Peridynamic differential operator and its applications

Madenci, Erdogan; Barut, A. O.; Futch, Michael J.

doi:10.1016/j.cma.2016.02.028

Cited by 306 publications

(138 citation statements)

References 18 publications

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“…Furthermore, the dilatation scalar state can be defined as θ(e) = γ (ωx)•e m , and the isotropic component of the extension scalar state is expressed as e i = θx 3 . In order to analyze the strain state, the peridynamic differential operator (PDDO) [Gu, Madenci and Zhang (2018); Madenci, Barut and Futch (2016); Madenci, Dorduncu, Barut et al (2017); Gu, Zhang and Madenci (2019)] is introduced to define the nonlocal deformation gradient tensor,…”

Section: Governing Equation and Constitutive Modelmentioning

confidence: 99%

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Coupled Digital Image Correlation and Peridynamics for Full-Field Deformation Measurement and Local Damage Prediction

Li¹,

Gu²,

Zhang³

et al. 2019

Computer Modeling in Engineering &Amp; Sciences

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Digital image correlation (DIC) measurement technique and peridynamics (PD) method have been applied in specific fields extensively owing to their respective advantages in obtaining full-field deformation and local failure of loaded materials and structures. This study provides a simple way to couple DIC measurements with PD simulations, which can circumvent the difficulties of DIC in dealing with discontinuous deformations. Taking the failure analysis of a compact tension specimen of aluminum alloy and a static three-point bending concrete beam as examples, the DIC experimental system firstly measures the full-field displacements, and then the PD simulation is applied on potential damage regions determined according to the correlation coefficients, to track the micro-crack evolution and macro-crack propagation. As results, the coupled DIC and PD approach can effectively measure the full-field displacement and the localized damage accumulation and crack propagation.

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Section: Governing Equation and Constitutive Modelmentioning

confidence: 99%

“…where the symbol ⊗ denotes the dyadic product of two vectors, N is the order of Taylor series expansion (TSE), the vector g is composed of PD functions described by Madenci et al [Madenci, Barut and Futch (2016); Madenci, Dorduncu, Barut et al (2017); Gu, Zhang and Madenci (2019)] as g (ξ; N ) = g 100…”

Section: Governing Equation and Constitutive Modelmentioning

confidence: 99%

Coupled Digital Image Correlation and Peridynamics for Full-Field Deformation Measurement and Local Damage Prediction

Li¹,

Gu²,

Zhang³

et al. 2019

Computer Modeling in Engineering &Amp; Sciences

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“…As derived in detail by Madenci et al , the PD differential operator can be expressed as

\begin{array}{l} left & \frac{\partial^{p_{1} + p_{2} + \cdot \cdot \cdot + p_{M}} f (x)}{\partial x_{1}^{p_{1}} \partial x_{2}^{p_{2}} \cdot \cdot \cdot \partial x_{M}^{p_{M}}} = \int_{H_{boldx}} f false(boldx + bold-italicξ false) g_{N}^{p_{1} p_{2} \cdot \cdot \cdot p_{M}} false(bold-italicξ false) d V \end{array}

in which p i denotes the order of differentiation with respect to variable x i with

i = 1, \dots, M

. The relative position vector between these points is defined as

ξ = x' - x

.…”

Section: Peridynamic Differential Operatormentioning

confidence: 99%

Numerical solution of linear and nonlinear partial differential equations using the peridynamic differential operator

Madenci

Dördüncü

Barut

et al. 2017

Numerical Methods Partial

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108

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This study presents numerical solutions to linear and nonlinear Partial Differential Equations (PDEs) by using the peridynamic differential operator. The solution process involves neither a derivative reduction process nor a special treatment to remove a jump discontinuity or a singularity. The peridynamic discretization can be both in time and space. The accuracy and robustness of this differential operator is demonstrated by considering challenging linear, nonlinear, and coupled PDEs subjected to Dirichlet and Neumann‐type boundary conditions. Their numerical solutions are achieved using either implicit or explicit methods. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1726–1753, 2017

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“…By using the PD differential operator derived by Madenci et al. , the displacement gradient tensor for a homogeneous deformation can be expressed in the form

\nabla u = H = \frac{tr (I)}{m} \int_{H_{x}} w ((), | {boldx}^{'} - x |) ([], boldu ({boldx}^{'}) - boldu (x)) \otimes ((), x^{'} - boldx) d V

\nabla u = H = \frac{tr (I)}{m} \int_{H_{x}} w ((), | {boldx}^{'} - x |) ({boldy}^{'} - y) \otimes ({boldx}^{'} - x) d V - I,

in which the weight function,

w (| {boldx}^{'} - x |)

, is specified as

w (false| x^{'} - boldx false|) = {((), \frac{δ}{false| x^{'} - boldx false|})}^{2}

and

m = \int_{H_{x}} w ((), | {boldx}^{'} - x |) | {boldx}^{'} - x

…”

Section: Peridynamic Integralsmentioning

confidence: 99%

“…It is derived based on the PD differential operator introduced by Madenci et al. . With the specific form of the weight function given in Eq.…”

Section: Peridynamic Integralsmentioning

confidence: 99%

Peridynamic integrals for strain invariants of homogeneous deformation

Madenci

2017

Z Angew Math Mech

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This study presents the peridynamic integrals. They enable the derivation of the peridynamic (nonlocal) form of the strain invariants. Therefore, the peridynamic form of the existing classical strain energy density functions can readily be constructed for linearly elastic and hyperelastic isotropic materials without any calibration. A general form of the force density vector is derived based on the strain energy density function that is expressed in terms of the first invariant of the right Cauchy-Green strain tensor and the Jacobian. In the case of linear elastic response for isotropic materials, the peridynamic force density vector is derived based on the classical form of the strain energy density function for three-and two-dimensional analysis. Also, a new form of the strain energy density function leads to a force density vector similar to that of bond-based peridynamics. Numerical results concern the verification of the peridynamic predictions with these force density vectors by considering a rectangular plate under uniform stretch.The peridynamic (PD) theory was introduced by Silling [1], and later generalized by Silling et al. [2]. The PD equation of motion is an integro-differential equation, and the integrand is free of the spatial derivatives of the displacement field. It permits damage nucleation at multiple damage sites in unspecified locations and its propagation along unguided paths with complex interactions.Conceptually rather simple, the bond-based PD introduced by Silling [1] is free of the assumption of small displacement gradients. However, it does not distinguish between dilatation and distortion. Therefore, it presents only one independent elastic constant; the second elastic constant has a fixed value depending on the analysis dimension. The corresponding strain energy density (SED) function is expressed in terms of the stretch between two material points, and one PD material parameter, referred to as the micro-modulus. This micro-modulus is determined through calibration against the classical SED by considering a simple loading condition, such as isotropic expansion. The determination of the micro-modulus for three-and two-dimensional analysis under plane strain and stress assumptions is described by Gerstle et al. [3]. Later, Bobaru et al. [4] proposed different forms of the micro-modulus function and discussed their effect on convergence of the computations. Recently, Chen et al. [5] presented a constructive approach for selecting the most appropriate kernel function in the bond-based PD equation of motion with respect to its convergence characteristics. The state-based PD introduced by Silling et al. [2] retains the two independent elastic constants such as the Young's modulus and Poisson's ratio. However, the determination of the force density-stretch relation requires an explicit form of the SED function. Silling et al.[2] suggested a SED function for three-dimensional analysis based on the deformation state and force state for a material referred to as a "linear peridynamic solid". T...

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Peridynamic differential operator and its applications

Cited by 306 publications

References 18 publications

Coupled Digital Image Correlation and Peridynamics for Full-Field Deformation Measurement and Local Damage Prediction

Coupled Digital Image Correlation and Peridynamics for Full-Field Deformation Measurement and Local Damage Prediction

Numerical solution of linear and nonlinear partial differential equations using the peridynamic differential operator

Peridynamic integrals for strain invariants of homogeneous deformation

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