_{1}

^{*}

A recent review publication presented an extensive and comprehensive assessment of the phenomenological relations of Poisson’s ratios (PRs) to the behavior and responses of contemporary materials under specific loading conditions. The present review and analysis paper is intended as a theoretical mechanics complement covering mathematical and physical modeling of a single original elastic and of six time and process (i.e. path and stress) dependent viscoelastic PR definitions as well as a seventh special path independent one. The implications and consequences of such models on material characterization are analyzed and summarized. Indeed, PRs based on experimentally obtained 2-D strains under distinct creep and/or relaxation processes exhibit radically different time responses for identical material specimen. These results confirm the PR’s implicit path dependence in addition to their separate intrinsic time reliance. Such non-uniqueness of viscoelastic PRs renders them unsuitable as universal material descriptors. Analytical formulations and experimental measurements also examine the physical impossibility of instantaneously achieving time independent loads or strains or their rates thus making certain PR definitions based on constant state variables, while mathematically valid, physically unrealistic and unachievable. A newly developed theoretical/experimental protocol for the determination of the time when loading patterns reach stead-state conditions based on strain accelerations demonstrates the capability to measure this time from experimental data. Due to the process dependent PRs, i.e. stress and stress history paths, the non-existence of a unique viscoelastic PR and of a universal elastic-viscoelastic correspondence principle or analogy (EVCP) in terms of PRs is demonstrated. Additionally and independently, the required double convolution integral construction of linear viscoelastic constitutive relations with the inclusion of PRs is cumbersome analytically and computationally needlessly highly CPU intensive. Furthermore, there is no theoretical fundamental hint as to what loading path is required to produce a unique universal viscoelastic PR definition necessary for formulating a PR based constitutive relation or an EVCP protocol. The analysis associated with an additional Class VII viscoelastic PR establishes it as a universal representation which is loading path and strain independent while still remaining time dependent. This Class PR can be the one used if it is desired to express constitutive relations in terms of PRs, subject to the caveat applying to all PR Classes regarding the CPU intensiveness in the time space due to triple product and double convolution integral constitutive relations. However, the use PRs is unnecessary since any set of material behavior can be uniquely and completely defined in terms of only moduli and/or compliances. The mathematical model of instantaneous initial loading paths, based on Heavi-side functions, is examined in detail and shown to lead to infinite velocities and accelerations. Additionally, even if non-instantaneous gradual loading functions are employed the resulting PRs are still load and load history dependent. Consequently, they represent specialized PR responses applicable and limited to those particular load and history combinations. Although the analyses contained herein are generalized to non-homogeneous linear viscoelastic materials, the main focus is on PR time and process dependence. The non-homogeneous material results and conclusions presented herein apply equally to homogeneous viscoelasticity and per se do not influence the results or conclusions of the analytical development regarding viscoelastic PRs. In short, these PR analyses apply to all linear viscoelastic material characterization.

A recent review publication [

Viscoelasticity had its origins with the seminal studies of Kelvin, Maxwell, Voigt, Boltzmann and Volterra during the period of 1865 to 1913 [

1) PR dependence on time and on stress and/or strain history

These are two separate phenomena. For instance, in linear viscoelasticity moduli, compliances, relaxation and creep functions are only time dependent, while all PRs except Class VII are both implicitly time dependent and stress history (path) dependent. The time dependence of viscoelastic PRs was firmly established experimentally over twenty years ago [

2) Contributions of the loading phase in determining material properties

The importance of starting loading phases on subsequent viscoelastic responses and consequently on material characterization (moduli/com- pliances) has been demonstrated through analytical simulations in [

In [

Finally, it is important to note that experimental determinations of viscoe-

Class | Name | Viscoelastic | Path Dependent | EVCP |
---|---|---|---|---|

Poisson’s Ratio | pendent | |||

I | Original [ | YES | NO^{I} | |

II | YES | YES, but limited | ||

1-D [ | to ^{II} | |||

III | LT or FT [ | YES | YES & NO^{III} | |

IV | Hencky [ | YES | NO, nonlinear^{IV} | |

V | Velocity [ | YES | NO^{V} | |

VI | YES | YES, but limited | ||

1-D [ | See Note VI below. | to ^{VI} | ||

VII | pseudo PR not | NO | YES^{VII} | |

based on strains |

NOTES: All PRs depend on time, stresses and stress-time histories, and these results apply equally to homogeneous and nonhomogeneous media. None of the PRs represent universal material properties as they are path dependent, i.e. ^{I}Not in proper form in FT space for EVCP. See ^{II}See the Section “The physics of time invariant stresses, etc.” for the physical difficulties associated with achieving constant strains or stresses or constant rates for^{III}Inverse FT has double convolution integral in ^{IV}Based on natural strains and, therefore, nonlinear. ^{V}Similar EVCP difficulties as Case I. ^{VI}If and only if^{VII}The Class VII PRs are path and strain independent and only functions of time.

lastic PR time functions are based on specific loading or strain histories and, consequently, are limited to those conditions and are non-exportable to other loadings, since viscoelastic PRs are path dependent and hence non-unique as seen in

The Class VII PR definition avoids loading path dependence problems, but still leads to double convolution integrals in the time space that are extremely CPU intensive.

A recent paper [

tension strains can reverse direction, a phenomenon which the authors denote by “negative stiffness.”

In [

However, as it is shown in [

Comment 1. Aside from Poisson’s great native genius, what made his namesake ratio universally accepted and most useful is that for all but seven decades of its two century existence the application were limited to elastic materials where for each of them it is a distinct constant. Unfortunately, nature was significantly less kind and far less magnanimous when it comes to viscoelasticity.

Detailed analyses and discussions of the above topics are presented in subsequent sections.

Elastic and viscoelastic constitutive relations based on moduli and/or compli- ances are derived from fundamental principles, i.e. the thermodynamic laws pertaining to energy and entropy [

In linear elastic materials moduli and compliances are time, path and stress independent and stress-strain relations involving PRs must be and are derived from these basic equations. Furthermore, in elasticity where PRs are constants and constitutive relations are algebraic, PRs are unique material properties and derivations are a straight forward matter.

In viscoelastic media where PRs are time, stress and stress history (path) de- pendent and where stress-strain relations involve time integrals multiple com- plications arise. For one, numerous PR definitions are possible and derivations may be carried out in either time space or in the integral transform space yielding in some instances mutually exclusive results and PR definitions. Secondly, since viscoelastic media dissipate energy loading histories take on prime importance and must be considered [

The PR analyses and formulations are generalized to non-homogeneous linear viscoelastic materials, however the main focus is on PR time and process dependence. The results apply equally to homogeneous viscoelastic media and PRs are path dependent for either homogeneous or non-homogeneous viscoelastic media.

Thus, while some of the viscoelastic PR models are defined in terms of con- sistent mathematical transient loading models, which are physically unattainable, they fail to translate into realistic and observable physical entities.

It will be shown that PRs, although extremely important in elastic material characterizations because there they are constants for each material, are not fundamental viscoelastic quantities in the same sense as relaxation moduli and creep compliances. Instead, they are multiply defined process dependent quan- tities as detailed in

Since elastic conditions form the ICs of any viscoelastic formulation and to properly establish EVCPs, it is necessary to first review pertinent elastic PR developments. Consider a Cartesian system with coordinates

Class | Time Space PR | FT Space PR |
---|---|---|

I | ||

II | ||

but step function physically unachievable | ||

III | ||

alternate equivalent definition (67) - (74): | see Eqs. (36) for lack of proper | |

equivalent physical relation in | ||

IV | ||

V | ||

VI | ||

for | same physical restriction as Class II | |

for | plus | |

the entire time interval | ||

VII | ||

this Class should be the PR of choice as it | is path independent and hence universal |

defined^{1} in [^{2} or

The anisotropic homogenous elastic constitutive relations read

with

In particular, for a 1-D loading

with the

However, for non-homogeneous elastic materials the usually constant 1-D PRs become spatial functions with values depending still on distinct material properties but also additionally on the position

For linear homogeneous elastic media, it can be readily proven that

It can be further demonstrated that for linear elastic media, the following hold true

and similar relations for the Lamé parameters (constants)

Alternate elastic constitutive relations are obtained by inverting the set (2) which yields

PR^{E} | K^{E} | E^{E} | G^{E} |
---|---|---|---|

0.5 | |||

0 | |||

−0.5 | |||

−1 |

In linear 1-D elasticity, the isotropic isothermal Hooke’s law simplifies to

where

Furthermore, in conservative linear systems Maxwell's reciprocal theorem [

Taking the FT of these linearized isotropic elastic constitutive relations yields

Comment 2. It should be noted that in most current viscoelastic materials, such as high polymers, rubbers, etc., the inequality (linearization term) in (11) and (12) is often violated because of low

The net effect of this linearization term is that linear elastic materials possess linear modulus/compliance constitutive relations, but potentially nonlinear ones when PR formulations are included.

However, the above linearization must be enforced since the EVCP depends on integral transforms, which can only be applied to linear systems. Additionally and separately, in order to formulate the EVCP in terms of PRs it is necessary to find a term by term and parameter by parameter viscoelastic match in the FT space to the elastic transformed relations (12), with a corresponding valid and plausible viscoelastic set of constitutive relations in the time space.

Alternately, the troublesome nonlinearity can be totally avoided if Hooke’s law is derived from fundamental deviatoric (shape change) and volumetric (volume changes) constitutive relations and by applying the three normal stresses simultaneously, to whit

Substitution of (14) into (13) yields

and one equation for each of the other two normal strains. Eq. (15) and its other two cousins are unambiguously linear and do not suffer from the nonlinearities of the PR relations (11). Furthermore in the latter formulation all three normal loads are applied simultaneously which is of prime importance in dissipate systems such as viscoelastic or plastic media.

Whereas in isotropic isothermal homogeneous linear elasticity the PR values are constants and material specific, the circumstances are far different for their vis- coelastic cousins under the same specifications.

Consider the general linear anisotropic isothermal viscoelastic relations [

Consequently, the Class I viscoelastic PR for an isothermal process when

Note that even in the simplest 1-D isotropic problem when only one stress derivative is present and if it is equal to a constant value^{3} for

Comment 3. In other instances when the stress or strain derivatives are time functions, they are “trapped” in the integrals and the PRs become time, stress and stress history dependent, i.e. process dependent [

In the very special case when

The relaxation moduli and creep compliances are as a matter of convenience generally represented by Prony series [

with similar expressions for

Fourier transform (FT) [

The FTs are equivalent to the two sided LTs as stated by the Rodrigues formula and provided all integrals exist these transforms are related to each other by [

Application of FTs converts the convolution integral isothermal constitutive relations (16) to

Class VI PR expressions are derived in [

The Elastic-Viscoelastic Correspondence Principle or Analogy (EVCP)^{4}

The EVCP, when applicable, is a powerful protocol for solving linear visco- elastic problems based on similarities between integral transform (IT) elastic and viscoelastic formulations and solutions. EVCP comes in two varieties: separation of variables [

The EVCP can be formally stated as:

1) Given an elastic body with the FT of its general solution

2) Then for viscoelastic convolution^{5} integral constitutive relations given by (16) and with identical boundary conditions, the viscoelastic solution is

and

are universally valid provided the transforms and their inverses exist.

Comment 4. The EVCP protocol [

In order to properly highlight the difficulties of generating an EVCP involving viscoelastic PRs, it is necessary to first review the process by which Eqs. (2), the 3-D general Hooke’s law with PRs is derived. Elasticity is a conservative system and its constitutive relations are algebraic. Whereas the viscoelastic system is a dissipative one with memory leading to integral differential constitutive relations in time, such as (16).

However, the linearization described in Section 3.3 must be enforced since the EVCP depends on integral transforms, which can only be applied to linear systems. Additionally and separately, in order to formulate the EVCP in terms of PRs it is necessary to find a term by term and parameter by parameter viscoelastic match in the FT space to the elastic transformed relations (12), with a corresponding valid and plausible viscoelastic set of constitutive relations in the time space.

In a non-conservative system such as a linear viscoelastic body, the luxury of choice of free sequencing of loads is no longer available, as different loading schedules result in distinct responses, i.e. Maxwel’s reciprocal theorem is inapp- licable. This is due to the material's memory, and consequent energy dissipation, which is expressed by integral type constitutive relations rather than the elastic algebraic ones. The viscoelastic ones also do not allow for the legitimate types of elastic algebraic approximations and manipulations of Eqs. (10) and (11), thus dictating the need for different approaches [

Since the correspondence principle has been proven inapplicable when PRs are involved [

but without PRs, the expressions

and Eqs. (25) are valid.

However, if

are valid subject to the limitations of the non-uniqueness of the viscoelastic PRs due to their stress and path (stress history) dependence. Difficulties with gen- erating a proper EVCP in terms of PRs stem from the following concepts, which will be established below in detail, see Eqs. (32).

1) The time and separate stress history dependence of the viscoelastic PRs, i.e. experimental conditions do not necessarily reflect in-service situations.

2) The inherent assemblage of individual material contributors in the elastic constitutive relations when PRs are involved calling for a triple product of ma- terial property parameters and stress tensors.

3) The viscoelastic PR triple product while obligatory in the Fourier transform space for establishing an EVCP has limited physical counterparts in the real time space nor any relation to the Boltzmann superposition principle [

In terms of moduli/compliances the 3-D Hooke’s law and the corresponding linear viscoelastic constitutive relations are both derivable from first principles [

However, when the isotropic Hooke’s law is assembled with elastic PRs, it is in an almost ad hoc manner of one direction at a time to yield^{6} [

Specifically, the pitfalls in producing a coherent EVCP arise from the triple product elastic combinations and their FTs^{7}

that requires a viscoelastic triple FT product counterpart

where ^{8}. The above four expressions indicate how the constitutive relations must be formulated in the real time space so that the desired FT triple products can be realized. If the isotropic 1-D elastic protocol of Eqs. (10) is used in assembling equivalent viscoelastic PR constitutive relations, then the constru- ction becomes

It then follows from this inverse heuristic approach that

There are three Classes of PRs that can produce the form (34), namely Class II, III and VI. However, neither Class II or VI produce a physically valid expre- ssion in the real time space. Class III, as pointed out in earlier, has only one convolution form out three possible ones that produce a physical counterpart (32b).

The definition of the Class VI viscoelastic PR leads to

and

Note that for the Class VI PRs the loading path is defined by prescribing it as

Some of the ambiguities arising from the Class III PRs can be resolved, i.e. removed, by defining a Class VII PR as

It is the only PR class that is entirely free from strain, stress and path restri- ctions since its definition is not based on the legacy Class I [

Comment 5. The Class VII is an artificially defined isothermal viscoelastic PR devoid of strain or stress associations and hence path independent while still remaining a time dependent universal martial property. It obeys the construction mandated by Eqs. (32) necessary for the existence of EVCPs. This Class PR should, therefore, be the one used if it is desired to express constitutive relations in terms of PRs. However, since this PR form nece- ssitates a priori knowledge of two compliances that completely characte- rize the isotropic viscoelastic medium, it is fair to ask why bother with the PR. Furthermore, since Class VII PRs do not involve strains they are not experimentally measurable.

At least three more than rhetorical questions remain:

1) What, if any, is the physical meaning of the Class VI PRs,

2) What is their relation to the strains?

3) What is the physical meaning, if any, of the double convolution integrals of Eqs. (32a) and (32c)?

The answers to all three questions are “none” indicating that they cannot be obtained directly through experimentally determined strains and stresses. Fur- thermore, due to the presence of a triple product in the above constitutive rela- tions these PRs are not a portal to EVCP formulations. Even though the Class VI PRs appear as pure universal material property descriptors their utility function is less than that of its parts, i.e. the compliances, since these PRs are first and foremost process dependent and, therefore, restricted and limited in scope in this case to time independent 1-D stresses

There is a certain irony accompanying the creation of the Class VI PR based on a time independent 1-D stress field

The intent is to produce a viscoelastic look alike to the structure of the elastic constitutive relations containing a mixture of moduli, stresses and the original Class I elastic PR [^{9}

For a viscoelastic medium,

with the real time convolution integral property

It should be noted that the Class VI PR is a special case of the general Class III PR with

In the viscoelastic constitutive relations there appears to be no clear protocol to derive directly in real time space either of the first two versions of double convolution integrals displayed in (32) that can be based on thermodynamic first principles or any mechanical or electrical simulation model generated under the three fundamental thermodynamic laws or on the Boltzmann superposition principle [

Comment 6. None of the five PR categories defined in [

The expressions (26) are valid in a very limited sense for Class II PRs

The following is a partial list of the cardinal difficulties that are directly attribu- table to their time dependence and that are associated with the use of viscoelastic PRs:

1) At least seven mutually independent PR definitions are available.

2) Difficulties stemming from the improper assembly of constitutive relations containing PRs in the time space resulting in combinations

for which no integral transform can obtained that leads to a proper EVCP expression. See Eqs. (33) for proper time space forms required to realize the needed double integral transform products and triple time function products (32).

3) Conversely independent assemblies in the integral transform space (FT and LT) lead to some combinations displayed in Eqs. (32) such that the triple pro- ducts have no physical counterpart in the real time space.

4) Additionally and separately viscoelastic PRs are also stress and stress history (i.e. path) dependent.

The Class II and VI PR formulations are predicated on the mathematically correct prescription of constant 1-D strain (II) or constant 1-D stress (VI), such that

where

The special

Consequently the Class II and VI mathematical models are well defined but loading (path) dependent. Unfortunately the actual physics of this loading problem are considerably less forgiving, as shown analytically in [

In addition, testing machine crossheads have inertia and require finite times to develop displacements and loads as demonstrated in [

The instantaneous loading concept can be modeled mathematically [

with four conditions on

The inherent difficulties associated with the

plus those of (45) and the more beneficial finite one

This effectively means that for the mathematical model, if the IC properly consists of an instantaneous impulse force

then it will propagate into the linear and nonlinear elastic and viscoelastic displacement solutions, such that

In view of consistency requirements one should expect no less than an infinite acceleration to produce an instantaneous impulse force

By contrast the quasi-static models differ from that dynamic ones essentially in the fact that after the completion of the time dependent loading cycle they re- main at a constant value. Let

With the mathematical model, one is again faced with a discontinuity due to the double valued Heaviside function at the temporal origin resulting in

Similar conclusions to those enunciated in Section 3.7.1 apply here as well.

Comment 7. In the final analysis the impulsive loading and sudden displa- cement initial problems are reminiscent of the non-existing sharp edged gusts, instantaneous penetration of atmospheric disturbances and zero time lift and drag buildups encountered in aeroelasticity. All such aerodynamically improbable physical models have been effectively corrected by the introduction of the three distinct delay functions of Theodorsen, Küssner and Wagner [

Refs. [

In Ref. [

sought, meaningful experiments need to be devised which can be solved analy- tically with symbolic values for the as yet unknown material parameters. Consi- der a “simple” 1-D tension or compression (without buckling) creep experi- ment (

The loading function

where

with

The time

The next two figure from [

Starting with the linear anisotropic viscoelastic constitutive relations (16) and specializing them to 1-D loading isothermal isotropic conditions yields

For a hypothetical constant stress

where

Eq. (58) is correct if and only if

If the loading cycle (55) is taken into account then Eq. (57) transforms into

and

Therefore, the error in strain responses generated by disregarding the starting stress transients is given by

and is shown in

However, a much more compelling case can be made for the linear relaxa- tion/creep functions or moduli/compliances, which are more pervasive and universal properties than specific strain responses [

The Class III PR integral transform definition,

which makes the expression independent of

with

for isotropic materials.

The normal stresses

Consequently, in all these instances the Class III PRs are also dependent on all non-zero stresses and on their histories as well as explicitly on time. Thus, different experimental determinations of PRs involving diverse stress and strain histories, lead to distinct PR functions for the same viscoelastic medium.

The neglect of the loading cycle in the period

Matching analyses for constant

Additionally, Category III PRs and their similar cousins carry with them an implicit contradiction. Consider a 1-D loading

when in fact it should read for

or

Eqs. (67) to (69) are identical if and only if the loading is 1-D and the stress is limited to

Furthermore, if the constitutive relations are of the linear non-convolution type then for the 1-D loading

and this PR is clearly implicitly stress and stress history (path) dependent, since generally

In summary, it is seen that

1) Eq. (63) is needed for the constructions of an EVCP in the transform space and its 1-D form is stress independent

2) Eq. (69) is the proper form of the constitutive relations in the real time space and is stress and stress history dependent

3) Proper care must be exercised in using and interpreting Eqs. (63) and (69) since they represent contradictory representations

Comment 8. While the preceding analyses establish Class III PR protocols that lead to the EVCP, they provide no clues as to what the proper accom- panying PR functions are. The PR analysis is built on the superposition of responses to 1-D loading(s) without specification as what it/they should be, i.e. constant stress, strain, their time derivatives, etc. It has been established that viscoelastic PRs, unlike moduli and compliances, are time, stress and stress history (path) dependent and hence non-unique. This ambiguity renders viscoelastic PRs unsuited for general material characterization. Consequently, constitutive relations and EVCPs based on universal path independent moduli and/or compliances remain the material descriptors of choice.

In

Similar differences in PR time function obtained in creep and relaxation were reported in [

Attempts at determining the time time

Comment 9. The great universal utility of elastic Poisson’s ratios due to their characteristically constant values for each homogeneous elastic mate- rial at a given temperature is lost in viscoelasticity. Conclusive experi- mental and analytical evidence undisputedly indicates that viscoelastic PRs are time, stress and stress history dependent and, therefore, not unique material descriptors. Consequently, they are not exportable or interchangeable from one loading condition to another.

Furthermore, the analysis demonstrates that the elastic-viscoelastic corres- pondence principle can only be expressed in terms of relaxation moduli or creep compliances and does not involve PRs. The equivalent triple product of Fourier transforms of PRs, moduli/compliances and strains/stresses necessary for the establishment of the EVCP, while mimicking Hooke's law, does not invert to a proper set of constitutive relations in the time space and, therefore, no EVCP involving PRs is possible.

Additionally and separately, analyses and experiments show that starting loading build ups, including achievable testing machine cross head speeds, are of significant importance to mandate their inclusion in material characterization protocols. Their neglect in the definitions of Class II and VI PRs, while mathematically defensible, are physically unattainable and unrealistic. Consequently, Class II and VI definitions only exist as idealized mathematical models but

cannot be duplicated physically for

Even if these two mathematical models were physically reproducible, the PRs so determined would only be applicable to the particular loading conditions under which they were experimentally determined. Consequently, the inherently specialized Class II and VI behaviors cannot be generalized and/or extended to any other loading processes since they produce process specific PRs as evidenced by

The pervasive advantages and disadvantages of viscoelastic PRs vis-à-vis the EVCP and their lack of universality are summarized in

For linear elastic and viscoelastic media:

1) Elastic PRs are of fundamental importance as linear material descriptors because they are constants for each material without regard as to what experi- mental protocol is used to determine their value. However, such is not the case of their linear viscoelastic counterparts as they are time, stress and stress history (path, process) dependent. Hence they have distinct values and functionality for each and every process. Consequently, experimentally obtained viscoelastic PRs do not match more complicated service loading conditions.

2) Elastic and viscoelastic PRs are inherently nonlinear, cf. (11) and (12). Their linear use is based on the explicit approximation of their linearity as governed by the ratio

3) All viscoelastic PRs are time dependent. Additionally, by definition PRs of all Classes, except VII, are time, stress and stress history (path, process) depen- dent and, therefore, non-unique, i.e. dependent on loading history. All consti- tutive relations involving PRs written in the time space are very computationally intensive compared to expressions involving only moduli or compliances.

4) The principal difference between PRs and moduli/compliances is that the former in all Classes except VII are an algebraic ratio of mutually perpendicular tension and/or compression strains, whereas the latter are ratios of stresses to their responding strains. This renders the PRs nonlinear functions of the two strains.

5) Unlike elastic PRs and whether in a homogeneous or non-homogeneous medium, all of the six Classes of viscoelastic Poisson’s ratios are represented by functions of time, stresses and stress histories. Due to their process dependency they are non-unique material descriptors and as such unsuitable for general material characterizations. Therefore, characterizations in terms of relaxation moduli and/or creep compliances are the characterization vehicles of choice as they represent universal functions with parameters specific to each material and while

Class | ADVANTAGES | DISADVANTAGES |
---|---|---|

I | none | does not lead to EVCP |

II | EVCP for | PR is restricted to process with |

only | a single time independent strain | |

III | leads to EVCP | equivalent PR in time space |

for some cases | must be properly defined | |

as PR is path dependent | ||

IV | none | nonlinear-no EVCP |

V | none | no EVCP |

VI | material dependent | no EVCP |

but only for | ||

VII | stress/strain independent | Eq. (32) makes use of |

all PRs very CPU intensive | ||

E, G, | stress independent unique | none |

K, C | material property functions, | |

isothermal EVCP |

Note: All PRs are time dependent. Additionally, by definition PRs of all Classes, except VII, are time, stress and stress history (path, process) dependent and, therefore, non-unique. All constitutive relations involving PRs written in the time space are very computationally intensive compared to expressions involving only moduli or compliances.

time dependent they are process independent.

6) The indisputable proof of the explicit time and path dependence of visco- elastic PRs can be found in experimental results, such as those of

7) Instantaneous non-zero loads, although mathematically rigorously defin- able by Dirac delta and/or Heaviside step functions, are physically unattainable. Consequently, the mathematically acceptable Class II and VI PR models are physically nonexistent and experimentally non-producible since they are solely based on one time independent stress or strain tensor.

8) Except in wave propagation problems, elastic loading histories are generally unimportant, since these are conservative materials without dissipation and memory. Viscoelastic materials, on the other hand, continuously dissipate ener- gy and loading histories are of prime significance to their responses.

9) When the proper loading cycles that are physically necessary to achieve constant values of one or more state variables are included in the analyses, sub- stantially different viscoelastic responses are achieved when compared to instan- taneous loading cycles.

10) Analyses establish that Class III PRs are capable to produce EVCP pro- tocols, on condition that physically acceptable double convolution integral constitutive relations are established (32b). However, these analyses fail to provide directions for establishing the unique, proper and necessary viscoelastic Class III PR functions of time, stress and stress history. Consequently, what process is to define the PRs “constant” strain, stress, their time derivatives, etc.? Hence, this inherent ambiguity renders viscoelastic path dependent non-unique PRs unsuited for general viscoelastic material characterization.

11) If viscoelastic PRs are to be used, then they must be obtained experi- mentally by loading and time processes that are exact detailed duplicates of the desired field conditions.

12) The Class VII is an artificially defined isothermal viscoelastic PR devoid of strain or stress associations and hence path independent while still remaining a time dependent universal material property descriptor. It obeys the construction mandated by Eqs. (32) necessary for the existence of EVCPs and does not suffer from the loading cycle problems associated with the other Classes. (See the next Item below.) However, its use in the time space is subject to intensive CPU use involving triple function products and double convolution integrals.

13) Since the viscoelastic PRs are time, stress and stress history dependent, unless the experimentally determined history exactly matches the application loading path no correspondence in properties can be established.

14) Some of the triple Fourier transform products, Eq. (32) necessary for establishing any PR based elastic viscoelastic correspondence principle have no physical real time counterpart constitutive relations, thus proving their nonexi- stence of the EVCP based on these PRs.

15) The evaluation of all the PR double convolution time integral constitutive relations stemming from the triple FT products of (32) are highly CPU intensive, thus impractical for use in analytical and/or finite element analyses. By contrast, constitutive relations without PRs but in terms of moduli or compliances lead to more efficient single convolution double product time integrals.

16) The validity of EVCPs without PRs based solely on moduli/compliances or on relaxation/creep functions remains unaffected and should be considered the universal unambiguous material characterization of choice.

17) If viscoelastic PRs are to be used in routine stress/strain analyses then their experimental determinations must exactly duplicate each and every actual service time and loading history.

18) In any viscoelastic formulation there is no need for the inclusion of PRs as all constitutive relation material properties (isotropic and anisotropic, homo- geneous and nonhomogeneous, linear and nonlinear) can be unambiguously characterized by their moduli and/or compliances thus removing any contro- versial descriptions due to the viscoelastic PR’s time and loading history dependence.

19) Finally, elastic solutions that are heavily dependent on PRs, such as thin plates, shells [

Hilton, H.H. (2017) Elastic and Viscoelastic Poisson’s Ratios: The Theoretical Mechanics Perspective. Materials Sciences and Applications, 8, 291-332. https://doi.org/10.4236/msa.2017.84021