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

### Abstract

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.

### Conclusions

• 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.

Please send me an e-mail if you wish to receive the full paper.

EPFL / 26 February 2001 / milan.jirasek@epfl.ch