Likely oscillatory motions of stochastic hyperelastic solids
Stochastic homogeneous hyperelastic solids are characterised by strain-energy functions where the parameters are random variables satisfying probability density functions. These models can propagate uncertainties from input data to output quantities of interest. To investigate the effect of probabilistic parameters on predicted mechanical responses, we study radial oscillations of cylindrical and spherical shells of stochastic incompressible isotropic hyperelastic material, formulated as quasi-equilibrated motions where the system is in equilibrium at every time instant. Additionally, we study finite shear oscillations of a cuboid, which are not quasi-equilibrated. We find that, for hyperelastic bodies of stochastic neo-Hookean or Mooney-Rivlin material, the amplitude and period of the oscillations follow probability distributions. Further, for cylindrical tubes and spherical shells, when an impulse surface traction is applied, there is a parameter interval where the oscillatory and non-oscillatory motions compete, in the sense that both have a chance to occur with a given probability. We refer to the dynamic evolution of these elastic systems, which exhibit inherent uncertainties due to the material properties, as “likely oscillatory motions”.
Key words: stochastic hyperelastic models, finite dynamic deformations, quasi-equilibrated motion, finite amplitude oscillations, incompressibility, applied probability.
“Denominetur motus talis, qualis omni momento temporis praebet configurationem capacem aequilibrii corporis iisdem viribus massalibus sollicitati, ‘motus quasi aequilibratus’. Generatim motus quasi aequilibratus non congruet legibus dynamicis et proinde motus verus corporis fieri non potest, manentibus iisdem viribus masalibus.” - C. Truesdell (1962) 
Motivated by numerous long-standing and modern engineering problems, oscillatory motions of cylindrical and spherical shells made of linear elastic material [55, 56, 53, 76] have generated a wide range of experimental, theoretical, and computational studies [5, 6, 7, 17, 27]. In contrast, time-dependent finite oscillations of cylindrical tubes and spherical shells of nonlinear hyperelastic material, relevant to the modelling of physical responses in many biological and synthetic systems [3, 9, 26, 38, 39, 40, 54], have been less investigated, and much of the work in finite nonlinear elasticity has focused on the static stability of pressurised shells [2, 16, 20, 21, 23, 29, 32, 34, 36, 57, 68, 79, 88, 109], or on wave-type solutions in infinite media [44, 75].
The governing equations for large amplitude oscillations of cylindrical tubes and spherical shells of homogeneous isotropic incompressible nonlinear hyperelastic material, formulated as special cases of quasi-equilibrated motions , were reviewed in . These are the class of motions for which the deformation field is circulation preserving, and at every time instant, the current configuration is a possible static configuration under the given forces. The free and forced axially symmetric radial oscillations of infinitely long, isotropic incompressible circular cylindrical tubes, with arbitrary wall thickness, were described for the first time in [50, 51]. In [41, 52, 105], free and forced oscillations of spherical shells were derived analogously. For the combined radial-axial large amplitude oscillations of hyperelastic cylindrical tubes, in , the surface tractions necessary to maintain the periodic motions were discussed, and the results were applied to a tube sealed at both ends and filled with an incompressible fluid. The dynamic deformation of cylindrical tubes of Mooney-Rivlin material in finite amplitude radial oscillation was obtained in [86, 84, 85]. For a hyperelastic spherical cavity of Mooney-Rivlin material, the solution to the nonlinear problem of large amplitude oscillations was computed numerically in . Theoretical and experimental studies of cylindrical and spherical shells of rubberlike material under external pressure were presented in . In , the finite amplitude radial oscillations of homogeneous isotropic incompressible hyperelastic spherical and cylindrical tubes under a constant pressure difference between the inner and the outer surface were analysed theoretically. The finite longitudinal, or ‘telescopic’, oscillations of infinitely long cylindrical tubes were studied in . In , the oscillatory motions of cylindrical and prismatic bodies of incompressible hyperelastic material under dynamic finite shear deformation were investigated. Other dynamic shear deformations were treated in , where it was emphasised that such shear motions were not quasi-equilibrated. In , the dynamic problem of axially symmetric oscillations of cylindrical tubes of transversely isotropic incompressible material, with radial transverse isotropy, was considered. The dynamic deformation of a longitudinally anisotropic thin-walled cylindrical tube under radial oscillations was obtained in . In , radial oscillations of non-homogeneous thick-walled cylindrical and spherical shells of neo-Hookean material, with a material constant varying continuously along the radial direction, were treated. In , asymmetric oscillations of homogeneous isotropic compressible hyperelastic cylindrical shells of arbitrary wall thickness under uniform radial dead-load traction were analysed using the theory of small deformations superposed on large elastic deformations, and the governing equations were solved numerically. In , the dynamic inflation of hyperelastic spherical membranes of Mooney-Rivlin material subjected to a uniform step pressure was studied, and the absence of damping in these models was discussed. It was concluded that, as the amplitude and period of oscillations are strongly influenced by the rate of internal pressure, if the pressure was suddenly imposed and the inflation process was short, then sustained oscillations due to the dominant elastic effects could be observed. However, for many systems under slowly increasing pressure, strong damping would generally preclude oscillations .
More recently, the dynamic response of incompressible hyperelastic cylindrical and spherical shells subjected to periodic loading was discussed in [77, 78]. Radial oscillations of cylindrical tubes and spherical shells of neo-Hookean , Mooney-Rivlin [67, 80], and Gent  hyperelastic materials were analysed in [14, 15], where it was concluded that, in general, both the amplitude and period of oscillations decrease when the stiffness of the material increases. The influence of material constitutive law on the dynamic behaviour of cylindrical and spherical shells was also examined in [8, 10, 82, 108], where the results for Yeoh  and Mooney-Rivlin material models were compared. In , the static and dynamic behaviour of circular cylindrical shells of homogeneous isotropic incompressible hyperelastic material modelling arterial walls were considered.
For the assessment and prediction of the mechanical responses of engineered and natural materials, additional challenges arise from the uncertainties in their elastic properties inferred from sparse and approximate observational data [31, 42, 49, 72, 74, 99]. For these materials, deterministic approaches, which are based on average data values, can greatly underestimate or overestimate their properties, and stochastic representations accounting also for data dispersion are needed to significantly improve assessment and predictions. In response to this challenge, stochastic elasticity is a fast developing field that combines nonlinear elasticity and stochastic theories in order to significantly improve model predictions by accounting for uncertainties in the mechanical responses of materials. Within this framework, stochastic hyperelastic materials are advanced phenomenological models described by a strain-energy density where the parameters are characterised by probability density functions, as constructed in [94, 95, 96, 97, 98] and . These models rely on the notion of entropy (or uncertainty) [87, 89] and on the maximum entropy principle for a discrete probability distribution [45, 46, 47], and are able to propagate uncertainties from input data to output quantities of interest . They are also suitable for incorporation into Bayesian methodologies  (see also ) for models selection and updates [66, 72, 81].
To study the effect of probabilistic model parameters on predicted mechanical responses, in [62, 63, 64, 65], for different bodies with simple geometries at finite strain deformations, it was shown explicitly that, in contrast to the deterministic elastic problem where a single critical value strictly separates the stable and unstable cases, for the stochastic problem, there is a probabilistic interval where the stable and unstable states always compete, in the sense that both have a quantifiable chance to be found. In addition, revisiting these problems from a novel perspective offered fresh opportunities for gaining new insights into the fundamental elastic solutions. Specific case studies included the classical problems of the Rivlin cube , the cavitation of a sphere under uniform tensile dead load , the inflation of pressurised spherical and cylindrical shells , and the rotation and perversion of anisotropic hyperelastic cylindrical tubes .
In this paper, we extend the stochastic framework developed in [63, 62, 64, 65] to the study of radial oscillations of cylindrical and spherical shells of stochastic incompressible isotropic hyperelastic material formulated as quasi-equilibrated motions, where the system is in equilibrium at every time instant, and also of finite shear oscillations of a cuboid, which are not quasi-equilibrated. We find that, for hyperelastic bodies of stochastic neo-Hookean or Mooney-Rivlin material, the amplitude and period of the oscillations follow probability distributions. Further, for cylindrical tubes and spherical shells, when an impulse surface traction is applied, there is a parameter interval where the oscillatory and non-oscillatory motions compete in the sense that both have a chance to occur with a given probability. We refer to the dynamic evolution of these elastic systems, which exhibit inherent uncertainties due to the material properties, as “likely oscillatory motions”.
In Section 2, we provide a summary of the stochastic elasticity prerequisites. Section 3 is devoted to the analysis of the oscillatory motion of a stochastic hyperelastic cuboid under dynamic generalised shear. This is followed in Sections 4 and 5 by the treatment of the radial oscillatory motions of stochastic cylindrical tubes and spherical shells with bounded wall thickness respectively. The limiting cases of thin- and infinitely thick-walled structures are also discussed. Less straight-forward detailed calculations inherent for these problems are deferred to the Appendix A. Concluding remarks are drawn in Section 6.
In this Section, we first recall the notion of (universal) quasi-equilibrated motion in finite elasticity, introduced in  and reviewed in , then summarise the stochastic finite elasticity framework developed in  and applied to static stability problems in [63, 62, 64, 65].
2.1 Quasi-equilibrated motion
For the large-strain time-dependent behaviour of an elastic solid, Cauchy’s laws of motion (balance laws of linear and angular momentum) are governed by the following Eulerian field equations [102, p. 40],
where is the motion of the given elastic solid, is the material density, which is assumed constant, is the body force, is the Cauchy stress tensor, and the superscript defines the transpose. To obtain possible dynamical solutions, one can solve Cauchy’s equation for particular motion or generalise known static solutions to the dynamical solution, using the notion of quasi-equilibrated motion:
[102, p. 208] A quasi-equilibrated motion is a motion, , of an incompressible homogeneous elastic solid subject to a given body force, , such that, for each value of , defines a static deformation that satisfies the equilibrium conditions under the body force .
[102, p. 208] A quasi-equilibrated motion, , of an incompressible homogeneous elastic solid subject to a given body force, , is dynamically possible, subject to the same body force, if and only if the motion is circulation preserving with a single-valued acceleration potential , i.e.,
For the condition (3) to be satisfied, it is necessary that
Then, the Cauchy stress tensor takes the form
where is the Cauchy stress for the equilibrium state at time and is the identity tensor. In this case, the stress field is determined by the present configuration alone. In particular, the shear stresses in the motion are the same as those of the equilibrium state at time .
2.2 Stochastic isotropic incompressible hyperelastic models
A stochastic homogeneous hyperelastic model is defined by a stochastic strain-energy function, for which the model parameters are random variables that satisfy standard probability laws [66, 94, 95, 96]. In this case, each model parameter is described in terms of its mean value and its variance, which contains information about the range of values about the mean value. As obtaining complete information about a random quantity in an elastic sample of material is rarely possible, the partial information provided by the mean value and the variance is commonly used in practice [19, 24, 42, 59, 69]. Here, we combine finite elasticity [33, 73, 102] and probability theory [37, 47], and rely on the following general hypotheses [66, 65, 63, 62]:
Material objectivity, stating that constitutive equations must be invariant under changes of frame of reference. This requires that the scalar strain-energy function, , depending only on the deformation gradient F, with respect to the reference configuration, is unaffected by a superimposed rigid-body transformation (which involves a change of position) after deformation, i.e., , where is a proper orthogonal tensor (rotation). Material objectivity is guaranteed by defining strain-energy functions in terms of invariants.
Material isotropy, requiring that the strain-energy function is unaffected by a superimposed rigid-body transformation prior to deformation, i.e., , where . For isotropic materials, the strain-energy function is a symmetric function of the principal stretches of F, i.e., .
Baker-Ericksen (BE) inequalities, which state that the greater principal (Cauchy) stress occurs in the direction of the greater principal stretch, are 
where and denote the principal stretches and the principal Cauchy stresses, respectively. The BE inequalities (6) take the equivalent form
Assumptions (A1)-(A3) are well-known principles in isotropic finite elasticity [33, 73, 102]. In particular, regarding the assumption (A3), we recall that, for a homogeneous hyperelastic body under uniaxial tension, the deformation is a simple extension in the direction of the tensile force if and only if the BE inequalities hold . Another important deformation is that of simple shear superposed on axial stretch, defined by
where and are the Cartesian coordinates for the reference (Lagrangian) and the current (Eulerian) configuration, respectively, and and are positive constants representing the shear parameter and the axial stretch ( for axial tension and for axial compression) respectively. For this deformation, the principal stretches, , satisfy
and, assuming that the material is incompressible, the associated principal Cauchy stresses take the form
is positive, i.e., , for all and . In the linear elastic limit, where and , the nonlinear shear modulus given by (11) converges to the classical shear modulus under infinitesimal deformation, i.e.,
Assumption (A4) then contains physically realistic expectations on the (positive) random shear modulus , which will be characterised by a suitable probability density function.
In the next Sections, we analyse the dynamic generalised shear deformation of a cuboid and the radially symmetric motion of cylindrical tube and spherical shells of stochastic isotropic incompressible hyperelastic material. One can regard a stochastic hyperelastic body as an ensemble of bodies with the same geometry where each individual body is made from a homogeneous isotropic incompressible hyperelastic material, with the elastic parameters drawn from known probability distributions. Then, for the individual hyperelastic bodies, the finite elasticity theory applies.
where and are random variables. The non-deterministic model (13) reduces to a stochastic neo-Hookean model if . If the random parameters and are replaced by their respective mean values, and , then the resulting mean hyperelastic model coincides with the usual deterministic one.
For the stochastic material model described by (13), the shear modulus in infinitesimal deformation is defined as . For this modulus, we set the following mathematical constraints, which are consistent with the assumption (A4) made in Section 2 ,
i.e., the mean value of the shear modulus is fixed and greater than zero, and the mean value of is fixed and finite. It follows that and are second-order random variables, i.e., they have finite mean and finite variance [90, 91]. Under the constraints (14), follows a Gamma probability distribution with hyperparameters and satisfying
where is the complete Gamma function
When and , we can define the auxiliary random variable 
such that . Then
where is the mean value, is the variance, and is the standard deviation of . The corresponding probability density function takes the form
where is the Beta function
Thus, for the random coefficients given by (19), the corresponding mean values take the form,
and the variances and covariance are, respectively,
Note that the random variables and are independent, depending on parameters and , respectively, whereas and are dependent variables as they both require to be defined. Explicit derivations of stochastic isotropic hyperelastic models calibrated to experimental data are presented in .
3 Generalised shear motion of a stochastic hyperelastic cuboid
The generalised shear motion of a cuboid of elastic material is described by 
where and are the Cartesian coordinates for the reference (Lagrangian, material) and current (Eulerian, spatial) configuration respectively, is a given constant, and , representing the displacement in the third direction, is a time-dependent function to be determined. Here, we assume that the edges of the cuboid are aligned with the directions of the Cartesian axes in the undeformed state.
This condition imposes very strict constraints on the motion. Yet, we will see that even though the generalised shear motion (28) is not quasi-equilibrated, exact solutions are still available, although these solutions are not universal [70, 104].
For the deformation (28), the gradient tensor is equal to
where and denote the partial first derivatives of with respect to and respectively. The corresponding left Cauchy-Green tensor is
and has the principal invariants
The associated Cauchy stress tensor takes the form [35, pp. 87-91]
where is the Lagrange multiplier for the incompressibility constraint (), and
are the nonlinear material parameters, with , given by (31).
3.1 Shear oscillations of a cuboid of stochastic neo-Hookean material
Then, by the equation of motion (1),
where and represent the second derivatives of with respect to and respectively. Hence, is independent of and .
We consider the undeformed cuboid to be long in the -direction, and impose an initial displacement and velocity . For the boundary condition, we distinguish the following two cases:
If we impose null normal Cauchy stresses, , on the faces perpendicular to the - and -directions, at all time, then is constant and .
If , as cannot be made point-wise zero, we denote the normal force acting on the cross-sections of area in the -direction at time by
and consider this force to be zero, i.e., at all time. Then, is independent of , and, by (35), it is also independent of and , hence, .
In both the above cases, (i) and (ii) respectively, by (35),
It remains to solve, by standard procedures, the linear wave equation (37), describing the propagation of waves, subject to the given initial and boundary conditions. To solve this equation, we let the shear stresses and , defined by (34), vanish at the sides, i.e.,
In this case, the general solution takes the form
These oscillations under the generalised shear motion (28) cannot be completely ‘free’, due to the non-zero tractions corresponding to the cases (i) and (ii) respectively. Note that the condition (29) is not satisfied.
As is a random variable, it follows that the speed of wave propagation, , is stochastic. Hence, both the period and the amplitude of the oscillations are stochastic. As an example, we consider the initial data and leading to and . In Figure 3, we illustrate the stochastic dynamic displacement on the edges when , , , , and is drawn from the Gamma distribution with hyperparameters and , as represented in Figure 1. We note from Figure 3 that, as we might expect, extremal probabilities always occur at the extreme displacement of the oscillations, i.e., when the cuboid is slowest. However, in between these probability maxima the variance grows over time. Thus, although the displacements are initially close (seen explicitly in the top of Figure 3 and by the tight distribution around the mean in the bottom of Figure 3), eventually, the phase difference dominates causing the displacements to diverge (top of Figure 3) and an increase in the variance of the distribution (bottom of Figure 3).
4 Quasi-equilibrated radial-axial motion of a stochastic hyperelastic cylindrical tube
For a circular cylindrical tube, the combined radial and axial motion is described by
where and are the cylindrical polar coordinates in the reference and current configuration, respectively, such that , and are the inner and outer radii in the undeformed state respectively, and are the inner and outer radius at time respectively, and is a given constant (when , the tube is everted, so that the inner surface becomes the outer surface). When , the time-dependent deformation (43) simplifies to that studied also in [14, 50, 51]. The case when is time-dependent was considered in .
The radial-axial motion (43) of the cylindrical tube is fully determined by the inner radius at time , which in turn is obtained from the initial conditions. Thus, the acceleration can be computed in terms of the acceleration on the inner surface. By the governing equations (43), the condition (4) is valid for , since
and the acceleration potential, , satisfies (3). Hence, this is a quasi-equilibrated motion, such that
The deformation gradient of (43), with respect to the polar coordinates , is equal to
the Cauchy-Green deformation tensor is
and the principal invariants take the form
Thus, the principal components of the equilibrium Cauchy stress tensor at time are
where is the Lagrangian multiplier for the incompressibility constraint (), and
are the nonlinear material parameters, with and given by (49).
As the stress components depend only on the radius , the system of equilibrium equations reduces to
corresponding to the combined deformation of simple shear superposed on axial stretch, described by (8), with shear parameter and stretch parameter . As shown in , this modulus is positive if the BE inequalities hold. In this case, the integrand is negative for and positive for . Using the first equation in (43), it is straightforward to show that (respectively, ) is equivalent to (respectively, ). When , the modulus defined by (55) coincides with the generalised shear modulus defined in [102, p. 174], and also in .
In this case, as , the three stress components defined by (54) are equal.
Next, for the cylindrical tube deforming by (43), we set the inner and outer radial pressures acting on the curvilinear surfaces and at time (measured per unit area in the present configuration), as and respectively [102, pp. 214-217]. Evaluating and , using (54), with and respectively, then subtracting the results, then gives
Setting the notation
we can rewrite
Then, we can express the equation (57) equivalently as follows,
For the cylindrical tube in finite dynamic deformation, we set
and find that is monotonically decreasing when and increasing when . This function will be useful in establishing whether the radial motion is oscillatory or not.
We also set the pressure impulse (suddenly applied pressure difference)
where is constant in time. Then, integrating (59) once gives
with defined by (62) and
where and are the initial conditions. By (64),
Physically, this system is analogous to the motion of a point mass with energy
The energy is , the potential is given by and the position-dependent mass is . Due to the constraints on the function , this system has simple dynamics. Depending on the constant , the system may have a static state or periodic motion. Indeed, the radial motion is periodic if and only if the following equation,
has exactly two distinct solutions, representing the amplitudes of the oscillation, and , such that . Then, by (58), the minimum and maximum radii of the inner surface in the oscillation are equal to and respectively, and by (66), the period of oscillation is equal to
Note that both the amplitudes and period of the oscillation are random variables described in terms of probability distributions.
4.1 Radial oscillations of a cylindrical tube of stochastic Mooney-Rivlin material
where . In this case, assuming that the nonlinear shear modulus has a uniform lower bound, i.e.,
for some constant , it follows that