A Novel Stress Tensor-Based Failure Criterion for Peridynamics ^†

Abstract

Peridynamic theory has recently shown to be a versatile tool for simulating complex phenomena related to the fracture and fragmentation of structural and composite materials. We introduce a novel failure criterion based on the classic stress tensor which takes inspiration from an approach proposed in the literature. Differently from the classic critical stretch-based failure criterion used in peridynamics, our approach takes into account the total elastic energy stored in the bond allowing to predict with more accuracy problems that involve mixed-mode I-II fracture. In order to show the effectiveness of the proposed failure criterion, a benchmark fracture problem is analyzed showing a good agreement with the experimental results and the numerical results obtained with other numerical methods.

1. Introduction

The prediction of failure due to the nucleation and consecutively propagation of a crack in brittle and quasi-brittle materials remains a challenge for the scientific community [1]. Upon all theories and numerical methods developed so far for dealing with phenomena related to crack propagation, the new nonlocal theory of continuum called Peridynamics [2] seems to be capable to treat such phenomena with extreme simplicity [3,4]. This theory is classified as nonlocal since it assumes that material points of the body can interact with each other if their distance is enclosed within a certain distance called horizon, which can be related to the length scale of the material and the phenomenon studied as well. The crack propagation is commonly introduced by breaking the bond, that is the interaction between particles, if its stretch is higher than a critical stretch related to the fracture energy of the material [5]. Such a failure criterion was mainly introduced for studying the crack propagation in mode I, indeed, it neglects the contribution of deviatoric part of the deformation energy to the total elastic energy stored in the bond. Such a drawback can be removed by adopting the correspondence material model introduced in [1] where the material model from the local theory of mechanics is combined with the nonlocal capabilities of peridynamics. According to this model, the classic failure criteria adopted in the classic theory mechanics can be used for deciding when a bond should be broken [6], by doing so both the deviatoric and volumetric strain during the fracture process are taken into account. After a brief overview of the state-based peridynamics in Section 2, we introduce in Section 3 a novel failure criterion based on the classic stress tensor which takes inspiration from the approach proposed in [6], then in Section 4 we simulate a classic benchmark fracture problem in order to validate our criterion followed by the conclusions in Section 5.

2. Overview of State-Based Peridynamics

The equation of motion defined by peridynamic theory [1] can be expressed at each material point $\mathit{x}$ (called source node) of the body at the time t as

$$\rho \left(\mathit{x}\right)\ddot{\mathit{y}}\left(\mathit{x},t\right)=\underset{H}{\overset{}{\int }}\left[\underset{\_}{T}\left[\mathit{x},t\right]〈\mathit{p}-\mathit{x}〉-\underset{\_}{T}\left[\mathit{p},t\right]〈\mathit{x}-\mathit{p}〉\right]d{V}_{\mathit{p}}+\mathit{b}(\mathit{x},t)$$

(1)

with $\rho \left(\mathit{x}\right)$ the mass density, $\mathit{y}$ the current position vector of the material point, $\underset{\_}{T}\left[\mathit{x},t\right]〈\mathit{p}-\mathit{x}〉$ the force state applied to the bond $〈\mathit{p}-\mathit{x}〉$ at the point $\mathit{x}$, $d{V}_{\mathit{p}}$ the infinitesimal volume associated to the material point $\mathit{p}$ (called family node), H the neighborhood of the point $\mathit{x}$ (a circle for 2D domain with center at point $\mathit{x}$) and $\mathit{b}$ body vector forces. Let us define the reference state $\underset{\_}{X}$, the displacement state $\underset{\_}{U}$ and the deformation state $\underset{\_}{Y}$ as

$$\begin{gathered} \underset{\_}{X}〈\mathit{p}-\mathit{x}〉=\mathit{p}-\mathit{x} \\ \underset{\_}{U}〈\mathit{p}-\mathit{x}〉=\mathit{u}\left(\mathit{p},t\right)-\mathit{u}\left(\mathit{x},t\right) \\ \underset{\_}{Y}〈\mathit{p}-\mathit{x}〉=\underset{\_}{X}〈\mathit{p}-\mathit{x}〉+\underset{\_}{U}〈\mathit{p}-\mathit{x}〉 \end{gathered}$$

(2)

therefore, the force state for the state-based peridynamic formulation is expressed as

$$\underset{\_}{T}\left[\mathit{x},t\right]〈\mathit{p}-\mathit{x}〉=\left\{\frac{2\left(2v-1\right)}{v-1}\left[\left(K+\frac{\mu }{9}\frac{{\left(v+1\right)}^2}{{\left(2v-1\right)}^2}\right)\theta -\frac{8\mu }{3q}\left(\underset{\_}{\omega e^d}\right)•\underset{\_}{x}\right]\frac{\underset{\_}{\omega x}}{q}+\frac{8\mu }{q}\underset{\_}{\omega e^d}\right\}·\frac{Y}{‖Y‖}$$

(3)

where $\theta$ is the volume dilatational of the region H, $\underset{\_}{e^d}=\underset{\_}{e}-\frac{\theta ‖X‖}{3}$ the scalar deviator state component of the bond elongation $\underset{\_}{e}=‖Y‖-‖X‖$, $\underset{\_}{\omega }$ the scalar state of the influence function, K the bulk modulus, µ the shear modulus, v Poisson’s ratio and q the weight of the region H. The local damage at the material point of the body is introduced by breaking irreversibly the bonds according to the failure criterion adopted, hence the damage can be computed as

$$\phi \left(x,t\right)=1-\frac{{\int }_H\lambda (\mathit{p}-\mathit{x},t){dV}_{\mathit{p}}}{{\int }_H{dV}_{\mathit{p}}}$$

(4)

with $\lambda (\mathit{p}-\mathit{x},t)$ a scalar-valued function which assumes the value of 1 if the bond is active and of 0 if the bond is broken. A failure criterion that is widely used for deciding when a bond has to be broken is based on the critical stretch, where the stretch is defined as $s=\underset{\_}{e}〈\mathit{p}-\mathit{x}〉/‖\underset{\_}{X}〈\mathit{p}-\mathit{x}〉‖$. Therefore, the bond will break when its stretch overcome the critical stretch which is related to the fracture energy of the material $G_0$ in mode I [5]. Equation (1) is commonly implemented numerically by means of a meshfree method, see [5] for more details.

3. Method of Failure Criterion

According to the correspondence model introduced in [1], a nonlocal approximation of the strain tensor can be expressed as

$$\overline{\mathit{F}}=\left(\underset{H}{\overset{}{\int }}\underset{\_}{\omega }〈\mathit{p}-\mathit{x}〉\underset{\_}{Y}〈\mathit{p}-\mathit{x}〉⨂(\mathit{p}-\mathit{x}){dV}_{p-x}\right){\left(\underset{H}{\overset{}{\int }}\underset{\_}{\omega }〈\mathit{p}-\mathit{x}〉(\mathit{p}-\mathit{x})⨂(\mathit{p}-\mathit{x}){dV}_{p-x}\right)}^{-1}$$

(5)

Consequently, Piola stress tensor $\overline{\mathit{\sigma }}$ can be expressed as $\overline{\mathit{\sigma }}=\widehat{\sigma }\left(\overline{\mathit{F}}\right)$ in which the function $\widehat{\sigma }$ represents any relationship between the stress tensor and the strain tensor. For this work we assume the hypothesis of isotropic and linear elastic material, therefore, such a function can be expressed for the case of 2D plane stress case as

$$\left[\begin{array}{c}{\sigma }_{xx}\\ {\sigma }_{yy}\\ {\sigma }_{xy}\end{array}\right]=\left[\begin{array}{ccc}K+\mu & K-\mu & 0\\ K-\mu & K+\mu & 0\\ 0& 0& \mu \end{array}\right]\left[\begin{array}{c}{\epsilon }_{xx}\\ {\epsilon }_{yy}\\ {\epsilon }_{xy}\end{array}\right]$$

(6)

Equations (5) and (6) allows to compute the state of stress at each material point of the body, therefore, as commonly done in classic mechanics our failure criterion will select a node of the grid as “potential” node to fail if a certain stress is higher than a critical stress which value depends on the type of failure criteria proposed in the classic mechanics literature. We highlight the term “potential” to remark that, differently from the classic theory of mechanics, peridynamics deals with bonds. Let us give insight into the criterion proposed in this work, first we employ the criterion named Maximum Normal Stress Criterion (MNSC) mainly used to predict the failure of brittle materials. According to MNSC, a material point (source node $\mathit{x}$) will be marked as failed if its maximum principal stress exceeds the uniaxial tensile strength ${\sigma }_t$ of the material, alternatively, the material point will be marked as failed if the minimum principal stress is lower than the uniaxial compressive strength ${\sigma }_c$ of the material. If the criterion is satisfied, we will evaluate the stress ${\sigma }_b$ at the family node $\mathit{p}$ along the direction of the bond $\mathit{\xi }=\mathit{p}-\mathit{x}$, recalling the Cauchy’s relation in matrix form, as

$$\left[\begin{array}{c}{\sigma }_{b,x}\\ {\sigma }_{b,y}\end{array}\right]=\left[\begin{array}{cc}{\sigma }_{xx}& {\sigma }_{xy}\\ {\sigma }_{xy}& {\sigma }_{yy}\end{array}\right]\left[\begin{array}{c}\frac{{\xi }_x}{‖\mathit{\xi }‖}\\ \frac{{\xi }_y}{‖\mathit{\xi }‖}\end{array}\right]$$

(7)

with ${\sigma }_{b,x}$, ${\sigma }_{b,y}$, ${\xi }_x$ and ${\xi }_y$ the corresponding components in the reference system (x, y). Hence, the breakage of the bond will be activated if ${\sigma }_b\ge {\sigma }_t$ or if ${\sigma }_b\le {\sigma }_c$.

4. Results

The benchmark problem concerns a pre-notched beam subjected to an impact load, the test is performed in [7]. Figure 1 shows the experimental setup of the specimen which is hit with a hammer on the upper side at the off-center of its length L.

We analyze the case for a value of the loading eccentricity $e=l/(L/2)=0.1$. The material is PMMA with the following mechanical properties: Young’s modulus E = 2.94 GPa, Poisson’s ratio ν = 0.3 and mass density $\rho$ = 1190 kg/m³. The grid is characterized by a grid spacing $\mathsf{\Delta }x=0.6$ mm, horizon $\delta =3$ mm and m-ratio = 5 while the time step is assumed to be 300 ns simulating the case under the assumption of plane stress condition. A tensile strength ${\sigma }_t=5$ MPa is assumed. Figure 2 shows the deformed beam with the contour plot of the damage, while in Figure 3 the simulated fracture pattern is compared with the experimental crack and the results obtained with other methods. It can be noticed as the slope of the simulated crack is in a good agreement with the experimental crack.

5. Conclusions

We propose a novel fracture criterion based on the stress tensor to be applied in the framework of peridynamic theory. The results show how this criterion is suitable for treating mixed-mode I-II fracture being capable to capture the fracture pattern observed in the experiment analyzed in this work.