The first part of this two-part study presents a new improved microplane constitutive model for concrete, representing the fourth version in the line of microplane models developed at Northwestern University. The constitutive law is characterized as a relation between the normal, volumetric, deviatoric, and shear stresses and strains on planes of various orientations, called the microplanes. The strain components on the microplanes are the projections of the continuum strain tensor, and the continuum stresses are obtained from the microplane stress components according to the principle of virtual work. The improvements include (1) a work-conjugate volumetric deviatoric split—the main improvement, facilitating physical interpretation of stress components; (2) additional horizontal boundaries (yield limits) for the normal and deviatoric microplane stress components, making it possible to control the curvature at the peaks of stress-strain curves; (3) an improved nonlinear frictional yield surface with plasticity asymptote; (4) a simpler and more effective fitting procedure with se- quential identification of material parameters; (5) a method to control the steepness and tail length of postpeak softening; and (6) damage modeling with a reduction of unloading stiffness and crack-closing boundary. The second part of this study, by Caner and Bazant, will present an algorithm for implementing the model in structural analysis programs and provide experimental verification and calibration by test data.

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

This paper represents Part II of a two-part study in which a new improved version of the microplane constitutive model for damage-plastic behavior of concrete in 3D is developed. In Part II, an explicit numerical algorithm for model M4 is formulated, the material parameters of model M4 are calibrated by optimum fitting of the basic test data available in the literature, and the model is verified by comparisons with these data. The data in which strain localization must have occurred are delocalized, and the size effect is filtered out from the data where necessary. Although model M4 contains many material parameters, all but four have fixed values for all types of concretes. Thus the user needs to adjust only four free material parameters to the data for a given concrete, for which a simple sequential identification procedure is developed. If the user's data consist only of the standard compression strength and the strain at uniaxial stress peak, the adjustment is explicit and immediate. Good agreement wi...

The formulation of microplane model M4 in Parts I and II is extended to rate dependence. Two types of rate effect in the nonlinear triaxial behavior of concrete are distinguished: (1) Rate dependence of fracturing (microcrack growth) associated with the activation energy of bond ruptures, and (2) creep (or viscoelasticity). Short-time linear creep (viscoelasticity) is approximated by a nonaging Maxwell spring-dashpot model calibrated so that its response at constant stress would be tangent to the compliance function of model B3 for a time delay characteristic of the problem at hand. An effective explicit algorithm for step-by-step finite-element analysis is formulated. The main reason that the rate dependence of fracturing must be taken into account is to simulate the sudden reversal of postpeak strain softening into hardening revealed by recent tests. The main reason that short-time creep (viscoelasticity) must be taken into account is to simulate the rate dependence of the initial and unloading stiffness. Good approximations of the rate effects observed in material testing are achieved. The model is suitable for finite-element analysis of impact, blast, earthquake, and short-time loads up to several hours duration.

The microplane material model for concrete, formulated mathematically in the preceding Part I, is here calibrated by material test data from all the typical laboratory tests taken from the literature. Then the model is verified by finite elements simulations of data for some characteristic tests with highly nonuniform strain fields. The scaling properties of model M7 are determined. With the volumetric stress effect taken from the previous load step, the M7 numerical algorithm is explicit, delivering in each load step the stress tensor from the strain tensor, with no iterative loop. This makes the model robust and suitable for large-scale finite element computations. There are 5 free, easily adjustable material parameters, which make it possible to match the given compressive strength, the corresponding strain, the given hydrostatic compression curve, and certain triaxial aspects. Besides, there are many fixed, hard-to-adjust, parameters, which can be taken the same for all concretes. The optimum values of material parameters are determined by fitting a particularly broad range of test results, including the important tests of compression-tension load cycles, mixed-mode fracture, tension-shear failure of double-edge-notched specimens, and vertex effect when axial compression is followed by torsion. Because of the lack of information on the material characteristic length or fracture energy, which can be obtained only by size effect tests on the same concrete, and on the precise boundary conditions and precise gauge locations, the finite element fitting of the present test data can hardly be expected to give better results than single-point simulations of specimens with approximately homogeneous strain states within the gauge length. Nevertheless, tensile test data with severe localization are delocalized on the basis of assumed material length. Model M7 is shown to fit a considerably broader range of test data than the preceding models M1–M6.Postprint (published version