김경종
23 KiB
line
| u_n | σ_t^I |
|---|---|
| 0 | σ_t^I |
| u_n | σ_t^I |
Figure 23.6.2–3 Postfailure stress-fracture energy curve.
Use the following option to specify the postfailure stress as a tabular function of the fracture energy:
*BRITTLE CRACKING, TYPE=GFI
Abaqus/CAE Usage: Property module: material editor: Mechanical→Brittle Cracking: Type: Displacement or GFI
Characteristic crack length
The implementation of the stress-displacement concept in a finite element model requires the definition of a characteristic length associated with a material point. The characteristic crack length is based on the element geometry and formulation: it is a typical length of a line across an element for a first-order element; it is half of the same typical length for a second-order element. For beams and trusses it is a characteristic length along the element axis. For membranes and shells it is a characteristic length in the reference surface. For axisymmetric elements it is a characteristic length in the r–z plane only. For cohesive elements it is equal to the constitutive thickness. We use this definition of the characteristic crack length because the direction in which cracks will occur 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. Elements that are as close to square as possible are, therefore, recommended unless you can predict the direction in which cracks will form. Alternatively, this mesh dependency could be reduced by directly specifying the characteristic length as a function of the element topology and material orientation in user subroutine VUCHARLENGTH (see “Defining the characteristic element length at a material point in Abaqus/Explicit” in “Material data definition,” Section 21.1.2).
Shear retention model
An important feature of the cracking model is that, whereas crack initiation is based on Mode I fracture only, postcracked behavior includes Mode II as well as Mode I. The Mode II shear behavior is based on the common observation that the shear behavior depends on the amount of crack opening. More specifically, the cracked shear modulus is reduced as the crack opens. Therefore, Abaqus/Explicit offers
a shear retention model in which the postcracked shear stiffness is defined as a function of the opening strain across the crack; the shear retention model must be defined in the cracking model, and zero shear retention should not be used.
In these models the dependence is defined by expressing the postcracking shear modulus, G _ { c } , a s a fraction of the uncracked shear modulus:
G _ {c} = \rho (e _ {n n} ^ {c k}) G,
where G is the shear modulus of the uncracked material and the shear retention factor, \rho ( e _ { n n } ^ { c k } ) , depends on the crack opening strain, e _ { n n } ^ { c k } . You can specify this dependence in piecewise linear form, as shown in Figure 23.6.2–4.
Figure 23.6.2–4 Piecewise linear form of the shear retention model.
Alternatively, shear retention can be defined in the power law form:
\rho (e _ {n n} ^ {c k}) = \left(1 - \frac {e _ {n n} ^ {c k}}{e _ {m a x} ^ {c k}}\right) ^ {p},
where p and e _ { m a x } ^ { c k } are material parameters. This form, shown in Figure 23.6.2–5, satisfies the requirements that \rho 1 as e _ { n n } ^ { c k } \ \to \ 0 (corresponding to the state before crack initiation) and \rho \to 0 as e _ { n n } ^ { c k } \to e _ { m a x } ^ { c k } (corresponding to complete loss of aggregate interlock). See “A cracking model for concrete and other brittle materials,” Section 4.5.3 of the Abaqus Theory Guide, for a discussion of how shear retention is calculated in the case of two or more cracks.
Input File Usage:
Use the following option to specify the piecewise linear form of the shear retention model:
\ast \mathrm { B R I T T L E ~ S H E A R } , \mathrm { T Y P E = R E T E N T I O N ~ F A C T O R }
Use the following option to specify the power law form of the shear retention model:
\mathrm { * B R I T I L E ~ S H E A R , ~ T Y P E { = } P O W E R ~ L A W }
line
| e_nn^ck | p=1 | p=2 | p=5 |
|---|---|---|---|
| 0 | 1 | 1 | 1 |
| e_ck_max | 0 | 0 | 0 |
Figure 23.6.2–5 Power law form of the shear retention model.
Abaqus/CAE Usage: Property module: material editor: Mechanical→Brittle Cracking: Suboptions→Brittle Shear Type: Retention Factor or Power Law
Calibration
One experiment, a uniaxial tension test, is required to calibrate the simplest version of the brittle cracking model. Other experiments may be required to gain accuracy in postfailure behavior.
Uniaxial tension test
This test is difficult to perform because it is necessary to have a very stiff testing machine to record the postcracking response. Quite often such equipment is not available; in this situation you must make an assumption about the tensile failure strength of the material and the postcracking response. For concrete the assumption usually made is that the tensile strength is 7–10% of the compressive strength. Uniaxial compression tests can be performed much more easily, so the compressive strength of concrete is usually known.
Postcracking tensile behavior
The values given for tension stiffening are a very important aspect of simulations using the Abaqus/Explicit brittle cracking model. The postcracking tensile response is highly dependent on the reinforcement present in the concrete. In simulations of unreinforced concrete, the tension stiffening models that are based on fracture energy concepts should be utilized. If reliable experimental data are not available, typical values that can be used were discussed before: common values of G _ { f } ^ { I } range from 40 N/m (0.22 lb/in) for a typical construction concrete (with a compressive strength of approximately 20 MPa, 2850 lb/in2 ) to 120 N/m (0.67 lb/in) for a high-strength concrete (with a compressive strength of approximately 40 MPa, 5700 lb/in2 ). In simulations of reinforced concrete the stress-strain tension stiffening model should be used; the amount of tension stiffening depends on the reinforcement present,
as discussed before. 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 about ten times the strain at failure. Since the strain at failure in standard concretes is typically 10−4 , this suggests that tension stiffening that reduces the stress to zero at a total strain of about 1 0 ^ { - 3 } is reasonable. This parameter should be calibrated to each particular case.
Postcracking shear behavior
Calibration of the postcracking shear behavior requires combined tension and shear experiments, which are difficult to perform. If such test data are not available, a reasonable starting point is to assume that the shear retention factor, , goes linearly to zero at the same crack opening strain used for the tension stiffening model.
Brittle failure criterion
You can define brittle failure of the material. When one, two, or all three local direct cracking strain (displacement) components at a material point reach the value defined as the failure strain (displacement), the material point fails and all the stress components are set to zero. If all of the material points in an element fail, the element is removed from the mesh. For example, removal of a first-order reducedintegration solid element takes place as soon as its only integration point fails. However, all throughthe-thickness integration points must fail before a shell element is removed from the mesh.
If the postfailure relation is defined in terms of stress versus strain, the failure strain must be given as the failure criterion. If the postfailure relation is defined in terms of stress versus displacement or stress versus fracture energy, the failure displacement must be given as the failure criterion. The failure strain (displacement) can be specified as a function of temperature and/or predefined field variables.
You can control how many cracks at a material point must fail before the material point is considered to have failed; the default is one crack. The number of cracks that must fail can only be one for beam and truss elements; it cannot be greater than two for plane stress and shell elements; and it cannot be greater than three otherwise.
Input File Usage: *BRITTLE FAILURE, CRACKS=n
Abaqus/CAE Usage: Property module: material editor:
Mechanical→Brittle Cracking: Suboptions→Brittle Failure and select
Failure Criteria: Unidirectional, Bidirectional, or Tridirectional to
indicate the number of cracks that must fail for the material point to fail.
Determining when to use the brittle failure criterion
The brittle failure criterion is a crude way of modeling failure in Abaqus/Explicit and should be used with care. The main motivation for including this capability is to help in computations where not removing an element that can no longer carry stress may lead to excessive distortion of that element and subsequent premature termination of the simulation. For example, in a monotonically loaded structure whose failure mechanism is expected to be dominated by a single tensile macrofracture (Mode I cracking), it may be reasonable to use the brittle failure criterion to remove elements. On the other hand, the fact that the brittle material loses its ability to carry tensile stress does not preclude it from withstanding compressive stress;
therefore, it may not be appropriate to remove elements if the material is expected to carry compressive loads after it has failed in tension. An example may be a shear wall subjected to cyclic loading as a result of some earthquake excitation; in this case cracks that develop completely under tensile stress will be able to carry compressive stress when load reversal takes place.
Thus, the effective use of the brittle failure criterion relies on you having some knowledge of the structural behavior and potential failure mechanism. The use of the brittle failure criterion based on an incorrect user assumption of the failure mechanism will generally result in an incorrect simulation.
Selecting the number of cracks that must fail before the material point is considered to have failed
When you define brittle failure, you can control how many cracks must open to beyond the failure value before a material point is considered to have failed. The default number of cracks (one) should be used for most structural applications where failure is dominated by Mode I type cracking. However, there are cases in which you should specify a higher number because multiple cracks need to form to develop the eventual failure mechanism. One example may be an unreinforced, deep concrete beam where the failure mechanism is dominated by shear; in this case it is possible that two cracks need to form at each material point for the shear failure mechanism to develop.
Again, the appropriate choice of the number of cracks that must fail relies on your knowledge of the structural and failure behaviors.
Using brittle failure with rebar
It is possible to use the brittle failure criterion in brittle cracking elements for which rebar are also defined; the obvious application is the modeling of reinforced concrete. When such elements fail according to the brittle failure criterion, the brittle cracking contribution to the element stress carrying capacity is removed but the rebar contribution to the element stress carrying capacity is not removed. However, if you also include shear failure in the rebar material definition, the rebar contribution to the element stress carrying capacity will also be removed if the shear failure criterion specified for the rebar is satisfied. This allows the modeling of progressive failure of an under-reinforced concrete structure where the concrete fails first followed by ductile failure of the reinforcement.
Elements
Abaqus/Explicit offers a variety of elements for use with the cracking model: truss; shell; two-dimensional beam; and plane stress, plane strain, axisymmetric, and three-dimensional continuum elements. The model cannot be used with pipe and three-dimensional beam elements. Plane triangular, triangular prism, and tetrahedral elements are not recommended for use in reinforced concrete analysis since these elements do not support the use of rebar.
Output
In addition to the standard output identifiers available in Abaqus/Explicit (see “Abaqus/Explicit output variable identifiers,” Section 4.2.2), the following output variables relate directly to material points that use the brittle cracking model:
| CKE | All cracking strain components. |
| CKLE | All cracking strain components in local crack axes. |
| CKEMAG | Cracking strain magnitude. |
| CKLS | All stress components in local crack axes. |
| CRACK | Crack orientations. |
| CKSTAT | Crack status of each crack. |
| STATUS | Status of element (brittle failure model). The status of an element is 1.0 if the element is active and 0.0 if the element is not. |
Additional reference
• Hillerborg, A., M. Modeer, and P. E. Petersson, “Analysis of Crack Formation and Crack Growth in Concrete by Means of Fracture Mechanics and Finite Elements,” Cement and Concrete Research, vol. 6, pp. 773–782, 1976.
23.6.3 CONCRETE DAMAGED PLASTICITY
Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE
References
• “Material library: overview,” Section 21.1.1
• “Inelastic behavior,” Section 23.1.1
• *CONCRETE DAMAGED PLASTICITY
• *CONCRETE TENSION STIFFENING
• *CONCRETE COMPRESSION HARDENING
• *CONCRETE TENSION DAMAGE
• *CONCRETE COMPRESSION DAMAGE
• “Defining concrete damaged plasticity” in “Defining plasticity,” Section 12.9.2 of the Abaqus/CAE User’s Guide, in the HTML version of this guide
Overview
The concrete damaged plasticity model in Abaqus:
• provides a general capability for modeling concrete and other quasi-brittle materials in all types of structures (beams, trusses, shells, and solids);
• uses concepts of isotropic damaged elasticity in combination with isotropic tensile and compressive plasticity to represent the inelastic behavior of concrete;
• can be used for plain concrete, even though it is intended primarily for the analysis of reinforced concrete structures;
• can be used with rebar to model concrete reinforcement;
• is designed for applications in which concrete is subjected to monotonic, cyclic, and/or dynamic loading under low confining pressures;
• consists of the combination of nonassociated multi-hardening plasticity and scalar (isotropic) damaged elasticity to describe the irreversible damage that occurs during the fracturing process;
• allows user control of stiffness recovery effects during cyclic load reversals;
• can be defined to be sensitive to the rate of straining;
• can be used in conjunction with a viscoplastic regularization of the constitutive equations in Abaqus/Standard to improve the convergence rate in the softening regime;
• requires that the elastic behavior of the material be isotropic and linear (see “Defining isotropic elasticity” in “Linear elastic behavior,” Section 22.2.1); and
• is defined in detail in “Damaged plasticity model for concrete and other quasi-brittle materials,” Section 4.5.2 of the Abaqus Theory Guide.
See “Inelastic behavior,” Section 23.1.1, for a discussion of the concrete models available in Abaqus.
Mechanical behavior
The 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, \tilde { \varepsilon } _ { t } ^ { p l } and \tilde { \varepsilon } _ { c } ^ { p l } , linked to failure mechanisms under tension and compression loading, respectively. We refer to \tilde { \varepsilon } _ { t } ^ { p l } and \tilde { \varepsilon } _ { c } ^ { p l } a s tensile and compressive equivalent plastic strains, respectively. The following sections discuss the main assumptions about the mechanical behavior of concrete.
Uniaxial tension and compression stress behavior
The model assumes that the uniaxial tensile and compressive response of concrete is characterized by damaged plasticity, as shown in Figure 23.6.3–1. Under uniaxial tension the stress-strain response follows a linear elastic relationship until the value of the failure stress, \sigma _ { t 0 } , 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, \sigma _ { c 0 } . In the plastic regime the response is typically characterized by stress hardening followed by strain softening beyond the ultimate stress, \sigma _ { c u } . 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,
\sigma_ {t} = \sigma_ {t} (\tilde {\varepsilon} _ {t} ^ {p l}, \dot {\tilde {\varepsilon}} _ {t} ^ {p l}, \theta , f _ {i}),
\sigma_ {c} = \sigma_ {c} (\tilde {\varepsilon} _ {c} ^ {p l}, \dot {\tilde {\varepsilon}} _ {c} ^ {p l}, \theta , f _ {i}),
where the subscripts t and c refer to tension and compression, respectively; \tilde { \varepsilon } _ { t } ^ { p l } and \tilde { \varepsilon } _ { c } ^ { p l } are the equivalent plastic strains, \boldsymbol { \dot { \tilde { \varepsilon } } } _ { t } ^ { p l } :pl and \dot { \tilde { \varepsilon } } _ { c } ^ { p l } :pl are the equivalent plastic strain rates, is the temperature, and f _ { i } , ( i = 1 , 2 , . . . ) （2 are other predefined field variables.
As shown in Figure 23.6.3–1, 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, d _ { t } and d _ { c } , which are assumed to be functions of the plastic strains, temperature, and field variables:
d _ {t} = d _ {t} (\tilde {\varepsilon} _ {t} ^ {p l}, \theta , f _ {i}); 0 \leq d _ {t} \leq 1,
d _ {c} = d _ {c} (\tilde {\varepsilon} _ {c} ^ {p l}, \theta , f _ {i}); 0 \leq d _ {c} \leq 1.
The damage variables can take values from zero, representing the undamaged material, to one, which represents total loss of strength.
line
| ε_t | σ_t |
|---|---|
| ε_t^pl | 0 |
| ε_t^el | σ_t0 |
| ε_t^el | (1 - d_t)E_0 |
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| ε_c | σ_c |
|---|---|
| ε̃_c^pl | σ_c0 |
| ε̃_c^el | σ_cu |
| ε_c | σ_c0 |
Figure 23.6.3–1 Response of concrete to uniaxial loading in tension (a) and compression (b).
If E _ { 0 } is the initial (undamaged) elastic stiffness of the material, the stress-strain relations under uniaxial tension and compression loading are, respectively:
\sigma_ {t} = (1 - d _ {t}) E _ {0} (\varepsilon_ {t} - \tilde {\varepsilon} _ {t} ^ {p l}),
\sigma_ {c} = (1 - d _ {c}) E _ {0} (\varepsilon_ {c} - \tilde {\varepsilon} _ {c} ^ {p l}).
We define the “effective” tensile and compressive cohesion stresses as
\begin{array}{l} \bar {\sigma} _ {t} = \frac {\sigma_ {t}}{(1 - d _ {t})} = E _ {0} (\varepsilon_ {t} - \tilde {\varepsilon} _ {t} ^ {p l}), \\ \bar {\sigma} _ {c} = \frac {\sigma_ {c}}{(1 - d _ {c})} = E _ {0} (\varepsilon_ {c} - \tilde {\varepsilon} _ {c} ^ {p l}). \\ \end{array}
The effective cohesion stresses determine the size of the yield (or failure) surface.
Uniaxial cyclic behavior
Under 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 d as
E = (1 - d) E _ {0},
where E _ { 0 } is the initial (undamaged) modulus of the material.
This expression holds both in the tensile ( \sigma _ { 1 1 } > 0 ) and the compressive ( \sigma _ { 1 1 } < 0 ) sides of the cycle. The stiffness degradation variable, d , is a function of the stress state and the uniaxial damage variables, d _ { t } and d _ { c } . For the uniaxial cyclic conditions Abaqus assumes that
(1 - d) = (1 - s _ {t} d _ {c}) (1 - s _ {c} d _ {t}),
where s _ { t } and s _ { c } are functions of the stress state that are introduced to model stiffness recovery effects associated with stress reversals. They are defined according to
\begin{array}{l} s _ {t} = 1 - w _ {t} r ^ {*} (\sigma_ {1 1}); 0 \leq w _ {t} \leq 1, \\ s _ {c} = 1 - w _ {c} (1 - r ^ {*} (\sigma_ {1 1})); \quad 0 \leq w _ {c} \leq 1, \\ \end{array}
where
r ^ {*} (\sigma_ {1 1}) = H (\sigma_ {1 1}) = \left\{ \begin{array}{l l} 1 & \mathrm{if} \quad \sigma_ {1 1} > 0 \\ 0 & \mathrm{if} \quad \sigma_ {1 1} < 0 \end{array} \right.
The weight factors w _ { t } and w _ { c } , which are assumed to be material properties, control the recovery of the tensile and compressive stiffness upon load reversal. To illustrate this, consider the example in Figure 23.6.3–2, where the load changes from tension to compression. Assume that there was no previous compressive damage (crushing) in the material; that is, \tilde { \varepsilon } _ { c } ^ { p l } = 0 and d _ { c } = 0 . Then