Interconversion between Relaxation Modulus and ...

INTERCONVERSION BETWEEN RELAXATION MODULUS COMPLIANCE FOR VISCOELASTIC SOLIDS

AND

CREEP

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By S. W. Park,1 Associate Member, ASCE, and Y. R. Kim,2 Member, ASCE ABSTRACT: Methods of interconversion between relaxation modulus and creep compliance for linear viscoelastic materials are discussed and illustrated using data from asphalt concrete. Existing methods of approximate interconversion are reviewed and compared for their approximating schemes. A new approximate interconversion scheme that uses the local log-log slope of the source function is introduced. The new scheme is based on the concept of equivalent time determined by rescaling the physical time. The rescaling factor, which can be interpreted as a shift factor on a logarithmic time axis, is dictated by the local slope of the source function on loglog scales. The unknown target function at a given time is obtained by taking the reciprocal of the source function evaluated at an equivalent time. Although the method is developed using a mathematical relationship based on the power-law representations of relaxation modulus and creep compliance, the method is not limited to material functions characterized by power-laws but can be applied to general, non-power-law material functions as long as the relevant material behaviors are broadband and smooth on logarithmic scales. The new method renders good results especially when the log-log slope of the source function varies smoothly with logarithmic time.

INTRODUCTION It is well known that all linear viscoelastic material functions are mathematically equivalent and each function contains essentially the same information on the relaxation and creep properties of the material. Therefore, a linear viscoelastic material function can be converted into other material functions through appropriate mathematical operations. The need for interconversion arises for different reasons. For instance, it is difficult to run a constant-strain relaxation test on stiff materials while a constant-stress creep test is easier to carry out. In this case, the relaxation modulus can be obtained from the creep compliance through an interconversion. As another example, a test with steady-state harmonic excitation usually renders more accurate material behavior than a test with transient excitation, especially when the applied excitation is of low amplitude. In this case, transient material functions can be obtained from a frequency-dependent material function through appropriate interconversions. A host of interconversion methods, either exact or approximate, has been proposed and discussed by others. An extensive treatment of the subject has been given by, for example, Ferry (1980) and Tschoegl (1989). Approximate interrelationships between the relaxation modulus and creep compliance have been developed by a number of authors including Leaderman (1958), Denby (1975), and Christensen (1982). Hopkins and Hamming (1957) presented a method based on the numerical evaluation of a convolution integral relating the relaxation modulus to the creep compliance. However, no particular work that compares the relative performance of these methods using actual material data has been brought to the writers’ attention. In this paper, some of the existing methods of interconversion between the relaxation modulus and creep compliance are reviewed and analyzed, and the performance of each method is assessed using available data from asphalt concrete. A new approximate interconversion scheme that uses the concept of 1 Res. Sci., School of Mech. Engrg., Georgia Inst. of Technol., Atlanta, GA 30332. 2 Assoc. Prof., Dept. of Civ. Engrg., North Carolina State Univ., Raleigh, NC 27695. Note. Associate Editor: Jan Olek. Discussion open until July 1, 1999. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on June 23, 1997. This paper is part of the Journal of Materials in Civil Engineering, Vol. 11, No. 1, February, 1999. 䉷ASCE, ISSN 0899-1561/99/0001-0076 – 0082/ $8.00 ⫹ $.50 per page. Paper No. 16053.

equivalent time (or logarithmic time shift) is introduced, and its performance is compared with other existing methods. According to the new method, the (unknown) target function at a given time is evaluated by taking the reciprocal of the (known) source function obtained at an equivalent (or rescaled) time. The method has its basis in the interrelationship between the relaxation modulus and creep compliance when both functions are represented by simple power laws. Data from uniaxial creep and relaxation tests on asphalt concrete are used in illustrating the existing and new interconversion methods. INTERCONVERSION BASED ON EXACT RELATIONSHIPS Relaxation modulus and creep compliance for a viscoelastic material are not each other’s reciprocal; instead, they are related, for instance, by a convolution integral (Ferry 1980)

冕

t

E(t ⫺ ␶)D(␶) d␶ = t

for t > 0

(1)

0

where E(t) = relaxation modulus; and D(t) = creep compliance. Taking the Laplace transform of (1), one finds 1 ¯ ¯ E(s)D(s) = 2 s

(2)

where ¯f(s) ⬅ 兰 ⬁0 f (t)e⫺st dt denotes the Laplace transform of f (t); and s = transform parameter. Integral equation (1) may be solved numerically using an appropriate numerical integration scheme. This approach is useful especially when an analytical form of the source function is not available but the function is represented only by a set of data. The range of integration is divided into subintervals that are small enough that the variation of the integrand within each subinterval can be approximated by an appropriate mean value. A recursive formula can be established from which the target function can be evaluated. Kim and Lee (1995) used this approach to find the relaxation modulus from experimentally determined creep compliance of asphalt concrete. Several different numerical integration schemes relative to (1) are reviewed and discussed in Appendix I. The relation (2) between Laplace-transformed relaxation and creep functions is useful in directly evaluating one transformed function when the other is known at a particular value of transform parameter. The target transient function at a particular time can then be evaluated by means of an appropriate Laplace transform inversion technique [e.g., Schapery (1962)].

76 / JOURNAL OF MATERIALS IN CIVIL ENGINEERING / FEBRUARY 1999

J. Mater. Civ. Eng. 1999.11:76-82.

APPROXIMATE INTERCONVERSION METHODS

D(t) ⬵

Several approximate methods of interconversion between transient relaxation and creep functions are available in the literature. Mostly, they are based on different schemes of adjustment of the elastic-like, reciprocal relationship between the modulus and compliance. A few of these methods are reviewed next. The mathematical property involved in the approximating scheme and the accuracy of each method are discussed and compared later.

E(t) ␲2t 2 E (t) ⫹ 4 2

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E(t)D(t) ⬵ 1

for t > 0

(3)

The relation (3) is not acceptable for many linear viscoelastic materials; however, for weakly viscoelastic materials, (3) provides a good approximation.

E(t)D(t) ⬵

(4)

D(t) = D1t n

(5)

where E1, D1, and n = positive constants. From (4) and (5), the following relation results (see Appendix II): E(t)D(t) =

sin n␲ n␲

(6)

Eq. (6) was first given by Leaderman (1958). It is to be noted that power-law functions (4) and (5) are graphically represented by straight lines on log-log scales, and the exponent n is identified with the (absolute) slope of these lines. When n approaches zero (i.e., for an elastic material), the right-hand side of (6) becomes unity. It has been known that (6) is a good approximate interrelationship between E(t) and D(t) when both functions do not exactly behave according to (4) and (5) but behave smoothly when plotted on log-log scales. In this case, n is the local loglog slope of the source function, that is n=

冏

冏

d log F(␶) d log ␶

Denby (1975), based on an approximation of a convolution integral relationship between the relaxation modulus and creep compliance such as (1), proposed the following interrelationship:

where F(␶) = source function, either E(␶) or D(␶); and 兩 兩 denotes the absolute value symbol. Note that n is now a function of t. The relationship (6) is very accurate in regions in which E(t) and D(t) are represented approximately by straight lines on log-log scales. Again, (6) is exact when both E(t) and D(t) are described by global power laws, such as (4) and (5). Interrelationship by Christensen (1982) Christensen (1982), by means of approximate relationships between the real and imaginary parts of a complex material function and between the transient function and the real part of the complex material function, developed an approximate interrelationship between the relaxation modulus and creep compliance of the following form [e.g., taking D(t) as the target function]:

1 n2␲ 2 1⫹ 6

(10)

where n is defined in (7). Comparison between (6), (9), and (10) It is shown that (6), (9), and (10) depend on n, the local slope of the source function on log-log scales. The behavior of each right-hand side of said equations as a function of n is shown graphically in Fig. 1. When expanded in series, the relations (6), (9), and (10) can be compared more directly. Using the Taylor series expansions, sin x = x ⫺ x3/3! ⫹ x5/5! ⫺ ⭈ ⭈ ⭈ and (1 ⫹ x)⫺1 = 1 ⫺ x ⫹ x2 ⫺ x3 ⫹ ⭈ ⭈ ⭈ (for 兩x兩 < 1), one can obtain the following relations: sin n␲ = n␲ =1⫺

n␲ ⫺

(n␲)3 (n␲)5 ⫹ ⫺ ⭈⭈⭈ 6 120 n␲

(n␲)2 (n␲)4 ⫹ ⫺ ⭈⭈⭈ 6 120

(11) 2

4

1 (n␲) (n␲) =1⫺ ⫹ ⫺ ⭈⭈⭈ n2␲2 4 16 1⫹ 4 1 (n␲)2 (n␲)4 ⫹ ⫺ ⭈⭈⭈ 2 2 = 1 ⫺ n␲ 6 36 1⫹ 6

(7)

at ␶=t

(9)

Interrelationship by Denby (1975)

E(t)D(t) =

E(t) = E1t ⫺n

1 n2␲ 2 1⫹ 4

A differential operator property, d (ln f ) = df/f, is used in deriving (9) from (8).

Power-Law-Based Interrelationship It is well known that, for many linear viscoelastic materials, the relaxation modulus and creep compliance are approximately represented by simple power laws over their transition zones. Consider the following forms of power laws for relaxation modulus and creep compliance, respectively:

(8)

2

The form (8) also applies when D(t) and E(t) are interchanged. It is informative to note that (8) can be transformed into the following alternative form in terms of n defined in (7)

Quasi-Elastic Interrelationship The most intuitive and crude interrelationship between the relaxation modulus and creep compliance is the one based on quasi-elastic approximation

再 冎 dE(t) dt

(12)

(13)

In view of Fig. 1 and according to (11) – (13), (6) is very close to (10) for small values of n (