Dispersion and isogeometric analyses of second‐order and fourth‐order implicit gradient‐enhanced plasticity models

Kolo, Isa; Borst, René de

doi:10.1002/nme.5749

Cited by 5 publications

(7 citation statements)

References 40 publications

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“…With further refinement, the band propagates smoothly. It is important to ensure that the presented results are similar to those obtained from a standard uniformly refined mesh in the literature [31,20]. A comparison is shown for the local and nonlocal effective plastic strain (figure 13).…”

Section: Gradient Plasticitymentioning

confidence: 76%

“…leads to the following weak forms for the equilibrium and nonlocal effective strain equations respectively [20,31]:…”

Section: Gradient Plasticitymentioning

confidence: 99%

“…The same order of interpolation is used for the interpolation of the displacements and the nonlocal plastic multiplier, i.e. N u and h both contain quadratic NURBS [31]. Next, the discretised variables are employed in the weak forms -eq.…”

Section: Gradient Plasticitymentioning

confidence: 99%

“…The relative error in energy norm is also plotted and it is clear that the error is highest at the elastic-plastic boundary. Finally, a localisation problem is considered using implicit gradient plasticity [31]. The problem is illustrated in figure 11.…”

Section: Classical Plasticitymentioning

confidence: 99%

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Strain-gradient elasticity and gradient-dependent plasticity with hierarchical refinement of NURBS

Kolo

Chen

Borst

2019

Finite Elements in Analysis and Design

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Higher-order strain-gradient models are relevant for engineering materials which exhibit size-dependent behaviour as observed from experiments. Typically, this class of models incorporate a length scale-related to micromechanical material properties-to capture size effects, remove stress singularities, or regularise an ill-posed boundary value problem resulting from localisation of deformation. The higher-order continuity requirement on shape functions can be met using NURBS discretisation, as is considered herein. However, NURBS have a tensor-product nature which makes selective refinement cumbersome. To maintain accuracy and efficiency in analysis, a finer mesh may be required, to capture a localisation band, certain geometrical features, or in regions with high gradients. This work presents straingradient elasticity and strain-gradient plasticity, both of second-order, with hierarchically refined NURBS. Refinement is performed based on a multi-level mesh with element-wise hierarchical basis functions interacting through an inter-level subdivison operator. This ensures a standard finite-element data structure. Suitable marking strategies have been used to select elements for refinement. The capability of the numerical schemes is demonstrated with two-dimensional examples.

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Section: Gradient Plasticitymentioning

confidence: 76%

“…leads to the following weak forms for the equilibrium and nonlocal effective strain equations respectively [20,31]:…”

Section: Gradient Plasticitymentioning

confidence: 99%

Section: Gradient Plasticitymentioning

confidence: 99%

Section: Classical Plasticitymentioning

confidence: 99%

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Strain-gradient elasticity and gradient-dependent plasticity with hierarchical refinement of NURBS

Kolo

Chen

Borst

2019

Finite Elements in Analysis and Design

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“…This observation is the underlying reason of the reported mesh sensitivity and the often erratic and unsatisfactory convergence behaviour of the equilibrium‐searching iterative procedure, which can be aggravated upon mesh refinement . A number of approaches have been proposed to repair this deficiency, including Cosserat plasticity, nonlocal plasticity, and gradient plasticity …”

Section: Introductionmentioning

confidence: 99%

On viscoplastic regularisation of strain‐softening rocks and soils

Borst

Duretz

2020

Num Anal Meth Geomechanics

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Summary Constitutive models for rocks and soils typically incorporate some form of strain softening. Moreover, many plasticity models for frictional materials use a nonassociated flow rule. Strain softening and nonassociated flow rules can cause loss of well‐posedness of the initial‐value problem, which can lead to a severe mesh dependence in simulations and poor convergence of the iterative solution procedure. The inclusion of viscosity, which is a common property of materials, seems a natural way to restore well‐posedness, but the mathematical properties of a rate‐dependent model, and therefore the effectiveness with respect to the removal of mesh dependence, can depend strongly on how the viscous element is incorporated. Herein, we show that rate‐dependent models, which are commonly applied to problems in the Earth's lithosphere, such as plate tectonics, are very different from the approach typically adopted for more shallow geotechnical engineering problems. We analyse the properties of these models under dynamic loadings, using dispersion analyses and one‐dimensional finite difference analyses, and complement them with two‐dimensional simulations of a typical strain localisation problem under quasi‐static loading conditions. Finally, we point out that a combined model, which features two viscous elements, may be the best way forward for modelling time‐dependent failure processes in the deeper layers of the Earth, since it not only enables modelling of the creep characteristics typical of long‐term behaviour but also regularises the initial/boundary‐value problem.

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The scaled boundary finite element method for dispersive wave propagation in higher‐order continua

Daneshyar

Sotoudeh

Ghaemian

2022

Numerical Meth Engineering

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The classical theory of elasticity relies on the perfect structure of matter. No matter how small a medium is, it always stays homogeneous-but the reality is different. Even though the classical continuum mechanics suffices well for describing phenomena that evolve at super-microscopic scales, it ceases to reproduce reasonable responses when the size effects are pronounced. It is due to the local constitutive equations of classical continuum mechanics, which are devoid of any length scales associated with the underlying microstructure.The gradient-dependent theory of elasticity redresses this shortcoming by incorporating the kinematic quantities of the corresponding representative volume element by enriching the differential equations with higher-order spatial and temporal derivatives. In this study, the scaled boundary finite element method is formulated for the equations of motion with higher-order inertia terms. To this end, the semi-discretized scaled boundary finite element equations of the medium are derived by introducing the scaled boundary transformation of geometry to the gradient-enriched equations of motion and applying the Galerkin method of weighted residuals. It is shown that the well-established available solution methods are incapable of handling the frequency-domain representation of the derived formulation. Accordingly, a numerical solution method based on the shooting technique is proposed. The solution procedure is formulated for general numerical integration schemes via the infinite Taylor series. The evolution of the impedance-diffusion matrix and contribution of inter-subdomain forces and tractions are extracted. Four different numerical integration methods are employed to describe the solution procedure. In addition, a comparison regarding their computational efficiency is presented. For the sake of verification, the scaled boundary finite element solutions of three numerical examples are compared with reference solutions that are obtained using finite element models with extremely fine meshes. The convergence trends are also presented for both h-and p-refinement methods. The results demonstrate the capability of the proposed formulation in reproducing accurate responses.

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Dispersion and isogeometric analyses of second‐order and fourth‐order implicit gradient‐enhanced plasticity models

Cited by 5 publications

References 40 publications

Strain-gradient elasticity and gradient-dependent plasticity with hierarchical refinement of NURBS

Strain-gradient elasticity and gradient-dependent plasticity with hierarchical refinement of NURBS

On viscoplastic regularisation of strain‐softening rocks and soils

The scaled boundary finite element method for dispersive wave propagation in higher‐order continua

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