Rend. Sem. Mat. Univ. Pol. Torino Vol. 58, 1 2000
Geom., Cont. and Micros., I
J. F. Ganghoffer NEW CONCEPTS IN NONLOCAL CONTINUUM MECHANICS
Abstract. A new theoretical framework in nonlocal mechanics is defined, based on the concept of influence functions between material points within the continuum.
The traditional idea of a fixed and isotropic representative volume is abandoned and the non-locality is introduced via an influence function, which defines a non-
local interaction between material points. The general framework developed is exemplified by the description of damage as a scalar internal variable : the local
damage rate at a given point can be expressed as a path integral involving the in- fluence functions and the values of the local rate of damage transported along each
path. The properties satisfied by the influence function are first evidenced and the influence function is given an explicit expression, using a path integration tech-
nique. The concept of a representative volume is further defined as an outcome of the stationarity of the internal entropy production with respect to the path. An im-
plicit equation which defines the representative volume is formulated. The strength of the nonlocal interaction is further incorporated into the space geometry, so that
a metric characteristic of a Riemanian space is coupled to the internal variable dis- tribution. It appears that the curvature characterises the strength of the nonlocal
interaction.
1. Introduction
Traditional continuum models in nonlocal mechanics usually rely on the assumption that the nonlocal variables are simply volume averages of the corresponding local variables over a fixed
and isotropic representative volume element around the considered material point, see e.g. [1, 2, 3, 4, 5]. Considerations based on micromechanical arguments however show that the size
of the representative volume, i.e. the extension of the interaction shall depend on the local variable distribution itself : in the work by M¨uhlhaus et al. [6], a Cosserat theory for granular
materials is elaborated, starting from a particulate model. The model predicts that the shear band thickness evolves with the shear strain. A micromechanical argument for nonlocal damage has
been advanced in [2]: the strain-softening damage due to distributed cracking is modelled by a periodic array of cracks. The results of the model show that the elastic part of the response shall
be local, whereas the damage recovered at the macro scale shall be nonlocal. Furthermore, the size of the averaging region is determined by the crack spacing.
During loading of the cracked body, the increment of the stress along one crack is the sum of the average stress increment over the crack length and the contributions of all other cracks :
1 1
S
i
x =
D 1
S
i
x E
+ Z
V
3
i j
x, ξ 1S
j
ξ d V ξ .
The interactions of a set of microcracks cancel out over a short distance, and this in turn deter- mines the size of the representative volume. The kernel 3x, ξ that determines the influence
113
114 J. F. Ganghoffer
between two cracks located at points x and ξ depends on both the radial and angular variables, and evolves with the current crack distribution pattern. Thus, the form of the influence function
shall depend upon the distribution of the internal variables at each time loading – step. The last integral is in fact a path integral, depending on the spatial distribution of the cracks. The path
that do effectively contribute to the local stress increment on the left-hand side of 1 change according to the evolution of the spatial pattern of cracks.
In Ganghoffer et al., a path integral formulation of the nonlocal interactions has been formu- lated, with damage as a focus. The scalar damage variable there represents the internal variable.
The new concepts advanced therein can be considered as an attempt to model in a phenomeno- logical manner the nonlocal interactions between defects in a solid material. In this contribution,
we only give the main thrust of the ideas developed in [7].
2. Path integral formulation of nonlocal mechanics
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