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Based on the experimental results and the finite element analysis, a constitutive model is proposed for two phase shape memory alloys by introducing a compensative volumetric strain into a constrained relationship between the two phases, accounting for the reduced constraint due to the growth of martensite band. The pseudoelasticity of NiTi shape memory alloy micro-tube, subjected to pure tension, is analyzed and compared with the experimental results. It can be seen that the pseudoelastic behavior, especially the phenomena of a stress drop during tension processes, can be well described with the proposed model. The proposed model separates the complicated constitutive behavior of a shape memory alloy (SMA) into simple responses arising respectively from its two phases, taking into account laminar microstructure, the thickness of martensite phase and the interaction between the two phases, and provides an easy but comprehensive method for the description of the constitutive behavior of SMAs under complex thermomechanical loading.

Shape memory alloys (SMAs) have been receiving increasing attention in recent years, due to their particular properties under thermomechanical loading, such as ferroelasticity, shape memory effect and pseudoelasticity. These properties are related to the martensitic phase transformation and are extensively used in many fields, such as aviation, national defense, instruments and medical devices,

With the improvement of experimental facilities, many new experimental phenomena have been discovered [

It is recognized that the macroscopic property of a material strongly depends on its microstructures. Great progress has been made in the constitutive relationships for SMAs in the past 20 years, and especially in recent years [

In this paper, the response of a NiTi SMA micro-tube subjected to pure tension is systematically investigated, taking into account the phase-transformation microstructures and their evolution. It is found that the physical and mechanical mechanism of the distinct stress drop can be attributed to the variation of the microstructure,

A two-phase constitutive model for SMAs is proposed based on the concept that an SMA is composed of austenite and martensite, and the constitutive behavior of the SMA is substantially a combination of that of each phase. With the specified ranges of stress and temperature, the behavior of martensite is assumed elastoplastic while that of austenite is assumed linearly elastic. In the following part, the discussion is restricted to the case of small deformation and the material is assumed plastically incompressible.

Iwan (1967) [

In _{r}_{r}_{r}_{r}^{p}^{(}^{r}^{)} is the generalized force conjugated with the ^{(}^{r}^{)} and the following inequality should be satisfied if any change occurs to ^{(}^{r}^{)} [

From

in which ^{(}^{r}^{)} is assumed to relate the response of the _{r}_{r}

where

where _{r}

in which

The combination of

where:

substituting

in which:

The deviatoric strain ^{e}^{p}^{T}

Making use of

For simplicity, the transformation lattice volume change is neglected in the present stage since for most SMAs it is negligible compared with the lattice shear deformation [

The differential of

Combining

where:

Keeping in mind that:

and:

one can derive the following equation from

where:

In this research, the constitutive model is proposed for SMAs based on experimental results and finite element analysis, taking into account laminar microstructure, the thickness of martensite phase and the interaction between the two phases. The representative volume element (RVE) of an SMA is shown in

Assuming that the in-plane strain components and out-of-plane stress components in both martensite and austenite in all lamellae are identical and equal respectively to the corresponding components of the overall strain and stress in the SMA, the conventional mixture theory gives:

where the superscripts

The other components of stress and strain can be determined by volume average as follows:

where ξ is the volume fraction of martensite.

The differential form of

with:

The constitutive model for martensitic phase

where:

in which
_{M}].

By the same way, one can easily obtain the following matrix:

where
_{A}].

Based on the study of the mechanical behavior, each phase responds when the two phases that coexist differ from the individual mechanical behavior of each phase. Volumetric strain compensation method is proposed to observe and study the effect of structure on the response of the mechanical behavior of each phase, while the hypothetical relation of strain compatibility is used to propose the constitutive relationship which can take into account the laminar spacing. Under the assumptions of small deformation—where isotropy is initially incompressible plastic—the simple mechanical model as in

Considering the constrained relationship between the two phases, one can use the following method to introduce the different effect of the constraint on the properties of the two phases.

in which κ is additional volumetric strain caused by different laminar spacing. This is the key point in this paper, which is quite different between this model and others. Additional volumetric strain is used to introduce the constrained relationship, which would be much more convenient and simple compared with other strains, while another model introduces the constrained relationship by the assumption that the in-plane strain has the following relationship [

where η is a factor of incompatible strain.

For soft phase, when it is tensed in the direction perpendicular to the lamina, the cross section of the soft phase will shrink more than the hard one, while the model assumes that the in-plane strain components should be the same. Inhibiting the shrinkage of the soft phase is in fact increasing the volumetric tensile stress in the soft phase, and it approximately works only when laminar spacing is small. However, when the laminar spacing is large, the in-plane constraint will decrease a lot, so much as to disappear. When it is tensed in parallel to the direction of the lamina, soft phase advance into the plastic deformation as the Poisson ratio of the plastic deformation is larger, so there is also a concern that the volumetric shrinkage was constrained.

The differential of

The parameter κ can be simply shown as:

where

The differential of

It can be obtained from

For martensitic phase,

where:

For austentic phase, one can easily obtain the following equation from

where:

in which κ and α are respectively corresponding to martensitic and austenitic material parameters from

For martensite:

that is:

in which:

By the same way, austenitic constitutive equation can be written as

where:

where _{1} and _{2} is decided by the difference of the material properties of the two phases. The value of _{1} and _{2} can be obtained from the finite element analysis, while _{1} and _{2} can contain information on laminar spacing so that it can introduce the effect of the laminar spacing. For SMA, in order to reflect that the effect of the hard and soft phase on incompatibility of two phases is different when their volumetric fraction change, generally speaking, there yields

when _{1}(h) = _{2}(1−h).

Taking into account the laminar microstructure, the dynamic change of the thickness of martensite and austenite during phase transformation and the constraints of the two phases, the in-plane strain coordinate and out-plane stress coordinate can be deduced, so that the constitutive relationships for SMAs which take into account the laminar microstructure can be obtained.

It can be obtained from

It can also be obtained from

Substituting

From the response formula of the martensite phase

Expanding

One can derive the following equation from

Substituting

Similarly, for austenite phase, one can easily obtain that:

Substituting

Among them: the concrete forms of
_{M}] of martensite in _{A}] of austenite in

Sun

Since the experiment was performed at room temperature and under quasistatic condition, the effect of temperature on the material properties can be neglected. In _{M}_{ij}_{e}_{ij}_{ij}^{1/2} is equivalent strain, and

The values of the material constants _{r}_{r}_{r} and a_{r} are fixed. So, the material constants _{r}_{r}

The numerical process is strain-controlled. For uniaxial tension—keeping in mind that the nucleation and propagation of a macroscopic martensite band was observed during the test under uniaxial tension, so its microstructure is simplified to be laminar—the relationship of the volume fraction of martensite ξ and the stretched strain is showed by

Using the constitutive model of strain compatibility based on volumetric compensation which is obtained in the previous chapter to compute mechanical behavior of a micro-tube under pure tensile: in the concrete computation _{1} = _{2} = 2, _{1} = 25, _{2} = 7.5 (material constants are listed in _{1} and _{2} have been assigned fixed values, it makes the computation locally different with the results of the experiment, and at the same time, it also makes _{1} and _{2} by finite element analysis in order to be better used for the computation of the proposed constitutive model.

A two-phase constitutive model for polycrystalline SMAs with laminar microstructure—based on the concept that a SMA is composed of austenite and martensite—and a new way to introduce the constraints of the two phases are proposed. The main characteristics of SMAs such as ferroelasticity and pseudoelasticity can be described with the proposed model and especially the phenomena of a stress drop during tension processes. This constitutive model can also explain the stress–strain relationship of other complicated structural materials like functionally graded material [

The responses of a NiTi SMA micro-tube subjected to pure tension were investigated and compared with the experimental results, taking into account the phase-transformation microstructures. The comparison between the calculated and experimental results shows satisfactory agreement.

The complicated constitutive behavior of a polycrystalline SMA is separated into the simple constitutive behavior of its two phases under the constraint with each other, which provides a simple but comprehensive description for the constitutive behavior of SMAs. The ratio of the in-plane strain components between martensite to austenite η is an important parameter in the model, which is the function of the stress state σ, temperature

The authors are grateful for support from the National Natural Foundation of China under Grant Nos. 90916009 and 11172336, and the Program for New Century Excellent Talents in University under Grant No. NCET-13-0634.

The authors declare no conflict of interest.

Mechanical model for the contribution of martensite.

A representative volume element of SMA, _{m}

The variation of tension–strain

The nominal tensile stress–strain curve of micro-tube.

The relationship between

Material constants.

Material | G (GPa) | _{1}; _{2}; _{3} (GPa) |
α_{1}; α_{2}; α_{3} | |
---|---|---|---|---|

Martensite | 18.45 | 0.167 | 50,000; 0; 0. | 500; 60; 5 |

Austenite | 13.8 | 0.167 | 16,500,000; 500,000; 0. | 30,000; 2000; 500 |