A plane stress model of bond-based Cosserat peridynamics and the effects of material parameters on crack patterns

doi:10.1016/j.enganabound.2020.11.011

Engineering Analysis with Boundary Elements

Volume 123, 1 February 2021, Pages 48-61

https://doi.org/10.1016/j.enganabound.2020.11.011 Get rights and content

Abstract

A plane stress model of bond-based Cosserat peridynamics is proposed, of which the constitutive equations of Cosserat continuum are implemented. The rotation of the proposed bond-based Cosserat peridynamic model is independent with the translational displacement, and the couple stress is considered in the material points’ interaction. The proposed model can degenerate into bond-based peridynamics and micropolar peridynamics under certain conditions. Several examples of crack propagation are designed to investigate the effect of material parameters on crack patterns. The numerical results are compared with other peridynamic models and experimental results to validate the proposed model. The numerical results show that the Cosserat shear modulus and the tangential stiffness of bond have an impact on crack propagation, including crack branching and crack pattern.

Introduction

Peridynamic (PD) theory proposed by Silling [1] of Sandia National Labs in 2000 is widely used in solving fracture and failure problems. As a non-local theory, material points in the body interact with each other within a certain distance called peridynamic horizon, the interactions between material points are called bonds. Unlike the differential form of equilibrium equation, the equilibrium equation in peridynamic theory is written in integral form, which makes it feasible to deal with discontinuous problems. With these merits, the peridynamic theory is widely used to solve fracture problems [2], [3]–4] and is used to establish a new method to solve partial differential equations [5,6].

Bond-based peridynamics is the first peridynamic model proposed by Silling [1], which involves only one parameter to describe the micro elastic bond stretch response. Therefore, the Poisson's ratio is restricted to be 1/3 in plane stress problems and 1/4 in both 3D and plane strain problems [7–9]. To solve the limitation of Poisson's ratio, Silling et al. [10] proposed the state-based peridynamic theory. The long-range forces of state-based peridynamics depend on all the bonds connected to them, and the number of material parameters equals to the ones in continuum mechanics. However, the state-based peridynamics is more complicated than bond-based peridynamics, and many researchers proposed the improved peridynamic models in bond-based peridynamic framework. Gerstle et al. [9] proposed the bond-based micropolar peridynamic theory, in which the pairwise moments are implemented to simulate linear elastic materials with varying Poisson's ratio. Diana and Casolo [11] proposed a generalized micropolar peridynamic model with shear deformability for linear and non-linear problems. Zheng et al. [12] proposed a new bond-based peridynamics in which the bond is subjected to axial and transverse pairwise forces, and the particle rotation angle is added to eliminate the additional bending moment.

It is noticed that the improved peridynamic models based on Cauchy continuum [9,11,12] take the rotation of DOFs of material points. However, the Cauchy continuum is not enough to describe the behaviors of the materials with independent rotation. This lies in the lacking of the rotational DOFs and the materials parameters corresponding to micro-deformation and micro-structure. To overcome such shortcomings, Cosserat continuum (micropolar continuum) is a feasible choice [13]. The Cosserat theory is characterized by independent rotation DOFs and the couple stress. This micro rotation is independent of the displacement of gradient-determined macro-rotation [14]. There are another two material parameters named internal length scale and Cosserat shear modulus of Cosserat continuum. The internal length scale is related to the width of the shear band and can remedy the ill-posed problem in the strain localization or bifurcation. Some researchers proposed a peridynamic model based on Cosserat continuum. Chowdhury et al. [15] proposed the state-based micropolar peridynamic theory, which introduced micro-rotational DOFs and brought the physically relevant material length scale into peridynamics. Chen et al. [16] proposed contact particle model of bond-based Cosserat peridynamics, in which a new failure criterion for bond broken is developed, and the relation between internal length scale and the peridynamic horizon is discussed.

Based on contact particle model of bond-based Cosserat peridynamics [16], this work develops a plane stress model of bond-based Cosserat peridynamics. Compared with the bond-based peridynamics, the contact particle model involves more parameters, for the description of size effects on both deformation and failure. Many researchers investigated the impact of Cosserat parameters. Khoei et al. [17] compared the classical and Cosserat theory and demonstrated the effect of internal length parameter. Xiu and Chu [14] investigated the load-displacement relationship and the strain localization behavior of granular materials. They examined the influence of microscopic parameters in the micromorphic model on macroscopic mechanical responses. To further develop the contact particle model and make it more widely used, the contact particle model based on plane stress condition is extended. Consider that the size effect on the failure should be analyzed under the peridynamic framework, the influence of material parameters of contact particle model on crack propagation is discussed through several numerical examples. The numerical results are compared with other peridynamic models, including bond-based peridynamics and micropolar peridynamics and experimental results to validate the proposed model.

The construction of this paper is as follows: Section 2 introduces the peridynamic theory, Section 3 gives the derivation of the contact particle model under plane stress condition, Section 4 discusses the effect of parameters on crack patterns through several numerical examples including single disc plate under compression, crack branching and single-disc plate crushing. Section 5 presents the discussions and conclusions.

Section snippets

Peridynamic theory of a continuum

In this section, the peridynamic theory is introduced. The acceleration of any particle at x in the reference configuration at time t is found from: $ρ \ddot{u} (x, t) = \int_{H_{x}} f (u (x^{'}, t) - u (x, t), x^{'} - x) dV x^{'} + b (x, t)$

H_x is a neighbourhood of x with a horizon radius of δ, whose value is related to the discrete scale Δx (δ = 3Δx normally). The horizon H_x of any material point is required to be constant usually, to improve the computational efficiency, Ren et al. [18,19] proposed the dual-horizon peridynamics in which

Cosserat continuum in plane stress problem

In contact particle model [16], the strain component in the z-direction of strain ε₀ is set to zero. This work derived a contact particle model of bond-based Cosserat peridynamics under the condition of plane stress.

Contrary to the Cauchy continuum, Cosserat continuum takes into account the independent micro-rotation θ.

In the plane stress problem, the equilibrium equations are: $L^{T} σ + f = 0$ $L^{T} = [\begin{matrix} \frac{\partial}{\partial x} & 0 & 0 & 0 & \frac{\partial}{\partial y} & 0 & 0 \\ 0 & \frac{\partial}{\partial y} & 0 & \frac{\partial}{\partial x} & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 1 & l_{c} \frac{\partial}{\partial x} & l_{c} \frac{\partial}{\partial y} \end{matrix}]$

The body force f is (f_x, f_y, m_z)^T, and the stress is written as: $σ = (σ_{xx}$

Single disc plate under uniaxial compression

The degradation of contact particle model has been discussed in Section 3.3, which shows that contact particle model can degrade into the micropolar peridynamics and the bond-based peridynamics under certain conditions. This section discusses the difference between the above three models. Consider a 2D disc plate with a radius of 40 mm and a thickness of 0.5 mm, a rigid square block with 20mm sides and the same thickness of the disc is formed at the upper and lower ends of the disc. The density

Conclusion

In this study, a contact particle model of plane stress problem is proposed, the contact particle model considers the rotation effect of material points and can degrade into bond-based peridynamics and micropolar peridynamics under certain conditions,

The numerical example of single disc plate under uniaxial compression is designed to investigate the connections between the above three PD models in crack patterns where the effect of G_c, l_c and k_t are discussed. In the example of a plate with an

Declaration of Competing Interest

None.

Acknowledgment

The authors are pleased to acknowledge the support of this work by the National Natural Science Foundation of China through contract/grant number 11772237, 11472196 and 11172216, and to acknowledge the open funds of the State Key Laboratory of Structural Analysis for Industrial Equipment (Dalian University of Technology) through contract/grant number GZ19110.

References (32)

SA Silling
Reformulation of elasticity theory for discontinuities and long-range forces
J Mech Phys Solids
(2000)
W Zhou et al.
Analyzing dynamic fracture process in fiber-reinforced composite materials with a peridynamic model
Eng Fract Mech
(2017)
J Wan et al.
Dependency of single- particle crushing patterns on discretization using peridynamics
Powder Technol
(2020)
A Yaghoobi et al.
Mesh less modeling framework for fiber reinforced concrete structures
Comput Struct
(2015)
Q.Z. Zhu et al.
Peridynamic formulations enriched with bond rotation effects
Int J Eng Sci
(2017)
W Gerstle et al.
Peridynamic modeling of concrete structures
Nucl Eng Des
(2007)
Z Chen et al.
Two Cosserat peridynamic models and numerical simulation of crack propagation
Eng Fract Mech
(2019)
A R Khoei et al.
3D finite element modeling of shear band localization via the micro-polar Cosserat continuum theory
Comput Mater Sci
(2010)
H. Ren et al.
Dual-horizon peridynamics: a stable solution to varying horizons
Comput Methods Appl Mech Eng
(2017)
SA Silling et al.
A meshfree method based on the peridynamic model of solid mechanics
Comput Struct
(2005)

R de Borst et al.

Localization in a Cosserat continuum under static and dynamic loading conditions

Comput Methods Appl Mech Eng

(1991)

XP Zhou et al.

A novel conjugated bond linear elastic model in bond-based peridynamics for fracture problems under dynamic loads

Eng Fract Mech

(2018)

WX Huang et al.

Analysis of plane Couette shear test of granular media in a Cosserat continuum approach

Mech Mater

(2014)

T. Rabczuk et al.

A non-local operator method for partial differential equations with application to electromagnetic waveguide problem

Comput Mater Contin

(2019)

H. Ren et al.

Non-local operator method with numerical integration for gradient solid

Comput Struct

(2020)

Y.D. Ha et al.

Studies of dynamic crack propagation and crack branching with peridynamics

Int J Fract

(2010)

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A plane stress model of bond-based Cosserat peridynamics and the effects of material parameters on crack patterns

Abstract

Introduction

Section snippets

Peridynamic theory of a continuum

Cosserat continuum in plane stress problem

Single disc plate under uniaxial compression

Conclusion

Declaration of Competing Interest

Acknowledgment

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Comput Struct

Int J Eng Sci

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Comput Mater Sci

Comput Methods Appl Mech Eng

Comput Struct

Comput Methods Appl Mech Eng

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Mech Mater

A non-local operator method for partial differential equations with application to electromagnetic waveguide problem

Comput Mater Contin

Non-local operator method with numerical integration for gradient solid

Comput Struct

Studies of dynamic crack propagation and crack branching with peridynamics

Int J Fract