Date of download: 11/4/2017 Copyright © ASME. All rights reserved. From: An Overview of Theories of Continuum Mechanics With Nonlocal Elastic Response and a General Framework for Conservative and Dissipative Systems Appl. Mech. Rev. 2017;69(3):030802-030802-18. doi:10.1115/1.4036723 Figure Legend: 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.

Date of download: 11/4/2017 Copyright © ASME. All rights reserved. From: An Overview of Theories of Continuum Mechanics With Nonlocal Elastic Response and a General Framework for Conservative and Dissipative Systems Appl. Mech. Rev. 2017;69(3):030802-030802-18. doi:10.1115/1.4036723 Figure Legend: 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.

Date of download: 11/4/2017 Copyright © ASME. All rights reserved. From: An Overview of Theories of Continuum Mechanics With Nonlocal Elastic Response and a General Framework for Conservative and Dissipative Systems Appl. Mech. Rev. 2017;69(3):030802-030802-18. doi:10.1115/1.4036723 Figure Legend: 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.

Date of download: 11/4/2017 Copyright © ASME. All rights reserved. From: An Overview of Theories of Continuum Mechanics With Nonlocal Elastic Response and a General Framework for Conservative and Dissipative Systems Appl. Mech. Rev. 2017;69(3):030802-030802-18. doi:10.1115/1.4036723 Figure Legend: (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.

Date of download: 11/4/2017 Copyright © ASME. All rights reserved. From: An Overview of Theories of Continuum Mechanics With Nonlocal Elastic Response and a General Framework for Conservative and Dissipative Systems Appl. Mech. Rev. 2017;69(3):030802-030802-18. doi:10.1115/1.4036723 Figure Legend: 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.