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Willam–Warnke yield criterion

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Three-parameter Willam-Warnke yield surface.

The Willam–Warnke yield criterion is a function that is used to predict when failure will occur in concrete and other cohesive-frictional materials such as rock, soil, and ceramics. This yield criterion has the functional form

f ( I 1 , J 2 , J 3 ) = 0 {\displaystyle f(I_{1},J_{2},J_{3})=0\,}

where I 1 {\displaystyle I_{1}} is the first invariant of the Cauchy stress tensor, and J 2 , J 3 {\displaystyle J_{2},J_{3}} are the second and third invariants of the deviatoric part of the Cauchy stress tensor. There are three material parameters ( σ c {\displaystyle \sigma _{c}} - the uniaxial compressive strength, σ t {\displaystyle \sigma _{t}} – the uniaxial tensile strength, σ b {\displaystyle \sigma _{b}} - the equibiaxial compressive strength) that have to be determined before the Willam-Warnke yield criterion may be applied to predict failure.

In terms of I 1 , J 2 , J 3 {\displaystyle I_{1},J_{2},J_{3}} , the Willam-Warnke yield criterion can be expressed as

f := J 2 + λ ( J 2 , J 3 )   ( I 1 3 B ) = 0 {\displaystyle f:={\sqrt {J_{2}}}+\lambda (J_{2},J_{3})~({\tfrac {I_{1}}{3}}-B)=0}

where λ {\displaystyle \lambda } is a function that depends on J 2 , J 3 {\displaystyle J_{2},J_{3}} and the three material parameters and B {\displaystyle B} depends only on the material parameters. The function λ {\displaystyle \lambda } can be interpreted as the friction angle which depends on the Lode angle ( θ {\displaystyle \theta } ). The quantity B {\displaystyle B} is interpreted as a cohesion pressure. The Willam-Warnke yield criterion may therefore be viewed as a combination of the Mohr–Coulomb and the Drucker–Prager yield criteria.

Willam-Warnke yield function

View of three-parameter Willam-Warnke yield surface in 3D space of principal stresses for σ c = 1 , σ t = 0.3 , σ b = 1.7 {\displaystyle \sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7}
Trace of the three-parameter Willam-Warnke yield surface in the σ 1 σ 2 {\displaystyle \sigma _{1}-\sigma _{2}} -plane for σ c = 1 , σ t = 0.3 , σ b = 1.7 {\displaystyle \sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7}

In the original paper, the three-parameter Willam-Warnke yield function was expressed as

f = 1 3 z   I 1 σ c + 2 5   1 r ( θ ) J 2 σ c 1 0 {\displaystyle f={\cfrac {1}{3z}}~{\cfrac {I_{1}}{\sigma _{c}}}+{\sqrt {\cfrac {2}{5}}}~{\cfrac {1}{r(\theta )}}{\cfrac {\sqrt {J_{2}}}{\sigma _{c}}}-1\leq 0}

where I 1 {\displaystyle I_{1}} is the first invariant of the stress tensor, J 2 {\displaystyle J_{2}} is the second invariant of the deviatoric part of the stress tensor, σ c {\displaystyle \sigma _{c}} is the yield stress in uniaxial compression, and θ {\displaystyle \theta } is the Lode angle given by

θ = 1 3 cos 1 ( 3 3 2   J 3 J 2 3 / 2 )   . {\displaystyle \theta ={\tfrac {1}{3}}\cos ^{-1}\left({\cfrac {3{\sqrt {3}}}{2}}~{\cfrac {J_{3}}{J_{2}^{3/2}}}\right)~.}

The locus of the boundary of the stress surface in the deviatoric stress plane is expressed in polar coordinates by the quantity r ( θ ) {\displaystyle r(\theta )} which is given by

r ( θ ) := u ( θ ) + v ( θ ) w ( θ ) {\displaystyle r(\theta ):={\cfrac {u(\theta )+v(\theta )}{w(\theta )}}}

where

u ( θ ) := 2   r c   ( r c 2 r t 2 )   cos θ v ( θ ) := r c   ( 2   r t r c ) 4   ( r c 2 r t 2 )   cos 2 θ + 5   r t 2 4   r t   r c w ( θ ) := 4 ( r c 2 r t 2 ) cos 2 θ + ( r c 2   r t ) 2 {\displaystyle {\begin{aligned}u(\theta ):=&2~r_{c}~(r_{c}^{2}-r_{t}^{2})~\cos \theta \\v(\theta ):=&r_{c}~(2~r_{t}-r_{c}){\sqrt {4~(r_{c}^{2}-r_{t}^{2})~\cos ^{2}\theta +5~r_{t}^{2}-4~r_{t}~r_{c}}}\\w(\theta ):=&4(r_{c}^{2}-r_{t}^{2})\cos ^{2}\theta +(r_{c}-2~r_{t})^{2}\end{aligned}}}

The quantities r t {\displaystyle r_{t}} and r c {\displaystyle r_{c}} describe the position vectors at the locations θ = 0 , 60 {\displaystyle \theta =0^{\circ },60^{\circ }} and can be expressed in terms of σ c , σ b , σ t {\displaystyle \sigma _{c},\sigma _{b},\sigma _{t}} as (here σ b {\displaystyle \sigma _{b}} is the failure stress under equi-biaxial compression and σ t {\displaystyle \sigma _{t}} is the failure stress under uniaxial tension)

r c := 6 5 [ σ b σ t 3 σ b σ t + σ c ( σ b σ t ) ]   ;     r t := 6 5 [ σ b σ t σ c ( 2 σ b + σ t ) ] {\displaystyle r_{c}:={\sqrt {\cfrac {6}{5}}}\left~;~~r_{t}:={\sqrt {\cfrac {6}{5}}}\left}

The parameter z {\displaystyle z} in the model is given by

z := σ b σ t σ c ( σ b σ t )   . {\displaystyle z:={\cfrac {\sigma _{b}\sigma _{t}}{\sigma _{c}(\sigma _{b}-\sigma _{t})}}~.}

The Haigh-Westergaard representation of the Willam-Warnke yield condition can be written as

f ( ξ , ρ , θ ) = 0 f := λ ¯ ( θ )   ρ + B ¯   ξ σ c 0 {\displaystyle f(\xi ,\rho ,\theta )=0\,\quad \equiv \quad f:={\bar {\lambda }}(\theta )~\rho +{\bar {B}}~\xi -\sigma _{c}\leq 0}

where

B ¯ := 1 3   z   ;     λ ¯ := 1 5   r ( θ )   . {\displaystyle {\bar {B}}:={\cfrac {1}{{\sqrt {3}}~z}}~;~~{\bar {\lambda }}:={\cfrac {1}{{\sqrt {5}}~r(\theta )}}~.}

Modified forms of the Willam-Warnke yield criterion

Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the π {\displaystyle \pi } -plane for σ c = 1 , σ t = 0.3 , σ b = 1.7 {\displaystyle \sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7}

An alternative form of the Willam-Warnke yield criterion in Haigh-Westergaard coordinates is the Ulm-Coussy-Bazant form:

f ( ξ , ρ , θ ) = 0 or f := ρ + λ ¯ ( θ )   ( ξ B ¯ ) = 0 {\displaystyle f(\xi ,\rho ,\theta )=0\,\quad {\text{or}}\quad f:=\rho +{\bar {\lambda }}(\theta )~\left(\xi -{\bar {B}}\right)=0}

where

λ ¯ := 2 3   u ( θ ) + v ( θ ) w ( θ )   ;     B ¯ := 1 3   [ σ b σ t σ b σ t ] {\displaystyle {\bar {\lambda }}:={\sqrt {\tfrac {2}{3}}}~{\cfrac {u(\theta )+v(\theta )}{w(\theta )}}~;~~{\bar {B}}:={\tfrac {1}{\sqrt {3}}}~\left}

and

r t := 3   ( σ b σ t ) 2 σ b σ t r c := 3   σ c   ( σ b σ t ) ( σ c + σ t ) σ b σ c σ t {\displaystyle {\begin{aligned}r_{t}:=&{\cfrac {{\sqrt {3}}~(\sigma _{b}-\sigma _{t})}{2\sigma _{b}-\sigma _{t}}}\\r_{c}:=&{\cfrac {{\sqrt {3}}~\sigma _{c}~(\sigma _{b}-\sigma _{t})}{(\sigma _{c}+\sigma _{t})\sigma _{b}-\sigma _{c}\sigma _{t}}}\end{aligned}}}

The quantities r c , r t {\displaystyle r_{c},r_{t}} are interpreted as friction coefficients. For the yield surface to be convex, the Willam-Warnke yield criterion requires that 2   r t r c r t / 2 {\displaystyle 2~r_{t}\geq r_{c}\geq r_{t}/2} and 0 θ π 3 {\displaystyle 0\leq \theta \leq {\cfrac {\pi }{3}}} .

View of Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in 3D space of principal stresses for σ c = 1 , σ t = 0.3 , σ b = 1.7 {\displaystyle \sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7}
Trace of the Ulm-Coussy-Bazant version of the three-parameter Willam-Warnke yield surface in the σ 1 σ 2 {\displaystyle \sigma _{1}-\sigma _{2}} -plane for σ c = 1 , σ t = 0.3 , σ b = 1.7 {\displaystyle \sigma _{c}=1,\sigma _{t}=0.3,\sigma _{b}=1.7}

See also

References

  1. Willam, K. J. and Warnke, E. P. (1975). "Constitutive models for the triaxial behavior of concrete." Proceedings of the International Assoc. for Bridge and Structural Engineering, vol 19, pp. 1–30.
  2. Ulm, F-J., Coussy, O., Bazant, Z. (1999) The ‘‘Chunnel’’ Fire. I: Chemoplastic softening in rapidly heated concrete. ASCE Journal of Engineering Mechanics, vol. 125, no. 3, pp. 272-282.
  • Chen, W. F. (1982). Plasticity in Reinforced Concrete. McGraw Hill. New York.

External links

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