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Nonlinear Finite Element Method Considering Martensite Plasticity For Shape Memory Alloy Structure

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Abstract

This work presents a nonlinear finite element method to simulate the macroscopic mechanical responses and the effects of martensite plasticity in a shape memory alloy (SMA) structure. A linear relationship formulation is adopted to express the influence of martensite plasticity on the inverse martensitic phase transition of SMA material. Incorporating with a trigonometric-type phase transition evolution law and an exponential-type plastic flow evolution law, an incremental mechanical model with two internal variables is supposed based on the macroscopic experimental phenomena. A nonlinear finite element equation is formulated and solved by the principle of virtual displacement and Newton-Raphson method respectively. By employing the proposed nonlinear finite element method, the uniform tensile bar and three-point bending beam are simulated and analyzed. Results illustrate that the presented nonlinear finite element method is suitable to act as an effective computational tool for the wide applications based on the SMA material considering the effects of martensite plasticity because all material constants related to the method can be obtained from macroscopic experiments.

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Abbreviations

a :

Nodal displacement matrix

A f :

Austenite finishing temperature, °C

A s :

Austenite starting temperature, °C

B :

Geometric matrix

C A :

Stress-temperature relation coefficient of austenite, MPa/°C

C M :

Stress-temperature relation coefficient of martensite, MPa/°C

\(\hat{\boldsymbol{D}}\) :

Tangent elastic matrix

D e :

Elastic matrix

E :

Elastic modulus, MPa

E a :

Austenite elastic modulus, MPa

E m :

Martensite elastic modulus, MPa

F p :

Direction vector of plastic incremental strain

F tr :

Direction vector of phase transition incremental strain

H p :

Plastic tangent modulus, MPa

H tr :

Phase transition tangent modulus, MPa

J :

Jacobian matrix

M s :

Martensite starting temperature, °C

P :

Nodal load matrix

T :

Ambient temperature, °C

α :

Thermal expansion coefficient, (°C)−1

ε :

Total strain

ε e :

Elastic strain

ε L :

Maximum phase transition strain

ε p :

Plastic strain

\({\bar \varepsilon ^{\rm{p}}}\) :

Equivalent plastic strain

ε pc :

Critical plastic strain

ε th :

Thermal strain

ε tr :

Phase transition strain

\({\bar \varepsilon ^{{\rm{tr}}}}\) :

Equivalent phase transition strain

Λ :

Thermal expansion vector

ξ :

Volume fraction of martensite

ξ s :

Volume fraction of stress-induced martensite or detwinned martensite

ξ us :

Volume fraction of unrecoverable stress-induced martensite or detwinned martensite

σ :

Stress, MPa

\(\bar \sigma \) :

Equivalent stress, MPa

σ af :

Austenite finishing stress, MPa

σ as :

Austenite starting stress, MPa

σ crf :

Initial martensite finishing stress, MPa

σ crs :

Initial martensite starting stress, MPa

σ mf :

Martensite finishing stress, MPa

σ ms :

Martensite starting stress, MPa

σ yc :

Yield stress, MPa

φ p :

Plastic potential

φ tr :

Phase transition potential

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Authors and Affiliations

  1. Department of Engineering Mechanics, College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao, Shandong, 266580, China

    Bo Zhou  (周博), Zetian Kang  (康泽天), Zhiyong Wang  (王志勇) & Shifeng Xue  (薛世峰)

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  1. Bo Zhou  (周博)
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  2. Zetian Kang  (康泽天)
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  3. Zhiyong Wang  (王志勇)
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  4. Shifeng Xue  (薛世峰)
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Corresponding author

Correspondence to Bo Zhou  (周博).

Additional information

Foundation item: the National Key Research and Development Program of China (No. 2017YFC0307604)

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Zhou, B., Kang, Z., Wang, Z. et al. Nonlinear Finite Element Method Considering Martensite Plasticity For Shape Memory Alloy Structure. J. Shanghai Jiaotong Univ. (Sci.) 26, 774–785 (2021). https://doi.org/10.1007/s12204-021-2327-z

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  • DOI: https://doi.org/10.1007/s12204-021-2327-z

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