biaxial stress-relaxation in skin

A.NNALS OF BIOMEDICAL ENGINEERING 4, 250-270 (1976)

Biaxial Stress-Relaxation in Skin

Y. LANIR

Department of Biomedical Engineering, Technion-Israel Institute of Technology, Haifa, Israel

Received June 9, 1976

Results of 58 biaxial and uniaxial stress-relaxation tests on flat specimens of abdominal rabbit skin are analyzed. It is shown that rabbit skin is a nonlinear viscoelastic material, i.e., the relaxation nmde of each stress component depends on the strain (or alternatively on the initial stress). The dependence of the relaxation mode on the initial stress is investigated in terms of a modified power law with three parameters (Eq. (15)) which are functions of the initial stress. The differences and similarities between uniaxial and biaxial stress relaxation tests and the effect of the tissue "preconditioning" is discussed. On the basis of the results a theory is developed for the uniaxial viscoelastic behavior of rabbit skin tissue when small incremental strains are superimposed on large constant deformation.

INTRODUCTION

The mechanical functioning of a living organism depends primarily on the rheological behavior of its cons t i tuents - - the various organs and tissues. I t is therefore of utmost importance that the rheological properties of tissues be well established. I t is a custom in the field of biomechanics to investigate the passive mechanical properties of tissues with the well-developed concepts of material sciences. Hence living tissues are discussed as being elastic, viscoelastic, plastic, etc., in the same way as industrial materials.

A lot of work has already been done along this line. I t is generally agreed that biological tissues (and soft tissues in particular) are anisotropic, nonlinear viscoelastic materials, i.e., their mechanical behavior is different in different directions, and the stress-strain relations are not linear and time dependent. The particular nature of the tissue's anisotropy, nonlinearity, and time dependence is not yet well established in many cases.

5Iost of the work on soft biological tissues has been confined to uniaxial investigations, much in the same way as in the case of metals and other industrial materials. Soft biological tissues differ, however, from metals in more than one way. In particular, they are capable of large reversible deformations compared to the infinitesimal reversible deformations in most metals. Under these conditions results of uniaxial tests cannot be generalized to multiaxial conditions. Since our organs function as three-dimensional bodies, we are thus required to investigate them under multiaxial conditions.

Copyright �9 1976 by Academic Press, Inc. All rights of reproduction in any form reserved.

250

BIAXIAL STRESS RELAXATION IN SKIN 251

In a previous report (Lanir and Fung, 1974b) it is shown that biaxial mechanical tests are needed to obtain comprehensive theological characterization of the skin tissue. Uniaxial tests are often very difficult to interpret. For example, the transverse configuration of a specimen of skin changes continuously during uniaxial stress relaxation; hence such a test cannot be regarded as a true relaxation test. Furthermore, in cyclic loading of a specimen subjected to uniaxial stress the relaxed length ("natural length") of the specimen is often found to change continuously, which is another feature attributable to the inability of the specimen to control its transverse dimension. In two-dimensional tests (with controlled transverse dimension or load), a unique relaxed state of the specimen can generally be obtained for each type of cyclic loading.

Most of the published results on stress relaxation of soft biological tissues are of uniaxial tests. I t is usually analyzed within the framework of the theory of linear viscoelasticity. Barbenel et al. (1973) reported on linear stress relaxation vs log(time) for several decades of time in torsional and uniaxial tension tests of human skin in vivo and in vitro. Similar results were obtained by Ridge and Wright (1964) for in vitro uniaxial tests of human skin, by Galford and McElhaney (1970) for scalp, brain and dura, by Buchthal and Kaiser (1951) for skeletal muscle in the resting state, by Yin and Fung (1971) for isolated ureteral segments of several mammals, and by Zatzman et al. (1954) in isolated segments of the common carotid arteries of d~)gs and human umbilical arteries. Kenedi et at. (1966) used a sum of exponential functions for the relaxation mode of human skin and costal cartilage. Fung (1972) proposed a quasi-linear viscoelastic theory for soft tissues in which the relaxation function K(~,I, ~2, t) is expressed as

K(X1, h2, t) = G(t)ace)()~l, X2), G(0) = 1, (1)

where G(t) is called the reduced relaxation function, which is a function of time only, and ~(~)(~,1, X)~ is the "elastic" response, which is a function of the strain (X~, X2) alone. K, G, and ~(~) are considered as tensors. G(t) is a tensor of rank 4, which is simplified to an isotropic tensor if the material is isotropic. ~,1, X2 are two principal stretch ratios. The third stretch ratio ha is determined by the con- dition of incompressibility, X~h2Xa = 1, which is assumed to hold for the skin. From Eq. (1) we have for the stress ~ in general,

f t

z(t) = / G(t -- t')~(~)(t')dt '. (2)

In posing Eq. (1) it is assumed that if normalized, then relaxation curves at different strain levels will form a single curve. This was shown to be valid in rabbit mesentery (Fung, 1972) and papillary muscle (Pinto and Fung, 1973). On the other hand, Goto and Kimoto (1966) found that in the case of the aorta and vena abdominalis of the toad, the stress-relaxation modes are strain dependent and do not normalize to a single curve. They suggested that at higher strain levels, the distribution function of relaxation times (H(T)) is shifted toward smaller values of r (associated with collagen), whereas at smaller strains, elastin (with higher v) is predominant. Sharma (1974) found that both axial and tangential

252 u LANIR

segments of steer aorta relax in a nonlinear exponential mode. The three constants of his model are strain dependent. Jenkins and Lit t le (1974) found tha t specimens of elastin (Ligamentum Nuchae) relax linearly with the logari thm of t ime only at a specified strain. They found tha t Fung's quasi-linear theory (Eq. (1)) can be applied to this tissue in the uniaxial case if G is made strain dependent :

G(e, t) = 1 + ~e 2(t) In (t), (3)

where e is the uniaxial strain and ~ is a constant. In the present work we wish to reduce and analyze the biaxial stress-relaxation

data obtained during a previous work (Lanir and Fung, 1974b). The results will be discussed in terms of possible implications on the rheological characterization and on the tissue's behavior under other types of mechanical tests.

METHODS

Analysis is done on experimental results of biaxial and uniaxial stress-relaxation tests of square specimens of rabbit abdominal skin. Details on the experimental setup and procedures can be found in the previous publication (Lanir and Fung, 1974a).

Altogether we have results from 143 relaxation tests on 11 specimens from different rabbits. Detailed analysis by computer is done on 58 more comprehensive tests of five specimens (42, 43, 44, 45, and 46). The results of these 58 experiments were originally recorded both on X - u plotters and computer cards by means of a P D P 8 / E laboratory minicomputor system. The data of the other 85 tests were manually reduced for purposes of comparison and checking of pre- l iminary conclusions.

Prior to each test the specimen was precondit ioned: the part icular test was performed several times until results converged. Stresses were allowed to relax for 10 rain with 5 rain intervals between successive tests.

PRELIMINARY OBSERVATIONS

Some general features of the experimental results of stress relaxation are discussed in a previous report (Lanir and Fung, 1974b). A brief summary will be presented here for the sake of completeness.

1. Stress relaxation at different levels of initial stress follow pat terns shown in Fig. 1. I t is readily seen tha t relaxation modes at different stress levels are different and do not reduce to a single curve. Hence the stress relaxation of the skin is nonlinear. This is not surprising if we recall the very high level of strains (stretch ratio of up to 2.0) in these tests.

2. Stimultaneous relaxation of two normal components of stress in a speei- men cannot be normalized to a single curve (see Lanir and Fung, 1974b, Fig. 11).

Thus the skin is an anisotropic, nonlinearly viscoelastic material. I ts theological characterization is very difficult. As a first step toward the construction of a con- s t i tut ive equation for the skin, we consider the stress relaxation under constant


Fx

&6.0

~7o

I Xx=1855 Xx= 1.583 Xx= 1198 Xx:12578

ooo, olo, o', ,'o ,~o = llme (rain)

FIG. 1. Patterns of stress relaxation at different levels of initial stress, (r (0). Note the "waviness" of the initial portions of the curves for low stress levels, and the nearly perfect straight line of curve I I I in the medimn and high time ranges.

strains in a biaxial stress field. We shall t ry to represent the stress relaxation da ta by a simple workable mathemat ica l model whose parameters are functions of strain (or stress). Such a ma themat ica l model mus t agree with the following observed features of relaxation modes.

3. When plot ted as stress vs log time, the curves are sigmoid shaped. They are gradual bo th at ve ry small and very high t ime ranges and steeper in between.

4. The overall steepness of the middle portions of the curves is directly related to the level of initial stress.

5. At low levels of stress the curves approach straight lines (curve III in Fig. 1).

6. In some cases of low initial stress, the stress will completely relax within the t ime range of our test (10 min). In a few cases there is even a slight increase of stress with t ime (curve IV in Fig. 1), the significance of which is doubtful because in all these cases the increase in stress is so small as to lie within the range of inaccuracy of the instruments . More evidence is needed before any conclusions can be reached. In the present work data of ascending portions of relaxation curves are disregarded.

THE MATHEMATICAL MODEL

Our final goal is to develop a comprehensive rheological characterizat ion of the skin tissue. This requires tha t the stress-relaxation da ta be represented in a ma themat ica l ly workable form. As in m a n y other cases, it was found tha t da ta can be represented equally well by several ma themat ica l models. Hence no physical significance should be a t tached to any model just on the basis of its adequacy in representing experimental results.

254 Y. LANIR

In our case we have a rectangular specimen biaxially stretched by applying loads in two mutual ly perpendicular directions such tha t one coincides with the direction of the length of the animal's body. Since no shear was observed (Lanir and Fung, 1974b), we shall assume tha t the directions of stretching are the principal directions for both the strain and stress tensors. In our relaxation tests, the specimen was stretched in one (the main) direction while its t ransverse dimension was either kept constant (by applying the necessary loads on i t - - biaxial relaxation), or it was free to change under no loads (uniaxial relaxation).

After several trials it was concluded tha t the relaxation of the two measured stress components in biaxial tests, and tha t of the single measured one (the main) in uniaxial tests can be adequately represented by the empirical formulas,

am(e~, t) -- o-,,,(e .... ~ ) 4. Eo-m(em, O) - am(e,., oo)-1/[-1 4. t/tm(em)] Nm('~), (4:)

and in biaxial tests,

at(era, t) = at(e .... ~ ) 4- I-at(era, 0) -- at(e,, oo)]/[-1 4- t/tt(em)-I Nt(~), (5)

where am, at, are the measured Lagrangian stress components in the main (m) and transverse (t) directions, em is the measured strain in the main direction, t is the t ime since the step change of em from zero ~o its current constant level, tin, C, N,~, and Nt are functions of em and therefore constants in each test.

Numerical reduction of our data yielded tha t generally Nm ~ Nt and tm ~ C. This result must be t rea ted with great caution for the following reason: The numerical values of the constant tm and N,. (or Nt and tt) in each da ta set are strongly dependent on am(e .... ~ ) [-or zt(em, ~ ) ] . At high e~, stresses continue to relax even after the four decades of t ime (0.001 + 10.0 rain) to which the experimental setup is capable. Hence in those cases, the tests were cut before stresses had fully relaxed. Under such conditions, one m ay arrive at an erroneous value of a(e~, ~ ) . I t was observed tha t small changes of z(e .... ~ ) in the numerical procedure cause substantial changes of the constant Nm and tm (or N~ and tt). Hence even if the error in evaluating a (e .... ~ ) is very small, the obtained values of Nm and t~ are questionable.

I t is thus felt tha t final conclusions concerning this point should be made only after results of longer relaxation data (5-6 decades) are available.

In the present work we shall pay a t tent ion mainly to the effect of strain (or stress) on the relaxation mode. In generalizing the empirical results of Eqs. (4) and (5) to a multiaxial rheological model, it will be assumed tha t N~ = Nt = N and t~ = tt = tr

The empirical relations (4) and (5) can be writ ten

am (era, t) = a~, (era, 0 ) . Cm (era, t), (6)

at (era, t) = at (era, 0 ) . Gt (era, t), where for each direction (m and t) we have

G(em, t) = B(em) 4- A(em)/[1 4- t/t~(em)-] ~Y(~ (7)

B(~,n) = a(~,,,, ~ ) / a ( e .... o),

A (~m) = 1 - B(~m). (8)


General izat ion of Eqs. (6)-(8) to three-dimensional condit ions can be done as follows :

z~.(e, t) = zkz(e, 0). Gijkl (e, t), (9)

where

Gi~'k~(e, t) = Bij-k~(e) + A ~ k ~ ( e ) / [ - 1 -~- t / t c (e ) ] ~*<e), i, j , k, l = 1, 2, 3. (10)

In Eqs. (9) and (10) e is the s train tensor, z~j. is the Lagrangian stress tensor, G~-k~, Bijk~, and A~j.k~ are 4-rank tensorial funct ions of the s t ra in ; N and to are scalar functions of the strain tensor.

In biaxial stretching of an incompressible material where no shear exists, we

have two independent components (principal values) of the strain: e~ and e2.

For this case we ean write

alj(el, e2, t) = a~,(ei, e~, O)G~k,(el, e2, t),

Glint(el, e2, t) = B ~ ( e ~ , e:) -~ A ~ ( e ~ , e~)/r l ~- tr e2)] N(e~,e~), i , j , lc, l = l , 2. (11)

I t is assumed t h a t the tensors B and A and the scalars N and to can be expressed as funct ions of polynomials of e~, e2, and coupled terms (Y. C. Fung, personal communicat ion) .

IO00D

I00.0

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I,C

0.1

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~ , ~ ,

l , ..... I ] It~ .... [ I ILLIk,,[ I II~, ,J I ll~L, 0.01 0.1 1.0 I0.0 I00.0

Time (rain) I000.0

FIG. 2. A typical comparison between experimental data (. in the x direct~ion and A in the y direction) and the model (Eq. (20)) for t >> to.

256 Y. LANIIr

SPECIMEN:~45

~ ~ . ~ TEST, R2

Xx= 1.655

. .-x~ Xy = LO00

= % d(O) = 96.76 g r/cm2

8x = 0.66

~ - Nx =0"2579

o g- in

o o

o o

o o

o o ~ d.o, d ,o ,Io r~.o t (rain)

FIG. 3. A typical comparison between experimental data (points) and the model (Eq. (15)).

I n o u r b i a x i a l t e s t s t h e t r a n s v e r s e s t r a i n (e2) v a n i s h e s . F o r t h i s case we sha l l

h a v e

O'll(el, t) = zll(el, 0)-{Bml(e~) + A~m(el)/[1 4- t/to(e~)3 ~(e')} + z~(e~, O) X {B11~2(el) -~ Al122(e1)//[-1 + t/tc(el)N(e~)}, (12)

a n d s i m i l a r l y fo r (r~ (ei, t).

TABLE 1

Weight of Male Rabbits and Dimensions of Specimens used in Stress-Relaxation Tests

Specimen Weight Ax ~ (cm~) ~ A y ~ (cm2) b h ~ (em) c (kg)

42 2.2 0.3515 0.3750 0.0955 43 2.8 0.5414 0.5568 0.1512 44 2.3 0.3399 0.3555 0.0952 45 2.6 0.5055 0.4903 0.1327 46 2.6 0.4554 0.4209 0.1236

a A x o: initial cross-sectional area in x direction. b AyO: initia/cross-sectional area in y direction. c h0: initial thickness of specimen.

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BIAXIAL STRESS RELAXATION IN SKIN

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By set t ing ~1 -- am, o-22 = z~, and

O'm(el, 0) = Oll(el, 0)[-Bllll(e:t) -~ Alm(e l )~ -[- o-22@1, 0)[-Bl122(el)

-~-Ai12e(el)3, (13) am(el, ~ ) = crll(el, O)Bml(e l ) -t- ~r2.2(el, O).Blr22(el),

we shall have

am(era, t) = z~(e,,, ~ ) + E~m(e .... 0) - ~,,(e .... ~ )3 /E1 + t/t~(e,,)3 ~v('m), (14)

which is analogous to Eq. (4). A similar result is obtained for zt(e .... t). In the following, Eq. (7) will be writ ten in a concise form as

G(e,, t) = B + A / ( 1 + t/tr N. (15)

For values of t << tc, G(e, t) is approx imate ly a constant. This agrees well with observed relaxation behavior a t t << 1.

For N << 1 we can expand G (t) as

a(e, t) = B + A{1 - ~u In (1 + t/tr + (A:~/2!)[-ln (1 + t/t~)~ ~ . . . }. (16)

For N < < I and ranges of t used in our tests (t < 10 rain), Eq. (16) can be reduced to

G(e~t) ~ (B + A) -- A . N . l n (1 + t/to), (17)

which agrees well with another feature of the relaxation data, namely tha t under low strain levels, the plot of G(t) vs log t (or in t) is a linear line.

For t >> tr Eq. (15) can be approximated by

~(e, t) ~ B + J / t 'v, (lS)

where A = A(to) ~'. Since N > 0, G(e, t) is "hyperbol ic ." Hence Eq. (15) repre- sents a "hyperbol ic" characterizat ion of the relaxation model.

A check on the adequacy of the proposed model can be obtained by plott ing log (dG/dt) vs log t. According to Eq. (15)

log ( - d G / d t ) = log (AN/t~) - (X + 1) log (1 + t/L,). (19)

Hence for t >> tr the above plot should give a s traight line :

log ( - -dG/dt) ~ log (ANto x) -- (N + 1) log t, (t >> t~). (20)

A corresponding plot of some experimental results is shown in Fig. 2. I t is readily seen tha t except for very low (t ~ t~) and very high ranges of t the da ta can be represented very adequate ly by a s traight line.

DATA REDUCTION

Direct calculation of the pa ramete r s B, N, and tr of the relaxation function G(e, t) for each tes t is often difficult because G(e, 0) in Eqs. (4) and (5) is either

FIG. 4. The linear viscoelastic functions for the modified power law (Eq. (15)) in small incremental deformations. (A) The spectrum H(r) of relaxation times, r. (B) The variations of the "storage modulus," G'; the "loss modulus," G ' ; the modulus, M; and the internal damping, tan a, vs frequency.

260 Y. LANIR

~ ~ ,~ __.~_~

2/' [ ~ , ~ } (.)'p

=

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unavailable or difficult to obtain due to the initial t ransient response of the system. Hence the following approach is used. Equat ions (6) and (15) are com- bined to yield

a(e, t) -- B ~ -]- A~ + t/tc) N, (21)

where

B ~ = B .a (0) = a ( ~ ) and A 0 = A.~(0) = a(0) -- a ( ~ ) . (22)

A set of parameters A ~ B ~ N, tc is searched tha t can "best fit" the experi- menta l results of a (t) vs t in the entire range of the experiment. The values of B and a(0) are then calculated from Eq. (22), and

~ ( 0 ) = B ~ ~- A 0. (23)

The first a t tempts to evaluate the constants A ~ B ~ N, and tc in Eq. (15) to fit the experimental da ta were carried out by a process of nonlinear minimization of an error of the least-squares type. I t was soon realized tha t the procedure was not always reliable. I t is well known tha t in a nonlinear minimization procedure the success depends on a proper choice of the initial values of the parameters. If the assumed initial values are too remote from their " t rue" values, the procedure may either diverge, or may lead to one of many possible sets of parameters tha t can fit the data reasonably well in the sense of localized relative minima. The answer may be physically unacceptable even though it appears adequate numerically.

Hence, in the present work, data reduction is done in two steps: First, a special search procedure is developed to evaluate the initial values of the parameters A ~ B ~ N, and to. Next, an improved evaluation of the parameters is obtained with an IB M library subroutine for nonlinear function minimization.

Several schemes to obtain initial parameter guess have been tried. After several a t tempts the following was adopted. To explain this procedure we note first tha t since the characteristic t ime to is found to be on the order of 0.001 to 0.01 min, da ta at t >> tr (say, t > 0.1 rain) can be approximated adquate ly by Eq. (18) or by

,~(e, t) = Bo + A ~ / t L t>> to. (24)

Hence the values of A ~ B ~ and N for each curve can be established for data at t > t*, where t* is on the order of 0.1 rain, by an iteration process in which the partial derivatives of an error of the least-squares type with respect to A ~ B ~ and N are equated to zero. Subsequently t~ is calculated to fit the exact model (Eq. (15)) from data at lower ranges of t (where t and t~ are of equal order of magnitude). Unfortunately, the experimental da ta in this t ime range are often unstable (curves I I and I I I in Fig. 1) due to transient response of the experimental system. To facilitate the calculation of t~, the original data, recorded on X - Y plotters, were arbitrari ly smoothed and digitized with a Gerber X - Y digitizer (with 0.001" resolution).

FIG. 5. Parameters of stress relaxation at different levels of a(0). (A, B) Specimen #42, kx = 1.000. Test R4 has different preconditioning than the rest. (C, D) Specimen #42, k~ = 1.000.

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In the next step the overall least-squares deviation of the mathematical model from the experimental data is evaluated for the entire range of t used in the experiment. This procedure is repeated for various values of t* until the "best fit" is obtained. The corresponding set of A ~ B ~ N, and tc is subsequently used as an initial guess in the computer l ibrary program mentioned above.

A typical comparison between the values given by the mathematical model and the raw data is given in Fig. 3.

RESULTS

Table 1 contains data on the weight of male rabbits and dimensions of the abdominal specimens on which most comprehensive tests were performed. In Table 2 the tests are listed in order of performance. The main and transverse stretch ratios of each test are listed as well. Uniaxial tests are noted by the letter S in the test 's name. Specimens # 4 3 - # 4 6 were tested by stretching the tissue in the x direction (the direction of the length of the body). Specimen # 4 2 was stretched in the x direction (tests R13/A-R17) and in the y direction as well (tests R4-R12). In "biaxial" tests the transverse dimension was kept at zero strain. In "uniaxial" tests the transverse edges of the specimen were free.

The preconditioning procedures used in the tests named above were the following: In specimen #42, test R4 was not preconditioned, and then the same test was repeated four times (R4, R5, R6, and R7) until repeated results (R7) were obtained. Subsequent tests were performed for 10 min with 5 rain intervals. In specimen #43, test R1 was not preconditioned, and then it was followed by three repetitions (R1, R1 /A and R1/B) until converging results were obtained. Prior to each subsequent test, the specimen was preconditioned again by perform- ing test R1 for 10 rain, followed by a 5 min, rest at equilibrium. In specimen #44 , a similar procedure was used with R 1 / C as the preconditioning test. R1/S was a uniaxial test (zy = 0). Specimen #45 was preconditioned four times with R1 and three times with R2, since it was realized tha t the uniaxial test RS1 altered the response of the tissue. Subsequent biaxial and uniaxial tests were carried out for 10 rain each with 5 rain rest. In specimen #46 , test R1 was repeated before each test for 10 rain followed by 5 rain rest.

In Figs. 4-7 the variat ion of ~ ( ~ ) , B, and N with respect to z(0) is shown. I t was found tha t it is simpler to relate B and N to a(0) ra ther than to relate it to the stretch ratio X (or the strain). The variat ion of tcvs ~ (0) was found to have no specific trend. I t is most probably due to the t ransient response of the experimental system which makes the values of to obtained quite unreliable. Fur ther discussion of the parameter to and its significance will have to wait until more accurate results are obtained.

DISCUSSION

The nonlinear viscoelastic features of skin are clearly manifested in Figs. 4-7. If skin were linearly viscoelastic, then B and N would be independent of a (0).

FIG. 6. Parameters of stress relaxation at different levels of a(0). (A, B) Specimen #43, h~ =1.000. Test R1 was not preconditioned. (C, D) Specimen #44. In test RIS, au = 0. In the rest of the tests k~ = 1.000.

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From the figures we see t ha t B and N seem to be monotonical ly decreasing functions when the initial stress r (0) is increased. In a few specimens there is a sharp drop of B and N at very low levels of r (0) (# 42, # 44, and # 45).

In a first approximat ion the var ia t ion of z ( ~ ) ~4th r (0) seems to follow a s t raight line if tests are similarily preconditioned.

Comparison of s imultaneous relaxation of two principal stresses in biaxial tes ts indicates t ha t they relax in different modes. The power N (indicating the ra te of stress relaxation) m a y be different between the relaxation mode of ~x and tha t of zy. In addit ion the rat io between the fully relaxed stress r (m) and the initial stress r is in general different. Nevertheless, as was ment ioned before ( "The Mathemat ica l Model") this result mus t be confirmed by future tests with longer t ime ranges.

The stress relaxation of the tissue is highly dependent on the preconditioning procedure imposed on the tissue. By comparing test R4 to test R7 in Fig. 5 (both done under the same stretch ratios except t ha t R7 was precondit ioned and R4 was not), we can see tha t preconditioning lowers the initial stress level

(0) for the same stretch ratio. At the same t ime it increased the equilibrium stress level r (m) and the value of the power N in our model. The same feature is seen (within deviations due to experimental errors) if we compare I l l to the rest, of the tests in Figs. 6A and 6B, R100 to the rest of the biaxial tes ts in Figs. 7A and 7B, and R7S to the other uniaxial tes ts in the same figure.

Uniaxial tests require uniaxial preconditioning. Uniaxial stress causes negat ive strain in the t ransverse direction, which m a y alter the t issue's response. In specimen # 4 5 (Figs. 7A and 7B) the tissue was properly precondit ioned both biaxially (with R2 /C) and uniaxially (with R1S and R2S/B) . Hence results of test R 1 / D in the t ransverse direction (y) do not follow the same pa t t e rn as the others, since the uniaxial tes t R1S altered the t issue's response in the y direction. A similar reason causes results of R100 in the y direction to differ considerably f rom those of R2/C. Specimen # 4 6 (Fig. 7C) was not uniaxially preconditioned. Uniaxial tes ts were performed in the order of increasing s t retch rat io (e.g., increasing negat ive strain in the t ransverse direction). This caused N to change abrup t ly f rom tes t to test, but seemed to have little effect on r ( ~ ) .

Thus a proper preconditioning requires tha t the highest and lowest strain levels occurring during the test ing procedure will be imposed initially several times, until repeat ing results are obtained.

By observing the results of biaxial and uniaxial tests of proper ly precondit ioned specimens (#44 in Figs. 6C and 6D, and # 4 5 in Figs. 7A and 7B) one sees tha t if compared on ~he basis of initial stress level, both uniaxial and biaxial tests yield similar values of equilibrium stress r ( ~ ) . The values of N are somewhat

FIG. 7. Parameters of stress relaxation at different levels of ~(0). (A, B) Specimen #45, ky = 1.000 in biaxial tests. r = 0 in uniaxial tests. In (A), notice the similarity of B~ and the slight difference of N~ between uniaxial and biaxial tests (denoted by the letter S in the name of the test). O : tests performed under different preconditioning or following high-level negative strain. (C) Specimen #46. X~ = 1.000 in tests R1/A and R1/B. r = 0 in the other tests. [] : tests performed under different preconditioning. Notice abrupt variation of Nx due to the effect of increasing levels of negative strain in uniaxial tests.

266 Y. LANIR

lower in uniaxial tests. This last mentioned property, if confirmed in fur ther studies, may play an impor tan t role in generalization of uniaxial stress relaxation data to multiaxial conditions. I t can facilitate the construction of consti tut ive equations for this tissue.

We can summarize these observations as follows.

1. After a step change of strain, the stresses in the skin tissue relax with time. The relaxation function depends on the initial stress (or strain) level. Thus the skin is nonlinearly viscoelastic.

2. The stress relaxation process can be represented by a "power" law as in Eqs. (4) and (5). The reduced relaxation function G(e, t) obtained in every experiment can be represented quite accurately by a modified power law with three parameters (Eq. (15)). Parameter B is related to the final equilibrium stress. Power N is related to the rate of stress relaxation.

3. To the first approximation, ~ ( ~ ) increases linearly with increasing initial stress ~ (0).

4. Constants B and N are monotonic decreasing functions of the initial stress z (0).

5. Different components of stress in the tissue relax in different modes, i.e., the relaxation function of the tissue is anisotropic.

6. A higher-frequency experimental system is needed to obtain more accurate measurements of the value of the third constant, to, which is related to the tissue's response immediately after step changes (or al ternat ively at higher frequencies).

7. Proper preconditioning is necessary to obtain repeatable results. In each loading cycle, no strain component should exceed the extrema used in the preconditioning process.

8. In a uniaxial test, stress will relax in a manner similar to biaxial tests. If compared on a basis of z (0) both types of tests will have the same ~ ( ~ ) . N will be somewhat lower in uniaxial tests.

9. Fur ther investigation of the biaxial t ime dependent behavior of skin tissue and more complex stretching schemes are needed to reveal the full range of nonlinear viscoelastic features of this tissue.

Behavior under Small Incremental Deformation

Since skin is nonlinearly viscoelastic, one cannot use the principle of super- position to predict stress response to arbi t rary stretching schemes by means of relaxation modes.

There are circumstances, however, in which we are interested in the response of the tissue to small perturbat ions superimposed on large deformations. The skin, as other soft tissues, though capable of large configurational changes, may experience only slight deviations from normal in vivo dimensions under a wide range of physiological conditions.

In such cases it is possible to use the theory of incremental linear viscoelasticity. We shall t rea t the case of uniaxial deformation. Generalization to two- and three-


dimensional deformations can be done upon obtaining further information on the anisotropic features of the relaxation mode, as discussed above.

If the tissue is subjected to a constant strain e*, and an additional small strain Ae is superimposed on e* at time t = 0, then the incremental stress is assumed to be

Aa(e* + Ae, t) = A(~(e* + Ae, o ) . a ( e * + Ae, t), (25) where

G(e* 4- be, t) = G(e*, t) = B(e*) 4- A(e*)/E1 + t/to(e*)~ v(~*). (26)

This is the incremental relaxation function. For sufficiently small values of Ae, Eq. (25) is linear in Ae. In this case Fung's (1972) hypothesis of quasi-linear viscoelasticity is valid, i.e.,

f ~ dz(e* + Ae, O) - d t ' , Ae ~< e* (27) Az(e* + Ae, t) = G(e*, t t') dt'

G(e*, t) can be considered as a linear viscoelastic relaxation function which is related to a spectrum of relaxation times EH (r) ~, and a complex modulus [-G* (~) 1. In Appendix 1, it is shown tha t the relaxation spectrum is

H (r) = EA*/F (N) ~ (to*It) x*. exp ( - t~*/r), (28)

where F (N) is the Gamma function and * denotes the value of the parameter at e = e*. The complex modulus G*(~) = G'(~) § i G ' ( ~ ) = M(~) exp (i5) has the real and imaginary parts

G' = B* + EA*/r(N*)](~to*)N*.~- .csc (~rN*)VN*(2~to*, 0), (29)

G" = - E A * / r ( N * ) ~ ( o ~ t J ) 2 v * . r . c s c (~rN*)V~+A~,(2~to*, 0), ~t~* > 0, (30)

M(~) = EG'(~) 2 + G"(o~)2~ tan ~ = G ' / G ' , (31)

where Vv is the Lommel function of two variables of order ,. Plots of H (t) vs log t, G', G ' , M, and tan ~ vs log ~ are given in Fig. 4a and

Fig. 4b.

APPENDIX I: DERIVATION OF EQS. (28)-(30)

H(r ) is defined by

G(t) = G ( ~ ) + (1 / r )H(r ) exp(- t , / r )d~- . (I-l)

From Eq. (26) we have, for the incremental relaxation mode G,

G(t) = B* + A*/(1 + t/tc*) N*, (I-2)

where B* = B(e*), A* = A (e*), tc* = to(e*), and N* = N(e*) . I t follows tha t

f0 (1/T)H(r) exp ( - - t / - r )d r A*/(1 t/to*) -Y*. + (!-3)

268 Y. LANII~

By setting ~, = 1 / r we get

/ , A*/(1 + t/to*) N* = Jo ( 1 / ~ ) H ( 1 / ' ) , ) exp (-- t~,)d 'y = 2~{ (1 / -y )H(1 /~ , ) } . (I-4)

Hence, H (r) can be found by the inverse Laplace transform of the lhs of Eq. (I-4) :

H (r) = (A*/F (N*)). (tr exp ( - to*/r) , (I-5)

where r (N*) is the Gamma function of N* defined by

I ' ( N * ) = t !v*-I exp ( - t ) d t , Re(N*) > 0. (1-6)

Smith (1958) dealt with linear viscoelastic materials exhibiting relaxation modes as in Eq. (I-2) and originally derived Eq. (1-5).

We can proceed to the dynamic viscoelastic functions. G' and G " are defined as

f0 G' (~o) = a ( ~ ) + o~ [G( t ) - a ( ~ )-] sin (cot)dt,

(1-7)

L G " (~o) = ~o [-G(t) -- G( ~ ) ~ cos (~ot)dt,

or alternatively,

f0 ~ G' (~o) -~ G( ~ ) + H (r)o~2r/ (1 + co2r2)dT,

fo G " ( @ = g ( r ) ~ o / ( 1 + o~2r2)dr.

Introduction of Eq. (I-2) to (I-7) will give

G'(co) = B* + co A * sin (~ot)/(1 + t/to*)a~*dt,

and

(I-8)

(1-9)

G"(o~) = oJ A * cos (cot)/(1 + t/to*)N*dt.

By setting X = ~o(t + to*) we get, from Eq. (I-9),

G'(co) = B * + A*(~otc*)N*Ecos (coto*).S(~oto*, 1 -- N * ) -- sin (~oto*)

�9 C(~otr 1 - N*)J, (I-10)

G " (~o) = A *(cotr (wto*) .C(r 1 -- N*) + sin (~ot~*).S(o~to*, 1 -- N * ) 7,


where

S(wtr 1 - N*) = j~ tc$

C(cotr 1 -- N*) = f ~

(1 /X N*) sin xdx,

(1 /X N*) cos xdx, Re(1 -- N*) < 1.

(I-11)

Alternative expressions for G' and G" can be found by introducing Eq. (I-2) and (I-5) to (I-8) and setting r = (1/x~). We get

G'(w) = B* + A*tr @v*-l.exp E - t J ~ x ~ ) / ( x ~ + 1)dx,

fo G"(~o) = A*to*N%N*-i/I '(N *) X N* exp (--t~*~ox)/(x 2 Jr 1)dx.

(I-12)

The

G'(,,) c"(~)

where

where

integrals in the above equations are Laplace transforms. Thus we have

= B * + [A*/P(N*)J(~to*)N*.~-.esc (~N*)VN.(2~to*, 0), (~-~3)

= --EA*/r(N*)~(o~to*)N*.~.csc (~N*)VI+N*(2~t~ 0), ~t~* > 0,

Vu is a Lommel function of two variables of order y defined by

V.y(W, Z) = cos ( W / 2 + Z2 /2W -t- vTr/2) + U_~+~(W, Z), (I-14)

U~(W, Z) = ~ (--1)"(W/2)~+~mJ~+2m(Z). m~O

J~ is the Bessel function of the first kind of order % For Z = 0 we have (Watson, 1958)

U'r(W, O) = ~ (--1)m(W/2)~+2m (I-15)

,~=t p (7 -k- 2m + 1)

Equat ion (I-13) is more suitable for computat ion than Eq. (I-10).

ACKNOWLEDGMENTS

The author wishes to thank Professor Y. C. Fung, Department of AMES-Bioengineering, University of California, San Diego, for his very helpful comments. The author wishes to thank Mrs. T Markuson for the drawings, and Mrs. D. Orlinsky for preparing the manuscript.

REFERENCES

Barbenel, J. C., Evans, J. H., and Finlay, J. B. Stress-strain time relations for soft connective tissues. In R. M. Kenedi (Ed.), Perspectives in biomedical engineering. London: Macmillan, 1973. Pp. 165-172.

Buchtal, F., and Kaiser, E. The rheology of the cross striated muscle fiber with particular reference to isotonic conditions. Det Kogelige Danske Videnskernes Selskab, Danish Biological Medicine 1951, 21, 318.

270 Y. LANIIZ

Fung, Y. C. Stress-strain history relations of soft tissues in simple elongation. In Y. C. Fung, N. Perrone, and M. Anliker (Eds.), Biomechanics-Its Foundations and Objectives. Englewood Cliffs, N. J. : Prentice-Hall, 1972. Pp. 181-208.

Galford, J. E., and McElhaney, J. H. A viscoelastic study of scalp, brain and dura. Journal of Biomechanics 1970, 3, 211-221.

Goto, M., and Kimoto, Y. Hysteresis and stress-relaxation of the blood vessels studied by a universal tensile instrument. Japanese Journal of Physiology 1966, 15, 169.

,Jenkins, R. B., and Little, R. Wm. A constitutive equation for parallel-fibered elastic tissue. Journal of Biomechanics 1974, 7, 397 402.

Kenedi, R. M., Gibson, T., Daly, C. H., and Abrahams, M. Biomechanical characteristics of human skin and costal cartilage. Federation Proceedings 1966, 25, 1084-1087.

Lanir, Y., and Fung, Y. C. Two-dimensional mechanical properties of rabbit skin. I. Experimental system. Journal of Biomechanics 1974a, 7, 29-34.

Lanir, Y., and Fung, Y. C. Two-dimensional mechanical properties of rabbit skin. II. Experi- mental results. Journal of Biomechanics 1974b, 7, 171-182.

Pinto, J. G., and Fung, Y. C. Mechanical properties of the heart muscle in passive state. Journal of Biomeehanics 1973, 6, 597-616.

Ridge, M. D., and Wright, V. A rheological study of skin. In R. M. Kenedi (Ed.), Biomechanics and related bioengineering topics. New York : Pergamon, 1964. Pp. 165-175.

Sharma, M. G. Viscoelastic behavior of conduct arteries. Biorheology 1974, 11, 279-291. Smith, T. L. Approximate equations for interconverting the various mechanical properties of

linear viscoelastic materials. Transactions of the Society of Rheology 1958, 2, 131. Watson, G. N. A treatise on the theory of Bessel functions. London/New York: Cambridge Uni-

versity Press, 1958. Yin, F. C. P., and Fung, Y. C. Mechanical properties of isolated mammalian uretral segnlents.

American Journal of Physiology 1971, 221, 1484-1493. Zatzman, M., Stacy, R. W., Randall, J., and Eberstein, A. Time course of stress relaxation in

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biaxial stress-relaxation in skin

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