International Journal of Solids and Structures, 38 (2001), 2921-2931
A THERMODYNAMICALLY CONSISTENT APPROACH TO MICROPLANE THEORY:
PART I. FREE ENERGY AND CONSISTENT MICROPLANE STRESSES
Ignacio Carol, Milan Jirásek, and
Zdenek P. Bazant
Microplane models are based on the assumption that the constitutive laws
of the material may be established between normal and shear components
of stress and strain on planes of generic orientation (so-called microplanes),
rather than between tensor components or their invariants. In the kinematically
constrained version of the model, it is assumed that the microplane strains
are projections of the strain tensor, and the stress tensor is obtained
by integrating stresses on microplanes of all orientations at a point.
Traditionally, microplane variables were defined intuitively, and the integral
relation for stresses was derived by application of the principle of virtual
work. In this paper, a new thermodynamic framework is proposed. A free-energy
potential is defined at the microplane level, such that its integral over
all orientations gives the standard macroscopic free energy. From this
simple assumption it is possible to introduce consistent microplane stresses
and their corresponding integral relation to the macroscopic stress tensor.
Based on this it is shown that, in spite of the excellent data fits achieved,
many existing formulations of microplane model were not guaranteed to be
fully thermodynamically compliant. A consequence is the lack of work-conjugacy
between some of the microplane stress and strain variables used, and the
danger of spurious energy dissipation/generation under certain load cycles.
The possibilities open by the new theoretical framework are developed further
in a Part II companion paper.
A new simple thermodynamically consistent framework is presented for the
formulation of microplane models. The main assumption is that the macroscopic
free energy may be obtained as the integral over all microplane orientations
of a microplane free-energy function, which depends on the microplane strains
and the internal variables. This assumption does not contradict most of
the early versions of microplane models for concrete with and without split
of normal components (M1 and M2), but leaves out the more recent M3 and
M4 models, for which the free energy of the various microplanes may not
be written in a decoupled form.
The new formulation leads to a consistent definition of the microplane
stresses which are conjugate to the microplane strains, and to the integral
form of the micro-macro equilibrium equation which applies to those stresses.
A comparison with the previous microplane models not precluded by the new
formulation leads to the conclusion that, while the earliest model without
split (M1) was correct, the following version of microplane model with
the split of normal components (M2) cannot be guaranteed to be thermodynamically
consistent. In that case, microplane stresses sigmaV and sigmaD are not
necessarily work-conjugates to their strain counterparts epsilonV and epsilonD,
and neither are in general their sums sigmaN=sigmaV+sigmaD and epsilonN=epsilonV+epsilonD.
The integral micro-macro relation for stresses does not coincide either
with the one obtained from the thermodynamic derivation, and the model
may not be guaranteed to be free of spurious dissipation or generation
of energy. Models with ``symmetric'' laws for the normal deviatoric component
(in the sense of symmetric behavior in tension and in compression) are
less sensitive to this problem.
In a companion paper, the new thermodynamic derivation is developed further
by applying standard concepts such as the Coleman method and Clausius-Duhem
inequality at both microplane and macroscopic levels, and the resulting
equations are illustrated with two example formulations of damage and plasticity.
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