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Review Article

An Overview of Theories of Continuum Mechanics With Nonlocal Elastic Response and a General Framework for Conservative and Dissipative Systems

[+] Author and Article Information
Arun R. Srinivasa

Holdredge/Paul Professor
Fellow ASME
Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77843-3123
e-mail asrinivasa@tamu.edu

J. N. Reddy

Oscar S. Wyatt Jr. Chair
Regents Professor
Distinguished Professor
Life Fellow ASME
Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77843-3123
e-mail: jnreddy@tamu.edu

Manuscript received September 20, 2016; final manuscript received April 22, 2017; published online June 6, 2017. Assoc. Editor: Pradeep R. Guduru.

Appl. Mech. Rev 69(3), 030802 (Jun 06, 2017) (18 pages) Paper No: AMR-16-1074; doi: 10.1115/1.4036723 History: Received September 20, 2016; Revised April 22, 2017

The aim of this review is to classify and provide a summary of the most widely used theories of continuum mechanics with nonlocal elastic response ranging from generalized continua to peridynamics showing, in broad outlines, the similarities and differences between them. We then show that, for elastic materials, these disparate approaches can be unified using a total energy-based methodology. While our primary focus is on elastic response, we show that a large class of local and nonlocal dissipative systems can also be unified by extending this methodology to a wide (but special) class of nonlocal dissipative continua. We hope that the paper may serve as a starting point for researchers for the development of novel nonlocal models.

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Figures

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Fig. 1

A taxonomy of purely mechanical continuum theories; models that incorporate additional length scales are of type III and type IV. We will focus attention only on the theories in the gray box.

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Fig. 2

The notion of the reference and current bond vectors in the peridynamic theory. A function Y[x]<ξ> maps a reference bond vector ξ to the current bond vector x′−x. The energy density functional at X, WX maps all the bond vectors at X to a scalar.

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Fig. 3

A schematic diagram of a material with an additional vector field: (a) reference, (b) bend, (c) splay, and (d) twist motions, which are independent of the motion of the body. The same motions are also accounted for in couple stress theories with axial vectors corresponding to the local rotation and are thus not independent of the motion of the body.

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Fig. 4

(a) A discrete system of particles (labeled i) moving with displacement ui from an initial configuration to the current configuration. Likewise for a continuum moving from its initial configuration to the current configuration specified by a displacement field u(X, t). In both cases, the Hamiltonian maps entire displacement and momentum density fields to a real number at each time instant.

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Fig. 5

A discrete beam figure with nodal positions ri and angles θi. It is possible to write the potential energy of the beamin terms of either the angles θi or the distances ||ri−ri−1||, ||ri+1−ri||, and ||ri+1−ri−1|| (by using the law of cosines). If we use the angles as the inputs, the corresponding generalized forces will be torques, which in turn can be written as “nonordinary” forces resulting in nonordinary state-based peridynamics (NSBP). On the other hand, if we use the distances, the corresponding forces will always be along the bond vectors and will result in “ordinary” state-based peridynamics (OSBP). This illustrates that whether it is NSBP or OSBP depends upon how the forces are resolved, which in turn depends upon the variables used in the potential energy.

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