Recent Improvements to Relea III of the
K&C Concrete Model
Joph M. Magallanes1, Youcai Wu1, L. Javier Malvar2, and John E. Crawford1
无忧购物
1Karagozian & Ca, Burbank, CA
2Naval Facilities Engineering Service Center, Port Hueneme, CA
Abstract
Recent improvements are made to Relea III of the Karagozian & Ca (K&C) concrete model. This three-invariant plasticity and damage-bad constitutive model is widely ud to model a number of materials, including normal and lightweight concrete, concrete masonry, and brick masonry, to compute the effects of quasi-static, blast, and impact loads on structures. This most recent version of the model, made available starting with LS-DYNA®v971 as *MAT_CONCRETE_DAMAGE_REL3, incorporates a number of improvements to the original model that are described in this paper. The model now exhibits: (a) an automatic input capability for generating the data for generic concrete materials and (b) methods to reduce mesh-dependencies due to strain-softening. A simple method is i
mplemented to regularize the fracture energy by internally scaling the damage function for the generic concrete model parameters. For ur-defined material parameters, a method is developed that can prerve fracture energy using the results of either single-element or multi-element simulations. Finally, concrete loading rate effects are discusd and guidance is provided on properly modeling such effects with the model.
1.Introduction
The K&C concrete (KCC) material model was first relead in 1994 [1-3]. This first version reprented a significant overhaul of the concrete material model (Model 16) in the DYNA3D finite element (FE) program. This overhaul included (1) adding a third, independent failure surface bad on a Willam-Warnke three-invariant formulation (a change from the original two-invariant formulation), (2) introducing a radial stress path for the strain rate enhancement algorithm, (3) adding a fracture energy dependent strain in tension, and (4) fixing veral major discrepancies in the original model.
The cond relea of the KCC model was relead in 1996 [4]. The Relea II model extended the previous model to include shear dilation (i.e., increa in volume due to shearing). The formulation i
欢乐的节日ntroduced in this version allowed the model to be partially associative, fully associative, or non-associative, facilitating a more accurate reprentation of the behavior of reinforced concrete structures. In addition, the strain rate effect algorithm was modified to allow for implementation of different strain rate enhancement factors, or dynamic increa factors (DIFs), in tension and compression [5].
This third relea of the model [6], made available in LS-DYNA starting with v971 as *MAT_CONCRETE_DAMAGE_REL3 [7-8], incorporates a number of improvements to the original formulation, which are described in this paper. First, the automatic input capability for generating the model parameters for “generic” concrete materials is discusd. Second, methods included in the model to reduce mesh-dependencies due to strain-softening are described. Finally, concrete loading rate effects are discusd and guidance is provided on properly modeling rate effects with the model.
The principal advantage that this model provides, in comparison with other constitutive models ud for concrete-like materials, is that it is relatively simple and numerically robust. It is capable of reproducing key concrete behaviors critical to blast and impact analys and is also quite easily calibrated to laboratory data. The modeling capabilities afforded by the improvements in Relea III
are evidenced in numerous recent studies in the literature [9-12]. The discussions prented in this paper are especially warranted in light of recent material testing conducted by K&C and others and to further detail features of the model.
2. Automatic Parameter Generation
Under sponsorship of various U.S. Department of Defen agencies, extensive mechanical characterization tests were completed for veral concrete materials. Material tests included unconfined uniaxial compression and tension tests, triaxial compression tests under various levels of confinement, hydrostatic compression tests, and a number of strain path tests. The types of material characterization tests are described in [13]. Default values for the KCC model were derived bad on the and other data derived from the literature (e.g., e [14]). The default concrete exhibited an average compressive strength, 'c f , slightly over 45 MPa (or 6,500 psi). To generalize the KCC model for other concrete materials, its parameters are adjusted using the concrete strength parameters to obtain the appropriate relationships between the concrete properties, e.g., relationships between tensile and compressive strength, and between bulk modulus and compressive strength, among others [15]. For the deviatoric strength, the KCC model us a simple function to characterize three independent failure surfaces that define the yield, maximum, and resid
ual strength of the material. Three parameters i a 0, i a 1, and i a 2 (9 parameters total for the three surfaces) define each of the failure surfaces:
()p a a p a p F i i i i ⋅++=210 (1) where, p is the pressure (i.e., mean normal stress) and i F is the i th of three failure surfaces.
For hardening, the plasticity surface ud in the model is interpolated between the yield and maximum surfaces bad on the value of the damage parameter, λ. For softening, a similar interpolation is performed between the maximum and residual surfaces.
The parameters in Eqn. (1) were calibrated to the original data and provide a best fit for that material; the are designated o i a 0, o i a 1, and o i a 2. The scaling coefficient, r , is defined as the
ratio of the interpolated 'c f (i.e., the material strength one desires to model) to the compressive
strength of the original material characterized in the laboratory, 'co f , or: ''co
c f f r = (2) For any lection of 'c f , the failure surface parameters are scale
d using r as follows: r a a o i i ⋅=00 (3a)
一生只爱一人o i i a a 11= (3b) a a o i i 22= (3c)
The failure surface obtained from the generic concrete is shown in Fig. 1a for three different concret
e compressive strengths. The Equation of State (EOS) is also scaled using r and is shown in Fig. 1b for the same three concrete compressive strengths. Triaxial compression
stress path (50 MPa)
Unconfined compression stress path
一磅肉(a) Maximum failure surface. (b) Equation of state. Fig. 1. Strength and volumetric respons obtained from the generic concrete model.
The tension softening parameter, 2b , is also scaled using simple relationships for concrete. The 2b model parameter, which controls the behavior of the model in strain softening, is adjusted to obtain the fracture energies (f G ) recommended by the CEB [16]. The are shown in Table 1. In Relea III, the fracture can be entered in one of two ways. First, a ur may directly enter 2b values, using Table 1 as guidance. Alternatively, the model can internally compute estimates for the fracture energies of Table 1 by entering 'c f and the maximum aggregate size (MAS ).
This automatic parameter generation capability was developed to provide analysts with a simple tool from which to model concrete when little is known other than the concrete’s compressive strength. There are, naturally, a number of limitations to this feature:
舒涵• Recent experimental studies have shown that concrete’s shear strength can be quite variable, especially for confining pressures greater than 50 MPa [13, 17-19]. Bad on the studies, a concrete’s failure surface can be affected by a number of other factors including the porosity and moisture content. Unfortunately, there is currently insufficient data from which to formulate functions for the other parameter effects.提纲的格式
• The volumetric respon (i.e., the EOS) for concrete is also variable. Although material characterization tests are expensive and time-consuming, calibrating the KCC model to specific material characterization data may be warranted in some situations.
• The superficial similarities between concrete and a number of concrete-like materials such as concrete and brick masonry tempt many to u the generic concrete fit to model masonry structures. Although the materials are indeed similar, the behaviors of concrete and brick masonry have subtle but important differences. The main difference is the lack of coar aggregate in the latter—coar aggregate tends to slow down crack propagation resulting, for example, in higher tensile strengths. Concrete masonry tends to behave more akin to
Table 1. Mode I tensile fracture energies for concrete bad on CEB recommendations.
lightweight concrete, where the coar aggregate is often weaker than the cement paste, allowing crack propagation through the aggregates. A recent study showed that the KCC model can provide excellent results if properly calibrated for the materials [20].
3.Strain-softening
In the prence of strain-softening, FE predictions will not provide objective solutions unless localization limiters are introduced. A number of methods have been propod including direct length scale techniques (e.g., the “crack band” fracture model [21-22]), introduction of artificial viscosity or rate dependency [23], non-local methods [24-25], or microplane methods [26]. All of the methods have some limitations: the crack band assumes a localization width (which could be the element size), the rate dependency is limited to a rate range and los effectiveness in the quasi-static limit, non-local methods rely on a zone of influence, and the microplane method is limited by the angle between microplanes. Some of the approaches (e.g. the last two) can be computationally intensive; hence a simple method was implemented in the model. The crack band method was implemented in two ways in Relea III: the first is ud for the generic concrete model and the cond is intended for advanced u of the model where sufficient data is available from which to directly calibrate the model parameters.
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For the generic concrete model, the crack band is assumed to occur within one element. The regularization is accomplished by internally scaling the softening branch of the damage function, ()λη, using the average size of each element. Results illustrating the effectiveness of the approach are shown in Fig. 2. Here the results of single element simulations loaded in unconfined uniaxial tension (UUT) under velocity control are shown for four different element sizes. On the left of the figure, the stress versus strain respon of the elements shows that for small elements, the softening is slow, while for larger elements, the softening is accelerated. On the right, the fracture energy, which is obtained by integrating the stress versus displacement respon for each element, is plotted as a function of the displacement. Despite a small error, the fracture energies for each of the elements are very clo to that in Table 1. Note that this regularization has limitations: it is not implemented when the element size exceeds 250 mm (the softening branch becomes vertical and the energy dissipated can no longer be reduced to match f G ), or when the elements are much smaller than the localization width (when the softening branch may approach a plastic limit, resulting possibly in excessive energy dissipation, e.g., in penetration problems).
Vertical
velocity
Fig. 2. Single element respon in UUT using the generic concrete model (40 MPa, 16 mm MAS).
For ur-defined material parameters, this regularization scheme may be insufficient if the parameters of ()λη are significantly different from the generic concrete material. The fracture energy regularization surface (FERS) approach was developed to provide means from which to prerve fracture energy in such cas. The FERS is a three-dimensional respon surface, which the KCC model can u to compute a value for 2b that considers the element size and the ur-input f G value. The FERS is defined as:
3222231111222221112112221102x x x x x x x x b ⋅+⋅+⋅+⋅+⋅⋅+⋅+⋅+=ββββββββ (4)
where, 1x is the average element size (internally calculated for each element in LS-DYNA), 2x is the desired f G value, and i β are the parameters of the FERS. In practice, one may execute
numerous single-element or multi-element LS-DYNA simulations, using various mesh sizes and values for 2b , and the i β parameters for Eqn (4) can be obtained using standard regression
methods to obtain suitable FERS parameters.
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