## Abstract

We present reasons, both experimental and mathematical, for why it is important to consider the radial deformation (thus the lateral traction-free boundary conditions as well) for phase transitions in a slender elastic cylinder. One of the main purposes of this paper is to derive an asymptotic model equation, which takes into account the radial deformation and satisfies the traction-free boundary conditions up to the right order. This is achieved from the three-dimensional field equations through a novel approach that combines a series expansion and an asymptotic expansion. An alternative approach based on Whitham's approach (well used in fluids) is also given. Then, we present some interesting analytic solutions for an infinite cylinder, including those that seem to describe the structure of a phase boundary, the nucleation process and the merge of two-phase boundaries. In particular, by considering the energy distribution based on the nucleation solution, it is revealed that the nucleation process is one of energy localizations, concentrations and separation.

## 1. Introduction

Applications of phase-transforming materials such as shape memory alloys (SMAs) and shape memory polymers are very broad. For example, they have been used to make satellite dampers, golf club heads and snake-like robots and so on. In particular, these materials have been used to design many minimal surgery devices (see Pelton *et al*. 1997). In the continuum-level modelling, phase-transforming materials may be modelled by strain–energy functions with multiple wells (Ball & James 1992) or in general by non-convex strain–energy functions. Systematic experiments have been carried out on uniaxial extensions of superelastic NiTi (a kind of SMAs) wires and strips (Shaw & Kyriakides 1995, 1997). Some key features of the measured engineering stress–strain curves during the loading process are (i) the nucleation stress occurs at a local maximum, which is significantly larger than the Maxwell stress; (ii) following the nucleation stress, there is a sharp stress drop; and (iii) afterwards, there is a stress plateau. These features were also observed in experiments done by others (Sun *et al*. 2000; Favier *et al*. 2001; Li & Sun 2002). In the authors' view, to justify that phase-transforming materials can be modelled by non-convex strain–energy functions, it is desirable (and probably necessary) to compare the mathematical solutions based on this type of energy functions with experimental results. However, it seems that not so many such comparisons are available in literature.

The difficulty could lie in the fact that we lack mathematical theories for mixed type equations that typically arise for non-convex energy functions. Analytical solutions for boundary-value problems are very few. In the classical paper of Ericksen (1975), analytical solutions were constructed for a static problem based on a pure one-dimensional stress model, which neglects the effect from other dimensions. Existing computational methods, such as finite-element methods, may not be directly used to obtain reliable numerical solutions because the well posedness of mixed-type equations is still an open problem (one does not even know how to impose the proper boundary conditions). Thus, results obtained by using packages that are designed for convex energy functions may not be completely trusted, especially the results near a phase boundary.

Therefore, analytical solutions for non-convex strain–energy functions could be very valuable. One of the purposes of this series of two papers is to provide a number of analytical solutions in a slender cylinder composed of a phase-transforming material with a non-convex strain–energy function. Comparisons with experimental results will also be made.

In the experiments, typically the radius-length ratio of the specimen is about 1/60. As a result, one might think that one can neglect the radial deformation and treat it as one-dimensional stress problem, so the solutions given by Ericksen (1975) can be used. However, we shall give several reasons, experimental and mathematical, to explain why, for phase transitions in a slender cylinder, it is essential to take into account both the radial deformation and the lateral surface traction-free boundary conditions.

In experiments (Shaw & Kyriakides 1995, 1997; Li & Sun 2002), it was observed that after the nucleation stress was reached the nucleation process began and accompanying with it there was a radial contraction (necking). Moreover, after the two-phase state was formed, the deformation was inhomogeneous, one part being thin and one part being thick. Thus, to model the nucleation process and the inhomogeneous deformation of different thicknesses, one should consider the radial deformation. Indeed, the solutions provided by Ericksen (1975) based on one-dimensional stress model cannot describe the nucleation process.

In the experiments on seven specimens with different diameters by Sun *et al*. (2000), some strong size effects were observed: the smaller the diameter is, the larger the nucleation stress is; also, the smaller the diameter is, the sharper the stress drop (taking place after the nucleation stress) is. Thus, to capture these size effects, one has to consider the diameter, which implies the consideration of the radial deformation.

Mathematically, one can also give justifications as to why one needs to consider the radial deformation for phase transitions in a slender cylinder. The explanation has been given in Dai (2004). Here, we give a brief recount. Consider the axial equilibrium equation (2.7). The presence of the last two terms is attributable to the radial deformation. For a phase-transforming material with a non-convex strain–energy function, typically the stress–strain curve has a peak-valley combination. In the loading process, as the external stress reaches the peak value (local maximum), the first term in equation (2.7) is exactly zero. Then the last two terms, no matter how small they are, are dominant terms, and cannot be neglected! This implies that there must be a radial deformation in the process of phase transformation (this is in agreement with experimental observations). Thus, to model phase transitions, the influence of the radial deformation should be taken into account.

If one treats the problem as a one-dimensional stress one, there is a discontinuous phase boundary between two phases (see Ericksen 1975), say, one phase with an axial strain value *γ*^{+} and another with *γ*^{−}. According to Love's relation, the radial displacements in the two phases are, respectively, *u=*−*mRγ*^{+} and *u=*−*mRγ*^{−} (where *m* is the Poisson's ratio). This implies that the shear strain *u*_{Z} is infinite at the phase boundary. However, the traction-free boundary conditions need the two stress components *Σ*_{rR} (depends on *u*_{Z}^{2}) and *Σ*_{rZ} (depends on *u*_{Z}, *u*_{Z}^{2} and *u*_{Z}^{3}) to be zero, which cannot be satisfied for an infinite *u*_{Z}. This once again gives evidence that one should consider the radial deformation. The above argument also implies that by imposing the traction-free boundary conditions, which force a finite shear strain, the phase boundary should be a smooth region and cannot be a discontinuous interface.

Owing to Reason 4, the deformations (with a smooth and narrow phase boundary) we consider here do not belong to the two-phase deformations studied by Gurtin (1983), which contain a surface of separation with jump discontinuities. Thus, the results established by Gurtin (1983) may not apply to the problems studied here.

Any one of the above four reasons justifies the importance of the radial deformation in phase transitions. In particular, from Reason 4, it can be seen the importance of the traction-free boundary conditions. In this series of two papers (referred to as parts I and II), we shall derive the asymptotic model equation for a slender cylinder thar takes into account the radial deformation as well as the traction-free boundary conditions. In part I, we also provide some analytical solutions in an infinite cylinder, from which some important information on the nucleation process and the structure of the phase boundary is extracted. In part II, analytical solutions for two boundary-value problems are obtained, and they seem to capture the key features observed in experiments.

Another important difference between the present one and many previous works (e.g. Ericksen 1975; Abeyaratne & Knowles 1990, 1991, 2000; Shaw & Kyriakides 1998) is that we use the full nonlinear stress–strain curve while the others mentioned above used a trilinear curve.

This paper is arranged as follows. In §2, we formulate the field equations by treating the slender cylinder as a three-dimensional object. In §3, we carry out a non-dimensionalization process to extract the important small variable and two small parameters which characterize this problem. Then we derive the asymptotic model equations in §4 through novel series and asymptotic expansions. By using Whitham's approach (well used for waves in fluid), we give an alterative derivation in §5. In §6, we give some interesting solutions in an infinite cylinder. Finally, some conclusions are drawn.

## 2. Three-dimensional field equations

We consider the axisymmetric deformation of a circular cylinder of radius *a*. We shall use the cylindrical coordinates (*r*, *θ*, *z*) and (*R*, *Θ*, *Z*) to represent a material point in the current and reference configurations, respectively. For an incompressible hyperelastic material, the strain–energy function *Φ* is a function of the first two invariants *I*_{1} and *I*_{2} of the left Cauchy–Green strain tensor; that is, *Φ*=*Φ*(*I*_{1}, *I*_{2}). We point out that although here, for convenience, we consider an incompressible hyperelastic material, our approach can be applied to study other types of materials as well. We suppose that *Φ* is non-convex in a pure one-dimensional stress problem such that phase transition can take place. The first Piola–Kirchhoff stress tensor *Σ* is given by(2.1)where *F* is the deformation gradient and *p* is the indeterminate pressure. The non-zero components of (*F*−*I*) are(2.2)where *u* and *w* are the radial and axial displacements, respectively.

If the strains are small (in experiments, the maximum value of the strain is about 8%), it is possible to expand the first Piola–Kirchhoff stress components in term of the strains up to any order. The formula containing terms up to the third-order material nonlinearity has been provided by Fu & Ogden (1999):(2.3)where *p*_{0} is the pressure value in equation (2.1) corresponding to zero strains, *p*^{*} is the incremental pressure and *a*_{jilk}^{1}, *a*_{jilknm}^{2} and *a*_{jilknmpq}^{3} are incremental elastic moduli, which can be calculated once a specific form of the strain–energy function is given. For example, for the strain–energy function(2.4)where *c*_{0}, *c*_{1}, *c*_{2} and *c*_{12} are material moduli, we find that(2.5)and the stress component *Σ*_{zZ} is given by(2.6)where the constant coefficients *a*_{1}, *a*_{2}, …, *a*_{11} can be expressed in terms of *c*_{1}, *c*_{2} and *c*_{12}. The other non-zero stress components can also be obtained but we omit their lengthy expressions for brevity. Owing to the complexity of calculations, we shall only work up to the third order material nonlinearity.

We point out that for other forms of strain–energy functions, equation (2.6) still holds and only the expressions of *a*_{1}, *a*_{2},…,*a*_{11} in terms of the material moduli are different. Thus, without loss of generality we shall work with equation (2.6) and other non-zero stress components obtained from the strain–energy function (2.4). Later on, in §5, we give the model equation for an arbitrary strain–energy function.

The field equations for static and axisymmetric problems are given by(2.7)(2.8)The incompressibility condition yields that(2.9)

We consider the case that the lateral surface of the cylinder is free of traction. Thus, we have the boundary conditions(2.10)Equations (2.7)–(2.9) provide three equations for three unknowns *u*, *w* and *p*^{*}.

## 3. Non-dimensional equations

Mathematically, it is too difficult to tackle equations (2.7)–(2.9) directly (if one substitutes the expressions of the stress components into eqation (2.7), the resulting nonlinear equation will be more than one page long). In this paper, we shall use the smallness of the radius-length ratio for a slender cylinder to derive the asymptotic model equation that takes into account the influences of the radial deformation and the traction-free boundary conditions. The approach is similar to that used in Dai & Huo (2002) and Dai & Fan (2004) for nonlinear dispersive waves in slender cylinders composed of standard materials. For that purpose, in this section, we first establish the proper non-dimensional equations with small parameters.

The first step is to introduce a very important transformation(3.1)The importance of equation (3.1) will be explained later.

We introduce the dimensionless quantities through the following scalings(3.2)where *l* is a characteristic length (in a boundary-value problem *l* can be the length of the cylinder), *h* is a characteristic axial displacement and *μ* is the material shear modulus. Substituting equation (3.2) into (2.7), (2.8) and (2.9), we obtain(3.3)(3.4)(3.5)where *ϵ*=*h*/*l* is a small parameter (equivalent to a small axial strain). Here and thereafter, we have dropped the tilde for convenience. The full forms of (3.3) and (3.4) are very lengthy and the interested readers can find them in appendix A. Substituting (3.2) into the traction-free boundary conditions (2.10), we have(3.6)(3.7)where *ν*=*a*^{2}/*l*^{2} is a small parameter for a slender cylinder.

If one had used the unknown *u* and the variable *R* instead of *v* and *s*, the resulting equations would have been 50% longer than (3.3) and (3.4) (cf. (A 1) and (A 2) in appendix A), which are already very lengthy. More importantly, we note that in the above equations the dependence on *R* and *a*/*l* is entirely through *s*(=*R*^{2}) and *ν=a*^{2}/*l*^{2}, respectively.

## 4. Model equation by series and asymptotic expansions

We note that *s* is also a small variable as 0 ≤*s*≤ν. Thus, from equations (3.3)–(3.7), we can see that the three unknowns *w*, *v* and *p*^{*} depend on the variable *z* and the small variable *s* and the small parameters *ϵ* and *ν*; that is,(4.1)

To go further, one might carry out an asymptotic expansion in terms of the small parameter *ν* or *ϵ*. However, such an approach cannot lead to a model equation that satisfies the traction-free boundary conditions up to the right asymptotic order. The right approach is to seek series expansions in terms of the small variable(4.2)

(4.3)

(4.4)

It should be noted that in the pure one-dimensional stress model, it is assumed that *V*_{0}=const. and *P*_{1}=*P*_{2}=*V*_{1}=*V*_{2}=*W*_{1}=*W*_{2}=0. So, here we are also considering the contributions due to the latter six quantities. We also point out that for (4.4) (or (4.2) or (4.3)) to be valid, it is not required that as . Rather, the underlying assumption is that *for fixed ϵ and ν*(for a small *ν*, this appears to be a very weak assumption).

Substituting equations (4.2)–(4.4) into equation (3.4) and equating the coefficient of *s*^{0} to zero yields(4.5)Similarly, substituting equations (4.2)–(4.4) into equation (3.3) and equating the coefficients of *s*^{0} and *s*^{1} yields(4.6)(4.7)The expressions of *H*_{1}, *H*_{2} and *H*_{3} are very lengthy, which are omitted for brevity. From the incompressibility condition (3.5), the vanishing of the coefficients of *s*^{0} and *s*^{1} leads to two equations(4.8)(4.9)Substituting equations (4.2)–(4.4) into the traction-free boundary conditions (3.6) and (3.7) (note that at the lateral surface *s=ν*, we cannot set zero the coefficients of *s*^{0} and *s*^{1}), we obtain(4.10)(4.11)where the lengthy expressions for *H*_{4}−*H*_{7} are omitted. In obtaining the above equations, we have neglected terms higher than (which is consistent with (2.3)).

Equations (4.5)–(4.11) are seven nonlinear ordinary differential equations, which are the governing equations for the seven unknowns *P*_{0}, *P*_{1}, *V*_{0}, *V*_{1}, *W*_{0}, *W*_{1} and *W*_{2}. Unfortunately, mathematically, it is still very difficult to tackle these seven nonlinear equations directly. To go further, we shall further use the smallness of the parameter *ϵ*.

From equation (4.8), we obtain(4.12)In the above equation, omitting terms higher than is as valid as omitting in equation (2.3) given in Fu & Ogden (1999). Using the above equation in equation (4.9), we can express *V*_{1} in terms of *V*_{0} and *W*_{1}. Then, from equations (4.5) and (4.7), we can express *P*_{1} and *W*_{2} in terms of *V*_{0}, *W*_{1} and *P*_{0}. Substituting the expressions for *W*_{0}, *V*_{1}, *P*_{0} and *W*_{2} into (4.6), (4.10) and (4.11), we obtain(4.13)(4.14)(4.15)Equations (4.13) (coming from the coefficient of *s*^{0} in the axial equilibrium equation), (4.14) and (4.15) (coming from the traction-free boundary conditions) provide three equations for three unknowns *P*_{0}, *V*_{0} and *W*_{1}.

We note that the terms with *νϵ* in equations (4.14) and (4.15) represent the coupling effects of the material nonlinearity and the geometrical size of the cylinder. To further simplify the analysis, we assume that (this implies that ). In experiments, the maximum displacement is about 8% *l* and *a* is about (1/60)*l*; thus, this is a very good approximation. In this case, the last terms in equations (4.14) and (4.15) can be neglected. As a result, from equation (4.14), we obtain(4.16)Substituting (4.16) into (4.13) and (4.15), we obtain(4.17)(4.18)We note that the above two equations come from the axial equilibrium equation (the coefficient of *s*^{0}) and zero tangential force at the lateral surface, the two most important relations for tension/extension problems in a *slender* cylinder.

With some simple manipulations, it is possible to eliminate *W*_{1zz} from equation (4.17) and (4.18), and as a result, we obtain an equation for the single unknown *V*_{0}:(4.19)By further using (4.12), we obtain the following equation for the axial strain *W*_{0z}:(4.20)where *d*_{1} and *d*_{2} are related to the material constants. In the case of (2.4), they are given by(4.21)

Integrating (4.20) once, we obtain(4.22)where *C* is an integration constant. To find out the physical meaning of *C*, we consider the resultant force *T* acting on the material cross-section that is planar and perpendicular to the cylinder axis in the reference configuration, and the formula is(4.23)By using equations (4.2)–(4.4) and the expressions of *V*_{0}, *V*_{1}, *W*_{1}, *P*_{0}, *P*_{1} in terms of *W*_{0z} in (2.6), it is possible to express *Σ*_{zZ} in terms of *W*_{0z}. Then, carrying out the integration in (4.23), we find that(4.24)Comparing equations (4.22) and (4.24), we have . Thus, we can rewrite (4.22) as(4.25)If we retain the original dimensional variable and let , we have(4.26)where(4.27)As *V*_{0}, *V*_{1}, *P*_{0}, *P*_{1}, *W*_{1} and *W*_{2} can be expressed in terms of *W*_{0z}, once *U* is found, all of these quantities can also be found. Thus, equations (4.26) provide the model equation for the present problem.

## 5. Whitham's approach

We can also obtain the governing equation through an approach used to study nonlinear waves in fluids (see Whitham 1974). The idea to obtain the model equation is to consider the terms that should be present. First, the dominant equation should be the axial equilibrium equation for a slender cylinder. For a model equation to be valid, we need it to be able to model the case of a uniform stretch (either in a complete low-strain phase or high-strain phase). For a given strain–energy function *Φ*(*I*_{1}, *I*_{2}), in the case of a uniform stretch with an axial strain *τ*_{0}, we can write . The equation governing *τ*_{0} is

Thus, in order that the model equation, which takes into account the radial deformation and traction-free conditions, can give the correct results for the case of a uniform stretch, that equation should contain the term

Other term(s) in the model equation should represent the influence due to the radial deformation and traction-free boundary conditions. To leading order, if one neglects the coupling effects of the material nonlinearity and the geometric size, that term should be a linear term. Therefore, to obtain that term, it is sufficient to work with linear three-dimensional field equations. In a recent paper (Dai & Huo 2002), we have worked with three-dimensional field equations with second-order material nonlinearity. Thus, as long as we only work with the linear parts in that paper, we can deduce the required term. Indeed, from eqns (3.6)–(3.10), (3.15) and (3.16) in Dai & Huo (2002), one can easily deduce the right term is . Thus, the correct model equation is(5.1)This is in agreement with the dynamical equation derived in Dai (2004).

One can check very easily that in the case of a uniform stretch with an axial strain *τ*_{0} for the strain–energy function given in (2.4),(5.2)Substituting equation (5.2) into (5.1), we obtain(5.3)which is the same as equation (4.20). This shows the agreement by two approaches.

The advantage of Whitham's approach is that one can write down the model equation quickly. However, the advantage of the approach given in §4 through series and asymptotic expansions is that it can lead to the precise relations how *W*_{1}, *W*_{2}, *V*_{0}, *V*_{1}, *P*_{0} and *P*_{1} depend on *W*_{0}, which Whitham's approach cannot provide.

It is also possible to deduce the model equation for an arbitrary strain–energy function *Φ*(*I*_{1}, *I*_{2}) based on the results given in §4. After some tedious calculations, one can express *I*_{1} and *I*_{2} in terms of *U*=*W*_{0Z}. Then, from (2.4), we find the average strain–energy function over a cross-section is given by(5.4)For a strain–energy function other than (2.4), only the coefficients in equation (2.6) and other stress components are different. As these equations are the starting points of analysis, equation (5.4) should hold for any strain–energy function, except that *D*_{1} and *D*_{2} are different for different strain–energy functions.

Suppose that the average stress of a cross-section is 3*μγ*. Then, for any strain–energy function, the average potential energy of a cross-section is(5.5)By the variational principle, we have the governing equation for *U*:(5.6)which is equivalent to (5.1).

From (5.4), it can be seen that by taking into account the radial deformation and traction-free boundary conditions, the average strain–energy function over a cross-section contains an extra term depending on the gradient, which appears to play the same role as the surface energy. It is because of this term that the average stress *γ* over a cross-section, as can be seen from (5.6), further depends on a strain-gradient term. Truskinovsky (1985) pioneered the idea of the regularizing augmentation, which involves directly adding terms (like a strain-gradient term) into the usual constitutive stress–strain relation. Although the end results appear to be the same, the mechanisms for this term in our approach and that of Truskinovsky are different. The regularized approach is intuitive and based on the phenomenological considerations but the physical basis for adding such terms is not fully understood (see Knowles 2002, p. 1173). More importantly, it is very difficult to determine the coefficients of these terms experimentally. Actually, the authors are not aware that anyone has determined the coefficient of the surface energy. Here, we have actually derived this strain-gradient term with an explicit coefficient. It turns out that the value of this coefficient (proportional to *a*^{2}) has a great influence on the nucleation stress and the number of solutions and so on (as found in part II).

## 6. Solutions in an infinitely long cylinder

Here and in part II, we shall consider a class of strain–energy functions *Φ*(*I*_{1}, *I*_{2}) such that is non-convex or equivalently that the stress −strain(*U*) curve has a peak-valley combination (see figure 1), a typical character of phase-transforming materials. For the form given in (5.2), this requires that(6.1)The peak stress value *γ*_{2}, the valley stress value *γ*_{1} and the Maxwell stress value *γ*_{m} can be expressed in terms of *D*_{1} and *D*_{2}(6.2)

Solutions in an infinitely long rod have been studied by Coleman (1983) and Coleman & Newman (1987). Their approach was to directly propose a one-dimensional constitutive relation (how the axial stress depends on the axial stretch). Tong *et al*. (2001) also adopted such an approach and they also gave some solutions based on a trilinear stress–strain curve. Here, we shall construct the solutions based on the asymptotic model equation (4.26), which is derived in a consistent manner. We present the results according to the different values of the dimensionless external stress. By integrating (4.26) once, we obtain(6.3)We consider only bounded solutions for an infinitely long cylinder, whose expressions can be obtained from the above equation by considering the behaviour of the right-hand side of equation (6.3) and through quadrature.

If one works with (5.6); that is, for a general strain–energy function, one has to use numerical computations to get the solution profiles. By using (4.26), one can obtain explicit solution expressions while the qualitative information revealed by them should also be valid for a general strain–energy function. 2. Here, we take the view that phase transitions can be modelled by non-convex strain–energy functions (Ericksen 1975; Ball & James 1992; Abeyaratne *et al.* 2001). Owing to the non-convex function used by us, all the non-trival solutions given below have the ‘multiphase’ character: some part is in the low-strain phase (corresponding to the part of the curve in figure 1 to the left of the stress peak *γ*_{2} and some part is in the high strain phase (corresponding to the part of the curve in figure 1 to the right of the valley stress *γ*_{1}).

*Case* (*a*): 0 ≤*γ*<γ_{1}

In this case, from equation (4.26), it is easy to see that there is a unique constant strain solution with *U*=*U*_{a}, where *U*_{a} is the unique positive root of(6.4)

*Case* (*b*): *γ=γ*_{1}

There are two solutions with constant strain values *U*_{a} and *U*_{c}, which are two positive roots of equation (6.4).

*Case* (*c*): *γ*_{1}<*γ* <γ_{m}(where *γ*_{m} is the Maxwell stress)

First, there are three solutions with constant strain values, which are three positive roots of (6.4).

In this case, that the right-hand side of equation (6.3) can have four roots (e.g. labelled in a decreasing order by: *α*_{1}, *g*_{1}, *g*_{2} and *α*_{2}). Correspondingly, there is also a family of periodic solutions, which is one-parameter dependent for a given *γ*. If we take that parameter to be *g*_{1}, the smallest strain value in one period, the solution expression is given by(6.5)where sn(.) is an elliptic function, *Z*_{0} is a constant phase shift and(6.6)

In the particular case that *g*_{2}=*α*_{2}, the above solution becomes a solitary-wave solution given by(6.7)where *g*_{1} _{min} is the corresponding value of *g*_{1} when *g*_{2}=*α*_{2}. Profiles of the slender cylinder corresponding to this solution are shown in figure 2 (here and thereafter, the vertical scale for the radial deformation is enlarged four times for clearness) for different *γ* values in a decreasing order. It appears that this solution describes the process of two-phase boundaries propagating into each other. When eventually they merge, a single high-strain phase is formed.

*Case* (*d*): *γ=γ*_{m}

First, there are three solutions with constant strain values, which are three positive roots of equation (6.4).

There is a kink/anti-kink solution given by(6.8)This appears to represent a two-phase solution, one phase with the high strain valueand another phase with the lower strain valueThe profile of the cylinder corresponding to a kink solution is shown in figure 3. In addition, from equation (6.8), it can be seen that the thickness of the phase boundary is inversely proportional to ; that is, proportional to *αa*, where ( upon using the last inequality in (6.1)). This is in agreement with the experimental observation: The thickness of the phase boundary is of the same order as the lateral thickness of the specimen; that is, the thinner the specimen is, the thinner the phase boundary is (Q. P. Sun 2004, private communication). The importance of the simple solution (6.8) is that it seems to give a description of the structure of the phase boundary. From (6.8), we can find that the gradient of *U* is inversely proportion to the radius *a*, and its maximum value is . The width between *U*_{Z}=±0.5*U*_{Zmax} is 0.88*αa*(<0.62*a*), which may be regarded as the observable width of the phase boundary.

In this case, there is also a family of one-parameter dependent periodic solutions, whose expression is still given by equation (6.5). Profiles of several periodic solutions are shown in figure 4. It can be seen that our asymptotic model permits multi-band (interface) solutions. We have calculated the total potential energy values per period of these four solutions shown in figure 4. From top to bottom, the values are −5.7825*μ*×10^{−4}, −5.7298*μ*×10^{−4}, and −5.6119*μ* × 10^{−4}. It can be seen that as the number of interfaces increases, the total potential energy value also increases. Thus, the configuration with fewer interfaces is more preferable.

*Case* (*e*): *γ*_{m}<γ<γ_{2}

First, there are three solutions with constant strain values, which are given by the three positive roots of (6.4).

There is an anti-solitary wave solution given by(6.9)whereand *g*_{2max} is the corresponding value of *g*_{2} when *g*_{1}=*α*_{1}. As observed in experiments (Shaw & Kyriakides 1995, 1997; Sun *et al*. 2000), in the nucleation process, the stress drops. Profiles of the cylinder corresponding to this solution for five different values of *γ* in a decreasing order are shown in figure 5. It can be seen that the solution (6.9) seems to describe the nucleation process: how locally a low strain is transformed into a high strain and how two interfaces are formed.

It will be interesting to see how the energy is distributed during the nucleation process. The local potential energy is given in (5.5). For a solution (e.g. (6.9)), both *U* and *U*_{Z} are functions of *Z*, and as a result is also a function of *Z*; that is, . We have also plotted the curves corresponding to the five solution profiles in figure 5. It can be seen that during the nucleation process, two localizations of the potential energy takes place near the middle. As the external stress drops, there are more and more concentrations of energy at these two localizations. In the meantime, the positions of these two localizations move apart. When the energy concentration process is completed, two interfaces are formed and become well separated from each other. Thus, from the point view of energy (based on a static problem), the nucleation process is a process of energy localizations, concentrations and separation.

In this case, there is also a family of one-parameter dependent periodic solutions. If we take the parameter to *g*_{2}, the maximum value of the strain *U* in one period, the solution expression is given by(6.10)

*Case* (*f*): *γ*=γ_{2}

There are two solutions with constant strain values, which are two positive roots of (6.4).

*Case* (*g*): *γ*>γ_{2}

There is a unique solution with a constant strain, which is given by the unique positive root of (6.4).

The stability issues of these solutions will not be addressed here for an infinitely long cylinder. In part II, when we consider two boundary-value problems, we shall use the energy criterion to address the preferred configurations.

## 7. Conclusions

Starting from the three-dimensional field equations, by a proper transformation and suitable scalings, we identify a small variable and two small parameters, which characterize the problem of phase transitions in a slender elastic cylinder. Then, through a series expansion in the small variable and an asymptotic expansion in the small parameter (measuring the material nonlinearity) and tedious calculations, we derive the proper asymptotic model equation that takes into account the influences of the radial deformation and traction-free boundary conditions. It turns out that this equation contains a higher-order derivative whose coefficient is proportional to *a*^{2}. This shows the importance of the lateral size of the specimen. An alternative derivation based on Whitham's approach is also presented. Then, we give the bounded solutions for an infinite cylinder. For example, a solution whose expression is represented by the elementary tanh function is obtained, and it appears that it describes a two-phase deformation with a smooth phase boundary. It is also found that the thickness of the phase boundary is proportional to the radius, which is in agreement with experimental observations. The authors are not aware that anyone has provided any analytical/numerical solution to describe the nucleation process. Here, we also give an interesting solution (based on a static problem), which appears to describe the nucleation process. From the point of view of the energy distribution, it is discovered that the nucleation process is one of energy localizations, concentrations and separation.

## Acknowledgments

The work described in this paper is fully supported by a grant from the Research Grants Council of the HKSAR, China (Project No. Cityu 1225/03E). The authors would like to thank Professors Y.B. Fu and Q.P. Sun for valuable discussions.

## Appendix A

The non-dimensional field equations (i.e. the full forms of equations (3.3) and (3.4)) are given below(A1)

(A2)

- Received September 24, 2004.
- Accepted August 1, 2005.

- © 2005 The Royal Society