김경종
364 lines
23 KiB
Markdown
364 lines
23 KiB
Markdown
<!-- source-page: 471 -->
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<details>
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<summary>text_image</summary>
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σt
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σt0
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E0
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(1-dt)E0
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wc = 1
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wc = 0
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εt
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</details>
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Figure 23.6.3–2 Illustration of the effect of the compression stiffness recovery parameter $w _ { c }$
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$$
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(1 - d) = (1 - s _ {c} d _ {t}) = (1 - (1 - w _ {c} (1 - r ^ {*})) d _ {t}).
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$$
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• In tension $( \sigma _ { 1 1 } > 0 ) , r ^ { * } = 1$ ; therefore, $d = d _ { t }$ as expected.
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• In compression $( \sigma _ { 1 1 } < 0 ) , r ^ { * } = 0$ , and $d = ( 1 - w _ { c } ) d _ { t }$ . If $w _ { c } ~ = ~ 1$ , then $d = 0 ;$ ; therefore, the material fully recovers the compressive stiffness (which in this case is the initial undamaged stiffness, $E = E _ { 0 } )$ . If, on the other hand, $w _ { c } = 0$ , then $d = d _ { t }$ and there is no stiffness recovery. Intermediate values of $w _ { c }$ result in partial recovery of the stiffness.
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# Multiaxial behavior
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The stress-strain relations for the general three-dimensional multiaxial condition are given by the scalar damage elasticity equation:
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$$
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\pmb {\sigma} = (1 - d) \mathbf {D} _ {0} ^ {e l}: (\pmb {\varepsilon} - \pmb {\varepsilon} ^ {p l}),
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$$
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where $\mathbf { D } _ { 0 } ^ { e l }$ is the initial (undamaged) elasticity matrix.
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The previous expression for the scalar stiffness degradation variable, $d ,$ is generalized to the multiaxial stress case by replacing the unit step function $r ^ { * } { \bigl ( } \sigma _ { 1 1 } { \bigr ) }$ with a multiaxial stress weight factor, $r ( \hat { \pmb { \sigma } } )$ , defined as
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$$
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r (\hat {\pmb {\sigma}}) = \frac {\sum_ {i = 1} ^ {3} \langle \hat {\sigma} _ {i} \rangle}{\sum_ {i = 1} ^ {3} | \hat {\sigma} _ {i} |}; 0 \leq r (\hat {\pmb {\sigma}}) \leq 1,
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$$
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where $\hat { \sigma } _ { i } ~ ( i = 1 , 2 , 3 )$ are the principal stress components. The Macauley bracket is defined by $\langle x \rangle = { \textstyle { \frac { 1 } { 2 } } } ( | x | + x )$ .
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See “Damaged plasticity model for concrete and other quasi-brittle materials,” Section 4.5.2 of the Abaqus Theory Guide, for further details of the constitutive model.
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# Reinforcement
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In 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.
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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.
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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 “Defining rebar as an element property,” Section 2.2.4, for more information regarding rebars.
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# Defining tension stiffening
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The 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.
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# Postfailure stress-strain relation
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In reinforced concrete the specification of postfailure behavior generally means giving the postfailure stress as a function of cracking strain, $\tilde { \varepsilon } _ { t } ^ { c k }$ . The cracking strain is defined as the total strain minus the elastic strain corresponding to the undamaged material; that is, $\tilde { \varepsilon } _ { t } ^ { c k } = \varepsilon _ { t } - \varepsilon _ { 0 t } ^ { e l }$ , where $\varepsilon _ { 0 t } ^ { e l } = \sigma _ { t } / E _ { 0 }$ , as illustrated in Figure 23.6.3–3. To avoid potential numerical problems, Abaqus enforces a lower limit on the postfailure stress equal to one-hundreth of the initial failure stress: $\sigma _ { t } \geq \sigma _ { t 0 } / 1 0 0$ .
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Tension stiffening data are given in terms of the cracking strain, $\tilde { \varepsilon } _ { t } ^ { c k }$ . When unloading data are available, the data are provided to Abaqus in terms of tensile damage curves, $d _ { t } - \tilde { \varepsilon } _ { t } ^ { c k }$ , as discussed below. Abaqus automatically converts the cracking strain values to plastic strain values using the relationship
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$$
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\tilde {\varepsilon} _ {t} ^ {p l} = \tilde {\varepsilon} _ {t} ^ {c k} - \frac {d _ {t}}{(1 - d _ {t})} \frac {\sigma_ {t}}{E _ {0}}.
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$$
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<details>
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<summary>line</summary>
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| ε_t | σ_t | Label |
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| ------- | ------- | ------------ |
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| ε̃_t^el | σ_t0 | E_0 |
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| ε̃_t^el | σ_t0 | (1 - d_t)E_0 |
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| ε̃_t^el | σ_t0 | E_0 |
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</details>
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Figure 23.6.3–3 Illustration of the definition of the cracking strain $\tilde { \varepsilon } _ { t } ^ { c k }$ used for the definition of tension stiffening data.
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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 $\tilde { \varepsilon } _ { t } ^ { p l } = \tilde { \varepsilon } _ { t } ^ { c \bar { k } }$ .
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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.
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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
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<!-- source-page: 474 -->
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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 $1 0 ^ { - 3 }$ is reasonable. This parameter should be calibrated to a particular case.
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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.
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Input File Usage: \*CONCRETE TENSION STIFFENING, TYPE=STRAIN (default)
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Concrete Damaged Plasticity: Tensile Behavior: Type: Strain
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# Fracture energy cracking criterion
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When 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, $G _ { f }$ , 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.
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This fracture energy cracking model can be invoked by specifying the postfailure stress as a tabular function of cracking displacement, as shown in Figure 23.6.3–4.
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<details>
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<summary>line</summary>
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| u_t^ck | σ_t |
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| ------ | --- |
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| 0 | 1.0 |
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| 1 | 0.5 |
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| 2 | 0.3 |
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| 3 | 0.2 |
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| 4 | 0.1 |
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</details>
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Figure 23.6.3–4 Postfailure stress-displacement curve.
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<!-- source-page: 475 -->
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Alternatively, the fracture energy, $G _ { f }$ , can be specified directly as a material property; in this case, define the failure stress, $\sigma _ { t 0 }$ , as a tabular function of the associated fracture energy. This model assumes a linear loss of strength after cracking, as shown in Figure 23.6.3–5.
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<details>
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<summary>text_image</summary>
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σₜ
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σₜ₀
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Gₕ
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uₜ₀ = 2Gₕ/σₜ₀
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uₜ
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</details>
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Figure 23.6.3–5 Postfailure stress-fracture energy curve.
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The cracking displacement at which complete loss of strength takes place is, therefore, $u _ { t 0 } = 2 G _ { f } / \sigma _ { t 0 }$ . Typical values of $G _ { f }$ 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 ).
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If tensile damage, $d _ { t }$ , is specified, Abaqus automatically converts the cracking displacement values to “plastic” displacement values using the relationship
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$$
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u _ {t} ^ {p l} = u _ {t} ^ {c k} - \frac {d _ {t}}{(1 - d _ {t})} \frac {\sigma_ {t} l _ {0}}{E _ {0}},
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$$
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where the specimen length, $l _ { 0 } .$ , is assumed to be one unit length, $l _ { 0 } = 1$ .
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# Implementation
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The 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 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. 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. Alternatively, this mesh dependency could be reduced by directly specifying the characteristic length as a function of the element topology and material orientation in
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user subroutine VUCHARLENGTH (see “Defining the characteristic element length at a material point in Abaqus/Explicit” in “Material data definition,” Section 21.1.2).
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<table><tr><td>Input File Usage:</td><td>Use the following option to specify the postfailure stress as a tabular function of displacement:*CONCRETE TENSION STIFFENING, TYPE=DISPLACEMENTUse the following option to specify the postfailure stress as a tabular function of the fracture energy:*CONCRETE TENSION STIFFENING, TYPE=GFI</td></tr></table>
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<table><tr><td rowspan="2">Abaqus/CAE Usage:</td><td>Property module: material editor: Mechanical→Plasticity→Concrete</td></tr><tr><td>Damaged Plasticity: Tensile Behavior: Type: Displacement or GFI</td></tr></table>
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# Defining compressive behavior
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You 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, $\tilde { \varepsilon } _ { c } ^ { i n }$ , 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.
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Hardening data are given in terms of an inelastic strain, $\tilde { \varepsilon } _ { c } ^ { i n }$ , instead of plastic strain, $\tilde { \varepsilon } _ { c } ^ { p l }$ . The compressive inelastic strain is defined as the total strain minus the elastic strain corresponding to the undamaged material, $\tilde { \varepsilon } _ { c } ^ { i n } = \varepsilon _ { c } - \varepsilon _ { 0 c } ^ { e l }$ , where $\varepsilon _ { 0 c } ^ { e l } = \sigma _ { c } / E _ { 0 }$ , as illustrated in Figure 23.6.3–6. Unloading data are provided to Abaqus in terms of compressive damage curves, $d _ { c } - \tilde { \varepsilon } _ { c } ^ { i n }$ , as discussed below. Abaqus automatically converts the inelastic strain values to plastic strain values using the relationship
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$$
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\tilde {\varepsilon} _ {c} ^ {p l} = \tilde {\varepsilon} _ {c} ^ {i n} - \frac {d _ {c}}{(1 - d _ {c})} \frac {\sigma_ {c}}{E _ {0}}.
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$$
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Abaqus will issue an error message if the calculated plastic strain values are negative and/or decreasing with increasing inelastic strain, which typically indicates that the compressive damage curves are incorrect. In the absence of compressive damage $\tilde { \varepsilon } _ { c } ^ { p l } = \tilde { \varepsilon } _ { c } ^ { i n }$ .
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Input File Usage: \*CONCRETE COMPRESSION HARDENING
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Concrete Damaged Plasticity: Compressive Behavior
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# Defining damage and stiffness recovery
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Damage, $d _ { t }$ and/or $d _ { c } .$ , can be specified in tabular form. (If damage is not specified, the model behaves as a plasticity model; consequently, $\tilde { \varepsilon } _ { t } ^ { p l } = \tilde { \varepsilon } _ { t } ^ { c k }$ and $\tilde { \varepsilon } _ { c } ^ { p l } = \tilde { \varepsilon } _ { c } ^ { i n } . )$
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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,
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<details>
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<summary>line</summary>
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| ε_c | σ_c |
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| ------- | ------- |
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| ε_c^in | σ_cu |
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| ε_c^el | σ_c0 |
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| ε_c^0c | σ_c0 |
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</details>
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Figure 23.6.3–6 Definition of the compressive inelastic (or crushing) strain $\tilde { \varepsilon } _ { c } ^ { i n }$ used for the definition of compression hardening data.
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$$
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d _ {t} | _ {t + \Delta t} = \max \left\{d _ {t} | _ {t}, d _ {t} (\tilde {\varepsilon} _ {t} ^ {p l} | _ {t + \Delta t}, \theta | _ {t + \Delta t}, f _ {i} | _ {t + \Delta t}) \right\},
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$$
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$$
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d _ {c} | _ {t + \Delta t} = \max \left\{d _ {c} | _ {t}, d _ {c} (\tilde {\varepsilon} _ {c} ^ {p l} | _ {t + \Delta t}, \theta | _ {t + \Delta t}, f _ {i} | _ {t + \Delta t}) \right\}.
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$$
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The choice of the damage properties is important since, generally, excessive damage may have a critical effect on the rate of convergence. It is recommended to avoid using values of the damage variables above 0.99, which corresponds to a 99% reduction of the stiffness.
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# Tensile damage
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You can define the uniaxial tension damage variable, $d _ { t }$ , as a tabular function of either cracking strain or cracking displacement.
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# Input File Usage:
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Use the following option to specify tensile damage as a function of cracking strain:
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$^ { \ast } \mathrm { C O N C R E T E ~ T E N S I O N ~ D A M A G E , ~ T Y P E = S T R A I N ~ ( d e f a u l t ) }$
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Use the following option to specify tensile damage as a function of cracking displacement:
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$^ { \ast } \mathrm { C O N C R E T E ~ T E N S I O N ~ D A M A G E } , \mathrm { T Y P E = D I S P L A C E M E N T }$
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<!-- source-page: 478 -->
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Concrete Damaged Plasticity: Tensile Behavior: Suboptions→Tension Damage: Type: Strain or Displacement
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# Compressive damage
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You can define the uniaxial compression damage variable, $d _ { c } ,$ , as a tabular function of inelastic (crushing) strain.
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Input File Usage: \*CONCRETE COMPRESSION DAMAGE
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Concrete Damaged Plasticity: Compressive Behavior: Suboptions→Compression Damage
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# Stiffness recovery
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As 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 $w _ { t }$ and $w _ { c }$ .
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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 $w _ { t } = 0$ and $w _ { c } = 1$ , is the default used by Abaqus. Figure 23.6.3–7 illustrates a uniaxial load cycle assuming the default behavior.
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Input File Usage: Use the following option to specify the compression stiffness recovery factor, $w _ { c } .$
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\*CONCRETE TENSION DAMAGE, COMPRESSION $\scriptstyle \mathrm { R E C O V E R Y } = w _ { c }$
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Use the following option to specify the tension stiffness recovery factor, $w _ { t } .$
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\*CONCRETE COMPRESSION DAMAGE, TENSION $\scriptstyle \mathrm { R E C O V E R Y } = w _ { t }$
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Concrete Damaged Plasticity:
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Tensile Behavior: Suboptions→Tension Damage: Compression recovery: $w _ { c }$
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Compressive Behavior: Suboptions→Compression Damage:
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Tension recovery: $w _ { t }$
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# Rate dependence
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The 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
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<!-- source-page: 479 -->
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<details>
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<summary>text_image</summary>
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w_t = 1
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w_t = 0
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(1-d_c)E_0
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(1-d_t)(1-d_c)E_0
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w_c = 0
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w_c = 1
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E_0
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(1-d_t)E_0
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E_0
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σ_t
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σ_t0
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</details>
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Figure 23.6.3–7 Uniaxial load cycle (tension-compression-tension) assuming default values for the stiffness recovery factors: $w _ { t } ~ = ~ 0$ and $w _ { c } ~ = ~ 1$ .
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(or displacement) rate, and you can specify compression hardening data as a tabular function of inelastic strain rate.
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Input File Usage: Use the following options:
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\*CONCRETE TENSION STIFFENING
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\*CONCRETE COMPRESSION HARDENING
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Concrete
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Damaged Plasticity:
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Tensile Behavior: Use strain-rate-dependent data
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Compressive Behavior: Use strain-rate-dependent data
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<!-- source-page: 480 -->
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# Concrete plasticity
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You can define flow potential, yield surface, and in Abaqus/Standard viscosity parameters for the concrete damaged plasticity material model.
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Input File Usage: \*CONCRETE DAMAGED PLASTICITY
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Abaqus/CAE Usage: Property module: material editor: Mechanical→Plasticity→Concrete Damaged Plasticity: Plasticity
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# Effective stress invariants
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The effective stress is defined as
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$$
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\bar {\pmb {\sigma}} = \mathbf {D} _ {0} ^ {e l}: (\pmb {\varepsilon} - \pmb {\varepsilon} ^ {p l}).
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$$
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The plastic flow potential function and the yield surface make use of two stress invariants of the effective stress tensor, namely the hydrostatic pressure stress,
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$$
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\bar {p} = - \frac {1}{3} \mathrm{trace} (\bar {\pmb {\sigma}}),
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$$
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and the Mises equivalent effective stress,
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$$
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\bar {q} = \sqrt {\frac {3}{2} (\bar {\bf S} : \bar {\bf S})},
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$$
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where is the effective stress deviator, defined as
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$$
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\bar {\mathbf {S}} = \bar {\boldsymbol {\sigma}} + \bar {p} \mathbf {I}.
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$$
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# Plastic flow
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The concrete damaged plasticity model assumes nonassociated potential plastic flow. The flow potential G used for this model is the Drucker-Prager hyperbolic function:
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$$
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G = \sqrt {(\epsilon \sigma_ {t 0} \tan \psi) ^ {2} + \bar {q} ^ {2}} - \bar {p} \tan \psi ,
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$$
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where
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$$
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\psi (\theta , f _ {i})
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$$
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$$
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\sigma_ {t 0} (\theta , f _ {i}) = \sigma_ {t} | _ {\tilde {\varepsilon} _ {t} ^ {p l} = 0, \dot {\tilde {\varepsilon}} _ {t} ^ {p l} = 0}
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||
$$
|
||
|
||
$$
|
||
\epsilon (\theta , f _ {i})
|
||
$$
|
||
|
||
is the dilation angle measured in the p–q plane at high confining pressure;
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||
|
||
is the uniaxial tensile stress at failure, taken from the userspecified tension stiffening data; and
|
||
|
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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).
|