Finite element analysis of stress concentration in pseudoelastic thin sheets considering phase transformation and plasticity

Abstract

Shape memory alloys (SMAs) are unique materials widely used in actuators, intelligent structures, and mechanical vibration absorbers due to their distinctive properties. Designing SMA mechanical components can present a challenge due to the presence of complex stress distributions field promoted by the combination of geometric discontinuities, such as holes or notches, and phase transformations that can induce stress redistribution. The use of stress concentration factors (K_t) is a traditional approach that provides a simple method for design calculations of SMA components. This study investigates the stress concentration present in 1–mm-thick pseudoelastic sheets with a hole and evaluates the stress concentration factors for this geometry. The effects of different hole diameters (3, 6, 9, and 12 mm) are investigated using a finite element model that incorporates pseudoelasticity and plasticity to evaluate the stresses, strains, and phase transformation fields. Experimental data, obtained from standardized specimens with identical heat treatment but different cyclic training strains (6.5 and 10%), are used to calibrate the constitutive model parameters. Numerical results are compared to experimental data showing good agreement. Stress concentration analysis reveals that, while during the initial loading phase K_t values are similar to the elastic analytical value, for higher stress levels both phase transformation and plasticity affect the value of K_t which presents a variation throughout the loading history. Finally, the training process was investigated showing that it can affect hysteresis and K_t values. Results indicate that the proposed methodology for estimating K_t values in SMAs can be a useful tool to assist the design of pseudoelastic components.

Introduction

Shape memory alloys (SMAs) are versatile materials classified as multifunctional, meaning they serve not only as structural components but also as sensors and actuators [1] as well as functional applications [2, 3–4]. These alloys are known for their ability to convert non-mechanical stimuli, such as thermal or magnetic inputs, into mechanical responses [1].

The primary effects in SMAs—shape memory and pseudoelasticity—are associated with phase transformation processes. Two main phases are present: austenite, which has a more ordered crystalline structure and is stable at high temperatures (above the final austenite transformation temperature, A_f) in a stress-free state, and martensite, which has a more complex crystalline structure and is stable at low temperatures (below the final martensite transformation temperature, M_f) in a stress-free state [1]. Martensite phase has two forms: The twinned one appears in the absence of mechanical loading, while the detwinned forms occur in the presence of mechanical loading. Additionally, in specific cases, a rhombohedral structure known as the R-phase can form. Among the various types of SMAs, nickel–titanium (NiTi) alloy is widely used [5]. This study focuses on the pseudoelastic effect in NiTi SMA thin sheets.

Pseudoelastic parts have austenite as the stable phase in a stress-free state for temperatures above the final austenite transformation temperature (A_f). Upon the application of mechanical loading, detwinned martensite forms, promoting a phase transformation strain that can be fully recovered during unloading, returning to the austenite phase [1].

Pseudoelastic phase transformation is a complex phenomenon, and its mechanical behavior needs to be understood in order to deal with stress concentration problems. There are several experimental techniques used to explore the phase transformation phenomena of this material. Delpueyo et al. [6] reviewed full-field techniques for thermomechanical characterization of SMAs, including methods such as infrared thermography (IRT), digital image correlation (DIC), grid method (GM), Moiré interferometry (MI), and heat source reconstruction (HSR). The study highlights new approaches using various techniques that have increasingly been employed to study the thermomechanical behavior of these alloys.

Regarding the study of stress concentration, numerical simulations have proven to be a useful technique for evaluating aspects of interest. Several recent studies deal with constitutive models able to represent the main behaviors related to shape memory alloys, such as phase transformation, tension–compression asymmetry, thermomechanical coupling, transformation-induced plasticity (TRIP), and plasticity [5, 7, 8–9] as well as functional and structural fatigue [10].

Transformation-induced plasticity (TRIP) is an important effect to be considered during the training phase and throughout the service life of shape memory alloys, where inelastic transformations occur during cyclic loading. This effect is responsible for residual stresses and strains, which influences the reduction of the component’s service life [7, 11].

Heat treatments can be used to promote changes in the microstructure and adjust the pseudoelastic behavior for a specific condition. Chen et al. [12] investigated geometrically graded pseudoelastic sheets with various heat treatments, focusing on stress–strain behavior and functional stability. Kong et al. [13] examined the combined effect of annealing temperature and cold working on the pseudoelastic behavior of NiTi SMA wires.

Some studies refer to specific behaviors of pseudoelastic components analyzing how they influence the monotonic and cyclic behaviors of the material. Li et al. [14, 15] focused on Lüders-type deformation in pseudoelastic thin sheets, concluding that Lüders bands induce high local strain rates, which affect the overall response and cause negative elastic strain contraction in other regions. Other studies in the literature deal with manufacturing [16] and measurement of mechanical properties [17, 18], showing the complex characteristics of this material.

The study of stress concentration factors (SCF) is an important tool for designing notched mechanical components, using traditional analytical models based on stress analysis, crack initiation, and propagation [19, 20]. Stress concentration is also a key parameter for analyzing damage behavior [21].

Several studies in the literature address the phenomenon of stress concentration in pseudoelastic thin sheets with holes. Murasawa et al. [22] used the Brinson constitutive model [23] to simulate the behavior of a 1-mm pseudoelastic sheet with double U-shaped notches using finite element analysis. Experimental stress–strain curves for both notched and unnotched parts were analyzed, revealing differences between experimental and numerical simulations, which were attributed to local phase transformation strain bands not described by the numerical model. Zhu et al. [24] studied NiTi thin plates with hole arrangements, using the Stebner–Brinson model to simulate stress distribution and martensite fractions. Their study highlighted the need to consider plasticity for an accurate analysis. A follow-up article [25] examined the effect of plasticity on pseudoelastic sheets with hole arrays in different configurations, showing that clustered holes lead to more distributed stresses, while dispersed holes create locally higher plasticity effects.

Shariat et al. [26] conducted numerical simulations of pseudoelastic sheets with holes of various sizes, spacings, and orientations, finding that deformation behavior depends on hole number and size. In a subsequent study [27], Shariat and Liu proposed a constitutive model calibrated with experimental data to obtain stress–strain curves for pseudoelastic and steel sheets with holes, concluding that increasing porosity reduces the elastic modulus of both austenite and martensite during loading and unloading. Larger hole sizes create greater stress gradients between holes and edges, with phase transformation redistributing stress near the holes. Shariat et al. [28] proposed an analytical model for describing the stress–strain curve of a sheet with a circular hole, which showed good agreement with experimental data. Xiao [29] studied the effect of double-edge semicircular notches on pseudoelastic NiTi plates, finding that deeper notches lead to greater localized strains and more pronounced phase transformation profiles.

Zitouni et al. [30] investigated functional fatigue in SMAs near drilled holes using DIC, concluding that the DIC technique is useful for evaluating functional fatigue and the transition to structural damage. Gu et al. [31] explored shape optimization of pseudoelastic sheets with holes, demonstrating that optimized configurations can significantly improve fatigue life. Hosseini and Ashrafi [32] performed finite element simulations using a constitutive model that accounts for martensitic transformation and plasticity to calculate the stress concentration factor in pseudoelastic sheets, revealing that local stresses near holes increase during phase transformation, potentially leading to plastic yielding and affecting functional fatigue.

Silva et al. [33, 34–35] used finite element modeling to study phase transformation-induced stress redistribution in thin sheets with stress concentrations under cyclic loading, proposing new criteria for calculating stress and strain concentration factors. Their studies indicated that heat treatment influences cyclic stress–strain behavior, such as the stabilized hysteretic loop achieved after training, due to its effect on critical stresses and temperatures.

The primary objective of this study is to investigate the stress concentration phenomena in a NiTi pseudoelastic 1-mm-thick sheet with different hole sizes and training strains. A numerical model based on the finite element method is presented to analyze stabilized stress–strain curves, considering the coupling between plasticity and martensitic phase transformation strain fields. Experimental data obtained from DSC and tensile tests were used to calibrate the model parameters. Two different stabilized stress–strain curve conditions are used to examine the effect of two different training conditions on the hysteresis (related to energy dissipation capacity) of the pseudoelastic thin sheet.

This paper is organized as follows: Sect. 2 presents the description of the experimental procedures used for the thermomechanical characterization of the material. Section 3 details the numerical model, constitutive model, experimental–numerical calibration, and numerical analysis. Section 4 presents the analysis of the stress concentration factors in thin plates with different hole diameters. The conclusions are presented in Sect. 5.

Experimental procedure

Thermomechanical characterization of the material is performed using differential scanning calorimetry (DSC) and tensile tests. Specimens were obtained from pseudoelastic equiatomic NiTi sheets (purchased from Nexmetal Corporation) with a length of 200 mm, a width of 100 mm, and a thickness of 1 mm. All the specimens were prepared by wire electrical discharge machining (EDM).

The pseudoelastic sheet showed reduced pseudoelastic behavior with a small hysteresis loop in the as-received condition, so it was necessary to perform a heat treatment to improve the pseudoelastic hysteresis throughout the cycles. Therefore, a heat treatment at 530 °C for 10 min followed by rapid quenching in water was performed. This condition was chosen after an experimental study considering different furnace temperatures and times.

Figure 1 shows a DSC analysis (Netzsch DSC200 F3) of a test specimen removed from the sheet after the heat treatment. The following transformation temperatures were obtained: A_s = −20.1 °C; A_f = 1.6 °C; R_s = −10.3 °C; R_f = −36.8 °C; M_s = −51.7 °C; and M_f = −77.3 °C, where A_s, A_f, R_s, R_f, M_s, M_f are, respectively, the austenite start transformation temperature, austenite finish transformation temperature, R-phase start transformation temperature, R-phase finish transformation temperature, martensite start transformation temperature, and martensite finish transformation temperature. For the DSC test, a temperature range between −150 °C and 120 °C was used, with a constant heating/cooling rate of 10 °C/min.

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Fig. 1

DSC result for the test specimen after the heat treatment at 530 °C for 10 min and cooling in water

Tensile test specimens were obtained from a pseudoelastic NiTi sheet according to ASTM E8 standard [36], with 1 mm thickness, 6 mm width, and a total length of 100 mm. Figure 2 shows a drawing with dimensions and a photograph of the test specimen used for the tensile tests.

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Fig. 2

ASTM E8 tensile specimen: a drawing with dimensions in mm, b picture of specimen after the heat treatment at 530 °C for 10 min and cooling in water

The specimen was tested on a uniaxial electromechanical testing machine Instron 5985 equipped with a 50-kN load cell, at room temperature. Displacement was applied through a sequence of quasi-static triangular waves, at a rate of 1% per minute. Strain was measured from a non-contacting video extensometer (Instron model AVE 2) using 25 mm between marks for initial length. The specimens loading program includes 3 loading stages. The first stage promotes the specimen training by applying ten cycles with a prescribed maximum strain during the loading and a minimum stress of 7 MPa during unloading. Two conditions are considered for the training maximum strain values during the loading stage: 6.5% and 10% (one specimen for each condition). After that, one cycle of prescribed strain was applied until a value of 15% is reached, followed by unloading until the stress 7 MPa is reached, a minimum stress limit is necessary to prevent the buckling of the test specimen. Finally, the last loading stage is applied until the fracture. The stress–strain curves of the complete test of the two conditions are presented in Fig. 3.

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Fig. 3

Cyclic stress–strain curve: a 6.5% training, b 10% training of standardized (ASTM E8) specimens after heat treatment

The evolution of the transformation-induced plasticity (TRIP) strain for the first 10 cycles of both specimens loading conditions is presented in Fig. 4. For this study, the cycles are considered stabilized when the difference in residual strain between 9 and 10th cycles is less than 5%. It is observed that the variation of the residual strain between the 9th and 10th cycles is equal to 4.2% and 1.0% for the 6.5% and 10% training specimens, respectively, which can be considered that a stable stress–strain hysteresis is reached, and the material is in a trained condition.

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Fig. 4

TRIP strain as a function of the number of cycles until stabilization for specimens trained with 6.5 and 10% strain

Numerical model

This section describes the methodology used to establish the numerical model. First, the constitutive model employed in the finite element analysis is presented. Subsequently, the determination of numerical parameters and the calibration between experimental and numerical results are discussed. Finally, the numerical analysis of plates with holes is presented.

Constitutive model

Several initiatives have been developed to propose constitutive models to describe the behavior of shape memory alloys. Some models have been proposed, with a special focus on numerical implementation, particularly the finite element method [5]. The literature review presented in [7] analyzes advanced SMA modeling techniques and highlights the importance of studying specific behaviors such as TRIP and plasticity [10, 37, 38–39], in addition to phase transformation, in this type of material.

To study the stress concentration phenomena of a pseudoelastic thin sheet with holes, a numerical model based on the finite element method is developed. The Auricchio model [40] was chosen to represent the pseudoelastic behavior of the material. It is a well-established model, which is available in several commercial softwares, such as Abaqus, Ansys, LS-Dyna [5, 8], and therefore can be applied in the study of complex geometries.

The model considers isothermal pseudoelasticity and utilizes the Drucker–Prager criterion to describe critical stresses for pseudoelastic transformation [1]. The model also incorporates the effect of plasticity after the martensitic transformation reaches the yield limit. The formulation, implemented in the computational package Abaqus [41], is based on the additive decomposition of strains, where the total strain is found by the sum of the elastic, phase transformation, and plastic strain components [5]:

1

$$\mathrm{\Delta }\epsilon =\mathrm{\Delta }{\epsilon }^{\mathit{el}}+\mathrm{\Delta }{\epsilon }^{\mathit{tr}}+\mathrm{\Delta }{\epsilon }^{\mathit{pl}}$$

where $\mathrm{\Delta }\epsilon$,$\mathrm{\Delta }{\epsilon }^{\mathit{el}}$, $\mathrm{\Delta }{\epsilon }^{\mathit{tr}}$ e $\mathrm{\Delta }{\epsilon }^{\mathit{pl}}$ are the total, elastic, phase transformation, and plastic strain increments, respectively.

The constitutive model considers that the elastic modulus and Poisson's ratio under intermediate transformation conditions obey the rule of mixtures:

2

$$E=E_A+\varsigma \left(E_M,-,E_A\right)$$

3

$$\nu ={\nu }_A+\varsigma \left({\nu }_M,-,{\nu }_A\right)$$where$E$,$E_A$, $E_M,\varsigma$,$\nu$,${\nu }_A$,${\nu }_M$ are, respectively, the resultant, austenitic, and martensitic elastic modulus, martensitic volume fraction, the resultant, austenitic, and martensitic Poisson ratio.

The initial and final critical transformation stresses during load (${\sigma }_{\mathit{tL}}^S$, ${\sigma }_{\mathit{tL}}^E$) and unload (${\sigma }_{\mathit{tU}}^s$, ${\sigma }_{\mathit{tU}}^E$), and transformation strain (${\epsilon }^L$) are required for the constitutive model.

The incremental phase transformation strain is calculated according to the rule:

4

$$\mathrm{\Delta }{\epsilon }^{\text{tr}}=\mathrm{\Delta }\varsigma \frac{\partial G^{\text{tr}}}{\partial \sigma }$$where $G^{\text{tr}}$ is the transformation flow potential, given in the form of the Drucker–Prager model:

5

$$G^{\text{tr}}=q-p\text{tan}\psi$$where p is the hydrostatic pressure and q of von Mises equivalent stress, defined as:

6

$$p=-\frac{1}{3}\text{tr}\sigma$$

7

$$q=\sqrt{\frac{3}{2}S:S}$$

The phase transformation surface ($F^{\mathit{tr}}$) is given by the Drucker–Prager equation and varies linearly with temperature:

8

$$F^{\text{tr}}=q-p\text{tan}\beta$$

Model calibration

A numerical model representing the tension test specimen was used to determine the parameters to be used in the finite element model. The constitutive model for isothermal pseudoelastic behavior has the possibility of including the hardening effect during the phase transformation process. The determination of model parameters involves a fitting process of the experimental data [42] considering stress–strain curves of the 10th cycle, the 15th cycle, and rupture cycle presented in Fig. 3. The parameters of the constitutive model obtained are presented in Table 1 (phase transformation) and Table 2 (plasticity).

Table 1. Parameters of the phase transformation constitutive model

Specimen	EA (MPa)	νA	EM (MPa)	νM	εr	${\sigma }_{\mathit{tL}}^S$(MPa)	${\sigma }_{\mathit{tL}}^E$(MPa)	${\sigma }_{\mathit{tU}}^S$(MPa)	${\sigma }_{\mathit{tU}}^E$(MPa)	${\sigma }_{\mathit{cL}}^S$(MPa)	T0 (K)	CA (MPa/K)	CM (MPa/K)
6.5% training	30,180	0.3	25,000	0.3	0.018	250	408	272	50	250	323	7	6.5
10% training	30,180	0.3	25,000	0.3	0.02	185	500	300	50	250	323	7	6.5

Table 2. Parameters of the plasticity model

6.5% training		10% training
Sy (MPa)	Total strain (mm/mm)	Sy (MPa)	Total strain (mm/mm)
745	0.08	673	0.05
880	0.10	853	0.10
1072	0.30	1030	0.30

Figure 5 shows a comparison between the stress–strain curves obtained from the numerical model and the experimental data, where a good correspondence is observed.

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Fig. 5

Comparison between the stress–strain curves obtained from the numerical model and the experimental data: a 6.5% training, b 10% training of standardized specimens after heat treatment

Numerical analysis of plates with holes

For the stress concentration study, a quasistatic finite element model of a plate with 20 mm width and 100 mm length containing a circular hole at its center was developed. Four different hole diameters are considered in the analysis: 3, 6, 9, and 12 mm. For model simplification, double symmetry was employed. The mesh shown in Fig. 6, obtained after a convergence analysis, uses quadratic plane stress elements (CPS8). A triangular (linear) loading–unloading prescribed displacement is applied at the upper edge of the sheet in the vertical (axial) direction, with an amplitude of 1 mm and a total time duration of 2 (dimensionless time). Displacement boundary conditions are applied to the left (fixed displacement in the horizontal direction) and lower (fixed displacement in the vertical direction) edges to represent the double symmetry condition.

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Fig. 6

Mesh of the finite element models with different diameters: a 3 mm, b 6 mm, c 9 mm, d 12 mm considering a quarter symmetry

Axial stress–strain curves for both training conditions considering phase transformation and plasticity (TR + PL) for all hole diameters obtained at the region of maximum axial stress are presented in Fig. 7. It can be noted that the conditions for 6.5% training strain cycles have lower strain values compared with the ones of the same diameter for 10% training. Higher values of residual stress after unloading are observed for 6.5% training condition, particularly in smaller diameters and similar values for diameter 12 mm.

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Fig. 7

Axial stress–strain curves for the local of maximum stress for plates with different hole diameters: a 6.5% training, b 10% training

The axial nominal stress–strain curves are presented in Fig. 8. It can be noted that the maximum nominal stress values for the 10% training condition are similar (for diameters 9 and 12) or lower (for diameters 3 and 6) when compared to 6.5% training condition.

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Fig. 8

Axial nominal stress–strain curves for plates with different hole diameters: a 6.5% training, b 10% training

Axial stress distribution for the models with 10% training at the end of the loading stage (dimensionless time equal to 1) is presented in Fig. 9. It is possible to observe that the stress values near the point of maximum stress are higher for parts with larger diameters. A similar behavior can be found for the results obtained for 6.5% training.

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Fig. 9

Axial stress distribution (in MPa) for plates with different hole diameters at the end of the loading stage: a 3 mm, b 6 mm, c 9 mm, d 12 mm

The distribution of martensitic volume fraction at the end of the loading (dimensionless time equal to 1) is present in Fig. 10. It can be observed that the region experiencing martensitic transformation expands as the hole diameter increases,; this is related to the stress distribution around the hole.

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Fig. 10

Martensitic volume fraction distribution for plates with different hole diameters at the end of the loading stage: a 3 mm, b 6 mm, c 9 mm, d 12 mm

Axial total strain at the end of the loading stage is present in Fig. 11. As the previous results, the strain distribution region around the hole increases as the diameter increases.

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Fig. 11

Axial total strain distribution for different diameters at the end of the loading stage: a 3 mm, b 6 mm, c 9 mm, d 12 mm

To study the influence of plasticity, two scenarios were analyzed: a case with phase transformation (TR) and a case with both phase transformation and plasticity (TR + PL). The plasticity behavior is included in the analysis for the (TR + PL) scenario by activating the plasticity constitutive model in the Abaqus finite element package. The stress–strain curves for both cases at the point of maximum axial stress are presented in Fig. 12 for a typical condition (10% training, 9 mm hole diameter, 1 mm displacement) [34]. It can be observed that the curve considering only phase transformation (TR) develops higher stress levels and lower deformation values. In the case with phase transformation and plasticity (TR + PL), the plastic strain contributes to amplifying the stress redistribution and a localized reduction in the stress is observed. Additionally, the curve of the case with plasticity exhibits residual compressive stress upon unloading, a characteristic behavior associated with displacement-controlled tests with the presence of plastic strain; this behavior is similar to the elastic–plastic behavior.

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Fig. 12

Stress–strain curves at region of maximum axial stress: case with phase transformation (TR) and case with phase transformation and plasticity (TR + PL) for plates with 9 mm hole diameter and 10% strain training

The distribution of total strain and the components of elastic, phase transformation, and plastic strain at the end of loading stage for the case considering phase transformation and plasticity are presented in Fig. 13. It can be noted that the phase transformation strain is lower than the elastic component. Furthermore, it is observed that the plastic strain (1.3%) is the smallest of the components analyzed and is confined to a very small region.

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Fig. 13

Axial strain distribution at the end of the loading stage for plates with different hole diameters: a total, b elastic, c phase transformation, d plastic

Figure 14 presents the time evolution of axial stress result (S22), martensite volume fraction (MVF), and axial plastic strain (PE22) during the loading stage (time equal to 1) at the edge of the hole for the typical condition, previously analyzed (diameter of 9 mm with 10% training strain). It can be noted that at time instant of 0.18, the phase transformation process begins, promoting the reduction of the stress rate. In time instant of 0.68, the plastic strain evolution initiates and promotes a further reduction of the stress rate.

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Fig. 14

Time evolution of the axial stress result (S22), martensite volume fraction (MVF), and the axial plastic strain (PE22) during loading stage at the edge of the hole (diameter 9 mm with 10% training strain)

Stress concentration factor

The classic stress concentration factor (SCF) for elastic behavior in a plate with a hole can be calculated using the analytical equation proposed in [43]:

9

$$K_t=3.00-3.13\left(\frac{2r}{D}\right)+3.66{\left(\frac{2r}{D}\right)}^2-1.53{\left(\frac{2r}{D}\right)}^3$$

For cases involving phase transformation and plasticity, an estimate for the SCF can be obtained using the following equation:

10

$$k_t=\frac{{\sigma }_{\text{max}}}{{\sigma }_{\text{nom}}}$$where ${\sigma }_{\mathit{max}}$ is the stress at the point of maximum stress, and ${\sigma }_{\mathit{nom}}$ is the nominal stress considering the net area, i.e., area of smallest cross section.

The evolution of the stress concentration factor throughout loading for the elastic analytical case (EA), estimated by Eq. (4), and for the cases considering phase transformation (TR) and phase transformation and plasticity (TR + PL) is shown in Fig. 15 for 6.5% and 10% training conditions.

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Fig. 15

Evolution of the stress concentration factor during loading for plates with different hole diameters: a 6.5% training, b 10% training

It can be observed that at the beginning of loading, while the phase transformation process has not yet initiated, and the specimen is fully austenitic, the value of the SCF is close to the one estimated by the analytical model for the elastic case. As the load increases, phase transformation initially develops in the region of maximum stress (time instant of 0.18 for diameter 9 mm with 10% training strain), resulting in martensite formation in that region, while the rest of the specimen remains in the austenitic phase. This phase transformation leads to stress redistribution in the region near the hole and, consequently, a reduction in the value of the SCF. When martensitic phase transformation develops in the rest of the specimen, the SCF starts to increase again. Yielding promotes a second stress redistribution (time instant of 0.68 for diameter 9 mm with 10% training strain), and the curves begin to differentiate. For the case considering only phase transformation, the SCF continues to increase, while the curve for the case considering phase transformation and plasticity shows a reduction in the SCF value after the onset of yielding.

Results shown in Fig. 15 indicate that plasticity leads to a reduction in the stress concentration factor due to stress redistribution caused by the plastic deformation near the hole. This result is consistent with results obtained by other authors [32]. It can be noted that in comparison with the 6.5% training, the TR + PL results for the 10% training condition show a decrease in SCF for hole diameters of 6, 9, and 12 mm after the plastic strain takes place. This behavior is associated with the yielding that occurs due to the larger stress levels reached for these three conditions.

The stress concentration evolution curves obtained from the numerical simulations shown in Fig. 15, indicate an influence of the training strain on the K_t factor value. Figure 7 shows that higher stress levels are observed for the 10% training level, and yielding occurs at a lower strain levels. Additionally, the variation in K_t is found to be comparatively smaller under this condition. This suggests that the training strain plays an important role in determining the mechanical behavior of the material, particularly in terms of stress distribution. At the end of loading stage, K_t values for the TR + PL curves are consistently lower than those for the TR curves due to the effect of plasticity.

For the case considering phase transformation and plasticity (TR + PL), the 6.5% training shows final K_t values at the end of the loading stage from 1.76 (D12 mm) to 1.98 (D3 mm), while for 10% training final K_t values at the end of the loading stage go from 1.74 (D12mm) to 2.26 (D3 mm). For the case considering only phase transformation (TR), larger values are observed: between 2.5 and 2.7 for 6.5% training and from 2.3 to 2.4 for 10% training.

In the same way as observed for an elastic condition, the diameter of the hole influences the K_t value, with larger diameters promoting lower K_t values. The phase transformation promotes the reduction of the K_t value in the beginning of the phase transformation process. A recovery of the K_t value is observed when a larger portion of the part experiences phase transformation, followed by a subsequent reduction after yielding.

The results show that the training process can also affect the K_t value, as the strain amplitude applied during training can modify the hysteresis and TRIP behaviors and, therefore, the stress distribution. When comparing the values obtained for 6.5 and 10% training deformation, the smallest difference between the K_t values is observed for the hole diameter of 12 mm (+ 1,15%), while the largest difference is observed for the diameter 3 mm (−12,44%).

It is important to mention that in displacement-controlled tests, the longitudinal force depends on specimen global stiffness. Also, the nominal stress is influenced by the remaining area (net area at the smaller cross section). Therefore, as the hole diameter increases the force necessary to reach a specific displacement is lower because of the lower global stiffness, but the net area is smaller, producing two concurrent effects. The result presented in Fig. 7 shows a decrease in the maximum nominal stress, as the hole diameter increases, indicating that the cross-sectional area has a greater influence.

Plasticity has a significant effect on the K_t factor, with plastic deformation leading to a reduction in the K_t value. This reduction is attributed to material softening following yielding, which attenuates the increase in K_t associated with phase transformation.

Conclusions

This study investigates the effect of stress concentration in pseudoelastic thin sheets with holes with a thickness of 1 mm. The study considered four different hole diameters and two distinct training histories. A finite element numerical model is used to analyze the stress and phase transformation fields and estimate stress concentration factors. An experimental procedure was developed to characterize the material and determine the constitutive model parameters. A subsequent finite element analysis was performed on a pseudoelastic thin plate with a hole to compare the effects of phase transformation and plasticity on the stress concentration factor (K_t).

The stress concentration factor exhibits complex behavior throughout the loading process due to coupling between phase transformation and plasticity, which redistributes the stress field. Initially, in the elastic region, a K_t value similar to the classic elastic analytical solution for a plate with a hole is observed. For higher stresses values, during phase transformation process, K_t first decreases, followed by a recovery, as a larger portion of the material undergoes phase transformation. Some regions that fully transform in detwinned martensite exhibit a second elastic behavior. After yielding, another reduction in K_t is observed due to stress redistribution.

The influence of training and stabilized stress–strain response on K_t is also analyzed. Results show that for the 6.5% strain training condition, the TR + PL case results in lower K_t values at the end of loading compared to the 10% strain condition. For hole diameters of 3, 6, and 9 mm, the following reductions are observed: 12.44%, 9.52%, and 3.68%, respectively. For the 12 mm diameter, a slight increase in K_t (1.15%) is observed. These results suggest that comparing the results of training with 6.5% and 10% strain, the 6.5% strain condition is more suitable for practical applications because it results in lower values of stress concentration factor (K_t).

Designing SMA components presents challenges due to the material’s complex behavior, which is further complicated by the redistribution of stresses from phase transformation and plasticity. These components are often subjected to cyclic loading, which can lead to structural and functional fatigue. The results indicate that the proposed methodology offers a useful tool for designing SMA components with stress concentration regions, as it enables the estimation of stress concentration factors.

Acknowledgements

The authors would like to acknowledge the support of CEFET/RJ and the Brazilian Research Agencies CNPq, Capes, Faperj, INCT-EIE of Smart Structures in Engineering and Finep.

Author contribution

The authors are solely responsible for the content of this work.

Data availability

The data sets generated during the current study are available from the corresponding author on reasonable request.

Declarations

Conflict of interest

The authors declare that one of the authors is an Associate Editor of the Journal of the Brazilian Society of Mechanical Sciences and Engineering. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Marcelo A. Savi

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References

1. Lagoudas, D. Shape memory alloys; 2008; Springer:

2. Teshome, FB; Peng, B; Oliveira, JP; Shen, J; Ao, S; Li, H; Chen, L; Tan, C; Song, X; Zhou, N; Zeng, Z. Role of Pd interlayer on NiTi to Ti6Al4V laser welded joints: microstructural evolution and strengthening mechanisms. Mater Des; 2023; [DOI: https://dx.doi.org/10.1016/j.matdes.2023.111845]

3. Cavaleiro, AJ; Oliveira, JP; Ramos, AS; Shen, J; Schell, N; Miranda, RMM; Vieira, MT. Multiscale characterization of NiTi shape memory alloy to Ti6Al4V dissimilar laser welded joints: reasons for inherent brittleness. Opt Laser Technol; 2025; [DOI: https://dx.doi.org/10.1016/j.optlastec.2024.111853]

4. Fu G, Liu X, Yi X, Zhang S, Cao X, Meng X, Gao Z, Wang, H (2023) Development of high-entropy shape-memory alloys: a review. Metals https://doi.org/10.3390/met13071279

5. Lecce, L. Shape memory alloy engineering for aerospace; 2015; Structural and Biomedical Applications, Elsevier:

6. Delpueyo, D; Jury, A; Balandraud, X; Grédiac, M. Applying full-field measurement techniques for the thermomechanical characterization of shape memory alloys: a review and classification. Shap Mem Pseudoelasticity; 2021; 7, pp. 462-490. [DOI: https://dx.doi.org/10.1007/s40830-021-00355-w]

7. Cisse, C; Zaki, W; Ben Zineb, T. A review of modeling techniques for advanced effects in shape memory alloy behavior. Smart Mater Struct; 2016; [DOI: https://dx.doi.org/10.1088/0964-1726/25/10/103001]

8. Mohammadgholipour A, Billah AM (2021) Mechanical properties and constitutive models of shape memory alloy for structural engineering: a review. J Intell Mater Syst Struct (2023). https://doi.org/10.1177/1045389X231185458

9. Paiva, A; Savi, MA; Braga, AMB; Pacheco, PMCL. A constitutive model for shape memory alloys considering tensile-compressive asymmetry and plasticity. Int J Solids Struct; 2005; [DOI: https://dx.doi.org/10.1016/j.ijsolstr.2004.11.006]

10. Dornelas, VM; Oliveira, SA; Savi, MA; Pacheco, PMCL. Numerical investigations of shape memory alloy fatigue. Metals; 2021; [DOI: https://dx.doi.org/10.3390/met11101558]

11. Oliveira, SA; Dornelas, VM; Savi, MA; Pacheco, PMCL; Paiva, A. A phenomenological description of shape memory alloy transformation induced plasticity. Meccanica; 2018; 3819632 [DOI: https://dx.doi.org/10.1007/s11012-018-0836-0]

12. Chen, W; Xi, R; Jiang, H; Li, X; Dong, G; Wang, X. Pseudoelasticity of geometrically graded NiTi shape memory alloys. Metals; 2023; [DOI: https://dx.doi.org/10.3390/met13091518]

13. Kong, X; Zhang, J; Ding, H; Liu, Q; Zhou, J; Qi, F; Wang, P. Combined effect of annealing temperature and pre-deformation on the pseudoelastic behaviors of NiTi shape memory alloys. J Mater Eng Perform; 2024; [DOI: https://dx.doi.org/10.1007/s11665-024-10077-6]

14. Li, Y; Shariat, BS; Yang, H; Liu, Y. Influences of extrinsic factors on the Lüders-type deformation behaviour of NiTi. Mater Sci Eng A; 2023; [DOI: https://dx.doi.org/10.1016/j.msea.2022.144418]

15. Li, Y; Shariat, BS; Yang, H; Sarkar, S; Zhang, J; Wang, Y; Favier, D; Liu, Y. A mechanical criterion for Lüders-type deformation of polycrystalline NiTi. Acta Mater; 2023; [DOI: https://dx.doi.org/10.1016/j.actamat.2022.118604]

16. Simões, JB; Cardins, HMA; Grassi, E; Araújo, CJ. A new way to obtain NiTi SMA pseudoelastic meshes: investment casting followed by hot rolling. J Braz Soc Mech Sci Eng; 2020; [DOI: https://dx.doi.org/10.1007/s40430-020-2203-z]

17. Di Leonardo, S; Pitarresi, G; Burriesci, G. Standard mechanical testing is inadequate for the mechanical characterization of shape-memory alloys: source of errors and a new corrective approach. Mater Des; 2022; [DOI: https://dx.doi.org/10.1016/j.matdes.2022.110538]

18. Shariat, BS; Liu, Y. Variation of Poisson’s ratio of NiTi during pseudoelastic deformation. Shape Mem Superelasticity; 2023; [DOI: https://dx.doi.org/10.1007/s40830-023-00464-8]

19. Hasan, MM; Baxevanis, T. Actuation fatigue life prediction of notched shape memory alloy members. J Appl Mech; 2019; [DOI: https://dx.doi.org/10.1115/1.4042994]

20. Simoes, M; Martínez-Pañeda, E. Phase field modelling of fracture and fatigue in shape memory alloys. Comput Methods Appl Mech Eng; 2021; 4169294 [DOI: https://dx.doi.org/10.1016/j.cma.2020.113504]

21. Phillips, FR; Wheeler, RW; Geltmacher, AB; Lagoudas, DC. Evolution of internal damage during actuation fatigue in shape memory alloys. Int J Fatigue; 2019; [DOI: https://dx.doi.org/10.1016/j.ijfatigue.2018.12.019]

22. Murasawa, G; Yoneyama, S; Miyata, K; Nishioka, A; Koda, T. Finite element analysis and experimental measurement of deformation behavior for NiTi plate with stress concentration part under uniaxial tensile loading. Strain; 2011; [DOI: https://dx.doi.org/10.1111/j.1475-1305.2009.00698.x]

23. Brinson, LC. One-dimensional constitutive behavior of shape memory alloys: thermomechanical derivation with non-constant material functions and redefined martensite internal variable. J Intell Mater Syst Struct; 1993; [DOI: https://dx.doi.org/10.1177/1045389X9300400213]

24. Zhu P, Stebner AP, Brinson C (2013) A numerical study of the coupling of elastic and transformation fields in pore arrays in shape memory alloy plates to advance porous structure design and optimization. Smart Mater Struct. https://doi.org/10.1088/0964-1726/22/9/094009

25. Zhu P, Stebner AP, Brinson C (2014) Plastic and transformation interactions of pores in shape memory alloy plates. Smart Mater Struct. https://doi.org/10.1088/0964-1726/23/10/104008

26. Shariat, BS; Liu, Y; Rio, G. Finite element computational modelling and experimental investigation of perforated NiTi plates under tension. Mater Res Bull; 2013; [DOI: https://dx.doi.org/10.1016/j.materresbull.2013.05.022]

27. Shariat, BS; Liu, Y; Rio, G. Numerical modelling of pseudoelastic behaviour of NiTi porous plates. J Intell Mater Syst Struct; 2014; [DOI: https://dx.doi.org/10.1177/1045389X13505783]

28. Shariat, BS; Liu, Y; Rio, G. Pseudoelastic behaviour of perforated NiTi shape memory plates under tension. Intermetallics; 2014; [DOI: https://dx.doi.org/10.1016/j.intermet.2014.02.016]

29. Xiao, Y; Zeng, P; Lei, L. Effect of double-edge semi-circular notches on the mechanical response of superelastic NiTi shape memory alloy: Experimental observations. J Strain Anal Eng Des; 2016; [DOI: https://dx.doi.org/10.1177/0309324716661460]

30. Zitouni, I; Abuzaid, W; Egilmez, M; Alkhader, M. Experimental assessment of the functional fatigue in biocompatible Ti67Zr19Nb11.5Sn2.5 shape memory alloy in the vicinity of drilled holes. J Mater Res Technol; 2023; [DOI: https://dx.doi.org/10.1016/j.jmrt.2023.10.143]

31. Gu, X; Cao, Y; Wang, JZ; Zhang, W; Moumni, Z. Shape optimization of SMA structures with respect to fatigue. Mater Des; 2020; [DOI: https://dx.doi.org/10.1016/j.matdes.2019.108456]

32. Hosseini, SM; Ashrafi, MJ. Stress concentration in shape memory alloy sheets with circular cavities: effects of transformation, saturation, and plasticity. J Strain Anal Eng Des; 2023; [DOI: https://dx.doi.org/10.1177/03093247221113755]

33. Silva BF, Souza LFG, Aguiar, RAA, Pacheco PMCL (2023) Analysis of stress concentration in pseudoelastic thin sheets using the finite element method. In: Proceedings COBEM (2023). https://doi.org/10.26678/ABCM.COBEM2023.COB2023-0759

34. Silva BF, Souza LFG, Aguiar RAA, Pacheco PMCL, (2024) Análise por Elementos Finitos do Fator de Concentração de Tensão em uma Chapa Fina Pseudoelástica com Transformação de Fase e Plasticidade, Proceedings CONEM 2024, (in Portuguese). DOI://https://doi.org/10.26678/ABCM.CONEM2024.CON24-1178.

35. Silva BF, Souza LFG, Aguiar RAA, Pacheco PMCL, (2024) Analysis of stress concentration in pseudoelastic plates using digital image correlation (DIC) and finite element method, Proceedings Mecsol 2024. DOI://https://doi.org/10.26678/ABCM.MECSOL2024.MSL24-0101.

36. ASTM E8 (2007) Standard test methods for tension testing of metallic materials, ASTM International

37. Oliveira, SA; Savi, MA; Zouain, N. A three-dimensional description of shape memory alloy thermomechanical behavior including plasticity. J Braz Soc Mech Sci Eng; 2016; [DOI: https://dx.doi.org/10.1007/s40430-015-0476-4]

38. Dornelas, VM; de Oliveira, SA; Savi, MA. A macroscopic description of shape memory alloy functional fatigue. Int J Mech Sci; 2020; [DOI: https://dx.doi.org/10.1016/j.ijmecsci.2019.105345]

39. Dornelas, VM; de Oliveira, SA; Savi, MA; Pacheco, PMCL; Souza, LFG. Fatigue on shape memory alloys: experimental observations and constitutive modeling. Int J Solids Struct; 2021; [DOI: https://dx.doi.org/10.1016/j.ijsolstr.2020.11.023]

40. Auricchio, F; Taylor, RL. Shape-Memory Alloys: Modeling and Numerical Simulations of the Finite-Strain Pseudoelastic Behavior. Comput Methods Appl Mech Eng; 1996; [DOI: https://dx.doi.org/10.1016/S0045-7825(96)01147-4]

41. ABAQUS. ABAQUS User Manuals, Version 2018

42. Adeodato, A; Vignoli, LL; Paiva, A; Monteiro, LLS; Pacheco, PMCL; Savi, MA. A shape memory alloy constitutive model with polynomial phase transformation kinetics. Shape Mem Superelasticity; 2022; [DOI: https://dx.doi.org/10.1007/s40830-022-00382-1]

43. Castro JTP, Meggiolaro MA (2009) Fatigue Techniques and Sizing Practices under Actual Service Loads. Createspace Independent Publishing Platform