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A comparative study of Maxwell viscoelasticity at large strains and rotations 1
2
Christoph E. Schrank*1, 2, Ali Karrech3, David A. Boutelier4, Klaus Regenauer-Lieb5 3
4
1) Queensland University of Technology, School of Earth, Environmental and Biological Sciences, 5
Brisbane, QLD, Australia 6
2) University of Western Australia, School of Earth and Environment, Crawley, WA, Australia 7
3) University of Western Australia, School of Civil, Environmental and Mining Engineering, 8
Crawley, WA, Australia 9
4) University of Newcastle, School of Environmental and Life Sciences, Callaghan, NSW, 10
Australia 11
5) University of New South Wales, School of Petroleum Engineering, Sydney, NSW, Australia 12
13
History 14
Accepted date: XX/YY/2017. Received date: XX/YY/2017; in original form date: XX/YY/2017 15
*Corresponding author: e-mail: christoph.schrank@qut.edu.au, tel.: (+61 7)3138 1583, ORCID: 16
0000-0001-9643-2382 17
18
Abbreviated title: Viscoelasticity at large transformations 19
Summary 20
We present a new Eulerian large-strain model for Maxwell viscoelasticity using a logarithmic co-21
rotational stress rate and the Hencky strain tensor. This model is compared to the small-strain model 22
without co-rotational terms and a formulation using the Jaumann stress rate. Homogeneous 23
isothermal simple shear is examined for Weissenberg numbers in the interval [0.1;10]. Significant 24
differences in shear-stress and energy evolution occur at Weissenberg numbers > 0.1 and shear 25
strains > 0.5. In this parameter range, the Maxwell-Jaumann model dissipates elastic energy 26
erroneously and thus should not be used. The small-strain model ignores finite transformations, 27
frame indifference, and self-consistency. As a result, it overestimates shear stresses compared to the 28
new model and entails significant errors in the energy budget. Our large-strain model provides an 29
energetically consistent approach to simulating non-coaxial viscoelastic deformation at large strains 30
and rotations. 31
Keywords 32
Fault zone rheology, creep and deformation, rheology: crust and lithosphere, rheology and friction 33
of fault zones 34
1. Introduction 35
The Maxwell body of a linear-elastic Hookean spring in series with a Newtonian dashpot is the 36
simplest rheological model for geological deformation (e.g., Bailey (2006)). It has been employed 37
to describe many deformation processes and structures, for example mantle convection (Beuchert 38
and Podladchikov, 2010, Mühlhaus and Regenauer-Lieb, 2005), Rayleigh-Taylor instabilities (Kaus 39
and Becker, 2007, Robin and Bailey, 2009), lithospheric deformation (Kaus and Podladchikov, 40
2006, Korenaga, 2007, Kusznir and Bott, 1977), postglacial rebound (Larsen et al., 2005, Peltier, 41
1976), shear zones (Branlund et al., 2000, Mancktelow, 2002, Sone and Uchide, 2016), and folds 42
(Liu et al., 2016, Mancktelow, 1999, Zhang et al., 2000). In spite of their simplicity, current 43
Maxwell models display well-known errors (e.g., Beuchert and Podladchikov, 2010, Mühlhaus and 44
Regenauer-Lieb, 2005) when the associated strains and, importantly, rotations are large. Such 45
conditions are met, for example, during plate motions (Tohver et al., 2010), in bending subducting 46
plates (Conrad and Hager, 1999), blocks of crust bounded by faults (Abels and Bischoff, 1999), 47
folds (Ramsay and Huber, 1987), shear zones (Ramsay, 1980), boudins (Mandal and Khan, 1991), 48
porphyroblasts (Johnson, 2009) and –clasts (Passchier and Simpson, 1986), and grains deforming 49
by diffusion creep (Wheeler, 2010). Large rotations pose a mathematical challenge when elasticity 50
is considered in the rheology of highly transformed materials (Dienes, 1979, Karrech et al., 2012, 51
Karrech et al., 2011, Moss, 1984, Bruhns et al., 2001a, Bruhns et al., 1999, Bruhns et al., 2001b, 52
Meyers et al., 2005, Xiao et al., 1997a, Xiao et al., 2006, Xiao et al., 1998a, Xiao et al., 1998b, 53
Colak, 2004, Lehmann, 1972a): one requires an objective formulation of the stress rate (time 54
derivative of stress). 55
If a volume element of a transforming body incurs a rigid-body rotation over an 56
infinitesimal time increment, its state of stress and strain should not change relative to a coordinate 57
system that co-rotates with this volume element. In other words, the notion of objectivity, or frame 58
indifference, requires that the co-rotational stress and strain increments become zero during an 59
instantaneous rigid-body rotation (see, for example, Chapter 1.6 of Chakrabarty (2006)). In an 60
Eulerian (stationary) reference frame, the objectivity requirement is commonly addressed with a so-61
called co-rotational objective stress rate, which contains spins that account for elastic rotations (for 62
an in-depth review, see Xiao et al., 2006). Here, we compare the self-consistent formulation of the 63
Eulerian finite-transformation model for Maxwell viscoelasticity (called FT model hereafter) to two 64
thermodynamically non-consistent Maxwell models, which are commonly used to model 65
geologically relevant strains (e.g., Kaus and Becker, 2007, Beuchert and Podladchikov, 2010, 66
Mancktelow, 1999, Zhang et al., 2000, Moresi et al., 2002): the small-strain formulation without 67
co-rotational terms, referred to as SS model, and the formulation using the Jaumann stress rate, 68
called MJ model hereafter. Self-consistency (i.e., adherence to thermodynamic principles) in the FT 69
model is achieved by using the logarithmic spin introduced by Xiao and co-workers in the context 70
of finite elastoplasticity (Bruhns, 2003, Bruhns et al., 1999, Xiao et al., 1997a, Xiao et al., 2006, 71
Xiao et al., 1998a, Xiao et al., 1998b). 72
The choice of the logarithmic spin in the new FT model is motivated by two reasons. First, 73
the logarithmic co-rotational rate of the Hencky strain was proven to be the only strain-rate 74
measure, which is always identical with the stretching (Bruhns et al., 1999, Xiao et al., 1997a, Xiao 75
et al., 1997b). The stretching, the symmetric part of the velocity gradient tensor, is the most 76
commonly used and kinematically most appealing measure of deformation rates (cf. chapter 2 of 77
Xiao et al., 1997b). Second, the logarithmic stress rate is the only choice, which delivers a self-78
consistent Eulerian constitutive model (Bruhns et al., 1999). Bruhns et al. (1999) prove self-79
consistency in the co-rotational Eulerian rate model through integrating the additive deformation-80
rate decomposition for the case of purely elastic stretching. If the rate formulation of the model is 81
exactly integrable to deliver a truly elastic response, i.e., fully reversible deformation without 82
energy dissipation, then the Eulerian finite-strain model is defined as (thermodynamically) self-83
consistent. Only models that use the logarithmic spin are self-consistent. 84
To analyse the relevance of the co-rotational logarithmic spin used in the FT model, 85
homogeneous 2D isothermal simple shear is examined because of its non-zero, skew-symmetric 86
spin and high conceptual importance for understanding shear zones (e.g., Ramsay, 1980, Ramsay 87
and Graham, 1970). The relative stress and energy states are evaluated for a shear-strain range of 0 88
to 10, providing the basis for comparison of the SS, MJ and FT models. The sections to come are 89
organised as follows: first, the need for a proper finite Maxwell model is established. Second, 90
analytical solutions for simple shear with the SS and MJ models are presented. Third, the FT model 91
is derived and its numerical solution described. Fourth, all three models are compared. Finally, the 92
results and their implications are discussed. 93
2. Maxwell viscoelasticity, co-rotational stress rates, and the need for a finite Maxwell model 94
The constitutive equation for Maxwell viscoelasticity of an incompressible material is: 95
96
Dij =12Lij + L
Tij( ) =
! ʹσ ij
2G+
ʹσ ij
2η (1), 97
98
whereDij is the stretching or deformation-rate tensor, defined as the symmetric part of the velocity-99
gradient tensor Lij = ∂vi / ∂x j with velocities vi and directions xi, the superscript T marks the matrix 100
transpose, the overdot denotes a derivative with respect to time t, G is the shear modulus of an 101
isotropic linear-elastic material, ʹσ ij is the deviator (indicated by the dash) of the Cauchy stress 102
tensor σ ij , and η is the Newtonian viscosity. In the limit of infinitesimal deformations, the 103
stretching tensor becomes the familiar strain-rate tensor for small strains. Equation (1) states that 104
elastic (first term) and viscous deformation rates (second term) are additive (serial coupling of an 105
elastic spring with a viscous dashpot). The term ʹ!σ ij is the stress rate, which highlights the need for 106
a co-rotational formulation in order to acknowledge rotations at finite strain. Formulations that 107
include geometrically admissible co-rotational terms are called “objective”. 108
Equation (1) is solved by direct integration for the SS model while ʹ!σ ij is replaced with rates 109
augmented by specific spins in the MJ and FT models (section 3). It is obvious that the SS model 110
ignores large rotations and strains and hence is not geometrically objective and not 111
thermodynamically self-consistent. Errors are expected at large non-coaxial deformations. The 112
magnitude of these errors can be estimated by comparing the SS model with other existing models 113
such as MJ and now FT. The MJ model employs a classical objective co-rotational stress rate, the 114
Jaumann rate (Jaumann, 1911). 115
Two important shortcomings of the MJ model under simple shear have been recognized in 116
the geodynamic literature previously (Mühlhaus and Regenauer-Lieb, 2005, Beuchert and 117
Podladchikov, 2010). To summarise these findings, we first recall the Maxwell relaxation time trel 118
and the Weissenberg number W: 119
120
(2a, b), 121
122
where !γ is a characteristic shear-strain rate, which in our study corresponds to the imposed simple-123
shear rate defined in Eq. (4). Deformation observed at time scales well below trel is dominated by 124
elastic contributions to the strain rate and stresses. After about five trel , deformation is essentially 125
viscous. The Weissenberg number W measures to which degree the material behavior is nonlinear 126
and how much of the current strain is recoverable (White, 1964, Dealy, 2010). Mühlhaus and 127
Regenauer-Lieb (2005) compared Maxwell models using the Jaumann and Green-Naghdi rates, two 128
objective stress rates from the continuum-mechanics community, with the SS model frequently used 129
in geological modeling. While these objective formulations contain the necessary geometric terms 130
that ensure frame indifference, they do not satisfy the self-consistency condition of Bruhns et al. 131
(1999). Consequently, for the same material properties, the Green-Naghdi and MJ models give 132
different stress-strain curves at shear strains > 1. The SS model (used in geology) and the Green-133
Naghdi model (used in some engineering applications, e.g., for car-crash simulations 134
(Abaqus/Explicit, 2015)) produce qualitatively similar results but the MJ model (used in other 135
engineering application, e.g., for calculations of the nonlinear creep of the steel carapace of a 136
nuclear reactor under elevated temperatures (Abaqus/Standard, 2009)) exhibits spurious softening 137
during numerical simulations of simple shear for W = 1. Mühlhaus and Regenauer-Lieb (2005) 138
showed, however, that this effect vanishes if non-linear creep and/or plasticity are added to the 139
constitutive model. Later, Beuchert and Podladchikov (2010) demonstrated numerically that the MJ 140
model exhibits unphysical shear-stress oscillations at W = 10. They also showed that the choice, and 141
even the neglect, of co-rotational terms become irrelevant at W ≤ 0.1. Therefore, two regimes of 142
intermediate (0.1 < W < 1) and high (W ≥ 1) Weissenberg numbers exist, in which one can expect 143
trel =ηG
W = γtrel
problems with the SS and MJ models. The question arises: where in the Earth may the intermediate 144
and high-W regimes occur? 145
Beuchert and Podladchikov (2010) and Mühlhaus and Regenauer-Lieb (2005) aiming for 146
modeling viscoelastic mantle convection argued that only the lithosphere exhibits sections with W 147
>> 0.1. At the tectonic time and length scales of interest to these studies, the lithosphere behaves 148
essentially elastic and hardly deforms. Hence, the problems of the MJ model, spurious softening 149
and stress oscillations, were predicted to be irrelevant in viscoelastic mantle simulations. This 150
argument is further supported by the notion that power-law flow and plasticity, both common in the 151
lithosphere (e.g., Karato, 2008), serve as stress limiters, which inhibit spurious softening and 152
oscillations (Beuchert and Podladchikov, 2010). 153
However, we argue that deformation problems within the lithosphere, in particular at length 154
scales ≤ 1 km, require the tackling of the problem of modeling Maxwell viscoelasticity in the 155
intermediate and high-W regimes properly. The shear moduli of most rock-forming minerals are 156
within the range of 10 to 100 GPa (Bass, 1995). The effective viscosity of lithospheric rocks varies 157
over several orders of magnitude due to its sensitivity to, e.g., material composition, temperature, 158
pressure, strain rate, stress, and grain size (e.g., Karato, 2008). We adopt lower and upper bounds of 159
1018 and 1025 Pas, respectively. Shear-strain rates also cover several orders of magnitude. A typical 160
lower bound for natural shear zones is 10-15 s-1 (Biermeier and Stüwe, 2003, Pfiffner and Ramsay, 161
1982). Estimates for the upper bound in nature are of order 10-10 s-1 for shear zones with widths ≤ 162
100 m (White and Mawer, 1992). Computing W for the above parameter ranges shows that a 163
significant subset (~ 35% to 50%) of this parameter space is within the critical regimes of W (Fig. 164
1). Moreover, one can envisage geological scenarios during which the internal deformation of the 165
lithosphere remains elastic, or transient viscoelastic, in large parts while the associated rotations 166
become large, for example, in subducting plates and blocks of crust rotating about a vertical axis. In 167
these cases, a proper objective treatment of large transformations, and elastic rotations in particular, 168
seems desirable. This effort should also involve stress advection (e.g., Truesdell, 1956), which is 169
irrelevant in the simple-shear case at hand. 170
171
3. Derivations: kinematics of 2D simple shear 172
We consider 2D plane-strain isothermal simple shear of a unit square, where x1 and x2 denote the 173
horizontal and vertical direction, respectively. The lower-left corner of the square is centered on the 174
origin of the coordinate system. Shear is parallel to x1. The boundary conditions are: 175
176
v1 x2 = 0( ) = v2 x2 = 0( ) = 0v1 x2 =1( ) = const. > 0v2 x2 =1( ) = 0∂v1∂x2
0 < x2 <1( ) = const. > 0
(3), 177
178
where vi denote the velocities in the respective directions. Thus, the velocity-gradient tensor Lij has 179
only one non-zero, constant entry: 180
181
L12 =∂v1∂x2
= !γ (4), 182
183
where is the constant shear-strain rate applied to the unit square. Hence, the deformation-rate 184
tensor Dij and the spin tensor Sij , the antisymmetric component of Lij , read: 185
186
γ
Dij =0!γ2
!γ2
0
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
Sij =12Lij − L
Tij( ) =
0!γ2
−!γ2
0
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥
(5a, b). 187
188
Equation (5b) expresses the internal rotations inherent to simple shear. Conceptually, the simple-189
shear model embodies an idealized plane-strain shear zone with homogeneous strain distribution. In 190
nature, shear zones generally display a heterogeneous strain pattern (Ramsay, 1980, Ramsay and 191
Graham, 1970). However, this heterogeneity is a function of the scale of observation. One can 192
identify length scales, at which deformation is, to first order, homogeneous. Hence, one may 193
consider the following analysis as an examination of a homogeneously deforming representative 194
elementary volume within an ideal shear zone, across which variations in parameters such as 195
temperature, composition, viscosity, fluid content, etc., are negligible. 196
3.1 Derivations: Maxwell model for small strain (SS model) 197
The SS model is solved by integrating Eq. (1) after substituting simple-shear kinematics (Eqs. (3), 198
(5a)), trel (Eq. (2a)), W (Eq. (2b)), and dimensionless time k = trel−1t , which yields: 199
200
σ12 k( ) =WG 1− e−k( )σ11 =σ 22 = 0
(6a, b). 201
202
The total deformational power per unit volume is: 203
204
!ETOT t( ) = ʹσ ij :Dij =σ12 !γ (7). 205
206
Integrating Eq. (7) over time with the initial condition ETOT t = 0( ) = 0 gives the total energy per unit 207
volume: 208
209
ETOT k( ) =W 2G k + e−k( )−1⎡⎣
⎤⎦ (8). 210
211
To decompose the total energy into elastic and viscous fractions, we use the additivity of the 212
deformation rates to introduce the elastic and viscous fraction of strain rate (e.g., Kaus and Becker, 213
2007): 214
215
1=!γel!γ+!γ vis!γ= fel + fvis (9), 216
217
where fel and fvis denote the elastic (subscript el) and viscous (subscript vis) fraction of the strain 218
rate, respectively. At infinite time, the shear stress is purely viscous: σ ∞vis =σ12 k→∞( ) =η !γ =WG . 219
Thus, the viscous and elastic stress fractions can be defined as: 220
221
fvis =σ12 k( )σ vis
∞=1− e−k
fel =1− fvis = e−k
(10a, b). 222
223
Inserting Eqs. (6), (9) and (10) into Eq. (7) and integrating over time yield the stored and viscous 224
energy per unit volume, again for zero initial energies: 225
226
Eel = !γ felσ12 dk =W2G 1
2+12e−2k − e−k
⎛
⎝⎜
⎞
⎠⎟ =
σ122
2G0
k
∫
Evis = !γ fvisσ12 dk =W2G 2e−k − 1
2e−2k + k − 3
2⎛
⎝⎜
⎞
⎠⎟
0
k
∫ (11a, b). 227
228
3.2 Derivations: Maxwell model with the Jaumann stress rate (MJ model) 229
The elastic stress rate in Eq. (1) ( ʹ!σ ij ) is now replaced by the Jaumann stress rate (Jaumann, 1911): 230
231
!τ ij = !σ ij − Sikσ kj +σ ikSkj (12). 232
233
This stress rate is equivalent to a rigid-body rotation of the stress tensor neglecting second and 234
third-order terms (Chakrabarty, 2006). Since in the problem at hand σ11 = −σ 22 ⇒ tr σ ij( ) = 0 (e.g., 235
Durban, 1990), the dashes for deviatoric quantities are dropped in the following.Substituting the 236
spin (Eq. (5b)), Eq. (12) reads: 237
238
!τ ij =!σ11 − !γσ12 !σ12 + !γσ11!σ 21 + !γσ11 !σ 22 + !γσ12
⎡
⎣⎢⎢
⎤
⎦⎥⎥
(13). 239
240
Inserting Eq. (13) into the constitutive model (Eq. (1)) with the deformation-rate tensor for simple 241
shear (Eq. (5a)) and substituting trel yield the system of coupled differential equations: 242
243
η !γ = trel !σ12 − !γ !σ 22( )+σ120 = trel !σ 22 + !γσ12( )+σ 22
(14). 244
245
The solution of Eq. (14) in terms of W and dimensionless time k is (Appendix 1): 246
σ12 k( ) = WG1+W 2 e−k W sin Wk( )− cos Wk( )⎡⎣ ⎤⎦+1⎡⎣ ⎤⎦
σ 22 k( ) = WG1+W 2 e−k W cos Wk( )+ sin Wk( )⎡⎣ ⎤⎦−W⎡⎣ ⎤⎦
(15a, b). 247
248
The first fraction in Eq. (15a), b is the steady-state viscous shear stress for the MJ model: 249
σ vis∞ =σ12 k→∞( ) =WG 1+W 2( )
−1. It differs from the steady-state viscous stress of the SS model 250
by the factor 1+W 2( )−1
. This factor highlights an important effect of the stress-rotation terms 251
included in the Jaumann stress rate: they decrease the maximum shear stress attained at steady state. 252
Using Eq. (7) and Eq. (15a), the total deformational energy per unit volume as function of k is: 253
254
ETOT k( ) =σ vis∞W k − e−k
W 2 +1W 2 −1( )cos Wk( )+ 2W sin Wk( )( )+W
2 −1W 2 +1
⎡
⎣⎢
⎤
⎦⎥(16). 255
256
Eq. (16) is also decomposed into elastic and viscous fractions. The elastic and viscous stress 257
fractions read: 258
259
(17a, b). 260
261
To conclude the energy decomposition, the time integrals of Eq. (11) need to be solved again after 262
substituting the appropriate expressions for the shear stress (Eq. (15a)) and elastic and viscous stress 263
fractions (Eq. (17)). The results are cumbersome because of the lengthy expressions resulting from 264
the integration of the trigonometric terms and are thus not shown here. Many mathematics software 265
packages can conduct this integration, and the interested user may find this more convenient. 266
3.3 Derivations: Maxwell model at finite strain (FT model) 267
The logarithmic finite-strain model requires slight modifications of Eqs. (1) and (12). The Cauchy 268
stress σ ij is replaced by the Kirchhoff stress Σij = Jσ ij where J is the determinant of the 269
deformation-gradient tensor. We use the Kirchhoff stress with more general applications in mind, 270
fel = e−k cos Wk( )−W sin Wk( )( )
fvis = e−k W sin Wk( )− cos Wk( )( )+1
even though J = 1 for simple shear. The co-rotational Cauchy stress rate !τ ij (Eq. (12)) is replaced by 271
the co-rotational rate of the Kirchhoff stress ⌣Σij : 272
273
⌣Σij = "Σij −ΩikΣkj + ΣikΩkj (18), 274
275
which involves a logarithmic spin of rotation discovered by Xiao and colleagues (Xiao et al., 276
1997a, Xiao et al., 1997b): 277
278
Ωij =Wij +λA + λBλA − λB
+2
lnλA − lnλB
⎛
⎝⎜
⎞
⎠⎟Pik
A( )DklPljB( )
A,B=1,A≠B
n∑ (19), 279
280
where A and B are dummy integers denoting the principal directions, λA is the Ath eigenvalue of the 281
left Cauchy-Green strain tensor, PikA( ) is its Ath eigenprojection (tensor product of an eigenvector 282
with itself), and n is the number of distinct eigenvalues. The sum vanishes naturally if the 283
eigenvalues are all equal. The spin of Eq. (19) contains a correction term compared to that of the 284
Jaumann formulation, which accounts for the stretching of the eigenprojections, not only their 285
rotation. The logarithmic spin produces a smooth, stable response under simple shear, unlike the 286
Jaumann spin, which produces aberrant oscillations (Lehmann, 1972b, Dienes, 1979). Moreover, 287
since the co-rotational logarithmic rate of the Hencky strain tensor coincides with the deformation-288
rate tensor, the former constitutes an ideal energy conjugate of the Kirchhoff stress (Bruhns, 2003). 289
Therefore, the Maxwell model stated in Eq. (1) involves another modification in the FT 290
model, in addition to inserting the stress rate of Eq. (18): the time derivative of the Eulerian 291
logarithmic Hencky strain tensor replaces the stretching. The associated system of differential 292
equations (Eq. (A9)) and the numerical method for integrating them are detailed in appendix 2. The 293
particular qualities, which render logarithmic kinematics the mathematically most consistent tool 294
for the constitutive behavior of geomaterials, are further discussed in great depth by Xiao et al. 295
(2006). 296
4. Results 297
We present the evolution of normalized shear stress σ12G−1 and the ratios of shear stress and 298
energies up to a shear strain of γ =10 for the intermediate- and high-W regimes. The former is 299
represented by the discrete set of solutions withW ∈ 0.1;0.2;...;0.9{ } , and the latter by the set 300
W ∈ 1;2;...;10{ } . 301
4.1 Shear stress evolution 302
The shear-stress curves start to diverge at γ ≈ 0.5 (Fig. 2). In general, shear stresses of the SS 303
model and those of the MJ model bracket shear stresses of the FT model. Their discrepancy 304
increases with increasing W. The spurious strain softening of the MJ model is significant for 0.5 ≤ 305
W ≤ 1 while stress oscillations are visible for all greater W. The SS and FT models exhibit 306
monotonously increasing shear stresses over the tested strain range. We continue with a detailed 307
comparison of relative shear stresses and energies, considering the intermediate- and high-W 308
regimes separately (Fig. 3). For the high-W regime, MJ results are omitted because their stress 309
oscillations are highly problematic (see Fig. 2b and appendix 3 for an in-depth analysis). 310
4.2 Intermediate-W regime 311
The SS model predicts higher stresses than the FT model (Fig. 3a). The maximum magnitude of this 312
discrepancy increases from ca. 1% at W = 0.1 and γ ≈ 0.5 to ca. 25% at W = 0.9 and γ ≈ 2.3 . At 313
γ ≤ 0.5 , the difference never exceeds 5%. The MJ model predicts lower stresses than the FT model 314
(Fig. 3a). The maximum discrepancy increases nonlinearly with shear strain and W and reaches 315
from <2% at W = 0.1 to ca. 42% W = 0.9 at γ =10 . The difference does not exceed 3% at γ ≤1 . 316
Since the total energy is the integral of the stress-strain curve, the evolutions of the relative 317
total energies resemble those of the relative shear stresses (Fig. 3b). The SS model computes larger 318
total energies than the FT model, ranging from ca. 1.5% excess at W = 0.1 and γ ≈1.3 to ca. 20% at 319
W = 0.9 and γ ≈ 3.8 . At γ =1 , the difference is <7%. Again, the MJ model computes lower 320
energies than the FT model with a maximum difference at γ =10 , being about 1% at W = 0.1 up to 321
ca. 32% at W = 0.9. At γ = 2 , the difference is <5% in all cases. 322
In the evolution of stored energy, the SS model predicts larger elastic energy than the FT 323
model, ranging from ca. 2% at W = 0.1 and γ ≈ 0.5 to about 56% at W = 0.9 and γ ≈ 2.3 (Fig. 3c). 324
The curve shapes mirror those of the shear-stress evolution because elastic energy and shear stress 325
are related by Eel = 0.5σ122G−1 (Eq. (11a)). The MJ model gives smaller energies than the FT-model 326
energies with the maximum difference appearing at γ =10 for W ≤ 0.7, ranging from ca. 5% at W = 327
0.1 to ~ 95% at W = 0.7. At larger W, negative ratios are obtained because the MJ theory computes 328
negative stored energy there. 329
In terms of dissipated viscous energy (Fig. 3d), the SS model first gives smaller and then 330
larger values than the FT model. The difference of the smaller values does not exceed 10% (at W = 331
0.9 and γ = 0.7 ). The difference of the lager values reaches ca. 35% at the largest W and smallest γ. 332
The MJ model exhibits the opposite behavior with considerably larger magnitudes of discrepancy, 333
attaining an excess of up to ~ 260% and a deficiency of ca. 30% for the largest W. 334
4.3 High-W regime 335
The overall trends of shear-stress evolution remain similar to those of the intermediate-W regime. 336
The SS model gives shear stresses higher than the FT model with a maximum discrepancy of ca. 337
30% at W = 1 to ca. 350% at W = 10 (Fig. 3e). This trend is reflected in the total-energy comparison 338
(Fig. 3f), which reveals a maximum difference of ~ 25% at W = 1 up to ~ 280% at W = 10. A 339
significant proportion of this energy error is due to problems with the computation of elastic energy 340
(Fig. 3g). The SS model computes larger stored energy than the FT model by up to ca. 50% at W = 341
1 up to more than one order of magnitude at W = 10. Accordingly, the differences in viscous-energy 342
evolution are smaller (Fig. 3h). The largest negative difference ranges from ca. 10% at W = 1 to 343
about 70% at W = 10; the largest positive difference covers a range of ~ 13% at W = 1 to ~ 65% at 344
W = 10. 345
5. Discussion 346
The three examined models for viscoelastic deformation differ markedly. The SS model ignores 347
large transformations, frame indifference, and self-consistency sensu Bruhns et al. (1999) a priori. 348
As a result, the FT and SS model differ most significantly during highly non-coaxial large 349
deformations in the intermediate- and high-W regimes. The MJ formulation performs worse in some 350
cases than the SS model in spite of its use of a geometrically correct objective stress rate. However, 351
it violates the self-consistency requirement, which incurs unphysical stress oscillations and negative 352
stored energy (Fig. 3c, see also appendix 3). This poses a serious problem for multi-physics 353
formulations with energy feedbacks that rely on accurately tracking cross-scale energy fluxes 354
through thermodynamic upscaling methods (Regenauer-Lieb et al., 2013, Regenauer-Lieb et al., 355
2014). The FT model provides an energetically consistent large-transformation model that satisfies 356
the objectivity and self-consistency requirements. We now proceed to discuss the applicability of 357
the examined Maxwell models in the light of our results. 358
With engineering applications in mind, our comparison reveals a range of W where all 359
discussed models deliver qualitatively similar results. At small strains and W < 0.1, the three tested 360
models are identical. Under these conditions, the SS model provides the simplest choice. At 361
intermediate W and γ <1 , the shear stresses of all three models are within ≤ +/- 10%. If this 362
accuracy is acceptable in this strain and W range, it does not matter practically, which model is 363
used. At γ >1 , the shear-stress deviations exceed 10% at W > 0.3 in the MJ model, and at W > 0.4 364
in the SS model. Nevertheless, the shear-stress overestimation of the SS model does not exceed 365
25% in the intermediate-W regime. One may perhaps envisage applications, where for the sake of 366
minimizing the computational burden, this overestimate is acceptable. The MJ model exhibits 367
spurious softening of the shear stresses at W ≥ 0.3 (Fig. 2a). This unphysical effect renders its use 368
problematic at and beyond this W. In the high-W regime, shear stresses in the SS model exceed 369
those of the FT model by ca. 10% at γ ≈ 0.5 and keep on increasing quickly while the MJ model 370
oscillates unrealistically (Fig. 2b). As it is clear that the formulation of the SS model is ill suited for 371
large transformation, the rapid increase and high stress magnitude of the SS model in high-W 372
regime cannot be considered acceptable. Therefore, we consider the FT model the best choice in the 373
high-W regime. 374
However, if energy consistency matters, the FT model is also better suited for most of the 375
intermediate-W regime. As emphasized in a review by Hobbs et al. (2011), the time evolution of the 376
stored energy of a coupled geological system holds the key to understanding dissipation and thus 377
the formation of localized structures. This notion is deeply rooted in thermodynamic principles 378
(Ziegler, 1977, Coussy, 2004). Therefore, if one wishes to tackle non-coaxial deformation 379
problems, where the mechanical energy of the system is coupled to non-mechanical dissipation 380
processes such as metamorphic processes or heat transfer, energy consistency is important. The SS 381
model overestimates stored mechanical energy in simple shear by well over 10% at W > 0.3. 382
Therefore, we recommend, as a pragmatic rule of thumb, to use the SS model in the intermediate-W 383
regime only in cases where energy consistency is not required or W ≤ 0.3. 384
The MJ model exhibits unphysical negative stored energy at W > 0.7, which obviously 385
violates thermodynamic principles. In fact, all MJ models exhibit negative time gradients in stored 386
energy in the examined W range (appendix 3, Fig. A1). Because the loading at the boundaries of the 387
homogeneously deforming unit square is never removed and material properties remain constant, 388
elastic strain and stored energy should not decrease. Therefore, the MJ model violates the self-389
consistency requirement and also the principle of energy conservation. The associated shear-stress 390
decrease becomes noticeable at W = 0.3 (Fig. 2a). Hence, the MJ model should not be used at W ≥ 391
0.3. The use of the MJ model is thus restricted to the W interval ]0.1;0.3[. This rather small 392
parameter range and the fundamental conceptual shortcomings of the MJ model lead to the question 393
if there is any practical merit in using it. 394
Our analysis suggests that the FT model should be the tool of choice for highly non-coaxial 395
finite viscoelastic deformation problems in the intermediate- and high-W regimes. This notion holds 396
true if one considers the concepts of frame indifference and self-consistency sensu Bruhns et al. 397
(1999) as meaningful physical constraints. Nevertheless, even though the FT model is objective and 398
energetically consistent, it remains to be seen if it can be falsified in laboratory experiments. 399
Experimental tests of the model with rotary rheometry and analogue materials are subject of 400
ongoing work. In addition to its unique shear-stress response, the FT model predicts non-zero 401
normal stresses in simple shear, which can also be observed and tested experimentally (Freeman 402
and Weissenberg, 1948). 403
404
6. Conclusions 405
We introduced a large-strain model for Maxwell viscoelasticity with a logarithmic co-rotational 406
stress rate (the “FT model”). An analysis of homogeneous isothermal simple shear with the FT 407
model compared to a small-strain formulation (the “SS model”) and a model using the Jaumann 408
stress rate (the “MJ model”) leads to the following key conclusions: 409
410
(1) At W ≤ 0.1, all models yield essentially identical results. 411
(2) At larger W, the models show increasing differences for γ > 0.5 . The SS model overestimates 412
shear stresses compared to the FT model while the MJ model exhibits an oscillatory response 413
underestimating the FT model. 414
(3) The MJ model violates the self-consistency condition resulting in stress oscillations and should 415
be disregarded. It does not deliver truly elastic behavior. 416
(4) In the intermediate-W regime, the shear-stress overestimates of the SS model may constitute 417
acceptable errors if energy consistency is not important. If energy consistency is desirable, the SS 418
model should not be used at W ≥ 0.3. 419
(5) In the high-W regime, stresses in the SS model become unacceptably large. The FT model 420
should be used in this domain. 421
422
The FT model constitutes a physically consistent Maxwell model for large non-coaxial 423
deformations, even at high Weissenberg numbers. It overcomes the conceptual limitations of the SS 424
model, which is limited to small transformations, not objective and not self-consistent. It also solves 425
the problem of the energetically aberrant oscillations of the MJ model. The FT model promises 426
particular usefulness for simulations of bending plates, rotating blocks of crust, and lithospheric 427
viscoelastic shear zones, especially those at the scale of hundreds of meters or smaller as well as 428
those involving strong heterogeneity of material properties due to, for example, layering or 429
inclusions. In such systems, high Weissenberg numbers and large rotations can be expected. 430
Acknowledgements 431
The authors gratefully acknowledge funding by the Australian Research Council via Discovery 432
project DP140103015. CS is grateful for support of the QUT High Performance Computing group. 433
We are very thankful for the thorough and most helpful reviews of Greg Houseman, an anonymous 434
reviewer, and the editor Jörg Renner. 435
436
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582
Figures 583
584
Fig. 1: Double-logarithmic contour plots of the decadic logarithm of W over those of viscosity and strain rate 585
for the (a) lower- and (b) upper-bound estimate of the shear modulus. The white dotted lines denote 586
parameter combinations with W = 0.1; the white dashed lines mark those with W = 1. Textboxes indicate the 587
models, which are valid in the respective parameter fields separated by those lines. 588
589
Fig. 2: Plots of normalized shear stress over shear strain for the SS, MJ, and FT models in the (a) 590
intermediate-W regime, and (b) the high-W regime. Each plot title indicates the respective W and the 591
dimensionless time kmax reached at final γ . Note that the results for W = 0.1 are omitted because the curves 592
cannot be distinguished at this scale of observation. 593
594Fig. 3: Top panel (light-grey canvas): Comparison of model results in the intermediate-W regime. (a) Shear-595
stress ratio, (b) total-energy ratio, (c) stored-energy ratio, and (d) viscous-energy ratio for the SS model 596
(solid lines) and the MJ model (dashed lines). Each sub-figure contains two plots: on the left, SS results are 597
divided by FT results; on the right, MJ results are divided by FT results. Curve symbols denote the respective 598
W, defined in the right panel of (d). Bottom panel (grey canvas): Comparison of the SS to the FT model in 599
the high-W regime. Results of the SS model are again divided by those of the FT model. The legend for the 600
respective value of W is shown in (e). 601
602
Appendix 603
1. Analytical solution of the MJ model 604
The MJ model for simple shear requires the solution of the following nonhomogeneous linear 605
system of ordinary differential equations: 606
607
η !γ = trel !σ12 − !γ !σ 22( )+σ120 = trel !σ 22 + !γσ12( )+σ 22
(A1). 608
609
For convenience, we insert the placeholders: 610
611
A = −trel−1
B = !γC =η !γtrel
−1
(A2 a, b, c), 612
613
which leads, after rearrangement, to: 614
615
!σ12 = Aσ12 + Bσ 22 +C!σ 22 = Aσ 22 − Bσ12
(A3 a, b). 616
617
The solution is: 618
619
σ12 t( ) = eAt sin Bt( )Q1 − cos Bt( )Q2⎡⎣ ⎤⎦−
ACA2 + B2
σ 22 t( ) = eAt sin Bt( )Q2 + cos Bt( )Q1⎡⎣ ⎤⎦−BC
A2 + B2
(A4 a, b). 620
621
From the initial conditions σ12 t = 0( ) =σ 22 t = 0( ) = 0 , the constants Q1 and Q2 are calculated: 622
Q1 =BC
A2 + B2
Q2 = −AC
A2 + B2
(A5 a, b). 623
624
Re-inserting all substitutions yields the final result: 625
626
σ12 t( ) =η !γ
trel−1 + trel !γ
2
⎛
⎝⎜
⎞
⎠⎟ e
−ttrel !γ sin !γt( )− trel−1 cos !γt( )⎡⎣ ⎤⎦+ trel
−1⎡
⎣⎢⎢
⎤
⎦⎥⎥
σ 22 t( ) =η !γ
trel−1 + trel !γ
2
⎛
⎝⎜
⎞
⎠⎟ e
−ttrel !γ cos !γt( )+ trel−1 sin !γt( )⎡⎣ ⎤⎦− !γ
⎡
⎣⎢⎢
⎤
⎦⎥⎥
(A6 a, b). 627
628
To compute the total energy, we integrate Eq. (7) after substituting Eq. (A6a) and W: 629
630
ETOT t( ) = GW !γW 2 +1
t − GW 2
W 2 +1( )2 e
−ttrel W 2 −1( )cos !γt( )+ 2W sin !γt( )⎡⎣
⎤⎦+Q3 (A7). 631
632
The integration constant Q3 is again determined from the initial condition ETOT t = 0( ) = 0 : 633
634
Q3 =GW 2 W 2 −1( )
W 2 +1( )2 (A8). 635
636
2. Numerical solution of FT model 637
From a numerical point of view, Ωij is completely defined at the beginning of each time increment, 638
given its purely kinematic nature. Similarly, to calculate the quantity −ΩikΣkj + ΣikΩkj , it is 639
customary to use the Kirchhoff stress of the beginning of the time increment in a material 640
integration module, the choice we opt for here. The only remaining quantity needed to obtain the 641
co-rotational rate of the Kirchhoff stress ⌣Σij is the time derivative !Σij . For the Maxwell model, the 642
time increment is: 643
644
Σij + trel !Σij = 2η !hij (A9), 645
646
where hij = 0.5lnbij is Hencky’s strain tensor, and bij is the left Cauchy-Green strain tensor (for 647
derivations of these tensors, see for example Karrech et al., 2011, Bruhns et al., 1999). This 648
equation is the modified version of the constitutive model in Eq. (1), involving the substitution of 649
the Kirchhoff stress, insertion of the Maxwell relaxation time, and replacement of the deformation-650
rate tensor with the time derivative of the Eulerian logarithmic Hencky strain tensor. We use the 651
central-difference scheme to approximate the stress and strain increments as: 652
653
!gij t = t0 +Δt2
⎛
⎝⎜
⎞
⎠⎟ =
ΔgijΔt
gij t = t0 +Δt2
⎛
⎝⎜
⎞
⎠⎟ = gij t = t0( )+
Δgij2
(A10), 654
655
where gij is some second-order tensor, gij t = t0( ) is its value at the beginning of the increment, and 656
Δgij is the change in gij over the time increment Δt . Hence, it can be seen that: 657
658
Δt2+ trel
⎛
⎝⎜
⎞
⎠⎟ΔΣij = 2ηΔhij −ΔtΣij t = t0( ) (A11). 659
660
Therefore, the Jacobian matrix describing the incremental constitutive behavior before co-rotational 661
correction is: 662
663
ΔΣij
Δ !hkl=
ηΔt2+ trel
δikδ jl +δilδ jk( ) (A12), 664
665
where !hkl is the Hencky strain tensor in Voigt notation, and δij is the Kronecker delta. Moreover, 666
the stress stored at the end of the increment is: 667
668
Σij t = t0 +Δt( ) = ΔΣij −ΩikΣkj + ΣikΩkj (A13). 669
670
The Jacobian matrix and the Kirchhoff stress at the end of the time increment expressed by Eqs. 671
(A12) and (A13), respectively, are sufficient to describe the constitutive behavior in a finite-element 672
code. We implemented the above formulation into the user routine UMAT (Abaqus/Standard, 2009) 673
to solve the FT model numerically. 674
675
3. Analysis of oscillations in the MJ model 676
The analytical solution in Eq. (15) explains the spurious softening and stress oscillations in the MJ 677
model (Fig. 2). Let us consider the expression for the shear stress in the following. It contains the 678
trigonometric term T12 =W sin Wk( )− cos Wk( ) , which is the obvious cause of the spurious 679
oscillations. The term T12 has a half-period of πW −1 . Its amplitude magnitude is: A = W 2 +1 . 680
The latter expression shows why T12 has no noticeable influence on the shear-stress response at W ≤ 681
0.1. In addition, T12 is multiplied with the damping factor e-k. Therefore, the oscillations are damped 682
at sufficiently large k ≥ 2π (Fig. A1). In the range of ten relaxation times, models with W < 1 683
exhibit spurious softening, while those at larger W show oscillations (Fig. A1). The gravity of this 684
fundamental problem becomes perhaps clearer when one considers the evolution of stored energy in 685
the MJ model. Eq. (15a) implies immediately that the elastic energy also contains trigonometric 686
terms and therefore is subject to oscillations. This can be seen in Fig. 3c, which reveals unphysical 687
negative stored energy. Due to the kinematics of the examined simple-shear case, the stored energy 688
in the system must not decrease but approach a constant maximum value of Eelmax = 0.5 σ vis
∞( )2G−1 in 689
the infinite-time limit. In other words, because constant-velocity boundary conditions are applied to 690
a homogeneous body with constant material properties, it should not relax stored stresses. However, 691
the elastic-energy solution for the MJ model exhibits negative gradients. The critical dimensionless 692
time kcrit, after which the time derivative of the elastic energy first becomes negative, is plotted for a 693
number of W in the right panel of Fig. A1. Beyond kcrit the MJ model is dissipative and violates the 694
first law of thermodynamics. Since the principal of energy conservation is at the heart of continuum 695
mechanics, we propose to abandon the MJ model. 696
697
Fig. A1: Left panel: Contour plot of e−k W sin Wk( )− cos Wk( )⎡⎣ ⎤⎦ in W-k space. Right panel: Plot of kcrit 698
over W. The time kcrit is the first point in time, at which ∂Eel / ∂k = 0 . For discussion, see text. 699
top related