The generalized viscoelastic spring

A spring/rod model is presented that describes one-dimensional behaviour of solids susceptible to large or small viscoelastic deformation. Derivation of its constitutive equation is underpinned by the fact that the internal energy, which the elastic part of deformation stores in the spring, changes in time with the observed strain as well as with some, unknown part of the strain-rate. The latter emerges through the action of a viscous flow potential and is the source of inelastic deformation. Thus, unlike its conventional viscoelasticity counterparts, the model does not postulate a priori a rule that relates strain with viscous flow formation. Instead, it considers that such a rule, as well as other important features of combined elastic and inelastic material response, should become known a posteriori through the solution of a relevant, well-posed boundary value problem. This paper begins with considerations compatible with large viscoelastic deformations and gradually progresses through simpler modelling situations. The latter also include the case of small viscoelastic strain that underpins formulation of classical, spring-dashpot viscoelastic models. In an example application, a relevant closed-form solution is obtained for a spring undergoing small viscoelastic deformation under the influence of a viscous flow potential which is quadratic in the stress.


Introduction
The subjects of elasticity, plasticity and viscoelasticity are generally considered as parts of continuum solid mechanics that deal with different kinds of solid material behaviour. In this context, most of the existing standard models of either plastic or viscoelastic material behaviour begin with some postulation that presupposes the manner in which the anticipated parts of elastic and inelastic deformation are assembled and enter the resulting constitutive rule. This observation may be felt more commonly associated with models met in the theory of plasticity. The model of multiplicative decomposition [1] is here referred to as one of the most successful relevant examples. Nevertheless, classical models in linear viscoelasticity also anticipate the manner in which the observed strain and its time derivative are assembled. The well-known and long-established method of appropriate combinations of elastic springs and 'inelastic' dampers/dashpots [2][3][4] is still considered a most successful of such (e.g. [5,6]). A recent plasticity model [7] follows a different approach and perceives the work hardening stage of a stretched deformable cube as a regime of material response in which the anticipated elastic and inelastic parts of deformation are initially both unknown; along with the rule that these are finally combined and form the observed total deformation. The model considers instead that these important features of the observed material response should become known through the solution of a relevant, well-posed mathematical boundary value problem. The manner in which this aim becomes possible in [7] is (i) by retaining the classical postulation that the inelastic part of deformation emerges through the reaction of a yield function, which also acts as a flow potential, and (ii) by properly accounting for the evident fact that the change of the energy, W, stored in the material depends on the change of both the observed deformation and the observed material flow. It is recalled that, upon removing the yield function involvement, the plasticity model proposed in [7] reduces naturally to, and collapses into its own hyperelasticity framework.
The present paper aims to demonstrate that the outlined new postulations [7] are similarly relevant and applicable to classical viscoelasticity situations, and, hence, to indicate that all three subjects of hyperelasticity, plasticity and viscoelasticity share a common theoretical background. With these aims in mind, attention here focuses onto one of the simplest possible models of one-dimensional deformable bodies, namely that of a cantilever viscoelastic cylindrical rod. Nevertheless, figure 1 describes schematically the existing close relationship between the linearly elastic spring and a linearly elastic cantilever rod subjected to tension (e.g. Sects. 23 and 30 in [8]), and serves as a useful reminder of the implied well-known mathematical equivalence.
It thus becomes initially clear that, wherever the term 'spring' is met in what follows, this may be replaced with the term 'rod' without further consequences, at least within the small deformation regime. In cases of large deformation, the cross-sectional area of a cylindrical rod is naturally expected to change with the deformation in accordance with well-known relevant rules. However, such a change of the cross-sectional area is here regarded of minor importance, as it influences only the relationship between the force and the relevant traction distribution applied externally on the relevant end boundary of the cantilever rod. Section 2 thus presents a new viscoelastic spring model that shares simultaneously properties of an elastic spring and an inelastic damper (dashpot). In strict terms, this essentially represents a one-dimensional deformable solid subjected to large viscoelastic deformation. Progressively simplifying considerations, afterwards enables this paper to proceed through simpler modelling situations and to finally establish the manner in which the presented model relates to its wellknown classical spring-dashpot counterparts.
Accordingly, §3 considers separately particular cases of infinitesimally small strain and succeeds to end with a relevant, geometrically linearized viscoelastic spring model. That model is applicable within the region of validity of the classical spring-dashpot models met in linear viscoelasticity. However, it is still substantially more general as, despite its small strain region of applicability, it makes use of a nonlinear constitutive equation.
The implied material nonlinearity does not prevent §4 to proceed and specify a special case in which a corresponding boundary/initial value problem admits an exact, closed-form solution. This is a case in which the internal energy function and the viscous flow potential of that geometrically linear viscoelastic spring are both quadratic functions of their arguments. Section 5 describes next the manner in which further approximations enable that small strain version of the present model to connect with its well-known Kelvin-Voight counterpart met in classical viscoelasticity. Finally, §6 summarizes important principal steps and considerations that underpin the model's novelty and originality, and briefly suggests directions for future relevant research and study. Figure 1 employs a single co-ordinate parameter, X, and illustrates schematically the known analogy between a linearly elastic spring, having elastic modulus K, and a linearly elastic cylindrical rod having Young's modulus E and cross-sectional area A. These elastic components are both considered of the same length, L, and are subjected to one-dimensional deformation through the action of the same external force. That force has magnitude f and is applied externally on the free end of the component, where X = L, along the direction of the depicted co-ordinate axis.

The generalized viscoelastic spring model
Interest here is directed towards a corresponding situation in which the observed response of the externally loaded spring is viscoelastic. In this context, it is worth recalling that such a kind of viscoelastic material response is observed very often in nature. Most polymers behave in this viscoelastic manner, while the Mullins effect is here also referred to as a well-known relevant example of rubber-type material behaviour (e.g. [9][10][11][12]).

(a) Preliminaries and viscous flow considerations
It is accordingly assumed that at its initial length L (at time t = t 0 , say) the rod of interest is unloaded and unstressed. Loading from that initial configuration takes off through application of a force: This is considered uniformly distributed over the area of the free end of the rod and may vary slowly in time. Moreover, it gives rise to immediate viscoelastic material response which, at t > t 0 , is described with the use of a current co-ordinate parameter, x; A is naturally expected to change at large deformation according to the known relevant rule. It is postulated that the observed one-dimensional deformation comprises simultaneous action and possible interaction of elastic deformation and viscous fluid flow.
The manner in which those parts of elastic and inelastic material response are merged and compose the observed total deformation is initially considered to be unknown. However, the royalsocietypublishing.org/journal/rspa Proc. R. Soc. A 476: mass of the viscoelastic spring is required to obey the standard continuity equation: where ρ and v represent current mass density and velocity, respectively; a dot denotes total differentiation with respect to time and a comma differentiation with respect to x. Denote with d v the rate of the inelastic deformation gradient, and note that, in general, this is different to its total deformation counterpart, v ,x ; namely d v = v ,x . It is postulated that there exists a viscous flow potential φ(σ ), where σ is the single Cauchy stress component acting along the spring/rod axis, and that this potential is the source of d v . The relevant viscous flow rule may then be postulated in the direct manner which is consistent with the general concept of a potential function; here, as well as in what follows, a prime denotes differentiation with respect to σ . Alternatively, adopt relevant plasticity postulations and assume that φ(σ ) is such that where κ is some non-negative parameter. Then, amend slightly a route employed already in classical plasticity [13,14] as well as in nonlinear viscoplasticity [15] by requiring from the rate of the dissipative work, to be stationary. The implied alteration refers to the fact that the stress is split into two parts (e.g. [16]) as follows: where the former part gives rise to recoverable work and the second contributes to the dissipative work rate ψ. The imposed requirement necessitates minimization of the expression where λ is a Lagrange multiplier, thus leading to If λ is considered constant, then this flow rule is seen as equivalent to (2.3) because, as becomes later more evident (see (2.18) and §3b), that constant can be absorbed by φ(σ ).

(b) Equilibrium and constitution
represent the rule that governs the spring deformation. In the light of the adopted notation, consider that the initial length, L, of the spring changes into , so that 0 ≤ x ≤ . The following boundary conditions thus hold: As was already mentioned, the elastic part of the deformation is initially considered indistinguishable from its inelastic counterpart. However, the rate of the internal energy that the deformation accumulates into the material is still governed by the usual rate-of-energy equation where W represents the internal energy per unit axial length stored into the implied, arbitrary part [0, x] of the deformed spring and ρ 0 is the initial density of the spring, at t = t 0 . The second term inside the integrand is the kinetic energy, while σ is evidently the only boundary traction acting externally on [0, x].
It is now recalled (e.g. [17]) that the material time derivative appearing in the left-hand side of (2.10) obeys the rule d dt Introduction of (2.11) into (2.10), followed by differentiation with respect to x, leads to It is next observed that, due to the implied energy dissipation, W changes with, and therefore depends not only on the deformation and its gradient but also on the rate of the inelastic deformation gradient, d v . In mathematical terms, this fact takes the form and leads toẆ A comparison of this result with (2.12) leads to which, due to the arbitrariness of both v and v ,x , requires While (2.17a) is recognized as the standard one-dimensional equation of motion, (2.17b) emerges as the constitutive equation sought for the present generalized viscoelastic spring model. On its own, the first term in the right-hand side of (2.17b) represents the constitutive equation of the corresponding hyperelastic spring. In the present viscoelastic spring case though, that term refers to the change of W with respect to the total, rather than to the still unknown elastic part of the deformation. In (2.17b), this is further accompanied by a term which is influenced heavily by the appearing features of inelastic deformation and viscous flow. It is revealing in this context to note that, if the Lagrange multiplier parameter appearing in (2.7) is considered constant, one obtainṡ and (2.17b) thus reduces to This makes it clearer that both (2.18) and its more general version (2.17b) are implicit in the stress and, also, differential rather than algebraic constitutive equations.
In summary, the presented viscoelastic spring model is alternatively regarded as an initial/boundary value problem that is described as follows: given the form of the flow potential introduced in (2.4) and the form of the internal energy density (2.14), determine three principal unknowns, namely the material density, ρ, the placement function, x, and the Cauchy stress, σ , in a manner that satisfies three principal equations, namely (2.2), (2.17a) and (2.18), subject to the boundary conditions (2.9) and the initial conditions:

c) Energy and dissipation considerations
In view of the general form (2.14) of the internal energy density, it is anticipated that represents the internal energy stored in a corresponding elastic spring subjected to the same loading and deformation conditions; the equality sign holds in the absence of deformation (F = 0). On the other hand, and in accordance with the definition (2.5), the energy dissipation is expected to be the following non-decreasing function of time: where σ D is to be determined through the solution of the boundary value problem of interest, and the equality sign holds in the special case that the spring behaves elastically (σ D = 0). Hence, by proposing the following connection between the internal energy densities of a pair of corresponding elastic and viscoelastic springs: This condition implies further that this generalized viscoelastic spring stops to represent a solid at the first instance, t = t F > t 0 , at which In this general form, the developed viscoelastic model is highly nonlinear, and its connection with specific boundary value problems must confront the usual analytical hurdles met in corresponding hyperelasticity problems. Moreover, as is seen in the next section, material nonlinearity is generally present even in the small strain regime. This additional mathematical difficulty is readily compatible with the viscoelastic nature of the solid of interest, as it is due to the fluid-type flow part of its material behaviour. Nevertheless, a further special case will later be identified as part of an application ( §4), where both geometric and material nonlinearities can be considered negligible. In that case, the model obtains a considerably simplified form that admits a closed-form solution and facilitates its connection, as well as its comparison, with its conventional counterparts ( §5).

Small strain
Potential connection of the present model with standard spring-dashpot models requires initially specialization of the former within the region of applicability of the latter, namely the region of infinitesimally small strain. Such a specialization begins with the introduction of the displacement: (3.1) and the subsequent approximation where the strain is considered much smaller than 1.
It is now recalled that in linear elasticity problems, where strains are considered infinitesimally small, the deformed and undeformed coordinate parameters are considered practically indistinguishable (x ≈ X) and the current mass density, ρ, indistinguishable from its initial counterpart, ρ 0 . However, in the present case, where the rod responds in a viscoelastic manner, potential differences between the current and the initial mass density might be found influential in applications. Therefore, such differences are not considered a priori negligible in what follows.
As ( In general, this is still a nonlinear differential equation for σ and stays such even in cases that its solution might be sought and found independently of the solution of the continuity and the equilibrium equations (2.2) and (2.17a).
The following connection is now made between the rate of inelastic deformation and the rate of small strain due to viscous flow: By virtue of (2.7), this implies that the inelastic part of small strain is Hence, the elastic parts of strain and displacement can, respectively, be calculated as follows: unknowns, ρ, u and σ , in a manner that satisfies the three principal equations (2.2), (2.17a) and (3.6), subject to the boundary conditions u| X=0 = 0, σ | X=L = T(t), (3.10) and the initial conditions ρ| t=0 = ρ 0 , u| t=0 = σ | t=0 = 0. (3.11) Following the solution of this initial/boundary value problem, the total strain and its viscoelastic part may be evaluated, a posteriori, through use of (3.3) and (3.8), respectively, while (3.9) will lead to the determination of the elastic parts of both the strain and the displacement.

(a) Quadratic form of the internal energy function
Foundation of linearized small strain elasticity models is based on purely quadratic forms of W, so that the resulting constitutive equation relates the stress and the strain in a linear manner. In the present viscoelastic spring case, the most general, purely quadratic form of (3.5a) is where E, η and ξ are appropriate material moduli whose values should enable consistency of the model with the internal energy conditions (2.20) and (2.23). It is seen that this form of W satisfies the initial condition (3.5b). Moreover, the form (3.12) of W enables the constitutive equation (3.6) to acquire the more specific form (3.13) or, by virtue of (2.7), σ = ρ ρ 0 Ee + ληφ + λφ σ e (ηe + λξ φ ) . (3.14) It is noted that, in the special case that η = ξ = 0, (3.12) reduces to its standard linear elasticity formŴ In that case, the constitutive equation (3.14) resembles closely its linearly elastic spring/rod counterpart, which, based on the aforementioned linear elasticity postulate, neglects the influence of the continuity equation (2.2) and implies that the material density is not affected by the deformation (ρ = ρ 0 ). It is thus seen that if this linear elasticity postulate is applied when η = ξ = 0, then (3.14) and (3.16) become identical. At the other extreme, by setting E = η = 0, one recognizes a constitutive model that exhibits purely fluid material behaviour. Such an observation is fully compatible with a basic rheology concept, according to which viscoelastic material description does not recognize or distinguish clear boundaries between solid and fluid material behaviour (e.g. [2][3][4][5]). That concept thus classifies the proposed generalized viscoelastic spring as a properly developed viscoelastic model, in the sense that at one extreme (η = ξ = 0) this resembles an elastic solid, while at the other (E = η = 0) a viscous fluid. However, a study of the purely fluid case (E = η = 0) is not pursued here, as it does not fulfil the energy and dissipation conditions set in §2c and thus is incompatible with the anticipated mechanical response of a solid spring.
It is also noted that the last term in the right-hand side of either (3.13) or (3.14) represents a constitutive equation that still preserves the nonlinear differential equation features observed earlier in (3.6). However, additional postulations concerning the choice of the viscous flow

(b) Quadratic form of the viscous flow potential
In this regard, consider next the special case of a viscous flow potential which is quadratic in σ , namely where G is some appropriate material modulus. Then, the viscous flow rule (2.7) yields As was anticipated earlier in §2, introduction of the new parameterĜ = λG enables incorporation of a constant value of λ into the form of the flow potential φ(σ ), and, hence, validates in that case a direct postulation of the flow rule (2.3). The constitutive equation (3.14) thus reduces to which, due to involvement of the ratioσ /ė is differential with respect to time, as well as nonlinear in σ and e. It is recalled that, in general, solution of (3.19) should be sought in conjunction with the kinematic relation (3.3), the equation of motion (2.17a) and the continuity equation (2.2), subject to the associated boundary conditions (3.10) and the initial conditions (3.11). Completion of the outlined solution will finally enable use of (3.8) and (3.9) and, hence, the determination of the remaining unknown deformation features, namely (3.20)

(c) Energy and dissipation considerations
The energy and power dissipation considerations discussed in §2c can also be applied in a posteriori manner, as soon as decomposition of stress into its two-part form (2.6) enables a similar split of the total work rate into its recoverable and dissipative parts. In more detail, introduction of (2.6) into (3.19) transforms the latter equation into the following: A necessary second condition that leads to precise determination of the involved couple of unknowns, σ R and σ D , is next established by balancing separately all terms in (3.21) that contribute to recoverable work. This condition is postulated through the observation that (3.21) should also hold true at time instances or stages of the deformation that may be dissipation-free Hence, a split of (3.21) into two parts containing terms that contribute to the creation of recoverable and dissipative work rate, respectively, leads tô It is noted that (3.22a) is a first-order nonlinear differential equation for σ R , and its solution may be attempted independently of (3.22b). The latter is a first-order linear differential equation for σ D , and, hence, its standard solution can be formed immediately after its first and last terms are fed by the solution of (3.22a).
After some suitable rearrangement, it can be verified that the forms of (3.19) and (3.22a) are essentially identical. This observation justifies, at least to a first approximation, the split of (3.21) that leads to (3.22) and suggests that σ and σ R might be sought as two different solutions of the same time-dependent differential equation. However, such a search for relevant solutions should still take place in conjunction with the equation of motion (2.17a) and the continuity equation (2.2). Hence, while the density, ρ, and the total strain, e, are unknown during the solution process of (3.19), their counterparts ρ and e e that emerge in equation (3.22a) are there considered determined and, therefore, known quantities.

Application: quasi-static viscoelastic deformation at small strain
Despite the mathematically complicated form of the constitutive equation (3.19), a closed-form solution of the outlined viscoelastic boundary value problem is still possible, at least in the case that small strain takes place in a quasi-static manner. Description of that solution begins with a displacement field of the form Here, the function α(t) is assumed to be such that the implied deformation is imposed in an adequately slow manner that justifies replacement of the equation of motion (2.17a) with its quasistatic version σ ,X = 0. (4.2) By virtue of (3.3), (4.1) leads next to the following results: These enable the continuity condition (2.2) to obtain the separable forṁ ρ + ρα(t) = 0, (4.4) and, hence, to give the material density of the viscoelastic spring as follows: The constitutive equation (3.19) attains then the following form of a first-order ordinary differential equation for σ : which is nonlinear due to the first term that appears in the square bracket. It is noted that, as neither e norė depends on X, equation (4.6) and, therefore, its potential solution for σ are also independent of X. Therefore, the quasi-static equation of motion (4.2) is already satisfied identically.
For small values of ξ , the nonlinear differential equation (4.6) may be solved asymptotically with the use of some standard perturbation technique. The first approximation to the implied asymptotic solution is evidently obtained by setting ξ = 0 and represents a special case in which the rate of inelastic strain does not influence on its own the internal energy function (3.12). That solution represents the deformation and material response of a generalized linearly viscoelastic spring, and, hence, deserves separate mention on its own merit.
(a) Particular case: the generalized linearly viscoelastic spring (ξ = 0) When ξ = 0, the quadratic form (3.12) of the internal energy function reduces to and is influenced by the rate of inelastic strain only through the coupling of the latter with the total strain. This form of W excludes effects associated with purely fluid constitutive behaviour but corresponds to a viscoelastic spring whose constitution equation (3.19) is generally still differential as well as nonlinear at small strain. However, the rather simple form of the displacement field (4.1) enables the reduced form (4.6) of the constitutive equation to acquire the following form of a linear first-order ordinary differential equation:σ + p(t)σ = q(t), (4.8) where Thus, when subjected to the initial condition (3.11c), equation (4.8) provides the stress in the following closed form: where, however, the noted integrations are generally expected to require numerical evaluation regardless of the form of the function α(t).

(b) Strictly quasi-static spring deformation
The linear form, is the optimal choice of α(t), as it produces no acceleration and thus reduces the equation of motion (2.17a) into its quasi-static counterpart (4.2). When combined with (4.11), the displacement distribution (4.1) thus gives rise to a strictly quasi-static deformation, which also satisfies the displacement boundary condition (3.10a). Moreover, solution of (2.2) suggests that the material density changes in time according to the rule When this is based on the quadratic internal energy function (4.7) and the quadratic flow potential (3.17), the choice (4.11) still gives rise to the stress distribution (4.10), where, however, it is Thus, the implied displacement and stress distributions represent an exact solution of this well-posed viscoelastic spring problem, provided that the external traction T(t) that appears in the final, yet to be used boundary condition (3.10b), is applied according to the rule (4.10). The remaining unknown features of this viscoelastic deformation, namely e v , e e and u e , may then be determined with the use of (3.20), where the noted integrations are expected to require numerical evaluation as well.
After e e and e v thus become known, the two-part decomposition (2.6) of σ also becomes possible through potential solution of the first-order ordinary differential equations (3.22). However, in the present case, where ξ = 0, both equations (3.22) attain linear forms, namelŷ (4.14) Either of these is a first-order linear ordinary equation and can evidently be brought into the standard form (4.8). Its closed-form solution can thus be described in the standard manner (4.10). More practical future investigations will thus require considerable further computational work that will not only accompany the outlined solution but will also enlighten matters associated with the subsequent split of the work rate into its recoverable and dissipative parts.

Relevance to conventional linear viscoelastic solids
Conventional linear viscoelasticity models are based on the simplification d v =ė, (5.1) which implies that the rate of viscous strain and the rate of total strain are considered a priori identical. In this special case, the small strain constitutive equation (3.13) of the present model reduces to the following: However, conventional viscoelasticity models do not make use of the continuity equation (2.2). They instead postulate a priori that the imposed small strain consideration is adequate to justify the approximation ρ ρ 0 = 1. which is quoted through (3.6) in a similar manner. Namely, by imposing the approximation (5.4) and disregarding the noted influence of the flow potential, φ.
It is now recalled that the constitutive equations of a standard elastic spring or a viscous damper (dashpot) are obtained through (5.5) by simply setting η = 0 or E = 0, respectively. Constitutive equations of conventional viscoelastic solids which are more complicated than that of the Kelvin-Voight solid may then still be built up in the manner employed in classical viscoelasticity. Namely, by (i) using appropriately several kinds of different combinations of one or more Kelvin-Voight solids with one or more springs and/or dashpots, and (ii) ignoring the concept of the internal energy function.
For instance, the well-known constitutive equation of Kelvin's version of the standard, threeparameter viscoelastic solid depicted in figure 2b, namely Kelvin-type of a three-parameter solid Figure 2. (a, b) Two classical models of a viscoelastic solid met commonly in the classical viscoelasticity literature; E, E 1 and E 2 represent elastic spring stiffness moduli while η is the modulus of damper viscosity.

Conclusion
The generalized viscoelastic spring model presented in this paper is directly relevant to the behaviour of one-dimensional solid components susceptible to large or small viscoelastic deformation. In its most general form, the large deformation version of the model requires the determination of three principal unknowns, namely, the material density, ρ, the current placement function, x, and the Cauchy stress, σ , in a manner that satisfies a set of three principal differential equations; namely, the continuity equation (2.2), the equation of motion (2.17a) and the constitutive equation (2.18). The corresponding small strain version of the model is essentially described in the same manner, with the displacement, u, replacing the placement function, and the small strain constitutive equation (3.6) replacing (2.18). Even in the small strain case though, the constitutive equation is, in general, a differential rather than an algebraic equation.
In either case, the derivation of the constitutive equation is underpinned by the fact that the internal energy stored in the viscoelastic spring material changes in time with and, therefore, depends on both the observed actual strain and the unknown part of the strain-rate that is due to the anticipated viscous flow. Creation of the implied unknown part of the strain-rate, which is evidently the source of the observed inelastic deformation, is accordingly related with the action of a viscous flow potential, which, in turn, depends on the magnitude of stress.
In this manner, the presented generalized viscoelastic spring model makes successful simultaneous use of two different fundamental concepts met in the theories of elasticity and plasticity, namely, the concepts of the internal energy density/function and that of the yield/flow potential. Hence, along with [7], this model is regarded as another piece of evidence in support of the claim that hyperelastic, plastic and viscoelastic material behaviour all have the same source and can accordingly be based on and studied within a common theoretical framework.
It has indeed been shown that, unlike its classical viscoelasticity counterparts, the present generalized model does not need to postulate a priori the manner in which the strain and its time rate are combined or related to form its constitutive equation. Instead, the model considers that this, as well as other important features of combined elastic and inelastic material behaviour, including the rate of energy dissipation, can become known a posteriori through the solution of a relevant, well-posed boundary value problem. The latter should always be completed by associating to the aforementioned set of three simultaneous differential equations an appropriate set of initial and boundary conditions. This paper developed the presented viscoelastic model around the classical example of a cantilever spring. It accordingly completed description of the implied boundary value problem by employing the boundary conditions (2.9) and the initial conditions (2.19) in the general case of large deformation, or their counterparts (3.10) and (3.11), respectively, in the case that deformation is infinitesimally small. In this context, considerable attention was also given to the closed-form solution obtained in §4 for a spring undergoing small viscoelastic strain under the influence of inelastic deformation underpinned by a quadratic form of the viscous flow potential. That exact, closed-form solution requires further consideration and study, and its computational implementation and treatment are regarded as a project of immediate priority and interest. Such a project should initially be expected to make better understood the relevance as well as the differences of this model with its classical spring-dashpot counterparts and, hence, to expose practical aspects of its anticipated theoretical superiority. The latter is, in brief, underpinned by the facts that the proposed generalized viscoelastic spring model enables (i) a posteriori determination of the rule that its elastic and inelastic deformation parts are assembled, (ii) consideration of the levels of energy stored in its viscoelastic material, and (iii) through time account for corresponding levels of energy dissipation.
Data accessibility. This article has no additional data. Competing interests. I declare I have no competing interests. Funding. The research work and output described in this publication received no funding from any research funding body.