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Mechanical behaviorThe model is a continuum, plasticity-based, damage model for concrete. It assumes that the main two failure mechanisms are tensile cracking and compressive crushing of the concrete material.
The evolution of the yield (or failure) surface is controlled by two hardening variables, and, linked to failure mechanisms under tension and compression loading, respectively. We refer to and as tensile and compressive equivalent plastic strains, respectively. The following sections discuss the main assumptions about the mechanical behavior of concrete. Under uniaxial tension the stress-strain response follows a linear elastic relationship until the value of the failure stress, is reached.
The failure stress corresponds to the onset of micro-cracking in the concrete material. Beyond the failure stress the formation of micro-cracks is represented macroscopically with a softening stress-strain response, which induces strain localization in the concrete structure. Under uniaxial compression the response is linear until the value of initial yield,.
In the plastic regime the response is typically characterized by stress hardening followed by strain softening beyond the ultimate stress,. This representation, although somewhat simplified, captures the main features of the response of concrete.It is assumed that the uniaxial stress-strain curves can be converted into stress versus plastic-strain curves. (This conversion is performed automatically by ABAQUS from the user-provided stress versus “inelastic” strain data, as explained below.) Thus. Where the subscripts t and c refer to tension and compression, respectively; and are the equivalent plastic strains, and are the equivalent plastic strain rates, is the temperature, and are other predefined field variables.As shown in, when the concrete specimen is unloaded from any point on the strain softening branch of the stress-strain curves, the unloading response is weakened: the elastic stiffness of the material appears to be damaged (or degraded).
The degradation of the elastic stiffness is characterized by two damage variables, and, which are assumed to be functions of the plastic strains, temperature, and field variables. Uniaxial cyclic behaviorUnder uniaxial cyclic loading conditions the degradation mechanisms are quite complex, involving the opening and closing of previously formed micro-cracks, as well as their interaction. Experimentally, it is observed that there is some recovery of the elastic stiffness as the load changes sign during a uniaxial cyclic test. The stiffness recovery effect, also known as the “unilateral effect,” is an important aspect of the concrete behavior under cyclic loading. The effect is usually more pronounced as the load changes from tension to compression, causing tensile cracks to close, which results in the recovery of the compressive stiffness.The concrete damaged plasticity model assumes that the reduction of the elastic modulus is given in terms of a scalar degradation variable as. ReinforcementIn ABAQUS reinforcement in concrete structures is typically provided by means of rebars, which are one-dimensional rods that can be defined singly or embedded in oriented surfaces. Rebars are typically used with metal plasticity models to describe the behavior of the rebar material and are superposed on a mesh of standard element types used to model the concrete.With this modeling approach, the concrete behavior is considered independently of the rebar.
Effects associated with the rebar/concrete interface, such as bond slip and dowel action, are modeled approximately by introducing some “tension stiffening” into the concrete modeling to simulate load transfer across cracks through the rebar. Details regarding tension stiffening are provided below.Defining the rebar can be tedious in complex problems, but it is important that this be done accurately since it may cause an analysis to fail due to lack of reinforcement in key regions of a model. See, for more information regarding rebars. Defining tension stiffeningThe postfailure behavior for direct straining is modeled with tension stiffening, which allows you to define the strain-softening behavior for cracked concrete. This behavior also allows for the effects of the reinforcement interaction with concrete to be simulated in a simple manner. Tension stiffening is required in the concrete damaged plasticity model. You can specify tension stiffening by means of a postfailure stress-strain relation or by applying a fracture energy cracking criterion.
ABAQUS will issue an error message if the calculated plastic strain values are negative and/or decreasing with increasing cracking strain, which typically indicates that the tensile damage curves are incorrect. In the absence of tensile damage.In cases with little or no reinforcement, the specification of a postfailure stress-strain relation introduces mesh sensitivity in the results, in the sense that the finite element predictions do not converge to a unique solution as the mesh is refined because mesh refinement leads to narrower crack bands. This problem typically occurs if cracking failure occurs only at localized regions in the structure and mesh refinement does not result in the formation of additional cracks.
If cracking failure is distributed evenly (either due to the effect of rebar or due to the presence of stabilizing elastic material, as in the case of plate bending), mesh sensitivity is less of a concern.In practical calculations for reinforced concrete, the mesh is usually such that each element contains rebars. The interaction between the rebars and the concrete tends to reduce the mesh sensitivity, provided that a reasonable amount of tension stiffening is introduced in the concrete model to simulate this interaction. This requires an estimate of the tension stiffening effect, which depends on such factors as the density of reinforcement, the quality of the bond between the rebar and the concrete, the relative size of the concrete aggregate compared to the rebar diameter, and the mesh. A reasonable starting point for relatively heavily reinforced concrete modeled with a fairly detailed mesh is to assume that the strain softening after failure reduces the stress linearly to zero at a total strain of about 10 times the strain at failure. The strain at failure in standard concretes is typically 10 4, which suggests that tension stiffening that reduces the stress to zero at a total strain of about 10 3 is reasonable. This parameter should be calibrated to a particular case.The choice of tension stiffening parameters is important since, generally, more tension stiffening makes it easier to obtain numerical solutions.
Too little tension stiffening will cause the local cracking failure in the concrete to introduce temporarily unstable behavior in the overall response of the model. Few practical designs exhibit such behavior, so that the presence of this type of response in the analysis model usually indicates that the tension stiffening is unreasonably low. Fracture energy cracking criterionWhen there is no reinforcement in significant regions of the model, the tension stiffening approach described above will introduce unreasonable mesh sensitivity into the results. However, it is generally accepted that Hillerborg's (1976) fracture energy proposal is adequate to allay the concern for many practical purposes. Hillerborg defines the energy required to open a unit area of crack, as a material parameter, using brittle fracture concepts. With this approach the concrete's brittle behavior is characterized by a stress-displacement response rather than a stress-strain response. Under tension a concrete specimen will crack across some section.
After it has been pulled apart sufficiently for most of the stress to be removed (so that the undamaged elastic strain is small), its length will be determined primarily by the opening at the crack. The opening does not depend on the specimen's length.This fracture energy cracking model can be invoked by specifying the postfailure stress as a tabular function of cracking displacement, as shown in. The cracking displacement at which complete loss of strength takes place is, therefore,.
Typical values of range from 40 N/m (0.22 lb/in) for a typical construction concrete (with a compressive strength of approximately 20 MPa, 2850 lb/in 2) to 120 N/m (0.67 lb/in) for a high-strength concrete (with a compressive strength of approximately 40 MPa, 5700 lb/in 2).If tensile damage, is specified, ABAQUS automatically converts the cracking displacement values to “plastic” displacement values using the relationship. ImplementationThe implementation of this stress-displacement concept in a finite element model requires the definition of a characteristic length associated with an integration point. The characteristic crack length is based on the element geometry: for beams and trusses we use the integration point length; for shell and planar elements we use the square root of the integration point area; for solid elements we use the cube root of the integration point volume.
This definition of the characteristic crack length is used because the direction in which cracking occurs is not known in advance. Therefore, elements with large aspect ratios will have rather different behavior depending on the direction in which they crack: some mesh sensitivity remains because of this effect, and elements that have aspect ratios close to one are recommended.
Defining compressive behaviorYou can define the stress-strain behavior of plain concrete in uniaxial compression outside the elastic range. Compressive stress data are provided as a tabular function of inelastic (or crushing) strain, and, if desired, strain rate, temperature, and field variables. Positive (absolute) values should be given for the compressive stress and strain. The stress-strain curve can be defined beyond the ultimate stress, into the strain-softening regime.Hardening data are given in terms of an inelastic strain, instead of plastic strain,. The compressive inelastic strain is defined as the total strain minus the elastic strain corresponding to the undamaged material, where, as illustrated in. Defining damage and stiffness recoveryDamage, and/or, can be specified in tabular form.
(If damage is not specified, the model behaves as a plasticity model; consequently, and.)In ABAQUS the damage variables are treated as non-decreasing material point quantities. At any increment during the analysis, the new value of each damage variable is obtained as the maximum between the value at the end of the previous increment and the value corresponding to the current state (interpolated from the user-specified tabular data); that is. Stiffness recoveryAs discussed above, stiffness recovery is an important aspect of the mechanical response of concrete under cyclic loading. ABAQUS allows direct user specification of the stiffness recovery factors and.The experimental observation in most quasi-brittle materials, including concrete, is that the compressive stiffness is recovered upon crack closure as the load changes from tension to compression. On the other hand, the tensile stiffness is not recovered as the load changes from compression to tension once crushing micro-cracks have developed.
This behavior, which corresponds to and, is the default used by ABAQUS. Illustrates a uniaxial load cycle assuming the default behavior. Rate dependenceThe rate-sensitive behavior of quasi-brittle materials is mainly connected to the retardation effects that high strain rates have on the growth of micro-cracks. The effect is usually more pronounced under tensile loading. As the strain rate increases, the stress-strain curves exhibit decreasing nonlinearity as well as an increase in the peak strength. You can specify tension stiffening as a tabular function of cracking strain (or displacement) rate, and you can specify compression hardening data as a tabular function of inelastic strain rate. Is a parameter, referred to as the eccentricity, that defines the rate at which the function approaches the asymptote (the flow potential tends to a straight line as the eccentricity tends to zero).This flow potential, which is continuous and smooth, ensures that the flow direction is always uniquely defined.
The function approaches the linear Drucker-Prager flow potential asymptotically at high confining pressure stress and intersects the hydrostatic pressure axis at 90°. See, for further discussion of this potential.The default flow potential eccentricity is, which implies that the material has almost the same dilation angle over a wide range of confining pressure stress values. Increasing the value of provides more curvature to the flow potential, implying that the dilation angle increases more rapidly as the confining pressure decreases.
Values of that are significantly less than the default value may lead to convergence problems if the material is subjected to low confining pressures because of the very tight curvature of the flow potential locally where it intersects the -axis. Nonassociated flowBecause plastic flow is nonassociated, the use of concrete damaged plasticity results in a nonsymmetric material stiffness matrix.
Therefore, to obtain an acceptable rate of convergence in ABAQUS/Standard, the unsymmetric matrix storage and solution scheme should be used. ABAQUS/Standard will automatically activate the unsymmetric solution scheme if concrete damaged plasticity is used in the analysis. If desired, you can turn off the unsymmetric solution scheme for a particular step (see ). Viscoplastic regularizationMaterial models exhibiting softening behavior and stiffness degradation often lead to severe convergence difficulties in implicit analysis programs, such as ABAQUS/Standard.
A common technique to overcome some of these convergence difficulties is the use of a viscoplastic regularization of the constitutive equations, which causes the consistent tangent stiffness of the softening material to become positive for sufficiently small time increments.The concrete damaged plasticity model can be regularized in ABAQUS/Standard using viscoplasticity by permitting stresses to be outside of the yield surface. We use a generalization of the Duvaut-Lions regularization, according to which the viscoplastic strain rate tensor, is defined as. Using the viscoplastic regularization with a small value for the viscosity parameter (small compared to the characteristic time increment) usually helps improve the rate of convergence of the model in the softening regime, without compromising results. The basic idea is that the solution of the viscoplastic system relaxes to that of the inviscid case as, where represents time. You can specify the value of the viscosity parameter as part of the concrete damaged plasticity material behavior definition. If the viscosity parameter is different from zero, output results of the plastic strain and stiffness degradation refer to the viscoplastic values,.
In ABAQUS/Standard the default value of the viscosity parameter is zero, so that no viscoplastic regularization is performed. Visualization of “crack directions”Unlike concrete models based on the smeared crack approach, the concrete damaged plasticity model does not have the notion of cracks developing at the material integration point.
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However, it is possible to introduce the concept of an effective crack direction with the purpose of obtaining a graphical visualization of the cracking patterns in the concrete structure. Different criteria can be adopted within the framework of scalar-damage plasticity for the definition of the direction of cracking. Following Lubliner et. al. (1989), we can assume that cracking initiates at points where the tensile equivalent plastic strain is greater than zero, and the maximum principal plastic strain is positive.
The direction of the vector normal to the crack plane is assumed to be parallel to the direction of the maximum principal plastic strain. This direction can be viewed in the Visualization module of ABAQUS/CAE. ElementsABAQUS offers a variety of elements for use with the concrete damaged plasticity model: truss, shell, plane stress, plane strain, generalized plane strain, axisymmetric, and three-dimensional elements.
Most beam elements can be used; however, beam elements in space that include shear stress caused by torsion (i.e., not thin-walled, open sections) and do not include hoop stress (i.e., not PIPE elements) cannot be used.For general shell analysis more than the default number of five integration points through the thickness of the shell should be used; nine thickness integration points are commonly used to model progressive failure of the concrete through the thickness with acceptable accuracy.
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