Strain Decomposition

For a state variable caused by non-monotonic loads, we can define its average value ⊔_m to be the center of the smallest hypersphere that encompasses all states ⊔ upon the cycle. The amplitude of a state variable is defined as ⊔_am = max|⊔−⊔_m| (see Fig. 2a). The effective Cauchy stress tensor is decomposed as:

Where σ'_m is the average (static) component of the stress tensor and σ_am is the amplitude cyclic component of the stress tensor. It is assumed that the cyclic component is relatively small compared to the average component, which reflects the fact that the stress conditions in the soil beneath a structure subjected to high-cyclic dynamic loading impacts. The strain tensor is similarly decomposed as:

where The cyclic component of the strain tensor undergoes short-term variations due to the low-amplitude cyclic component of the stress tensor and is controlled by the elastic behavior of the material. Under stress-controlled, fully drained conditions, as a result, the stress loops are closed and the material reacts with not perfectly closed strain loops, and thus the strain accumulates. The average part of the strain tensor gradually increases as the cycle number increases (see Fig. 2c). In the following subsections, a new accumulation model will be derived that captures only the average response of the stress-strain behavior under cyclic loading, reflecting the accumulation of secondary deformation proceeding cycle by cycle.

Fig. 2.

Accumulation of secondary deformation proceeding cycle by cycle under stress-controlled conditions: stress cycles leave strain accumulation.

As a start, compression stress and compression strain are used as positive values, whereas tensile stress and tensile strain are negative. The average strain rate ε̇ _m is separated into an elastic (recoverable) strain rate ε̇ ^e_m and an inelastic (irrecoverable) strain rate ε̇ ⁱ_m as:

Where the general term of inelastic strain rate is decomposed into plastic and the accumulated strain rate is denoted by superscripts p and acc, respectively. Note that by separating plastic and accumulated strain rate, we are emphasizing rate independent (for stress paths touching the yield surface) and rate dependent cyclic plasticity, respectively. In the context of cyclic loading, “rate” means a derivative with respect to the number of cycles N, i.e., ⊔ ̇= d⊔/dN instead of ⊔ ̇= d⊔/dt, in which the accumulation of secondary deformations over a number of load cycles N can be regarded as a continuous process (pseudo-creep), and the discrete number of load cycles N is treated as a “smoothed” continuous time-like variable. For the sake of simplicity, we neglect the plastic strain rate caused by the harmonic loading in this study, then:

A separate elastic relationship is suggested to relate the elastic strain rate ε̇ ^e_m in Eq. (4) to the average stress rate σ'_m as:

where E is a fourth-order elastic stiffness tensor.

An accumulated strain rate ε̇ _m^acc in Eq. (4) is calculated from a flow rule that was originally proposed by Perzyna [26] as:

Where γ is referred to as the fluidity parameter, φ is an overstress function of the yield function f, and g is a plastic potential function. In our model, a scaling function S is used instead of γ⟨φ(f)⟩, which will be calculated by means of accumulated volumetric strain. ∂g ∂ó′^m denotes the direction of strain accumulation. The presented results of cyclic triaxial tests on sands demonstrate that the direction of strain accumulation depends essentially on the average stress ratio, while a slight influence of the stress loop (span, shape, polarization), the void ratio, the loading frequency, the static preloading, and the grain size distribution curve is observed when the cycles are performed [6]. These experimental findings prove that the cyclic flow rule can be well approximated by the flow rules of constitutive models for monotonic loading. Therefore, the direction of strain accumulation can be treated separately from the scale function S in Eq. (6).

Volumetric Compaction Relationship for 1D Stressing

Yin first introduced a logarithmic hyperbolic function to describe the long-term creep characteristics of clay [27]. In this study, we suggest a similar logarithmic hyperbolic function with limited accumulated deformation for fitting the nonlinear accumulation behavior of granular soils under high-cyclic loading. It is expressed as:

where ψ and ε^acc_vml are two constant parameters. ε^acc_vml is the limit accumulated strain, and ψ is a fitting parameter. From Eq. (7), it is also found that when N = ∞, ε^acc_vm = ε^acc_vml. This implies that there is a limit for the accumulated volume strain. With the increase of the cycle number, the volume strain due to accumulation will never exceed this volume strain limit of ε^acc_vml. That is to say, after a specific number of load cycles, the material response turns into a shakedown. However, this function is generally reasonable, as the utilization of granular locking during the process of progressive material compaction serves to prevent the unlimited expansion of the associated accumulated deformation.

Differentiating Eq. (7) leads to the accumulation rate with respect to N:

This is the so-called “time” hardening formulation. Another formulation can be obtained by eliminating time (the number of cycles) from Eq. (7). This brings in the strain and leads to the so-called strain hardening formulation, in which the accumulation rate in a variable stress situation depends on the stress (through the parameters that will be discussed in the next section). In this method, Eq. (8) is written as:

Three-Dimensional Generalization

In this study, an associated plastic flow rule is employed, where the plastic potential function g in Equation (6) is equivalent to the flow-surface function f, which depends on stress level and hardening history. Based on the results of drained cyclic triaxial tests, the flow rule of the modified Cam-clay model can well approximate the measured ε̇ _vm^acc/ε̇ _qm^acc ratios [16]. We consider an elliptical plastic potential like that used in the modified Cam-clay model:

where M is the slope of the critical state line in the p-q space and controls the shape of the plastic potential. M = M_c = 6sinφ∕(3−sinφ) in compression, and M = M_e = 6sinφ∕(3+sinφ) in extension; φ is the internal friction angle. The value of p_m0 is the hydrostatic pressure where an elliptical locus meets the p axis. If a stress state is known, p_m0 can be found from Eq. (10):

According to Eq. (6), the accumulated volumetric and deviatoric strain rates, ε̇ _vm and ε̇ _qm^acc, respectively, are related to the normal plastic potential at the current stress state as:

The scaling function S can be calculated by Eqs. (9) and (12) as:

The general expression for the accumulated strain rates is given by:

via Eqs. (6) and (14). The description of the model is now complete, and the accumulation model can be summarized as:

where the accumulation rates ε̇ _m^acc are calculated from Eq. (15).

The components of the accumulation rates contributing to volumetric and deviatoric strains are generated by the first and second terms, respectively, of the differential form of the plastic potential. For the elliptical plastical potential function in Eq. (6), these forms are:

The derivatives of the stress invariants are:

and the direction of strain accumulation is obtained from:

Cyclic Triaxial Tests

The verification of the proposed model will be performed by employing the cyclic triaxial test data. The tests are done on freshly dense fine sand (ρ_s = 2.63 g/cm³, e_min = 0.624, e_max = 0.925, d₅₀ = 0.23 mm, U = d₆₀/d₁₀ = 1.85, φ_c = 33.2). The stress conditions for the axisymmetric specimen have been schematized in Fig. 3. Under triaxial conditions, the soil sample is confined by a hydrostatic stress σ_c, while an additional vertical stress σ_v is applied. The vertical stress is composed of a static part σ_stat and a cyclic part σ_cyc, which is characterized by a periodic sinusoidal vibration with an amplitude σ_am and a frequency f. Apparently, the principal stress, σ₁, results from the summation of the confining pressure σ_c and the vertically applied stress, σ_v, while the principal stress σ₃, is equal to the confining pressure. The stress state is characterized by an average part with the hydrostatic pressure p_m = σ_c + σ_sat/3, the deviatoric stress q_m = σ_sat, a cyclic part with the amplitude of the hydrostatic pressure p_am = σ_am/3, and the amplitude of the deviatoric stress q_am = σ_am (see Fig. 3). The cyclic stress ratio,

is used. An axial deformation ε₁ and a radial deformationε₃ are observed, whereas the strain invariants reduce to:

Fig. 3.

Stress state in p-q space for a cyclic triaxial test: An average stress (described by p_m and q_m) is superposed by a cyclic portion (described by p_am and q_am).

Determination of the Elastic Stiffness E

The elastic stiffness E interrelates the trends of stress and strain evolution in Eq. (16). If the elastic deformation of the soil is assumed to be isotropic, there are only two constants (or modules) as:

Where I is a fourth-order identity tensor which is defined as I_ijkl=(1/2)(δ_ikδ_jl+δ_ilδ_jk) and δ is a second-order Kronecker unit tensor with a Kronecker’s symbol δ_ij, ⨂ denotes tensor product. ν is the Poisson’s ratio. The shear modulus G can be deduced as:

For an axisymmetric stress state, it is convenient to rewrite eq. (16) with Roscoe’s invariants as:

Where K is a bulk modulus and G is a shear modulus. In a drained test with stress-controlled cycles, the ratio of the rates of volumetric and deviatoric portions corresponds to the well-known formula of the modified Cam-clay model as:

For a stress-controlled isotropic stress state, q_m = 0, q̇ _m = 0. Under undrained conditions, ε̇ _vm = 0, Eq. (27) takes the form of pore pressure accumulation (isotropic relaxation) as:

Under drained conditions, ṗ '_m = 0, Eq. (27) takes the form of volumetric strain accumulation as:

The bulk modulus, K, can be obtained from the ratio of the rate of pore pressure accumulation, u̇ , in an undrained cyclic triaxial test to the rate of volumetric strain accumulation, ε̇ _vm^acc, in a drained cyclic triaxial test with similar initial stress, void ratio, and cyclic loading.

In combination with a constant Poisson’s ratio ν, the shear modulus G can be calculated from Eq. (26).

For evaluation of bulk modulus, K = u̇ /ε̇ _vm^acc, 12 drained and undrained cyclic triaxial tests (see Table 1) at different stress levels p_m and stress ratio η_m are carried out. All tests were performed with uniform, medium-coarse to coarse quartz sand. All samples are prepared with a relative density of about 60%. The mean grain diameter d₅₀ is equal to 0.56mm and the maximum void ratio e_max and minimum void ratio e_min are equal to 0.942 and 0.573, respectively. The critical friction angle φ_c is equal to 37.7. For each pair of drained and undrained cyclic tests, the bulk modulus, K, is calculated from Eq. (31). In all tests (drained and undrained), the first cycle may be irregular and may generate much more deformation than subsequent ones. The proposed accumulation model predicts only the accumulation due to the subsequent regular cycles. As for numerical calculations with the proposed accumulation model, we will calculate the first cycle using a conventional implicit constitutive model. So the first cycle is not included in the evaluation of bulk modulus. In all diagrams, N = 1 refers to the first regular cycle. The rates u̇ and ε̇ _vm^acc are calculated for 20 load cycle increments. Fig. 4 presents the bulk modulus K versus average hydrostatic p_m. The obvious pressure-dependence of K can be approximated by:

Fig. 4.

Bulk modulus K determined by the comparison with the data from the cyclic triaxial tests as a function of hydrostatic pressure p_m.

With K_ref = 145 MPa and n = 0.51 (the thick solid line in Fig. 4). The p_atm is the standard atmospheric pressure, i.e., p_atm= 100 kPa. Furthermore, a constant Poisson’s ratio ν = 0.3 is assumed, which is appropriate for medium density compacted sands.

Determination of the Accumulation

Model Parameters ε^acc_vml, Ψ

As mentioned above, ε^acc_vml is the limit of volume strain accumulation in Eq. (7). The determination of ε^acc_vml is not an easy task because it is impossible to run a test of an infinite cycle. However, ε^acc_vml will have an upper bound value. In an extreme case, assuming the void ratio of a soil element (or specimen) all becomes the minimum void ratio e_min under accumulation at infinite cycle, the upper bound limit is ε^acc_{vml, uper} = e_min/(1 + e_min). Since e_min = 0.573, the upper limit value of volume strain accumulation ε^acc_{vml, uper} = 1.342 for medium dense compacted sands in this paper. In case no long-term accumulation data are available, the limit value of volume strain accumulation can be estimated from curve-fitting test data.

For curve fitting and determination of ε_vm^acc and ψ, Eq. (7) can be written as:

The measured data of volume strain accumulation against the number of cycles is used to calculate the relationship between the normalized ratio ln(1 + N)/ ε_vm^acc and ln(1 + N). For isotropic stress, it is found that the data points of ln(1 + N)/ε_vm^acc against ln(1 + N) are almost on a straight line (see Fig. 5). Comparing the best-fitting equation with Eq. (33), over a load cycle increment [N, N + ΔN], ε_vm^acc and ψ can be computed as:

Fig. 5.

Comparison of measured data and fitting curve by a straight line obtained from the normalized ratio ln(1 + N)/ε_vm ^acc to ln (1+N) for tests at p_m = 100 kPa, 200 kPa, 300 kPa respectively.

Fig. 6.

Relationship of accumulation model paramters a) ε_vml ^acc and b) ψ to the isotropic stress p_m scale on a straight line respectively.

Fig. 7.

Experimental results of cyclic triaxial tests versus predictions of model: accumulation of a) volumetric deformation, ε_vm acc and (b) deviatoric deformation, ε_qm ^acc over 10⁶ cycles under axi-symmetrical stress conditions at hydrostatic pressure p_m = 100 kPa, cyclic stress ratio ζ_am = 0.3, and avegage stress ratio η_m = 0.5, 0.75, 1.0 respectively.

ratio ζ_am = 0.3, and avegage stress ratio η_m = 0.5, 0.75, 1.0 respectively. and

For isotropic stress of 100 kPa, computing the data points of ln(1 + N)/ε_vm^acc against ln(1 + N) over a load increment [100, 1000] with Eqs. (34) and (35), it is found that 1∕ψ = 129.86 and ψ = 0.0077; 1/ε_vm^acc = 29.925 and 1/ε_vm^acc = 0.0332. The same procedure has been used to compute eqs. (34) and (35) from the test data for isotropic stresses of 200 and 300 kPa. The values of ψ and ε^acc_vml for these three stresses are presented in Table 2. The fitted lines are shown in Fig. 5. In general, Eq. (33) fits the test data very well.

Examining the data in Table 2, it is found that both ε^acc_vml and ψ decrease with isotropic stress. The relationship of ε^acc_vml and ψ to the isotropic stress p_m scale is plotted in Fig. 6. It is seen from the figures that the data points are almost on a straight line. By best fitting these data points, it is found that:

Fig. 8.

Direction of strain accumulation as a function of the mean stress ratio from the flow rule of the modified Cam clay model is compared with the experimental results from cyclic triaxial tests in incremental form a) Ω = Δ ε^acc vm/εacc ^qm and b) reciprocal value 1/Ω.

Where ε^acc_vmlu = 0.0715 and ψ = 0.0065 are in correspondence with p_u = 1 kPa (unit stress), p_atm is the standard atmospheric pressure, i.e., p_atm = 100 kPa. Fig. 6 (or Eqs. (36) and (37)) shows that the two parameters ε^acc_vml and ψ are not constant, but decrease as the isotropic stress p_m increases with ε^acc_{vml, ref} = 0.01, α = 1 and ψ_ref = 0.001, β = 1.

Simulation of Drained Cyclic Triaxial Compression Tests

With the values of K, ε^acc_vml, and ψ given by eqs. (32), (36) and (37), the proposed model has been fully calibrated for describing the accumulation of secondary deformation in granular soils under high-cyclic loading. Fig. 7 shows a comparison between experiments and corresponding predictions concerning the uniaxial accumulation of volume and deviatoric deformations of sands subjected to different average stresses (p_m, q_m) = (100, 50), (100, 75), and (100, 100) kPa with a constant cyclic stress ratio ζ_am = 0.3. It was found that the numerical results match the experimental data well. Both permanent volumetric strain and deviatoric strain increase monotonically with the number of cycles, though predictions during the first load cycles appear to be somewhat inaccurate. However, this discrepancy is less important because the number of load cycles commonly associated with foundation deterioration lies far beyond one thousand.

Fig. 8 shows the direction of strain accumulation from the cyclic triaxial tests in incremental form Ω = Δ ε^acc_vm/ε^acc_qm. Both the measured directions of strain accumulation and the flow rule of the modified Cam-clay model are given in Fig. 8. It is seen that the flow rule of the modified Cam-clay model, i.e., a model for monotonic loading, approximates well the direction of strain accumulation under cyclic loading. That is to say, cyclic loading with average stress induces the accumulation of strain in the direction of average stress with the increase of the number of load cycles. The same conclusions as mentioned above were reported by Chang & Whitman [28].