The propagation of phase transition wave (macroscopic phase boundary) under impact in materials attracts more and more attention recently^[1]. Chen and Lagoudas^{[2, 3]} studied the longitudinal stress wave propagation along a semi-infinite rod of shape memory alloy (SMA) under impulsive loads. Berezovski and Maugin^[4] discussed the propagation of stress induced phase transition fronts in inhomogeneous thermo-elastic solids. Recently,Tang et al.^{[5, 6, 7, 8, 9]} investigated more systematically the propagation features of one-dimensional (1D) phase transition waves induced by thermo-elastic martensitic transition. Some novel phenomena such as bifurcation and acceleration of the unloading phase boundaries were found in their research and an approach of making functionally graded material based on the irreversible phase transition under shock loading was further proposed. However,the works mentioned above only dealt with 1D longitudinal waves and the research regarding the combined stress wave propagation in phase transition materials has rarely been reported.

nsition materials has rarely been reported. SMA is one of the newest functional materials. The shape memory effect and the pseudoelastic effect of SMA are due to the austenite to martensite phase transition under certain stress and temperature. NiTi SMA has good shape memory effect if the atomic ratio of Ti and Ni is about 1:1. Recently,the tension-compression asymmetry of NiTi is found in axial tension and compression experiments^{[10, 11, 12, 13, 14, 15]}. Qidwai and Lagoudas^[16] attributed this asymmetry to the effect of hydrostatic pressure stress and presented a thermodynamic constitutive relation which can predict the tension-compression asymmetry of SMA. Most studies have shown that the phase transition in solid is affected by both hydrostatic pressure and deviator stress besides temperature^{[1, 5]}. Recently,Guo et al. ^[17] proposed a 3D criterion by considering both hydrostatic pressure and deviatoric stress effects. The critical phase transition surface predicted by this criterion is a conic surface in the principal stress space,demonstrating the asymmetric property in tension and compression states,which is well agreed with the experiment results. The existing combined stress wave theory is based on the conventional elasto-plasticity that assumes the hydrostatic pressure does not affect the yield surface. Hence,how does the hydrostatic pressure influence the propagation features of combined stress waves with phase transition? Is there any new phenomenon? These are worth exploring.

In this paper,according to the research method of combined stress wave in elastic-plastic materials^[18],the characteristic equations and eigen solutions of the combined phase transition stress waves in NiTi SMA thin walled tubes are established by simplify the phase transition criterion of Guo et al. ^[17]. The features of the combined phase transition stress waves from the austenite phase region to the mixed phase region are described and discussed. Some new phenomena which differ from those of conventional elastic-plastic materials are found.

Consider a semi-infinite thin-walled tube with phase transition subject to the combined longitudinal and torsion pulses load at the free end as shown in Fig. 1. Assume that the thickness of the tube is far thinner than its radius. Then,the radial inertial effect and stress inhomogeneity can be ignored,and there are 1D plane waves propagating along the tube. Therefore,the longitudinal and circumferential particle velocities u and v,the longitudinal stress and strain σ and ε,the shear stress and strain τ and γ are all the functions of the Lagrangian coordinate x and the time t .

Fig. 1Geometry of thin-walled tube.

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The longitudinal and circumferential continuity equations in the tube can be written as

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Conservation of the longitudinal and circumferential momentums in the tube gives

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where ρ is the mass density.

It is well-known that phase transition can make the discontinuity both in volume and shape of a material. Take the tensile stress and strain with positive values. Then,the stress σ_ij and the transition strain discontinuity ε^pt_ij can be decomposed into volumetric and deviatoric components as follows:

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where ,s_ij,ε^pt_v = ε^pt_kk,and γ^pt_ij are the hydrostatic pressure,deviatoric stress tensor, volumetric strain,and deviatoric strain tensor caused by phase transition,respectively.

Recently,based on the viewpoint of energy^[17],a 3D criterion for “stress induced” phase transition is proposed by considering both hydrostatic pressure and deviatoric stress effects as follows:

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where is the von Mises effective stress, is the effective deviatoric strain of transition,Φ(ξ) is a function of the volume fraction ξ of the transformed phase and temperature T.

Since ε^pt_v and γ^pt_eff are constants for a certain phase transition at T,Eq. (5) means that both p and σ_eff affect the critical condition of transition as reported in numerous references. However, it can be divided into two types: (i) p is the leading factor for a lot of high pressure phase transitions; (ii) σ_eff is the leading factor for most thermo-elastic martensitic phase transitions in which the main deformation is distortion with small volume change. We shall limit the present study to the second type of transition. Thus,the critical phase transition surface predicted by Eq. (5) appears as a conic surface in the principal stress space as shown in Fig. 2. For ε^pt_v > 0 (volume expansion during transition),the asymmetric property in the tension and compression states differs from the conventional cylindrical yield surface in the same space. For ε^pt_v < 0,the conic surface will have the opposite direction.

Fig. 2Sketch for critical phase transition surface in principal stress space for εptv > 0.

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For the case of a tube subjected to combined longitudinal and torsion loading denoted as (σ,τ),we can get p = − σ/3 and from Eq. (3). Then,Eq. (5) can be expressed as

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Let and ,where α means the ratio of the volumetric change respect to the shape change caused by phase transition,and φ means a kind of nominal phase transition stress. Equation (6) can be written to a standard elliptical form with mathematical transformation

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where .

The critical surface of phase transition based on Eq. (7) is an ellipse in the στ-plane,but its center (denoted as Ω in Fig. 3) shifts left or right on the σ axis depending on the sign of ε^pt_v . If the volume of the material during transition is expanding,as the austenitic to martensitic phase transition of NiTi alloy,ε^pt_v is positive whileβis negative,and the center of the initial phase transition ellipse will move leftwards (see Fig. 3(a)). If the volume of the phase transition is contractive,the opposite results will be obtained (see Fig. 3(b)),showing the tension-compression asymmetry. Especially,if the volume change is zero during the phase transition,Eq. (7) degenerates to the von-Mises yield function. For simplicity,the parameters α and φ are set to be constant in this work,and their values can be determined from the experimental results.

Fig. 3Initial phase transition ellipse in στ-plane.

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The shape memory effect and pseudo-elastic effect of SMA are due to the austenite to martensite phase transition. A sketch of the simple tension stress-strain curves for both shape memory effect and pseudo-elastic effect is shown in Fig. 4. In the present work only the loading curve OAB is involved. Lines OA and AB denote the elastic stage in the austenite phase and the hardening stage in mixed phase,respectively,which is similar to the loading curve of elastic-plastic materials. Therefore,the incremental constitutive relation for the loading curve O → A → B under combined compression and shear stresses can be obtained according to the isotropic linearly work-hardening elastic-plastic model.

Fig. 4Schematic stress-strain curves for shape memory effect and pseudo-elastic effect of SMA.

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The total longitudinal and shear strain increments are considered as the sum of the elastic and phase transition parts

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where the subscripts “e” and “pt” identify the elastic and phase transition,respectively. The elastic part of strain can be given by Hooke’s law as

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where E is the Young’s modulus,and µ is the shear modulus of austenite phase. The phase transition part of strain is

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where f(σ,τ) is the phase transition function,andλis a positive scalar. Taking the criterion of Eq. (7) as the phase transition function yields

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If isotropic linearly work-hardening is assumed,there is a one to one correspondence between k and the phase transition work W^pt. Thus,

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The phase transition work rate will be

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From Eqs. (7),(10),(12) and the relation ,the function is found to satisfy

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where σ_v = σ −βk,and k˙ can be obtained from Eq. (7). The function W^pt(k) can be determined from the stress-strain curve of a simple tension/compression test or a pure shear test. Let σ = ϕ(ε) be the stress-strain curve after the critical point of phase transition for a simple tension/compression. Then,

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where g(σ) = ϕ ′(ε) denotes the slope of the stress-strain curve.

For a simple tension or compression,Eq. (7) reduces to σ = θ_vk,where θ_v = (β + θ)k for tension and θ_v = (β − θ)k for compression. Using the relationship ,Eq. (14) becomes

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Substituting Eqs. (9),(10),and (13) into Eq. (8) and considering Eq. (15),the governing equation can be obtained in a matrix form as follows:

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where

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For the elastic state g(θ_vk) = E,we can get S(k) = 0 from Eq. (17); while in the phase transition state,S(k) ≠ 0 since g(θ_vk) = E_M where E_M is the slope of the hardening stage in the mixed phase.

The governing system of phase transition wave in a thin-walled tube with phase transition derived in the previous part can be written in the following tensor form:

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where

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The first-order partial differential equation (18) is a quasi-linear symmetric hyperbolic system, which can be solved by the characteristic theory^[19]. The characteristic wave speed c of Eq. (18) can be determined by the roots of the determinant

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Using the method of block matrix,Eq. (20) becomes

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Thus, is the eigenvalue of matrix G,and the roots of Eq. (20) are

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The wave speeds denoted byc_sandc_fare obtained by taking the plus and minus signs in Eq. (22),and they represent the “slow” and “fast” phase transition wave speeds,respectively, which can also be called the phase transition SSW or FSW caused by the coupling of longitudinal and torsional stresses. In the elastic region (i.e.,S = 0),c_f is reduced to be the elastic longitudinal velocity c₀,andc_sis the elastic shear velocity c₂,where and .

Equation (18) admits a simple wave solution in which w is a function of c only. According to the theory of simple waves^[20],dw is proportional to the right eigenvector r defined by

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This will result in the following equation for the stress path in the στ-plane for the simple wave solution^[21]:

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Thus,the stress paths in the στ-space can be obtained by numerical integration according to Eq. (24).

Taking NiTi SMA as an example,the material parameters for austenite to martensite transition at room temperature are listed in Table 1.

Table 1. NiTi SMA material parameters

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As mentioned previously,the parameters α and φ for NiTi SMA can be obtained from the experiments. From the experimental data of Ti-49.75Ni(at.%) polycrystal under biaxial proportional loading given by Lexcellent and Blanc^[24],the initial phase transition stress is σ^t_Ms = 370.0 MPa under tensile loading and σ^c_Ms = −510.0 MPa under compression loading. By using Eq. (6),it yields

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Thus,we have

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Using the data of Eq. (26),the initial phase transition surface in the σ₁σ₂-plane determined by Eq. (6) can be plotted. The ellipse predication of the criterion shows good agreement with the experiment data^[24] as shown in Fig. 5.

Fig. 5Comparison of initial phase transition surface in σ1σ2-plane predication of criterion and experiment data.

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Austenite to martensite phase transition of NiTi SMA shows shape changes mainly,together with small volume changes. Taking the hydrostatic pressure together with the deviatoric stress into consideration,the critical criterion of phase transition is established. The prediction is in good agreement with the experimental data. From the loading-unloading pseudoelasticity stress-strain curve of Ti-50.8%Ni (at.%) SMA polycrystal under pure torsion given by Zhu et al.^[25],γ^pt_eff is about 4%. Thus,using the parameter α determined by Eq.(26),we have ε^pt_v = 0.477γ^pt_eff,which means that the volume change predicted by the criterion is about 1.91% during the martensite phase transition. Actually,from the lattice parameters of austenite to martensite phase transition of Ti-49.2%Ni (at.%) SMA tested by Kudoh et al.^[26] (cubic austenite: a₀ = 0.301 nm and monoclinic martensite: a_m = 0.289 8 nm,b_m = 0.410 8 nm, c_m = 0.464 6 nm,β_m = 97.78 ◦),the volume change is a_mb_mc_m sin β_m/2(a₀)³ − 1,and the volume change is about 0.476%. Therefore,it can be found that the volume change predicted by the criterion is larger than the value calculated by the lattice parameters,but the initial phase transition stress predicted by the criterion shows good agreement with the experiment data. Guo et al.^[17] thought that the difference of the component of NiTi SMA led to the difference of the volume change,while the lattice parameters associated with the components of the alloys^[26]. Till now,there has not been a full investigation of the parameters during the martensite phase transition of NiTi SMA. Therefore,we use the data of Eq. (26) in the qualitative research of the combined phase transition waves in NiTi SMA.

From the value of the parameter α determined by Eq. (26),all the parameters of Eq. (7) can be obtained. The results are listed in Table 2. The initial and subsequent phase transition ellipses of NiTi SMA can be plotted in Fig. 6.

Table 2. Parameters of criterion for phase transition of NiTi SMA

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Fig. 6Phase transition ellipses and stress paths of simple waves in στ-plane.

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In the στ-stress plane,Eq. (24) can describe different stress paths,as shown in Fig. 6. It can be seen that the vertical coordinate has been shifted rightwards since the volume is expanding during the transition for NiTi SMA.

The initial transition ellipse divides the plane into an elastic region and a transition region. In the elastic region,the disturbance can propagate only with two elastic wave speeds: the longitudinal wave speed = 3 142.6 m/s and the shear wave speed = 1 949.0 m/s,which correspond to the horizontal and vertical path lines,respectively. In the transition region,the disturbance will propagate with the “slow” phase transition wave speed c_s or the “fast” phase transition wave speed c_f. The solid lines and dashed lines in Fig. 6 represent the stress paths forc_sand c_f,respectively. The stress paths ofc_semanate from the initial phase transition surface while the stress paths for c_f are perpendicular to the stress paths of c_s.The wave speeds ofc_sandc_fare decreasing along the arrow direction as shown in Fig. 6. For c_f,the dot line MN changes the arrow direction,where MN is a trace of the points with the maximum shear stress on the initial and subsequent transition ellipses.

The transition region can be further divided into three zones by the line MN and τ-axis as shown in Fig. 6,where Zone 2 between the line MN and the vertical coordinate τ is special for the phase transition material since the line MN coincides with the vertical coordinate for conventional elastic-plastic materials. A stress path from the elastic region into Zone 3 or Zone 1 in the transition region is similar to that of the elastic-plastic combined stress wave,in which several basic stress paths were studied by Clifton^[27] and Ting^[28]. However,the stress path from the elastic region into Zone 2 in the transition region has not been investigated yet in the elastic-plastic combined stress wave theory. Several typical loading paths for combined phase transition waves related to Zone 2 which might be interesting are discussed in detail below.

(i) Considering that a tube,initially at rest and unstressed,is suddenly subjected to a constant shear stress τ = 512.0 MPa at the end x = 0,which means that the initial and final stress states are from the original point O (0,0) in the elastic region to point C (0,512.0) on the τ axis in the transition region as shown in Fig. 7. According to the present combined stress wave theory,the stress path ofc_semanated from B′ on the τ axis does not pass through point C. Therefore,the stress path from O to C in the στ-plane should be

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as shown in Fig. 7. That means that the stress must first arrive the compression stress state A along path OA with c₀,then to the point B with c₂,finally arrive the loading stress state C with c_s (the speeds range from 434.1 m/s to 432.8 m/s),and can even reach the tension-torsion stress state D if the final loading stress state is D. The wave solution in the xt-plane and the combined stress wave profile at x = 100 mm of the tube are shown in Fig. 8. Under the twist step-loading,it can be found that it generates a longitudinal elastic compression wave first,and then an elastic torsion wave followed by a “slow” phase transition wave with the compression stress reduced to zero and the shear stress increased to the final stress state C in the meantime. The variation of the longitudinal stress is an unexpected result which will not occur during the step-loading of conventional elastic-plastic tubes.

Fig. 7Stress path of NiTi thin-walled tube loaded by step twist.

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Fig. 8Wave solution of NiTi thin-walled tube loaded by step twist.

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(ii) As a second example,we consider a case in which the tube is initially stressed statically by a shear stress τ₀ which is greater than or equal to τ_Ms (the start stress of phase transition by simple torsion). Here,we suppose that the final loading stress state is H( −481.3,350.0)(MPa). Thus,the stress paths as shown in Fig. 9 are

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Fig. 9Stress path of pre-twisted NiTi thin-walled tube loaded by combined step compression and twist.

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The most remarkable thing is that the stress must first arrive the point F through the c₀ path because of the shift of the center of the subsequent phase transition ellipse; E and F are symmetrical with the minor axis of the subsequent ellipse here. Then,the final stress state H is got through the c_f path (the speeds range from 3 122.7 m/s to 2 756.9 m/s) and the c_s path (the speeds range from 568.1 m/s to 520.2 m/s). The solution in the xt-plane and the combined stress wave profile at x = 100 mm of the tube are shown in Fig. 10.

Fig. 10Wave solution of pre-twisted NiTi thin-walled tube loaded by combined step compression and twist.

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(i) The incremental constitutive relations with combined stresses for phase transition wave propagation in a thin-walled tube are derived based on the phase transition criterion considering both the hydrostatic pressure and the deviatoric stress. It is found that the centers of the initial and subsequent phase transition ellipses shift along the σ-axis in the στ-plane due to the tension-compression asymmetry induced by the hydrostatic pressure,which is different from the conventional elastic-plastic materials.

(ii) The governing equation of wave propagation based on the incremental constitutive relations with combined stresses is established,which offers the wave solutions of the “fast” and “slow” phase transition waves under combined axial and circumferential stresses in the phase transition region.

(iii) For a NiTi thin-walled tube (α > 0) initially subjected to the step-twist loading on the τ-axis in the transition region,it will first generate an elastic compression wave followed by an elastic shear wave and slow transition waves. This phenomenon is not observed for the conventional elastic-plastic material whose yield condition is independent of the hydrostatic pressure (α = 0).

(iv) The criterion of the phase transition in this paper is similar to the criterion of J₂-I₁ proposed by Qidwai and Lagoudas^[16] and the Drucker-Prager type criterion introduced by Auricchio et al.^[29]. If only the stress terms are taken into account,the criterion of the present work is similar to the Drucker-Prager yield function which includes the effect of the hydrostatic pressure on the yield behavior and is frequently used in geotechnics. Thus,the present results for thermo-elastic phase transition material can also be used to the Drucker-Prager type materials such as geomaterial.

(v) It must be mentioned that shock waves may occur when the stress is suddenly loaded to the second phase (martensite phase) beyond the point B in Fig. 4,which will not be discussed here and need further investigation.