Introduction
Polycrystalline ice exhibits both brittle and ductile behavior. At low strain rates, strength increases with strain rate, following a power law. In constant strain-rate tests, the maximum stress is reached at strain close to 10−2 (Reference Mellor and ColeMellor and Cole, 1982). At high strain rates (typically higher than 10−3s−1 in uniaxial compression), the material is purely brittle and the failure stress is relatively insensitive to strain rate (Reference ColeCole, 1987). However, the failure mode is highly sensitive to the ice/end-cap interface conditions (Reference Lasonde, Gies, Schulson, Saeki and HirayamaLasonde and others, 1988); it has been reported to be either axial splitting, shear faulting or explosive failure (Reference SchulsonSchulson, 1991). Fracture occurs at very low strains (Reference SchulsonSchulson, 1987). In the intermediate range of strain rates, microcracking occurs and a peak stress is observed. Strain in excess of 5 × 10−2 can be reached without macroscopic failure (Reference Meyssonnier, Duval, Axelsson and FranssonMeyssonnicr and Duval, 1989; Reference Stone, Jordaan, Jones, McKenna, Axelsson and FranssonStone and others, 1989). Ice exhibits a semi-ductile behavior.
The strain rate marking the ductile-brittle transition depends on stress state, temperature and grain-size (Reference Schulson and CannonSchulson and Cannon, 1984; Reference ColeCole, 1987). By inhibiting cracking activity, a moderate confining pressure shifts the ductile-brittle transition towards higher strain rates and the strength of ice increases (Reference JonesJones, 1978, Reference Jones1982; Reference Murrell, Sammonds and RistMurrell and others, 1990).
The paper is concerned with the strength of freshwater isotropic granular ice under uniaxial and triaxial compression, in the range of strain rate where ice exhibits a semi-ductile behavior. It focuses on the influence of microcracks on ice deformation.
Continuum-damage mechanics provides a suitable framework for the development of constitutive models describing the progressive degradation of the mechanical properties of ice undergoing homogeneously distributed microcracking. It has only recently been introduced into the field of ice mechanics (Reference SjölindSjölind, 1987; Reference Karr and ChoiKarr and Choi, 1989; Reference McKenna, Meyssonnier, Jordaan, Sinha, Sodhi and ChungMcKenna and others, 1989; Reference Meyssonnier, Duval, Axelsson and FranssonMeyssonnier and Duval, 1989). Damage models incorporate kinetic equations describing the damage evolution, and equations giving the mechanical behavior of the material at a given state of damage.
Concerning the damage growth, recent work has shown the important role of the delayed clastic strain in the nucleation process (Reference SinhaSinha, 1984; Reference SchulsonSchulson, 1987; Reference Kalifa, Duval, Ricard, Sinha, Sodhi and ChungKalifa and others, 1989). Besides, observations using either visual or accoustic emission techniques have shown that maximum cracking activity occurs before the min-imum creep rate (or the peak stress for constant strain-rate tests) is reached, and that further evolution, especially the development of heterogeneities, depends on the stress level (or strain rate) (Reference GoldGold, 1970a, Reference Goldb; Reference St Lawrence and ColeSt Lawrence and Cole, 1982; Reference Cole, St Lawrence, Hardy and LcightonCole and St Lawrence, 1984; Reference ColeCole, 1986, Reference Cole1991). This was also observed under triaxial compression (Reference Murrell, Sammonds and RistMurrell and others, 1991). However, many more quantitative results on the crack population under uniaxial and triaxial compression arc required to model the evolution of damage with the stress-strain history.
Concerning the mechanical response of damaged ice, two recent works have focused on the semi-ductile be-havior of granular ice. Reference Meyssonnier, Duval, Axelsson and FranssonMeyssonnier and Duval (1989) have shown that the creep rate of damaged ice could be 500 times higher than that of virgin ice, but that the exponent of the power law (n = 3) was unchanged. These results were discussed by using the continuum-damage mechanics approach. Reference SinhaSinha (1988) incorporated the effect of cracking in his non-linear viscoelastic creep equation. He assumed that the delayed elastic and clastic strain components were not affected by damage. The effect of cracks on the viscous strain rate was analysed by adopting Reference WeertmanWeertman’s (1969) model for dilute concentration of non-interacting cracks.
The present work describes two main series of experiments, both focused on the semi-ductile behavior of granular ice. In the first one, we carried out constant cross-head velocity tests under confining pressure. The critical stress and strain at the peak stress were determined, and the structural state of ice at this stage was characterized. The aim of the second scries was to study the influence of damage on the behavior law, in uniaxial compression, at a strain rate of 5 × 10−4 s−1. Tests were halted at various levels of damage and recovery strain components were measured at unloading. Besides these two series, constant strain-rate tests and creep tests were performed on virgin specimens in order to determine the mechanical constants of the ice used.
Before describing our experimental procedure and discussing the results, we will present the rhcological method with which we described the behavior of virgin ice, as well as the concept we used to incorporate the effect of damage on the constitutive equations.
Constitutive Equations and Effect of Damage
In most models describing the creep behavior of polycrystalline ice, strain (ε) is assumed as the sum of an instantaneous elastic component εe, and two time-dependent components: the delayed clastic strain (or visco-elastic strain) εd and the viscous strain εv. Several rhcological models have been proposed to describe the creep behavior of polycrystalline ice (Reference Le Gac, Duval and TrydeLe Gac and Duval, 1980; Reference Ashby and DuvalAshby and Duval, 1985; Reference Sunder and WuShyam Sunder and Wu, 1989). A simple rheological model, showing both transient and steady-state creep, is the Burgers body: a Maxwell element and a Kelvin-Voigt clement in series (Fig. 1). The Burgers body, characterized by four parameters, has been adopted in this paper. A more realistic model should include several Kelvin-Voigt elements as shown in Figure 1.
Rheological model used to represent the creep behavior of granular ice.
The constitutive equation associated with the Burgers body is:
where 
is visco-elastic modulus, η is viscosity, Ee is Young’s modulus, A and n are the constants of Glen’s power law. Since this model does not incorporate the influence of the microstructure or temperature, the parameters of the equation have to be determined for every case. To take into account the effect of damage on the constitutive equations, we used the concept of effective stress: a damaged volume of material under the applied stress σ shows the same strain response as the undamaged one submitted to the effective stress 
(Reference ChabocheChaboche, 1988).
In the general case, the effective stress tensor 
is related to the applied stress tensor σ through the relationship:
where D is the damage tensor. When damage is isotropic, D becomes a scalar variable D. Although damage is not isotropic in ice, we assumed that D is a scalar. When there is no damage, D = 0 and 
When failure occurs, D = 1.
Moreover, by using a single damage variable, we assumed that the effect of damage is the same for the clastic, delayed clastic and viscous strain components.
Experimental Procedure
Specimens
All tests were carried out at −10.0±0.2°C on fairly bubble-free, isotropic granular ice (Ih ice) made at the laboratory. Specimens exhibited a mean grain-size of either 2 mm (called in this paper “fine”-grained ice) or 5 mm (“coarse”-grained ice). The average grain-size (dg) was estimated using the relationship (Reference DieterDieter, 1976):
where Na is the number of grains per unit area. The specimens were machined to obtain cylinders of 40 mm diameter for triaxial tests and 60 mm for uniaxial tests. Their height was twice their diameter. For triaxial tests, the specimen dimensions were limited by the capability of the testing machine and by the geometry of the confinement cell. In all tests, the end caps were smooth stainless steel platens, and the specimens were not bonded to them. Before testing, the specimens were pre-loaded at ≈0.5 MPa for 10 min in order to smooth the ends of the specimens. The very small deformation induced by this pre-loading was neglected.
Triaxial Tests
The triaxial tests were performed on a 40 kN electromechanical press provided with a displacement control mechanism of a platen, and which supplied a uniaxial stress to a specimen. The stiffness of the frame was measured as 0.23 MN mm−1. An isotropic pressure, obtained by the compression of a perfectly transparent silicon oil in a confinement cell, was superimposed. This cell included a compensation piston connected to the main piston in order to ensure a constant fluid pressure. A more detailed description of this cell was given in a previous paper (Reference Kalifa, Duval, Ricard, Sinha, Sodhi and ChungKalifa and others, 1989). The walls of the cell were provided with four windows that enabled visual monitoring of the specimens during the tests, an important feature for cracking studies. Cracks were rendered visible by the reflection of a light source on their faces.
The machine was equipped with a strain-gauge load cell providing a measurement of the axial load. The main stresses σ1, σ2, σ3 on the specimen are:
where F is the axial force, S the cross-sectional area of the specimen and Pc the confining pressure. Note that compressive stresses arc treated as positive, and that σ1 is the greatest stress. An LVDT transducer fixed to the cap of the confinement cell measured the displacement of the main piston (dp ), from which we approximated the axial strain of the specimen: ε = dp/h where h is the nominal specimen height.
The cross-head velocity was varied so that the nominal strain rate ranged between 2.5 × 10−5 and 5 × 10−3 s−1. However, the actual strain rate did not remain constant during the tests: the initial strain rate was smaller than the nominal value, which was reached close to the peak stress. For each strain rate, the confining pressure was varied between 0 and 10 × 0.2 MPa. It was monitored on a dial and no variation was noticed during the tests.
Visual monitoring of the samples during the tests gave us a general idea of the initiation and the evolution of cracking activity. Special attention was given to cracking homogeneity and to ice/end-cap interface effects. A quantitative characterization of the structural state of ice at the peak stress was subsequently obtained by systematically stopping the tests at this event and immediately analyzing sections taken from the specimens. This analysis was performed only on coarse-grained specimens. The peak stress was identified by direct observation of the force measurement. When this step was reached, the confining pressure was immediately and slowly reduced to zero in order to avoid crack closure. A quick release of confining pressure would have produced new cracks. This undesired specimen alteration was apparently negligible if the procedure was done carefully. Three sections of a thickness of 2 mm were taken from the central one-third of the specimens: two were horizontal (perpendicular to the σ1 axis) and the third one was vertical. This analysis provided qualitative results on orientation, length and nuclcation site of the cracks, which were discussed in a previous paper (Reference Kalifa, Duval, Ricard, Sinha, Sodhi and ChungKalifa and others, 1989). Crack density was measured on the horizontal sections and observed through a microscope under polarized light. We counted the cracks intersecting the upper surface of the section and a straight-line segment. This operation was repeated eight times, and 40–60 cracks were counted on each section. We then deduced a linear density of cracks ρL:
where Nc is the number of cracks counted in the section and L is the total segment length. To calculate the number of cracks per grain (ρg), we assumed that the cracks had a mean length equal to the average grain diameter (dg). Then,
Note that this calculation was done with no regard to the crack-nuclcation sites. We preferred to use this procedure rather than to estimate a superficial crack density, because the latter was high enough to enable crack linkage. Then, the distinction between a single crack and two linked cracks was ambiguous.
Uniaxial Tests
The uniaxial test series was performed on a second machine: a closed-loop controled machine with a capacity of 0.5 MN and a theoretical rigidity of 1.68 MN mm−1. This machine was equipped with a piezoelectric load cell providing a measurement of the axial load. Strain was measured with LVDTs, using two methods: the transducer measured either the relative displacement of the platens or was fixed directly on the central part of the specimen. A light melting freezing process assured the attachment of the grips on the surface of the specimen. The grips were connected together during this procedure in order to prevent misalignment and removed before testing.
For both triaxial and uniaxial tests, the analog outputs of all the transducers were recorded on a strip-chart recorder. Stress-strain curves were then generated by taking data points from these records and using a least-squares fitting method.
Uniaxial tests were conducted at a constant piston velocity. Since the height of the specimen varied slightly from test to test, the actual strain rate was also altered. The average strain rates are, for fine-grained ice 
and for coarse-grained ice 
.
The tests were stopped at various stages on the stress-strain path, before or after the peak stress. Strain recovery after unloading was measured. Since delayed elastic strain takes a very long time to recover fully, strain was expressed as
where 
is the non-elastic or complementary strain. However, a lower bound of εd and an upper bound of εv were estimated. Before each test, a “stress-impulse” test was performed in order to measure Young’s modulus of virgin ice. The specimen was loaded at 10−3s−1, then unloaded very fast. The maximum stress σf never exceeded 3 MPa in order to prevent cracking. As shown by the strain measurement, the specimen exhibited a linear elastic behavior. Young’s modulus Ee was then deduced:
where εf is the deformation instantaneously recovered at unloading.
Creep Tests
Creep tests were carried out on virgin ice in order to determine visco-elastic and viscoplastic constants. A third apparatus was used for these tests: the constant load was provided by means of dead weights hung at the extremity of a lever. The strain was measured directly on the specimen. The applied load was between 1.5 and 1.7 MPa and, after 2–2.5 h, the secondary creep was virtually reached. Assuming that the secondary creep rate 
is given by Glen’s law:
we deduced A from the tests. As well, the viscous deformation was calculated as
where tf is test duration. The linear elastic strain (εe) was also measured at unloading. The delayed elastic component (εd) was merely deduced:
where εf is the total strain at the end of the test. A visco-elastic modulus Ed was then calculated:
Results and Analysis
Mechanical Constants of Virgin Ice
Table 1 summarizes the average mechanical constants oí virgin ice for both grain-sizes. Elastic moduli for each test arc given in Table 2. The large standard deviation of the clastic moduli may be partly due to the method of measurement: very small strains arc involved in the stress-impulse tests and, since only one transducer was fixed on the specimen, a small non-parallelism of the ends could induce a large inaccuracy. Also, one notes that strain measurements made on the specimen gave a larger value of the modulus. The effect of grain-size is not noticeable; it is within the standard deviation. The low value of the experimental moduli compared to the theoretical value (9.5 GPa) may also be a result of the procedure for making the granular ice; the very small bubbles it contains may reduce the elastic modulus.
Experimental values of the mechanical constants used in the constitutive equation (Equation (3)), for both grain-sizes
Results for uniaxial tests. The second column indicates the type of strain measurement: “platens” means that strain was measured between the platens, “specimen” means that the transducer was fixed on the central part of the specimen. In the other columns, the subscript p is used for measurement at the peak stress, while the subscript f is related to the end of the tests. As well, εe and εc are the elastic and complementary components, respectively, at the end of the tests
The tests gave exactly the same visco-elastic modulus for both grain-sizes. The variation of η with grain-size was also very small.
Uniaxial Tests
Test conditions and results arc summarized in Table 2.
For each grain-size, a composite σ–ε curve combining all six tests was generated, using a least-squares fitting method. The curves arc plotted in Figure 2. The equations are:
where σ is expressed in MPa and e in units of millistrain.
Composite stress-strain curves for uniaxial tests, generated using the analog records of all the tests (six tests for each grain-size).
The visco-elastic viscosity was calculated for both grain-sizes, substituting Equations (13) in Equation (1), and integrating the latter between t = 0 and t = 1s. These times were chosen according to the fact that the viscous strain is negligible during the first second of loading.
The strain components were measured at unloading. The linear clastic strain and the complementary strain are represented as a function of strain in Figures 3 and 4, respectively. Since there was no evident grain-size effect, we plotted both sets of data on the same graphs. The equations of the fitting curves (using a least-squares method) are:
where ε, εe and εc are in millistrain. One notes that the linear clastic strain passes a maximum at ε ≈ 3 × 10−3, which is very close to the strain at the peak stress. Therefore, part of the elastic deformation is released after the peak stress and transferred to the complementary component, as shown by the increase in εc. From the data points, it is possible to give an estimate of the delayed elastic and viscous components at the peak stress:
Uniaxial tests: elastic component versus total strain. The polynomial fitting uses data points for both grain-sizes.
Uniaxial teats: complementary component versus total strain. The polynomial fitting uses data points for both grain-sizes.
Triaxial Tests
Experimental conditions and results are summarized in Tables 3 and 4.
Results related to the peak stress for triaxial tests on fine-grained ice (dg = 2 mm)
Results related lo the peak stress for triaxial tests on coarse-grained ice (ds = 5 mm). The number of cracks per grain was deduced from the last two columns by dividing each number by the average grain diameter (dg)
In all experiments, the first cracks appeared at critical stress and strain between 2 and 4 times lower than those corresponding to the peak stress. Consistent with previous work (Reference ColeCole, 1986), their sizes were slightly smaller than those of grains. Further increase in stress did not result in visible crack growth, but other similar microcracks were nucleated. At strain rates higher than 10−4 s−1, high cracking activity made the specimens opaque.
When specimens did not fail, stress-strain curves were characterized by a peak, the width of which increased with decreasing strain rate and increasing confining pressure. For
increased with strain rate following a power law, as shown in Figure 5, where σp is plotted versus the nominal strain rate on a log-log scale diagram. In uniaxial compression, for 
failure occurred along shear bands, and strength was independent of strain rate, giving prominence to a brittle behavior. Careful examination of the rupture surfaces showed that they followed the grain boundaries. This has been confirmed by quantitative investigations performed on the thick sections (Reference Kalifa, Duval, Ricard, Sinha, Sodhi and ChungKalifa and others, 1989), and was also reported by Reference ColeCole (1987). The application of a confining pressure led to homogeneous cracking activity and prevented specimen failure. In Figure 5, the plateau observed in uniaxial compression disappeared in triaxial compression.
Triaxial tests: maximum deviatoric stress versus strain rate for various confining pressures, (a) fiue-grained ice; (b) coarsegrained ice.
In Figure 6, σp is plotted versus σ3 for the various strain rates tested and for both grain-sizes. The rate of strength increase with σ3 increased strongly with strain rate. At 5 × 10−5 s−1, σp increased from about 4 to 7 MPa while the confining pressure was varied from 0 to 10 MPa. At 5 × 10−3 s−1, the same variation in confining pressure made σp increase from 10 to 20 MPa. These results are in good agreement with previous work (Reference JonesJones, 1978, Reference Jones1982; Reference Murrell, Sammonds and RistMurrell and others, 1991).
Triaxial tests: maximum deviatoric stress versus confining pressure for various strain rates, (a) Fine-grained ice; (b) coarsegrained ice.
Note that, consistent with observations in uniaxial compression by Reference Schulson and CannonSchulson and Cannon (1984) and Reference ColeCole (1987), the strength of ice decreased slightly with increasing grain-size. This was also true in triaxial compression.
In Figure 7, crack density (as a number of cracks per grain) is plotted versus σ3 for various strain rates. The ends of the vertical bars correspond to the crack density on the two horizontal sections made for each specimen. At low strain rates (< 10−4 s−1), an increase in confining pressure resulted in a striking decrease in crack density at the peak stress; it passed from 1 to 0.3 cracks per grain when Pc was increased from 0 to 10 MPa. At higher confining pressures, strain at the peak stress is of the order of 10−2 (Tables 3 and 4), and grains exhibited a visible permanent deformation (at least in fine-grained ice). One has to be reminded that 10−2 is the critical strain for the onset of dynamic rccrystallization (corresponding to the beginning of tertiary creep) in creep tests on ice. This indicates that ice has an essentially ductile behaviour, with very little influence of cracks. This assertion is supported by the weak increase in strength with confining pressure; Reference Jones and ChewJones and Chew (1983) demonstrated that confining pressure has little effect on viscous deformation processes.
Triaxial tests: crack density at the peak stress versus confining pressure for various strain rates. Measurements were only done for coarse-grained ice.
For
Discussion
The triaxial tests we performed showed that the application of a confining pressure shifts the ductile brittle transition towards higher strain rates: strength kept increasing with strain rate beyond 10−3s−1 instead of reaching a plateau; localized damage and sudden failure were replaced by homogeneous damage and semi-ductile behavior. The latter was characterized by a critical crack density at the peak stress of 1 crack per grain. Qualitative observations have shown that damage is anisotropic: cracks are preferentially oriented parallel to the most compressive stress axis, although this trend weakens with increasing confining pressure (Reference Kalifa, Duval, Ricard, Sinha, Sodhi and ChungKalifa and others, 1989). We also observed that confining pressure tends to close cracks. Reference Jaeger and CookJaeger and Cook (1984) calculated that only a small normal load is required for a penny-shaped crack to close. The consequences are: (a) a low alteration of the elastic modulus (Reference Jaeger and CookJaeger and Cook, 1984); as a result, the relative alteration of both visco-elastic and viscoplastic properties is higher, (b) An increase in the friction forces between crack faces, reducing the effective shear stresses they undergo; as a result, for the same applied stress field, the local stresses at the crack tips are lower and the propensity of cracks to propagate as well as to interact is smaller.
Assuming that the elastic modulus (Ee) is not altered by damage, the clastic component at the peak stress is 
. In Figure 8, histograms show εe and εc as a function of strain rate and confining pressure. This calculation underestimates slightly εe especially at Pc = 0 MPa. However, it is clear that, in spite of the increase in peak stress with confining pressure, inducing an increase in εe, the ratio εc/εe increases with confining pressure.
Estimation of the evolution, of strain components at the peak stress with confining pressure and strain rate, for coarse-grained ice. Equation (17) urns used in conjunction with strain measurements.
Using the concept of effective stress and assuming that ice behaves according to the rheological model of Figure 1, we constructed 
and εc – ε plots for the uniaxial compression tests. For these, we substituted σ for 
in Equation (1) and solved the new equation, 
being the variable. The details of the calculations are given in the Appendix. The mechanical parameters appearing in the equations are those given in Table 1. Results in uniaxial compression at 5 × 10−4 s−1 are given in Figures 9 and 10 for fine grains and coarse grains, respectively. The curves σ – ε and 
(Figs 9a and 10a) mingle in the first part, before damage takes place. This confirms that the mechanical parameters of virgin ice were correctly estimated. When the effect of damage becomes significant, the curves diverge.
Comparison between experiment and theory, using the concept of effective stress, for fine-grained ice tested in uniaxial compression. In graph, (a), experimental stress and effective stress are plotted versus strain. In graph, (b), the elastic component is concerned: data points are plotted in conjunction with, a fitting curve; the curve εe(σ) is calculated using the model, and the curve εe(σ) is followed by that of an undamaged specimen which has undergone the stress of σ(t). Graph (c) is related to the complementary component.
Comparison between experiment and theory, using the concept of effective stress, for coarse-grained ice. tested in uniaxial compression. Presentation is the same as in Figure 9.
In Figures 9b and 10b, we plotted εe calculated with the model versus ε, in conjunction with the curve εe – ε which would be followed by an undamaged specimen undergoing σ(t). Also plotted arc the data points. The equation of the fitting curve is Equation (14a). Similar plots are given for the complementary strain εc in Figures 9c and 10c. The first observation is that the decrease in elastic strain observed after the peak stress is not represented by the model. However, for fine-grained ice, the data fit fairly well with the undamaged specimen curve. Thus, fine-grained ice exhibited a very slight alteration of the clastic modulus with damage. This alteration is larger for coarse-grained ice: data points are above the curve for the undamaged specimen after the peak stress, pointing out a slight decrease in elastic modulus. In Figures 9c and 10c, the complementary strain given by the model is smaller than the one actually exhibited by specimens, especially after the peak stress.
Therefore, under the conditions of the study, the model overestimates the alteration of the elastic modulus and underestimates the alteration of the visco-elastic and viscoplastic properties. Below the ductile-brittle transition, the relative importance of damage on elastic and non-elastic properties depends on strain rate. Moreover, the high plastic anisotropy of ice monocrystals may emphasize the role of cracks on the non-elastic behaviour of the polycrystal; cracks are probably nucleated preferentially in those grains undergoing the highest internal stresses, i.e. that are badly oriented for basal glide. Crack nucleation should make stress distribution more homogeneous, hence stress should rise in uncracked grains. As a result, complementary deformation of the specimen should increase.
This preliminary study shows that the effect of damage on the deformation of isotropic granular ice cannot be described by introducing the effective stress directly in the rheological equation. The results seem to support the assumption made by Reference Karr and ChoiKarr and Choi (1989) that damage affects the elastic, delayed elastic and viscous strain components independently.
Conclusions
Uniaxial and triaxial compression tests have been carried out at –10°C in order to study the semi-ductile behavior of granular ice. We have shown that the role of confining pressure is to inhibit crack growth and crack interaction, and to shift the ductile-brittle transition towards higher strain rates. For
The evolution of the clastic and the complementary strain components with the stress-strain history was in-vestigated in uniaxial compression tests at 
by measuring recovery strain during relaxation. We applied the concept of effective stress to incorporate the effect of damage on these components. Although tests are preliminary, we have shown that the use of a single damage variable is not satisfactory. More tests are needed before establishing how microcracks specifically affect the elastic, delayed elastic and viscous strain components. The next step will be the formulation of damage evolution.
Acknowledgements
This work was supported by the Centre National de la Recherche Scientifique, Département des Sciences Physiques pour l–Ingénieur (France), and the Ministère de la Recherche, Mission Scientifique et Technique (France). We wish to thank Ε. M. Schulson and S.J. Jones for constructive comments on this work. We are grateful to M. Ricard, who designed the triaxial cell and provided technical assistance during the tests. We are also very grateful to the Institute for Marine Dynamics (St. John–s, Newfoundland, Canada) where part of this work was written.
The accuracy of references in the text and in this list is the responsibility of the authors, to whom queries should be addressed.
Appendix Calculation of the Effective Stress and of the Corresponding Strain Components
The principle of strain equivalence adopted here states that the strain response of the damaged specimen is derived by substituting the applied stress σ with the effective stress 
in the constitutive equation for the virgin specimen, i.e. Equation (1).
To solve this equation numerically, we split the strain history ε(t) into n intervals of constant length Δε=10–5. n is given by 
where εf is the strain at the end of the test. The duration Δti
of the ith time interval [ti–1, ti] is a function of strain rate. In the uniaxial tests, strain rate was a constant and Δti
is merely 
. At the instant ti,
Equation (1) is
or
We make the assumption that the effective stress is constant over the ith interval and equal to σi = σ(ti). We can then write differently each strain component. The elastic component is:
The delayed elastic component can be expressed as:
Then,
Finally, the viscous component is:
Substituting Equations (A3), (A5) and (A6) into Equation (A2), and using the definition of Δε
This is an equation of the third degree in 
. The condition at t0
= 0 is 
is different from 0 when a pre-load was applied before starting the test. Equation (A7) has one real solution, which is found by an approximation method. Then, 
is replaced in Equations (A3), (A5) and (A6) to give the values of the strain components at t = ti.
