- Email: [email protected]
Non-linear viscoelastic response of ice
Non-linear viscoelastic response of ice L. W. Morland
School of Mathematics, University of East Anglia, Norwich NR4 7TJ U.K. The duality of constant stress and constant displacement-rate response of ice in uni-axial stress tests is used to infer a first order differential, non-linear viscoelastic, relation which describes qualitatively both responses through primary, secondary and tertiary creeps. It is the most simple model which reflects this observed transient behaviour. The dominant infhtence of temperature is incorporated through a rate factor, Extrapolation to three-dimensional response is far from unique, even with an isotropy assumption, and highlights the sparsity of multi-axial data to establish a satisfactory engineering model.
INTRODUCTION Coherent masses of ice, in the sea or on land, are now an important part of the world's engineering environment. The manner in which ice deforms, and fails, under mechanical loads and changing thermal conditions is an essential component of cold regions engineering. The effects of ice contact on structures involves coupled stress analyses in which both ice and structure are responding to the loads, in manners governed by their respective material properties - constitutive descriptions. Any analysis of such couples responses, even in idealised geometries and with the power of high speed computation, requires the most simple constitutive model for the ice consistent with the main features of its known behaviour. Time scales of applications vary widely, from dynamic problems in which wave propagation effects are important, to slower quasi-static problems in which particle momentum associated with the deformation is negligible though rigid body momentum may be significant. A long-time scale viscous fluid model appropriate to glacier flow over years to thousands of years is not applicable to engineering contact problems. Ice is essentially viscoelastic over the time scale of seconds to hours, or months, depending on temperature. It exhibits infinitesimal elastic strains on application of a step load, followed by creep to small, but not infinitesimal, strains. The response is highly non-linear even at small strains, and the time scales, or rates, are strongly influenced by temperature. I present here a non-linear viscoelastic model of differential type, in fact a first order differential relation, which has the minimal ingredients necessary to model key features of laboratory uni-axial test data, and which is sufficient to provide good qualitative and quantitative correlation. The basis for the formulation lies in the responses in uni-axial stress tests both to applied constant stress (creep test) and to applied constant displacementPaper accepted October 1990. Discussion closes April 1992. 9 1991 Elsevier Science Publishers Ltd
254
Applied Ocean Research, 1991, VoL 13, No. 5
rate, as illustrated by Mellor t, with further constructions related to the duality of these response conjectured by Mellor. Early discussions of the non-linear creep response, Sinha 2"3, Morland 4, Gold and Sinha 5 focussed only on the constant stress response, and inferred relations are entirely inconsistent with the constant displacement rate response. Subsequently, recognising that the dual response must be satisfied, Morland and Spring 6'7, Spring and Morland 8"9 and Williams 1~ constructed both differential and integral type laws for both fluid and solid behaviour, which exhibit the required qualitative features of both responses. Very recently, Sunder and Wu 11"12 have constructed uni-axial and multi-axial orthotropic differential type models under the restriction of infinitesimal strain which describe both responses in primary and secondary creep, but do not attempt to encompass tertiary creep. Their approach through internal variables of state requires further differential relations to determine the evolution of the new variables, adding more degrees of freedom to the model but at the expense of equation complexity - a system of three coupled first order differential equations for uni-axial response. My aim is to show that one first order differential equation can describe the qualitative features of both response in uni-axial stress, including tertiary creep, and that there are many ways to extrapolate to threedimensional behaviour at finite strain, isotropic or anisotropic, when multi-axial data is available. The dominant temperature influence is incorporated through a rate factor. Data and models reflect the response of a small homogeneous sample. Non-homogenity on the large scale, if recognised, could be incorporated into numerical applications by prescribing non-homogeneous material functions and parameters in the common differential law. The present differential law is an alternative to that constructed for a solid response by Spring and Morland 8, with a reduced shape, and fewer terms. It therefore implies a stronger duality of constant stress and constant displacement-rate responses. It is, of course, a continuum
Non-linear viscoelastic response o f ice: L. IV. Morland
model for coherent ice which determines stresses and deformations under prescribed loading histories. Fracture, and post-fracture behaviour, require new theory, but the continuum model is necessary to predict fracture under a given fracture criterion, and to show the pattern of stress development prior to fracture.
_
1 i
UNIAXIAL R E S P O N S E Consider the idealisation of a constant temperature laboratory uni-axial load test on an ice sample in which load and contraction (displacement) are measured as functions of time t. Let ~ denote the nominal axial compressive stress (force per unit initial cross-section), and e denote the axial compressive engineering strain (contraction per unit initial length). The constant load test measures e(t) when a step function stress ~" is applied and maintained constant - a creep test. There is an instantaneous infinitesimal strain e o given by (1)
e o = a l e o,
where E o is the initial (Young's) modulus, followed by a primary decelerating creep from an initial strain-rate b(0) to a minimum strain-rate b,, at time t,, and strain e,,, and then by an accelerating tertiary creep in which k increases. The minimum strain-rate is termed secondary creep. Figure 1, from Meilor ~, illustrates this response in the form of a strain-rate b dependence on strain e, noting that e is a strictly monotonic increasing function of t (b > 0). This figure shows two curves to indicate the effects of different constant ~', namely, that strain-rates increase with ~ at a given strain. The instantaneous strain eo is depicted as a small gap, which may be negligible in comparison with e,, (a factor 1 0 - 2 a t 6-~ 106Nm- 2). Dashed lines show the expected extrapolation (if a coherent uni-axial load configuration could be maintained) into finite strain which would require ~--* 0 as e ~ 1. The limit is discussed below. A constant contraction-rate test measures ~ as a function of e = rt at constant E = r. Figure 2, from
,\
I
\+I/"
"\ ,
, I I e
r
I
"---/
I ,
o
e~
1
~ e
Fig. 2. Stress dependence on strain at constant strain-rate.
Mellor t, shows two curves at different values of r to indicate the effect of increasing strain-rate, namely, that increasing r increases the stress at a given strain. Corresponding to primary creep the stress-strain slope decreases continuously to zero at a strain e~t = rtM where the stress # has a maximum 6at, associated with secondary creep, then corresponding to tertiary creep the slope takes increasing negative values as the stress decreases. Again the dashed lines show expected continuum extrapolation into finite strain. The viscous model assumed for the long-time shear response in glacier flow relates the natural strain-rate to the Cauchy stress, but commonly adopts the secondary creep response, that is, the minimum strain-rate Era(#), instead of the long-time asymptotic of tertiary creep. With the common incompressibility assumption, the axial natural strain-rate ~ and Cauchy stress a in uni-axial stress, measured positive in tension, are related to the engineering measures by E = --E/(I -- e),
t'~(6) < O,
e 0
1
a = --(1 -- e)6".
(2)
Thus, as e ~ I at constant 6", a ~ 0, which implies k --* 0 for a viscous relation, so that ~--* 0 more rapidly, like the product L(I -- ~). As e ---, 1 at constant E = r,/~ ~ - 0% implying that a ~ - oo and hence # ~ oo like the product - a ( 1 - e ) - t . These limiting behaviours are reflected in the dashed line extrapolations in Figs I and 2. However, Mellor x suggests that the tertiary limit and secondary creep may be little different at the low shear stresses in typical glacier flows. The time t m to minimum strain-rate at constant stress is a rapidly decreasing function of 0, and the time tat to maximum stress at constant strain-rate r is a rapidly decreasing function of r. Thus
, _
/t'
1 m
Fig. 1. Strain-rate dependence on strain at constant stress.
e
t'M(r) < O,
(3)
and the rapid variation implies a highly non-linear viscoelastic response even at small strain. Linear viscoelastic response requires t., and tat to be respectively
Applied Ocean Research, 1991, Vol. 13, No. 5
255
Non-linear viscoelastic response o f ice: L. IV. Morland
independent of 6. and r. In addition, both t,, and t M are rapidly decreasing functions of temperature Twhen the same constant stress and constant strain-rate tests are performed at different constant temperatures. Temperature influence is discussed later. Now Mellor ~ has conjectured that the constant stress and constant strain-rate response each fully determine the (uni-axial) mechanical properties. That is, the two responses, for all constant 6- and all constant r, contain equivalent information. However, a particular shape of relation may be correlated with one of the responses for a discrete set of results and then predict an entirely incorrect picture of the second response. For example, Morland 4 constructs a second order differential relation for shear strain-rate in terms of stress based solely on primary creep response, which at constant strain-rate implies constant stress. Mellor ~drew attention to the two observed responses which must both be described by a valid constitutive model. The subsequent Morland and Spring 6"7 and Spring and Morland 8.9 constructions permitted both shapes of response shown in Figs 1 and 2. The models, differential and integral types, contained response functions neither completely determined by a family of creep tests nor by a family ofconstant strain-rate tests, and with differing amounts of information required from the second set of responses; that is, the Mellor duality is not complete. A re-examination of first order differential relations of solid type, treated by Spring and Morland s, in conjunction with two explicit Mellor 1 duality conjectures, now shows that two of the response functions of a differential relation with three response functions of single argument is determined by a family of creep tests while the third is determined by the maximum stress at each constant strain-rate. This is surely the most simple model describing both types of response qualitatively and satisfying the major quantitative correlations through the imposed duality conditions. The adopted Meilor I conjectures, reinforced by Mellor and Cole 13A4 are as follows. First, the major duality condition asserts that the minimum strain-rate at a given constant stress is the constant strain-rate which delivers that stress as the maximum stress, and, vice-versa, that the maximum stress at a given constant strain-rate is the constant stress which delivers that strain-rate as the minimum strain-rate. These duality conditions are expressed by bm[6.M(r)] = r,
6.MEe,,(6.)]= 6..
(4)
The second conjecture is that the strain at minimum strain-rate at constant stress is independent of the stress, and that the strain at maximum stress at constant strain-rate is independent of strain-rate, and these two constant strains are equal, thus e,, = e~t = constant.
(5)
This constant strain is approximately one per cent, small but not infinitesimal; that is em = eM ~ 0.01.
256
Applied Ocean Research, 1991, Vol. 13, No. 5
(6)
A third conjecture is that the initial stress-strain slope at constant strain-rate is independent of strain-rate, but using the stress-strain jump conditions deduced by Spring and Morland 8, for the proposed form of differential relation, the initial slope is identical to the Young's modulus E o defined by the initial jump condition (1) which is not related to a strain-rate, so this conjecture is satisfied. Approximate elastic analyses use effective moduli distinct from initial values governing the instantaneous response, in an attempt to average the creep effects, but these will be in serious error when application time scales are of order tm or t M, which vary over a very wide range as stress and strain-rate levels change. UNI-AXIAL RELATION The earlier viscoelastic constructions 6.7,8.9 started from frame indifferent tensor relations of appropriate shapes so that restriction to uni-axial stress permitted at least qualitative correlation with constant stress and constant strain-rate responses. It is simpler to develop directly a uni-axial differential relation to match qualitatively the responses shown in Figs 1 and 2 and satisfy the duality conditions (4) and (5). However, the extrapolation to a three-dimensional constitutive law remains a complex problem. An instantaneous (elastic) response to a step load is given by relation (1) in terms of the initial modulus Eo. The subsequent primary-secondary-tertiary creep, illustrated by Fig. 1, requires that the strain e increases monotonically with time through decelerating, stationary and accelerating phases while e is still small. The model must therefore yield a differential equation for e(t) at constant stress #. Similarly, at constant strain-rate, illustrated in Fig. 2, the model must yield a differential equation for 6(0, since an explicit law for #(t) in terms ofe(t) and b(t) = r will result in the strain eM at maximum stress and the initial stress-strain slope varying with r, contrary to the Mellor conjectures. The Spring and Morland 8 formulation results in the uni-axial first order differential relation
6. + , ( )6 = ,p( )~ + n( )e,
(7)
where the response functions ~, ~p, f~ can depend on 6" and e. In the constant stress test # = 6.oH(t),
(8)
where H(t) is the Heaviside unit function, the relation becomes the first order differential equation ~0( )k = f~( )e = 6"0, (t > 0)
(9)
subject to the initial jump condition e(0 + ) = 6.o~o/~O o
(10)
where (1)o, (Po are values at 6 = 0, e = 0; that is, Eo = ~Po/qbo9
(11)
In the constant strain-rate test = rt,
(12)
the relation (7) becomes the first order differential equation
Non-lhzear viscoelastic response o f ice: L. IV. M o r l a n d r )~ + 6. = ~o( )r = f~( )rt
(13)
with the initial condition 3.(0)=0. If ~ 0 then e(0+) -- 0 for a step load, and the algebraic relation (13) shows that the maximum stress 6.M will occur when
I l -- r -Oq9 ~ -- rt -~ld6. ~ -~f = rzOq~ -~e + r2t ~e = 0 ,
(14)
which depends on r. Similarly, the initial slope r - ~d6./dt evaluated at t = 6 = e = 0 also depends on r. These consequences, not noted in reference s, demand that ~- is present in the relation (7). The construction of restricted forms of ~(), ~() and ~ ( ) from constant stress and constant strain-rate responses was discussed in reference 8, together with predictions for some simple multi-axial load histories. However, re-examination of the differential equation (9) for e(t) at constant stress shows that non-uniform solutions are obtained iffl( ) ~ 0 , then re-examination of the reduced differential equation (13) for 3.(0 at constant strain-rate shows that the initial stress-strain slope is ~po/Oo, which is independent of r. An alternative simpler model which does not have the f~( )e term will now be constructed, which has the strain eM at maximum stress at constant strain-rate independent of r and which satisfies the duality conditions (4) and (5). Consider the first order differential relation, written with unit coefficient for the strain-rate
= ~
+ o(e)hO),
(15)
where the E( ) dependence, on 6 and e in general, is not specified at this stage, and the non-rate term is separable in 6. and e. The functions E(), (), h( ) are all positive, and by the initial conditions (10) and (1 I), E = Eo at 6. = 0, e = 0. This may be interpreted as a strain-rate decomposition = be +/~. (16) into elastic and inelastic parts governed by the hypoelastic and creep relations ~ = b/E( ), ~ = g(e)h(6.). (17) Sunder and Wu ~~ introduce a decomposition into three parts with rate dependencies on two further internal variables, resulting in three coupled first order differential relations in place of the single relation (15). They present motivation and mechanisms, but not extending to tertiary creep, and it is not clear that gross response follows such mechanisms. Here the emphasis is on simplicity since non-linear viscoelastic response is not a routine model for analysis or computation of initial and boundary-value problems. It is demonstrated that the major features can be realised by the model (15). In the constant stress test (8), the relation (15) becomes = gCe)h(Oo),
(18)
subject to the initial condition e(O + ) = 3.olEo, (19) based on the conditions (10) and (I 1). For the constant strain-rate test (12), the relation (15) becomes
b
r = ~)
+ g(rt)h(6.),
(20)
subject to 3.(0) = 0. Now the relation (18) implies that
gmh(&o)
(21)
g,, = g(e,,), = constant, ~ 1
(22)
~m(3.0) =
where in view of the property (5) and accommodating the scale factor in h(3.). Thus h(#o) is simply the minimum strainrate dependence on the stress 0o, defining the secondary creep or viscous law. A variety of experimental data was analysed by Smith and Morland ~s, not in any agreement, and empirical functions h(~') based on chosen data were constructed. Key features are h(O) = O, h'(6.) > O, h"(#) > 0.
(23)
This direct interpretation of h(6.) hinges on the assertion that e,,, is independent of 3.. It now follows from the relation (18) that (24)
a(e) = e(e)/h(3.o),
primary, secondary, tertiary strain-rate dependence on strain e is identical for all constant stress levels, and the response curves at 3.~ and 3"2 are simply scaled at each e by the ratio h(6.0h/(#2), equivalently the ratio of their minimum strain-rates. While qualitatively acceptable, and correct by construction for the important secondary creep, this property has not been verified for detailed data sets over the entire strain range. Given the difficulties and inconsistencies associated with ice testing, such a property may remain in doubt, but the model is attractive for its simplicity and overall correlations. The primary, secondary, tertiary creep properties are described by O<%e < em:
g'(e) < O,
e =em:
g'(e,,) = 0,
e > e,,:
g'(e) > 0,
(25)
where g(0) is an extrapolation of g(eo) as eo ~ 0 (3.0 ~ 0). The constant strain-rate response is now governed by the differential equation (20) for 6.(t) = ~-(e/r), where g(e), h(a) are prescribed but E is still unspecified other than its initial value E o. Immediately, 6(0)= r/Eo > 0 as required, but it must be shown that O <.%t < eM/r: t=tM=eM/r: t > e~t/r:
b > O,
~=0,
(26)
~'<0,
where eM = e,,; that is, the solution 6(0 of the differential equation (20) increases monotonically to its maximum 6.M at t M = enJr, then decreases monotonically. The properties (25) and (26) apply only to ranges of e before the extrapolated dashed line behaviours shown in Figs 1 and 2 are realised. Consider the differential equation (20) expressed in terms of the stress-strain response 6.(e) for each r as shown in Fig. 2, thus 1 d6. - 1 - r-lg(e)h(3.), E( ) de
3-(0)= 0.
(27)
Suppose the first stationary point (maximum) of the solution 6.(e) occurs at e = e~ < e,,, where g'(e) < 0, then differentiating equation (27) shows that (d26./de 2) = - - r - l E ( )h(6.)g'(e 0 > 0 at e = e x, which is a contradiction to (d#/de) becoming zero from positive values. Hence the first maximum occurs at et >/e~t, provided that g(e)h(3.) actually reaches the value r in the
Applied Ocean Research, 1991, Vol. 13, No. 5
257
Non-lhlear viscoelastic response o f ice: L. IV. Morland appropriate range of e. It will be shown that a response function E(6.) can be constructed which yields a solution with (d6./de) = 0 at e = eM for all r. Given that (d6./de) = 0 at e = e n at a stress 6. = 6"M(r), then independent of E(6.), using the properties (5) and (22) hE6'M(r)] = r.
(28)
The monotonic property (23) shows there is a unique inverse maximum stress function 6.,,(r) satisfying OM(0) = 0,
6'i~t(r)> 0.
(29)
However, the initial value problem (27) for a prescribed E(6.) has a unique solution 6.(e,r) with 6.(eM,r) ~ #M(r) in general; that is, (O6./de):~ 0 at e = e M. Since the first stationary point (maximum) is at e~ >eM, and g(el) > g(eM) = 1, then if 6.(e~,r) = 6.~(r), it follows that h[~'l(r)] < r which implies that 6.1(r) < 6.n(r). Hence, since (d6./de) > 0 in 0 ~< e < e 1, then 6.(eM,r ) < 6.M(r). In particular, if 6.*(e,r) is the solution of the initial value problem (27) for E ( ~ ) ~ Eo, then
6.*(eM, r) = 6.*t(r ) < 6.M(r).
(30)
The construction of E(6.) requires that 6.*t(r) is strictly increasing except perhaps at a finite set of values r. Suppose that for e < e2, (O6"*(e,r)/Or) < 0, then differentiating equation (27) with respect to r for E(6.)=~E shows that (a26.*/&Oe) > 0 for e > e2: that is, 6.* decreases and its slope increases with r. Since there is a common slope E o at e = 0 for all r, and 6.*(0) = 0, this is not possible, Thus (O6.*/Or) > 0 for some range 0 ~< e > e 3 where e 3 = e3(r). If (06.*/8r) = 0 at e a, then (026*/8rOe) > 0 at e a so that (06.*lOt) becomes positive again for e > %. There remains the possibility that (O6.*/dr):~O at e = e M for some range [r~, r2] of r. However, this implies that 8(86.*/Oe)/Or > 0 at e = e~t on [r l, r2] which, since (O6.*/Or) > 0 for e > e n, implies that the solution curves for r = r~ and r = r 2 continue to diverge so that 6"*(eM,r2) > 6'*(eM, r0, contradicting the uniform 6.*(eM, r) proposition. Thus -,t
~rM (r) > 0, (31) except perhaps at a finite set of discrete values r. The differential equation (27) can now be written
E o d6. E(6.) de
O6.*(e, r) 8e '
6.(0)= O,
(32)
f~
(33)
(34)
using the definition (28). By the inversion (28),
E(6.) = Eoht(6.)6.*~[h(6.)].
258
Applied Ocean Research, 1991, Vol. 13, No. 5
t~ = 6. = 6. M [~,.(6.)],
(38)
is confirmed. Thus a minimal differential relation (15) with three response functions 9(e), h(6.), E(6) satisfies the major duality relations (4) and (5), correlates with the minimum strain-rate as a function of the constant stress and the complete strain-rate function of strain at one constant stress, and with the maximum stress as a function of the constant strain-rate. TEMPERATURE INFLUENCE If the ice volume change is linearly thermoelastic under change of pressure p = - (l/3)tr(a) and temperature T, where a is the Cauchy stress tensor, then the infinitesimal dilatation is given by
0 = --h'(p-po) + ~h'(T-To),
(39)
where Po, To are the initial pressure and temperature and h,e are the compressibility and thermal expansion coefficient respectively. The second of the identities (2) neglects 0 compared to unity, but the limit e--* 1 conclusions hold generally. The viscoelastic creep is essentially a shear property and is strongly temperature dependent in its rate effects, though initial moduli (here E o and K-~) are relatively insensitive to temperature changes over practical ranges. The most simple model then is to postulate thermorheologically simple response in which temperature influences only the time scale of response at a given stress ~6'4. If T(X, t), t >1 O, is the temperature history of a given material element X, a pseudo-time ((X, t) for the element X is defined by
O_f = a[T(X, t)], ((X, 0) = 0, Ot
(40)
or equivalently
((X, t) = f l a[T(X, t')dt',
(41)
(35)
(36)
if(T) > 0 (T ~ T,,),
(42)
where Tm is the melt temperature. For each element X the time t is replaced by the pseudo-time ((X, t) in the viscoelastsic relation so that a material time derivative of any variable q(X, t) is replaced by a derivative with respect to (; that is, if q(X, t) = O(X, O,
Ot
Note, that by the relation (33) since 6.M(r)~ 0 as r-~ 0. -@t -t that aM(r)--*aM(r ) as r ~ 0 , and h'(6.M)6.1~l(r)=~l, so E(0) ~ E o as 6. ~ 0 in the expression (35), as required. Finally, by the relations (21), (22) and (28), b,.[6.n(r)] = h[6.M(r)] = r,
(37)
so by the monotonicity (23), the second of the duality conditions (4), namely
a(To) = 1,
for a solution with 6.(eM, r) = 6.M(r), which determines the required E(6.) explicitly by EE6.M(r)] = Eoh'E6.u(r)]6.*~(r),
h(#) = b,,(6.) = h(6.),
where a(T) is a positive rate factor with properties
and hence
" Eod6. E(6.) = 6"*t(r)
confirming the first of the duality conditions (41), and if 0M[bm(6')] = 6 say, then by the relations (28), (21) and (22),
*--=
O~
a i r ( x , t)].
(43)
In essence, as T increases so that a(T) increases, the rate of change of q on the real time scale t is greater than that on the pseudo-time scale ( which is expressed by the modified differential relation. The uni-axial relation (15) now becomes
E(#)O = a = a(T)E(#)h(6.)g(e).
(44)
Non-linear viscoelastic response of ice: L. Iv. Morland If T is uniform, a(T) is simply a scale factor. If temperature depends only on particle reference position X, which is unsteady with respect to spatial position unless displacements are infinitesimal, then the relation (15) reflects temperature induced non-homogeneity of the mechanical response. For a general temperature field T(X, t) the coefficient a(T) has explicit t dependence. In dynamic impact problems the simpler approximation T(X) will be satisfactory. Note that the gradient a'(T) is most pronounced near melting T,,, so that uniform a(T) may be adequate for very cold ice. In view of the time contraction as T and a(T) increase, the response time parameters t,,~) and tM(r) are also decreasing functions of temperature as noted earlier, rapidly decreasing near melting. Empirical functions a(T) over 251K--*273K and 212K ~ 273K were constructed by Smith and Morland ts based on Mellor and Testa 17 data, which show a(t) decreasing by factors of 0.014 and 3.64 x 10 -s respectively. CONSTITUTIVE LAWS Constitutive laws describing general three-dimensional deformation must be tensor relations (co-ordinate independent) satisfying frame indifference (objectivity); see, for example, TruesdelP 8. Here the variables are stress, strain, stress-rate and strain-rate, for which there are various tensor measures offering different advantages. In addition, the laws may be restricted by particular material symmetries (isotropies) in a given reference configuration. Generalisation of the uni-axial relation (44) coupled with the isotropic dilatation relation (39) involves both directional assumptions and choices of tensor variables, resulting in entirely different multi-axial responses from models consistent with the relations (44) and (39). Conventional multi-axial tests yield very little extra information to test these generalisations, Morland and Earle ~9, so a significant experimental and theoretical research programme is necessary to establish convincing laws. Simple theoretical generalisations can be explored to compare predictions for basic loading configurations and highlight particular features. If x and X are spatial and referential particle position with components xi, X~ (1 = 1, 2, 3) in rectangular coordinates Ox~, the deformation gradient F and displacement gradient H (both with respect to material coordinates) are given by
which are essentially non-linear in the displacement gradient. The linear approximations E = ~ = 89 + H z)
are valid only when each displacement gradient is infinitesimal; that is, both strain and rotation are infinitesimal:
II n II << t, it E it << I, [I to [1 = 11~(H - H T) II << 1.
F o = 3X i
where 6o is the Kronecker delta and 1 is the unit tensor. Common strain measures are C=FrF=U
z,
B=FF r=V2,
(46) E =
89
-
1),
=
-
H I = e 1, E = (I + ~et), H2 = e2, E2
e2(1 + 89
and the rotation is zero The rate of strain D with .respect to the current configuration, which measures rates of stretch of line elements and rates of angle decrease between line elements, and the current spin or vorticity W are the symmetric and anti-symmetric parts ofthe spatial velocity gradient L: L,j = ~dvi (i, j = 1, 2, 3), L = D + W,
(52)
where v(x, t) is the spatial velocity field. There follows the identity F" = LF,
(53)
but no simple relations between I21 and D and between R and W. In the uni-axial configuration the principal components are Fi = I + e~, L i = Di: b, = {1 + e,)O,,
(i = 1, 2, 3),
(54)
which implies the strain-rate relation (2) with DI = k. A referential strain-rate is I~ = F r D F ,
(55)
which yields the uni-axial result/~t = (1 + el}~r Tractions per unit current cross-section are measured by the Cauchy stress tr, while tractions per unit reference cross-section are measured by the Piola-Kirchoff stress g" given by b = 3tr(Fr) - 1, j ___detF = Po/P
(56)
where p is density and Po is reference density. In uni-axial stress J = (1 + el)(l + e:) 2
(57)
which yields the stress relation (2) for ~t = - t r , a t = tr, el = - e when J = 1 to impose incompressibility. Constitutive laws are conveniently expressed as relations between frame indifferent tensors or between frame invariant tensors, defined respectively by A* = QAQ r and A* = A,
(58)
1),
where F has the unique polar decompositions F = RU = VR
(47)
such that R is the rotation of the principal stretch axes and U, V are positive-definite symmetric tensors measuring the stretches of line elements and changes of angle (shearing) between line elements. It follows that E=-~(HrH+H+Hr),e=~(HH
(50)
For strains of a few per-cent in the tertiary creep range, the approximation (50) may not be satisfactory. In the uniaxial stress configuration with axial extension gradient e t = - e and equal lateral extension gradients e 2 = e a, the principal strain components are
?~1 = dal/(1 + el), ~A'! = ~ij -4- Hi./, F = I q- H, (i, j : 1, 2, 3), (45)
(49)
r+H+Hr),
(48)
where A* represents the components of A following a rotation Q(t) of the co-ordinate frame V, B, e, D and a are frame indifferent while U, C, E, E, FrS, RT# are frame invariant, and F* = QF, R* = QR, W* = QWQ T + QQT, 8" = QS.
(59)
Frame indifferent stress-rates are the Jaumann rate b = b - Wa + aW,
Applied Ocean Research, 1991, VoL 13, No. 5
(60)
259
Non-linear viscoelastic response o f ice: L. IV. Morland involving the spin W, the polar rate introduced by Dienes 2~ (61)
~" = b q- o ' R R r -- R R T a ,
involving the rotation R and rotation-rate 1~, and a rate 5" = b + a L + L Ta
(62)
introduced by Spring and Morland 8, while (FrS") 9 and (Rrs")" are frame invariant. Dienes 2~ notes that using the Jaumann rate (60) in an isotropic hypoelastic model of simple shear gives physically unreasonable response, while the polar rate (61) gives sensible response; the rate (62) is also satisfactory. Generalisation of the uni-axial relation (44) imposes parallelism between the chosen strain-rate and stress-rate tensors, and a tensor direction associated with the non-rate term. A realistic approach is to adopt the infinitesimal dilatation law (39) and seek an isotropic deviatoric viscoelastic relation to recover the form (44) in uni-axial stress. It seems sensible to assume that the current deviatoric strain-rate D' and current deviatoric stress o' are parallel, where D' = D -- (1/3)01, 0 = J/J, 0' = a + pl,
(63)
noting that ~:r may not be negligible near I = 0 where d --- do = 0(Eo). By comparison with the starting form (15) for el, this relation is in terms of Cauchy stress o, including the argument 1 of d and h, and there is the extra factor (1 + et) in the stress-rate term. In terms of 6"t, the relation (70) can be written [1 -- (1 + el)n(I)6"l]~ 1 = (1 +
+ (2/3)~(e01~(I)~1 I = (2/9)(1 + e02612, ~(el) = (1 + e02O(K),
1 da--Z= (1 + e O- t { 1 - r-tg(eOh(al)}, G(at) de1 at(0) = 0,
or
b' = b" + a'W - Wa',
(64)
based on the rates (61) and (60), which both have the properties a'1 = ~ (uni-axial stress).
(65)
The two definitions (64) involve rotation-rate from reference and current states respectively, so the latter is more appropriate to the use of D' and a'; they have different tensor directions! A convenient measure of shear stress magnitude is the invariant = ~312 2 (uni-axial stress).
(66)
Strain is necessarily measured from the reference state, and a convenient measure of shear strain magnitude is the principal invariant K = 89
2 - tr(C2)} - 3,
(~(I)D' = d-' + a(T)~l(K)G(l)l~(l)a'.
(68)
In uni-axial stress, with the infinitesimal 0 approximation, J = 1,(1 + e2) 2 : (1 --F et) -1, K=(1
+ex) -2 + 2(1+
= ln(1 + et),
2D2 = -- Dl + (1/3)~61,
REFERENCES 1
2 3 4 5
7 8
(70) 9
where
H(I) = [(2/3) + (l/9)Kd(l)]/d(l),
260
Applied Ocean Research, 1991, Vol. 13, No. 5
(71)
(75)
I am very grateful to the Ophthalmic Unit at the West Norwich Hospital whose skills and care made completion of this manuscript possible.
and the deviatoric relation (68) at T = TO reduces to kt = (1 + el)H(I)b I + (2/3)(1 + e~)O(K)l~(I)al,
~(0) = 0,
ACKNOWLEDGEMENT
6
(69)
~(~.)= 9(et),
then the previous form is obtained in terms of ~.
a t = (1 + el)6-1,
e l ) - 3,
(74)
and given that a t has a maximum aM(r) at e 1 = e M = constant = e,, for all r, the construction of G(a 0 is similar to the previous construction of E(a). Introduce a new strain variable
(67)
which is identically zero in uniform dilatation, and in the reference state. A possible deviatoric relation is then
(73)
but now the separable form (15) is even more distorted. For letl << 1 with 01 = 6, the form (15) is recovered. Suppose though, that the uni-axial responses have the same interpretations in terms of al as in 6"1 before, then the relation (70) for constant a I requires that (2/3)1~(I)a I = h(a 0 is simply bm expressed as a function of a~, where e,, is constant, independent of a~, and (1 + eOO(K ) = 0(e0, with 9(e,,) = 1,is the shape of one chosen strain-rate response in terms of strain. Define H(I) = G(aO, then the constant strain-rate ~ = r differential equation (27) has the modified form
b' = b' + a ' R R r -- l~Rra '
I = 89
(72)
where
and choose a parallel frame indifferent deviatoric stress rate
tr(~') = 0,
eO2n(I)~
10
Mellor, M. Mechanical properties of polycrystalline ice, in Physics and Mechanics of Ice (Ed. TD'de, P.), 2 ! 7-245, Proceedinos of the IUTAM S)'mp., Copenhagen, Denmark, 1979. Springer-Verlag, Berlin, 1980 Sinha, N. K. Rheology of columnar grained ice, Exp. Afech., 18, 464-470, 1978 Sinha, N. K. Short term theology of polycrystalline ice, J. Glacial., 21,457-473, 1978 Morland, L. W. Constitutive laws for ice, Cold Re9. Sci. Tech., l, 101-108, 1979 Gold, L. W. and Sinha, N. K. The rheological behaviour of ice at small strain, Physics and Mechanics of Ice (Ed. Tryde, P.), 117-128, Proceedings of the IUTAM Symp., Copenhagen, Denmark, 1979. Springer-Verlag, Berlin, 1980 Morland, L. W. and Spring, U. Viscoelastic fluid relation for the deformation of ice, Cold Re9. Sci. Tech., 4, 255-268, 1981 Morland, L. W. and Spring, U. Single integral representations for nonlinear viscoelastic solids, Mech. Materials, 1, 161-170, 1982 Spring, U. and Morland, L. W. Viscoelastic solid relation for the deformation of ice, Cold Re9. Sci. Tech., 5, 221-234, 1982 Spring, U. and Morland, L. W. Integral representations for the deformation of ice, Cold Re 9. Sci. Tech. 6, 185-193, 1983 Williams, H. T. Single integral representations in ice mechanics, Cold Reg. Sci. Tech, 9, 89-95, 1984
Non-lhtear viscoelastic response o f ice: L. IV. M o r l a n d 11 12 !3 14 15
Sunder, S. S. and Wu, M. S. A differential flow model for polycrystalline ice, Cold Reg. Sci. Tech., 16, 45--62, 1989 Sunder, S. S. and Wu, M. S. A multi-axial differential model of flow in orthotropic polycrystalline ice, Cold Reg. Sci. Tech., 16, 223-235, 1989 Mellor, M. and Cole, D. M. Deformation and failure of ice under constant stress or constant strain-rate, Cold Reg. Sci. Tech., 5 201-219, 1982 Mellor, M. and Cole, D. M. Stress/strain/time relations for ice under uni-axial compression, Cold Reg. Sei. Tech., 6, 207-230, 1983 Smith, G. D. and Morland, L. W. Viscous relations for the steady creep of polycrystalline ice, Cold Reg. Sci. Tech., 5, 141-150, 1981
16 17 18 19
Morland, L. W. and Lee, E. H. Stress analysis for linear viscoelastic materials with temperature variation, Trans Soc. Rheol., 4, 233-263, 1960 Mellor, M. and Testa, R. Effects of temperature on the creep of ice, J. 61aciol., 8, 131-145, 1969 Truesdell, C. "/he Elements of Continuum Mechanics, SpringerVerlag, Berlin, 1966 Morland, L. W. and Earle, E. N. Correlation of shear behaviour of ice with bi-axial stress response, 183-192, 3, Proceedings of
the 7th Int. Conf. on Port and Ocean Engineering under Arctic Conditions, Helsinki, Finland (Ed. Technical Research Centre of 20
Finland) 1983 Dienes, J. K. Theory of deformation, Part I: Kinematics, /os Alamos Report LA-11063-MS, 1, 1987
Applied Ocean Research, 1991, Vol. l & No. 8
261
School of Mathematics, University of East Anglia, Norwich NR4 7TJ U.K. The duality of constant stress and constant displacement-rate response of ice in uni-axial stress tests is used to infer a first order differential, non-linear viscoelastic, relation which describes qualitatively both responses through primary, secondary and tertiary creeps. It is the most simple model which reflects this observed transient behaviour. The dominant infhtence of temperature is incorporated through a rate factor, Extrapolation to three-dimensional response is far from unique, even with an isotropy assumption, and highlights the sparsity of multi-axial data to establish a satisfactory engineering model.
INTRODUCTION Coherent masses of ice, in the sea or on land, are now an important part of the world's engineering environment. The manner in which ice deforms, and fails, under mechanical loads and changing thermal conditions is an essential component of cold regions engineering. The effects of ice contact on structures involves coupled stress analyses in which both ice and structure are responding to the loads, in manners governed by their respective material properties - constitutive descriptions. Any analysis of such couples responses, even in idealised geometries and with the power of high speed computation, requires the most simple constitutive model for the ice consistent with the main features of its known behaviour. Time scales of applications vary widely, from dynamic problems in which wave propagation effects are important, to slower quasi-static problems in which particle momentum associated with the deformation is negligible though rigid body momentum may be significant. A long-time scale viscous fluid model appropriate to glacier flow over years to thousands of years is not applicable to engineering contact problems. Ice is essentially viscoelastic over the time scale of seconds to hours, or months, depending on temperature. It exhibits infinitesimal elastic strains on application of a step load, followed by creep to small, but not infinitesimal, strains. The response is highly non-linear even at small strains, and the time scales, or rates, are strongly influenced by temperature. I present here a non-linear viscoelastic model of differential type, in fact a first order differential relation, which has the minimal ingredients necessary to model key features of laboratory uni-axial test data, and which is sufficient to provide good qualitative and quantitative correlation. The basis for the formulation lies in the responses in uni-axial stress tests both to applied constant stress (creep test) and to applied constant displacementPaper accepted October 1990. Discussion closes April 1992. 9 1991 Elsevier Science Publishers Ltd
254
Applied Ocean Research, 1991, VoL 13, No. 5
rate, as illustrated by Mellor t, with further constructions related to the duality of these response conjectured by Mellor. Early discussions of the non-linear creep response, Sinha 2"3, Morland 4, Gold and Sinha 5 focussed only on the constant stress response, and inferred relations are entirely inconsistent with the constant displacement rate response. Subsequently, recognising that the dual response must be satisfied, Morland and Spring 6'7, Spring and Morland 8"9 and Williams 1~ constructed both differential and integral type laws for both fluid and solid behaviour, which exhibit the required qualitative features of both responses. Very recently, Sunder and Wu 11"12 have constructed uni-axial and multi-axial orthotropic differential type models under the restriction of infinitesimal strain which describe both responses in primary and secondary creep, but do not attempt to encompass tertiary creep. Their approach through internal variables of state requires further differential relations to determine the evolution of the new variables, adding more degrees of freedom to the model but at the expense of equation complexity - a system of three coupled first order differential equations for uni-axial response. My aim is to show that one first order differential equation can describe the qualitative features of both response in uni-axial stress, including tertiary creep, and that there are many ways to extrapolate to threedimensional behaviour at finite strain, isotropic or anisotropic, when multi-axial data is available. The dominant temperature influence is incorporated through a rate factor. Data and models reflect the response of a small homogeneous sample. Non-homogenity on the large scale, if recognised, could be incorporated into numerical applications by prescribing non-homogeneous material functions and parameters in the common differential law. The present differential law is an alternative to that constructed for a solid response by Spring and Morland 8, with a reduced shape, and fewer terms. It therefore implies a stronger duality of constant stress and constant displacement-rate responses. It is, of course, a continuum
Non-linear viscoelastic response o f ice: L. IV. Morland
model for coherent ice which determines stresses and deformations under prescribed loading histories. Fracture, and post-fracture behaviour, require new theory, but the continuum model is necessary to predict fracture under a given fracture criterion, and to show the pattern of stress development prior to fracture.
_
1 i
UNIAXIAL R E S P O N S E Consider the idealisation of a constant temperature laboratory uni-axial load test on an ice sample in which load and contraction (displacement) are measured as functions of time t. Let ~ denote the nominal axial compressive stress (force per unit initial cross-section), and e denote the axial compressive engineering strain (contraction per unit initial length). The constant load test measures e(t) when a step function stress ~" is applied and maintained constant - a creep test. There is an instantaneous infinitesimal strain e o given by (1)
e o = a l e o,
where E o is the initial (Young's) modulus, followed by a primary decelerating creep from an initial strain-rate b(0) to a minimum strain-rate b,, at time t,, and strain e,,, and then by an accelerating tertiary creep in which k increases. The minimum strain-rate is termed secondary creep. Figure 1, from Meilor ~, illustrates this response in the form of a strain-rate b dependence on strain e, noting that e is a strictly monotonic increasing function of t (b > 0). This figure shows two curves to indicate the effects of different constant ~', namely, that strain-rates increase with ~ at a given strain. The instantaneous strain eo is depicted as a small gap, which may be negligible in comparison with e,, (a factor 1 0 - 2 a t 6-~ 106Nm- 2). Dashed lines show the expected extrapolation (if a coherent uni-axial load configuration could be maintained) into finite strain which would require ~--* 0 as e ~ 1. The limit is discussed below. A constant contraction-rate test measures ~ as a function of e = rt at constant E = r. Figure 2, from
,\
I
\+I/"
"\ ,
, I I e
r
I
"---/
I ,
o
e~
1
~ e
Fig. 2. Stress dependence on strain at constant strain-rate.
Mellor t, shows two curves at different values of r to indicate the effect of increasing strain-rate, namely, that increasing r increases the stress at a given strain. Corresponding to primary creep the stress-strain slope decreases continuously to zero at a strain e~t = rtM where the stress # has a maximum 6at, associated with secondary creep, then corresponding to tertiary creep the slope takes increasing negative values as the stress decreases. Again the dashed lines show expected continuum extrapolation into finite strain. The viscous model assumed for the long-time shear response in glacier flow relates the natural strain-rate to the Cauchy stress, but commonly adopts the secondary creep response, that is, the minimum strain-rate Era(#), instead of the long-time asymptotic of tertiary creep. With the common incompressibility assumption, the axial natural strain-rate ~ and Cauchy stress a in uni-axial stress, measured positive in tension, are related to the engineering measures by E = --E/(I -- e),
t'~(6) < O,
e 0
1
a = --(1 -- e)6".
(2)
Thus, as e ~ I at constant 6", a ~ 0, which implies k --* 0 for a viscous relation, so that ~--* 0 more rapidly, like the product L(I -- ~). As e ---, 1 at constant E = r,/~ ~ - 0% implying that a ~ - oo and hence # ~ oo like the product - a ( 1 - e ) - t . These limiting behaviours are reflected in the dashed line extrapolations in Figs I and 2. However, Mellor x suggests that the tertiary limit and secondary creep may be little different at the low shear stresses in typical glacier flows. The time t m to minimum strain-rate at constant stress is a rapidly decreasing function of 0, and the time tat to maximum stress at constant strain-rate r is a rapidly decreasing function of r. Thus
, _
/t'
1 m
Fig. 1. Strain-rate dependence on strain at constant stress.
e
t'M(r) < O,
(3)
and the rapid variation implies a highly non-linear viscoelastic response even at small strain. Linear viscoelastic response requires t., and tat to be respectively
Applied Ocean Research, 1991, Vol. 13, No. 5
255
Non-linear viscoelastic response o f ice: L. IV. Morland
independent of 6. and r. In addition, both t,, and t M are rapidly decreasing functions of temperature Twhen the same constant stress and constant strain-rate tests are performed at different constant temperatures. Temperature influence is discussed later. Now Mellor ~ has conjectured that the constant stress and constant strain-rate response each fully determine the (uni-axial) mechanical properties. That is, the two responses, for all constant 6- and all constant r, contain equivalent information. However, a particular shape of relation may be correlated with one of the responses for a discrete set of results and then predict an entirely incorrect picture of the second response. For example, Morland 4 constructs a second order differential relation for shear strain-rate in terms of stress based solely on primary creep response, which at constant strain-rate implies constant stress. Mellor ~drew attention to the two observed responses which must both be described by a valid constitutive model. The subsequent Morland and Spring 6"7 and Spring and Morland 8.9 constructions permitted both shapes of response shown in Figs 1 and 2. The models, differential and integral types, contained response functions neither completely determined by a family of creep tests nor by a family ofconstant strain-rate tests, and with differing amounts of information required from the second set of responses; that is, the Mellor duality is not complete. A re-examination of first order differential relations of solid type, treated by Spring and Morland s, in conjunction with two explicit Mellor 1 duality conjectures, now shows that two of the response functions of a differential relation with three response functions of single argument is determined by a family of creep tests while the third is determined by the maximum stress at each constant strain-rate. This is surely the most simple model describing both types of response qualitatively and satisfying the major quantitative correlations through the imposed duality conditions. The adopted Meilor I conjectures, reinforced by Mellor and Cole 13A4 are as follows. First, the major duality condition asserts that the minimum strain-rate at a given constant stress is the constant strain-rate which delivers that stress as the maximum stress, and, vice-versa, that the maximum stress at a given constant strain-rate is the constant stress which delivers that strain-rate as the minimum strain-rate. These duality conditions are expressed by bm[6.M(r)] = r,
6.MEe,,(6.)]= 6..
(4)
The second conjecture is that the strain at minimum strain-rate at constant stress is independent of the stress, and that the strain at maximum stress at constant strain-rate is independent of strain-rate, and these two constant strains are equal, thus e,, = e~t = constant.
(5)
This constant strain is approximately one per cent, small but not infinitesimal; that is em = eM ~ 0.01.
256
Applied Ocean Research, 1991, Vol. 13, No. 5
(6)
A third conjecture is that the initial stress-strain slope at constant strain-rate is independent of strain-rate, but using the stress-strain jump conditions deduced by Spring and Morland 8, for the proposed form of differential relation, the initial slope is identical to the Young's modulus E o defined by the initial jump condition (1) which is not related to a strain-rate, so this conjecture is satisfied. Approximate elastic analyses use effective moduli distinct from initial values governing the instantaneous response, in an attempt to average the creep effects, but these will be in serious error when application time scales are of order tm or t M, which vary over a very wide range as stress and strain-rate levels change. UNI-AXIAL RELATION The earlier viscoelastic constructions 6.7,8.9 started from frame indifferent tensor relations of appropriate shapes so that restriction to uni-axial stress permitted at least qualitative correlation with constant stress and constant strain-rate responses. It is simpler to develop directly a uni-axial differential relation to match qualitatively the responses shown in Figs 1 and 2 and satisfy the duality conditions (4) and (5). However, the extrapolation to a three-dimensional constitutive law remains a complex problem. An instantaneous (elastic) response to a step load is given by relation (1) in terms of the initial modulus Eo. The subsequent primary-secondary-tertiary creep, illustrated by Fig. 1, requires that the strain e increases monotonically with time through decelerating, stationary and accelerating phases while e is still small. The model must therefore yield a differential equation for e(t) at constant stress #. Similarly, at constant strain-rate, illustrated in Fig. 2, the model must yield a differential equation for 6(0, since an explicit law for #(t) in terms ofe(t) and b(t) = r will result in the strain eM at maximum stress and the initial stress-strain slope varying with r, contrary to the Mellor conjectures. The Spring and Morland 8 formulation results in the uni-axial first order differential relation
6. + , ( )6 = ,p( )~ + n( )e,
(7)
where the response functions ~, ~p, f~ can depend on 6" and e. In the constant stress test # = 6.oH(t),
(8)
where H(t) is the Heaviside unit function, the relation becomes the first order differential equation ~0( )k = f~( )e = 6"0, (t > 0)
(9)
subject to the initial jump condition e(0 + ) = 6.o~o/~O o
(10)
where (1)o, (Po are values at 6 = 0, e = 0; that is, Eo = ~Po/qbo9
(11)
In the constant strain-rate test = rt,
(12)
the relation (7) becomes the first order differential equation
Non-lhzear viscoelastic response o f ice: L. IV. M o r l a n d r )~ + 6. = ~o( )r = f~( )rt
(13)
with the initial condition 3.(0)=0. If ~ 0 then e(0+) -- 0 for a step load, and the algebraic relation (13) shows that the maximum stress 6.M will occur when
I l -- r -Oq9 ~ -- rt -~ld6. ~ -~f = rzOq~ -~e + r2t ~e = 0 ,
(14)
which depends on r. Similarly, the initial slope r - ~d6./dt evaluated at t = 6 = e = 0 also depends on r. These consequences, not noted in reference s, demand that ~- is present in the relation (7). The construction of restricted forms of ~(), ~() and ~ ( ) from constant stress and constant strain-rate responses was discussed in reference 8, together with predictions for some simple multi-axial load histories. However, re-examination of the differential equation (9) for e(t) at constant stress shows that non-uniform solutions are obtained iffl( ) ~ 0 , then re-examination of the reduced differential equation (13) for 3.(0 at constant strain-rate shows that the initial stress-strain slope is ~po/Oo, which is independent of r. An alternative simpler model which does not have the f~( )e term will now be constructed, which has the strain eM at maximum stress at constant strain-rate independent of r and which satisfies the duality conditions (4) and (5). Consider the first order differential relation, written with unit coefficient for the strain-rate
= ~
+ o(e)hO),
(15)
where the E( ) dependence, on 6 and e in general, is not specified at this stage, and the non-rate term is separable in 6. and e. The functions E(), (), h( ) are all positive, and by the initial conditions (10) and (1 I), E = Eo at 6. = 0, e = 0. This may be interpreted as a strain-rate decomposition = be +/~. (16) into elastic and inelastic parts governed by the hypoelastic and creep relations ~ = b/E( ), ~ = g(e)h(6.). (17) Sunder and Wu ~~ introduce a decomposition into three parts with rate dependencies on two further internal variables, resulting in three coupled first order differential relations in place of the single relation (15). They present motivation and mechanisms, but not extending to tertiary creep, and it is not clear that gross response follows such mechanisms. Here the emphasis is on simplicity since non-linear viscoelastic response is not a routine model for analysis or computation of initial and boundary-value problems. It is demonstrated that the major features can be realised by the model (15). In the constant stress test (8), the relation (15) becomes = gCe)h(Oo),
(18)
subject to the initial condition e(O + ) = 3.olEo, (19) based on the conditions (10) and (I 1). For the constant strain-rate test (12), the relation (15) becomes
b
r = ~)
+ g(rt)h(6.),
(20)
subject to 3.(0) = 0. Now the relation (18) implies that
gmh(&o)
(21)
g,, = g(e,,), = constant, ~ 1
(22)
~m(3.0) =
where in view of the property (5) and accommodating the scale factor in h(3.). Thus h(#o) is simply the minimum strainrate dependence on the stress 0o, defining the secondary creep or viscous law. A variety of experimental data was analysed by Smith and Morland ~s, not in any agreement, and empirical functions h(~') based on chosen data were constructed. Key features are h(O) = O, h'(6.) > O, h"(#) > 0.
(23)
This direct interpretation of h(6.) hinges on the assertion that e,,, is independent of 3.. It now follows from the relation (18) that (24)
a(e) = e(e)/h(3.o),
primary, secondary, tertiary strain-rate dependence on strain e is identical for all constant stress levels, and the response curves at 3.~ and 3"2 are simply scaled at each e by the ratio h(6.0h/(#2), equivalently the ratio of their minimum strain-rates. While qualitatively acceptable, and correct by construction for the important secondary creep, this property has not been verified for detailed data sets over the entire strain range. Given the difficulties and inconsistencies associated with ice testing, such a property may remain in doubt, but the model is attractive for its simplicity and overall correlations. The primary, secondary, tertiary creep properties are described by O<%e < em:
g'(e) < O,
e =em:
g'(e,,) = 0,
e > e,,:
g'(e) > 0,
(25)
where g(0) is an extrapolation of g(eo) as eo ~ 0 (3.0 ~ 0). The constant strain-rate response is now governed by the differential equation (20) for 6.(t) = ~-(e/r), where g(e), h(a) are prescribed but E is still unspecified other than its initial value E o. Immediately, 6(0)= r/Eo > 0 as required, but it must be shown that O <.%t < eM/r: t=tM=eM/r: t > e~t/r:
b > O,
~=0,
(26)
~'<0,
where eM = e,,; that is, the solution 6(0 of the differential equation (20) increases monotonically to its maximum 6.M at t M = enJr, then decreases monotonically. The properties (25) and (26) apply only to ranges of e before the extrapolated dashed line behaviours shown in Figs 1 and 2 are realised. Consider the differential equation (20) expressed in terms of the stress-strain response 6.(e) for each r as shown in Fig. 2, thus 1 d6. - 1 - r-lg(e)h(3.), E( ) de
3-(0)= 0.
(27)
Suppose the first stationary point (maximum) of the solution 6.(e) occurs at e = e~ < e,,, where g'(e) < 0, then differentiating equation (27) shows that (d26./de 2) = - - r - l E ( )h(6.)g'(e 0 > 0 at e = e x, which is a contradiction to (d#/de) becoming zero from positive values. Hence the first maximum occurs at et >/e~t, provided that g(e)h(3.) actually reaches the value r in the
Applied Ocean Research, 1991, Vol. 13, No. 5
257
Non-lhlear viscoelastic response o f ice: L. IV. Morland appropriate range of e. It will be shown that a response function E(6.) can be constructed which yields a solution with (d6./de) = 0 at e = eM for all r. Given that (d6./de) = 0 at e = e n at a stress 6. = 6"M(r), then independent of E(6.), using the properties (5) and (22) hE6'M(r)] = r.
(28)
The monotonic property (23) shows there is a unique inverse maximum stress function 6.,,(r) satisfying OM(0) = 0,
6'i~t(r)> 0.
(29)
However, the initial value problem (27) for a prescribed E(6.) has a unique solution 6.(e,r) with 6.(eM,r) ~ #M(r) in general; that is, (O6./de):~ 0 at e = e M. Since the first stationary point (maximum) is at e~ >eM, and g(el) > g(eM) = 1, then if 6.(e~,r) = 6.~(r), it follows that h[~'l(r)] < r which implies that 6.1(r) < 6.n(r). Hence, since (d6./de) > 0 in 0 ~< e < e 1, then 6.(eM,r ) < 6.M(r). In particular, if 6.*(e,r) is the solution of the initial value problem (27) for E ( ~ ) ~ Eo, then
6.*(eM, r) = 6.*t(r ) < 6.M(r).
(30)
The construction of E(6.) requires that 6.*t(r) is strictly increasing except perhaps at a finite set of values r. Suppose that for e < e2, (O6"*(e,r)/Or) < 0, then differentiating equation (27) with respect to r for E(6.)=~E shows that (a26.*/&Oe) > 0 for e > e2: that is, 6.* decreases and its slope increases with r. Since there is a common slope E o at e = 0 for all r, and 6.*(0) = 0, this is not possible, Thus (O6.*/Or) > 0 for some range 0 ~< e > e 3 where e 3 = e3(r). If (06.*/8r) = 0 at e a, then (026*/8rOe) > 0 at e a so that (06.*lOt) becomes positive again for e > %. There remains the possibility that (O6.*/dr):~O at e = e M for some range [r~, r2] of r. However, this implies that 8(86.*/Oe)/Or > 0 at e = e~t on [r l, r2] which, since (O6.*/Or) > 0 for e > e n, implies that the solution curves for r = r~ and r = r 2 continue to diverge so that 6"*(eM,r2) > 6'*(eM, r0, contradicting the uniform 6.*(eM, r) proposition. Thus -,t
~rM (r) > 0, (31) except perhaps at a finite set of discrete values r. The differential equation (27) can now be written
E o d6. E(6.) de
O6.*(e, r) 8e '
6.(0)= O,
(32)
f~
(33)
(34)
using the definition (28). By the inversion (28),
E(6.) = Eoht(6.)6.*~[h(6.)].
258
Applied Ocean Research, 1991, Vol. 13, No. 5
t~ = 6. = 6. M [~,.(6.)],
(38)
is confirmed. Thus a minimal differential relation (15) with three response functions 9(e), h(6.), E(6) satisfies the major duality relations (4) and (5), correlates with the minimum strain-rate as a function of the constant stress and the complete strain-rate function of strain at one constant stress, and with the maximum stress as a function of the constant strain-rate. TEMPERATURE INFLUENCE If the ice volume change is linearly thermoelastic under change of pressure p = - (l/3)tr(a) and temperature T, where a is the Cauchy stress tensor, then the infinitesimal dilatation is given by
0 = --h'(p-po) + ~h'(T-To),
(39)
where Po, To are the initial pressure and temperature and h,e are the compressibility and thermal expansion coefficient respectively. The second of the identities (2) neglects 0 compared to unity, but the limit e--* 1 conclusions hold generally. The viscoelastic creep is essentially a shear property and is strongly temperature dependent in its rate effects, though initial moduli (here E o and K-~) are relatively insensitive to temperature changes over practical ranges. The most simple model then is to postulate thermorheologically simple response in which temperature influences only the time scale of response at a given stress ~6'4. If T(X, t), t >1 O, is the temperature history of a given material element X, a pseudo-time ((X, t) for the element X is defined by
O_f = a[T(X, t)], ((X, 0) = 0, Ot
(40)
or equivalently
((X, t) = f l a[T(X, t')dt',
(41)
(35)
(36)
if(T) > 0 (T ~ T,,),
(42)
where Tm is the melt temperature. For each element X the time t is replaced by the pseudo-time ((X, t) in the viscoelastsic relation so that a material time derivative of any variable q(X, t) is replaced by a derivative with respect to (; that is, if q(X, t) = O(X, O,
Ot
Note, that by the relation (33) since 6.M(r)~ 0 as r-~ 0. -@t -t that aM(r)--*aM(r ) as r ~ 0 , and h'(6.M)6.1~l(r)=~l, so E(0) ~ E o as 6. ~ 0 in the expression (35), as required. Finally, by the relations (21), (22) and (28), b,.[6.n(r)] = h[6.M(r)] = r,
(37)
so by the monotonicity (23), the second of the duality conditions (4), namely
a(To) = 1,
for a solution with 6.(eM, r) = 6.M(r), which determines the required E(6.) explicitly by EE6.M(r)] = Eoh'E6.u(r)]6.*~(r),
h(#) = b,,(6.) = h(6.),
where a(T) is a positive rate factor with properties
and hence
" Eod6. E(6.) = 6"*t(r)
confirming the first of the duality conditions (41), and if 0M[bm(6')] = 6 say, then by the relations (28), (21) and (22),
*--=
O~
a i r ( x , t)].
(43)
In essence, as T increases so that a(T) increases, the rate of change of q on the real time scale t is greater than that on the pseudo-time scale ( which is expressed by the modified differential relation. The uni-axial relation (15) now becomes
E(#)O = a = a(T)E(#)h(6.)g(e).
(44)
Non-linear viscoelastic response of ice: L. Iv. Morland If T is uniform, a(T) is simply a scale factor. If temperature depends only on particle reference position X, which is unsteady with respect to spatial position unless displacements are infinitesimal, then the relation (15) reflects temperature induced non-homogeneity of the mechanical response. For a general temperature field T(X, t) the coefficient a(T) has explicit t dependence. In dynamic impact problems the simpler approximation T(X) will be satisfactory. Note that the gradient a'(T) is most pronounced near melting T,,, so that uniform a(T) may be adequate for very cold ice. In view of the time contraction as T and a(T) increase, the response time parameters t,,~) and tM(r) are also decreasing functions of temperature as noted earlier, rapidly decreasing near melting. Empirical functions a(T) over 251K--*273K and 212K ~ 273K were constructed by Smith and Morland ts based on Mellor and Testa 17 data, which show a(t) decreasing by factors of 0.014 and 3.64 x 10 -s respectively. CONSTITUTIVE LAWS Constitutive laws describing general three-dimensional deformation must be tensor relations (co-ordinate independent) satisfying frame indifference (objectivity); see, for example, TruesdelP 8. Here the variables are stress, strain, stress-rate and strain-rate, for which there are various tensor measures offering different advantages. In addition, the laws may be restricted by particular material symmetries (isotropies) in a given reference configuration. Generalisation of the uni-axial relation (44) coupled with the isotropic dilatation relation (39) involves both directional assumptions and choices of tensor variables, resulting in entirely different multi-axial responses from models consistent with the relations (44) and (39). Conventional multi-axial tests yield very little extra information to test these generalisations, Morland and Earle ~9, so a significant experimental and theoretical research programme is necessary to establish convincing laws. Simple theoretical generalisations can be explored to compare predictions for basic loading configurations and highlight particular features. If x and X are spatial and referential particle position with components xi, X~ (1 = 1, 2, 3) in rectangular coordinates Ox~, the deformation gradient F and displacement gradient H (both with respect to material coordinates) are given by
which are essentially non-linear in the displacement gradient. The linear approximations E = ~ = 89 + H z)
are valid only when each displacement gradient is infinitesimal; that is, both strain and rotation are infinitesimal:
II n II << t, it E it << I, [I to [1 = 11~(H - H T) II << 1.
F o = 3X i
where 6o is the Kronecker delta and 1 is the unit tensor. Common strain measures are C=FrF=U
z,
B=FF r=V2,
(46) E =
89
-
1),
=
-
H I = e 1, E = (I + ~et), H2 = e2, E2
e2(1 + 89
and the rotation is zero The rate of strain D with .respect to the current configuration, which measures rates of stretch of line elements and rates of angle decrease between line elements, and the current spin or vorticity W are the symmetric and anti-symmetric parts ofthe spatial velocity gradient L: L,j = ~dvi (i, j = 1, 2, 3), L = D + W,
(52)
where v(x, t) is the spatial velocity field. There follows the identity F" = LF,
(53)
but no simple relations between I21 and D and between R and W. In the uni-axial configuration the principal components are Fi = I + e~, L i = Di: b, = {1 + e,)O,,
(i = 1, 2, 3),
(54)
which implies the strain-rate relation (2) with DI = k. A referential strain-rate is I~ = F r D F ,
(55)
which yields the uni-axial result/~t = (1 + el}~r Tractions per unit current cross-section are measured by the Cauchy stress tr, while tractions per unit reference cross-section are measured by the Piola-Kirchoff stress g" given by b = 3tr(Fr) - 1, j ___detF = Po/P
(56)
where p is density and Po is reference density. In uni-axial stress J = (1 + el)(l + e:) 2
(57)
which yields the stress relation (2) for ~t = - t r , a t = tr, el = - e when J = 1 to impose incompressibility. Constitutive laws are conveniently expressed as relations between frame indifferent tensors or between frame invariant tensors, defined respectively by A* = QAQ r and A* = A,
(58)
1),
where F has the unique polar decompositions F = RU = VR
(47)
such that R is the rotation of the principal stretch axes and U, V are positive-definite symmetric tensors measuring the stretches of line elements and changes of angle (shearing) between line elements. It follows that E=-~(HrH+H+Hr),e=~(HH
(50)
For strains of a few per-cent in the tertiary creep range, the approximation (50) may not be satisfactory. In the uniaxial stress configuration with axial extension gradient e t = - e and equal lateral extension gradients e 2 = e a, the principal strain components are
?~1 = dal/(1 + el), ~A'! = ~ij -4- Hi./, F = I q- H, (i, j : 1, 2, 3), (45)
(49)
r+H+Hr),
(48)
where A* represents the components of A following a rotation Q(t) of the co-ordinate frame V, B, e, D and a are frame indifferent while U, C, E, E, FrS, RT# are frame invariant, and F* = QF, R* = QR, W* = QWQ T + QQT, 8" = QS.
(59)
Frame indifferent stress-rates are the Jaumann rate b = b - Wa + aW,
Applied Ocean Research, 1991, VoL 13, No. 5
(60)
259
Non-linear viscoelastic response o f ice: L. IV. Morland involving the spin W, the polar rate introduced by Dienes 2~ (61)
~" = b q- o ' R R r -- R R T a ,
involving the rotation R and rotation-rate 1~, and a rate 5" = b + a L + L Ta
(62)
introduced by Spring and Morland 8, while (FrS") 9 and (Rrs")" are frame invariant. Dienes 2~ notes that using the Jaumann rate (60) in an isotropic hypoelastic model of simple shear gives physically unreasonable response, while the polar rate (61) gives sensible response; the rate (62) is also satisfactory. Generalisation of the uni-axial relation (44) imposes parallelism between the chosen strain-rate and stress-rate tensors, and a tensor direction associated with the non-rate term. A realistic approach is to adopt the infinitesimal dilatation law (39) and seek an isotropic deviatoric viscoelastic relation to recover the form (44) in uni-axial stress. It seems sensible to assume that the current deviatoric strain-rate D' and current deviatoric stress o' are parallel, where D' = D -- (1/3)01, 0 = J/J, 0' = a + pl,
(63)
noting that ~:r may not be negligible near I = 0 where d --- do = 0(Eo). By comparison with the starting form (15) for el, this relation is in terms of Cauchy stress o, including the argument 1 of d and h, and there is the extra factor (1 + et) in the stress-rate term. In terms of 6"t, the relation (70) can be written [1 -- (1 + el)n(I)6"l]~ 1 = (1 +
+ (2/3)~(e01~(I)~1 I = (2/9)(1 + e02612, ~(el) = (1 + e02O(K),
1 da--Z= (1 + e O- t { 1 - r-tg(eOh(al)}, G(at) de1 at(0) = 0,
or
b' = b" + a'W - Wa',
(64)
based on the rates (61) and (60), which both have the properties a'1 = ~ (uni-axial stress).
(65)
The two definitions (64) involve rotation-rate from reference and current states respectively, so the latter is more appropriate to the use of D' and a'; they have different tensor directions! A convenient measure of shear stress magnitude is the invariant = ~312 2 (uni-axial stress).
(66)
Strain is necessarily measured from the reference state, and a convenient measure of shear strain magnitude is the principal invariant K = 89
2 - tr(C2)} - 3,
(~(I)D' = d-' + a(T)~l(K)G(l)l~(l)a'.
(68)
In uni-axial stress, with the infinitesimal 0 approximation, J = 1,(1 + e2) 2 : (1 --F et) -1, K=(1
+ex) -2 + 2(1+
= ln(1 + et),
2D2 = -- Dl + (1/3)~61,
REFERENCES 1
2 3 4 5
7 8
(70) 9
where
H(I) = [(2/3) + (l/9)Kd(l)]/d(l),
260
Applied Ocean Research, 1991, Vol. 13, No. 5
(71)
(75)
I am very grateful to the Ophthalmic Unit at the West Norwich Hospital whose skills and care made completion of this manuscript possible.
and the deviatoric relation (68) at T = TO reduces to kt = (1 + el)H(I)b I + (2/3)(1 + e~)O(K)l~(I)al,
~(0) = 0,
ACKNOWLEDGEMENT
6
(69)
~(~.)= 9(et),
then the previous form is obtained in terms of ~.
a t = (1 + el)6-1,
e l ) - 3,
(74)
and given that a t has a maximum aM(r) at e 1 = e M = constant = e,, for all r, the construction of G(a 0 is similar to the previous construction of E(a). Introduce a new strain variable
(67)
which is identically zero in uniform dilatation, and in the reference state. A possible deviatoric relation is then
(73)
but now the separable form (15) is even more distorted. For letl << 1 with 01 = 6, the form (15) is recovered. Suppose though, that the uni-axial responses have the same interpretations in terms of al as in 6"1 before, then the relation (70) for constant a I requires that (2/3)1~(I)a I = h(a 0 is simply bm expressed as a function of a~, where e,, is constant, independent of a~, and (1 + eOO(K ) = 0(e0, with 9(e,,) = 1,is the shape of one chosen strain-rate response in terms of strain. Define H(I) = G(aO, then the constant strain-rate ~ = r differential equation (27) has the modified form
b' = b' + a ' R R r -- l~Rra '
I = 89
(72)
where
and choose a parallel frame indifferent deviatoric stress rate
tr(~') = 0,
eO2n(I)~
10
Mellor, M. Mechanical properties of polycrystalline ice, in Physics and Mechanics of Ice (Ed. TD'de, P.), 2 ! 7-245, Proceedinos of the IUTAM S)'mp., Copenhagen, Denmark, 1979. Springer-Verlag, Berlin, 1980 Sinha, N. K. Rheology of columnar grained ice, Exp. Afech., 18, 464-470, 1978 Sinha, N. K. Short term theology of polycrystalline ice, J. Glacial., 21,457-473, 1978 Morland, L. W. Constitutive laws for ice, Cold Re9. Sci. Tech., l, 101-108, 1979 Gold, L. W. and Sinha, N. K. The rheological behaviour of ice at small strain, Physics and Mechanics of Ice (Ed. Tryde, P.), 117-128, Proceedings of the IUTAM Symp., Copenhagen, Denmark, 1979. Springer-Verlag, Berlin, 1980 Morland, L. W. and Spring, U. Viscoelastic fluid relation for the deformation of ice, Cold Re9. Sci. Tech., 4, 255-268, 1981 Morland, L. W. and Spring, U. Single integral representations for nonlinear viscoelastic solids, Mech. Materials, 1, 161-170, 1982 Spring, U. and Morland, L. W. Viscoelastic solid relation for the deformation of ice, Cold Re9. Sci. Tech., 5, 221-234, 1982 Spring, U. and Morland, L. W. Integral representations for the deformation of ice, Cold Re 9. Sci. Tech. 6, 185-193, 1983 Williams, H. T. Single integral representations in ice mechanics, Cold Reg. Sci. Tech, 9, 89-95, 1984
Non-lhtear viscoelastic response o f ice: L. IV. M o r l a n d 11 12 !3 14 15
Sunder, S. S. and Wu, M. S. A differential flow model for polycrystalline ice, Cold Reg. Sci. Tech., 16, 45--62, 1989 Sunder, S. S. and Wu, M. S. A multi-axial differential model of flow in orthotropic polycrystalline ice, Cold Reg. Sci. Tech., 16, 223-235, 1989 Mellor, M. and Cole, D. M. Deformation and failure of ice under constant stress or constant strain-rate, Cold Reg. Sci. Tech., 5 201-219, 1982 Mellor, M. and Cole, D. M. Stress/strain/time relations for ice under uni-axial compression, Cold Reg. Sei. Tech., 6, 207-230, 1983 Smith, G. D. and Morland, L. W. Viscous relations for the steady creep of polycrystalline ice, Cold Reg. Sci. Tech., 5, 141-150, 1981
16 17 18 19
Morland, L. W. and Lee, E. H. Stress analysis for linear viscoelastic materials with temperature variation, Trans Soc. Rheol., 4, 233-263, 1960 Mellor, M. and Testa, R. Effects of temperature on the creep of ice, J. 61aciol., 8, 131-145, 1969 Truesdell, C. "/he Elements of Continuum Mechanics, SpringerVerlag, Berlin, 1966 Morland, L. W. and Earle, E. N. Correlation of shear behaviour of ice with bi-axial stress response, 183-192, 3, Proceedings of
the 7th Int. Conf. on Port and Ocean Engineering under Arctic Conditions, Helsinki, Finland (Ed. Technical Research Centre of 20
Finland) 1983 Dienes, J. K. Theory of deformation, Part I: Kinematics, /os Alamos Report LA-11063-MS, 1, 1987
Applied Ocean Research, 1991, Vol. l & No. 8
261