Finite Element Method of Functionally Graded Shape Memory Alloy Based on UMAT

Abstract

Functionally graded shape memory alloy (FG-SMA) is widely used in practical engineering regions due to it possessing the excellent properties of both FG material and SMA material. In this paper, the incremental constitutive equation of SMA was established by using the concept of a shape memory factor. On this basis, the secondary development function of the ABAQUS software 2023 was used to write the user-defined material subroutine (UMAT). The phase transformation and mechanical behavior of transverse and axial FG NiTi SMA cantilever beams under concentrated load at free ends were numerically simulated by discrete modeling. Numerical results show that the stress and shape memory factor were distributed asymmetrically along the thickness direction of the transverse FG-SMA cantilever beam, while the stress and the shape memory factor distributed symmetrically along the thickness direction of the cross section of the axial FG-SMA cantilever beam. The bearing capacity of the axial FG-SMA cantilever beam is stronger than the SMA homogeneous cantilever beam, but weaker than the transverse FG-SMA cantilever beam. The load-bearing capacity of the transverse FG-SMA cantilever beam is twice that of the axial FG-SMA cantilever beam under the same functionally graded parameters and deflection conditions. The discrete modeling method of FG-SMA beams proposed in this paper can simulate the phase transformation and mechanical behavior of an FG-SMA beam well, which provides a reference for the practical application and numerical calculation of FG-SMA structures.

1. Introduction

Shape memory alloys (SMAs) are promising candidates as materials in the fields of automotive, aerospace, biomedical, industrial process control, electronic instrumentation, telecommunications, and military applications because of their ability to undergo a structural displacive phase transformation [1,2], known as a martensitic transformation [3], which occurs between austenite of a crystallographic more-ordered phase (stable at high temperature and low stress) and martensite of a crystallographic less-ordered phase (stable at low temperature and high stress). Owing to exhibiting a reversible martensitic transformation, SMAs can recover their original shapes from large deformations well beyond their elastic strain limits upon unloading or heating, which are referred to as pseudo-elasticity and the shape memory effect, respectively [4,5]. There are four characteristic temperatures, namely, the martensitic starting temperature M_s, martensitic finishing temperature M_f, austenitic starting temperature A_s and austenitic finishing temperature A_f, in an SMA at a free-stress state, and they are generally satisfied with the relationship M_f < M_s < A_s < A_f [6]. A functionally graded material (FGM) is a new type of heterogeneous material with gradient changes in material properties or geometric structures along a specific direction. Compared with homogeneous materials, FGMs have the advantages of not only eliminating stress concentration and reducing the residual stress, but also enhancing the connection strength [7,8,9]. Due to having dual characteristics of functionally graded materials and shape memory alloy materials [10], functionally graded shape memory alloys (FG-SMA) are widely used in national defense, aerospace, micro-electromechanical and other high-tech fields.

At present, experts and scholars have carried out experimental research on the preparation and mechanical properties of FG-SMAs. Shariat et al. [11] elaborated on the concept and the experimental and thermodynamic behavior characteristics of NiTi FG-SMA, as well as the preparation technology of the FG-SMA. In addition, some experts have prepared functionally graded shape memory alloy materials through different methods [12,13,14,15] and studied the thermodynamic properties of FG-SMA structures through experimental methods such as X-ray diffraction, transmission electron microscopy, atomic force microscopy and resistance testing [16,17,18]. On this basis, some experts and scholars have described and predicted the mechanical properties of FG-SMAs by constructing mechanical models and have attempted to provide theoretical solutions [19,20,21,22,23,24]. Due to the difficulty of efficiently solving the nonlinear models mentioned above and the complexity of the FG-SMA structure, it is difficult for these models to exert their advantages in practical engineering applications. Therefore, it is necessary to develop a numerical calculation method for practical engineering applications to solve the computational difficulties of FG-SMA materials in the engineering application process. In recent years, with the increasing application of SMAs, some experts and scholars have studied the thermodynamic behavior of SMA structures under complex loads by establishing a finite element method [25,26,27]. However, the above finite element numerical calculation method is only applicable to homogeneous SMA structures and cannot effectively calculate the thermodynamic behavior of FG-SMA structures with material or structural parameters continuously changing in a specific direction.

To the best of the researchers’ knowledge, most of the existing studies have focused on the preparation and performance testing of FG-SMAs and the finite element numerical simulation of SMA homogeneous structures, and there are few numerical simulations on the deformation behavior of FG-SMA structures under the actions of different types of loads. In this paper, an incremental constitutive model and shape memory evolution equation of an SMA are derived using the concept of a shape memory factor, and the UMAT subroutine is written. Then, the FG-SMA beam structure is layered/segmented along the graded direction. The deformation and phase transformation behavior of an FG-SMA beam after loading are numerically simulated by establishing a simplified model. The functionally graded effect of the FG-SMA beam is investigated, which provides a calculation basis and reference for the design and application of FG-SMA structures.

2. Shape Memory Evolution Equation

The shape memory factor is defined to describe the thermo-mechanical processes of super elasticity and shape memory effect of an SMA [28,29]. The shape memory factor is a scalar with a range of variation between 0 and 1. By using the shape memory factor, the thermodynamic processes of SMA superelasticity and shape memory effects can be intuitively described. For superelasticity, during the loading process, when the material stress increases to the critical value, the material begins to produce shape memory strain, and the shape memory factor gradually increases from 0. When the shape memory factor increases to 1, the shape memory strain reaches its maximum value. During the unloading process, when the material stress decreases to the critical value, the shape memory strain begins to gradually recover, and the shape memory factor gradually decreases. When the shape memory factor decreases to 0, the shape memory strain fully recovers. The relationship between the shape memory factor and martensite volume fraction is [2]:

$$\xi ={\xi }_0+(1-{\xi }_0)\eta$$

(1)

where ξ is the martensitic volume fraction, ξ₀ is the initial value of martensitic volume fraction and η denotes the shape memory factor.

Based on the isotropic assumption, when the temperature T satisfies T > M_s, it is assumed that the initial state of the SMA is completely in the austenitic phase, for the positive phase transformation process $({\sigma }_{\mathrm{ms}}\le {\sigma }_{\mathrm{eq}}\le {\sigma }_{\mathrm{mf}})$ [2]:

$$\eta =\frac{3}{4}[\frac{2({\sigma }_{\mathrm{eq}}-{\sigma }_{\mathrm{ms}})+{\sigma }_{\mathrm{scr}}-{\sigma }_{\mathrm{fcr}}}{{\sigma }_{\mathrm{scr}}+{\sigma }_{\mathrm{fcr}}}]-\frac{1}{4}{[\frac{2({\sigma }_{\mathrm{eq}}-{\sigma }_{\mathrm{ms}})+{\sigma }_{\mathrm{scr}}-{\sigma }_{\mathrm{fcr}}}{{\sigma }_{\mathrm{scr}}+{\sigma }_{\mathrm{fcr}}}]}^3+\frac{1}{2}$$

(2)

Among them,

$${\sigma }_{\mathrm{eq}}=\sqrt{\frac{3}{2}\sigma {’}_{ij}\sigma {’}_{ij}}$$

(3)

represents the equivalent stress [2],

$$\sigma {’}_{ij}={\sigma }_{ij}-\frac{1}{3}{\sigma }_{mm}$$

(4)

denotes the deviatoric stress tensor, and σ_ms and σ_mf represent the starting and finishing stress of the martensitic phase transformation, respectively. The relationship between σ_ms, σ_mf and the ambient temperature T can be described as [30]:

$$\left\{\begin{array}{l}\begin{array}{cc}{\sigma }_{\mathrm{ms}}={\sigma }_{\mathrm{scr}}+C_{\mathrm{M}}(T-M_{\mathrm{s}}),& \end{array}\\ \begin{array}{cc}{\sigma }_{\mathrm{mf}}={\sigma }_{\mathrm{fcr}}+C_{\mathrm{M}}(T-M_{\mathrm{s}})& \end{array}\end{array}\right.$$

(5)

where σ_scr and σ_fcr represent the initial values of the martensitic starting stress and martensitic finishing stress of the SMA, respectively. C_M is the temperature influence coefficient of the critical stress of the martensitic phase transformation, and M_s represents the martensitic starting temperature.

When the ambient temperature T > A_s, for the inverse phase transition process $({\sigma }_{\mathrm{af}}<{\sigma }_{\mathrm{eq}}<{\sigma }_{\mathrm{as}})$ [2]:

$$\eta ={\eta }_0\left\{\frac{3}{4}[\frac{{\sigma }_{\mathrm{eq}}-{\sigma }_{\mathrm{af}}}{C_{\mathrm{A}}(A_{\mathrm{f}}-A_{\mathrm{s}})}-\frac{1}{2}]-\right.\left.\frac{1}{4}{[\frac{{\sigma }_{\mathrm{eq}}-{\sigma }_{\mathrm{af}}}{C_{\mathrm{A}}(A_{\mathrm{f}}-A_{\mathrm{s}})}-\frac{1}{2}]}^3+\frac{1}{2}\right\}$$

(6)

where

$$\left\{\begin{array}{c}{\sigma }_{\mathrm{as}}=C_{\mathrm{A}}(\mathrm{T}-A_{\mathrm{s}}) ,\hfill \\ {\sigma }_{\mathrm{af}}=C_{\mathrm{A}}(\mathrm{T}-A_{\mathrm{f}})\hfill \end{array}\right.$$

(7)

respectively denote the austenstic starting stress and austenstic finishing stress [30]. η₀ represents the initial value of the shape memory factor before the onset of the martensitic reverse phase transformation. C_A is the temperature influence coefficient of the critical stress of the martensitic reverse phase transformation. A_s and A_f represent the starting and finishing temperatures of the martensitic reverse phase transformation, respectively.

3. Incremental Constitutive Equation of SMA

According to the basic principles of thermodynamics, without considering the plastic strain, the strain tensor of an SMA can be expressed with the internal variable (shape memory factor η), temperature and stress tensor as [31,32]:

$${\epsilon }_{ij}=\epsilon \left(\eta ,\mathsf{{\rm T}},{\sigma }_{kl}\right)={\epsilon }_{ij}^{\mathrm{tr}}+{\epsilon }_{ij}^{\mathrm{th}}+{\epsilon }_{ij}^{\mathrm{e}}$$

(8)

where ε_ij represents the strain tensor component, and T and σ_kl represent the temperature and stress tensor components, respectively. ${\epsilon }_{ij}^{\mathrm{tr}}$, ${\epsilon }_{ij}^{\mathrm{th}}$ and ${\epsilon }_{ij}^{\mathrm{e}}$ represent the phase transformation strain tensor components, thermal strain tensor components and elastic strain tensor components, respectively. The differential operation is carried out at both ends of Equation (8), and it yields [2]:

$$\mathrm{d}{\epsilon }_{ij}=\mathrm{d}{\epsilon }_{ij}^{\mathrm{tr}}+\mathrm{d}{\epsilon }_{ij}^{\mathrm{th}}+\mathrm{d}{\epsilon }_{ij}^{\mathrm{e}}=\frac{\partial {\epsilon }_{ij}}{\partial \eta }\mathrm{d}\eta +{\Theta }_{ij}\mathrm{d}T+S_{ijkl}\mathrm{d}{\sigma }_{kl}$$

(9)

where dT represents the temperature increment and dσ_kl represents the incremental stress tensor components.

$${\Theta }_{ij}=\frac{\partial {\epsilon }_{ij}}{\partial T}$$

(10)

is the thermal expansion tensor component of the SMA [2].

$$S_{ijkl}=\frac{\partial {\epsilon }_{ij}}{\partial {\sigma }_{kl}}$$

(11)

Is the flexibility tensor component of the SMA [2]. S_ijkl and Θ_ij are functions of the martensite volume fraction ξ, namely:

$$\left\{\begin{array}{c}\begin{array}{cc}{S}_{ijkl}=(1-\xi )S_{ijkl}^{\mathrm{A}}+\xi S_{ijkl}^{\mathrm{M}}& ,\end{array}\\ \begin{array}{cc}{\Theta }_{ij}=(1-\xi ){\alpha }^{\mathrm{A}}+\xi {\alpha }^{\mathrm{M}}& .\end{array}\end{array}\right.$$

(12)

Substituting Equation (1) into the above formula yields:

$$\left\{\begin{array}{c}\begin{array}{cc}{S}_{ijkl}=S_{ijkl}^{\mathrm{A}}+[{\xi }_0+\left(1-{\xi }_0\right)\eta ](S_{ijkl}^{\mathrm{M}}-S_{ijkl}^{\mathrm{A}})& ,\end{array}\\ \begin{array}{cc}{\Theta }_{ij}={\alpha }^{\mathrm{A}}+[{\xi }_0+\left(1-{\xi }_0\right)\eta ]({\alpha }^{\mathrm{M}}-{\alpha }^{\mathrm{A}}).& \end{array}\end{array}\right.$$

(13)

where the superscripts “A” and “M” represent austenite and martensite, respectively.

For the isotropic SMA material, the phase change flow occurs during the phase change process, and the relationship between the increment of the phase change strain component and the increment of the shape memory factor is [2]:

$$\mathrm{d}{\epsilon }_{ij}^{\mathrm{tr}}={\epsilon }_{\mathrm{L}}{\Lambda }_{ij}\mathrm{d}\xi ={\epsilon }_{\mathrm{L}}{\Lambda }_{ij}\left(1-{\xi }_0\right)\mathrm{d}\eta$$

(14)

where ε_L represents the maximum phase transformation strain that the SMA can produce in a uniaxial tensile test and Λ_ij represents the directional component of the phase transformation strain.

According to the phase change flow rule, during the phase change of the SMA, the component of phase change strain direction is [6]:

$${\Lambda }_{ij}=\left\{\begin{array}{c}\frac{3\sigma {’}_{ij}}{2{\sigma }_{\mathrm{eq}}} \mathrm{positive} \mathrm{phase} \mathrm{transformation} ,\hfill \\ \frac{{\epsilon }_{ij}^{\mathrm{tr}}}{{\overline{\epsilon }}^{\mathrm{tr}}} \mathrm{reverse} \mathrm{phase} \mathrm{transformation}\hfill \end{array}\right.$$

(15)

in which

$${\overline{\epsilon }}^{\mathrm{tr}}=\sqrt{\frac{2}{3}{\epsilon }_{ij}^{\mathrm{tr}}{\epsilon }_{ij}^{\mathrm{tr}}}$$

(16)

denotes the equivalent phase transformation strain. Substituting (13) and (14) into (9) yields:

$$\mathrm{d}{\sigma }_{kl}=C_{ijkl}\left[\mathrm{d}{\epsilon }_{ij}-{\epsilon }_{\mathrm{L}}{\Lambda }_{ij}\left(1-{\xi }_0\right)\mathrm{d}\eta -{\Theta }_{ij}\mathrm{d}T\right]$$

(17)

where C_ijkl denotes the stiffness tensor component [6].

4. UMAT Subroutine Flow

The flowchart of the UMAT user subroutine calculation is shown in Figure 1. Before the calculation, the state variables are defined, and also the material parameters and the initial values of the state variables are read from ABAQUS. For applications in problems and some numerical methods, many powerful mathematical techniques have been used in recent years by researchers [33,34,35,36]. An effective orthogonal spline collocation (OSC) method [37] was considered and compared, and we have chosen the following iterative method for finite element numerical calculation programming. By dividing the whole loading and unloading process into several incremental steps, the material parameters are considered to be constant during each incremental step. Throughout the loading and unloading process, the state variables satisfy the following equation:

$${\Phi }^{(\mathrm{n}+1)}={\Phi }^{(\mathrm{n})}+\mathrm{d}{\Phi }^{(\mathrm{n})}$$

(18)

where Φ represents the state variable, while Φ⁽ⁿ⁾ and dΦ⁽ⁿ⁾ are the initial and incremental values of the state variable at the nth incremental step, respectively.

From Equations (5) and (7), the starting and finishing stresses of the martensitic phase transformation, ${\sigma }_{\mathrm{ms}}^{(\mathrm{n})}$ and ${\sigma }_{\mathrm{mf}}^{(\mathrm{n})}$, along with the starting and finishing stresses of the austenstic phase transformation, ${\sigma }_{\mathrm{as}}^{(\mathrm{n})}$ and ${\sigma }_{\mathrm{af}}^{(\mathrm{n})}$, are obtained for the temperature T(n) = T(n − 1) + dT. Assuming that the initial phase of the SMA is austenite and the initial loading process is austenitic linear elastic loading without a martensitic phase transformation, then ξ₀ = 0, η⁽ⁿ⁾ = 0, dη⁽ⁿ⁾ = 0. Substituting the above conditions into Equations (13), (17) and (18) yields:

$$\left\{\begin{array}{c}\mathrm{d}{\sigma }_{kl}^{(\mathrm{n})}=C_{ijkl}^{\mathrm{A}}(\mathrm{d}{\epsilon }_{ij}-{\alpha }^{\mathrm{A}}\mathrm{d}T),\\ {\sigma }_{kl}^{(\mathrm{n}+1)}={\sigma }_{kl}^{(\mathrm{n})}+\mathrm{d}{\sigma }_{kl}^{(\mathrm{n})}\end{array}\right.$$

(19)

where dε_ij and dT denote the incremental components of strain and temperature within each incremental step, respectively.

The SMA undergoes a martensitic phase transformation process, and then the stress state results at the (n_th) step are obtained and substituted into Equations (2)–(4) and (18), which yields:

$$\left\{\begin{array}{c}\begin{array}{c}{\eta }^{(\mathrm{n})}=\frac{3}{4}[\frac{2({\sigma }_{\mathrm{eq}}^{(\mathrm{n})}-{\sigma }_{\mathrm{ms}}^{(\mathrm{n})})+{\sigma }_{\mathrm{scr}}-{\sigma }_{\mathrm{fcr}}}{{\sigma }_{\mathrm{scr}}+{\sigma }_{\mathrm{fcr}}}]-\frac{1}{4}{[\frac{2({\sigma }_{\mathrm{eq}}^{(\mathrm{n})}-{\sigma }_{\mathrm{ms}}^{(\mathrm{n})})+{\sigma }_{\mathrm{scr}}-{\sigma }_{\mathrm{fcr}}}{{\sigma }_{\mathrm{scr}}+{\sigma }_{\mathrm{fcr}}}]}^3+\frac{1}{2} ,\end{array}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{d}{\eta }^{(\mathrm{n}-1)}={\eta }^{(\mathrm{n})}-{\eta }^{(\mathrm{n}-1)}\hfill \end{array}\right.$$

(20)

and substituting the results of Equation (20) into Equations (13)–(17) yields:

$$\mathrm{d}{\epsilon }_{ij}^{\mathrm{tr}(\mathrm{n}-1)}={\epsilon }_{\mathrm{L}}{\Lambda }_{ij}^{(\mathrm{n}-1)}\mathrm{d}{\eta }^{(\mathrm{n}-1)}$$

(21)

$$\mathrm{d}{\sigma }_{ij}^{(\mathrm{n})}=C_{ijkl}^{(\mathrm{n})}(\mathrm{d}{\epsilon }_{ij}-\mathrm{d}{\epsilon }_{ij}^{\mathrm{tr}(\mathrm{n}-1)}-{\Theta }_{ij}^{(\mathrm{n})}\mathrm{d}T)$$

(22)

The state variables are updated according to Equation (18) until the martensitic phase transformation of the SMA is complete; then, the loading process follows the martensitic linear elastic loading law.

During the unloading process, the iterative method is also used. Furthermore, the unloading process is divided into three stages, namely, the martensite linear elastic unloading stage, the martensite inverse phase transformation stage and the austenite linear elastic unloading stage. The initial phase of the SMA is martensite and the initial unloading process is martensite linear elastic unloading without martensite inverse phase transformation, and thus ξ₀ = 1, η⁽ⁿ⁾ = 1, dη⁽ⁿ⁾ = 0. Substituting the above conditions into Equations (13), (17) and (18) yields:

$$\left\{\begin{array}{c}\mathrm{d}{\sigma }_{kl}^{(\mathrm{n})}=C_{ijkl}^{\mathrm{M}}(\mathrm{d}{\epsilon }_{ij}-{\alpha }^{\mathrm{M}}\mathrm{d}T),\\ {\sigma }_{kl}^{(\mathrm{n}+1)}={\sigma }_{kl}^{(\mathrm{n})}+\mathrm{d}{\sigma }_{kl}^{(\mathrm{n})}\end{array}\right.$$

(23)

The SMA undergoes a martensitic inverse phase transformation process, and η₀ = 1 until the martensitic inverse phase transformation of the SMA is complete. Reading the stress state at the (n_th) step and substituting the results into Equations (3)–(6) and (18) yields:

$$\left\{\begin{array}{c}{\eta }^{(\mathrm{n})}=\frac{3}{4}[\frac{{\sigma }_{\mathrm{eq}}^{(\mathrm{n})}-{\sigma }_{\mathrm{af}}^{(\mathrm{n})}}{C_{\mathrm{A}}(A_{\mathrm{f}}-A_{\mathrm{s}})}-\frac{1}{2}]-\frac{1}{4}{[\frac{{\sigma }_{\mathrm{eq}}^{(\mathrm{n})}-{\sigma }_{\mathrm{af}}^{(\mathrm{n})}}{C_{\mathrm{A}}(A_{\mathrm{f}}-A_{\mathrm{s}})}-\frac{1}{2}]}^3+\frac{1}{2},\\ \mathrm{d}{\eta }^{(\mathrm{n}-1)}={\eta }^{(\mathrm{n})}-{\eta }^{(\mathrm{n}-1)}\end{array}\right.$$

(24)

Equation (24) is substituted into Equations (15), (21) and (22), and the state variables are updated according to Equation (18) until the end of the inverse phase transformation, when the SMA is completely transformed into austenite; then, the unloading process follows the austenite linear elastic unloading law.

5. Finite Element Discrete Modeling Method

Consider FG-SMA structures: the material properties are assumed to vary continuously in a specific direction and obey a graded distribution, which makes it impossible to numerically model them realistically with a finite element method. To accomplish this, a discrete modeling method is proposed for the effective numerical simulation of the FG-SMA structures as follows:

(1)

Full-scale modeling of FG-SMA structure;

(2)

The FG-SMA structure is stratified or segmented along the graded direction to establish a discrete model of the FG-SMA structure;

(3)

The material parameters of each layer or segment in the FG-SMA discrete structure are derived from the coordinates (e.g., surface coordinates, geometrical midplane coordinates, etc.) and within a single layer segment, the material parameters are kept homogeneous;

(4)

Between adjacent layers or segments, deformation maintains continuous consistency;

(5)

A finite element model of the FG-SMA discrete structure is numerically established and solved.

6. Models and Material Parameters

6.1. Models and Boundary Conditions

As shown in Figure 2 and Figure 3, a rectangular section beam of length L, width b and constant thickness h is considered defined in the Cartesian coordinate system (0 ≤ x ≤ L, −b/2 ≤ y ≤ b/2, −h/2 ≤ z ≤ h/2) and subjected to an end-concentrated force of magnitude P in the z-direction. In order to facilitate the modeling, a transverse FG-SMA cantilever beam is discretized and divided into 10 layers along the thickness direction, as shown in Figure 2, and an axial FG-SMA cantilever beam is discretized and divided into 20 segments along its length, as shown in Figure 3. The material parameters, i.e., the SMA austenitic modulus of elasticity E_a, martensitic modulus of elasticity E_m, phase transformation starting stress σ_ms, phase transformation finishing stress σ_mf and maximum phase transformation strain ε_L, are kept constant within a single layer/segment.

For the transverse FG-SMA cantilever beam in Figure 2, the material parameters vary along the graded direction according to the following pattern:

$$\Psi (x)={\Psi }_{}^0[(e-1)(1-\frac{z}{h})+1]$$

(25)

where Ψ(x) represents the material parameter, Ψ⁰ represents the initial value of the material parameter, e is the functionally graded parameter and z represents the coordinate value of the upper surface of the single layer of the cantilever beam.

For the axial FG-SMA cantilever beam in Figure 3, the material parameters vary along the graded direction according to the following pattern:

$$\Psi (x)={\Psi }_{}^0[(e-1)\frac{x}{h}+1]$$

(26)

where x denotes the coordinate value of the left surface of each segment of the cantilever beam.

The FG-SMA finite element discrete model is established, in which full displacement constraints are applied to the fixed end of the cantilever beam, a concentrated force load P is applied to the free end, and the cell type is a C3D20R solid cell.

6.2. Material Parameters

The material parameters of the FG-SMA cantilever beams are shown in

Table 1. Material parameters of the FG-SMA cantilever beam [23].

${\mathit{E}}_{\mathbf{a}}^{\mathbf{0}}$/GPa	${\mathit{E}}_{\mathbf{m}}^{\mathbf{0}}$/GPa	μ	${\mathsf{\epsilon }}_{\mathbf{L}}^{\mathbf{0}}$
70	30	0.43	0.069
${\mathit{C}}_{\mathbf{A}}^{\mathbf{0}}$/(MPa/°C)	${\mathit{C}}_{\mathbf{M}}^{\mathbf{0}}$/(MPa/°C)	${\mathsf{\sigma }}_{\mathbf{scr}}^{\mathbf{0}}$/MPa	${\mathsf{\sigma }}_{\mathbf{fcr}}^{\mathbf{0}}$/MPa
8	13	100	170
${\mathit{M}}_{\mathbf{f}}^{\mathbf{0}}$/°C	${\mathit{M}}_{\mathbf{s}}^{\mathbf{0}}$/°C	${\mathit{A}}_{\mathbf{s}}^{\mathbf{0}}$/°C	${\mathit{A}}_{\mathbf{f}}^{\mathbf{0}}$/°C
7	15	45	56

E_a—Austenitic elastic modulus, E_m—Martensite elastic modulus, μ—Poisson’s ratio of SMA, ε_L—Maximum transformation strain, M_f—Martensite finish temperature, M_s—Martensite start temperature, A_s—Austenitic start temperature, A_f—Austenitic finish temperature, σ_scr—Martensite initial start stress, σ_fcr—Martensite initial finish stress, C_A—Austenitic stress influence coefficient, C_M—Martensite stress influence coefficient.

, and the initial state of the cantilever beams is the austenitic phase with an ambient temperature of 60 °C.

7. Numerical Simulation

In this section, the mechanical and martensitic transformation behaviors of SMA cantilever beams are numerically analyzed. Numerical results for the bending and shape memory behaviors of SMA homogeneous cantilever beams are presented in Figure 4, Figure 5 and Figure 6.

Figure 4 shows the stress distribution in the cross section of the SMA homogeneous cantilever beam suffering the concentrated force with different applied loads P (while e = 1, and T = 60 °C). As can be seen from Figure 4, the stress distribution in the cantilever beam section is symmetrical along the thickness direction, and the maximum values of the stresses appear on the upper and lower surfaces of the fixed end. With the gradual increase of the load, the internal stress in the cantilever beam cross section increases gradually.

Figure 5 shows the shape memory factor distribution in the cross section of the SMA homogeneous cantilever beam when the applied concentrated force load P is 3.0 kN, 3.5 kN and 4.0 kN (while e = 1 and T = 60 °C). As can be seen from the results in the figure, the martensitic phase transformation has not yet occurred within the cross section when the load P is 3.0 kN. When the load P is 3.5 kN, a partially martensitic phase transformation occurs in the cross section of the cantilever beam. When the load P is 4.0 kN, a complete martensitic phase transformation occurs in the cross section of the cantilever beam. The martensitic phase transformation distribution in the cantilever beam section is symmetrical along the thickness direction in the loading process. It should be noted that the complete martensitic phase transformation does not occur at the fixed end of the cantilever beam, but at a distance of about 10 mm from the fixed end when the load P is 4.0 kN. In addition, the maximum depth of the complete martensitic transformation zone is 3 mm.

Figure 6 shows the deflection w of SMA homogeneous beams suffering the concentrated force load with different applied loads P (while e = 1 and T = 60 °C). As can be seen from the curve in Figure 6, when the concentrated force load P is set to be 3.0 kN, 3.5 kN and 4.0 kN, the corresponding maximum deflection values at the free end of the FG-SMA cantilever beam are 16.96 mm, 22.11 mm and 52.00 mm, respectively. The results in the figure show that the deflection of the beam increases gradually with the incrementing of the applied load. The change in the increase rate shows a nonlinear trend, especially when the applied load P is 4.0 kN, which is due to the occurrence of a martensitic phase transformation in the FG-SMA cantilever beam with the incrementing of the applied load.

Numerical results for the bending and shape memory behaviors of transverse FG-SMA cantilever beams are presented in Figure 7, Figure 8, Figure 9 and Figure 10. Figure 7 shows the stress distribution in the cross section of the transverse FG-SMA cantilever beam suffering the concentrated force with different applied loads P (while e = 5, and T = 60 °C). As can be seen from Figure 7, the absolute value of the stress in the cross section of the transverse FG-SMA cantilever beam is asymmetrically distributed along the thickness direction. The maximum value of the stress occurs on the upper surface of the fixed end, which is due to the graded change of the material parameters along the thickness direction.

Figure 8 shows the shape memory factor distribution in the cross section of the transverse FG-SMA cantilever beam when the applied concentrated force load P is 6.0 kN, 8.0 kN and 10.0 kN (while e = 5 and T = 60 °C). From the results in the figure, it can be seen that martensitic phase transformation has not yet occurred within the cross section when the load P is 6.0 kN. When the load P is 8.0 kN, a partially martensitic phase transformation occurs in the cross section of the cantilever beam. When the load P is 10.0 kN, a complete martensitic phase transformation occurs in the cross section of the cantilever beam. It should be noted that the martensitic phase transformation distribution in the cantilever beam section is asymmetrical along the thickness direction in the loading process. The maximum value of the shape memory factor occurs at the lower surface of the fixed end, which is due to the martensite phase transformation starting stresses in the lowest layer of the cantilever beam being smaller.

Figure 9 shows the deflection w of the transverse FG-SMA cantilever beam suffering a concentrated force load with different applied loads P (while e = 1 and T = 60 °C). As can be seen from the curve in Figure 9, when the concentrated force load P is set to be 6.0 kN, 8.0 kN and 10.0 kN, the corresponding maximum deflection values at the free end of the FG-SMA cantilever beam are 14.10 mm, 20.66 mm and 56.57 mm, respectively. The change of the increase rate shows a nonlinear trend, especially when the applied load P is 10.0 kN, which is due to the occurrence of a martensitic phase transformation in the FG-SMA cantilever beam with the incrementing of the applied loads.

Figure 10 shows the comparison of the maximum deflection of the transverse FG-SMA cantilever beam and SMA homogeneous beam suffering a concentrated force load with different applied loads P (while e = 5 and T = 60 °C). As can be seen from the curves in the figure, the load–deflection curves of both the transverse FG-SMA cantilever beam and the SMA homogeneous beam exhibit nonlinear characteristics. A comparison of the curve results in the figure indicates that the maximum deflection of the transverse FG-SMA cantilever beam is smaller under the same load. With the same maximum deflection deformation, the load applied to the transverse FG-SMA beam is three times that of the SMA homogeneous beam, which indicates that the transverse FG-SMA cantilever beam has a stronger load-bearing capacity.

Numerical results for the bending and shape memory behaviors of axial FG-SMA cantilever beams are presented in Figure 11, Figure 12 and Figure 13.

Figure 11 shows the stress distribution in the cross section of the axial FG-SMA cantilever beam suffering ta concentrated force with different applied loads P (while e = 5 and T = 60 °C). As can be seen from Figure 11, similar to the SMA homogeneous beam, the stress distribution in the axial FG-SMA cantilever beam is symmetrical along the thickness direction, and the maximum values of the stresses appear on the upper and lower surfaces of the fixed end. It should be noted that the stress distribution position within the cross section of an axial FG-SMA cantilever beam is different from that of an SMA homogeneous beam. Compared with the results in Figure 4, the stress distribution in the axial FG-SMA cantilever beam is concentrated at the fixed-end position, while the stress distribution in the SMA homogeneous beam is more pronounced at the position far from the fixed end. The variation of material parameters along the axial direction of the axial FG-SMA cantilever beam results in a change in the stress distribution under the same external load, which can effectively guide the structural design of axial FG-SMAs.

Figure 12 shows the shape memory factor distribution in the cross section of the axial FG-SMA cantilever beam when the applied concentrated force load P is 3.0 kN, 4.0 kN and 5.0 kN, respectively (while e = 1 and T = 60 °C). From the results in the figure, it can be seen that martensitic phase transformation has not yet occurred within the cross section when the load P is 3.0 kN. When the load P is 4.0 kN, a partially martensitic phase transformation occurs in the cross section of the cantilever beam. When the load P is 5.0 kN, a complete martensitic phase transformation occurs in the cross section of the cantilever beam. Similar to the SMA homogeneous beam, the martensitic phase transformation distribution in the cantilever beam section is symmetrical along the thickness direction in the loading process. When the load P is 5.0 kN, a complete martensitic phase transformation zone occurs at a distance of about 7.5 mm from the fixed end, and the maximum depth of the complete martensitic phase transformation zone is 6 mm. Compared with the results in Figure 5, the complete martensitic phase transformation zone in the axial FG-SMA cantilever beam is closer to the fixed end, which indirectly reflects the influence of material parameter changes along the axial direction on the phase transformation behavior of axial FG-SMA structures.

Figure 13 shows the deflection w of the axial FG-SMA cantilever beam suffering a concentrated force load with different applied loads P (while e = 1 and T = 60 °C). As can be seen from the curve in Figure 13, when the concentrated force load P is set to be 3.0 kN, 4.0 kN and 5.0 kN, the corresponding maximum deflection values at the free end of the FG-SMA cantilever beam are 10.23 mm, 23.85 mm and 50.85 mm, respectively. Compared with the curve results of the SMA homogeneous beam in Figure 6, the maximum deflection of the axial FG-SMA cantilever beam is smaller under the same load, indicating an enhanced bearing capacity.

Figure 14 shows the comparison of the maximum deflections of the FG-SMA cantilever beam and SMA homogeneous beam suffering a concentrated force load with different applied loads P (while e = 5 and T = 60 °C). As can be seen from the curves in the figure, the load–deflection curves of both the FG-SMA cantilever beam and the SMA homogeneous beam exhibit nonlinear characteristics. A comparison of the curve results in the figure indicates that the maximum deflection of the SMA homogeneous cantilever beam is the smallest under the same load. With the same maximum deflection deformation, the load on the transverse FG-SMA beam is the largest, followed by the load on the axial FG-SMA beam, and the load on the SMA homogeneous beam is the smallest. This indicates that the FG-SMA cantilever beam has a stronger load-bearing capacity, and compared to the axial FG-SMA cantilever beam, the transverse FG-SMA cantilever beam has greater stiffness.

8. Conclusions

Based on the concept of the shape memory factor, the incremental intrinsic equation and shape memory evolution equation of shape memory alloy (SMA) were derived, and a nonlinear finite element numerical calculation method for FG-SMA structures was constructed. By numerically simulating the mechanical and phase transformation behavior of FG-SMA cantilever beams during loading and comparing the numerical results with those of SMA homogeneous beams, the following conclusions can be drawn:

(1)

The phase transformation and mechanical response in the cross section of the axial FG-SMA cantilever beam and SMA homogeneous beam are symmetrically distributed along the thickness direction, but those of the transverse FG-SMA cantilever beam are asymmetric along the thickness direction. The FG-SMA cantilever beam has a stronger load-bearing capacity, and compared to the axial FG-SMA cantilever beam, the transverse FG-SMA cantilever beam has greater stiffness.

(2)

It should be noted that for the SMA homogeneous cantilever beam subjected to concentrated loads at the free end, the martensitic phase transformation begins not at the fixed end of the cantilever beam during the loading process, but at a distance of about 10 mm from the fixed end. The martensitic phase transformation in the axial FG-SMA cantilever beam begins closer to the fixed end, which indirectly reflects the influence of material parameters along the axial direction on the phase transformation behavior of the axial FG-SMA structure.

(3)

Compared with the SMA homogeneous beam, the stress distribution in the axial FG-SMA cantilever beam is concentrated at the fixed-end position, while the stress distribution in the SMA homogeneous beam is more pronounced at the position far from the fixed end. The variation of the material parameters of the axial FG-SMA cantilever beam leads to changes in stress distribution under the same external load, which can effectively guide the structural design of axial FG-SMA structures.

9. Novelty and Application

In this work, we proposed an ABAQUS user-defined material subroutine (UMAT) by using the incremental constitutive equation of an SMA and established a discrete modeling method to numerically simulate the phase transformation and mechanical behaviors of an FG-SMA based on the finite element method. The simulation results are compared with the relevant results of an SMA homogeneous beam, including the distribution of stress and the shape memory factor, as well as the bearing capacity.