Research PaperShear band static evolution by spatially mobilized plane criterion based Drucker-Prager model and numerical manifold method
Introduction
The development of shear band is accompanied with highly localized deformation which initially emerges from a homogeneous deforming displacement field. Associated with the loss of material strength, the shear band induces tangential relative sliding which causes the failure of geo-materials (Armero and Linder, 2008, Gravouil et al., 2002, Rudnicki and Rice, 1975, Hobbs and Ord, 1989, Pijaudier-Cabot and Bažant, 1987). Actually, the shear band is a strain localization band with a finite width. Therefore, the classic finite element method which uses the classic rate-independent plastic model fails to accurately predict the localization phenomenon.
To tackle the problem above, some efforts have been made including using an elastic-viscoplastic material model (Loret and Prevost, 1990, Loret et al., 1995, Prevost and Loret, 1990), incorporating a length scale into the plastic evolution equations (Pietruszczak and Óz, 1981), and also other theories including nonlocal integral method (Chen et al., 2000, Di Prisco and Imposimato, 2003), high gradient formulation (Swaddiwudhipong and Poh, 2019), and the Cosserat theorem (Ehlers and Volk, 1999, Khoei and Karimi, 2008). Nevertheless, the introduction of a length scale into the governing equation does not guarantee to accurately capture the jump in the displacement gradient without the refinement of mesh in the area of shear band. Deb and Prevost (Deb et al., 1996) used adaptive meshing to accurately capture the shear band behavior by refining the mesh in the area of the weak discontinuity, which, however, causes high computation cost. Considering that the shear band width is very small compared with other length scales and that the calibration of the length scale requires special experiment approaches (Wolf et al., 2003), some researchers preferred to treat the width of shear band as zero (Belytschko et al., 1988, Borja, 2000, Oliver et al., 1999, Regueiro and Borja, 2001, Simo et al., 1993) so that the kinematics of the weak discontinuity in strain field were reduced to those of strong discontinuity in displacement field. In this way, numerical computation is simplified by avoiding both the required mesh refinement near the discontinuity as well as the need for a more complex plastic model (Sanborn and Prévost, 2011). Hence, in this study the shear band is treated as strong discontinuities with zero width.
By assuming the distribution of the plastic multiplier takes on the form of Dirac-delta function, researchers (Simo et al., 1993, Sanborn and Prévost, 2011, Oliver, 1996, Borja, 2002, Armero and Garikipati, 1996, Armero, 2009, Borja and Regueiro, 2001) derived that the onset of the strong discontinuity strain localization occurred when the acoustic tensor became singular. Leroy (Leroy and Ortiz, 1989) and Oliver (Oliver, 1996) regarded the searching algorithm for the localization of associative plasticity as the constrained optimization problem. And this algorithm was extended, by Sanborn (Sanborn and Prévost, 2011) and Wells (Wells and Sluys, 2001), to the non-associative plasticity. Compared with the numerical algorithm, scholars (Rudnicki and Rice, 1975, Oliver et al., 1999, Armero and Garikipati, 1996, Borja and Regueiro, 2001, Runesson et al., 1991, Ottosen and Runesson, 1991) also derived the analytical solution of the initiation condition. This study revises the Drucker-Prager model by the SMP criterion (Sun et al., 1999, Matsuoka et al., 1999, Matsuoka and Nakai, 1974, Matsuoka and Nakai, 1985) of which the influence on the evolution of shear band is studied, which is not covered by existing literatures. Since it is still in progress to derive the closed-form solution of the localization condition for the SMP-based Drucker-Prager model, the numerical algorithm is used in this study.
To advance the solution after the strong discontinuity is inserted into the mesh, it is necessary to formulate a contact constitutive model to account for the behavior of the contact interfaces. Cohesive laws (Samaniego and Belytschko, 2005, Rabczuk et al., 2007, Rabczuk and Samaniego, 2008, Zi and Belytschko, 2003) and frictional laws (Yu, 2010, Khoei and Nikbakht, 2007, Kim et al., 2007, Liu and Borja, 2008) are the two primary approaches to govern the post-localization behavior along the shear band. The contact in NMM is governed by the Mohr-Coulomb frictional law. Inherited from DDA (discontinuous deformation analysis), the contact in NMM is treated as a collection of contact pairs. Each contact pair could be in one of the three contact state: fixed, slipping, or open. And the selection and the shifting of the contact state is determined by the open-close iteration (Shi, 1992). This study revises the cohesion when the contact is in slipping state so that the decaying of the cohesion is realized with the growth of the sliding. Besides, when the contact state is shifted to open state from fixed state or slipping state, the contact stiffness suddenly becomes zero. The big change in contact stiffness deters the convergence of a nonlinear problem (Sanborn and Prévost, 2011). Therefore, this paper proposes to construct the contact matrix in open state in three different cases to alleviate the change in stiffness.
In (Liu and Borja, 2008), Borja compared the performance of the Newton method and the LATIN (large time increment) method in solving the nonlinear constitutive integration. In this study, the constitutive integration of continuum could be reduced to the MiCP (mixed complementarity problem) which is solved by the GSPC (Gauss-Seidel based projection contraction) algorithm (Zheng et al., 2020). Compared with Newton method, GSPC possesses high accuracy and it could be easily implemented because the inverse of Jacobian matrix is not needed. GSPC is obtained by improving the projection contraction algorithm (PCA) (He, 1997) by Gauss-Seidel iterative method. PCA has achieved much success in dealing with complementarity related problems including the simulation of static growth of multiple cracks (Zheng et al., 2015), the force method DDA (Zheng et al., 2016), and the constitutive integration for multiple yield surfaces problem (Zheng et al., 2020).
The evolution of shear band has been simulated by various numerical methods such as the assumed enhanced strain method (AES) (Simo et al., 1993, Borja and Regueiro, 2001), the extended finite element method (XFEM) (Sanborn and Prévost, 2011, Mikaeili and Liu, 2018, Liu, 2015), the numerical manifold method (Wu and Wong, 2013); etc. In AES, the enhancement equation is introduced to account for the localized elements and the elastic stiffness matrix is weakened with the extra unknowns condensed out (Borja and Regueiro, 2001). In XFEM, jump function is introduced for the elements containing discontinuities and the level set method is used to locate the shear band. The shortcoming of the XFEM is that the explicit representation of the discontinuity surface is needed by means of the level set, which adds complexity to this method. To circumvent this deficiency, cracking particles method (CPM) is proposed by Rabczuk in (Rabczuk and Belytschko, 2007, Rabczuk and Belytschko, 2010). CPM, a simplified meshfree method, simulates the growth of the discontinuity by the activation of a series of discontinuous surfaces which are defined at the corresponding particles. A major advantage of CPM is that this method could trace complex evolution of discontinuities because no continuity of the evolution path is necessary. In this paper by NMM, the shear band is treated as physical boundary, by cutting the mesh. Hence, similar to CPM, no explicit representation of the discontinuity surface is needed in NMM. Further, the critical advantage of NMM is that the displacement interpolation possesses high accuracy because the interpolation is provided by the regularly deployed the star nodes associated with each physical patch. While in XFEM or AES, the interpolation accuracy would be weakened because the element nodes used to interpolate would be distorted and severely irregular with the big growth of the shear band.
This paper is organized as follows. Section 2 briefly introduces the numerical manifold method (NMM) used in the simulation of shear bands. Section 3 revises the Drucker-Prager model by the SMP criterion with recourse to the transformed stress method, of which the constitutive integration can be reduced to the mixed complementarity problem. And how the shear band initiates and evolves is briefly introduced in this section. Section 4 modifies the contact treatment in NMM. Section 5 establishes the NMM formulation for the shear band of strain localization. Also, the displacement loading method is extended to the material nonlinearity accompanied with contact problem. Section 6 studies two typical examples, the soil block and the soil slope under displacement loading, by the proposed approach. And some conclusions are reached in Section 7.
Section snippets
Brief introduction to NMM
Fig. 1, representing a problem domain with a shear band inside, is used to introduce the basic concepts of NMM involved in this study. More aspects of NMM could be found in Yang et al., 2020, Yang et al., 2018, Yang et al., 2016, Wu et al., 2019a, Wu et al., 2019b, Wu et al., 2020, Zheng and Xu, 2014.The problem domain is denoted by and the shear band is . Mathematical cover can be generated by any type of meshes as long as these meshes cover . In this case, the triangular finite element
The constitutive model and the initiation and evolution of shear band
The original Drucker-Prager (D-P) constitutive model is isotropic in terms of strength on the deviatoric plane, causing the strength of soil in general states not to be properly described. To overcome the deficiency, the transformed stress method (Sun et al., 1999) is applied to incorporate the SMP criterion into the D-P model. Subsequently, how the strain localization is initiated and evolved is briefly introduced.
Analysis for the contact interfaces
Inherited from DDA (discontinuous deformation analysis), the contact between the interfaces in NMM is treated as a collection of contact pairs. The contact is governed by one dimensional Mohr-Coulomb frictional law. The contact state for each contact pair could be in fixed state, slipping state or open state. For clarity, a brief introduction of contact state is given here. A contact pair comprises a vertex and edge. For an incremental load step, contact of fixed state means the vertex is fixed
Governing equations
For the convenience of presentation, it is specified here that all symbols used in Section 5 are in matrix or vector forms, even though some letters has already been used as tensors in Section 3.
The weak form of the boundary value problem with displacement discontinuities reads
Here is the variation operator. is the stress vector. is the strain vector. , and are respectively the prescribed body force vector in the
Numerical examples
In this section, two numerical examples are presented to demonstrate the capability of the new approach to capture the static evolution of shear band. Comparisons have been made between the influence of SMP-based DP model and the original DP model on the formation of shear band. Note that all the material parameters except and used in the following two examples are based on those in (Borja, 2000, Regueiro and Borja, 2001, Mikaeili and Liu, 2018).
Conclusions and future work
This paper studies the effect of the SMP criterion on the formation of shear band and the load–displacement curves by using the transformed stress method to revise the D-P model by the SMP criterion. As expected, the results show that the DP model overestimates the yield strength in plane strain state than that by the SMP based criterion. Due to the direction of the major compressive stress, the shear band paths are very close for both criterions in the soil block example. But in the soil slope
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The support of the National Natural Science Foundation of China (Grant Nos. 51538001 and 52079002) is gratefully acknowledged.
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