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  1. Material
  2. Material1D
  3. Concrete

ConcreteK4

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Last updated 1 year ago

A Uniaxial Plastic-Damage Concrete Model

References

Syntax

material ConcreteK4 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) [12]
# (1) int, unique material tag
# (2) double, elastic modulus (+), typical: 30000 (in MPa)
# (3) double, crack closing stiffness ratio (+), typical: ~0.1
# (4) double, tensile hardening ratio (+), typical: [0,1]
# (5) double, compressive hardening ratio (+), typical: [0,1]
# (6) double, tensile strength (+), typical: 3 (in MPa)
# (7) double, crush (compressive) strength (+), typical: 40 (in MPa)
# (8) double, crush (compressive) strain (+), typical: 0.002
# (9) double, yield (compressive) strength ratio (<1), typical: ~0.7
# (10) double, reference tensile zeta
# (11) double, reference compressive zeta
# [12] double, density, default: 0.0

Theory

There is no official name proposed by the authors. The name K4 comes from the fact that the four authors' last names all start with the letter K.

The model implements the proposed hardening laws and crack closing mechanism. But the damage evolution is different from the original formulation. In specific, the original linear damage evolution is replaced by an exponential law that would alleviate potential numerical issues, as the numerical world does not like exact zeros.

Additive Split

σ=(1−d)σˉ.\sigma=\left(1-d\right)\bar{\sigma}.σ=(1−d)σˉ.

Yield Function

F={Ft,σˉ>0,Fc,otherwise,F=\left\{ \begin{array}{ll} F_t,&\bar{\sigma}>0,\\ F_c,&\text{otherwise}, \end{array} \right.F={Ft​,Fc​,​σˉ>0,otherwise,​

with

Ft=σˉ−σˉy,t,Fc=−σˉ−σˉy,c.F_t=\bar{\sigma}-\bar{\sigma}_{y,t},\qquad F_c=-\bar{\sigma}-\bar{\sigma}_{y,c}.Ft​=σˉ−σˉy,t​,Fc​=−σˉ−σˉy,c​.

Flow Rule

εp˙=γ signσˉ.\dot{\varepsilon^p}=\gamma~\text{sign}{\bar{\sigma}}.εp˙=γ signσˉ.

Hardening Laws

σˉy,t=ft+htkt,σˉy,c={fy+hckc,kc⩽k0,fy+hck0+hd(kc−k0),otherwise.\bar{\sigma}_{y,t}=f_t+h_tk_t,\qquad \bar{\sigma}_{y,c}=\left\{ \begin{array}{ll} f_y+h_ck_c,&k_c\leqslant{}k_0,\\ f_y+h_ck_0+h_d\left(k_c-k_0\right),&\text{otherwise}. \end{array} \right.σˉy,t​=ft​+ht​kt​,σˉy,c​={fy​+hc​kc​,fy​+hc​k0​+hd​(kc​−k0​),​kc​⩽k0​,otherwise.​

The tensile strength ftf_tft​ is controlled by (6). The hardening modulus hth_tht​ is computed as (4) times elastic modulus.

The crush strength fcf_cfc​ (7) can be expressed as fc=fy+hck0f_c=f_y+h_ck_0fc​=fy​+hc​k0​ where fyf_yfy​ is (9) times fcf_cfc​. k0k_0k0​ is strain when crush strength is reached, this is controlled by (8). The hardening modulus hch_chc​ is automatically computed. The hardening modulus hdh_dhd​ is computed as (5) times elastic modulus.

Damage

dt=1−exp⁡(−kter,t),dc=1−exp⁡(−kcer,c).d_t=1-\exp\left(-\dfrac{k_t}{e_{r,t}}\right),\qquad d_c=1-\exp\left(-\dfrac{k_c}{e_{r,c}}\right).dt​=1−exp(−er,t​kt​​),dc​=1−exp(−er,c​kc​​).

The reference strain er,te_{r,t}er,t​ and er,ce_{r,c}er,c​ are used to control the damage evolution. They can be expressed as

er,t=ftζtE,er,c=fcζcE,e_{r,t}=\dfrac{f_t}{\zeta_tE},\qquad e_{r,c}=\dfrac{f_c}{\zeta_cE},er,t​=ζt​Eft​​,er,c​=ζc​Efc​​,

in which ζt\zeta_tζt​ and ζc\zeta_cζc​ are (10) and (11) respectively.

Examples

Effect of Crack Closing Stiffness Ratio

The crack closing stiffness ratio (3) controls the slope of the segment where crack opening is gradually closed.

Effect of Tensile Reference Strain

Basic model:

material ConcreteK4 1 3E4 .05 .01 .01 3. 10. 2E-3 .7 1. 1.

Effect of Compressive Reference Strain

Basic model:

material ConcreteK4 1 3E4 .05 .01 .01 3. 10. 2E-3 .7 1. 1.

A Cyclic Example

material ConcreteK4 1 3E4 .05 .01 .01 3. 10. 2E-3 .7 5. 5.

materialTest1D 1 -1E-5 300 90 150 90 150 90 150 90 150 90 150 90 150 90 150
10.1061/(ASCE)ST.1943-541X.0002592
Constitutive Modelling Cookbook
crack closing ratio
crack closing ratio
tensile reference strain
compressive reference strain
cyclic