Likely equilibria of stochastic hyperelastic spherical shells and tubes
Abstract
In large deformations, internally pressurised elastic spherical shells and tubes may undergo a limitpoint, or inflation, instability manifested by a rapid transition in which their radii suddenly increase. The possible existence of such an instability depends on the material constitutive model. Here, we revisit this problem in the context of stochastic incompressible hyperelastic materials, and ask the question: what is the probability distribution of stable radially symmetric inflation, such that the internal pressure always increases as the radial stretch increases? For the classic elastic problem, involving isotropic incompressible materials, there is a critical parameter value that strictly separates the cases where inflation instability can occur or not. By contrast, for the stochastic problem, we show that the inherent variability of the probabilistic parameters implies that there is always competition between the two cases. To illustrate this, we draw on published experimental data for rubber, and derive the probability distribution of the corresponding random shear modulus to predict the inflation responses for a spherical shell and a cylindrical tube made of a material characterised by this parameter.
Key words: stochastic hyperelastic model, spherical shell, cylindrical tube, finite inflation, limitpoint instability, probabilities.
1 Introduction
The idealised model of an internally pressurised hollow cylinder or sphere is instructive as it applies to many structures from living cells to blood vessels to aircraft fuselages [14, 46]. For these structures to be serviceable they must be able to withstand and function at a certain level of internal pressure without damage. The finite symmetric inflation and stretching of a cylindrical tube of homogeneous isotropic incompressible hyperelastic material was initially studied by Rivlin (1949) [33]. For an elastic spherical shell, the finite radially symmetric inflation was first investigated by Green and Shield (1950) [15] (see also [2, 37]). A general theory of possible qualitative behaviours for both the elastic tubes and the spherical shells was developed by Carroll (1987) [7], who showed that, depending on the particular material and initial geometry, the internal pressure may increase monotonically, or it may increase and then decrease, or it may increase, decrease, and then increase again. This formed the basis for further studies where these deformations were examined for different material constitutive laws [30, 13, 48], and opened the way to the modelling of more complex phenomena [16, 26, 47]. Recently, localised bulging instabilities in an inflated isotropic hyperelastic tube of arbitrary thickness, which were also observed in some materials, provided that the tube is sufficiently long [12], were modelled and analysed within the framework of nonlinear elasticity in [11].
Clearly, the behaviour of a structure depends on the inextricable relation between its material properties and its geometry. It is therefore of the utmost importance to use suitable constitutive models as confirmed by experience and experiments. Unavoidably, uncertainties are attached to material properties. For natural and industrial elastic materials, uncertainties in the mechanical responses generally arise from the inherent microstructural inhomogeneity, sampletosample intrinsic variability, and lack of data, which are sparse, inferred from indirect measurements, and contaminated by noise [6, 10, 19, 31]. Within the nonlinear elasticity field theory, which is based on average data values and covers the simplest case where internal forces only depend on the current deformation of the material and not on its history, hyperelastic materials are the class of material models described by a strainenergy function with respect to the reference configuration [14, 32, 45]. For these materials, boundary value problems can be cast as variational problems, which provide powerful methods for obtaining approximate solutions, and can also be used to generate finite element methods for computer simulations. The tradition so far has been to only consider average values to fit deterministic models. More recently, the use of the information about uncertainties and the variability in the acquired data in nonlinear elasticity has been proposed by the introduction of stochastic hyperelastic models which are characterised by a stochastic strainenergy function for which the model parameters are random variables that satisfy standard probability laws [28, 40, 41, 42, 43]. These models are based on the notion of entropy (or uncertainty) defined by Shannon (1948) [35, 36], and employ the maximum entropy principle for a discrete probability distribution introduced by Jaynes (1957) [20, 21, 22].
Stochastic approaches are continually growing in importance as a tool in many disciplines, such as materials science, engineering, and biomechanics, where understanding the variability in the mechanical behaviour of materials is critical. However, by contrast to the tremendous development and variety of stochastic finite element methods, which have been proposed and implemented extensively in recent years [3, 4, 17, 18], at the fundamental level, there is very little understanding of the uncertainties in the mechanical properties of elastic materials under large strains. In this case, the question arises [29]: what is the influence of the stochastic model parameters on the predicted elastic responses?
In recognition of the fact that a crucial part of assessing the elasticity of materials is to quantify the uncertainties in their mechanical responses under large deformations, in the present study, we determine the probability distribution of stable deformation for spherical shells and cylindrical tubes of stochastic isotropic hyperelastic material under radially symmetric inflation. For the purely elastic problem, involving isotropic incompressible materials, there is a critical parameter value that strictly separates the cases where inflation instability can occur or not. However, for the stochastic problem, we find that, due to the probabilistic nature of the material law, there is always competition between the two cases. Therefore, we can no longer talk about ‘equilibria’ but ‘likely equilibria’ obtained under a given internal pressure with a given probability.
Our stochastic elastic setting provides a general mathematical framework applicable to a class of stochastic hyperelastic materials for which similar results can be obtained. As a specific example, we refer to the experimental data for vulcanised rubber of Rivlin and Saunders (1951) [34], from which we derive the probability distribution of the random shear modulus, and predict the inflation responses for a spherical shell and a cylindrical tube made of a material characterised by this parameter.
We begin, in section 2, with a detailed description of the stochastic elastic framework. Then, in sections 3 and 4, for stochastic spherical and tubes, respectively, first, we review the solution to the elastic problem under radially symmetric inflation, then we recast the problem in the stochastic setting and find the probabilistic solution. Concluding remarks and a further outlook are provided in section 5.
2 Stochastic isotropic hyperelastic materials
We briefly recall that a hyperelastic material is characterised by a strainenergy function that depends on the deformation gradient tensor, , with respect to the reference configuration [14, 32, 45]. It is typically characterised by a set of deterministic model parameters, which then contribute to defining the constant elastic moduli under small strains, or the nonlinear elastic moduli, which are functions of the deformation, under large strains [27].
By contrast, a stochastic hyperelastic material is described by a stochastic strainenergy function, which represents an ensemble of elastic strainenergy functions that are similar in form, but for which the model parameters are random variables that satisfy standard probability laws [28, 40, 41, 42]. Specifically, each model parameter is described in terms of its statistical moments that represent partial information in the order of importance. In practice, the first and second moments, namely the mean value and the variance, respectively, usually suffice for the approximation [8, 19, 25]. Here, we combine information theory with finite elasticity, and rely on the following assumptions [28, 29]:

Material objectivity: the principle of material objectivity (frame indifference) states that constitutive equations must be invariant under changes of frame of reference. It requires that the scalar strainenergy function is unaffected by a superimposed rigidbody transformation (which involves a change of position) after deformation, i.e. , where is a proper orthogonal tensor (rotation). Material objectivity is guaranteed by considering strainenergy functions defined in terms of invariants.

Material isotropy: the principle of isotropy requires that the scalar strainenergy function is unaffected by a superimposed rigidbody transformation prior to deformation, i.e. where . For isotropic materials, the strainenergy function is a symmetric function of the principal stretches of F, , , , i.e. .

BakerEricksen inequalities: in addition to the fundamental principles of objectivity and material symmetry, in order for the behaviour of a hyperelastic material to be physically realistic, there are some universally accepted constraints on the constitutive equations. Specifically, for a hyperelastic body, the BakerEricksen (BE) inequalities state that the greater principal stress occurs in the direction of the greater principal stretch [5]. In particular, under uniaxial tension, the deformation is a simple extension in the direction of the tensile force if and only if the BE inequalities hold [24]. Under these mechanical constraints, the shear modulus of the material is positive [27].
Assumptions (A1)(A3) are longstanding principles in isotropic finite elasticity [14, 32, 45], while (A4) concerns physically realistic expectations on the random shear modulus, which will be described by a probability distribution.
In particular, we confine our attention to a class of stochastic incompressible hyperelastic materials described by the constitutive law [42, 28],
(1) 
where and are deterministic constants, and and are random parameters. Consistent with the purely elastic theory [27], the random shear modulus for infinitesimal deformations of the stochastic model (1) is defined as .
For these materials, condition (A4) is guaranteed by the following constraints on the expected values [40, 41, 42, 28, 29]:
(2) 
These expected values are then used to find the maximum likelihood probability for the random shear modulus, , with mean value , and standard deviation , defined as the square root of the variance, . Specifically, under the constraints (2), follows a Gamma probability distribution [38, 39], with hyperparameters and satisfying
(3) 
The corresponding probability density function takes the form [1, 23]
(4) 
where is the complete Gamma function
(5) 
When and , we can define the auxiliary random variables [28]
(6) 
such that . Then, under the following constraints [40, 41, 42, 28],
(7) 
the random variable follows a standard Beta distribution [1, 23], with hyperparameters and satisfying
(8) 
where is the mean value, is the standard deviation, and is the variance of . The corresponding probability density function takes the form
(9) 
where is the Beta function
(10) 
Then, for the random coefficients and , the corresponding mean values are
(11) 
and the variances and covariance are, respectively,
(12)  
(13)  
(14) 
We further note that, when , assuming that the standard deviation, , is constant, by (3), . Next, defining , the probability density function (4) takes the form
Then, the limit of the above function as is equal to
Hence, the Gamma probability density function (4) is approximated by a normal (Gaussian) density function
(15) 
where is a random normal variable with mean value and standard deviation .
When , the probability distribution (4) reduces to an exponential distribution,
(16) 
In this case, the mean value, and standard deviation, , defined by (3), take comparable values. This situation may arise, for example, when the sampled data contain a lot of noise.
2.1 Rubberlike material
For rubberlike material, the first experimental data in large deformations were reported by Rivlin and Saunders (1951) [34]. In light of these data, and assuming that, under sufficiently small deformations, the rubberlike material can be described by the stochastic hyperelastic model (1), we derive the probability distribution for the random shear modulus, , for this material (see figure 1). We note that, here, we make this assumption in order to obtain some probability distributions based on real data measurements, without attempting to calibrate a specific hyperelastic model to the given data. For hyperelastic models calibrated to experimental data for rubberlike material, under large deformations, see, for example, [44, 9, 27] and the references therein.
The chosen data values are recorded in table 3. The Gamma probability distribution fitted to the shear modulus data, together with the normal distribution derived from the Gamma distribution, and also the standard normal distribution fitted to the data, are represented in figure 1. Note the similarity between the Gamma and normal distributions in this case. For each probability distribution, the mean value, , and standard deviation, , are recorded in table 3. For the normal distribution derived from the Gamma distribution, the mean value and standard deviation are given by (3), as for the Gamma distribution. For the auxiliary random variable , the mean value, , and standard deviation, , are provided in table 3.
1.90  1.80  1.70  1.90  1.80  

1.07  1.25  1.39  1.02  1.09  
(kg/cm)  1.77  1.89  2.01  1.68  1.76 
(kg/cm)  0.20  0.23  0.21  0.29  0.21 
(kg/cm)  3.94  4.24  4.44  3.94  3.94 
Probability density function (pdf)  

Gamma pdf fitted to data  
Normal pdf derived from Gamma pdf      
Normal pdf fitted to data     
Probability density function (pdf)  
Beta pdf fitted to data 
Next, we analyse the radially symmetric inflation of a spherical shell and of a cylindrical tube of stochastic hyperelastic material defined by (1), with the random shear modulus, , following either the Gamma or the normal probability distribution fitted to the given data. We can regard the stochastic spherical shell (or tube) as an ensemble of shells (tubes) with the same geometry, such that each shell (tube) is made from a single homogeneous isotropic incompressible hyperelastic material, with the elastic parameters distributed according to the given probability density function. Then, for each individual shell (tube), the finite elasticity theory applies. The question is: what is the probability distribution of stable radially symmetric inflation, such that the internal pressure always increases as the radial stretch increases?
3 Stochastic incompressible spherical shell
First, we consider a spherical shell of stochastic hyperelastic material described by (1), subject to the following radially symmetric deformation (figure 3),
(17) 
where and are the spherical polar coordinates in the reference and the current configuration, respectively, such that , and is to be determined.
The deformation gradient is , with
(18) 
where , , represent the radial, tangential, and azimuthal stretch ratios, respectively, and denotes differentiation of with respect to .
The radial equation of equilibrium is
(19) 
or equivalently,
(20) 
where is the first PiolaKirchhoff stress tensor. For an incompressible material,
Denoting
(21) 
where , we obtain
(22) 
Next, setting the external pressure (at ) equal to zero, by (20) and (22), the internal pressure (at ) is equal to
(23) 
where and are the stretch ratios for the inner and outer radii, respectively.
Due to material incompressibility, the material volume in the spherical shell is conserved. Hence, , and
(24) 
i.e. the internal pressure is a function of the inner stretch ratio .
As for the purely elastic shell, for the stochastic spherical shell, a limitpoint instability occurs if there is a change in the monotonicity of , defined by (23), as a function of . When the spherical shell is thin, i.e. , we can approximate the internal pressure as follows [14, p. 443],
(25) 
and find the critical value of where a limitpoint instability occurs by solving for .
3.1 Purely elastic shell
In the purely elastic case, for a spherical shell of hyperelastic material defined by the strainenergy function
(26) 
where and are positive constants, and is the shear modulus, the function (21) is equal to
(27) 
Then, the internal pressure given by (25) takes the form
(28) 
and the equation is equivalent to
(29) 
Specifically, if , then, when , the internal pressure increases to a maximum value then decreases, otherwise the internal pressure increases monotonically. If , then (29) is equivalent to
(30) 
where .
In particular, for a spherical shell of MooneyRivlin material, with and , the minimum value of , such that inflation instability occurs, is the minimum value of the following function,
(31) 
i.e. . In this case:

When , the internal pressure increases to a maximum, then decreases to a minimum, then increases again;

When , the internal pressure is always increasing.
3.2 Stochastic elastic shell
For a spherical shell of stochastic MooneyRivlin material, characterised by the strainenergy function (1) with and , the probability distribution of stable inflation, such that internal pressure monotonically increases as the radial stretch increases, is
(32) 
where if the random shear modulus, , follows the Gamma distribution, with and , and if , follows the normal distribution, with and (see table 3).
The probability distribution of inflation instability occurring, such that the internal pressure begins to decrease, is
(33) 
The distributions given by equations (32)(33) are approximated numerically in figure 5 (blue lines for and red lines for ). Specifically, was divided into steps, then for each value of , random values of were numerically generated from a specified Gamma (or normal) distribution and compared with the inequalities defining the two intervals for values of . For the purely elastic shell, which is based on the mean value of the shear modulus, , the critical value of strictly separates the cases where inflation instability can occur or not. For the stochastic problem, for the same critical value, there is, by definition, exactly chance of a randomly chosen shell for which inflation is stable, (blue solid or dashed line if the shear modulus is Gamma or normal distributed, respectively) and chance of a randomly chosen shell, such that a limitpoint instability occurs (red solid or dashed line). To increase the probability of stable inflation (), one must consider sufficiently small values of , below the expected critical value, whereas a limitpoint instability is certain to occur () only if the model reduces to the neoHookean case. However, the inherent variability in the probabilistic system means that there will also exist events where there is competition between the two cases.
To illustrate this, in figure 5, we show the probability distribution of the normalised internal pressure as a function of the inner stretch , when follows a Gamma distribution with and , and follows a Beta distribution with and (see table 3). In this case, , and instability is expected to occur. Nevertheless, the probability distribution suggests that there is also around 10% chance that the inflation is stable.
4 Stochastic incompressible cylindrical tube
Next, a cylindrical tube of stochastic hyperelastic material given by (1) is deformed through the combined effects of radially symmetric inflation and axial extension [33], as follows (figure 7),
(34) 
where and are the cylindrical polar coordinates in the reference and the current configuration, respectively, such that , is a given constant (when , the tube is inverted, so that the inner surface becomes the outer surface), and is to be determined.
The deformation gradient is , with
(35) 
where , , are the radial, tangential, and longitudinal stretch ratios, respectively.
For the cylindrical tube, the radial equation of equilibrium is given by (19), and is equivalent to
(36) 
Denoting
(37) 
where and , we obtain
(38) 
Setting the external pressure (at ) equal to zero, by (36) and (38), the internal pressure (at ) is equal to
(39) 
where and are the stretches for the inner and outer radii, respectively. As the material volume of the tube is conserved, , we have
(40) 
and the internal pressure is a function of the inner stretch ratio .
For the cylindrical tube, a limitpoint instability occurs if there is a change in the monotonicity of , given by (39), as a function of . Assuming that the tube is thin, i.e. , we approximate the internal pressure as [32, p. 290]
(41) 
and find the point of instability by solving for .
4.1 Purely elastic tube
In the elastic case, for a cylindrical tube of hyperelastic material defined by the strainenergy function (26), the corresponding function (37) takes the form
(42) 
Then, the internal pressure given by (41) is equal to
(43) 
and the equation becomes
(44) 
In this case, if , then, when , the internal pressure increases to a maximum value then decreases, otherwise the internal pressure is always increasing. If , then (44) is equivalent to
(45) 
where .
Next, we specialise to the case where , , and , for which (45) takes the form
(46) 
Then, the minimum value of , such that inflation instability occurs, is the minimum of the function
i.e. .
4.2 Stochastic elastic tube
For a cylindrical tube of stochastic hyperelastic material described by the strainenergy function (1) with and , the probability distribution of stable inflation, such that the internal pressure always increases as the radial stretch increases, is
(47) 
where if the random shear modulus, , follows the Gamma distribution, with and , or if , follows the normal distribution, with and (see table 3).
The probability distribution of inflation instability occurring is
(48) 
The numerical approximations for each distribution given by (47)(48) are shown in figure 9 (blue lines for and red lines for ). For the purely elastic tube, the critical value, , strictly divides the cases of inflation instability occurring or not. However, in the stochastic case, to increase the chance that inflation is always stable (), one must take sufficiently small values of the parameter , below the expected critical point, while instability is guaranteed () only for the stochastic neoHookean tube.
As an example, in figure 9, we show the probability distribution of the normalised internal pressure as a function of the inner stretch , when follows a Gamma distribution with and , and follows a Beta distribution with and (see table 3). Hence, , and instability is expected to occur. However, the probability distribution suggests there is also around 5% chance that the inflation is stable.
5 Conclusion
For hyperelastic spherical shells and cylindrical tubes under symmetric finite inflation, we showed that, when material parameters are random variables, there is always competition between monotonic expansion and limitpoint instability. Specifically, by contrast to the purely elastic problem, where there is a deterministic critical value that strictly separate the cases where either the radially symmetric inflation is stable or a limitpoint instability occurs, for the stochastic problem, there are probabilistic intervals for the model parameters, where there is a quantifiable chance for both the stable and unstable states to be found.
For numerical illustration, we considered experimental data for rubberlike material, and derived the probability distribution of the corresponding random shear modulus to predict the inflation behaviour of internally pressurised hollow cylinders and spheres made of a material characterised by this parameter. The general framework provided by our stochastic elastic setting is applicable to a class of stochastic hyperelastic materials for which similar results can be obtained.
This study addresses the need for a deeper understanding of the influence and sources of randomness in natural and industrial materials where mathematical models that take into account the variability in the observed mechanical responses are crucial.
Data access.
The datasets supporting this article have been included in the main text.
Authors contribution.
All authors contributed equally to all aspects of this article and gave final approval for publication.
Competing interests.
The authors declare that they have no competing interests.
Funding.
The support for Alain Goriely by the Engineering and Physical Sciences Research Council of Great Britain under research grant EP/R020205/1 is gratefully acknowledged.
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