dropping by 15.97%. This reduction is attributed to the restricted movement of martensitic interfaces and twin boundaries, demonstrating the time-dependent nature of these internal mechanisms. A higher decrease was observed between lower frequencies (0.5–1 Hz) and higher ones (1–10 Hz), suggesting a non-linear response to frequency changes.

In the isothermal analysis during the B2–B19′ phase transformation, a substantial reduction in IF_Total was observed, decreasing by 79.02% over a period of 30 min. The IF_Tr vanishes as the phase transformation stabilises, leaving the IF_Int to dominate. This stabilisation indicates the cessation of irreversible interphase movements, making the material’s inherent damping behaviour more pronounced.

The (IF_PT + IF_Int)_B2–B19′ increases with applied strain, with more pronounced effects at lower frequencies due to the more unrestrained movement of martensitic interfaces.

The iterative decomposition method effectively separated the IF_Total into IF_Int and IF_Tr, enabling accurate calculation of the transformed volume fraction n(T). The method converged after eight iterations with minimal error (below 1E−5), and the predicted IF spectrum closely matched experimental results with a peak error of 3%. This demonstrates the method’s reliability for precisely analysing IF contributions in SMAs.

These findings contribute significantly to understanding the damping mechanisms in NiTiCu SMAs and provide a reliable approach for characterising their behaviour across different operational conditions.

Footnotes

ORCID iD

Rajeshkannan Radhamani

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

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

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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