## Abstract

Finding the complete set of stability conditions of an elastic half-space has been an open problem ever since Biot (Biot 1963 *Appl. Sci. Res.* **12**, 168–182 (doi:10.1007/BF03184638)) first studied the surface instability of half-spaces by seeking solutions of the incremental equilibrium equations. Towards solving this problem, a method based on the energy stability criterion is developed in the present work. A variational problem of minimizing the elastic energy associated with a half-space is formulated. The second variation condition is derived and is converted to an eigenvalue problem. For a half-space of neo-Hookean materials, the eigenvalue problem is solved, which leads to complete descriptions of stability and instability regions in the deformation space.

## 1. Introduction

The problem of surface instability of an elastic half-space was first studied by Biot [1]. In an exposition of the methodology used in his study, Biot writes: ‘A general theory … of small incremental deformations in the vicinity of a state of initial stress. … was applied to the problem of surface instability of an elastic half-space with an initial compression in a direction parallel with the surface. It was found that the surface may become unstable and exhibit a spontaneous waviness beyond a critical value of the compression stress’ [1, pp. 169–170]. For a half-space of the incompressible neo-Hookean material, Biot has derived the incremental equilibrium equations, and sought a solution which is sinusoidal along a direction parallel to the surface and decreases exponentially with depth. This leads to a characteristic equation in terms of the initial strains. In the case when the initial compression is realized by maintaining the two principal stretches at λ and 1, respectively, Biot finds that the characteristic equation is satisfied if (eqn (4.3) in [1])

The problem of surface instability of half-spaces has unique and interesting features. A half-space, of which the boundary consists of a single plane, possesses a distinct geometric simplicity. The semi-infinite extent of a half-space makes it suitable to model materials and structures that have large dimensions in space. In contrast to other types of instabilities, surface instability manifests itself in the form of deformations primarily confined to a narrow region near the surface. Owing to these unique features, the problem of surface instability has attracted the interests of researchers in the past five decades, and has been extended to other geometry and material systems. Only a small sample of the representative works is given here [2–18]. The method of incremental solutions used by Biot, and by virtually all other researchers in studying surface instability, is based on the adjacent equilibrium stability criterion. This stability criterion asserts that a primary equilibrium state, such as the initial compressed state of an elastic half-space, becomes unstable when there are other equilibrium states nearby. While the incremental solution method and the underlying adjacent equilibrium stability criterion are widely used, they leave some issues of fundamental importance unsolved or even unaddressed:

— an incremental solution, such as that found by Biot [1], does not necessarily correspond to an equilibrium state near the primary equilibrium state. In the bifurcation theory, the existence of an increment solution is a necessary condition for the primary equilibrium state to be a bifurcation point. This condition is nevertheless not sufficient. A verification of the existence of the bifurcation solution branches, and therefore the adjacent equilibrium states, requires an extensive nonlinear bifurcation analysis;

— the adjacent equilibrium stability criterion lacks rigorous physical and mathematical bases. It does not provide clear reasoning on the distinction between the primary state and the adjacent states, as the designation of the primary state is often just an equilibrium solution that happens to be known to the researcher. More importantly, the adjacent equilibrium stability criterion does not provide a logical argument on why the primary state becomes unstable when adjacent equilibrium states exist. Why can not a primary state be initially unstable and become stable when adjacent equilibrium states exist?

— a bifurcation analysis, within which the adjacent equilibrium stability criterion inheres, focuses only on equilibrium states during variation of one or more bifurcation parameters, such as the compression strain in [1]. It cannot explain why the primary state is stable prior to the bifurcation point, and why the primary state remains unstable after passing the bifurcation point. At best the adjacent equilibrium stability criterion can provide information on the conditions under which a physical system changes its stability during a process (such as loading), while the ultimate purpose of a stability analysis is to determine whether various states of the system are stable.

This work presents a stability analysis to resolve the above issues by using the energy stability criterion. This criterion asserts that an equilibrium state is stable if an appropriate potential energy functional assumes, at the equilibrium state, a minimum in a class of kinematically admissible (not necessarily equilibrium) states. The energy stability criterion can be derived from more basic physical principles such as the second law of thermodynamics, as well as from the dynamic stability criteria that capture the essence of instability phenomena. By using the energy stability criterion, we derive the complete set of stability conditions for all homogeneous deformations of an elastic half-space, and thus solve an open problem since the pioneering work of Biot [1] on the surface instability of half-spaces.

It is noted that the energy stability criterion has been successfully used to study a large number of elastic stability problems. However, to our knowledge, it has not been used to study the surface instability of half-spaces. This is perhaps due to the difficulty associated with minimizing an energy functional that is infinite. To overcome this difficulty, we devise a method in this work, which is applicable to other problems involving unbounded domains.

It is worth noting that some researchers use the argument of the energy criterion to draw conclusion that a bifurcation solution is stable by showing that it has a lower energy than the primary solution. Such conclusions are incomplete to say the least. Indeed, to conclude the stability of an equilibrium solution by the energy criterion, one must show that the solution has a lower energy than *all* kinematically admissible states, not just the primary equilibrium state.

It is also worth noting that while many researchers have confused a bifurcation analysis with a stability analysis, others have studied and recognized the distinction between them. For example, Ericksen & Toupin [19], and Hill [20] have shown that the existence of an incremental solution at an equilibrium state implies that the second variation of the energy functional is not positive definite at the equilibrium state. This one-way implication, however, cannot establish either stability or instability of an equilibrium state at which there exists no incremental solution. In other words, no conclusion can be drawn concerning the stability of the primary state before or after the critical value of the compression is reached.

Another important development that is highly relevant to the present work is the connection between surface instability and the non-existence of surface waves. In an early study of surface waves propagating in a semi-infinite body, Hayes & Rivlin find that ‘such waves may not be propagated in a principal direction unless stringent conditions are imposed on the strain-energy function’ [21], p. 353. This issue has been further studied by a number of researchers. Most notably, Dowaikh & Ogden [22] study the propagation of infinitesimal surface waves on a half-space of incompressible isotropic material subject to a pure homogeneous underlying deformation. They derive the conditions under which a surface wave can propagate along a principal direction with a non-zero speed. The existence of surface waves is interpreted as ensuring that the underlying deformation is stable. Striking similarity between the stability conditions found by Dowaikh and Ogden and those in the present work suggests important connections between energy minimization and existence of waves.

Biot’s work on surface instability of half-spaces is already considered as a classic in the literature of mechanics, joining other great works in elastic stability, such as that of Euler [23] for the determination of buckling loads of axially compressed columns. It is interesting to note that the derivation of Euler’s critical loads, which can now be found in undergraduate textbooks, uses the idea of the adjacent equilibrium stability criterion. As such, it does not provide a complete understanding of the instability phenomenon. Often, it is taken for granted that a straight column is stable before reaching the critical load, and becomes unstable after passing the critical load. The answers to these and other questions concerning Euler’s stability analysis came years later with the use of the energy stability criterion, which was developed from the work of Lagrange [24] and Dirichlet [25]. Retrospectively, the study of surface instability of half-spaces, initiated by Biot’s work, is now at a stage where a stability analysis using the energy criterion is in order.

This paper is organized as follows. Section 2 is devoted to the basic formulation of the minimization problem for an elastic half-space. An immediate difficulty arises in that the elastic energy functional for a deformed half-space is infinite, which renders the standard methods in the calculus of variations inapplicable. We overcome this difficulty by considering a sequence of sub-bodies that approaches the half-space, and minimizing the energy for each sub-body. Passing to the limit leads to a definition for the energy of a deformed half-space to be a minimum. The first and second variation conditions for this minimization problem are derived in §3. In the derivation of the second variation condition, which is a quadratic integral inequality over the half-space, it is important to choose appropriate boundary conditions at infinity to ensure the convergence of the quadratic integral. The solution of the second variation condition is studied in §4. By minimizing the second variation in a suitable set of variations, the quadratic integral inequality is shown to be equivalent to an eigenvalue problem of a system of partial differential equations. The second variation condition holds if and only if all eigenvalues are non-negative. This eigenvalue problem is solved in §5 by using the Fourier transform, which reduces the partial differential equations to ordinary differential equations. In §6, the solution to the eigenvalue problem is found for a half-space of neo-Hookean material. The solution gives the stability region and the instability region in the space of the principal stretches. This provides unambiguous answers to the questions raised above. Among other things, it shows that the point given by (1.1) is on the boundary of the stability region and the instability region. An equilibrium solution branch under increasing compression is stable before reaching the bifurcation point, becomes unstable at the bifurcation point, and remains unstable thereafter. A summary and possible future research are given in §7.

## 2. Minimum energy principle for elastic half-spaces

We consider an elastic body that occupies, in a reference configuration, a half-space represented by

A deformation of the body can be expressed by a smooth function **x** is assumed to satisfy, in addition to (2.3), the boundary condition
**x**_{0} faster than 1/|**X**|^{2}. It is noted that (2.4) is stronger than the boundary condition
**x**_{0} is defined on the entire half-space B and satisfies (2.3). As a result, **x**_{0} itself is kinematically admissible and can be identified with a deformation whose stability is being examined, such as a homogeneous deformation with prescribed principal stretches.

The elastic body is assumed to be homogeneous, and endowed with a strain-energy function
**F**.

In this work, we formulate an energy stability criterion for the stability of elastic half-spaces. The energy stability criterion, which can be traced back to Lagrange and Dirichlet, has been widely used to study the stabilities of systems of particles or bounded continua. It asserts that a deformation **x** is stable if a properly defined potential energy functional at **x** is not greater than the potential energy at other kinematically admissible deformations in an appropriate neighbourhood of **x**. For a bounded elastic body, the potential energy functional consists of the strain energy stored in the deformed body and the potential energy associated with a loading device. If the loading device prescribes the deformation on the boundary, the potential energy is taken to be zero. Such a loading device is called a hard loading device, as opposed to a soft loading device which prescribes the traction on the boundary. In this work, being consistent with the boundary condition (2.4) (or (2.5)), we adopt an extension of a hard loading device to the half-space, and assume that the loading device does zero work when the elastic half-space deforms from **x** to another kinematically admissible deformation. As a side note, an extension of a soft loading device to the half-space is far less obvious. For a bounded body, a soft loading device prescribes the traction applied on the boundary. For the half-space, the notion of boundary has to be extended to the material points at infinity. It is not clear how to prescribe the traction at infinity from outset.

From the above discussion, the potential energy consists of only the strain energy stored in the deformed body. Such a potential energy, however, is infinite in general, as B defined by (2.1) is unbounded. Using the energy stability criterion to study the stability of the half-space then faces the difficulty of comparing infinite energies.

The concept of improper integrals over unbounded domains suggests that one overcomes this difficulty by comparing the energies of a sequence of sub-bodies that approaches the half-space B. Precisely, we define a sub-body **x** in the sub-body

We now introduce the energy stability criterion for deformations of the half-space. Given two kinematically admissible deformations **x** and **x** is not greater than that associated with *R*>0, such as
*r*>*R*. Furthermore, deformation **x** is said to be stable if its energy is not greater than that of each kinematically admissible deformation in a neighbourhood of **x**. A neighbourhood of **x** is the set of all kinematically admissible deformations *ϵ*>0. Here, ∥⋅∥ is a properly defined norm for the space of deformation functions.

## 3. First and second variation conditions

The energy stability criterion formulated above does not constitute a standard minimization problem in calculus of variations, as the energy functional to be minimized is not defined on a fixed region, but on a sequence of hemispherical regions that approach the half-space. We now derive the first and second variation conditions for this minimization problem.

Let a kinematically admissible deformation **x** be given that satisfies the constraint of incompressibility (2.3) and the boundary condition (2.4). Suppose that the deformation **x** is stable. That is, there exists an *ϵ*>0, such that for each kinematically admissible deformation *ϵ*-neighbourhood of **x** defined by (2.10) there is an *R* so that
*r*>*R*.

The derivation of the first variation condition is straightforward. By the definition of stability, the deformation **x** being stable implies that inequality (3.1) holds for each kinematically admissible deformation *r*>*R*. In particular, (3.1) holds for all kinematically admissible deformations that satisfy
*R*. This is a constrained minimization problem on the bounded domain *p* a hydrostatic pressure required by the constraint of incompressibility, and again *X*_{3}=0, corresponding to a free traction boundary condition. Equations (3.3) and (3.4) must hold for an arbitrary *R*. This leads to the following first variation condition for minimizing the potential energy in the half-space:

To derive the second variation condition for the given deformation **x**, let a kinematically admissible deformation

where the last term is of higher order in **n** is the unit outward normal to **u** that satisfy *r*.

This leads to the following second variation condition for the stability of the half-space:
**u** that satisfy the constraint

In the next section, we solve the quadratic integral inequality (3.11) by seeking minima of the integral in a compact subset of the class of admissible displacements **u**.

## 4. An eigenvalue problem associated with the second variation condition

The integral in the second variation condition (3.11) is a continuous functional of the displacement **u**. A natural way to determine whether (3.11) holds for all **u** is to examine whether the minimum, if it exists, of the functional is non-negative. However, such a minimum may not exist. This suggests us to identify a compact subset of the domain of the functional, on which the minimum exists, and on which the positive definiteness of the functional remains unchanged. It can be readily verified that inequality (3.11) holds for all kinematically admissible displacements **u** if and only if it holds for all kinematically admissible displacements that satisfy an additional normalization condition
**u** that satisfy (3.12), (3.13) and (4.1), with the equality in (4.2) holding for a particular **u**. Therefore, inequality (3.11) holds if and only if *μ*≥0.

The particular **u**, at which the equality in (4.2) holds, can be found by minimizing the integral subject to the constraint (3.12) and the normalization condition (4.1). The first variation condition of this constrained minimization problem leads to the following boundary-value problem:
*γ* and *μ* are the Lagrange multipliers associated with the constraints (3.12) and (4.1), respectively. Here, we have restated equations (3.12), (3.13) and (4.1) for a complete presentation of the boundary-value problem. Also, the notation has been so chosen that the constant Lagrange multiplier *μ* is in connection with the minimum value of the integral in (4.2). Indeed, taking the inner product of (4.3) and **u**, integrating the resulting equation on B, and using (4.4)–(4.7), we find that at a solution of the boundary-value problem (4.3)–(4.7), the value of the integral in (4.2) is exactly *μ*.

We thus treat (4.3)–(4.7) as an eigenvalue problem associated with minimizing the second variation in (3.11), with *μ* being the eigenvalue. The above arguments lead to the conclusion that the second variation condition (3.11) is satisfied if and only if *μ*≥0 for all solutions of the eigenvalue problem (4.3)–(4.7).

## 5. Solutions of the eigenvalue problem for homogeneous deformations

A deformation **x** is homogeneous if the deformation gradient is independent of **X**. An admissible homogeneous deformation **x** of a homogeneous elastic body satisfies the equilibrium equation (3.5) and the boundary conditions (2.4) and (3.6) if and only if the hydrostatic pressure *p* is a constant and **x** is identical to the prescribed far-field deformation **x**_{0}. In this case, the eigenvalue problem (4.3)–(4.7) consists of a system of linear partial differential equations with constant coefficients, and can be solved by using the Fourier transform.

A solution pair (**u**,*γ*) of (4.3)–(4.7) admits its two-dimensional Fourier transform *X*_{1}−*X*_{2} plane:
*X*_{3}:

As **F** is constant, the Fourier transform of (4.4) yields
*p* and the derivatives of *W* are constant, the Fourier transform of (4.3), with help of (5.6), gives
*ξ*=|** ξ**|. Via the Fourier transform, we have effectively reduced the partial differential equations (4.4) and (4.3) to an equivalent system of ordinary differential equations (5.6) and (5.7) with variable

*X*

_{3}, the Fourier transform variables

**being parameters.**

*ξ*Similarly, the Fourier transform of the boundary condition (4.6) gives

Equations (5.6) and (5.7) can be converted to a system of six first-order linear ordinary differential equations with constant coefficients, which possesses six linearly independent solutions. These solutions can be found by using the standard method. Let
**a**, *b* and *m* are complex vector and scalars to be determined. Substituting (5.10) into (5.6) and (5.7) results in the following system of algebraic equations:

The above equations are a system of homogeneous linear algebraic equations for **a** and *b*. It has a non-trivial solution if and only if the determinant of the coefficient matrix vanishes. It can be shown, by using the symmetry of ∂^{2}*W*/∂**F**^{2}, that the determinant is a cubic function of *m*^{2}. This leads to three pairs of solutions, each pair consisting of two opposite numbers. Hence, equations (5.11) and (5.12) have six independent solutions
*m*_{j} have positive real parts, and the other three negative real parts. Substituting the solutions (5.13) into (5.10) gives six linearly independent solutions of (5.6) and (5.7). Only three of them with positive real parts of *m* satisfy the boundary condition (5.9). We thus form the following solutions of (5.6) and (5.7) that satisfy the boundary condition (5.9):
*C*_{j},*j*=1,2,3 are complex coefficients that are dependent on ** ξ**. It remains to satisfy the boundary conditions (5.8). Upon substitution of (5.14) into (5.8), we arrive at a system of three algebraic equations for

*C*

_{j}. The condition under which these equations have non-trivial solutions then determines the eigenvalues

*μ*, and hence the stability of the deformation

**x**according to the sign of

*μ*. Moreover, these non-trivial solutions lead to the Fourier transforms of the displacements for the minimum of the potential energy of the half-space.

## 6. Half-space of a neo-Hookean material under bi-axial loading

We now consider the stability of homogeneous deformations of a half-space composed of a neo-Hookean material. The strain-energy function of a neo-Hookean material is given by
*c* is a positive material constant that corresponds to the shear modulus of infinitesimal deformation. We then have
**I**_{4} is the identity fourth-order tensor. The equilibrium equation (3.5) and the traction boundary condition (3.6) reduce to

The half-space is assumed to be subjected to bi-axial loading with the far-field deformation **x**_{0} given by
_{1} and λ_{2} are the stretches in the directions of *X*_{1} and *X*_{2}, respectively. A solution, which corresponds to a homogeneous deformation, of the equilibrium equation (6.3), the constraint (2.3) and the boundary conditions (2.4) and (6.4) is given by

To examine the stability of this homogeneous deformation, we solve the system of algebraic equations (5.11) and (5.12), the latter now reducing to
^{1} of (5.11) and (6.9) are
*m* are used to form the solution (5.14) which is now written explicitly as

Substituting (6.11) and (6.12) into the boundary condition (5.8) and making use of (6.2), (6.7) and (6.8), we find that
** ζ**,

**and**

*η***e**

_{3}are mutually orthogonal, the coefficient of each term in (6.13) must vanish. That is,

*μ*=

*c*, equations (6.14)–(6.16) have a solution

*C*

_{2}=

*C*

_{3}=0 and

*C*

_{1}being an arbitrary function of

*ξ*

_{1}and

*ξ*

_{2}. This solution, however, plays no role in determining the stability of the homogeneous deformation, as this deformation is stable if the eigenvalue

*μ*is positive for

*each and every*solution. Similarly, when

*μ*=2

*c*, there is a solution

*C*

_{1}=0, and

*C*

_{2}and

*C*

_{3}being arbitrary on the set

*η*=

*ξ*and vanishing otherwise. This solution again plays no role in determining the stability of the homogeneous deformation.

It now suffices to consider the remaining solutions of (6.14)–(6.16), for which *μ*≠*c* and *μ*≠2*c*. These solutions are given by
**C** is an arbitrary complex-valued vector and the function *f*: **u**,*γ*) of the boundary-value problem (4.3)–(4.7) are given by
**u**,*γ*) is then given by the inverse Fourier transforms of *g*, **v** and *ψ* are the inverse Fourier transforms of *f*,

For prescribed stretches (λ_{1},λ_{2}), the solution (**u**,*γ*) given by (6.23) is non-zero if and only if *g*, and therefore *f*, is non-zero. It then follows from (6.18) that **u** is non-zero if and only if the set *μ*, as well as λ_{1} and λ_{2}. It then follows that the homogeneous deformation (6.6) is unstable if the set J is non-empty for some negative *μ*, that is, if the equation *J*(*ξ*_{1},*ξ*_{2})=0 has non-zero solutions for some negative *μ*. The latter equation can be written as
*μ*, and admits solutions
*μ*_{1}<*μ*_{2}. Hence, equation (6.24) has solutions for a negative *μ* if and only if there exists *α* that satisfies (6.26) and
*α*_{0}≅0.2956 is the only positive solution of the equation
_{1},λ_{2}) that satisfies
*α* that satisfies (6.26) is greater than *α*_{0} and hence renders *μ*_{1} positive, that is, the second variation is non-negative for all admissible displacements. We thus conclude that the region of stability for the homogeneous deformation (6.6) of a half-space of neo-Hookean materials is given by (6.32).

It is worth pointing out that *α*_{0} is the square of the Biot’s critical value of stretch for the two-dimensional compression. Indeed, a review of [1] shows that the Biot’s critical value is obtained from an equation that is identical to (6.30) with *α* being replaced by λ^{2}, the square of stretch. It is also worth mentioning that inequality (6.28) has appeared in the paper by Dowaikh & Ogden [22] as the condition of non-existence of surface waves along a principal direction under plane strain λ_{2}=1. Indeed, the reversed inequality (6.28), with *α* being replaced by λ^{2}, is identical to (7.12) in [22], that ensures the existence of a surface wave when the surface is free of traction.

The instability and stability regions in the λ_{1}–λ_{2} plane are plotted in figure 2. It is observed that Biot’s result (1.1) corresponds to the point (λ_{1},λ_{2})=(1,0.5437) or (0.5437,1) on the boundary of the stability region and the instability region, which signifies a change of stability. It also provides definite and clear answers to the questions raised in the introduction section. Indeed, it shows that the undeformed state (λ_{1},λ_{2})=(1,1) is stable. Along a loading path of increasing compression, the homogeneous deformations are stable until reaching a boundary point between the stability region and the instability region. When the compression increases further from that point, the homogeneous deformations remain unstable. Possible ensuing events include the half-space reaching an inhomogeneous stable equilibrium state, or entering a dynamic process without settling down to a stable equilibrium.

At each point (λ_{1},λ_{2}) in the region of instability, there exist multiple kinematically admissible displacements **u** which render *μ*_{1}, and therefore the second variation, negative. We wish to identify the displacements that correspond to the smallest negative eigenvalue, and hence correspond to the lowest energy. Such displacements provide characterizations of the surface deformation when the homogeneous deformation becomes unstable. It is observed from (6.26) and (6.27) that *μ*_{1} is an increasing function of *α*, and attains the minimum value when _{1}<λ_{2}. It then follows from (6.25) that *α* takes the minimum value *ξ*_{1}≠0 and *ξ*_{2}=0. In other words, the set J, on which function *f*(*ξ*_{1},*ξ*_{2}) for the smallest eigenvalue *μ*_{1} can be non-zero, is given by
*f* is non-zero only on set J, it must be a generalized function (distribution) in order for its inverse Fourier transform, and therefore the displacement **u**, to be a non-zero function. The most general form of *f* is then
*δ* is the Dirac *δ*-function. Substituting (6.34) into (6.20) and taking the inverse Fourier transform, we find that the displacement **u** corresponding to the smallest negative eigenvalue *μ*_{1} is given by

Some important observations can be made. As we have shown, a displacement **u** that minimizes the second variation must be an eigenfunction of the eigenvalue problem (4.3)–(4.7), with the corresponding eigenvalue being the minimum of the second variation. When λ_{1}<λ_{2}, such a displacement is given by (6.35), which is independent of *X*_{2}. Also, it follows from (6.10), (6.21) and (6.35) that the *X*_{2}-components of **u** vanish. Therefore, the displacement **u** in (6.35) corresponds to a plane strain state for which the displacement component *u*_{2} is zero and the remaining two displacement components *u*_{1} and *u*_{3} are independent of *X*_{2}. It is noted that the inequality λ_{1}<λ_{2} implies that the larger in-plane compression of the half-space occurs in the *X*_{1} direction, which is precisely the direction of the aforementioned plane strain. This conclusion is consistent with the common belief that buckling should occur in the direction of maximum compression.

It is noted that the displacement **u** in (6.35) involves an arbitrary function *ϕ*. The normalization condition (4.5) imposes, on *ϕ*, a condition which can be found by using Barseval’s theorem. In addition, the far-field boundary condition (4.7) requires that **u** approaches zero rapidly as *ϕ* that meets certain regularity conditions. It is interesting to investigate whether the boundary condition (4.7) can be replaced by a physically reasonable condition, and how this boundary condition affects *ϕ* and the eigenfunctions.

## 7. Summary

This paper solves a long-standing open problem of determining the complete set of stability conditions for a half-space composed of incompressible neo-Hookean materials. By using the energy stability criterion, we determine the region of stability and the region of instability in the space of principal stretches. The salient features of the analysis include:

— The conventional energy stability criterion is extended to a problem with an unbounded domain, the half-space. To overcome the difficulty of analysing a functional of infinite value, we minimize the energy functional over a sequence of bounded domains that approach the half-space.

— The first and second variation conditions of this non-conventional variational problem are derived. A pertaining technical issue is to identify the proper boundary conditions at infinity to ensure the convergence of the second variation.

— The second variation condition, which is a quadratic integral inequality and serves as a stability condition, is solved by minimizing the second variation itself. As the full space of variations is not compact, the minimum of the second variation may not exist. We thus deduce the second variation on a compact subset, which ensures the existence of the minimum. The Euler–Lagrange equation in minimizing the second variation leads to an eigenvalue problem of a system of partial differential equations. The minimum of the second variation corresponds to an eigenvalue.

— The eigenvalue problem is solved by using the Fourier transform, which reduces the partial differential equations to a system of ordinary differential equations. As an eigenfunction may not be integrable, its Fourier transform could be a generalized function (distribution).

— For given principal stretches of the half-space, all the eigenvalues are found, and the sign of the smallest eigenvalue determines the stability of the half-space at the given stretches. This gives the complete set of stability conditions in the space of the principal stretches.

— By analysing the eigenfunctions and their inverse Fourier transform, it is found that the displacement that induces instability of the half-space occurs in the direction of the largest compression.

Further research of surface instability of half-spaces could be taken in several directions. A possible extension of the present work is to study the stability of half-spaces of other type elastic materials. With the methodology developed in this work, it is even possible to derive stability conditions for elastic half-spaces with a strain-energy function of general form. Such stability conditions, expected in the form of inequalities involving the derivatives of the strain-energy function, will provide important insight into the interconnections between material behaviours and the constitutive functions. A related remark is that the critical value of the compression (1.1) predicted by Biot [1] has been challenged or criticized based on the claims that it does not agree with the experiments conducted by Gent & Cho [5] and others. Such claims, however, lack merit as the neo-Hookean model, on which Biot’s work is based, may not be a reasonable model for the materials in the experiments.

Another possible direction is to study the minima of the energy functional in the function spaces that involve singular functions. In the present work, only smooth functions are considered. This is thus unable to detect possible instabilities induced by non-smooth perturbations. It has been reported that instability can be induced by crease-type deformations that may involve singular strain fields. The existing works on crease instability, however, have not considered singular functions.

Yet another direction is to seek absolute minima of the energy functional. The present work concerns the relative minima, by considering small perturbations (in the C^{1} norm) from the ground state. Namely, the energy of an equilibrium state is compared with the energies of the neighbouring states to determine the stability of the equilibrium state. The stability conditions so obtained pertain to the local stability, which excludes possible instabilities due to the states that have lower energy than the equilibrium state but are not in a neighbourhood of the equilibrium state. The work of Biot [1], and of others, based on the adjacent equilibrium stability criterion is subjected to the same limitation. On a related note, efforts have been made to study the crease surface instability by using a post bifurcation analysis. This approach, potentially being able to reach states not in a neighbourhood of the ground state, could be problematic as a crease deformation may not be connected to the ground state via a bifurcation solution branch of equilibrium deformations.

## Data accessibility

The research reported here is purely analytical and thus does not have any associated experimental data.

## Authors' contributions

All the authors contributed to the research and to preparing the manuscript. They all gave final approval for submission.

## Competing interests

We declare we have no competing interests.

## Funding

We received no funding for this study.

## Acknowledgements

The authors thank anonymous reviewers for their comments and suggestions on an earlier version of the manuscript.

## Footnotes

↵1 Here and henceforth we assume that

*ξ*≠*η*. The case where*ξ*=*η*corresponds to λ_{1}=λ_{2}=1. The stability of this undeformed state can be deduced from that of the neighbouring states, since the smallest*μ*depends on λ_{1}and λ_{2}continuously.

- Received December 7, 2017.
- Accepted April 9, 2018.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.