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E. Smith - One of the best experts on this subject based on the ideXlab platform.
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The size of the fully developed Softening zone associated with a crack in a strain-Softening material. III, The coupled effects of Softening zone force Law and matrix fracture resistance
International Journal of Engineering Science, 2003Co-Authors: E. SmithAbstract:Abstract Earlier theoretical work (Parts I and II) has been concerned with the determination of the fully developed Softening zone size in a strain-Softening material for (a) a semi-infinite crack in a remotely loaded infinite solid, and (b) a long crack in a double cantilever beam specimen. The matrix was assumed to have zero fracture resistance and in both cases it was shown that the fully developed Softening zone size, though being critically dependent on the maximum stress and displacement within the Softening zone, was relatively insensitive to the details of the Softening Law. The present paper broadens the scope of this conclusion to include the more general, and indeed more practically appropriate, case where the matrix has a finite fracture resistance.
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The non-uniqueness of the maximum load J-value with a cohesive zone model of an elastic/Softening solid
Engineering Fracture Mechanics, 1995Co-Authors: E. SmithAbstract:Theoretical analyses, based on an idealised Softening Law representation, show the extent to which the maximum load J-value is non-unique for an elastic Softening solid having a positive geometry. The non-uniqueness arises as a consequence of the load maximum occurring prior to the attainment of a fully developed Softening zone.
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The elastically equivalent Softening zone size for an elastic-Softening material: II. A simple piece-wise Softening Law
Mechanics of Materials, 1994Co-Authors: E. SmithAbstract:Abstract The paper determines the elastically equivalent Softening zone size RE for an elastic-Softening material when there is a semi-infinite crack in a remotely loaded infinite solid; the parameter RE plays a central role in size effect expressions that are used to correlate the maximum loads that can be sustained by solids having different dimensions. The stress (p) versus displacement (v) Softening Law considered is that for which p = pc for o K IC = E O G I , where Eo is the reduced modulus and GI = λpcδc is the contribution to the specific fracture energy arising from the initially rapid Softening region. Analysis of a specific model demonstrates the viability of this approach by showing that there is consistency with the proposed size effect expressions based on RE.
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The compressive failure of notched composites: the effects of geometry and microbuckling Softening Law
International Journal of Fracture, 1994Co-Authors: E. SmithAbstract:The compressive failure of notched composites is modelled from the basis of the understanding that has been reached with regard to the behaviour of cracked elastic Softening solids that are subjected to tensile loadings. Particular attention is given to the effects of geometrical parameters and the Softening Law describing the behaviour of the microbuckling damage zone, and it is shown how the peak load can be related to a solid's geometrical parameters and the damage zone Softening Law via an analytical expression. The predictions are shown to be consistent with Sutcliffe and Fleck's numerical and experimental results for the compressive failure of centre notched panels of carbon fibre-epoxy laminates.
J Mendez - One of the best experts on this subject based on the ideXlab platform.
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crystal plasticity modeling of slip activity in ti 6al 4v under high cycle fatigue loading
International Journal of Plasticity, 2009Co-Authors: Florent Bridier, David L Mcdowell, P Villechaise, J MendezAbstract:Abstract Deformation micromechanisms of a Ti–6Al–4V alloy under fatigue loading at room temperature are studied using a three-dimensional crystal plasticity constitutive model. The model employs a minimum set of fitting parameters based on experimental data for Ti–6Al–4V. Single slip is strongly favored through a Softening Law that affects mainly the driving force for slip on the first activated slip system. Cyclic deformation behavior at the macroscopic scale and at the local scale of grains is analyzed through the simulation of 20 cycles of fatigue on a polycrystalline structure of 900 randomly oriented grains. The progressive activation of slip (basal, prismatic, and pyramidal) is analyzed and compared to experimental observations.
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Crystal plasticity modeling of slip activity in Ti–6Al–4V under high cycle fatigue loading
International Journal of Plasticity, 2009Co-Authors: Florent Bridier, David L Mcdowell, P Villechaise, J MendezAbstract:Abstract Deformation micromechanisms of a Ti–6Al–4V alloy under fatigue loading at room temperature are studied using a three-dimensional crystal plasticity constitutive model. The model employs a minimum set of fitting parameters based on experimental data for Ti–6Al–4V. Single slip is strongly favored through a Softening Law that affects mainly the driving force for slip on the first activated slip system. Cyclic deformation behavior at the macroscopic scale and at the local scale of grains is analyzed through the simulation of 20 cycles of fatigue on a polycrystalline structure of 900 randomly oriented grains. The progressive activation of slip (basal, prismatic, and pyramidal) is analyzed and compared to experimental observations.
Florent Bridier - One of the best experts on this subject based on the ideXlab platform.
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crystal plasticity modeling of slip activity in ti 6al 4v under high cycle fatigue loading
International Journal of Plasticity, 2009Co-Authors: Florent Bridier, David L Mcdowell, P Villechaise, J MendezAbstract:Abstract Deformation micromechanisms of a Ti–6Al–4V alloy under fatigue loading at room temperature are studied using a three-dimensional crystal plasticity constitutive model. The model employs a minimum set of fitting parameters based on experimental data for Ti–6Al–4V. Single slip is strongly favored through a Softening Law that affects mainly the driving force for slip on the first activated slip system. Cyclic deformation behavior at the macroscopic scale and at the local scale of grains is analyzed through the simulation of 20 cycles of fatigue on a polycrystalline structure of 900 randomly oriented grains. The progressive activation of slip (basal, prismatic, and pyramidal) is analyzed and compared to experimental observations.
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Crystal plasticity modeling of slip activity in Ti–6Al–4V under high cycle fatigue loading
International Journal of Plasticity, 2009Co-Authors: Florent Bridier, David L Mcdowell, P Villechaise, J MendezAbstract:Abstract Deformation micromechanisms of a Ti–6Al–4V alloy under fatigue loading at room temperature are studied using a three-dimensional crystal plasticity constitutive model. The model employs a minimum set of fitting parameters based on experimental data for Ti–6Al–4V. Single slip is strongly favored through a Softening Law that affects mainly the driving force for slip on the first activated slip system. Cyclic deformation behavior at the macroscopic scale and at the local scale of grains is analyzed through the simulation of 20 cycles of fatigue on a polycrystalline structure of 900 randomly oriented grains. The progressive activation of slip (basal, prismatic, and pyramidal) is analyzed and compared to experimental observations.
Milan Jirásek - One of the best experts on this subject based on the ideXlab platform.
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localization properties of strain Softening gradient plasticity models part ii theories with gradients of internal variables
International Journal of Solids and Structures, 2009Co-Authors: Milan Jirásek, Simon RolshovenAbstract:The second part of this paper compares and evaluates enhancements of the conventional plasticity theory by gradients of internal variables. Attention is focused on their performance as localization limiters. Both explicit and implicit gradient formulations are considered. It is shown that certain models suffer by serious mathematical deficiencies that would complicate their numerical implementation. Some other models are appropriate only at early stages of the Softening process but later exhibit locking accompanied by a spurious expansion of the localized plastic zone. The comparative study indicates that a convenient and robust tool for regularized modeling of the entire localization process is provided by the implicit gradient approach combined with a suitable form of the hardening/Softening Law.
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On regularized plasticity models for strain-Softening materials
2001Co-Authors: Simon Rolshoven, Milan JirásekAbstract:The paper analyzes and compares several regularization techniques for Softening plasticity. It is shown that a basic nonlocal plasticity model with a nonlocal cumulative plastic strain in the Softening Law provides only a partial regularization. As an alternative, a refined nonlocal plasticity model with a combination of local and nonlocal cumulative plastic strain is investigated in detail. An efficient numerical algorithm solving the nonlocal consistency condition is outlined and a convergence proof is given. Furthermore, the behavior of the refined nonlocal model is compared to gradient plasticity with a Softening Law dependent on the gradient of the Softening variable. The differences between plastic strain profiles localized inside the body and at the boundary are investigated and the cotTespondence between the boundary conditions in the gradient formulation and the rescaling of the weight function in the nonlocal formulation is discussed. A physical interpretation of the attractive or repulsive character of the boundary layer is suggested.
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Nonlocal models for damage and fracture: Comparison of approaches
International Journal of Solids and Structures, 1998Co-Authors: Milan JirásekAbstract:Abstract The paper analyzes nonlocal constitutive models used in simulations of damage and fracture processes of quasibrittle materials. A number of nonlocal formulations found in the literature are classified according to the type of variable subjected to nonlocal averaging. Analytical and numerical solutions of a simple one-dimensional localization problem are presented. It is shown that some of the formulations inevitably lead to residual stresses even at very late stages of the deformation process and, consequently, they are not capable of modeling complete separation in a widely open macroscopic crack. The mechanisms leading to this specific type of stress locking are explained based on a theoretical analysis of the nonlocal constitutive equations. It is also pointed out that the nonlocal approach distorts the shape of the stress-strain diagram, which has to be taken into account when designing an appropriate local Softening Law.
Zdeněk P Bažant - One of the best experts on this subject based on the ideXlab platform.
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eigenvalue method for computing size effect of cohesive cracks with residual stress with application to kink bands in composites
International Journal of Engineering Science, 2003Co-Authors: Goangseup Zi, Zdeněk P BažantAbstract:Abstract The previously developed eigenvalue method for computing the size effect of cohesive crack model is extended to the cohesive crack model with a finite residual stress. In this model, the structure size for which a specified relative length of kink-band corresponds to the maximum load is obtained as an eigenvalue of a homogeneous Fredholm integral equation. This new method is direct and much more efficient than the classical finite element approach in which the entire load-deflection history must be computed to obtain the maximum load. A secondary purpose of the paper is to apply the new method to the effect of structure size on the compressive strength of unidirectional fiber–polymer composites failing by propagation of kink-band with fiber microbuckling. The kink-band is simulated by a cohesive crack model with a linear compressive Softening Law and a finite residual stress. The simulation shows that the specimens tested have a negative–positive geometry, i.e., the energy release rate of the kink-band for a unit load first decreases but at a certain length of propagation begins to increase. Finally the effect of shape of the Softening Law of cohesive crack on the size effect curve is studied by using the new eigenvalue method. It is shown that, for a negative–positive geometry, the size effect on the peak load depends on the entire Softening curve if the specimens is not too small.
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Cohesive Crack Model with Rate-Dependent Opening and Viscoelasticity: II. Numerical Algorithm, Behavior and Size Effect
International Journal of Fracture, 1997Co-Authors: Zdeněk P BažantAbstract:In the preceding companion paper (Bažant and Li, 1995), the solution of an aging viscoelastic Law was structure containing a cohesive crack with a rate-dependent stress-displacement Softening Law was reduced to a system of one-dimensional integro-differential equations involving compliance functions for points on the crack faces and the load point. An effective numerical algorithm for solving these equations, which dramatically reduces the computer time compared to the general two-dimensional finite element solution, is presented. The behavior of the model for various loading conditions is studied. It is shown that the model can closely reproduce the available experimental data from fracture tests with different loading rates spanning several orders of magnitude, and tests with sudden changes of the loading rate. Influence of the loading rate on the size effect and brittleness is also analyzed and is shown to agree with experiments.
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Cohesive Crack with Rate-Dependent Opening and Viscoelasticity: I. Mathematical Model and Scaling
International Journal of Fracture, 1997Co-Authors: Zdeněk P BažantAbstract:The time dependence of fracture has two sources: (1) the viscoelasticity of material behavior in the bulk of the structure, and (2) the rate process of the breakage of bonds in the fracture process zone which causes the Softening Law for the crack opening to be rate-dependent. The objective of this study is to clarify the differences between these two influences and their role in the size effect on the nominal strength of stucture. Previously developed theories of time-dependent cohesive crack growth in a viscoelastic material with or without aging are extended to a general compliance formulation of the cohesive crack model applicable to structures such as concrete structures, in which the fracture process zone (cohesive zone) is large, i.e., cannot be neglected in comparison to the structure dimensions. To deal with a large process zone interacting with the structure boundaries, a boundary integral formulation of the cohesive crack model in terms of the compliance functions for loads applied anywhere on the crack surfaces is introduced. Since an unopened cohesive crack (crack of zero width) transmits stresses and is equivalent to no crack at all, it is assumed that at the outset there exists such a crack, extending along the entire future crack path (which must be known). Thus it is unnecessary to deal mathematically with a moving crack tip, which keeps the formulation simple because the compliance functions for the surface points of such an imagined preexisting unopened crack do not change as the actual front of the opened part of the cohesive crack advances. First the compliance formulation of the cohesive crack model is generalized for aging viscoelastic material behavior, using the elastic-viscoelastic analog (correspondence principle). The formulation is then enriched by a rate-dependent Softening Law based on the activation energy theory for the rate process of bond ruptures on the atomic level, which was recently proposed and validated for concrete but is also applicable to polymers, rocks and ceramics, and can be applied to ice if the nonlinear creep of ice is approximated by linear viscoelasticity. Some implications for the characteristic length, scaling and size effect are also discussed. The problems of numerical algorithm, size effect, roles of the different sources of time dependence and rate effect, and experimental verification are left for a subsequent companion paper.
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Cohesive crack modeling of influence of sudden changes in loading rate on concrete fracture
Engineering Fracture Mechanics, 1995Co-Authors: S. Tandon, Katherine T. Faber, Zdeněk P BažantAbstract:The results of an experimental study of a sudden change in loading rate on the fracture behavior of normal- and high-strength concrete specimens of three different sizes are reported. Geometrically similar three-point bend specimens were subjected to either a sudden 1000-fold increase or a 10-fold decrease of the loading rate. It was observed that for a large increase of the loading rate, the post-peak Softening can be reversed to hardening followed by a second peak of the stress-strain diagram. A sudden decrease of the loading rate initially causes, a steeper Softening slope of this diagram. The results are similar for normal and high strength concrete specimens. The viscoelastic cohesive crack model with the rate-dependent Softening Law is used to model the experimental results.
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Eigenvalue analysis of size effect for cohesive crack model
International Journal of Fracture, 1994Co-Authors: Zdeněk P BažantAbstract:The paper analyses the effect of structure size on the nominal strength of the structure that is implied by the cohesive (or fictitious) crack model proposed for concrete by Hillerborg et al. A new method to calculate the maximum load of geometrically similar structures of different sizes without calculating the entire load-deflection curves is presented. The problem is reduced to a matrix eigenvalue problem, in which the structure size for which the maximum load occurs at the given (relative) length of the cohesive crack is obtained as the smallest eigenvalue. Subsequently, the maximum load, nominal strength and load-point displacement are calculated from the matrix equilibrium equation. The nonlinearity of the Softening stress-displacement Law is handled by iteration. For a linear Softening Law, the eigenvalue problem is linear and independent of the matrix equilibrium equation, and the peak load can then be obtained without solving the equilibrium equation. The effect of the shape of the Softening Law is studied, and it is found that the size effect curve is not very sensitive to it. The generalized size effect Law proposed earlier by Bažant, which describes a transition between the horizontal and inclined asymptotes of strength theory and linear elastic fracture mechanics, is found to fit the numerical results very well. Finally some implications for the determination of fracture energy from the size effect tests are discussed. The results are of interest for quasibrittle materials such as concrete, rocks, sea ice and modern tough ceramics.