Showing papers by "Zdenek P. Bazant published in 1988"

Nonlocal Continuum Damage, Localization Instability and Convergence

Abstract: A recent nonlocal damage formulation, in which the spatially averaged quantity was the energy dissipated due to strain-softening, is extended to a more general form in which the strain remains local while any variable that controls strain-softening is nonlocal. In contrast to the original imbricate nonlocal model for strain-softening, the stresses which figure in the constitutive relation satisfy the differential equations of equilibrium and boundary conditions of the usual classical form, and no zero-energy spurious modes of instability are encountered. However, the field operator for the present formulation is in general nonsymmetric, although not for the elastic part of response. It is shown that the energy dissipation and damage cannot localize into regions of vanishing volume. The static strain-localization instability, whose solution is reduced to an integral equation, is found to be controlled by the characteristic length of the material introduced in the averaging rule. The calculated static stability limits are close to those obtained in the previous nonlocal studies, as well as to those obtained by the crack band model in which the continuum is treated as local but the minimum size of the strain-softening region (localization region) is prescribed as a localization limiter. Furthermore, the rate of convergence of static finite-element solutions with nonlocal damage is studied and is found to be of a power type, almost quadratric. A smooth weighting function in the averaging operator is found to lead to a much better convergence than unsmooth functions.

Microplane Model for Brittle-Plastic Material: I. Theory

Abstract: A generalized microplane model for brittle-plastic heterogeneous materials such as concrete, which describes not only tensile cracking but also nonlinear triaxial response in compression and shear, is presented. The constitutive properties are characterized separately on planes of various orientations within the material, called the microplanes, on which there are only few stress and strain components and no tensorial, requirements need to be observed. These requirements are satisfied automatically by integration over all spatial directions. The state of each microplane is characterized by normal deviatoric and volumetric strains and shear strain, which makes it possible to match any Poisson ratio. The microplane strains are assumed to be the resolved components of the macroscopic strain tensor. The central assumption is that on the microplane level the stress-strain diagrams for monotonic loading are path-independent and that all the path dependence on the macrolevel is due to unloading, which happens selectively on microplanes of some orientations. The response on the microplane is assumed to depend on the lateral normal strain which does no work. in consequence, the incremental elastic moduli tensor is nonsymmetric, which is necessary to model friction and dilatancy. This tensor is also generally anistropic and fully populated (i.e., none of its elements can be prescribed as zero). The model involves many fewer free material parameters than the existing comprehensive macroscopic phenomenologic constitutive models for concrete.

Microplane Model for Brittle‐Plastic Material: II. Verification

Abstract: The general microplane model formulated in Part I of this study is calibrated and verified by comparison with numerous nonlinear triaxial test data from the literature and good agreement is achieved. The model involves many fewer material parameters than the previous phenomenologic macroscopic models. Numerical implementation is also considered.

Effect of Temperature and Humidity on Fracture Energy of Concrete

Abstract: Fracture experiments were conducted at temperatures from 20 to 200 C (68 to 392 F) to determine the dependence of the Mode I fracture energy of concrete on temperature as well as the specific water content. The fracture energy values were determined by testing geometrically similar specimens of sizes in the ratio 1:2:4:8 and then applying Bazant's size effect law. Three-point bend specimens and eccentric compression specimmns are found to yield approximately the same fracture energies, regardless of temperature. To describe the temperature dependence of fracture energy, a recently derived simple formula based on the activation energy theory (rate process theory) is used and verified by test results. The temperature effect is determined both for concrete predried in an oven and for wet (saturated) concrete. By interpolation, an approximate formula for the effect of moisture content on fracture energy is also obtained. This effect is found to be small at room temperature but large at temperatures close to 100 C (212 F).

Size effect in pullout tests

Abstract: The results of tests of the pullout strength of reinforcing bars embedded in concrete are reported. The test specimens are 1.5, 3, and 6-in. cubes with geometrically similar bars. The results are found to be consistent with Bazant's size effect law for the nominal stress at softening failures due to disturbed cracking. Based on the size effect law, an approximate formula predicting pullout strength is develoed.

Softening Instability: Part I—Localization Into a Planar Band

Abstract: Distributed damage such as cracking in heterogeneous brittle materials may be ap­ proximately described by a strain-softening continuum. To make analytical solu­ tions feasible, the continuum is assumed to be local but localization of softening strain into a region of vanishing volume is precluded by requiring that the softening region, assumed to be in a state of homogeneous strain, must have a certain minimum thickness which is a material property. Exact conditions of stability of an initially uniform strain field against strain localization are obtained for the case of an infinite layer in which the strain localizes into an infinite planar band. First, the problem is solved for small strain. Then a linearized incremental solution is obtained taking into account geometrical nonlinearity of strain. The stability condition is shown to depend on the ratio of the layer thickness to the softening band thickness. It is found that if this ratio is not too large compared to J, the state of homogeneous strain may be stable well into the softening range. Part II of this study applies Eshelby's theorem to determine the conditions of localization into ellipsoidal regions in infinite space, and also solves localization into circular or spherical regions in finite bodies.

Softening Instability: Part II—Localization Into Ellipsoidal Regions

Abstract: Extending the preceding study of exact solutions for finite-size strain-softening regions in layers and infinite space, exact solution of localization instability is ob­ tained for the localization of strain into an ellipsoidal region in an infinite solid. The solution exploits Eshelby's theorem for eigenstrains in elliptical inclusions in an in­ finite elastic solid. The special cases of localization of strain into a spherical region in three dimensions and into a circular region in two dimensions are further solved for finite solids-spheres in 3D and circles in 2D. The solutions show that even if the body is infinite the localization into finite regions of such shapes cannot take place at the start of strain-softening (a state corresponding to the peak of the stress-strain diagram) but at a finite strain-softening slope. If the size of the body relative to the size of the softening region is decreased and the boundary is restrained, homogeneous strain-softening remains stable into a larger strain. The results also can be used as checks for finite element programs for strain-softening. The present solutions determine only stability of equilibration states but not bifurcations of the equilibrium path.