Constitutive Modelling Flashcards
Linear-elastic vs Elasto-plastic
-L-E cannot simulate failure. Also assume E to be constant but it decreases with strain so elasto plastic normally applies
E-P:
Uniaxially loaded linearly elastic perfectly plastic bar
-L-E implies perfect recovery of strains
-PP implies no increase in stress with unloading elastic
* Constant E and unloading // to L-E curve
* Only partial recovery
Uniaxially loaded linearly elastic with strain hardening
- sigma exceeds sigma_y with a nonlinear increase
- Continuous increase up to ultimate failure
- Resembles an oedometer test
Uniaxially loaded linearly elastic with strain stiffening
- sigma falls under sigma_y with a nonlinear increase continuous decrease up to ultimate failure
- resembles shear box test for dense sand which softens as it tends to the residual strength
Concepts for extension to general stress strain space
-Required since there are 6 stresses and strains rather than 1
-Concepts necessary
*Coincidence of axis
*Hardening/softening behaviour
*Yield function
*Plastic potential func.
Coincidence of axis
Direction of accumalated stress and incremental increase in strains, elastic strains in L-E and plastic ones after, coincide
Yield function
F({sigma},{k}) = 0
- No longer defined by one stress
- F=0 is elasto-plastic , <0 is elastic and no state for >0 exists
- k is the state parameter (Hardening softening rules)
- Constant means perfectly plastic
- Increasing means hardening
- Decreasing means softening
Plastic potential function
P({sigma},{m})=0
-Applied in the flow rule to get the direction of the plastic strain so m isnt important
delta epsilon_i_p = constant . partial dP/d.sigma_i
-epsilon_p is always perp. to the PP surface
-coincidence of axis allows sigma_i and delta epsilon_i_p to be plotted on the same exis
- Associated plasticity means PPF and YF re the same and the yield condition applies
2D rep. of yeild function
- hardening characteristics
Perfectly-plastic
- Yield function remains stationary
- No stress in y still leads to epsilon_y due to Poissons ratio for applied sigma_h
- Strain hardening
* Isotropic means yield function changes size
* Kinematic means the field function surface moves
* epsilon_p changes as the hardening happens
Invariants
Principle stresses independent of axis unlike sigma_x, sigma_y, sigma_z
- Mean effective stress
-Deviator stress = 1/root(6) . root(sum(difference if the three stresses squared))
-Lodes angle = Tan inv(1/root(3) .
2(sigma2’ - sigma3’)/(sigma1’-sigma3’ - 1))
Significance:
- Mean effective stress is the distance from the deviator plane to the origin when scaled by root(3)
-Deviator stress is the distance form stress state P to the centre of the deviator plane when scaled by root(2)
-Lodes angle is the position of P measured clockwise form the horizontal
-Commonly used to interpret experimentals
Space diagonal
All three stresses are equal
Deviatoric plane
plane perp. to the plane diagonal
Triaxial compression vs extension
Compression
-sigma1’ = sigma_a’
-sigma2’ = ‘sigma3’ = sigma_r’
-lodes angel = - 30 degrees
Extension
-sigma3’ = sigma_a’
-sigma1’ = ‘sigma2’ = sigma_r’
-lodes angel = 30 degrees
Strain invariants
-Volumetric strain (epsilon_vol) = sum of the three
-Deviatoric strain (epsilon_d) = 2/root(6) . root(sum(difference of the strains squared))
Trescas failure
F = sigma1 - sigma2 - 2Su = J.cos(theta) - Su = 0
- Yield surface is a regular hexagon on the deviatoric plane and since it independent o p’ is remains constant
-Amplitude of all stress states are Su
PPF:
-coincidence of axis allows J/epsilon_d and P/epsilon_vol plotted together
-the change in volumetric plastic strain is zero for associated plasticity
-Tresca is perfectly plastic
