A new framework in nonlocal mechanics

Ganghoffer, J.F.; Borst, René de

doi:10.1016/s0020-7225(99)00030-0

Cited by 32 publications

(18 citation statements)

References 24 publications

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“…By inserting (22) into (25), at the limit when x → 0 (14) is fully restored and this enables us to affirm that the two models presented in this paper are totally equivalent and that (20) may be used as a discretized version of (10). Summing up, the elastic equilibrium 1D problem is governed by equilibrium, compatibility and a constitutive law, respectively, as…”

Section: The Equivalent Point-spring Model Of the Non-local Barmentioning

confidence: 55%

“…In fact arbitrary kinematic boundary conditions may be enforced in (20), as usual, simply by putting in the first or the last component of the vector u the corresponding assigned value. Mechanical boundary conditions may be simply enforced by assigning prescribed values of the applied load to the pertinent elements of the vector f.…”

Section: The Equivalent Point-spring Model Of the Non-local Barmentioning

confidence: 99%

“…Let us assume the following values of the bar geometrical and mechanical parameters: A = 1 cm 2 , E = 2.1 × 10 6 daNcm −2 , L = 100 cm, F = 100 daN. For a given number m of volume elements V j , (20) is solved to determine the displacement field. Based on the latter all the response variables may be derived.…”

Section: Exponential Attenuation Functionmentioning

confidence: 99%

“…In this context several contributions may be found in different branches of continuum mechanics up to the midseventies [15][16][17], when the lack of certain important features in the model led researchers to abandon this strategy. Recently the integral model of non-local elasticity has been widely used to smooth the stress field at crack tips [18][19][20] or in the context of damage and plasticity [12,13], often resorting to thermodynamic considerations [21].…”

Section: Introductionmentioning

confidence: 99%

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Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory

Paola¹,

Failla

Zingales³

2009

J Elasticity

126

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In this paper the physically-based approach to non-local elasticity theory is introduced. It is formulated by reverting the continuum to an ensemble of interacting volume elements. Interactions between adjacent elements are classical contact forces while long-range interactions between non-adjacent elements are modelled as distance-decaying central body forces. The latter are proportional to the relative displacements rather than to the strain field as in the Eringen model and subsequent developments. At the limit the displacement field is found to be governed by an integro-differential equation, solved by a simple discretization procedure suggested by the underlying mechanical model itself, with corresponding static boundary conditions enforced in a quite simple form. It is then shown that the constitutive law of the proposed model coalesces with the Eringen constitutive law for an unbounded domain under suitable assumptions, whereas it remains substantially different for a bounded domain. Thermodynamic consistency of the model also has been investigated in detail and some numerical applications are presented for different parameters and different functional forms for the decay of the long range forces. For simplicity, the problem is formulated for a 1D continuum while the general formulation for a 3D elastic solid has been reported in the appendix.

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Section: The Equivalent Point-spring Model Of the Non-local Barmentioning

confidence: 55%

Section: The Equivalent Point-spring Model Of the Non-local Barmentioning

confidence: 99%

Section: Exponential Attenuation Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory

Paola¹,

Failla

Zingales³

2009

J Elasticity

126

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“…A coupled elastoplastic-damage model including the gradient of damage is proposed by Nedjar [19]. Recently, a general theoretical framework in non-local mechanics was outlined by Gangho er and de Borst [20].…”

Section: Introductionmentioning

confidence: 99%

Theory and numerics of a thermodynamically consistent framework for geometrically linear gradient plasticity

Liebe

Steinmann

2001

Numerical Meth Engineering

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SUMMARYThe paper presents the theory and the numerics of a thermodynamically consistent formulation of gradient plasticity at small strains. Starting from the classical local continuum formulation, which fails to produce physically meaningful and numerically converging results within localization computations, a thermodynamically motivated gradient plasticity formulation is envisioned. The model is based on an assumption for the Helmholtz free energy incorporating the gradient of the internal history variable, a yield condition and the postulate of maximum dissipation resulting in an associated structure. As a result the driving force conjugated to the hardening evolution is identiÿed as the quasi-non-local drag stress which incorporates besides the strictly local drag stress essentially the divergence of a vectorial hardening ux. At the numerical side, besides the balance of linear momentum, the algorithmic consistency condition has to be solved in weak form. Thereby, the crucial issue is the determination of the active constraints exhibiting plastic loading which is solved by an active set search algorithm borrowed from convex non-linear programming. Moreover, di erent discretization techniques are proposed in order to compare the FE-performance in local plasticity with the advocated gradient formulation both for hardening and softening.

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The finite element method for the mechanically based model of non‐local continuum

Zingales

Paola

Inzerillo

2011

Numerical Meth Engineering

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SUMMARYIn this paper the finite element method (FEM) for the mechanically based non-local elastic continuum model is proposed. In such a model each volume element of the domain is considered mutually interacting with the others, beside classical interactions involved by the Cauchy stress field, by means of central body forces that are monotonically decreasing with their inter-distance and proportional to the product of the interacting volume elements. The constitutive relations of the long-range interactions involve the product of the relative displacement of the centroids of volume elements by a proper, distance-decaying function, which accounts for the decrement of the long-range interactions as long as distance increases. In this study, the elastic problem involving long-range central interactions for isotropic elastic continuum will be solved with the aid of the FEM. The accuracy of the solution obtained with the proposed FEM code is compared with other solutions obtained with Galerkins' approximation as well as with finite difference method. Moreover, a parametric study regarding the effect of the material length scale in the mechanically based model and in the Kröner-Eringen non-local elasticity has been investigated for a plane elasticity problem.

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A new framework in nonlocal mechanics

Cited by 32 publications

References 24 publications

Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory

Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory

Theory and numerics of a thermodynamically consistent framework for geometrically linear gradient plasticity

The finite element method for the mechanically based model of non‐local continuum

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