# Interconversions Between Linear Viscoelastic Functions by Using the Relaxation-Creep Duality Representation

Interconversions Between Linear Viscoelastic Functions by Using the Relaxation-Creep Duality Representation

Abstract

The interconversions between relaxation moduli and creep compliances including stretch, shear, bulk parts, and time-dependent Poisson’s ratio are derived by using the relaxation-creep duality representation. The relaxation-creep duality representation for the viscoelastic functions introduced in this paper is composed by an exponential function which characterizes the relaxation behavior and a complementary exponential function which characterizes the creep one. All viscoelastic functions can be represented as the same form. The new sets of coefficients between viscoelastic functions obey the elastic-like interconversions, and do not involve the characteristic times. The relationships of characteristic times between those functions are also derived. These interconversion formulas can then be calculated easily. Three literatures are referred to calculate the consistency of the viscoelastic functions via the new interconversions introduced in this work. The Young’s relaxation modulus in one literature is not consistent to the shear one in another literature. By assuming a constant bulk modulus, the modified Young’s relaxation modulus and time-dependent Poisson’s ratio that derived by the new interconversions can meet the measured curves and can be consistent to the shear creep compliance in the literatures.

1. Introduction

Typical electronic package is composed by different components, including molding compound, solder joints, copper traces, polymers, and silicon. Each material has its own constitutive behavior, like elasticity, plasticity, and viscoelasticity [1]. Viscoelasticity is an interesting property that combines both effects of elasticity and viscosity. The property leads the material behavior not only like a solid, but also a fluid [2].

Sham et al. [3] studied the numerical analysis of electronic integrated circuit package. They found that the elastic model overestimates the stress concentrations in the underfill fillet around the die corner than the viscoelastic model, and concluded that the viscoelastic nature of underfill material should be properly taken into account if the failure of solder joints and the lifetime of the package are to be accurately estimated. The effects of underfill and its material models, including constant elastic model, temperature dependent elastic model, and viscoelastic model, on thermomechanical behaviors of flip chip package were compared by Chen et al. [4]. It is found that the viscoelastic model gives comparatively large plastic strain range, big displacements in the shear direction, and sequentially low solder joint lifetime. Thus it is important to use the viscoelastic model to describe the material behavior of underfill rather than the pure elastic one.

On the other hand, most mold compounds exhibit viscoelastic behavior even at room temperature, even though these polymers are filled as much as 90% by weight with silica filler particles. Kim et al. [5] studied the warpage mechanism by simulating a strip type packaging for plastic grid array, and compared the elastic and viscoelastic models for molding compound and substrate. They found that the relaxation behaviors of the molding compound and the substrate materials had significant effect on the warpage development. The viscoelastic calculations presented non-monotonic warpage development, and predicted the bigger deformations than the elastic solutions. Miyake et al. [6] also found that the viscoelastic calculation with relaxation of shear modulus and bulk modulus is the most appropriate method for predicting the warpage for the thin small outline packages, and the elastic method may result in a false.

Other materials that often used in semiconductor industry were also studied. Chen et al. [7] studied a series of tensile and three-point bending tests at various temperatures and loading rates using a commercial polymethyl methacrylate (PMMA). They concluded that based on their calculated apparent activation energies, it is suggested that the failure process of thermoplastic polymers follows a viscoelastic process, either glass or transition. Inoue et al. [8] analyzed the influential factors in determining the adhesive strength of anisotropic conductive films (ACF) joints. They found the peel strength of the joints exhibits a significant dependence on test speed, due to the viscoelasticity of the binder. All the tests suggested that the viscoelastic property should be considered for the semiconductor materials.

Various viscoelastic constitutive models allow us to analyze the time-dependent relaxation or creep behavior of materials. The commonly used models are the generalized Maxwell and the generalized Voigt models [2]. The generalized Maxwell model, which also called the Weichert model, is connected by many Maxwell elements in parallel, while the generalized Voigt model, which also called the Kelvin model, is connected by many Voigt elements in series. The successful point is to introduce the multiple characteristic times, thus both generalized models can describe the complicated curves. However, the induced difficulty is that the relationship between relaxation modulus and creep compliance is hard to find. One way is to do a measurement for the relaxation modulus [7], and to do another measurement for the creep compliance [8] independently. The two sets of fitted parameters thus have no relationship. Moreover, for the 3-dimentional cases, the relaxation moduli and creep compliances include stretch, shear, and bulk parts. The Poisson’s ratio of viscoelastic material is also a time-dependent property [9]. The relationships between stretch, shear, bulk moduli/compliances, as well as time-dependent Poisson’s ratio in generalized Maxwell and generalized Voigt models are very complex, even can’t be solved analytically.

The numerical methods for the viscoelastic interconversions were studied in abundance, but only focused on the integral relationship between the relaxation modulus and creep compliance. As early as 1957, Hopkins and Hamming [10] analyzed the numerical interconversion between creep and relaxation in the time domain. They divided the range of time domain into a finite number of intervals and applied the trapezoid rule to carry out the integration within each interval. Knoff and Hopkins [11] improved this method, and found that the better approximation is obtained by the assumption that both functions can be assumed to be linear within a series of increasing time intervals which do not change as the calculation progresses. Baumgaertel and Winter [12] treated the relation between discrete retardation and relaxation spectra in the Laplace transform domain (s-domain). The relation of coefficients between relaxation and retardation functions was determined analytically. However, the interconversion forms are very complex and thus hard to calculate. Other numerical methods for the interconversions between relaxation and creep functions can be found in the literatures [13-18]. Note that the interconversions for other viscoelastic functions, for example, Young’s relaxation modulus and time-dependent Poisson’s ratio, have not found in the literatures.

In this work, we attempt to analytically derive the relationships between relaxation moduli and creep compliances by using the relaxation-creep duality representation, including stretch, shear, bulk parts, and time-dependent Poisson’s ratio. The new sets of coefficients between viscoelastic functions obey the elastic-like interconversions, and do not involve the characteristic times. The relationships of characteristic times between those functions are also derived. These interconversion formulas are very easy to calculate. Once the parameters of a viscoelastic function can be obtained, other functions can also be obtained via these relationships.

The outline of this paper is summarized as below. Section 1 gives the introduction to the viscoelastic interconversion problem. The relaxation-creep duality representation and the interconversions between viscoelastic functions are given in the Section 2. Section 3 presents the applications of the new interconversions to the different measurements of PMMA, and describes how to obtain the reasonable viscoelastic parameters. The conclusion is concluded in Section 4. The symbol descriptions are summarized in Table 1.

2. Viscoelastic model

2.1 Linear viscoelastic constitutive equations

For an isotropic elastic material, the constitutive equation for stress and strain can be written as the Hooke’s law [1]:

, (1)

with initial conditions

, . (2)

One can define the deviatoric stress and deviatoric strain,

, , (3)

so that the shear part and dilatational part can be separated:

, , (4)

where the Einstein summation convention is used. The shear part is useful for the later discussed viscoelasticity in which the Newtonian shear viscosity is added to the shear component [19].

The linear viscoelastic correspondence principle states that [9], if an elastic solution to a stress analysis problem is known, substitution of the appropriate complex-plane transforms for the elastic quantities supplies the viscoelastic solution to the same problem in the transformed plane. Thus, the stress-strain relations of the theory of elasticity become relations between the Laplace-Carson transforms of the stress and strain in which the moduli and compliances are replaced by the corresponding relaxances and retardances. The corresponding quantities for moduli and Poisson’s ratio are

, , (5)

, (6)

, (7)

and for the compliances are

, , , (8)

where the Laplace-Carson transform is used,

. (9)

The consistent equations of Eqs. (6) and (7) can be immediately obtained by the convolution theory, respectively,

, (10)

. (11)

The relaxances and retardances have the relationships:

, , , (12)

which corresponding to the consistent equations between relaxation and creep functions,

, , . (13)

The quantities , , and are called the stretch, bulk, and shear relaxances, respectively, while the quantities , , and are called the stretch, bulk, and shear retardances, respectively. By using the inverse Laplace transform for the relaxances and retardances, the corresponding relaxation and creep functions can then be obtained.

Several definitions of viscoelastic Poisson’s ratio can be found in the literatures [20]. The viscoelastic Poisson’s ratio obtained from the viscoelastic correspondence principle (Eq. (7)) coincides the Class II definition of Hilton’s classification. Other definitions may lead to the nonlinear dependence on stresses and on loading histories even in linear viscoelastic media [21]. Thus the time-dependent Poisson’s ratio obtained from the viscoelastic corresponding principle, i.e., the Class II definition of Hilton’s classification, is adopted here. Due to the creep-like property of time-dependent Poisson’s ratio, the quantity is called the Poisson’s retardance [9].

Appling the linear viscoelastic correspondence principle to the elastic constitutive equations, one can get

, . (14)

The time-dependent viscoelastic constitutive equations can be obtained by using the inverse Laplace transform,

, , (15)

where the dot denotes the time-differential operation with respective to . The above formal equations are called the Boltzmann causal integral equations (BCIEs) with memory kernels and , which define how the stresses at time t depend on the earlier history of the strain rates via the memory kernels [22].

Another useful representation of viscoelasticity is to use the retardances to represent the constitutive equations,

, . (16)

The corresponding BCIEs are

, . (17)

2.2 Relaxation-creep duality representation for the 2nd-order time-differential constitutive equation

A linear thermodynamic derivation was given by Biot [23] for representation of a system having viscoelasticity by means of a potential and dissipation function familiar in Lagrangian mechanics. The viscoelastic problem can be viewed as an eigen-value problem, and the general solutions of relaxation moduli and creep compliances can be calculated as the sum of the exponential decay functions. By using the Laplace transform for the sum of these functions, the relaxances and retardances can be represented as the rational polynomial functions. Using the viscoelastic corresponding principle and taking the inverse Laplace transform, the high-order time-differential constitutive equation (TDCE) can then be obtained.

The general 2nd-order TDCE can be written as

, (18)

, (19)

where are the coefficients, and K is the pure-elastic bulk modulus. Since the viscoelastic bulk modulus changes much less with time than the shear modulus, the viscous effect of bulk modulus is here assumed to be neglected [24].

By taking the viscoelastic corresponding principle, one can obtain the shear relaxance,

, (20)

where

, , , (21)

, . (22)

The inverse of characteristic times, , are the roots of the 2nd-order polynomial in the denominator. Eqs. (22) are called the Vieta’s formulas [25], which link the coefficients of a polynomial to sums and products of its roots. Note that the characteristic times are all positive real numbers. Hereafter the Vieta’s formulas will be used to represent the polynomials. The characteristic times are determined by the coefficients of the deviate stress and its differentials. The independent parameters can be chosen as . The shear relaxation modulus with relaxation-creep duality form can be restored by taking the inverse Laplace transform,

, (23)

where

, , (24)

, . (25)

The fractions and have the relationships,

, , (26)

and the complementary exponential function is defined as

. (27)

The relaxation-creep duality representation is described by five parameters: the instantaneous shear modulus, ; the modulating shear modulus, , to modulate the fractions ; the permanent shear modulus, ; and characteristic times . The function is a constant function if ; a relaxation modulus if ; and a creep compliance if . The duality of relaxation and creep is described by the exponential and complementary exponential function pair, where the relaxation contribution is characterized by the exponential function while the creep contribution is characterized by the complementary one.

2.3 Interconversions between viscoelastic functions

The interconversions between viscoelastic functions can be derived by using the relationships between relaxances and retardances. For the shear retardance, it can be obtained from the inverse of shear relaxance,

, (28)

where

, , , (29)

, . (30)

The shear creep compliance can then be restored as the relaxation-creep duality form by using the inverse Laplace transform,

, (31)

where

, , (32)

, . (33)

The fractions and also have the relationships,

, . (34)

One can see that if , then , i.e., the relaxation property of shear modulus guarantees the creep property of shear compliance.

If the shear relaxation and creep functions are both expressed as the Prony series,

, , (35)

where the coefficients of Prony series are

, , , . (36)

One can prove that the inverse relationships are not available between and . Instead, in the relaxation-creep duality representation, the inverse relationships between and are hold.

For the Young’s relaxation modulus, by using Eq. (6), the same form of stretch relaxance can be obtained,

, (37)

where

, , , (38)

, . (39)

The Young’s relaxation modulus can then be restored as the relaxation-creep duality form by using the inverse Laplace transform,

, (40)

where

, , (41)

, . (42)

The fractions and also have the relationships,

, . (43)

One can see that if , then , i.e., the relaxation property of shear modulus guarantees the relaxation property of Young’s modulus.

Similar to the shear creep compliance, the stretch retardance can be obtained from the inverse of stretch relaxance,

, (44)

where

, , , (45)

, . (46)

One can see that the characteristic times of stretch creep compliance are same as the ones of shear creep compliance. It means that the creep trends are synchronous for both stretch and shear creep compliances, but the relaxation trends are not synchronous for both stretch and shear relaxation moduli. The reason of the synchrony between stretch and shear creep compliances can be understood from the inverse of Eq. (6). The denominators of both stretch and shear retardances are same as , thus having the same characteristic times. Note that the synchrony between stretch and shear creep compliances is still hold for any order TDCE.

The stretch creep compliance can then be restored as the relaxation-creep duality form by using the inverse Laplace transform,

, (47)

where

, , (48)

, . (49)

The fractions and still have the relationships,

, . (50)

The time-dependent Poisson’s ratio can also be obtained by using Eq. (7),

, (51)

where

, , , (52)

, . (53)

One can see that the characteristic times of time-dependent Poisson’s ratio are same as the ones of Young’s relaxation modulus. It means that the trends of Young’s relaxation modulus and time-dependent Poisson’s ratio are synchronous. The reason of the synchrony between Young’s relaxation modulus and time-dependent Poisson’s ratio is due to the denominators of Eqs. (6) and (7) are same as , thus having the same characteristic times. Note that the synchrony between Young’s relaxation modulus and time-dependent Poisson’s ratio is still hold for any order TDCE.

Again, if is assumed, then , i.e., the relaxation property of shear modulus guarantees the creep property of time-dependent Poisson’s ratio.

The time-dependent Poisson’s ratio can then be restored as the relaxation-creep duality form by using the inverse Laplace transform,

, (54)

where

, , (55)

, . (56)

The fractions and also have the relationships,

, . (57)

From the above derivations, the relationships between elastic constants are hold for the viscoelastic coefficients in the relaxation-creep duality representation. All the viscoelastic functions have the same form, which includes the instantaneous, modulating, and permanent constants, and two characteristic times. These features are the intrinsic properties of the general 2nd-order TDCE. Note that the above viscoelastic functions are all consistent to each other.

2.4 Relaxation-creep duality representation for the Nth-order time-differential constitutive equation

The general Nth-order TDCE can be written as

, (58)

, (59)

where are the coefficients, and K is again the elastic bulk modulus. By taking the viscoelastic corresponding principle, one can get the shear relaxance,

, (60)

where

, ,…, , (61)

, ,…, . (62)

The inverse of characteristic times, , are the roots of the Nth-order polynomial in the denominator, and the Vieta’s formulas (Eqs. (62)) are also used. The characteristic times are determined by the coefficients of the deviate stress and its differentials. The independent parameters can be chosen as . The shear relaxation modulus with relaxation-creep duality form can be restore by taking the inverse Laplace transform,

, (63)

where are the fractions that calculated by , , , and (see Appendix A for the detailed calculation), and have the property,

. (64)

Other viscoelastic functions can be represented as the same form. For example, the Young’s relaxance can be written as

, (65)

where

, ,…, , (66)

,…, . (67)

From Eqs. (66), it can be found that all the modular constants observe the interconversions like the elastic one. The roots finding of the polynomial of degree 5 or greater for can’t be solved analytically since there can be no general formula, involving only arithmetic operations and radicals, that expresses the roots of a polynomial of degree 5 or greater in terms of its coefficients. However, one can use the Newton-Raphson method or other numerical methods to obtain the approximated solutions. Once the relationships of modulating constants and characteristic times between any two viscoelastic functions are solved, the unknown viscoelastic function can then be obtained from the known one with a constant bulk modulus.

3. Viscoelasticity measurements of PMMA and their relationships

In order to show the power of the derived interconversions, three literatures are referred. The first literature is the work of Lu et al. [26], where the uniaxial and shear relaxation, and Poisson creep, as well as their conversion to bulk relaxation for PMMA were studied. In their work, the strip specimens were used to measure the uniaxial modulus and Poisson’s ratio in relaxation with an image moiré method, while the tubular specimens were used to measure the uniaxial and the shear relaxation moduli as monitored with the digital image correlation method. The second literature is the work of Lu et al. [8] in which the measurement of creep compliance of PMMA was performed by nanoindentation with Berkovich indenter. And the third literature is the work of Huang et al. [7] in which the measurement of Young’s relaxation modulus of PMMA was also studied by nanoindentation with Berkovich indenter.

The fitting result of Young’s relaxation modulus in the third literature [7] was shown as,

, (68)

where the units of time and modulus are s and GPa, respectively; and the superscript P indicates the coefficients of Prony series. The instantaneous stretch modulus can be calculated as,

. (69)

For the shear creep compliance in the second literature, the fitting result was,

, (70)

where the units of time and compliance are s and GPa-1, respectively; and the superscript P still indicates the coefficients of Prony series. The permanent shear compliance can be calculated as,

. (71)

To obtain the shear relaxation modulus, the fractions are firstly calculated,

, , . (72)

The results can be checked by .

The next step is to calculate the fractions from the fractions as shown in the appendix,

, , . (73)

The results can be checked by .

The shear modulating compliances can then be obtained,

, (74)

. (75)

Finally the shear relaxation modulus and its parameters can be derived as,

, (76)

, (77)

, (78)

where the units of time and modulus are s and GPa, respectively.

To see the reasonability of the calculated shear relaxation modulus, one can compare the logarithm value of with the Fig. 15 in the first literature [26],

. (79)

It can be seen that this value coincides with the initial value of the shear relaxation modulus in the Fig. 15 of Ref. [26]. However, the calculated permanent modulus is higher than the measured value in the Fig. 15 of Ref. [26]. The reason is that in the Fig. 6 of Ref. [8], the fitting limit was about 750 nm, thus the long-time behavior was not characterized well. Hereafter, we will focus on the short-time behavior.

Once the shear and stretch relaxation moduli are obtained, the bulk modulus and time-dependent Poisson’s ratio can then be estimated. Before to do it, the modulating moduli should be calculated firstly,

. (80)

The parameters for the time-dependent Poisson’s ratio can then be calculated as,

, . (81)

It can be seen that the instantaneous value of time-dependent Poisson’s ratio is very small. Furthermore, the permanent value is just 0.266, which means the function is a creep-like one, and the maximal value does not exceed 0.3 as adopted in the Refs. [7] and [8].

On the other hand, the bulk modulus can be calculated from the stretch and shear constants as,

. (82)

The four values are not equal due to the Young’s relaxation modulus and shear relaxation modulus are not consistent. However, if we take the average of these four values as the constant bulk modulus, i.e.,

, (83)

the modified Young’s relaxation modulus and its parameters can then be obtained from the constant bulk modulus and the above shear relaxation function,

, (84)

, (85)

. (86)

The function versus time is drawn in the Fig. 1. One can see that the modified Young’s relaxation modulus is very close to the original one as shown in the Fig. 5 of Ref. [7]. The modified Young’s relaxation modulus can not only meet the request of the constant bulk modulus, but also be consistent with the shear creep compliance that measured in the Ref. [8].

If we further calculate the parameters of time-dependent Poisson’s ratio, we can find,

, . (87)

The time-dependent Poisson’s ratio is still smaller than 0.3 by seeing the instantaneous and permanent values.

To search the reasonable value of the above parameters, one may refer to the Fig. 9 of Ref. [26]. The initial bulk modulus shown in the figure is about . If one takes the value as the constant bulk modulus, one can see that the value along the time is always within the error bounds. Thus the constant bulk modulus and the above shear relaxation modulus are used to re-calculate the reasonable Young’s relaxation modulus and time-dependent Poisson’s ratio. The results are shown as below,

, (88)

, . (89)

The modified Young’s relaxation modulus is shown in the Fig. 1. Compare the Fig. 1 to the Fig. 5 of Ref. [7], one can see that the modified Young’s relaxation modulus is more close to their conventional test, i.e., the Ref. [26]. Furthermore, the instantaneous Poisson’s ratio is also close to the result of the Fig. 6 of the same reference. Thus, by using the constant bulk modulus and the fitted shear creep compliance from the Ref. [8], the calculated Young’s modulus as well as the time-dependent Poisson’s ratio can be consistent with the results of the Ref. [26].

Due to the long-time behavior was not characterized well in the Ref. [8], the long-time behaviors of the calculated viscoelastic functions are also not described well. However, one may predict that if the permanent Poisson’s ratio is 1/2, the permanent shear and stretch moduli should be zero as shown in the 3rd equation of Eqs. (52) and the 3rd equation of Eqs. (38). Such a material is a thermoplastic material as the PMMA should be. The mathematical form of the shear creep compliance can be referred to the Appendix B.

4. Conclusion

In this work, the relaxation-creep duality representation for the viscoelastic functions is introduced, and the interconversions between stretch, bulk, and shear relaxation and creep functions, as well as time-dependent Poisson’s ratio are also derived. The relaxation-creep duality representation is composed by an exponential function and a complementary exponential one. One characterizes the relaxation behavior, another characterizes the creep behavior. If the coefficient of the exponential function is great or less than the one of the complementary exponential function, then the viscoelastic function will be a relaxation-type or creep-type function, respectively. Thus the time-dependent Poisson’s ratio is a creep-type viscoelastic function due to the instantaneous Poisson’s ratio is less than the permanent one.

The relaxation-creep duality representation of Nth-order time-differential constitutive equation of viscoelastic materials has independent parameters: an instantaneous constant, the modulating constants, a permanent constant, and the N characteristic times. Including the bulk modulus, the number of independent parameters of an Nth-order time-differential viscoelastic material is . The interconversions of instantaneous, modulating, and permanent values between stretch, bulk, shear relaxation moduli and creep compliances, and Poisson’s ratio are similar to the interconversions between the pure-elastic constants. And these values have clearly physical meanings to characterize the viscoelastic curves. The elastic-like constants including the instantaneous, modulating, and permanent constants are very useful since they obey the usual relationships between the pure-elastic constants. These relationships of elastic-like constants are independent to the characteristic times. On the contrary, the relationships of Prony coefficients between the different viscoelastic functions are not simple, and hard to find the analytic form.

Three literatures are referred to calculate the consistency of the viscoelastic functions via the new interconversions introduced in this work. By assuming a constant bulk modulus, the modified Young’s relaxation modulus can meet the measured curve. If one further requires the reasonable value of the time-dependent Poisson’s ratio, the bulk modulus estimated from the Ref. [26] can be used, and the calculated results lead to the consistency between viscoelastic functions obtained from the Refs. [26] and [8].

The relaxation moduli as well as creep compliances can be obtained when the viscoelastic parameters are properly measured or calculated. The material behaviors can then be understood by the features of viscoelastic functions. By using the viscoelastic correspondence principle, the time-dependent constitutive equations can be written down. Solving the time-dependent constitutive equations by the analytic calculation or numerical simulation, the mechanics of this viscoelastic material can then be determined.

Appendix A: Calculation of coefficients

Let be an arbitrary viscoelastic function, the relaxation-creep duality representation for the function can be written as

, (A1)

where and are the instantaneous and permanent constants, respectively; are the characteristic times; and are the coefficients which will be calculated in this appendix.

Representing the viscoelastic function in the Laplace transform domain,

, (A2)

where are the modulating constants, and are the numerator part and denominator part of the rational function , respectively. One can factorize to the form,

. (A3)

The constants are the coefficients of Prony series, and have the relationships to as

. (A4)

Taking the inverse Laplace transform, the Prony series representation of this function can be obtained.

Expanding the Eq. (A3) and comparing to the Eq. (A2), one can obtain the relations,

, (A5)

, (A6)

, (A7)

…,

, (A8)

. (A9)

Defining the fractions as

, ,…, , (A10)

one can solve the coefficients of Prony series as

, (A11)

where

, (A12)

. (A13)

Dividing by , the fractions can then be obtained. Note that the relationships between and are linear. One can use the matrix form to solve the from , or to solve the from inversely. The correctness of the results can be checked by

, . (A14)

Another convenient calculation is the residue method [27]. Recall Eqs. (A2) and (A3), the coefficients of Prony series are just the residues,

, (A15)

where the l'Hôpital's rule is used, and

. (A16)

However, the coefficients of Prony series between different viscoelastic functions have no simple interconversions. Instead, the interconversions of modulating constants between different viscoelastic functions obey the relations similar to the pure-elastic constants.

Appendix B: Degenerated viscoelastic functions

For a thermoplastic material, the permanent shear or stretch modulus is usually set to zero, and the corresponding permanent compliance is thus infinite. The creep compliance in this situation thus evolves linearly as time. We call such viscoelastic functions as degenerated. In this appendix, the relationship between the relaxation modulus with zero permanent modulus and the degenerated compliance with infinite permanent compliance is introduced.

Using the 2nd-order time-dependent constitutive equation as an example, and setting the permanent shear modulus as zero, one can see that the shear retardance is as the form,

, (B1)

where

, , (B2)

, . (B3)

The shear creep compliance can thus be solved by taking the inverse Laplace transform,

, (B4)

where

. (B5)

One can see that the shear creep compliance includes a linear term, a relaxation term, and a creep term. The quantity is not same as , but a finite permanent compliance of the creep function without the linear term. The actual value of is infinite due to the linear term. It should be noted that the number of independent parameters is four because of the zero permanent modulus.

Acknowledgement

The authors would like to thank Assistant Professor Tz-Cheng Chiu (Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan) for giving advising.

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Symbol Description

σij Stress component

εij Strain component

Sij Deviatoric stress component

eij Deviatoric strain component

E Young’s modulus

ν Poisson’s ratio

G Shear modulus

K Bulk modulus

t Time

s Laplace transform parameter