A novel small punch creep test to determine norton creep properties of LPBF Ti64 alloy

Laser Powder Bed Fusion (LPBF) Ti-6Al-4V (Ti64)alloy is increasingly favored in additive manufacturing (AM) for complex, lightweight, and high-strength components. However, the inherent microstructural heterogeneity in LPBF materials necessitates specialized characterization methods to ensure reliable performance in service conditions. The Small Punch Creep Test (SPCT) has gained recognition as an innovative and efficient technique for assessing materials creep behavior, particularly beneficial for AM components allowing easier sampling and local characterization of microstructure heterogeneities. Although standards such as CWA 15627 and EN 10371 provide empirical correlations between SPCT and uniaxial creep test (UCT) results, the complex deformation mechanisms in LPBF materials, including anisotropy, porosity, and phase evolution, necessitate material-specific calibration for accurate interpretation. This study proposes a direct methodology to derive the Norton creep parameters of LPBF Ti64 alloy from both Small Punch Test (SPT) and SPCT data, integrating a new representative stress-strain method with inverse finite element analysis (FEA). By first extracting elasto-plastic properties from SPT using the representative stress-strain method, the model enables efficient extraction of Norton creep parameters through inverse analysis of SPCT results. Experimental validation on LPBF Ti64 alloy demonstrates strong agreement with properties derived from SPCT, verifying the accuracy and practical applicability of the proposed method for AM components.

1 Introduction

The reliable performance and longevity of engineering components depend on accurate characterization of mechanical properties and ongoing assessment of material degradation under operating conditions (Shi and Jar, 2024b; Omacht et al., 2025; Shi and Jar, 2025). Conventional mechanical tests, such as uniaxial tensile and uniaxial creep tests (UCT), remain the benchmark for reliable property determination, offering well-defined stress states and direct evaluation of yield strength, ductility, and Norton creep parameters (Shi and Jar, 2022; 2024a; Tan and Ben Jar, 2022; Zheng et al., 2022; 2023; 2024). However, they are often destructive, require large, uniform specimens, and are impractical for in-service components where structural integrity must be preserved or for characterizing specific microstructural regions in complex geometries. Non-destructive evaluation techniques, such as ultrasonic testing and magnetic field analysis, excel at detecting defects; however, they are limited in quantifying time-dependent degradation mechanisms, such as creep, which governs long-term performance during service life (Reis et al., 2005).

To address these limitations, Small Sample Test Techniques (SSTT) have emerged as effective alternatives, enabling localized property evaluation from minimal material volumes without impairing component integrity (Zhong et al., 2024). SSTT methods are particularly valuable for site-specific assessment in weld heat-affected zones, protective coatings, or thin-walled structures. Among SSTT, the Small Punch Test (SPT), developed in the early 1980s for the nuclear industries, has become a versatile approach for evaluating yield, creep, and fracture properties using miniature disk specimens (Lucon et al., 2021), particularly when conventional testing is impractical. It is widely recognized as an important tool for assessing the residual life and structural integrity of in-service components. With time, SPT has been extended to numerous materials including nickel-based alloys, Al alloys, Mg alloys, gold alloys, stainless steels, carbon steels, and Ti alloys (Melkior et al., 2023), which has been widely applied to alternative industrial sectors such as petrochemical, aerospace, power plant and automotive (Arunkumar, 2021). However, SPT often relies on empirical correlations, which can lack consistency and necessitate material-specific calibration. The hybrid experimental–numerical method, which combines test data with finite element (FE) simulations, is widely adopted; however, it requires extensive iteration to achieve accuracy, making it computationally demanding and unsuitable for rapid processing of large datasets. The energy principle-based method proposed by Peng et al. (2018) offers a more direct route; however, it struggles with the complex, multiaxial stress state inherent in SPT specimens (Zhou et al., 2024).

The Small Punch Creep Test (SPCT), the load-controlled variant of SPT, is now standardized and widely applied to power-plant steels and nickel-based superalloys (Dobeš and Milička, 2008; Dobeš and Dymáček, 2016; Arunkumar, 2021). However, direct SPCT–UCT equivalence is complicated by the multiaxial, non-uniform stress–strain state of SPCT specimens. Standards such as CWA 15627 and EN 10371 provide empirical conversion relationships (Li et al., 2024); however, these are material-dependent and less reliable for microstructurally heterogeneous materials, such as those produced by additive manufacturing (AM).

These limitations are particularly relevant for Ti-6Al-4V (Ti64), the most widely used titanium alloy, often called the workhorse of the titanium industry. Ti64 combines low density, metallurgical stability, high strength to weight ratio, corrosion resistance, and biocompatibility (Reis et al., 2005; Sen et al., 2007), making it indispensable for aerospace structures, biomedical implants, marine equipment, defense hardware, and chemical processing components (Tjong and Mai, 2008; Zhang et al., 2017; Teixeira et al., 2020). Traditionally produced by cast-and-wrought processing, Ti64 is now increasingly manufactured using AM techniques, particularly Laser Powder Bed Fusion (LPBF), a form of metal 3D printing, to produce complex, lightweight geometries with minimal material waste. The rapid solidification inherent to LPBF produces a fine, metastable martensitic α′ microstructure, in contrast to the equilibrium α+β structure from conventional routes. This martensite decomposes toward α+β microstructure during thermal or thermo-mechanical exposure, with kinetics strongly influenced by local thermal history and stress state. LPBF also induces strong anisotropy and microstructural variation. While SPT has been used to characterize tensile properties in both wrought and AM Ti64, creep behavior of AM Ti64 has only been studied via UCT, and limited SPCT studies on LPBF Ti64 have been reported (Lalé et al., 2023).

This paper proposes an SPCT-based methodology, integrated with a representative stress–strain analysis of SPT and inverse finite element analysis, to determine the Norton creep parameters of LPBF Ti64. The procedure first extracts elasto-plastic properties from SPT, then fits SPCT deflection–time curves at 450 °C across multiple stress levels via inverse analysis to obtain the Norton model parameters directly from miniature disks. Validation on LPBF Ti64 shows primary, secondary, and tertiary creep in SPCT and yields Norton parameters that closely match UCT; a double-logarithmic comparison confirms strong correlation. The proposed framework delivers a standardized, accurate, and scalable pathway for high-temperature creep evaluation of LPBF Ti64 and provides a powerful tool for broader deployment of SPCT in AM alloy characterization.

2 Materials

The LPBF Ti64 specimens used in this study was produced using a 3DSystems ProX200 machine with a laser power of 300 W, scan speed of 1800 mm/s, hatch spacing of 85 μm, layer thickness of 50 μm, and spot diameter of 70 µm using a Ti64 building plate measuring 140 × 140 mm (Lalé, 2024; Dumontet, 2019). The microstructure of the material in this study was investigated using a Jeol 7800F scanning electron microscope (SEM) at Centre de Microcaractérisation Raimond Castaing, Toulouse, France, with further experimental details provided in previous work (Lalé et al., 2023; Lalé and Viguier, 2023). The microstructure of the LPBF Ti64 is shown in Figure 1. It was observed that microstructure is composed exclusively of α′ martensite laths, in agreement with previous studies (Villa et al., 2020; Mahmud et al., 2021; Wang et al., 2021; Shi et al., 2023). Figure 1 also confirms the observation from Yang et al. (Yang et al., 2016), that is, microstructure of LPBF Ti64 contains primary, secondary, tertiary, and quaternary martensite laths. Each of these types of needles has a characteristic size that has been reported by Yang et al. in Table 1. Lalé investigated the microstructures of Ti64 produced by two additive manufacturing machines and found that resulting microstructures from the two machines consisted exclusively of α′ martensite laths with the same morphology (Lalé, 2024). Cho carried out heat treatments on two martensitic microstructures of Ti64: one produced by LPBF and another induced by water quenching of a forged material. The SEM micrographs from Cho’s study revealed that the martensite plates from quenching exhibit a significantly higher elongation due to slower cooling rates (Cho, 2018). It is important to note that the martensitic morphology of AM Ti64 is governed by the cooling rate (Motyka, 2021). Shi et al. systematically explained the variation in martensitic transformation with cooling rate (Shi et al., 2023). Motyka offers a comprehensive review of the current understanding of martensite formation and decomposition processes in two-phase titanium alloys (Motyka, 2021). In the present study, detailed information on the feeding direction and scan pattern of the LPBF process was not recorded. Therefore, the analysis of anisotropy is limited to building-direction orientation.

Figure 1

Figure 1. Backscattered electron SEM micrograph of the material produced by 3DSystems ProX200 in this study.

Table 1

Table 1. Order of magnitude of characteristic sizes of different types of martensite created during the manufacturing and cooling of LPBF Ti64 (Yang et al., 2016).

The SPT is a displacement-controlled technique in which a ball is pressed into a miniature disk specimen clamped at its edges. The test produces a characteristic load–deflection curve that reflects elastic bending, plastic bending, membrane stretching, plastic instability, and eventual fracture. From this curve, mechanical parameters such as yield strength and ductility can be indirectly determined using appropriate models. In contrast, the Small Punch Creep Test (SPCT) is a load-controlled variant of SPT, designed to evaluate time-dependent creep deformation. Under constant load at elevated temperature, the disk specimen undergoes primary, secondary, and tertiary creep, and the resulting deflection–time curve provides access to creep strain rates and Norton creep parameters. Both techniques are particularly advantageous for additive manufacturing materials, as they require only sub-millimeter-thick specimens and enable characterization of localized microstructural regions. In this paper, specimens of SPT and SPCT were extracted from a 10 mm diameter cylinder and machined to a thickness of 0.5 mm. UCT specimens were machined from a 20 mm diameter cylinder, which were conducted for comparison with the results obtained from SPCT (Lalé et al., 2023). Both types of specimens were oriented along the building direction, with a porosity below 0.5%. Figures 2a,b show the schematic of specimen geometries and test setups for the small punch (SP) and tensile tests, respectively. Figure 2a depicts schematic testing setups for both SPT and SPCT. Figure 2b highlights thermocouple positions and the laser extensometer strain measurement setup. Red dots indicate thermocouple positions and laser extensometer used to measure the displacement. The purposes of SPT are twofold. Firstly, the SPT curve is used to determine the initial deflection at the applied creep load for the high-temperature SPCT. More importantly, a new representative stress-strain method was developed to derive tensile properties by using the SPT curve, detailed in section 3. The SPCTs and UCTs were both performed at 450 °C. It is important to notice that many researchers investigated the creep behavior on Ti64 at a temperature of 450 °C, which is representative of its typical high-temperature creep response (Badea et al., 2014; Ochonogor et al., 2017; Viespoli et al., 2020). SPT was performed at 400 °C to evaluate the material’s high-temperature strength below the creep threshold, whereas SPCTs were conducted at 450 °C to capture the time-dependent deformation behavior in the active creep regime. The UCTs were conducted under the specified stresses of 400, 500, and 600 MPa. Constant loads were controlled by the machine to conduct the SPCTs, where the load values ($F_{initial}$) were determined from the initial stress values (${\sigma }_{initial}$) and the calculation can be found in our previous work (Lalé et al., 2023), which is shown in Equation 1:

$F_{initial}={\sigma }_{initial}*3,33*k_{SP}\frac{r^{1,2}{h}_0}{{\left(0,5*D\right)}^{0,2}}(1)$

where $D$ is the receiving hole diameter, $h_0$ the initial sample thickness and $r$ is the punch radius. The initial stress values for SPCT are 400 and 450 MPa. Images of specimens before and after the experiments can be found in the previous work (Lalé, 2024).

Figure 2

Figure 2. Schematic testing setups for (a) both SPT and SPCT, and (b) the UCT specimen gbometry indicating thermocouple positions and the laser extensometer strain measurement arrangement (unit: mm).

3 Methods

The method used in this study is described briefly, and complete details can be found in our previous work (Xie et al., 2025). As shown in Figure 3, the method begins with the SPT data on LPBF Ti64 to determine the elasto-plastic parameters (yield strength: $S_y$, strain hardening exponent: $n$, hardening constant: $K$, Young’s modulus: $E$) using a representative stress–strain method, followed by the SPCTs to generate a comprehensive dataset for the inverse algorithm. Finally, a two-dimensional finite element model of the SPCT is employed to predict the deflection–time response, enabling the application of the inverse approach of SPCT to effectively determine the Norton creep law parameters $A$ and $m$ of LBPF Ti64. In the Norton creep law shown in Figure 3, ${\dot{\epsilon }}_{\mathit{min}}$ and $\sigma$ represent minimum creep strain rates and applied stress of the creep behavior, respectively. MATLAB’s lsqnonlin optimization function was used to minimize the difference between experimental and simulated SPCT curves. The calculated Norton creep parameters were compared against those obtained from UCT to verify the reliability of the proposed approach.

Figure 3

Figure 3. Flowchart of the method used in this study.

3.1 Representative stress-strain method

In 1951, Tabor introduced the concept of representative stress and strain (Xie et al., 2025). This concept aims to simplify a complex contact stress problem into a representative stress and strain issue at specific points. This theoretical framework establishes an equivalent mapping model between multiaxial stress fields and uniaxial constitutive relationships, thereby transforming complex contact mechanics problems into parameterized analytical formulations at characteristic points. The approach provides a new paradigm for nondestructive evaluation of material mechanical properties. The representative stress-strain method to SPT facilitates the determination of elasto-plastic properties, enabling a streamlined prediction of Norton creep law parameters by the inverse approach of SPCT. The central concept of representative stress-strain method lies in constructing a quantitative correlation between indentation responses—such as load, penetration depth, and residual impression geometry—and intrinsic material parameters including yield strength, elastic modulus, and strain-hardening exponent. Continuous refinements have been made to improve the physical definition of the representative strain parameter, leading to more accurate modeling of elasto-plastic deformation. For instance, Ahn and Kwon (Zhou et al., 2024) introduced a shear strain gradient-based analytical framework, extending the this approach by accounting for strain gradient effects within the contact zone, thereby improving precision in materials with size-dependent strengthening. The formulas for the representative stress ${\sigma }_r$ and strain ${\epsilon }_r$ are as follows (Zhou et al., 2024),

${\sigma }_r=\frac{F}{\pi {a_c}^2\mathrm{\Psi }}=\frac{P_m}{\mathrm{\Psi }}(2)$

${\epsilon }_r=\alpha \frac{a_c}{\sqrt{R^2-{a_c}^2}}(3)$

here, $\mathrm{\Psi }$ and $\alpha$ are the coefficients of representative stress and strain, respectively, R represents the radius of spherical indenter, F denotes the load, P_m corresponds to the mean contact pressure, and a_c signifies the contact radius.

To adapt the representative stress-strain method into SPT, it is necessary to resolve the representative coefficients, i.e., $\mathrm{\Psi }$, $\alpha$ and $a_c$. The ball indenter is assumed to be in rigid contact with the specimen. Dimensional analysis is employed to establish the dimensionless formulations of the aforementioned coefficients. Initially, it is assumed that the indenter of SPT is in rigid contact with the specimen, thus excluding the elastic modulus and Poisson’s ratio of the indenter from the calculations. Under this assumption, the load F applied by the small punch is considered as a function of the following independent parameters:

$F=f\left(E,\nu ,S_y,n,\delta ,R\right)(4)$

here, E is the Young’s modulus, v is the Poisson’s ratio, S_y is the yield stress, n is the strain hardening exponent, δ is the specimen deflection. Equation 4 can also be rephrased in the following manner based on Equations 2, 3,

$F=f\left(E,\nu ,\sigma ,n,a_c,R\right)(5)$

Here, σ represents the stress at the selected contact point from the finite element analysis, which is derived from the simulation results of ANSYS. In this study, the very first contact point between the indenter and the sample is designated as the selected point, thus, the stress at this contact point is defined as σ. Here, the deflection δ is was replaced by the contact radius $a_c$, and the yield strength S_y is replaced by the σ, representing the stress at the contact point. This substitution allows the derivation of the dimensionless function Π₀. The relationship between $\sigma$ and $S_y$ can be described by Equation 10. The relationship between δ and contact radius $a_c$ can be described by Equation 16. Based on the $\mathrm{\Pi }$ theorem, Equation 5 can be reformulated into Equation 6, ensuring that the dimensions on both sides remain consistent.

$F=\pi a_c^2\sigma {\mathrm{\Pi }}_0\left(\frac{E}{\sigma },n,\nu ,\frac{a_c}{R}\right)(6)$

${\mathrm{\Pi }}_0$ is a dimensionless function. For steels, v = 0.3 and Equation 6 can be reformulated accordingly,

$\frac{F}{\pi a_c^2\sigma }=\frac{P_m}{\sigma }=\mathrm{\Psi }={\mathrm{\Pi }}_1\left(\frac{E}{\sigma },n,\frac{\delta }{R}\right)(7)$

here, $\mathrm{\Psi }$ is the dimensionless function of $\frac{E}{\sigma }$, n, and $\frac{\mathrm{\delta }}{R}$. Similarly, $\alpha$, and $\frac{a_c}{R}$ can also be the dimensionless functions of $\frac{E}{\sigma }$, n, and $\frac{\mathrm{\delta }}{R}$ based on Equation 3,

$\mathrm{\alpha }=\frac{{\epsilon }_r}{a_c/\sqrt{R^2-a_c^2}}={\mathrm{\Pi }}_2\left(\frac{E}{\sigma },n,\frac{\mathrm{\delta }}{R}\right)(8)$

$\frac{a_c}{R}={\mathrm{\Pi }}_3\left(\frac{E}{\sigma },n,\frac{\mathrm{\delta }}{R}\right)(9)$

In Equations 5–9, σ represents the stress at the selected contact point from the finite element analysis, which is derived from the simulation results of ANSYS. In this study, the very first contact point between the indenter and the sample is designated as the representative point; thus, the stress at this contact point is defined as the representative stress, σ.

A three-parameter power law function is used as the isotropic elasto-plastic constitutive model. The primary rationale for selecting this model lies in its ability to adhere to the power-law form within the ANSYS framework, ensuring a robust representation of material behavior under yield conditions.

$\sigma =S_y{\left(1+\frac{E_T}{S_y}{\epsilon }_p\right)}^n(10)$

where ε_p is the plastic strain, E_T is materials’ Young’s modulus at elevated temperature that is subject to a reduction, which can be calculated by the following equation:

$E_T={\chi }_{sT}E(11)$

where, ${\chi }_{sT}$ is the reduction coefficient,

${\chi }_{sT}=\left\{\begin{array}{l}\frac{7T_s-4780}{6T_s-4760}20℃\le T_s<600℃\\ \frac{1000-T_s}{6T_s-2800}600℃\le T_s\le 1000℃\end{array}\right.(12)$

where, T_s is the working temperature (unit: $℃$). At 600 °C, the reduction coefficient is calculated to be 0.5 by Equation 12. For the sake of convenience, the Young’s modulus of steels at room temperature is usually taken as 210 GPa. Therefore, by the reduction coefficient the Young’s modulus of steels is 105 GPa at 600 °C.

In this study, ANSYS and MATLAB were employed to establish a comprehensive finite element analysis (FEA) solution platform, facilitating the compilation of a robust database through forward FEA. The details of FEA can be found in our previous work (Xie et al., 2025). As presented in Table 2, the material properties for FEA are systematically varied by modifying the yield stress (Sy) and the power law exponent (n), resulting in a comprehensive database comprising 100 distinct material models. The simulations are performed under a maximum deflection of 1.5 mm, with each material model yielding 30 output increments. Key output parameters encompass deflection, load, contact radius, as well as stress, strain, hardening index, and elastic modulus at the contact interface between the indenter and specimen.

Table 2

Table 2. Material properties for FEA.

In Equation 7, the coefficient Ψ is identified as a dimensionless function that correlates with the parameters $\frac{E}{\sigma }$, n, and $\frac{\mathrm{\delta }}{R}$. An initial exploration of the relationship between Ψ and E/σ reveals distribution of Ψ and the corresponding fitting curves as functions of $\frac{E}{\sigma }$, which is shown in Equation 13

$\mathrm{\Psi }=a_1*\left[1-\mathit{exp}\left(-b_1*\frac{E}{\sigma }\right)\right](13)$

where a₁ and b₁ are fitting coefficients.

In addition, variation of the $\mathrm{\Psi }$ as a function of δ and n across the designated ranges can be determined. The parameter δ varies from 0.05 mm to 1.5 mm in increments of 0.05 mm, resulting in 30 distinct values. Simultaneously, n is assessed from 0.05 to 0.5, also at intervals of 0.05, producing 10 corresponding values and culminating in a total of 300 combinations. Therefore, the following equation can be derived by curve fitting,

$\mathrm{\Psi }=a_2*\left[1-b_2*\mathit{tan}h\left(c_2*\delta \right)\right](\mathrm{14a})$

where coefficients a₂, b₂ and c₂ are fitting coefficients and all dependent on n:

$a_2=0.6157*n+1.9911(\mathrm{14b})$

$b_2=-0.0682*n+0.6717(\mathrm{14c})$

$c_2=8.8034*n+3.6026(\mathrm{14d})$

The methodology for ascertaining the coefficient α aligns closely with the process employed for defining the coefficient $\mathrm{\Psi }$. It is important to note that the coefficient α is concurrently affected by both $\delta$ and n. Curve fitting indicates that the optimal expression for the coefficient α exhibits minimal sensitivity to fluctuations in $\delta$ and n. As a result, an average value has been utilized for subsequent analysis, as illustrated in Equation 15,

$\alpha =0.114(15)$

The analysis for variation of a_c as a function of $\delta$ and n reveals that the power law function represented by Equation 16,

$a_c=a^{\prime }*{\delta }^{b^{\prime }}(16)$

where coefficients a’ and b’ are defined as fitting coefficients contingent upon the strain hardening exponent n. The logarithmic transformation of both sides of Equation 16 facilitates a regression analysis, thereby allowing for the calculation of these coefficients. As n varies with 10 distinct values, ten corresponding sets of a’ and b’ are derived. Following this, the fitting curves are employed to establish linear expressions representing a’ and b’ as functions of n, which are shown in Equations 17, 18.

$a^{\prime }=-0.1419*n+0.9463(17)$

$b^{\prime }=0.2428*n+0.3956(18)$

Therefore, $\mathrm{\Psi },$ $\alpha ,$ and $a_c$ in the representative stress-strain method can be determined based on Equations 14a, 15, 16, respectively. Equation 10 can be used to express the relationship between $\sigma$ and $S_y$. Equation 16 can be used to express the relationship between δ and contact radius $a_c$. Figure 4 shows the flow chart of representative stress-strain method to obtain elasto-plastic properties using SPT (Xie et al., 2025). In summary, representative stress-strain method to obtain the elasto-plastic properties using SPT includes the following steps: The first step involves extracting the load-deflection curve from SPT at an elevated temperature. The second step determines Young’s modulus (E_T) at working temperature using Equation 11. In the third step, the initial value n₀ = 0.05, which is then substituted into Equations 14–16 to calculate the coefficients, $\mathrm{\Psi }$, α and a_c, as functions of deflection $\delta$. The fourth step involves substituting these coefficients into Equations 2, 3 to calculated the corresponding representative stress, ${\sigma }_r$,and strain, ${\epsilon }_r$, values. In the fifth step, a fitting is performed using the Hollomon model on the representative stress and strain points obtained in the fourth step, yielding the hardening exponent (n) and coefficient (K). The reason to apply the Hollomon intrinsic model is grounded in the principles of the representative stress-strain method (Xie et al., 2025). Finally, if the fitted value of n_i at the ith step differs from n_i-1 at the (i-1)th step, it is updated and the calculations are repeated until the convergence of n_i and n_i-1 is achieved. The yield stress is then determined by locating the intersection point of the linear elastic modulus (E_T) line at the plastic strain of 0.2% and the representative stress-strain curve.

Figure 4

Figure 4. Flowchart of representative stress-strain method to obtain elasto-plastic properties using SPT (Xie et al., 2025).

3.2 Inverse SPCT method for Norton creep properties

Inverse approach developed in our previous work was utilized for predicting Norton creep properties from SPCT (Xie et al., 2025). During the initial loading phase of SPCT, significant elasto-plastic deformation and initial deflection occur, which substantially impacts subsequent creep deformation. Therefore, to improve the accuracy of creep properties predicted from SPCT, it is essential to determine the elasto-plastic properties and initial deflection (${\mathrm{\delta }}_0$) during the initial stage of SPCT.

Firstly, the initial deflection is ascertained from the SPT load-deflection curve corresponding to the applied creep load. Secondly, finite element modelling of SPCT is performed. The elasto-plastic component of constitutive model in SPCT is defined by Equation 10. The material’s elastic modulus is calculated using Equation 11. The yield stress S_y and strain hardening exponent n are derived from the SPT curve based on the representative stress-strain method. Furthermore, as the Norton creep function describes the secondary stage of creep behavior, only the first and second stages of the SPCT curve are utilized in predicting the Norton creep properties via the inverse approach of SPCT. To mitigate the effects of excessive plastic deformation on creep, the maximum load for the SPCT is calculated using the Chakrabarty model (Xie et al., 2025) in accordance with CWA 15267, which is shown in Equation 19:

$\mathrm{F}/\sigma =3.332a^{-0.202}{R}^{1.192}{t}_0(19)$

The SPCT outlined in this study utilized a hole with radius a = 2 mm, a specimen thickness t₀ = 0.5 mm, and a punch radius R = 1.25 mm, thus with a calculated maximum creep load F ≤ 1.891S_y.

The Norton creep constitutive model is widely utilized to describe the steady-state creep behavior of materials during the second stage of creep. The minimum creep strain rate can be determined using the uniaxial form of the Norton creep law, which is shown in Equation 20

${\dot{\epsilon }}_{\mathit{min}}=A{{\sigma }_{initial}}^m(20)$

where, ${\dot{\mathrm{\epsilon }}}_{\min }$ is the minimal creep strain rate, A represents the Norton coefficient, m denotes the stress exponent, and ${\sigma }_{initial}$ is the applied initial stress for creep tests. When utilizing the inverse approach of SPCT to predict Norton creep properties, it is essential to specify the initial values of A and m, as well as ranges for these two parameters. Typically, the stress exponent m is set within a range of 3–20, while the Norton coefficient A can vary significantly from 10⁻¹⁰ to 10⁻⁵⁰. In the inverse approach of SPCT, the initial value for A is set at 10⁻¹⁵, with m starting at 10.

Figure 5 shows the flowchart of obtaining the Norton creep properties using inverse approach of SPCT (Xie et al., 2025). To summarize, the algorithm for predicting Norton creep properties using both SPT and SPCT is as follows: The first step involves inputting the first and second stages of SPCT curves as well as ${\mathrm{\delta }}_0$ into MATLAB. The second step establishes a joint MATLAB-ANSYS solving platform, where elasto-plastic parameters obtained from high-temperature SPT curves are incorporated into the finite element model of SPCT, alongside the initial values and ranges for the Norton coefficient A and stress exponent m. In the third step, the MATLAB optimization function lsqnonlin is employed to automatically select a new pair of parameters, A and m, which are then input into the finite element model of SPCT. The fourth step conducts finite element simulations based on the updated A and m values, and extracts the deflection-time curves from the results. Finally, in the fifth step, the finite element results of SPCT deflection-time curves are compared with the experimental data under corresponding load levels; the sum of squared residuals is fed back into the iteration process. If the minimization does not reach the threshold, a new pair of A and m is selected, and steps from two to five are repeated until the error between the simulation and experimental deflection-time curves meets the specified threshold, thereby determining the final values of A and m. Further details can be found in our previous work (Xie et al., 2025).

Figure 5

Figure 5. Flowchart of obtaining the Norton creep properties using inverse approach of SPCT (Xie et al., 2025).

4 Results & discussion

Figure 6a shows the engineering stress–engineering strain curve obtained from a uniaxial tensile test on LPBF Ti64 at 400 °C. The ultimate tensile strength is 872 MPa at the strain of 0.025. Figure 6b shows the sample curve of the SPT on LPBF Ti64 at 400 °C. The curve exhibits an initial linear increase in force with displacement, corresponding to elastic bending, followed by a nonlinear rise due to plastic deformation. As displacement increases further, the curve reaches a peak force of approximately 1287 N at around 0.96 mm, indicating the plastic instability, after which the force gradually decreases, signifying damage evolution and eventual material failure. From the literature (Lucon et al., 2021; Torres and Gordon, 2021), the force–displacement diagram obtained from the SPT can be clearly divided into five successive regions with increasing displacement: elastic bending, plastic bending, membrane stretching (local bending), plastic instability, and final material failure. However, the force exhibits fluctuations caused by internal fracture in the displacement range of 0.45–0.50 mm, which is consistent with the results from Janča et al. (2016). Lalé also proposed that LPBF materials exhibit high yield stress; however, they possess low ductility (Lalé, 2024). By analyzing the SPT data, the study first delineates the creep loads observed within the SPT curve and then pinpoints the corresponding initial deflection values located along the abscissa axis of the load–deflection curves, which were indicated by the triangles in Figure 6b.

Figure 6

Figure 6. (a) Engineering stress versus engineering strain curves for tensile test on LPBF Ti64 at 400 °C (b) force versus laser displacement curves for SPT on LPBF Ti64 at 400 °C, and 400 MPa and 450 MPa initial stress conditions are highlighted using green and red triangles, respectively; (c) strain versus time curves for UCTs on LPBF Ti64 at 450 °C under 400, 500 and 600 MPa; (d) deflection versus time curves for SPCT on LPBF Ti64 at 450 °C under 400 and 450 MPa.

Figure 6c shows the strain-time curves obtained from the UCTs on LPBF Ti64 under applied stresses of 400, 500, and 600 MPa. These curves exhibit the distinct stages of creep deformation: an initial primary phase characterized by a decreasing strain rate, a secondary phase with a steady minimum strain rate, and a tertiary phase in which the strain rate accelerates rapidly, culminating in material failure, consistent with results reported in the literature (Sato et al., 2006). At 600 MPa, premature failure is observed, whereas at 400 MPa, the material shows a prolonged secondary creep stage with significantly delayed rupture. The slopes of the strain-time curves for Ti64 at various stress levels are calculated, yielding the minimum creep strain rates for the secondary phase. Table 3 summarizes the results of these minimum creep strain rates under different creep stresses at 450 °C.

Table 3

Table 3. Summary of the minimum creep strain rates under different creep stresses at 450 °C.

Figure 6d shows the deflection (δ) versus time curves from SPCT on LPBF Ti64 under applied loads corresponding to initial stresses of 400 and 450 MPa. These curves exhibit the classic characteristics of SPCT curves: an initial primary stage with a decreasing displacement rate, a secondary stage with a nearly constant rate, and a tertiary stage marked by rapid displacement acceleration leading to specimen rupture, consistent with results reported in the literature (Feng et al., 2015). Higher applied stresses result in shorter primary and secondary stages and earlier onset of tertiary creep. Figure 6c,d shows much lower time to rupture for UCT, indicating shorter primary and secondary stages compared to SPCT. The longer rupture time observed in SPCT compared with UCT under the same nominal stress is attributed to the multiaxial stress state and geometric constraint introduced by the ball indenter. The local effective stress in the SPCT specimen is reduced due to bending and shear stresses, extending the apparent creep life, especially in the primary and secondary stages.

Figure 7 presents the fitted Norton creep law parameters using the minimum strain rate versus stress following the standard tests of UCT on LPBF Ti64 at 450 °C. The determined experimental Norton creep law parameters are fitted using the Norton creep law, ${\dot{\epsilon }}_{\mathit{min}}=A{\sigma }^{\mathrm{m}}$, yielding the values of Norton creep parameters, where $A=2.3462\times {10}^{-18}$, $m=5.5238$. The use of three stress levels (400 MPa, 500 MPa, and 600 MPa) was found to be sufficient for accurately fitting the Norton creep law, providing reliable estimates of the creep parameters for LPBF LPBF Ti64. Further increasing the number of stress points would not significantly improve the precision of the fitted parameters, as evidenced by the good agreement between the experimental data and the fitted model.

Figure 7

Figure 7. Norton creep law parameters obtained from UCT of LPBF Ti64 at 450 °C.

Figure 8 shows the elasto-plastic parameters of LPBF Ti64 determined using the SPT-based representative stress–strain method. The red points represent the calculated representative stress–strain data, while the black curve corresponds to the Hollomon model fit. The blue points and the red line indicate the 0.2% elastic offset method used to determine the yield strength. The fitting yields $S_y=706$ MPa, $n=0.1219$, $K=1247$ MPa, $E=94956$ MPa. These values are consistent with the results in the literature listed in Table 4 (Lalé and Viguier, 2023; Lalé, 2024), although the red points in Figure 8 exhibit a non-monotonic trend, first increasing and then decreasing. This stress decrease might be related to the first rupture of the SPT reported by Janča et al. (2016). Further research is warranted to elucidate this issue more comprehensively. It is important to note that the values of Young’s modulus of Ti64 decrease linearly with increase of temperature (Yang et al., 2021).

Figure 8

Figure 8. Elasto-plastic parameters of LPBF-Ti64 determined by SPT-based representative stress–strain method (red points are the representative stress-strain points, black points are the Hollomon fitting data points, and the red solid line represents the 0.2% elastic offset method).

Table 4

Table 4. Summary of the elasto-plastic parameters of LPBF Ti64 determined using the SPT-based representative stress–strain method and values from literature (Lalé and Viguier, 2023; Lalé, 2024).

Figure 9a,b present sample curves illustrating the fitting of the first and second stages obtained from SPCT under 400 MPa and 450 MPa, respectively, via inverse FEA. The measured deflection–time data (blue curve) are compared with FEA predictions (red curve), showing excellent agreement throughout the creep duration. The optimized Norton creep law parameters obtained via the inverse approach, determined under 400 MPa and 450 MPa, respectively, are also shown in Figure 9a,b.

Figure 9

Figure 9. Sample curves for the inverse approach of SPCT to predict Norton creep law parameters of LPBF Ti64 at 450 °C under (a)400 MPa and (b)450 MPa (red lines represent data from FEA and blue lines are the measured data from experiments).

Figure 10 compares the Norton creep parameters obtained from UCTs and SPCTs for LPBF Ti64 at 450 °C, plotted in double logarithmic form. It should be noted that the axes in Figure 10 are plotted as log (stress) and log (minimum strain rate), while the axes in Figure 5 are plotted as the stress and minimum strain rate. It should be noticed that differences in A of orders of magnitude can be compensated easily by m, and that these differences are negligible, due to its high sensitivity. The slopes of the fitted lines correspond to $m$, while the intercepts represent $A$. The average of the optimized Norton creep law parameters obtained via the inverse approach, determined under 400 MPa and 450 MPa are used. The close alignment of the UCT (black line) and SPCT (red line) results indicates well agreement between the two testing methods in characterizing creep behavior. This close alignment demonstrates that SPCT, combined with inverse analysis, can reproduce creep constants with accuracy comparable to standard uniaxial testing. The method therefore enables creep characterization using miniature samples, with the added advantage of being applicable to localized or anisotropic regions typical of LPBF materials. Nevertheless, limitations remain. For example, non-monotonic behavior in SPT data may reflect premature rupture events, and SPCT sensitivity to local defects can influence creep lifetimes. Furthermore, anisotropy arising from feeding patterns was not captured in this study, and future work should integrate these factors. Despite these constraints, the overall results highlight the accuracy, efficiency, and practical advantages of SPCT for creep assessment in additively manufactured alloys.

Figure 10

Figure 10. Comparison of the Norton creep parameters under double logarithmic form between UCT and SPCT for LPBF Ti64 at 450 °C (black line represents data from UCT and red line represents data from the present SPCT).

5 Conclusion

This study developed and validated an approach for determining the Norton creep parameters of LPBF Ti64 by integrating SPT and SPCT. Representative stress–strain method applied to SPT data enabled accurate determination of elasto-plastic parameters ($S_y$, $n$, $K$, and $E$) of LPBF Ti64, with values in good agreement with those reported in the literature. SPCT creep curves at 450 °C exhibited distinct primary, secondary, and tertiary stages, consistent with UCT observations, with higher stresses accelerating the transition to tertiary creep. Inverse FEA analysis of SPCT data provided optimized Norton parameters that closely matched those obtained from UCTs, confirming the reliability of the proposed approach. Comparison of SPCT and UCT results in double logarithmic form showed strong correlation, supporting SPCT as a robust alternative to UCT for creep characterization of LPBF Ti64. Overall, this work provides a standardized, efficient, and accurate pathway for high-temperature Norton creep evaluation of LPBF Ti64.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

FS: Formal Analysis, Writing – original draft, Writing – review and editing. ML: Conceptualization, Data curation, Methodology, Resources, Writing – review and editing. FY: Conceptualization, Formal Analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review and editing. MA: Conceptualization, Investigation, Methodology, Resources, Validation, Writing – review and editing. BV: Conceptualization, Data curation, Investigation, Methodology, Resources, Supervision, Validation, Writing – review and editing. YL: Conceptualization, Investigation, Methodology, Resources, Supervision, Validation, Visualization, Writing – review and editing.

Funding

The authors declare that financial support was received for the research and/or publication of this article. This study was supported by Ningbo Key R&D Programme Project (2023Z036), National Natural Science Foundation of China (51971113), and Natural Science Foundation of Zhejiang Province (LY21A020002).

Acknowledgements

The authors gratefully acknowledge Petr Dymáček from IPM Brno for his insightful comments and constructive feedback on this study.

Conflict of interest

Author YL was employed by DNV-GL Energy (former KEMA, retired).

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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The authors declare that no Generative AI was used in the creation of this manuscript.

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