Research Article
Initial Yield Surface of Shape Memory Alloy in Stress Induced Martensitic Phase Transformation
JJ Zhu1*,WM Huang2, and KM Liew3
1School of Civil Engineering and Architecture, Wuyi University, China
2School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore
3Department of Architecture and Civil Engineering , City University of Hong Kong, Kowloon, Hong Kong
JJ Zhu, School of Civil Engineering and Architecture, Wuyi University, China.
Received Date: August 28, 2018; Published Date: September 24, 2018
Abstract
Based on the critical conditions for nucleation and the actual eigenstrain of a given shape memory alloy in the stress induced martensitic phase transformation, the closed-form expression of its corresponding initial yield (nucleation) surface is obtained. Consequently, the underline mechanism for non-symmetry in the stress vs. strain relationship between uniaxial tension and uniaxial compression experimentally observed in many SMAs is revealed. Unlike that in many traditional criteria, such as the Von Mises and Tresca criteria, in which the yield surfaces are functions of stress only, the yield surface derived here is closely related to the crystal structures before and after the phase transformation. The initial yield surface of a polycrystalline CuZnAl shape memory alloy tube under combined loading of tension and torsion predicted by the formula developed here agrees well with the experimental results reported in the literature.
Keywords: Shape memory alloy; Phase transformation; Microstructure; Yield surface
Introduction
The thermomechanism behind shape memory phenomenon in Shape Memory Alloy (SMA) is a significant amount of shear strain upon phase transformation. It is well known that such phase transformation can be induced either by temperature (known as thermally induced phase transformation) or by stress (stress induced phase transformation). As a result of this unique shape memory behavior, SMAs have been used in various applications for some years [1].
So far, there are still a few questions on the behavior of SMA waiting to be solved. Asymmetry in tension and compression of SMAs is one among others [2-7]. Experiments under combined loads of tension and torsion have been carried out on NiTi [8] and CuZnAl [9]. From these tests, it is observed that, in general, the yield (transformation start stress in stress induced phase transformation) surface of SMA cannot be described either by Von Mises criterion or by Tresca criterion. Therefore, it turns to be an interesting phenomenon for further investigation.
In this paper, based on the critical condition of nucleation and corresponding eigenstrain in austenite to martensite phase transformation, the initial yield surface of SMA is investigated. The resulting yield surface of polycrystalline CuZnAl (in tube shape) under combining tension and torsion loads is compared with the experimental result reported in the literature.
Phase Transformation Eigen-strain and Critical Condition for Nucleation
It is assumed here that the austenite finish temperature of a polycrystalline SMA is Af . In the absence of external stress, at temperature T (T > Af ) this SMA is 100% austenite. Provided that a stress state Σ is applied on it and then gradually increased. As long as Σ reaches a certain level, stress induced phase transformation starts. This is called nucleation in material science. The resulting martensite is known as martensite Habit Plane Variant (HPV).
In some SMAs, for instance, CuZnAl, the martensite HPV is Lattice Correspondence Variant (LCV). But in some other SMAs, such as NiTi and CuAlNi, HPV is formed by a pair of twinned LCVs. Readers may refer to reference by Saburi and Nenno [10] for details of the relation among martensite variant, HPV and LCV.
Let us take a Representative Volume Element (RVE) from this
polycrystalline SMA. Assume that this RVE includes N grains. Under
a given stress state Σ , in grain k (1 ≤ k ≤ N ), up to 24 martensite
HPV may be produced. Provided that the phase transformation
eigenstrain of l th martensite HPV ( 1 ≤ l ≤ 24) is given by , the
work done by external stress to produce this martensite HPV may
be expressed as [11]
![Click here to view Large Equation 1 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E001.png)
Therefore, the thermodynamic driving force (Π) can be written as
![Click here to view Large Equation 2 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E002.png)
Here, U(T) is the thermodynamic driving force due to chemical free energy and interaction energy etc. In the literature, U(T) are presented in slightly different forms [12-28]. Despite the difference, the fact is that U(T) depends only on temperature T and volume fraction of martensite variant (but not on applied stress Σ ). Well known Clausius-Clapyron equation [11,29,30] can be derived from Equation (2). Thus, at a given temperature T, the condition for nucleation is given by
![Click here to view Large Equation 3 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E003.png)
An equivalent form may be written as
![Click here to view Large Equation 4 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E004.png)
Recall previous assumption, this SMA is initially pure austenite. The surface of nucleation stress is termed initial yield surface in this paper. If yield just starts (nucleation), only one martensite HPV is induced. Thus, the nucleation criterion may be written as
![Click here to view Large Equation 5 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E005.png)
where the maximum is over all possible phase transformation eigenstrains of martensite HPV, i.e., overall k (1 ≤ k ≤ N ) and l ( 1 ≤ l ≤ 24 ). Equation (5) can be reduced to
![Click here to view Large Equation 6 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E006.png)
Equations similar to Eqs. (2) & (6) have been applied in studies on both single crystalline and polycrystalline SMAs. To name a few, papers [31-36] are based on similar principle.
In martensite phase transformation, deformation of habit plane variants may be shown as Figure 1, where n is normal vector of habit plane, and b is shape strain vector. In general, the transformation eigenstrain of SMA includes not only shear deformation, but also a small dilatation. Therefore, shape strain vector b is normally not perpendicular to unit normal vector of habit plane n . Decompose vector b into two parts: one is in n direction, and the other is vertical to n , i.e.,
![Click here to view Large Equation 7 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E007.png)
![Click here to view Large Figure 1 irispublishers-openaccess-engineering-sciences](../images/irispublishers-openaccess-engineering-sciences.ID.000505.G001.png)
It is known that both g and ε depend only on lattice parameters (i.e. microstructures) of austenite and martensite, but not on the orientation of martensite HPV. Suppose that the deformation of martensite HPV is F , then
![Click here to view Large Equation 8 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E008.png)
in which I is identity tensor. The right Cauchy-Green strain tensor of martensite HPV is given by (Ball and James 1987)
![Click here to view Large Equation 9 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E009.png)
The phase transformation eigenstrain of martensite HPV may be written as
![Click here to view Large Equation 10 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E010.png)
Substituting Eq. (5) into Eq. (10) yield
![Click here to view Large Equation 11 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E011.png)
From Eq. (7)
![Click here to view Large Equation 12 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E012.png)
Substituting Eq. (12) into Eq. (11) yields
![Click here to view Large Equation 13 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E013.png)
Substituting Eq. (7) into Eq. (13) yield
![Click here to view Large Equation 14 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E014.png)
Let
![Click here to view Large Equation 15 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E015.png)
and
![Click here to view Large Equation 16 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E016.png)
Eq. (14)-(16) yield
![Click here to view Large Equation 17 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E017.png)
It is noted that in Eqs. (11), (13) or (17), the second term is nonlinear term. In previous literatures [12,18] among others, it is believed that the nonlinear terms can be ignored, thus Eq. (13) reduce to
![Click here to view Large Equation 18 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E018.png)
Or equivalently Eq. (16) reduce to
![Click here to view Large Equation 19 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E019.png)
From the point view of calculation, the approximate value of Eigen-strain phase transformation, Eq. (18) is acceptable. In present paper, we focused on the shape of yield surface, as we will see late, yield surface is sensitive on parameter Θ, thus the nonlinear term of phase transformation eigen-strain plays an important ruler in yield surface, ignore nonlinear terms may result in a wrong yield surface. In some of literature [34,35] the work done by external stress is calculated by
![Click here to view Large Equation 20 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E020.png)
In fact, from Eq. (18) we also obtain
![Click here to view Large Equation 21 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E021.png)
It is apparently that (F - I) is not frame independent (i.e. it is not a tenser), thus, it cannot be taken as strain measurement. In both of above two approximate calculations, the nonlinear terms are ignored. It may result in a wrong yield surface.
The maximum of Eq. (6) is over all possible pairs of n and m (n2 = 1, m2 = 1 and n ⋅m = 0 ). Substituting Eq. (17) into Eq. (6), the nucleation condition becomes
![Click here to view Large Equation 22 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E022.png)
Subsequently,
![Click here to view Large Equation 23 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E023.png)
If taking rectangular Cartesian coordinate system o − e1 e2 e3 along the directions of principal stresses σ1 ,σ2and σ3, (σ1 ≥σ2≥σ3 then
![Click here to view Large Equation 24 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E024.png)
Let
![Click here to view Large Equation 25 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E025.png)
and
![Click here to view Large Equation 26 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E026.png)
From Eqs. (23)-(26)
![Click here to view Large Equation 27 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E027.png)
where n and m satisfy the following conditions,
![Click here to view Large Equation 28 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E028.png)
From Eqs. (27)-(30)
![Click here to view Large Equation 29 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E029.png)
Define
![Click here to view Large Equation 30 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E030.png)
It is apparent that to find the maximum of Π is equivalent to find the maximum of W , and also equivalent to find the maximum of Λ .
Define
![Click here to view Large Equation 31 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E031.png)
From Eqs. (31)-(34)
![Click here to view Large Equation 32 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E032.png)
Hence, the original problem turns to find the maximum of Eq. (35) under the constrain conditions given by Eqs. (28)-(30). Introduce Lagrange multiply fractions λ1, 2λ, 3λ and
![Click here to view Large Equation 33 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E033.png)
The condition for maximum Γ is
![Click here to view Large Equation 34 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E034.png)
i.e.,
![Click here to view Large Equation 35 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E035.png)
By solving simultaneous Eqs. (28)-(30) & (38)-(40), and substituting the solution into Eq. (36), the maximum Γ can be obtained.
Let
![Click here to view Large Equation 36 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E036.png)
It can be proved that the maximum value of Γ does not exist, if any two of A , B and C are non-singular.
For instance, if detA ≠ 0 and detB ≠ 0 , Eqs. (38) & (39) give
![Click here to view Large Equation 37 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E037.png)
From Eqs. (29) & (30)
![Click here to view Large Equation 38 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E038.png)
It is apparent that Eqs. (44) & (45) cannot result in the maximum Γ .
Maximum Λ
From the discussion above, we know that there are four possible ways to get maximum Λ , i.e. A, B, and C are all singular, or only one of A, B and C is non-singular. All these four cases are studied in following four subsections respectively.
Matrixes A , B and C are all singular
In this case,
By substituting Eqs. (41)-(43) into Eq. (46), one can see that if Eq. (46) has solution, either α = 1 or α = 0 . In both cases
![Click here to view Large Equation 39 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E039.png)
From Eqs. (47) and (38)-(40)
![Click here to view Large Equation 40 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E040.png)
If α = 1: From Eq. (48), Eqs. (28)-(30) yield
![Click here to view Large Equation 41 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E041.png)
Re-writing the second formula of Eq. (49) gives
![Click here to view Large Equation 42 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E042.png)
Subsequently the first formula in Eq. (49) becomes
![Click here to view Large Equation 43 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E043.png)
and the third formula in Eq. (49) turns to be
![Click here to view Large Equation 44 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E044.png)
Hence, the results of Eq. (52) are
![Click here to view Large Equation 45 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E045.png)
Substituting Eqs. (48) & (50) into Eq. (35) yields
![Click here to view Large Equation 46 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E046.png)
Substituting x = x1 into Eq. (54) yields
![Click here to view Large Equation 47 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E047.png)
Substituting x = x2 into Eq. (54) yields
![Click here to view Large Equation 48 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E048.png)
It is noticed that Λ2 > Λ1. Thus, the maximum Λ is
![Click here to view Large Equation 49 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E049.png)
If α = 0 : According to Eq. (48), Eqs. (28)-(30) can be re-written as
![Click here to view Large Equation 50 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E050.png)
The second formula in Eq. (58) may be rewritten as
![Click here to view Large Equation 51 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E051.png)
Therefore, the first formula in Eq. (58) becomes
![Click here to view Large Equation 52 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E052.png)
and the third formula in Eq. (58) is
![Click here to view Large Equation 53 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E053.png)
The results of Eq. (61) are
![Click here to view Large Equation 54 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E054.png)
Substituting Eqs. (48) & (59) into Eq. (35) gives
![Click here to view Large Equation 55 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E055.png)
Substituting of y = y1 into Eq. (54) gives
![Click here to view Large Equation 56 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E056.png)
Substituting of y = y2 into Eq. (54) gives
![Click here to view Large Equation 57 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E057.png)
As Λ2 > Λ1 , the maximum Λ is
![Click here to view Large Equation 58 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E058.png)
This is same as Eq. (57). So the maximum Λ for both α = 0 and α = 1 is
![Click here to view Large Equation 59 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E059.png)
Matrix is non-singular
In this case both B and C are singular, but
![Click here to view Large Equation 60 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E060.png)
From Eqs. (68) & (38)
![Click here to view Large Equation 61 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E061.png)
Equation (35) becomes
![Click here to view Large Equation 62 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E062.png)
From
![Click here to view Large Equation 63 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E063.png)
λ1a and λ2 may be presented as functions of λ3
![Click here to view Large Equation 64 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E064.png)
Substituting Eq. (72) into Eqs. (39)-(40) yields
![Click here to view Large Equation 65 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E065.png)
Substituting Eqs. (69) & (73) into Eqs. (28)-(30) yields
![Click here to view Large Equation 66 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E066.png)
Solving Eq. (74) results in the following eight groups of solution for variables n2 ,n3 and λ3
![Click here to view Large Equation 67 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E067.png)
and
![Click here to view Large Equation 68 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E068.png)
Using Eqs. (73), (75) & (76), m2 and m3 can be solved. Substituting these eight groups of solution one by one into Eq. (70), and then comparing them, the maximum Λ is obtained as
![Click here to view Large Equation 69 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E069.png)
Matrix B is non-singular
In this case both A and C are singular, but
![Click here to view Large Equation 70 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E070.png)
Eqs. (78) & (39) yields
![Click here to view Large Equation 71 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E071.png)
Equation (35) becomes
![Click here to view Large Equation 72 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E072.png)
Since
![Click here to view Large Equation 73 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E073.png)
one may express λ1 and λ2as functions of λ3, i.e.,
![Click here to view Large Equation 74 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E074.png)
Substituting Eq. (82) into Eqs. (38) & (40) yields
![Click here to view Large Equation 75 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E075.png)
Substituting Eqs. (79) & (83) into Eqs. (28)-(30) yields
![Click here to view Large Equation 76 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E076.png)
Eight groups of solution for variables n1,n3 and λ3 are resulted from Eq. (84). There are
![Click here to view Large Equation 77 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E077.png)
and
![Click here to view Large Equation 78 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E078.png)
m1 and mn3 can be solved from Eq. (83). Substituting these eight groups of solution one after another into Eq. (80), we get the maximum Λ as
![Click here to view Large Equation 79 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E079.png)
Matrix C is non-singular
In this case, both A and B are singular, but
![Click here to view Large Equation 80 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E080.png)
From Eqs. (88) & (40), we obtain
![Click here to view Large Equation 81 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E081.png)
As
![Click here to view Large Equation 82 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E082.png)
λ1 and λ2 can be expressed as functions of λ3 ,
![Click here to view Large Equation 83 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E083.png)
Substituting Eq. (91) into Eqs. (38)-(39) yields
![Click here to view Large Equation 84 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E084.png)
Substituting Eqs. (79) & (83) into Eqs. (28)-(30) yields
![Click here to view Large Equation 85 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E085.png)
By solving Eq. (93), eight groups of n1, n2 and λ3 are resulted. They are
![Click here to view Large Equation 86 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E086.png)
From Eq. (92), m1 and m2 are obtained. Substituting these eight groups into Eq. (35), the maximum of Λ is given by
![Click here to view Large Equation 87 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E087.png)
Yield Surface
As shown previously, if α = 1 or α = 0 ,
![Click here to view Large Equation 88 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E088.png)
On the other hand, if α ≠ 1 and α ≠ 0 , according to Eqs. (77), (87) & (96)
![Click here to view Large Equation 89 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E089.png)
Therefore, for whatever the case,
![Click here to view Large Equation 90 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E090.png)
It is surprising to see that Λmax has nothing to do with α. It means that Λmax is independent on the applied load.
From Eq. (32), we haveM
![Click here to view Large Equation 91 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E091.png)
Substituting Eq. (34) and Eq. (99) into Eq. (100) we have
![Click here to view Large Equation 92 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E092.png)
Since the nucleation condition is given by
![Click here to view Large Equation 93 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E093.png)
i.e. the yield criterion reads
![Click here to view Large Equation 94 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E094.png)
Denote
![Click here to view Large Equation 95 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E095.png)
then Eq. (103) may be expressed as
![Click here to view Large Equation 96 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E096.png)
Denote
![Click here to view Large Equation 97 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E097.png)
then, Eq. (105) can be written as
![Click here to view Large Equation 98 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E098.png)
If ε = 0 (i.e. no dilation), Eq. (16) becomes
![Click here to view Large Equation 99 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E099.png)
Substituting Eq. (107) into Eq. (106) we have
![Click here to view Large Equation 100 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E100.png)
Notice that g2 << 1, thus κ ≈ 0 . If κ = 0 then Eq. (107) is reduced to
![Click here to view Large Equation 101 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E101.png)
This is Tresca criterion.
We can express the resolved shear stress as
![Click here to view Large Equation 102 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E102.png)
If ε = 0, by substituting Eq. (23) into Eq. (6), Eq. (6) reduces to
![Click here to view Large Equation 103 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E103.png)
Equation (98) is in a similar form as Schmid law [37]. It indicates that Tresca criterion that is empirical in nature can be derived from Schmid law.
Let
![Click here to view Large Equation 104 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E104.png)
Equation (108) may be expressed in terms of σ0 and α as
![Click here to view Large Equation 105 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E105.png)
where 0 <α < 1 and σ1≥σ2≥σ3.
In order to obtain a complete yield surface, the whole principal stress space is divided into six sections as shown in Figure 2. By swapping the positions of σ1, σ2 and σ3 , the yield surfaces in other five sections can be produced from yield surface in one section that is available.
![Click here to view Large Figure 2 irispublishers-openaccess-engineering-sciences](../images/irispublishers-openaccess-engineering-sciences.ID.000503.G002.png)
Case Study
The case studied here is CuZnAl polycrystal. The transformation is from cubic to monoclinic ( DO3 →6M ). Its lattice parameters are measured by Chakravorty & Wayman [38] as a0 = 0.5996 (nm ) , a = 0.4553 (nm ), b = 0.5452 (nm ), c = 3.8977 (nm ), and θ = 87.5o . The phase transformation eigenstrains have been calculated in literature [21,26,39,40]. Unit normal vector of habit plane n and shape strain vector b have been calculated in reference paper [21,26] based on Eqs. (14) & (15). They are
![Click here to view Large Equation 106 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E106.png)
where
![Click here to view Large Equation 107 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E107.png)
From Eq. (7) we have
![Click here to view Large Equation 108 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E108.png)
and
![Click here to view Large Equation 109 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E109.png)
Substituting Eqs. (115)-(116) into Eqs. (118) & (119) result in ε = -0.0036 and g = 0.1866 . Substituting g and ε into Eq. (16), we have Θ = 0.0138 , From Eqs. (34), we have μ = 0.074 . From Eq. (106), we have κ = 0.1378 .
Yield surface can be produced by substituting the value of κ into Eq. (107). Resulting yield surface against Treseca yield surface is shown in Figure 3.
![Click here to view Large Figure 3 irispublishers-openaccess-engineering-sciences](../images/irispublishers-openaccess-engineering-sciences.ID.000505.G003.png)
From Eq. (103) it is known that if Θ = 0 , the yield surface reduces to Treseca yield surface. The non-symmetry of materials in tension and compression is dependent upon the difference between real yield surface and Treseca yield surface, therefore it sensitively relates with parameter Θ . From Eq. (16) it is known that Θ contain two parts, first part ε result from volume variation during phase transformation, and second part result from nonlinear property of phase transformation Eigen-strain. Numerical calculation shows that nonlinear term of phase transformation Eigen-strain sometime may large than volume variation during phase transformation. Thus, it cannot be ignored.
It is noticed that, if Θ ≠ 0 , the size of yield surface projected in π -plane depends on the hydraulic press σ0. Yield surface of CuZnAl in biaxial stress (σ2 = 0) plane is plotted in Figure 4. The numerical result is obtained by using the scheme proposed in reference paper [34]. 29791 grains are used in the calculation. And the transformation strain in numerical study is calculated by (UTU − I)/ 2 instead of (F − I). Here, U is taken from reference paper [39].
![Click here to view Large Figure 4 irispublishers-openaccess-engineering-sciences](../images/irispublishers-openaccess-engineering-sciences.ID.000505.G004.png)
The yield strength of a thin polycrystalline CuZnAl round tube under combining tension and torsion loads has been tested and reported in reference paper [9]. Let σ stands for extension stress and τ for shear stress. The corresponding principal stresses are
![Click here to view Large Equation 110 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E110.png)
Substituting Eq. (120) into Eq. (107), the initial yield surface becomes
![Click here to view Large Equation 111 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E111.png)
If κ = 0 , Eq. (121) turns to be Tresca yield surface,
![Click here to view Large Equation 112 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E112.png)
On the other hand, standard Von Mises yield surface is given by
![Click here to view Large Equation 113 irispublishers-openaccess-engineering-sciences](../tables/irispublishers-openaccess-engineering-sciences.ID.000505.E113.png)
![Click here to view Large Figure 5 irispublishers-openaccess-engineering-sciences](../images/irispublishers-openaccess-engineering-sciences.ID.000505.G005.png)
The yield surfaces by Eqs. (121)-(123) are plotted against the experimental results in Figure 5. As we can see, except one point which corresponding to pure torsion, the proposed surface is closer to the testing result than other surfaces.
Conclusion
Based on the critical condition for nucleation and eigenstrain in stress induced martensitic transformation, the initial yield (nucleation) surface of SMA is derived. A simple close form solution is given. This method provides not only a simple way to explain the non-symmetry in tension and compression of many SMAs, but also a convenient way to predict the yield surface of SMA. The result differs from tradition criterion, such as Von Mises and Tresca, in two aspects. First, it is in terms of not only stress but also parameters of microstructure. Second, it is solely derived from crystal parameters. It shows a clear and strong relationship between crystalline structure and yield surface. Yield surface not only dependent up volume variation but also dependent upon nonlinear term of shear deformation. The resulting initial yield surface of CuZnAl thin tube under tension plus torsion load agrees well with the experimental result reported in the literature.
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JJ Zhu, WM Huang, KM Liew. Initial Yield Surface of Shape Memory Alloy in Stress Induced Martensitic Phase Transformation. Glob J Eng Sci. 1(1): 2018. GJES.MS.ID.000505.
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Yield surface, Shape memory alloy, Martensitic phase transformation, Phase transformation, Microstructure, Thermomechanism, Eigen-strain, Critical condition for nucleation, martensite habit plane variant, Maximum Λ, π – plane, Tresca yield, Crystalline structure, Parameter, Polycrystalline, Eigen-strain, Tresca criterion, Cauchy-Green strain tensor of martensite, Lattice parameters, Temperature
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