Critical Assessment of Nonlocal Strain-Softening Methods in Biaxial Compression

Summersgill, F. C.; Kontoe, S.; Potts, D. M.

doi:10.1061/(ASCE)GM.1943-5622.0000852

Open access

Technical Papers

Jan 16, 2017

Critical Assessment of Nonlocal Strain-Softening Methods in Biaxial Compression

Authors: F. C. Summersgill [email protected], S. Kontoe [email protected], and D. M. Potts [email protected]Author Affiliations

Publication: International Journal of Geomechanics

Volume 17, Issue 7

https://doi.org/10.1061/(ASCE)GM.1943-5622.0000852

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Abstract

When modeling the development of slip surfaces in strain-softening soils, the FEM can suffer from mesh dependency and a lack of convergence of the analysis. The calculation of strains using a nonlocal method avoids the independent softening of a solitary point that can cause these symptoms. Nonlocal methods calculate the strain at a point with reference to the strains at calculation points surrounding it. In this study, a biaxial compression analysis is used to compare the original and two modified nonlocal methods with local strain-softening analysis. Three nonlocal parameters are introduced, and their influence on the nonlocal strain calculation is investigated. Meshes with different discretizations permit a comparison of the mesh dependency of these methods and a sensitivity analysis of nonlocal parameters. All the nonlocal methods are found to have significantly less mesh dependency than the local method. Both modified nonlocal methods exhibit less mesh dependency than the original approach, but one of the two modified methods, the overnonlocal method, is shown to be unstable because the mesh is refined for certain values of the nonlocal parameters.

Introduction

When modeling the development of slip surfaces in strain-softening soils, the FEM can suffer from mesh dependency and a lack of convergence of the analysis (Bažant and Jirásek 2002; Galavi and Schweiger 2010; Vermeer and Brinkgreve 1994).

Numerical convergence issues arise because the inclusion of a strain-related soil property (i.e., strength) causes the governing partial differential equations for a static analysis to change from elliptic, in which all points in the domain are subject to a change at once, to hyperbolic, in which the changes are applied following a specified condition. This causes an ill-posed boundary value problem (Vardoulakis and Sulem 1995; Lu et al. 2012).

In the continuum of a typical finite-element analysis, the strain associated with strength softening is calculated using the displacement information computed at the nodes of the elements. The relative location of these nodes affects both the shear band thickness and the direction of its development (Galavi and Schweiger 2010). There can be a large difference, i.e., a high gradient, between displacement and therefore strain at neighboring nodes. This sudden change in strain is reflected in the areas of material that are modeled as undergoing strain softening. For a locally defined strain-softening model, the degree of material softening is linked to the current strain at that point only. The strain calculated at any point is calculated from the displacement value assigned to that point given the current values of stress and the previous history of deformation (Bažant and Jirásek 2002). This can lead to one calculation point in the mesh independently experiencing high strain and a corresponding reduction in strength for this level of strain. The point can appear to have softened excessively in relation to the surrounding points (Vermeer and Brinkgreve 1994; Bažant and Jirásek 2002). The reduction in strength reduces the load and stress that can be carried by the material at that location, and this excess load and stress will be redistributed to the neighboring material. This localizes the area of strain softening and links softening geometrically to the mesh (Vermeer and Brinkgreve 1994). Through this localization shear band formation and growth can align to the nodes of the mesh. Meshes with different size elements or arrangements of elements will create a slip surface extending in different directions and with different thicknesses (Summersgill 2015).

The local concept of yielding and strain softening does not reflect the reality of slip surface formation and growth. In reality, plastic deformation, of which strain softening is one form, is nonlocal in character. It arises from the accumulation of dislocations and plastic flow from their movements (Eringen 1981). The growth of shear bands is not decided by the stress and strain at the location of the highest strain but by the release of energy from the volume surrounding that point. Furthermore, in a continuum model, such as the FEM, it is not the ultimate strain value at the center point of the slip surface that is calculated, but the average strain value within the representative volume, i.e., between two degrees of freedom (Bažant and Jirásek 2002). The heterogeneity caused by shear bands at a resolution below the distance between two degrees of freedom should be represented by an appropriate value and gradient between these two points (Bažant and Jirásek 2002).

Nonlocal models have been implemented in several numerical codes as a regularization tool for slip surface or fracture formation. The nonlocal approach assumes that as microstructure evolves at one point, it naturally influences the surrounding points (Di Prisco et al. 2002). These models do not alter the fundamental governing equations; instead, they introduce the calculation of strain as a nonlocal variable by spatially averaging the local variables (Lu et al. 2009). This makes the approach of nonlocal strain regularization more straightforward for implementation into an existing finite-element code.

The present study aims to provide a systematic comparison of three existing nonlocal methods and a conventional local strain-softening approach with particular emphasis on mesh dependency and nonlocal parameter selection. In this study, a biaxial compression analysis of both undrained and drained numerical simulations is used to compare the original and two modified nonlocal methods with a local strain-softening analysis. Three nonlocal parameters are introduced, and their influence on the nonlocal strain calculation is investigated. Analyses with meshes with different discretizations permit a comparison of the mesh dependency of these methods and a sensitivity analysis of the nonlocal parameters.

Nonlocal Method

A nonlocal method provides a means of regulating strain calculations to avoid excessive softening and a means of relating strain at a point and, therefore, strain softening, to strains in the surrounding area. This reduces the mesh dependence and potential numerical convergence issues associated with strain-softening analyses (Summersgill 2015).

A nonlocal model is defined as a model with an averaging integral and characteristic or defined length (DL) (Bažant and Jirásek 2002). A fully nonlocal model will treat both stress and strain as nonlocal components. However, when the model is used as a regularization tool, it is common to adopt a partial nonlocal constitutive relation in which only the nonlocal plastic strains control the softening (Galavi and Schweiger 2010). This also makes the model easier to implement in existing finite-element codes, because the nonlocal calculation can be performed as an additional calculation after the initial local calculation, with minimal modification to the existing calculations or governing equations. This format aids the calculation of nonlocal strain by providing an initial complete strain distribution for the first computation of nonlocal strain (Vermeer and Brinkgreve 1994).

Original Formulation

One of the first formulations for a nonlocal strain model is presented in Eq. (1). This was proposed by Eringen (1981) for strain-hardening applications and by Bažant et al. (1984) as a strain-softening damage model

ε^{p *} (x_{n}) = \frac{1}{V_{ω}} ∭ [ω (x_{n}^{'}) ε^{p} (x_{n}^{'})] d x_{1}^{'} d x_{2}^{'} d x_{3}^{'}

(1)

where ε^p = accumulated plastic deviatoric strain; * = nonlocal parameter; and x_n is the point at which the calculation of the nonlocal strain, ε^p*, is required, whereas

x_{n}^{'}

refers to all the surrounding locations, i.e., the location of reference strains. Therefore, ε^p(

x_{n}^{'}

) equals the reference strain at the reference location. The weighting function, ω(

x_{n}^{'}

) is defined for all the reference locations, but it is centered at the location x_n.

The weighting function is the Gaussian or normal distribution, which is shown in Eq. (2) and in Fig. 1. This introduces an additional parameter, the DL. A larger DL will create a wider and shorter weighting distribution, thus, affecting the rate of softening, as shown in Fig. 2(a). The effect of DL on the nonlocal plastic strain distribution can be seen in Fig. 3(a), which has been derived for a local strain equal to 1 over a distance of 0.5 m centered at the calculation point, i.e., from −0.25 to 0.25 m (note that the same calculation is used in all panels of this figure).

ω (x_{n}^{'}) = \frac{1}{D L \sqrt{π}} e x p [- \frac{{(x_{n}^{'} - x_{n})}^{T} (x_{n}^{'} - x_{n})}{{D L}^{2}}]

(2)

Fig. 1. Weighting distribution: original and G&S

Fig. 2. Variation of DL changes in the weighting function: (a) original; (b) G&S

Fig. 3. Variation in nonlocal strain with a change in DL: (a) original; (b) G&S; (c) overnonlocal; (d) DL = 1.0 all nonlocal methods

The weighting function is chosen so that it will not alter a uniform field of strain.

The integral of the weighting function in the three dimensions x₁, x₂, and x₃ is referred to as the reference volume, V_ω, as shown in Eq. (3). This is used to normalize the calculation of the nonlocal strain and for the Gaussian distribution is equal to approximately 1. The latter attribute of the function ensures that a uniform field of strain would remain unmodified.

V_{ω} = ∭ ω (x_{n}^{'}) d x_{1}^{'} d x_{2}^{'} d x_{3}^{'}

(3)

This method, which will be referred to as the original method in this paper, has been shown to have low mesh dependency when applied to strain-softening analyses (Jostad and Grimstad 2011). However, the softening of the material still occurs dominantly at the center point of the slip surface. This is where the weighting function has its maximum and therefore where the largest nonlocal strain is calculated (Galavi and Schweiger 2010; Vermeer and Brinkgreve 1994; Lu et al. 2012; Jostad and Grimstad 2011). Excessive softening could still potentially occur at this point, and the spread of the slip surface would therefore be constrained. In light of this, modifications have been made to the original nonlocal method to address the central high concentration of strain. Two modified methods are presented and evaluated below, the G&S method and the overnonlocal method.

G&S Method

This modified method was proposed by Galavi and Schweiger (2010) and will be referred to as the G&S method. The fundamental idea underpinning this method is that the development of the slip surface is influenced by the directly surrounding areas and not by the concentrated strain at the center of the slip surface. Therefore, this method proposes a modified weighting function, as shown in Fig. 1 and Eq. (4), which limits the central concentration of strains without introducing any new parameters. The greatest contributions of strains to the nonlocal calculation now stem from points that lay a little to each side of the calculation point, and the strain contribution at the calculation point is actually zero. The calculated nonlocal strain is, therefore, a little lower with a wider distribution, as shown in Fig. 1. The G&S weighting function is influenced by the size of DL in a similar fashion to the original weighting function, as shown in Figs. 2(a and b), respectively. This in turn leads to a similar effect of DL on the nonlocal plastic strain distribution, which is shown in Fig. 3(b). The integral of the weighting function is still equal to 1; therefore, the condition of zero alteration of a uniform field is again satisfied.

ω (x_{n}^{'}) = \frac{\sqrt{{(x_{n}^{'} - x_{n})}^{T} (x_{n}^{'} - x_{n})}}{{D L}^{2}} \exp [- \frac{{(x_{n}^{'} - x_{n})}^{T} (x_{n}^{'} - x_{n})}{{D L}^{2}}]

(4)

Overnonlocal Method

Vermeer and Brinkgreve (1994) proposed the overnonlocal method, which alters the nonlocal strain formulation, as shown in Eq. (5). The overnonlocal method adopts a weighting function that prevents the formation of a concentrated peak, providing a more uniform value of strain across the slip surface. This introduces a new parameter, α, which is used to provide a contribution from the local strain at the point of calculation and to increase the nonlocal contribution to the calculation of nonlocal strain. This increases the width of the slip surface compared with the original method and reduces the concentration of strain at the center, as shown in Fig. 3(d), for α = 1.5 and DL = 1.0.

In Eq. (5), when α is greater than 1, the local strain contribution is negative. This significantly reduces the value of nonlocal strains calculated over the areas in which local strain is high and increases the nonlocal strain value calculated in the areas immediately adjacent to the local strain distribution. If α = 1.0, then the formula reverts to the original nonlocal formulation. If α is less than 1.0, then the local strain part of the equation contributes by increasing the nonlocal strain to be greater than the local strain, which contradicts the aims of the formulation. The alpha parameter, α, must always be greater than 1.0 (Vermeer and Brinkgreve 1994).

For certain combinations of local strain input, such as the α value and DL, there is a negative nonlocal strain. This is shown in Fig. 3(c), in which for an α of 1.5 three values of DL are examined. For DL = 0.5, the nonlocal strain is always positive, but for DL = 1.0 and 1.5, there are negative values around the local strain input. The influence of α on the nonlocal strain calculation is compared with the other two methods in Fig. 4, with DL = 1.0. This figure shows that an increase in α increases the nonlocal strain outside of the area of local strain input and decreases the nonlocal strain within the area of local strain. The influence of the α parameter will be further investigated in biaxial compression simulations.

ε^{p *} (x_{n}) = (1 - α) ε^{p} (x_{n}) + \frac{α}{V} ∭ [ω (x_{n}^{'}) ε^{p} (x_{n}^{'})] d x_{1}^{'} d x_{2}^{'} d x_{3}^{'}

(5)

Fig. 4. Variation in nonlocal strain for a change in α

Biaxial Compression Analyses

Biaxial compression tests have been widely used (Vermeer and Brinkgreve 1994; Lu et al 2012; Jostad and Grimstad 2011) to assess the mesh dependence of nonlocal methods because they can impose severe strain localization in the numerical model. The biaxial compression analyses compress a quadrilateral mesh from two opposing sides, while leaving the two other opposing sides free to deform. This creates a bifurcation and a slip surface, which permits an assessment of the ability of local and nonlocal strain-softening models to regulate slip surface formation for a simple analysis.

The patterns of slip surfaces in biaxial compression analyses can vary from one to three or four bifurcations, depending on the dimensions of the mesh and the boundary conditions imposed. The preferential development of one or more slip surfaces will affect the load versus displacement response. In the analyses presented here, one-quarter of a full problem with a square mesh is modeled, as shown in Fig. 5. This provides analyses that form a single slip surface each time, and this enables a direct comparison of the nonlocal models under consistent conditions. It should be noted that analyses of the full problem (i.e., not making use of any symmetry in geometry) with the same boundary conditions leads to exactly the same response as the quarter model presented in this paper (Summersgill 2015). The initial conditions consist of a vertical stress of 50 kPa imposed on the top horizontal boundary and a 100-kPa horizontal stress applied along the right lateral boundary. It was found to be unnecessary to include the stiff platens used by Galavi and Schweiger (2010) to introduce inhomogeneous strain. Restricting the horizontal displacement of the top horizontal mesh boundary in Fig. 5(b) and imposing an equal vertical displacement to all nodes along this boundary was sufficient for a slip surface to form.

Fig. 5. Boundary conditions for biaxial compression

The analyses presented compare the mesh dependence of the strain-softening models by performing all analyses with three meshes containing different square element sizes. All the meshes are 1 × 1 m containing 100 elements (10 × 10), 400 elements (20 × 20), or 1,600 elements (40 × 40), as shown in Fig. 6. An 80 × 80 mesh was used only for the investigation of the α parameter of the overnonlocal method, which contains 0.0125 × 0.0125 m square elements in a 1-m square mesh. In all cases, plane strain eight-noded isoparametric elements with reduced Gaussian integration were used. This integration scheme was also used to evaluate the integrals in Eqs. (1), (3), and (5).

Fig. 6. Mesh layouts for biaxial compression: (a) 100 elements, 10 × 10, 0.1 × 0.1 m; (b) 400 elements, 20 × 20, 0.05 × 0.05 m; (c) 1,600 elements, 40 × 40, 0.025 × 0.025 m

The three nonlocal strain-softening methods were implemented in the finite-element code called the Imperial College finite-element program (ICFEP) (Potts and Zdravković 1999), and compared with existing local strain-softening models. The latter are variants of the Tresca and Mohr-Coulomb models in which the limits for peak and residual strength can be specified by a value of the percentage of plastic deviatoric strain (E). Results for the local method are compared with the three nonlocal methods, including two sets of overnonlocal analyses, with α = 1.5 and 2.0, following values used by Jostad and Grimstad (2011). An accelerated modified Newton-Raphson scheme with a substepping stress point algorithm was used to solve the finite-element equations (Potts and Ganendra 1994).

Both undrained and drained strain-softening analyses were performed. For the undrained analyses a Tresca failure criterion is adopted. The soil has a peak undrained strength of 100 kPa at 0% plastic deviatoric strain and a residual undrained strength of 50 kPa at 15% plastic deviatoric strain. The drained analyses used a Mohr-Coulomb failure criterion. Similarly, the drained soil has a peak strength of φ' = 25° at 0% plastic deviatoric strain, which reduces linearly to a residual strength of φ' = 10° at 15% plastic deviatoric strain, while zero cohesional strength and zero dilation are specified. The effect of the angle of dilation is investigated separately and discussed later. The stiffness of soil is constant with a Young’s modulus, E = 50 MPa, Poisson’s ratio, μ = 0.49 and 0.2 for undrained and drained conditions, respectively.

For the nonlocal methods DL = 0.1 m is used for the analyses, unless otherwise stated. This value was chosen because it is the length of the largest element used in the analyses. This value allows the local strains surrounding the point of calculation to contribute significantly to the nonlocal strain calculations. If the value chosen for DL is too small, the nonlocal strain calculation will be very similar to the local strain input; therefore, the calculation process would be redundant. The influence of the DL parameter on nonlocal strain is investigated and discussed later.

A radius of influence (RI) can also be specified to limit the number of elements used in the calculation of the nonlocal strain. The weighting function is an exponential function based on distance; therefore, the contribution of strain diminishes rapidly with distance, as can be seen in Fig. 1. The exclusion of strains at a distance greater than four times DL will alter the outcome of the nonlocal strain calculation very little, but it will improve numerical efficiency. A specified RI of 0.4 m was therefore considered as appropriate for these analyses. The sensitivity of the analysis on the RI is further investigated in the last part of this study.

Discussion of Results

Undrained Analyses

Fig. 7 presents the reaction load on the top of the mesh for the applied vertical displacement for all considered approaches and spatial discretizations. Clearly, the load-displacement curves of all the nonlocal strain-softening methods show significantly less mesh dependency than the local method. Because the strain limits controlling strain softening are independent of the element size in the local method, it is expected for a smaller element size to result in earlier and faster softening of the material, which is in agreement with the findings of previous studies (Conte et al 2010; Schädlich 2012). The residual plateau is reached later for all of the nonlocal analyses, indicating that they delay the softening of the material compared with the local analysis. The overnonlocal method with α = 1.5 shows the least mesh dependency [Fig. 7(c)] followed by the G&S method [Fig. 7(d)]. However, when a value of α = 2.0 is used for the overnonlocal method, the 40 × 40 mesh produces premature softening of the material compared with the analyses using meshes with larger elements, but with otherwise identical arrangements [Fig. 7(e)].

Fig. 8 compares the contour plots of accumulated plastic deviatoric strain for the local and nonlocal G&S strain-softening methods for the three mesh discretizations. The width of the slip surface is less dependent on element size for the nonlocal analyses compared with the local analyses. The slip surface width reduces with element size for the local analysis, whereas the width of the largest contour remains similar for all three meshes when the nonlocal G&S method is used.

Fig. 8. Strain distribution contour plot for undrained analyses

The regularization of the plastic strains is shown by a diagonal cross section through the mesh of the distribution of accumulated local plastic deviatoric strains plotted for each strain-softening method in Fig. 9. The nonlocal methods do not permit a high concentration of strain to develop compared with the local method. This is the case for all the nonlocal analyses, except the overnonlocal with α = 2.0 for the 40 × 40 mesh in Fig. 9(e). The strain distribution for Fig. 9(e) is more comparable to the local strain-softening results in Fig. 9(a). The influence of the α parameter is further investigated in a later section of this study.

Drained Analyses

The undrained biaxial compression analyses, presented in the previous section, were repeated using drained strength parameters. The local strain-softening results are compared with the results for the three nonlocal methods in Figs. 10 and 11. The behavior is similar to the undrained analyses with a reduction in mesh dependence for the nonlocal methods. The original nonlocal and overnonlocal with α = 1.5 showed greater mesh dependency than the equivalent undrained analyses. The G&S nonlocal method demonstrated the least mesh dependency overall. The combination of DL = 0.1 m, α = 2.0, and the 40 × 40 mesh for the overnonlocal method again resulted in a sudden reduction in the reaction load, which is identified as premature softening in the previous section [Fig. 10(d)].

Fig. 11. Strain variation over diagonal cross section for drained analyses

Furthermore, the slip surface formation is very different for the case of the local strain method compared with the nonlocal methods. As shown in the diagonal cross section of Fig. 11(b), two distinct slip surfaces can be identified in the plastic deviatoric strain distribution plot of the local strain method. The differences in the predicted failure mechanisms between the local strain and some of the nonlocal approaches are also illustrated in the contour plots of accumulated local plastic deviatoric strain for the 40 × 40 configuration in Figs. 12(a–c) and in the incremental displacement vectors of Figs. 12(d and e). The plots in Fig. 12(c and e) demonstrate that the failure mechanism predicted by the nonlocal G&S method, in terms of contour plots and displacement vectors, respectively, produces a well constrained slip surface. This was consistent for different mesh discretizations.

Fig. 12. Strain distribution contour plot for drained analyses

One soil property that influences the location and shape of the slip surface in a stiff clay cutting is the angle of dilation, ψ' (Potts et al. 1997). The nonassociated results (i.e., ψ' = 0) for G&S nonlocal analyses with a DL of 0.1 m and RI of 0.4 m are compared with a set of associated analyses (i.e., ψ' = φ') in Fig. 13. The associated analyses were performed with ψ' equal to φ' at both peak and residual limits and both softening at the same rate. The associated results show the same low mesh dependence in addition to a lower imposed displacement to reach residual loading, as shown in Fig. 13(a). The strain distributions for these two sets of analyses show higher strain values than the nonassociated results, as shown in Fig. 13(b). For each mesh, the shape of the associated results is a stretched version of the corresponding nonassociated results. The dilation angle has not caused a change in the width of the slip surface formed, only an increase in the strains developed.

Fig. 13. Influence of the dilation angle: (a) load versus displacement; (b) strain distribution over a cross section

Investigation of Nonlocal Parameters

Three nonlocal parameters are introduced for the nonlocal strain-softening methods. The influence of these parameters on local strain calculations was presented for a simple local strain input in Figs. 3 and 4. A biaxial compression analysis provides a more complex situation in which it is important to evaluate the influence of these parameters.

The α parameter is only used in the overnonlocal modified strain-softening method. The DL is a required input for all the nonlocal methods and influences the shape of the weighting function used to calculate nonlocal strain, as shown in Fig. 2, and, consequently, the softening rate. The RI is an optional parameter that improves the numerical efficiency of analyses. The parameters DL and RI are investigated using the nonlocal G&S method, because it was demonstrated in the previous sections that this method had the least overall mesh dependency.

α Parameter

As previously discussed, the α parameter controls the distribution of nonlocal strain for the overnonlocal method. This is further investigated parametrically in this section, examining a range of values of α > 1.0. Because a premature softening and high concentration of strain was previously observed for the overnonlocal results for α = 2.0 and the finest mesh, further undrained analyses for α = 1.75 and 2.25 were performed for the same discretization. The load-displacement responses for α = 1.75, 2.0, and 2.25 are shown in Fig. 14. The result for α = 2.25 and 40 × 40 mesh was unstable, but the response for α = 1.75 and 40 × 40 mesh was stable. However, when an α = 1.75 was applied to a finer mesh with 80 × 80 elements of 0.0125 m, the load-displacement response did exhibit premature softening. This is reflected in the distribution of strain over a diagonal cross section of the mesh for the three analyses with the finest mesh for each α value, which is shown in Fig. 15. There is a high value of strain concentrated in a smaller area similar in shape to the local strain-softening method results of Fig. 9(b). The expected result for a nonlocal strain-softening method is a wider spread and lower nonlocal strain value, as shown for the 10 × 10 and 20 × 20 mesh results in Fig. 15. The investigation on the α parameter suggests that there is uncertainty in trusting the outcome of analyses using the overnonlocal method when using a high α and fine mesh.

Fig. 14. Load versus displacement for variation with α

Fig. 15. Strain variation over diagonal cross section for variation in α

DL

The DL defines the shape of the weighting function used in the calculation of nonlocal strain (Fig. 2). A higher DL is expected to result in a wider slip surface with a lower maximum nonlocal strain. This affects the rate of strain softening, because the same input of local strain will result in a lower nonlocal strain for a larger DL and, therefore, a slower rate of softening.

Two sets of biaxial compression analyses were performed to assess the influence of the DL, in both undrained and drained conditions. The examined values of DL are restricted by the size of the elements in the meshes used. The minimum value used for DL should be equal to the maximum element size found in the meshes. This permits the nonlocal calculations to have local reference strains at an appropriate distance from the calculation point for them to influence the nonlocal strain. It should be noted that the local strains are plastic deviatoric strains, which are calculated at the elements’ Gauss points. Without some local strain values providing a meaningful contribution to the nonlocal strain calculated, the nonlocal strain value would be very similar to the local strain, making the process of calculating it redundant. The values for DL were chosen for a ratio of the element length to DL of 1:1, 1:2, 1:4, and 1:8.

The results for the undrained and drained analyses are presented in Fig. 16 in terms of load-displacement curves. They confirm that as the DL increases the imposed displacement required to reach a residual reaction load increases, indicating a reduction in the softening rate. The change in DL seems to control only the softening rate, because the peak and residual reaction loads are not affected. The undrained results for the reaction load versus imposed displacement show consistent and almost mesh-independent behavior. For the undrained case only a minor mesh dependency can be depicted for DL = 0.1, which is related to the use of a 1:1 ratio. In the drained analyses, the results for a DL = 0.05 are not the same and require further investigation. The distributions of strains for a diagonal cross section of the drained analyses are presented in Fig. 17 by the mesh used. Clearly, there is a reduction in the peak nonlocal plastic deviatoric strain as DL increases. Furthermore, the mesh has an influence on the width of the slip surface; as element size decreases so does the slip surface width for the same value of DL. To identify a suitable ratio of element length to DL, the nonconforming results are reexamined. The results for the 20 × 20 mesh and DL equal to the element size (ratio 1:1) show the formation of two slip surfaces. Therefore the 1:1 ratio leads to a mesh dependency in both the undrained and drained analyses. Given the difference exhibited in these analyses, a ratio of at least 1:2 would be preferable for the maximum element size to DL.

Fig. 16. Variation of DL: (a) undrained; (b) drained

Fig. 17. Variation of DL for drained analyses, presented by DL value

RI

The RI is an optional parameter that limits the distance from the calculation point of the neighboring strains that will be included in the nonlocal strain calculation. A variety of analyses were performed to validate the use of a RI value four times the DL, which is used in the analyses presented earlier. These analyses evaluate the RI value for a 1 × 1-m mesh for both drained and undrained soil properties, using DL values of 0.1 m or 0.2 m and mesh discretizations of 10 × 10 and 20 × 20 elements. The results for a set of drained analyses using a 20 × 20 mesh and DL = 0.1 m are presented in Fig. 18. This set is representative of all the RI investigations presented in Summersgill (2015). The value of RI has some influence on the results for a ratio of 1:1 and 1:2 of DL to RI. For a value of RI three times DL or greater, the reaction load response and strain distribution is almost indistinguishable from an analysis performed with no RI specified, i.e., in which all neighboring strains contribute to the calculation of a nonlocal strain no matter how far away they are from the position for which the nonlocal strain is being evaluated.

Fig. 18. Variation of RI, drained analyses, DL = 0.1 m with 20 × 20 mesh: (a) load versus displacement; (b) strain distribution over a cross section

The time-saving that RI provides makes the use of this parameter a sensible decision. For the analysis presented in Fig. 18, the RI = 0.3 m analysis is performed in 28% of the time of an analysis with no RI specified (i.e., RI equal to infinity). An RI = 0.4 m still provides a time-saving, requiring 41% of the time for an analysis with no RI specified. These results are for a simple mesh and relatively simple problem. The potential time-saving for a more complex mesh or analysis could be significant.

Conclusions

This paper assesses the performance of three nonlocal strain-softening methods (namely, the original formulation of Eringen 1981, the overnonlocal of Vermeer and Brinkgreve 1994, and the G&S of Galavi and Schweiger 2010) with a comparison to each other and to a local strain-softening method. The mesh dependency of each of the methods is evaluated with biaxial compression analyses that use three meshes with different element discretizations. All the nonlocal methods are found to have significantly lower mesh dependency than the examined local strain-softening method, for both drained and undrained strength parameters. Given the lower mesh dependency, the nonlocal strain-softening method is a more viable option for boundary value problems with a range of element sizes and shapes. Both modified methods (i.e., the overnonlocal and the G&S methods) exhibit less mesh dependency than the original nonlocal approach. However, a parametric study on the performance of the overnonlocal method for a range of α values showed that the method becomes unstable for high values of α and fine meshes. Therefore, among the considered approaches, the G&S method provides the best compromise between low mesh dependency and consistency of results.

A sensitivity investigation of the results for the nonlocal parameters of DL and RI was also undertaken. The results show that the DL parameter influences the rate of strain softening, whereas its value is also constrained by the size of the elements in the mesh used. The DL must be sufficiently large that the local strains referenced for calculating the nonlocal strain will contribute significantly to the nonlocal calculation. The selection of nonlocal parameters, DL and RI, required for use in the nonlocal G&S method should consider both the size of the elements in the mesh used and the impact of DL on the rate of strain softening.

Acknowledgments

This work was part of the doctoral research of the first author at Imperial College London through an Industrial Cooperative Award in Science & Technology (CASE) jointly funded by the Engineering and Physical Sciences Research Council (EPSRC) and Geotechnical Consulting Group, LLP. This support is gratefully acknowledged.

References

Bažant, Z., Belytschko, T., and Chang, T. (1984). “Continuum model for strain softening.” J. Eng. Mech., 1666–1692.

Abstract

Introduction

Nonlocal Method

Original Formulation

G&S Method

Overnonlocal Method

Biaxial Compression Analyses

Discussion of Results

Undrained Analyses

Drained Analyses

Investigation of Nonlocal Parameters

α Parameter

DL

RI

Conclusions

Acknowledgments

References

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