An alternative form of energy density demonstrating the severe strain-stiffening in thin spherical and cylindrical shells

Md. Moonim Lteefi , Deepk Kumr , , Somnth Srngi

a Department of Mechanical Engineering, Indian Institute of Technology Patna, Bihar, 801103, India

b Department of Mechanical Engineering, Maulana Azad National Institute of Technology Bhopal, Madhya Pradesh, 462003, India

Keywords: Hyperelastic material Strain-stiffening Constitutive modeling Strain energy function

ABSTRACT The present article investigates an elastic instability phenomenon for internally pressurized spherical thin balloons and thin cylindrical tubes composed of incompressible hyperelastic material. A mathematical model is formulated by proposing a new strain energy density function. In the family of limited elas- tic materials, many material models exhibit strain-stiffening. However, they fail to predict severe strain- stiffening in a moderate range of deformations in the stress-strain relations. The proposed energy func- tion contains three material parameters and shows substantially improved stain stiffening properties than the limited elastic material models. The model is further applied to explore the elastic instability phe- nomenon in spherical and cylindrical shells. The findings are compared with other existing models and validated with experimental results. The model shows better agreement with experimental results and exhibits a substantial strain-stiffening effect than the current models.

In the current scenario, rubbers and rubber-like materials have attracted many researchers for modern soft engineering and med- ical field applications [1–3] . Since the 1940s, enormous progress has been achieved in developing hyperelastic material modeling to characterize the stress-strain response at large deformations. The significant results have been obtained in incompressible hypere- lastic material modeling and have also been experimentally con- firmed [4,5] . This remarkable success projected a considerable light on the physical behavior of rubber-like materials. However, the theory of elasticity for hyperelastic materials subjected to large deformations is highly nonlinear. Therefore, so many mathemati- cal difficulties are still encountered at the current time. In gen- eral, rubber-like materials are assumed to be an incompressible isotropic hyperelastic materials. A preliminary step toward com- plete modeling is analyzing their elastic properties and nonlin- ear stress-strain characteristics [6,7] . In line with that, the uni- axial and biaxial loading tests on such materials reveal a non- linear stress-strain response with higher extensibility in the low- stress range and progressively lower extensibility at large strain. This phenomenon is well-known as ”strain-hardening”or ”strain- stiffening”.

In short, hyperelasticity refers to a stress-deformation response that is derivable from an elastic-free energy potential. It is typically used for materials that experience large elastic deformation. Ap- plications for elastomers such as vulcanized rubber and synthetic polymers, along with some biological materials, often fall into this category. To characterize such materials’ behavior, it is a common practice that a suitable constitutive relation must be developed us- ing an appropriate strain energy function. The strain energy func- tion must be expressed to describe the high deformability, recov- erability after deformation, and nonlinear load-deformation behav- ior for a particular material class. Most of the existing material models are phenomenologically based [8,9] . These phenomenologi- cal models correlate with the experimental data to determine their material parameters. Based on the “strain-stiffening”phenomenon, all the material models in nonlinear elasticity were divided into two leading families: “power law”and “l(fā)imiting chain extensibil- ity”models. Inthiscontext, Knowles[10] proposeda prototype of the strain energy for incompressible isotropic materials belonging to the first family (power-law models). The corresponding energy density expression is given as follows

whereμrepresents the shear modulus andb,nare the dimension- less positive material constants. In addition,I1= tr B = tr C denotes the first principal invariant of either left B or right C Cauchy-Green deformation tensor. Physically, the first principal invariantI1signi- fies the net resultant of the strain tensor. The model Eq. (1) also physically represents the hardening phenomenon in simple shear forn>1 . On lettingn→ ∞ in Eq. (1) , this recovers another strain energy density form given as follows

which was introduced by Fung (1967) specifically used in biolog- ical materials. Next, Boyce [11] described the same belonging to the second family (models with strain-stiffening). Both families of these material models have their own molecular basis.

Some of the few materials similar to rubbers were consid- ered in the studies [12–14] . The studies observed a rapid strain- stiffening phenomenon in biological materials, even at moderate strains. The classical models, such as the Mooney-Rivlin [15] , Gent [16] and HorganSaccomandi[17] models, failtopredict thisstrain- stiffening at moderate strain in these biological materials. In this context, researchers seek a new material model that can pre- dict such strain-stiffening phenomenon in varieties of materials at moderate as well as large strains. One of the simplest strain-energy on chain extensibility and strain strain-stiffening effects at large strain only was proposed by Gent [16] , which is given as follows

whereμrepresents the shear modulus andJmdenotes the limit- ing value forI1?3 . On lettingJm→ ∞ in the above Gent model Eq. (3) , we may recover the Neo-Hookean model given as follows

Horgan and Saccomandi [18] proposed an alternative limiting chain extensibility model, which can predict the chain extensibility and strain-stiffening effects at large strains only. This model [18] in- cludes a dependence on the second invariant as well, given as fol- lows

whereμrepresents shear modulus for infinitesimal deformations. In addition, the parameterJdenotes an alternate limiting chain extensibility parameter defined in the Horgan-Saccomandi model [17,19–21] analogous toJmin the Gent model. Mathematically,Jfollows the constraint depending on both the firstI1 as well as the secondI2principal invariants of Cauchy-Green deformation tensor as. On the other hand,Jmfollows the constraint depending on the first invariantI1 only given asJm>I1?3 . Addi- tionally, the parameterJhas the constraint based on the square value of the maximum principal stretch referred to a locking stretch allowed by the finite extensibility of the polymeric chain network given by max

The present article proposes a newly amended form of the free energy density function for an incompressible isotropic hyperelas- tic material. The motivating key concerns of such materials that undergo severe strain-stiffening at moderate and large strains. We also examine the instabilities of internally pressurized spherical and cylindrical shells considered in [23–25] for special classes of constitutive models that give rise to severe strain-stiffening in their stress response curves at a moderate range of strain.

We summarize the theory of hyperelasticity [23,26] and pro- pose a new amended energy density function to study the elastic deformation behavior of an incompressible isotropic hyperelastic material in an isothermal loading condition.

Consider a point X in the reference configurationβ0, which is displaced during deformation in a new position vector x in the cur- rent configurationβ. Then, the deformation gradient tensor F may be defined as F =. For a hyperelastic material, the stress tensor is obtained from the principle of virtual work using the free en- ergy density potential functionWthat can be expressed by using the principal invariants of deformation gradient tensor F as follows

By applying the mechanical energy principle in an isothermal de- formation condition, the total mechanical energy rate for any part of a body is balanced by the real mechanical power. Following the second law of thermodynamics-based classical continuum me- chanics theory [23,26–29] , we may arrive at the constitutive equa- tion for a hyperelastic material as follows

where in T represents the Cauchy stress tensor and B , C denote the left and right Cauchy-Green deformation tensors, respectively. For an isotropic finite deformation, we may consider one of the Cauchy-Green deformation tensors B or C as an independent vari- able. However, their principal invariants remain the same, but prin- cipal directions may change for any given deformation. Now, the strain energy density functionWmay be represented in the in- variant form of any one of the deformation gradient tensors as follows

whereinI1,I2,I3are the first, second, and third invariants of any one of these deformation gradient tensors B or C . In addition, these invariants can also be expressed in terms of the corresponding principal stretchesλ1,λ2andλ3as follows

Following the theory of elasticity [23,26] , the Cauchy stress tensor T foranincompressibleisotropichyperelasticmaterialisgivenas follows

wherepis the Lagrange multiplier associated with the incompress- ibility constraint. However, it is sometimes convenient to use the nominal or engineering stress S based on the reference configura- tion of the body given by

From the phenomenological point of view, the non-Gaussian distribution of the polymeric network of concern can be divided into two categories: (i) free energy density models with chain ex- tensibility parameter and (ii) model with strain-hardening param- eter. Our emphasis here is to generalize both the limiting chain ex- tensibility and strain-hardening concepts in a single physics-based phenomenological model. In the same context, a newly amended form of energy density function that is derived based on the gov- erning postulates [30] along with the polyconvexity [31] of the en- ergy function is proposed as

whereμis the shear modulus andI1 ,I2 are the first and second invariants of left Cauchy-Green deformation tensor B . In addition,Jandnhere denote the limiting chain extensibility and hardening parameters, respectively. One may recover the neoHookean model 4 from the proposed model at its ground staten= 1 andJ→ ∞ Therefore, the above-amended energy function Eq. (12) contains a total of three material parameters. The second invariant depen- dency and hardening parameternin the proposed energy func- tion Eq. (12) may ensure the sensitivity in varieties of material responses reflected in strain-stiffening at moderate as well as large strains. Hence, the proposed energy density model Eq. (12) might yield an overall more accurate reflection of the stress-stretch re- sponses in other varieties of rubber-like materials as well.

The objective of this paragraph is to examine the predictive ca- pability of the proposed energy density Eq. (12) among other ex- isting energy densities by Knowles (Eq. (1)), Gent (Eq. (3)) and Horgan Saccomandi (Eq. (5)) by comparing all with the Treloar [32] experimental data for vulcanized natural rubber. The com- parison is made for both uniaxial tension and equibiaxial ten- sion deformations. In this context, the uniaxial and biaxial stress field expressions are derived for the energy densities Eqs. (1) , (3) and (5) including ours Eq. (12) by using the stress field Eqs. (10) and (11) . The corresponding uniaxial and biaxial stress field equations are given as follows

whereSuniaxialin Eq. (13) represents the uniaxial engineering stress. However,Sbiaxialin Eq. (14) represents the biaxial engineer- ing stress. To compare both uniaxial tension and equibiaxial ten- sion deformations for above energy densities with Treloar [32] ex- perimental data, we plot the stress-stretch responses Eqs. (13) and (14) as shown in Figs. 1 and 2 , respectively. The correspond- ing material parameters are preserved constant (mentioned in the Figs. 1 and 2) for both uniaxial tension and equibiaxial tension de- formations.

As shown in Figs. 1 and 2 , all the energy densities by Knowles (Eq. (1)), Gent (Eq. (3)), Horgan Saccomandi (Eq. (5)) and ours Eq. (12) are able to describe both uniaxial and equibiaxial tensile tests conducted by Treloar [32] for vulcanized natural rubber. The stress-stretch curves in the plots at large stretch rise much more rapidly from the proposed model. The reason behind this is the in- corporation of an additional hardening parameter absent in other models. The importance of this hardening parameter lies in the soft materials undergoing severe strain-stiffening at both moder- ate and large strains. Thus, the proposed model exhibits a better strain-stiffening effect and would be a better choice to model the same.

A classical problem on a thin spherical shell’s inflation under severe strain-stiffening is presented. Here, inflation is defined as the difference between atmospheric and applied pressure. The pro- posed energy density function Eq. (12) is used to analyze the elas- tic instability curves for varying stiffening and chain extensibility parameters.

Consider an elastomeric spherical balloon made of an incom- pressible isotropic rubber-like material with the system coordi- nates(r0,Θ,φ)in the reference configurationβ0and(r,θ,φ)in the current configurationβ. The geometrical description of this spherical balloon with an undeformed radiusr0and thick- nesst0<

Fig. 1. Comparison of the proposed model for uniaxial engineering stress response against existing models along with the experimental data.

Fig. 2. Comparison of proposed model for biaxial engineering stress response against existing models alon with the experimental data.

Fig. 3. A thin membrane rubber-like material-based spherical balloon in the unde- formed and deformed states.

Fig. 4. Comparison of the proposed model against Gent and Horgan-Saccomandi models along with the Beatty experimental data [26] .

whereλθ,λφare the principal stretches in the spherical surface andλris the principal stretch in the thickness direction.

On assuming that the balloon remains spherical on inflation, we have the relation between inflation pressurePand stretchλ=r/r0from the literature [33,34] for an incompressible isotropic hypere- lastic material as

where?denotes the ratio between the undeformed thicknesst0and radiusr0. In the literature, several authors examined the elas- tic instability of thin spherical shells through the classical relation- ship between the inflation pressurePand the stretchλ=r/r0given in Eq. (16) for a different variety of strain-energy densities. In line with that, we also examine and test the same using the same classical relationship Eq. (16) for a new energy density func- tion Eq. (12) . For our simplification, we may define a normalized dimensionless pressureP*given as follows

whereμis the shear modulus and?denotes the ratio between the undeformed thicknesst0and radiusr0. To explore the above classical relation Eq. (16) for the elastic instabilities of thin spher- ical shells, we redefine the same for few standard energy functions, including ours similar to Beatty [26] given as follows

statically. The stretches on the thin-shell spherical balloon may be considered as constant in all directions. Therefore, we have a ho-

For the same problem, we made a comparison in Fig. 4 on the above-redefined relations Eq. (18) and fit the same to match the extreme points of an experimental pressure test values inves- tigated by Beatty [26] . We observe that the inflation curve for the proposed model only reaches its experimental peak value of the balloon as compared to the other models. However, the Gent model and the Horgan-Saccomandi model neither matches the maximum or minimum pressure test values. The slope of the given inflation curves here for both the Gent model and the Horgan- Saccomandi model approaches an almost constant at certain infla- tion pressure. Hence, Gent model and Horgan-Saccomandi models do not predict an indefinite enormous bursting pressure at a fi- nite stretch value. Therefore, these models can not be an appro- priate choice to model an elastic instability phenomenon in strain hardening of the material at considerable strains. Nonetheless, the proposed model also does not provide a qualitative depiction of an overall balloon inflation problem. But, the proposed model predicts an indefinitely large bursting pressure situation at a finite value of stretch. Therefore, it can be a better choice among others to model an elastic deformation of rubber-like materials undergoing severe strain-stiffening phenomena at moderate and large stretch level.

Fig. 5. Normalized pressure versus stretch plot for the proposed model in spherical balloon inflation at different Jand n .

Fig. 6. A thin membrane rubber-like material-based cylindrical tube in the unde- formed and deformed states.

In Fig. 5 , we plot the normalized pressureP*defined in Eq. (18) for the proposed model Eq. (12) asP*(λ,J,n)to empha- size the dependence on the chain extensibility parameterJas well as strain hardening parametern. We here note that the inflation curves rise to a maximum, decrease to a minimum, and then in- crease to infinity. The inflation curves also rise much more rapidly at a moderate stretch as we increase the chain extensibility param- eterJand decrease strain hardening parametern. This shows that the proposed energy function Eq. (12) can be an alternative bet- ter choice among other chain stiffening models like Gent (Eq. (3)) and Horgan-Saccomandi (Eq. (5)) to predict the critical pressure in elastic instability of rubber-like materials undergoing severe strain- stiffening at both moderate and large deformations. This elastic in- stability phenomenon occurs due to the fluid inside that causes a bursting force onto the balloon walls and the critical stresses are induced in the balloon.

A thin cylindrical shell’s inflation under severe strain-stiffening is presented. The inflation is defined here the difference between atmospheric and applied pressure. The proposed energy density function Eq. (12) is reused to analyze the elastic instability curves for varying stiffening and chain extensibility parameters.

Consider an elastomeric cylindrical tube made of an incom- pressible isotropic rubber-like material. The system coordinates in the reference configurationβ0and current configurationβare taken as(r0,Θ,Z)and(r,θ,z), respectively. The cylindrical tube has an undeformed inner radiusr0 , lengthl0and thicknesst0<

whereλr,λθandλzare the principal stretches in the radial, cir- cumferential and axial direction of the cylinder.

In the literature, Gent [35] studied a classical problem on the inflation of a circular cylindrical thin shell unconstrained in length. The problem was discussed for the Gent energy function Eq. (3) . Holzapfel [12] specifically for biomechanics-based applications also studied a similar problem in arteries. We also examine the elas- tic instability of the same cylindrical thin shell undergoing severe strain-stiffening at considerable strain here using the proposed en- ergy density Eq. (12) . In this context, the principal Cauchy stress components for the geometry shown in Fig. 6 are obtained from the stress field Eq. (10) as follows

wherepis the Lagrange multiplier associated with hydrostatic pressure. On assuming the thin shell approximationTrr= 0 and applying the same in above equation Eq. (20) , we obtain the hoop/circumferential and axial/longitudinal principal stresses given as follows

For an applied inflation pressureP, the radial and axial equilibrium equations for a cylindrical thin tube yield

wheret0 andr0 represent the wall thickness and radius of the cylindrical tube at its undeformed configuration, respectively. The above relation Eq. (22) also represents a well-known resultTθθ= 2Tzzin pressure vessel theory as well. On usingTθθ= 2Tzz, one may obtain a relation between the circumferential and longitudi- nal stretches asλθ=λθ(λz)that is valid for any arbitrary strain energy density functionW. Coming to the objective of the prob- lem, we may define a normalized dimensionless pressureP*for our simplification given as follows

whereμis the shear modulus andr0,t0represent the geometri- cal dimensions of the tube. The above relation Eq. (23) shows a relation between normalized dimensionless inflation pressure, cir- cumferential and longitudinal principal stretches for any arbitrary energy density functionW. On using equations Eqs. (21) , (22) and (23) , we may derive the normalized inflation pressureP*in terms of the hoop stretchλ=r/r0 and axial stretchλzfor the proposed energy density function Eq. (12) given as follows

whereI1 =f(λ,λz)andI2=f(λ,λz)denote the first and second invariants of left Cauchy-Green deformation tensor B . On usingTθθ= 2Tzz, we obtainλθ=λθ(λz)for the proposed energy den- sity function Eq. (12) . From this obtained relationλθ=λθ(λz), we may express the principal hoop and axial stretches of the tube in terms of an auxiliary variable called volume expansion ratiov=λ2λzgiven as follows

wherev≥1 can be considered as a physical parameter to analyse the elastic instability in cylindrical thin shell. Also, a pair of the above equations Eq. (25) can be viewed as a parameteric relation between the principal hoop stretchλand axial stretchλzof the tube.

We compare existing models with the analytical findings of the proposed model along with Alexander experimental data [36] in Fig. 7 . With a mathematical simplicity and two structurally rel- evant model parametersJandn, we note that our model only matches the thin cylindrical shell data and appears to provide an even better description of the data. Therefore, it ought to pro- vide more accurate solutions for the forgoing considered boundary value deformation problems.

On using Eqs. (24) and (25) , we may now define the normal- ized pressureP*as a function ofP*(λ,J,n)to emphasize the de- pendency on the chain extensibility parameterJand strain hard- ening parametern, collectively. On variations of these materials parameters, the rubber-like materials get softened or hardened ac- cordingly. These variations can be identified through the volume expansion instability limits of the material, as shown in Fig. 8 .

Fig. 7. Comparison of proposed model against Gent and Horgan-Saccomandi mod- els along with the Alexander’s experimental data [36] .

Fig. 8. Normalized pressure versus volume expression ratio plot for the proposed model in cylindrical tube inflation at different Jand n .

Herein, we observe an increase in the volume expansion in- stability limit of the material with an increase in the chain ex- tensibility parameterJor a decrease in the hardening parametern. This is because of the high entropy level among the polymeric chains, which are not aligned. Due to less space for the network’s movement in the undeformed state, it requires considerable pres- sure at the beginning of deformation. Once inflation gets started, most of the monomers in the network become aligned in the di- rection of stretching and get free space for their movement. Now, the stretch level continuously increases even at low pressure and reaches its maximum value. If stretching is continued, the poly- meric bonds and bond angles require more considerable energy to change the chains’ configurations. This phenomenon is termed the strain-stiffening of hyperelastic material. Then, no further in- crement of stretch level happens even on an increment of infla- tion pressure, and its corresponding stretch is termed the lock- ing stretch. While material gets strain stiffened, the chain network starts to break, and the polymeric chain’s entropy level increases. This will hinder the movement of other polymeric chains in the polymeric network. Further, there is almost no increment of the stretch level while the pressure reaches its infinite level. This phe- nomenon ensures the bursting of a hyperelastic thin shell.

This work presents a phenomenological constitutive model for an incompressible isotropic hyperelastic material. The proposed model Eq. (12) reflects the locking stretch and chain stiffening phenomena at moderate as well as large strains and also robust in fitting the experimental data by Beatty [26] and Treloar [32] for a vulcanized natural rubber. The model Eq. (12) is also used to study an elastic instability phenomenon in spherical and cylindrical thin shells undergoing severe strain-stiffening at considerable deforma- tion range. Our model Eq. (12) is compared with other theoretical models to predict the bursting pressure in a thin spherical balloon and observed that our model predicts a better match with experi- mental data.

Parametric studies are performed by varying the limiting chain extensibility parameterJand the hardening parametern, which play an important role in the severe strain-stiffening phenomenon of elastic materials. It is observed that an increase in the chain extensibility parameterJand a decrease inthe hardening param- eterncan be used to enhance the volume expansion instability limit of the hyperelastic material. Finally, the mathematical sim- plicity of the proposed model Eq. (12) , which contains just three material constants, may facilitate the analytic solutions for various engineering problems in the rubber industry.

Declaration of Competing Interest

The authors declares that there is no conflict of interests re- garding the publication of this paper.