Microplane Model M7 for Plain Concrete. I: Formulation

Abstract

Mathematical modeling of the nonlinear triaxial behavior and damage of such a complex material as concrete has been a long-standing challenge in which progress has been made only in gradual increments. The goal of this study is a realistic and robust material model for explicit finite-element programs for concrete structures that computes the stress tensor from the given strain tensor and some history variables. The microplane models, which use a constitutive equation in a vectorial rather than tensorial form and are semimultiscale by virtue of capturing interactions among phenomena of different orientation, can serve this goal effectively. This paper presents a new concrete microplane model, M7, which achieves this goal much better than the previous versions M1-M6 developed at Northwestern University since 1985. The basic mathematical structure of M7 is logically correlated to thermodynamic potentials for the elastic regime, the tensile and compressive damage regimes, and the frictional slip regime. Given that the volumetric-deviatoric (V-D) split of strains is inevitable for distinguishing between compression failures at low and high confinement, the key idea is to apply the V-D split only to the microplane compressive stress-strain boundaries (or strain-dependent yield limits), the sum of which is compared with the total normal stress from the microplane constitutive relation. This avoids the use of the V-D split of the elastic strains and of the tensile stress-strain boundary, which caused various troubles in M3-M6 such as excessive lateral strains and stress locking in far postpeak uniaxial extension, poor representation of unloading and loading cycles, and inability to represent high dilatancy under postpeak compression in lower-strength concretes. Moreover, the differences between high hydrostatic compression and compressive uniaxial strain are accurately captured by considering the compressive volumetric boundary as dependent on the principal strain difference. The model is verified extensively in the companion paper.

Multi-Laminate Rate-Dependent Modelling of Static and Dynamic Concrete Behavior through Damage Formulation

“…Some great advantages of multi-plane models were reviewed in Bazant et al [28] and Brocca and Bazant [29]. The history of the trend of development in multi-plane models for modeling concrete behavior has been briefly explained in the Caner and Bazant's [30] literature. In the Sadrnejad's [31] literature, basic formulations of multi-laminate framework have been well explained.…”

“…This effect has been experimentally shown in Caner et al [32], which is very important under dynamic loads. Furthermore, unlike tensorial form, in order to simulate the nonlinear material behavior of concrete which significantly occurs both in the service and the failure ranges [22], the strain dependence of the yield limit can be easily taken into account in the vectorial component from one generic sampling plane [22,30]. Unlike the multi-laminate models in which the projection of macroscopic stresses on the sampling planes ((static constraints)) is usually used, the micro-plane models use the projection of macroscopic strains (kinematic constraints) [33].…”

“…It is very important to mention that the vertex effects (i.e., the inelastic deformation increments are parallel to the current loading surface in the stress space) are fundamentally overcome in the multi-plane models which represent multi-surface loading with some vectorial limit surfaces at each sampling plane [30]. Because of the principles of tensorial macro models which have one or a few loading surfaces, this effect is essentially overlooked [30]. This effect has been experimentally shown in Caner et al [32], which is very important under dynamic loads.…”

“…In numerical computations, this integral is approximated according to some of the optimal Gaussian integration formulas for a spherical surface, which replaces the integral by a weighted sum over a finite number of sampling planes [30]. This approximation may be written as follows:…”

Fracture and Size Effect on Strength of Plain Concrete Disks under Biaxial Flexure Analyzed by Microplane Model M7

“…A conceptual advantage is that the constitutive law is expressed in a vectorial form rather than a tensorial form (Bazant and Oh 1985). The microplane model has been proven capable of accurately capturing the damage, fracture behavior, and size effect in various quasi-brittle materials, which exhibit postpeak softening damage because of distributed cracking Caner and Bazant 2013).…”