## Abstract

The mechanical and functional responses of shape memory alloys (SMAs), which are often used in small volume applications, can be evaluated using instrumented indentation tests. However, deciphering the indentation test results in SMAs can be complicated due to the combined effects of the non-uniform state of stress underneath the indenter and stress-induced phase transformation. To address this issue, an expanding cavity model (ECM) applicable to spherical indentation of SMAs is developed in this work based on an analytical solution for an internally pressurized hollow sphere. Analytical expressions for key indentation parameters such as the mean contact pressure and size of the transforming zone are obtained, whose validity is evaluated by recourse to finite element simulations and published experimental data for a Ni–Ti alloy. It is shown that the ECM predicts the above parameters reasonably well for indentation strains varying from 0.01 to 0.04. Also, a method is proposed to determine the critical stress required to initiate phase transformation under uniaxial compression based on the application of the ECM to interpret the indentation stress–strain response.

## 1 Introduction

Shape memory alloys (SMAs) are a unique class of materials that are able to regain their original shape after deformation when subjected to a suitable thermo-mechanical cycle. A completely reversible stress-induced austenite to martensite phase transformation (SIMT), which can accommodate large strains, is responsible for this shape recovery [14]. If an SMA is deformed above the critical stress required to initiate SIMT at temperatures greater than its austenite finish temperature (Af), large and fully recoverable strain can result. This phenomenon is called superelasticity. However, if any plastic deformation occurs, it can reduce the extent of reverse transformation and hence strain recovery. In this case, the remnant strain is accompanied by the presence of residual untransformed martensite.

SMAs are widely used as micro-actuators, cardiovascular stents, and in the development of MEMS-based devices [3]. Since in most of the applications SMAs are used as small devices, it is important to probe and characterize the mechanical response at small scales. Instrumented indentation tests are ideal for this purpose owing to their ability to examine small volumes of material in a non-destructive manner [57]. The hardness, measured from these tests, is an important material property. In the context of a spherical indenter, variation of the mean contact pressure, pm, with indentation strain, ∊ind (taken as 0.2ac/R, where ac is the contact radius and R is the radius of the indenter), called the indentation stress–strain curve, reveals important information about bulk material properties. For example, the yield strength under uniaxial compression is related to pm as pm ≃ 3σy for rigid-perfectly plastic materials [8]. However, interpreting the indentation response of SMAs is a challenging task as the non-homogeneous stress state underneath the indenter can invoke a complex interaction between SIMT and plastic deformation [912]. Therefore, theoretical studies and numerical modeling are essential to obtain useful information about elastic and transformation properties from indentation test results [13].

The expanding cavity model (ECM), originally put forward by Marsh [14] and further developed by Johnson [15], is widely used to predict the indentation stress–strain curve [15,16] owing to its simplicity. It is derived on the principle that the deformation occurring beneath a blunt indenter is similar to that obtained in a hollow sphere subjected to pressure along its inner surface. Thus, the contact surface on indentation is assumed to behave like an expanding hemispherical cavity. Furthermore, it is assumed that the plastically yielded region outside the contact zone is hemispherical. By imposing the condition that the radial dilatation of the core during an increment in indentation depth accommodates the material displaced by the indenter, the size of the plastically yielded region is determined, which in turn is used to obtain pm [15].

Apart from the indentation stress–strain curve, ECM can also predict the size of the plastic zone well and qualitatively describe the stress variation [17] beneath the indenter, although there are some well-known limitations in this regard. These pertain to the assumed hemispherical shape of the inelastic zone boundary, radial displacement of core boundary, and violation of traction free condition outside the contact zone on the specimen boundary since the stress field pertaining to an internally pressurized spherical cavity is employed [1820]. A modified form of ECM [18] was successfully utilized to describe the indentation response of other materials like metallic glasses that show a pressure-dependent yield response [21]. It was demonstrated that the above-modified ECM predicts an increase in mean contact pressure with pressure sensitivity index similar to experimental observations [22,23].

The main objective of the present work is to develop an ECM suitable for SMAs. For this purpose, firstly the stress and displacement fields prevailing in a SMA hollow sphere subjected to monotonically increasing internal pressure are determined using an isotropic constitutive model that represents SIMT [24] assuming incompressible deformation. Some of the features of this solution are qualitatively similar to those pertaining to an internally pressurized hollow SMA cylinder [25]. However, unlike in Ref. [25], the elastic moduli of the austenite and martensite phases are taken to be different in the present hollow sphere solution. The above analytical solution is then validated by performing finite element (FE) simulations with material properties pertaining to a Ni–Ti alloy [24]. It is then applied to spherical indentation of SMAs and expressions for pm and the size of the transformation zone rto as functions of ac are derived pertaining to the stage when SIMT is ongoing at the core boundary. FE simulations of the spherical indentation behavior are then conducted at three representative temperatures using the same material properties as in the analysis of internally pressurized hollow sphere to validate the ECM predictions. The results show that pm and rto, as given by ECM, agree well with the FE simulations for indentation strains in the range of 0.01–0.04. Based on the above results, a method is then proposed to determine the transformation stress under uniaxial compression from indentation tests. The indentation stress–strain variation predicted by ECM is also shown to corroborate well with published experimental data [9] for a Ni–Ti alloy in the above range of ∊ind.

## 2 Analytical Solution for Shape Memory Alloy Hollow Sphere Subjected to Internal Pressure

A SMA hollow sphere, initially in austenite (A) phase, of internal radius a and external radius b, subjected to monotonically increasing internal pressure, pc, at a fixed temperature above martensite start temperature, Ms, is considered. The sphere is assumed to be governed by the isotropic constitutive model proposed in Refs. [24,26] which embodies both the elastic and phase transformation response of a polycrystalline SMA alloy that is not sharply textured. The transformation condition is taken to be based on the Von Mises equivalent stress and transformation hardening as well as plastic yielding in both the austenite and martensite phases is ignored. Analytical expressions for the stress and displacement fields corresponding to the above problem are derived in this section.

### 2.1 Transformation Function for Spherically Symmetric Stress State.

The transformation function used is based on the form proposed by Hartl and Lagoudas [24] and is given as
$ϕ(σ,θ,ξ)=σ:mt+p$
(1)
Here,
$mt=32Hcσ′σ¯,whereσ¯=32σ′:σ′$
(2)
Also,
$p=12σ:S~:σ+C1$
(3)
where
$C1=ρη~0θ−ρu~0$
(4)
In the above equations, $σ′=σ−13(σ:1)1$, where 1 is the identity tensor, represents the deviatoric stress and $σ¯$ is the Mises equivalent stress. Furthermore, mt is the transformation flow tensor associated with forward transformation (which is taken to be proportional to σ′), p is a thermodynamic variable (see Ref. [24]), and Hc is the transformation strain under uniaxial compression. Also, θ is the absolute temperature, ξ is the martensite volume fraction, and $ρη~0$ and $ρu~0$ are the difference in specific entropy and internal energy, respectively, between the martensite (M) and austenite (A) phases. Thus,
$ρη~0=ρη~0M−ρη~0Aandρu~0=ρu~0M−ρu~0A$
(5)
Furthermore, $S~=SM−SA$ is the difference in the fourth-order compliance tensor between the M and A phases.
The transformation condition for the onset of forward (A to M) transformation is written as [24,27]
$ϕ−Yt=0,ξ˙>0,0≤ξ≤1$
(6)
where Yt is a temperature independent material threshold parameter, which controls initiation and progress of phase transformation. It must be noted that since only the solution for monotonically increasing internal pressure at fixed temperature is sought, the condition for reverse phase transformation is not considered here. On using Eqs. (1) and (2), the transformation function becomes
$ϕ=Hcσ¯+p$
(7)
Within the framework of the above isotropic constitutive model, the hollow sphere subjected to internal pressure can be assumed to experience a spherically symmetric stress state, which, in spherical co-ordinates (ρ, α, ϕ), can be represented as (σρ, σα = σϕ). It can be shown that for this stress state, the Mises equivalent stress is given by
$σ¯=∣σρ−σα∣$
(8)
Guided by the solution for an elastic-plastic hollow sphere subjected to internal pressure [28], it is expected that σρ < σα. Thus, the normal components of mt in spherical co-ordinates can be expressed as
$mt={−Hc,Hc2,Hc2}$
(9)
The components of the total infinitesimal strain tensor under monotonic loading can be written as
$ϵρ=ϵρe+ϵρt=1E(ξ)(σρ−2ν(ξ)σα)−ξHcϵα=ϵϕ=ϵαe+ϵαt=1E(ξ)((1−ν(ξ))σα−ν(ξ)σρ)+ξHc2$
(10)
where e and t are elastic and transformation strains, respectively. Also, ξ(ρ) is determined from the compatibility condition [29] given by
$dϵαdρ+ϵα−ϵρρ=0$
(11)
It is assumed that
$1E(ξ)=1EA+ξE~,where$
(12a)

$1E~=1EM−1EA$
(12b)
and ν(ξ) = νM = νA (constant). Thus, the elastic strain components are expressed as
${ϵρeϵαeϵϕe}=(1EA+ξE~)[1−ν−ν−ν1−ν−ν−ν1]{σρσασϕ}$
(13)
The compliance matrix corresponding to A phase, [SA], and the difference in compliance between M and A phases, $[S~]$, are therefore given as
$[SA]=1EA[1−ν−ν−ν1−ν−ν−ν1]and,[S~]=1E~[1−ν−ν−ν1−ν−ν−ν1]$
(14)
On using (14) in (3) and then in Eq. (7), the transformation condition for the spherically symmetric stress state becomes
$Hc(σα−σρ)+12E~[σρ2−4νσρσα+2(1−ν)σα2]+C1−Yt=0$
(15)
In order to simplify further development, the material is assumed to be elastically incompressible (i.e., $ν=12$). The transformation criterion can then be expressed after some reduction as
$σα−σρ=C2−HcE~=σ0$
(16)
where
$C2=(HcE~)2+2E~(Yt−C1)$
(17)

It must be noted that Eq. (16) is similar to the yield condition corresponding to a spherically symmetric stress state satisfying the Von Mises yield criterion. However, there are some key differences with respect to a SMA hollow sphere undergoing SIMT. First, σ0 (which actually coincides with the transformation stress under uniaxial compression) is an increasing function of temperature through the dependence of C1 on θ (see Eq. (4)). Second, and more importantly, 0 ≤ ξ ≤ 1, which implies that Eq. (15) or (16) apply only in the range of ρ over which phase transformation is occurring (rti ≤ ρ ≤ rto). Thus, the solution corresponding to three stages need to be obtained. The first stage corresponds to the entire sphere remaining in A phase. The second stage pertains to SIMT occurring adjacent to the inner surface of the sphere, while the third stage applies to the situation where it is fully completed at ρ = a (i.e., a fully transformed martensite core develops at the inner surface). Thus, the third stage corresponds to rto > rti > a. These three stages are considered successively below.

### 2.2 Stage I: Entire Sphere in the Austenite Phase.

It is expected that for applied cavity pressure pc less than a value p1, the sphere will be fully in the A phase. In this pressure regime (pc < p1), the stresses prevailing in the sphere are given as [29]
$σρ=pc[1−(a/b)3][−(aρ)3+(ab)3]and$
(18a)

$σα=σϕ=pc[1−(a/b)3][12(aρ)3+(ab)3]$
(18b)
Also, for the case of $ν=12$, the radial displacement uρ is given by
$uρ=34pcEA(a3[1−(a/b)3])(1ρ2)$
(19)
The pressure p1 at which transformation commences at the inner surface is obtained by enforcing the condition (σα − σρ)|ρ = a = σ0 as
$p1=23σ0(1−(a/b)3)$
(20)

### 2.3 Stage II: Partially Transformed Martensite at the Inner Surface.

It is expected that there will exist a range of core pressure p1 < pc < p2, during which SIMT is ongoing at the inner surface (ρ = a). However, fully transformed martensite is not yet formed at the core (i.e., 0 < ξ < 1 at ρ = a). Thus, for p1 < pc < p2, there are two domains of deformation: (i) a transforming region (a<ρ < rto) in which 0 < ξ < 1 and (ii) an elastic region (rto < ρ < b) in the austenite phase.

#### 2.3.1 Stresses and Displacements in the Austenite Phase.

The stresses and radial displacement for rto ≤ ρ ≤ b are similar to Eqs. (18) and (19). However, the boundary conditions, σρ|ρ = b = 0 and $(σα−σρ)∣ρ=rto=σ0$, need to be satisfied. This gives the expressions for stresses and radial uρ as
$σρ=2σ03[−(rtoρ)3+(rtob)3]σα=2σ03[12(rtoρ)3+(rtob)3],anduρ=σ0rto2EA(rtoρ)2$
(21)

#### 2.3.2 Stresses and Displacement in the Currently Transforming Region.

Next, to find expressions for stresses in the range a ≤ ρ < rto, the equilibrium equation is invoked
$dσρdρ=2σα−σρρ$
(22)
Equations (22) and (16) can be solved simultaneously to determine σρ and σα. Furthermore, on enforcing the continuity of σρ across ρ = rto, one obtains
$σρ=−2σ0[ln(rtoρ)+13(1−(rtob)3)]andσα=σ0[−2ln(rtoρ)+13(1+2(rtob)3)]$
(23)
By applying the boundary condition σρ|ρ = a = −pc, the relation between pc and rto is determined as
$pc=2σ0[ln(rtoa)+13(1−(rtob)3)]$
(24)
The above non-linear equation must be solved iteratively to obtain rto for a given pc. However, for b/a ≫ 1 and rto/b ≪ 1, the expression for rto can be written approximately as
$rto≃aexp[pc2σ0−13]$
(25)
The expression for ξ(ρ) is determined by solving the compatibility equation (11) along with Eq. (10) for the case $ν=12$ as
$ξ=Dρ3+C3,whereC3=σ0EA((σ0/E~)+Hc)$
(26)
and D is an integration constant. It is determined by applying the condition ξ(ρ = rto) = 0 which gives
$ξ=1C4(−1+(rtoρ)3),whereC4=1C3$
(27)
On using Eq. (10) along with Eq. (27) and the strain-displacement equation ∊α = uρ/ρ, the radial displacement for the case of ν = 1/2 is obtained as
$uρ=σ0rto2EA(rtoρ)2$
(28)
Stage II would prevail till a core pressure p2 is attained when ξ = 1 at ρ = a. This gives the radial extent of the transforming region as
$rto=(1+C4)1/3a$
(29)
On substituting the above value of rto in Eq. (24), one gets p2 as
$p2=2σ03[ln(1+C4)+(1−(1+C4)(a/b)3)]$
(30)

### 2.4 Stage III: Fully Transformed Martensite at the Inner Surface.

For pc > p2, there will be three deformation domains: (i) a martensite elastic regime in which ξ = 1 for a ≤ ρ < rti, (ii) partially transformed region in which 0 ≤ ξ < 1 for rti < ρ < rto, and (iii) austenite elastic region (rto < ρ < b) in which ξ = 0. Since the stresses and displacements in the second and third domains will be the same as those given by Eqs. (21), (23), and (28), respectively, attention will be restricted below only to the first domain. The inner radius rti of the second domain mentioned above is ascertained by setting ξ = 1 in Eq. (27) as
$rti=rto(1+C4)1/3$
(31)

#### 2.4.1 Stresses and Displacements in the Martensite Core.

The strain components in the martensite core are given by Eq. (10) with ξ = 1. By using the strain-displacement relations in the above equation, one gets
$σρ−2νσα=EMuρ′+EMHcand$
(32a)

$σα(1−ν)−νσρ=EMuρ/ρ−EMHc/2$
(32b)
Solving Eqs. (32a) and (32b) simultaneously gives
$σρ=EM(1−ν)(1+ν)(1−2ν)uρ′+2EMν(1+ν)(1−2ν)uρρ+EMHc(1+ν)and$
(33)

$σα=EMν(1+ν)(1−2ν)uρ′+EM(1+ν)(1−2ν)uρρ+EM(ν−1/2)Hc(1+ν)(1−2ν)$
(34)
Now, substituting Eq. (33) and (34) in the equilibrium equation (22), one obtains the governing differential equation for uρ as
$uρ″+2uρ′ρ−2uρρ2=κρ$
(35)
where
$κ=−3Hc(1−2ν)(1−ν)$
(36)
Solving for uρ, from Eq. (35) and using (33)(34), the stresses and displacements can be expressed as
$σρ=−A¯ρ3+B¯−EMHc(1−ν)lnρ$
(37a)

$σα=A¯2ρ3+B¯−EMHc2(1−ν)(1+2lnρ)$
(37b)
and
$uρ=(1+ν)2EMA¯ρ2+(1−2ν)EMB¯ρ+κρlnρ3$
(37c)
The constants $A¯$ and $B¯$ are obtained by applying the boundary condition at ρ = a and continuity of uρ at ρ = rti. Thus, the following general expressions for σρ, σα, and uρ are determined for $ν=12$:
$σρ=2σ03(EMEA)[(rtoa)3−(rtoρ)3]−2EMHcln(ρa)−pc$
(38)

$σα=2σ03(EMEA)[(rtoa)3+12(rtoρ)3]−2EMHcln(ρa)−EMHc−pc$
(39)
and
$uρ=σ02EArto(rtoρ)2$
(40)
Finally, by using continuity of σρ at ρ = rti and Eq. (31), the core pressure pc is related to rto as
$pc=2σ03[1−(rtob)3−(EMEA)(1+C4−(rtoa)3)+ln(1+C4)]−2EMHcln(rtoa)+2EMHc3ln(1+C4)$
(41)
Thus, to find rto for pc > p2, the above equation has to be solved iteratively. Once rto is obtained, the stresses and radial displacement in the martensite core can be computed from Eqs. (38)(40).

## 3 Finite Element Analysis of an Internally Pressurized Shape Memory Alloy Hollow Sphere

In this section, FE analysis of an internally pressurized hollow sphere is conducted using a rate-dependent version of the SMA constitutive model of Hartl and Lagoudas [24] along with an axisymmetric formulation [10,30]. However, a small value of the rate exponent is chosen in the computations to obtain nearly rate independent response. The advantage of using a rate-dependent version is to facilitate easy numerical implementation by applying the rate tangent formulation [30]. The stress and ξ variations across the thickness of the sphere, as well as the evolution of transformation zone size (rto) with pc, are computed and compared with the analytical solution derived in Sec. 2.

### 3.1 Modeling Details.

In Fig. 1, one-quarter of a section of the hollow sphere, subjected to internal pressure pc, in a diametral (rz) plane is shown. The sphere, with b/a chosen as 4, is modeled using four-noded axisymmetric elements based on the B-bar formulation [31] to treat nearly incompressible deformation.

A highly refined mesh with elements of size ∼0.0025a is used near the inner surface to provide reliable estimates of stresses and rto. The material properties of the alloy considered are listed in Table 1. These pertain to a Ni–Ti alloy [24], except that the Poisson’s ratio ν of both A and M phases is taken as 0.495 to compare the results with the analytical solution discussed in Sec. 2. The simulations are conducted at three temperatures, namely, θ = 403 K (θ > Af), θ = 383 K (θ ∼ Af), and θ = 353 K (θ ≪ Af) to establish the validity of the analytical solution in different temperature regimes. For all three temperatures, the initial phase is taken to be 100% austenite. Furthermore, in order to understand the effect of Poisson’s ratio on the stress variations, additional FE simulations of an internally pressurized hollow sphere are conducted taking ν = 0.33 for θ = 353 and 403 K. In Table 2, the values of the constants C1, C2, C4, and σ0 as well as the critical cavity pressures p1 and p2 occurring in the analytical solution (Sec. 2) corresponding to the chosen material properties are summarized for four temperatures. It should be noted that σ0, p1, and p2 increase strongly with θ. Thus, while σ0 = 235.5 MPa and p2 = 575.5 MPa at θ = 353 K, they enhance to 703 MPa and 1370 MPa at θ = 418 K. Indeed, the σ0 values given in Table 2 are in close agreement with the transformation stress levels corresponding to uniaxial compression at the respective temperatures (see Refs. [24,30]).

### 3.2 Results and Discussion

#### 3.2.1 Variation of Cavity Pressure With Displacement.

The normalized cavity pressure, (pc/EA), is plotted against normalized radial displacement (uρ/a) at the inner surface of the sphere in Fig. 2 for θ = 353, 383, and 403 K, along with the corresponding analytical solution. In addition to numerical solutions based on ν = 0.495 for all three temperatures, variations pertaining to ν = 0.33 for θ = 353 and 403 K are also displayed.

It can be seen that there is good agreement between the numerical and analytical solutions for all three θ values. At all the temperatures, there is an initial regime where pc shows a linear variation with displacement, followed by a non-linear regime with varying slope. The linear regime is attributed to the fact that initially, the deformation is due to elastic austenite response. The deviation from linearity occurs due to the onset of phase transformation at the inner surface of the sphere. On closer inspection, it can be seen that the cavity pressure at which the deviation occurs for any given temperature matches well with the corresponding p1 value listed in Table 2. Also, the stiffening behavior observed at later stages can be attributed to elastic deformation of the transformed martensite adjacent to the inner surface. Indeed, the cavity pressures at which this stiffening sets in corroborate well with the p2 values indicated in Table 2. It can be further noted from this figure that pc/EA versus uρ/a variations for ν = 0.33 differ only marginally from the analytical and numerical solutions based on incompressible elastic behavior.

#### 3.2.2 Transformation Zone Size.

In Fig. 3, the evolution of the normalized transformation zone size (rto/a) with pc/EA is plotted for three temperatures. It should be noted that the transformation zone size corresponding to the FE solution is calculated as the radial extent beyond which ξ < 0.03, while the analytical estimate of rto is obtained by solving equations (24) or (41) iteratively for a given pc. The analytical variations for rto/a versus pc/EA agree well with the FE results for all three θ. Also, it can be observed that transformation begins at lower cavity pressures (i.e., p1) at smaller θ (see also Table 2) and rto evolves more rapidly with pc. By contrast, at higher θ, especially for θ > Af, it increases more gradually with pc (see curve pertaining to θ = 403 K) suggesting that the kinetics for phase transformation in the sphere is much slower at higher θ. However, irrespective of temperature, the evolution of rto/a slows down at higher pc (greater than p2), after the fully transformed martensite core starts to develop (at ρ = a).

#### 3.2.3 Radial Variation of ξ and Stresses.

The variation of martensite volume fraction, ξ, with ρ/a is plotted in Figs. 4(a) and 4(b) for θ = 353 K, corresponding to pc/EA = 0.004 (p1 < pc < p2) and pc/EA = 0.0116 (pc > p2), respectively. Figure 4(a) shows that for p1 < pc < p2, ξ decays smoothly from ρ = a to ρ = rto (∼1.4a) beyond which it is 0. On the other hand, Fig. 4(b) indicates that for pc > p2, ξ = 1 over a radial extent from ρ = a to ρ = rti (∼1.2a), corresponding to fully transformed martensite core and thereafter decreases smoothly to 0 at ρ = rto (∼3.1a). It can be observed from both Figs. 4(a) and 4(b) that the analytical solution predicts the variation of ξ accurately. Similar plots are shown in Figs. 5(a) and 5(b) corresponding to two pc values which fall within the regimes p1 < pc < p2 and pc > p2 for θ = 403 K, where again good agreement is observed between the analytical and numerical solutions. On comparing Figs. 4 and 5, it can be noted that the levels of pc needed to generate similar radial variations of ξ which increase with θ.

In Figs. 6(a)6(c), the variations of normalized stresses σρ/EA and σα/EA with ρ/a for θ = 353 K corresponding to the three cavity pressure ranges (pc < p1, p1 < pc < p2, and pc > p2, respectively) are shown. Similar variations are presented in Figs. 7(a)7(c) for θ = 403 K. FE analysis results based on both ν = 0.495 and 0.33 are displayed in these figures. Again, the analytical and numerical solutions for the radial variation of stress components display close agreement irrespective of θ and pc. Also, it is observed that, in all the three pressure regimes and at both temperatures, the stress variations for ν = 0.33 match well with those obtained assuming incompressible elastic response. This further reinforces the validity of assuming elastic incompressibility in deriving the analytical solution.

Figures 6 and 7 show that σρ varies smoothly from −pc on the inner surface (ρ = a) to 0 at the outer surface (ρ = b). On the other hand, although σα/EA variations are continuous, they show kinks at ρ = rto and ρ = rti (for pc > p1). Moreover, while σα at ρ = a is tensile for pc < p1 (see Figs. 6(a) and 7(a)), it turns compressive in the core pressure range p1 < pc < p2 (see Figs. 6(b) and 7(b)). This is due to maintenance of the transformation condition σα − σρ = σ0 at ρ = a in the latter regime of pc. In this regime, maximum (tensile) σα is attained at the edge of the transforming zone (ρ = rto). However, with increase in pc beyond p2, martensite core develops at ρ = a and σα once again becomes tensile at ρ = a (refer Figs. 6(c) and 7(c)). Thus, the variation of σα/EA with ρ/a reflects the different deformation domains occurring for an internally pressurized SMA sphere.

## 4 Expanding Cavity Model Applied to Indentation Analysis

Following Marsh [14], Johnson [15], and Narasimhan [18], it is assumed that the contact surface during indentation is encompassed by a hemispherical core of radius ac (refer Fig. 8) within which a hydrostatic compressive stress pc is presumed to act (i.e., for ρ ≤ ac). Thus, the indented material is assumed to behave like a pressurized spherical cavity and the stresses and displacements outside the core boundary (ρ > ac) can be reasonably approximated by the analytical solution derived in Sec. 2 corresponding to b/a → ∞. However, previous studies [15,18] have shown that while the expanding cavity model predicts the variation of mean contact pressure with contact radius ac (or indentation stress–strain curve) well, it has some limitations with respect to describing the detailed stress variations. For example, vanishing normal traction on the specimen surface outside the contact zone is not captured by the model (see Ref. [18]). In addition, as will be seen below, for indentation of SMAs, the utility of the model is restricted only to the range of contact radius ac over which phase transformation is ongoing in the core. So attention is confined here only to predict the mean contact pressure and size of the transforming zone rto over a limited range of ac. Notwithstanding this limitation, the utility of the model to deduce some key material parameters of SMAs from indentation analysis will be illustrated.

By considering the analytical solution pertaining to stage II (Sec. 2.3) and taking a < rtob, the cavity pressure given by Eq. (24) can be approximated as
$pc≃2σ03+2σ0ln(rtoac)$
(42)
The ECM of indentation further assumes [15] that the radial displacement of particles at the core boundary (ρ = ac) accommodates the volume of material displaced by the indenter during an increment of penetration (refer Fig. 8). For a conical indenter with semi-cone angle π/2 − β, this implies (see Fig. 8) that
$2πac2duρ(ac)=πac2dactanβ$
(43)
From Eq. (21), one obtains increment in radial displacement at the core boundary as
$duρ(ac)=3σ02EAac2rto2drto$
(44)
On substituting Eq. (44) into Eq. (43), it follows that
$d(rto3)dac=EAσ0ac2tanβ$
(45)
For a spherical indenter, tan β is taken as ac/R as suggested by Johnson [32] to obtain
$d(rto3)dac=EAσ0ac3R$
(46)
Equation (46) needs to be integrated from the stage of commencement of transformation, ac = ac0. With reference to a cylindrical co-ordinate system (r, θ, z) centered at the point of first contact (so that z-axis coincides with the axis of the indenter) as shown in Fig. 8, the transformation condition for points along z-axis can be written similar to Eq. (16) for $ν=12$ as
$σrr−σzz=σ0$
(47)
On substituting the elastic Hertz solution for σrr and σzz corresponding to a spherical indenter along the z-axis, when the material is in the initial austenite phase, it can be shown that the transformation condition Eq. (47) will be first satisfied (see [18]) at EAaco/(σ0R) ∼ 2.3. On integrating Eq. (46) from ac = aco, the transformation zone size rto can be related to ac in normalized form as
$(r~to3)=14[(a~c)4−(a~co)4]+(a~co)3$
(48)
where $a~c=EAac/(σ0R)$ and $r~to=EArto/(σ0R)$. On using $a~co≃2.3$ as noted above, one gets
$(rtoac)3=a~c4+5.2a~c3$
(49)
Finally, the core pressure is obtained by substituting Eq. (49) into (42) as
$pc≃2σ03[1+ln(a~c4+5.2a~c3)]$
(50)
The variation of pc/EA with indentation strain is shown in Fig. 9 for three temperatures. It can be observed that at a given ∊ind, pc enhances with θ which is qualitatively similar to the effect of temperature on cavity pressure for the hollow sphere at a given uρ(a)/a (refer Fig. 2). It must be noted that while the stress state beneath the indenter is not purely hydrostatic, it can be approximated as suggested by Johnson [32] with reference to a cylindrical co-ordinate system (r, θ, z) centered at point of first contact (see Fig. 8) as
$σzz=−(pc+σ^);σrr=σθθ≃−(pc−σ^/2)$
(51)
Here, $σ^$ is determined by substituting the above equations into the transformation condition Eq. (47) as $σ^=2σ0/3$. Thus, the mean contact pressure, pm, is obtained from the magnitude of σzz beneath the indenter (Eq. (51)) as
$pm=2σ03[2+ln(a~c4+5.2a~c3)]$
(52)

## 5 Finite Element Analysis of Spherical Indentation

In order to validate the expressions for the size of transformation zone rto and the mean contact pressure pm given by Eqs. (49) and (52) and also to ascertain the range of indentation strain ∊ind over which these equations are valid, finite element simulation of spherical indentation is conducted. The constitutive model of Hartl and Lagoudas [24], which was used in the analysis of internally pressurized hollow sphere in Sec. 3, is also employed in these simulations. The simulations are conducted with an axisymmetric formulation taking the spherical indenter as rigid similar to the analysis performed in Ref. [10].

### 5.1 Modeling Aspects.

The axisymmetric finite element mesh employed in the present computations is similar to that used in Ref. [10] and is composed of 12,498 four-noded quadrilateral elements based on the B-bar formulation [31,33]. It is highly refined close to the contact surface with element size ∼0.002R, (where R is the radius of the indenter), to accurately resolve the steep stress gradients in this region. Also, the specimen dimensions are sufficiently large to suppress boundary effects. The contact between the spherical indenter and the SMA specimen is assumed as frictionless and modeled using gap elements which are based on penalty formulation [34]. The SMA material properties listed in Table 1 are used, except for ν which is taken as 0.33 in the simulations that are conducted corresponding to θ = 353, 383, and 403 K. Furthermore, plastic yielding and transformation hardening are ignored. In order to investigate the effect of ν on the ECM solution discussed in Sec. 4 (which was actually for $ν=12$), an additional simulation is performed for θ = 353 K taking ν = 0.42.

### 5.2 Results and Discussion

#### 5.2.1 Evolution of the Transformation Zone Size.

In Fig. 10, the variations of the normalized size of transformation $r~to$ with ∊ind are plotted for three temperatures. Here, the analytical variations for $r~to$ versus ∊ind based on ECM are described by Eq. (49), while those given by the numerical solutions are deduced as the distance along the indentation axis at which ξ decreases to a small value of 0.03. It is observed that ECM predicts the transformation zone size accurately for ∊ind in the range of 0.01–0.04. For ∊ind < 0.01, the transformed region is too small and elastic response of the austenite phase controls the deformation. On the other hand, for ∊ind > 0.04, the region composed of fully transformed martensite (ξ = 1) extends beyond the core (of radius ac), thus rendering the ECM solution invalid. It can be seen from Fig. 10 that $r~to$ decreases with enhancement in θ at a given ∊ind implying slower evolution of the transformed zone.

#### 5.2.2 Mean Contact Pressure.

The normalized mean contact pressure (pm/EA) is plotted against indentation strain ∊ind for θ = 353, 383, and 403 K in Fig. 11(a). In this figure, the numerical data are shown by symbols, with the “$⋄$” symbols pertaining to ν = 0.33 and the “×” symbols corresponding to the additional computation performed for θ = 353 K taking ν = 0.42. The dash-dot curves depict the variations given by ECM, Eq. (52). It can be seen that ECM predicts the indentation stress–strain response well within the range of ∊ind from about 0.01–0.04. The additional numerical results corresponding to ν = 0.42 for θ = 353 K also match well with the ECM solution which indicates that there is not much sensitivity to ν.

As already mentioned, the validity of the ECM solution is restricted to 0.01 < ∊ind < 0.04. Nevertheless, the applicability of ECM in the above range of ∊ind can be exploited to infer σ0 (which coincides with the transformation stress under uniaxial compression) as a function of θ. To demonstrate this, curves of the type pm = A + B ln(ac/R) are fitted to the numerical data for all three θ values (in the above noted range of ∊ind) and are shown by solid line curves in Fig. 11(a). It can be seen that the fitted curves are in good agreement with the actual variations predicted by ECM (compare solid line and dash-dot line curves). On employing the value of the constant A determined from the fits which should correspond to 2σ0(2 + ln(EA/4σ0))/3 (as per Eq. (52)), σ0 can be estimated by an iterative (Newton–Raphson) approach.1 The σ0 values thus estimated from the fits to the numerical data are 256, 425, and 580 MPa for θ = 353, 383, and 403 K, respectively. These match reasonably well with the corresponding analytical values of σ0 given in Table 2.

Further inverse analysis can be carried out by employing σ0 thus determined over a range of θ along with Eq. (16) and expressions for C1 and C2 (see Eqs. (4) and (17)) to infer Hc and Yt (given $E~,ρη~0$, and $ρu~0$ for the SMA). Thus, both the transformation stress under uniaxial compression σ0 (for different θ) and transformation strain Hc can be estimated by fitting Eq. (52) to experimentally obtained data for indentation stress versus strain variations (pm versus ∊ind). This demonstrates the utility of the ECM solution for determining some important material parameters via an inverse approach from indentation test data. Such an inverse approach was applied by Bardia and Narasimhan [35] to ascertain pressure sensitivity index for polymer materials by combining ECM with experimental results.

In order to make contact with experiments, the indentation stress–strain response predicted by ECM is compared with published data obtained from spherical indentation tests performed by Wood and Clyne [9] on a Ni–Ti SMA in Fig. 11(b). However, the Ni–Ti alloy tested in Ref. [9] shows considerable transformation hardening. Thus, at a temperature around Af (∼100 °C), the stress required to start SIMT for this alloy under uniaxial compression, $σos$, is about 280 MPa, whereas that at the end of transformation, $σof$, is around 475 MPa. Hence, in Fig. 11(b), the ECM solution for variation of pm with ∊ind (given by Eq. (52)) based on both $σos$ and $σof$ indicated above are plotted along with the experimental data extracted from the indentation load–displacement plots pertaining to θ ∼ Af (Fig. 6 of Ref. [9]). To this end, the contact radius is calculated using the geometric relation $ac=2Rh−h2$, where h is the indentation depth and R is the indenter radius as suggested in Ref. [36]. It can be seen from this figure that for small indentation strain (∊ind ∼ 0.012), the experimental data for pm matches reasonably well with the ECM solution pertaining to $σos$. With increasing values of ∊ind, it gradually increases owing to SIMT progressing underneath the indenter and attains the variation given by ECM based on $σof$ at ∊ind ∼ 0.036. This implies that SIMT is complete over a large portion of material beneath the indenter at this stage. Thus, Fig. 11(b) demonstrates that the experimental data for pm versus ∊ind pertaining to this Ni–Ti alloy lies within the bounding curves based on $σos$ and $σof$, predicted by ECM, for ∊ind in the range from 0.01 to about 0.036. This corroborates with the conclusions derived based on comparison of ECM solution with finite element analysis results.

## 6 Summary and Conclusions

Analytical expressions for the radial variation and evolution of stress, martensite volume fraction, and displacement for an internally pressurized SMA hollow sphere governed by the constitutive equations proposed given in Ref. [24] have been derived. However, plastic yielding is ignored and elastic incompressibility (i.e., ν = 0.5) is assumed. FE simulations of the above problem have also been conducted at three temperatures ranging from well below to above Af to validate the analytical solution. The ECM has then been applied to derive analytical expressions for the mean contact pressure and size of the transformed zone obtained during indentation. Again, finite element simulations of spherical indentation of SMAs have been carried out at three temperatures to validate the predictions of ECM. The following are the important conclusions of this work.

• There are three distinct stages of deformation for an internally pressurized hollow SMA sphere. In stage I, the entire sphere is in the initial austenite phase. In stage II, phase transformation is ongoing in the inner surface of the sphere while in stage III, fully transformed martensite core develops adjacent to the inner surface.

• The evolution of cavity pressure with radial displacement and size of the transformed zone (rto) with pc obtained from FE analysis of the internally pressurized hollow sphere agree well with their corresponding analytical solutions for all θ considered. At a given radial displacement of the inner surface, pc enhances with temperature and correlates with higher transformation stress under uniaxial tension or compression (σ0). By contrast, rto evolves faster with cavity pressure as θ decreases.

• The radial distribution of stresses σρ and σα computed from the FE simulations based on both compressible and incompressible elastic behavior also match well with the analytical solution corresponding to all three stages and at all temperatures considered. While σρ varies smoothly from −pc on the inner surface to 0 at the outer surface of the sphere, σα shows kinks corresponding to both the inner and outer boundaries (ρ = rti, rto) of the transforming region. Furthermore, while σα is tensile at the inner surface of the sphere during stage I, it becomes compressive during stage II and reverts back to tension in stage III. In stage II, tensile peak in σα is attained at ρ = rto.

• It is found that the expression for the mean contact pressure and the size of the transformed region obtained from the ECM solution as applied to indentation match well with the FE results for indentation strains in the range of 0.01–0.04, irrespective of temperature from well below to above Af. For ∊ind < 0.01, the size of the transformed region is very small and the response is predominantly elastic in A-phase. By contrast, for ∊ind > 0.04, the fully transformed martensite zone (ξ = 1) extend beyond the core (ρ = a) thus restricting the validity of the ECM solution.

• At a given indentation strain, the mean contact pressure pm enhances with temperature, which is traced to higher pc and σ0. On the other hand, the size of the transformed zone reduces with an increase in temperature at a given ∊ind implying sluggish development of phase transformation.

• By fitting plots of the mean contact pressure versus the indentation strain to an equation of the form pm = A + B ln(ac/R), the transformation stress σ0 is deduced with reasonable accuracy. With further inverse analysis, it would also be possible to estimate Hc and Yt by applying the proposed ECM to experimental indentation stress–strain data over a range of temperatures.

• The indentation stress–strain variation predicted by ECM has also been compared with experimental data for a Ni–Ti alloy [9]. It is found that the experimental data lie between the pm versus ∊ind curves given by ECM corresponding to $σ0s$ and $σ0f$ for this alloy, thereby validating the latter.

In closing, it must be mentioned that plastic yielding has been ignored while deriving the ECM solution. This is because SIMT would precede plastic slip as long as the temperature is not significantly above Af. Furthermore, it is important to note that the ECM solution derived in this work is valid only up to indentation strain of 0.04, which would correspond to normalized indentation depth h/R of 0.02. Recent finite element simulations by Anuja et al. [10,12] have shown that at such low normalized indentation depths, plastic strains beneath the indenter are negligible and therefore would have little influence on the indentation response irrespective of temperature.

## Footnote

1

It was found that more reliable estimates of σ0 can be obtained by using A rather than B due to some scatter in pm which is attributed to errors in precisely ascertaining ac.

## Acknowledgment

R. Narasimhan would like to gratefully acknowledge the Science and Engineering Research Board (Government of India) for financial support under the JC Bose Fellowship scheme.

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