Model Tensile Cracking in Soil with Coupled Meshless-FEM Method

Abstract

An adaptive, coupled method based on radial point interpolation meshless method (RPIM) and finite element method (FEM) is proposed for the simulation of tensile cracking in soil structure. Potential crack area in the initial FEM mesh is converted to RPIM nodes and refined for crack analysis. A crack tracking approach is adapted to model complex crack surface and eliminate node distribution bias. Numerical example is presented to verify the proposed method.

1 Introduction

Tensile cracking is a common problem in geotechnical structure, which may leads to catastrophic failure. Smeared crack model, which is based on continuum mechanics, models cracking by a band in which the material is weakened according to specific constitutive model (Rots and Blaauwendraad 1989). Although widely used, some problems still remain unsolved in smeared crack model. The crack is element-wise, thus the influence of element size on the cracking path is inevitable. As a consequence, accurate crack simulation requires very fine mesh. However the local mesh refinement remains difficult in the FEM, especially in three dimensions. The smeared crack model also suffers the shortcoming of mesh bias.

In meshless methods, only particle information is needed. Hence, crack simulation and adaptive analysis is relatively simple (Belytschko et al. 1996). Because of the low efficiency of meshless methods, its coupling with FEM is a natural way to make use of the advantages of these two methods. Among many meshless methods, the radial point interpolation method (RPIM) can be coupled with FEM easily, because its shape function is constructed by point interpolation (Wang and Liu 2002).

In this paper, an adaptive coupled RPIM-FEM method is proposed to simulate tensile cracking. The post cracking behavior of geo-material is modeled by an isotropic damage model. To overcome mesh bias, a crack tracking algorithm is introduced, making it possible to apply the proposed method to complicated problems.

2 Smeared Crack with Damage Model

An isotropic damage model is used to model the degradation of cracked material, which has the following constitutive equation

$$\varvec{\sigma}= (1 - d)\bar{\varvec{\sigma }} = (1 - d)\varvec{D}:\varvec{\varepsilon}$$

(1)

where $\bar{\varvec{\sigma }}$ is the effective stress tensor, D the isotropic linear elastic constitutive tensor, ε the total strain and d the damage index. To split the tensile contribution from total effective stress, an equivalent stress is defined by Cervera and Chiumenti (2006)

$$\tau = \left\langle {\bar{\sigma }_{1} } \right\rangle$$

(2)

where $\bar{\sigma }_{1}$ is the first principal effective stress, and the symbol $\left\langle \cdot \right\rangle$ is Macaulay brackets. The damage criterion is defined as

$$F = F(\tau ,r) = \tau - r \le 0$$

(3)

where r is an internal stress-like variable. The evolution of r is given as

$$r = \hbox{max} \left\{ {\sigma_{t} ,\;\hbox{max} (\tau )} \right\}$$

(4)

where $\sigma_{\varvec{t}}$ is the uniaxial tensile strength. The damage index d is defined as a function of the internal variable r

$$d(r) = 1 - \frac{{\sigma_{t} }}{r}\exp ( - 2H_{s} \frac{{r - \sigma_{t} }}{r})$$

(5)

where H _s is a constant dependent on the mesh size used in the simulation. By the concept of fracture energy, H _s can be computed by the following equation

$$H_{s} = \frac{{\sigma_{t}^{2} l_{ch} }}{{2EG_{f} - \sigma_{t}^{2} l_{ch} }}$$

(6)

where E is the Young’s modulus, G _f is the fracture energy. l _ch is the characteristic length of the mesh used in the simulation.

3 The Coupled RPIM and FEM Method

In RPIM, a field function $u(\varvec{x})$ is approximated by radial basis functions $R_{i} (\varvec{x})$ and a set of polynomial basis functions $P_{j} (\varvec{x})$ (Liu et al. 2005)

$$u(\varvec{x}) = \sum\limits_{i = 1}^{n} {R_{i} (\varvec{x})a_{i} } + \sum\limits_{j = 1}^{m} {P_{j} (\varvec{x})b_{j} } = \varvec{R}^{\text{T}} (\varvec{x})\varvec{a} + \varvec{P}^{\text{T}} (\varvec{x})\varvec{b}$$

(7)

where n is the number of nodes in the influence domain of the sample point x, m is the number of polynomial basis function, a _i and b _j are the corresponding coefficients. By interpolating all the nodal values in the influence domain, the field function can be expressed as

$$u(\varvec{x}) = [\varvec{R}^{\text{T}} (\varvec{x})\quad \varvec{P}^{\text{T}} (\varvec{x})]\varvec{G}^{ - 1} \left( {\begin{array}{*{20}c} {\varvec{u}_{e} } \\ {\mathbf{0}} \\ \end{array} } \right) = \varvec{N}(\varvec{x})\varvec{u}_{e} = \sum\limits_{i = 1}^{n} {N_{i} u_{i} }$$

(8)

where G is the interpolating matrix, u _e is the nodal value vector, N(x) is the resulting shape function vector composed of N _i (x), i = 1, 2, …, n.

Constructed by point interpolation, the shape functions of RPIM have the same Kronecker delta property as the FEM shape function. With this property, the coupling of RPIM and FEM is direct and simple. The problem domain can be divided into two subdomains, $\Omega_{\text{RPIM}}$ and $\Omega_{{\text{FEM}}}$, connected by a set of interface nodes belonging to both sides. Due to the Kronecker delta property of both RPIM and FEM, C ⁰ continuity is achieved naturally along the interface.

To distinguish the RPIM region and FEM region, automatic identification of the potential crack zone is needed. The criterion is as follows:

$$\bar{\sigma }_{1} > 0.85\sigma_{t}$$

(9)

In the distinguished potential crack zone, the original finite elements are converted to RPIM region. For better simulation of crack propagation, the RPIM area is automatically refined. More details about the adaptive coupling method can be found in Yuan et al. (2014).

4 Crack Tracking

The smeared crack suffers the drawback of mesh bias, that is, its solution is affected by the mesh discretization (Rots and Blaauwendraad 1989). The similar pathological dependency of solution is also found in meshless methods when smeared crack is adapted. A lot of remeshing methods and complicated constitutive models have been proposed to solve this problem. Recently Oliver et al. (2004) proposed a crack tracking method, which is based on the global stress status, to handle the mesh bias problem. In our coupled meshless-FEM method, this global tracking method is implemented to simulate complicated curved cracking.

The global crack tracking method assumes that at arbitrary location x, there is a potential crack surface depicted by a small crack plane s(x). The potential crack propagation paths can be viewed as a collection of surfaces, which are the envelopes of the patches of the assumed crack planes. The collection of crack surfaces can also, be considered as the level contours of a scalar function $\phi (\varvec{x})$. By this assumption any level contour $S_{i} = \left\{ {\varvec{x} \in\Omega \left| {\phi (\varvec{x}) = \phi_{i} } \right.} \right\}$ represents a potential crack path, as shown in Fig. 1. If we can find an appropriate field function, the crack propagation can be tracked easily.

Fig. 1

Global crack tracking

According to the assumption, the level contours are the crack envelopes, comprised of patches of crack plane. As a result, at any location, the gradient of $\phi (\varvec{x})$ is perpendicular to the crack plane, i.e., $\varvec{N}_{crk} = \nabla \phi /\left\| {\nabla \phi } \right\|$. If the crack surfaces are considered as equal-potential surfaces, then the function is very similar to a potential function, like temperature or water pressure. By this concept, the function can be obtained by the following approach.

In the local coordinate defined on crack plane, we have $\varvec{T}_{crk} \cdot \varvec{N}_{crk} = 0$. Substituting $\varvec{N}_{crk} = \nabla \phi /\left\| {\nabla \phi } \right\|$ into previous equation and pre-multiplying $\varvec{T}_{crk}$, it yields

$$\varvec{T}_{crk} \otimes \varvec{T}_{crk} \cdot \nabla \phi = 0$$

(10)

Define a flow $\varvec{q} = - \varvec{D}_{con} \cdot \nabla \phi$, where $\varvec{D}_{con} = \varvec{T}_{crk} \otimes \varvec{T}_{crk}$ can be considered as a conductivity matrix. Therefore, the field function can be obtained by solving the following thermal-conduct like problem

$$\begin{aligned} \nabla \cdot\varvec{q} & = \nabla \cdot( - \varvec{D}_{con} \cdot\nabla \phi ) = 0\quad {\text{in}}\;\Omega \\ q_{v} & = \varvec{q} \cdot \varvec{v} = 0\quad \text{in}\;\partial \Omega \\ \end{aligned}$$

(11)

where v is the outward unit vector normal to the boundary $\partial \Omega$.

The above thermal-conduct like problem is linear and well posed, and can be solved with the same coupled FEM-RPIM method used to obtain displacement. In crack analysis, the crack tracking can be carried out once per step or every several steps. Once the value of $\phi (\varvec{x})$ is obtained, the position of a crack can be identified. If an influence domain of an integral point is cracked, then the further propagation path of the crack is known to us: it lays on the iso-line of $\phi (\varvec{x})$ which crosses this very influence of domain.

5 Numerical Example

A well-studied shear beam test is chosen to verify the proposed method. The geometry and configuration is shown in Fig. 2. The following material properties are used: Young’s module E = 40 MPa, Poisson’s ratio $\nu = 0. 3$, tensile strength $\sigma_{t} = 6.0 \times 10^{5}\; {\text{Pa}}$ and mode I fracture energy $G_{f} = 25\; {\text{J/m}}^{ 2}$.

Fig. 2

Geometry of the shear beam (unit: mm)

Two simulations, with crack tracking and without crack tracking, are performed. In the analysis with crack tracking, the additional thermal-conduct like problem is solved in each step. Initially, a coarse FEM mesh is used. Figure 3 gives the result of crack tracking and damage index at the last step. Clearly, the crack tracking method captures the potential crack path well. With crack tracking, the crack starts from the notch and propagates towards the upper edge of the beam in a curved path.

Fig. 3

Crack simulation with tracking: a crack tracking; b damage index

The adaptive coupling process is clearly shown in the deformed geometries given in Fig. 4. The refined RPIM region starts from the notch and extends upwards, followed by the cracking. The node distribution in the RPIM region is very dense; therefore an accurate representation of the cracking can be obtained. With tracking the crack propagates along a curved path, while without tracking it goes along the alignment of the node distribution, which is obviously unphysical. The load displacement curve is given in Fig. 5.

Fig. 4

Deformed mesh: a with crack tracking; b without crack tracking

Fig. 5

Load versus displacement at the loading point

6 Conclusion

A coupled RPIM-FEM method is proposed for the simulation of tensile crack. With adaptive refinement and crack tracking it can simulate cracking problem with complicated propagation path. The computational cost of this method is relatively low because meshless method is only used in the crack region. The implementation of the method is quite simple and its extension to three dimensions is straightforward. Numerical example shows that the proposed method can be used to analysis the tensile cracking in geo-material.