Lecture in Nonlinear FEM on. the Building- and Civil Engineering sectors 8.th. semester for

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1 Lecture in Nonlinear FEM on the Building- and Civil Engineering sectors 8.th. semester for the Building- and Civil Engineering, B8k, and Mechanical Engineering, B8m AALBORG UNIVERSITY ESBJERG, DENMARK ***************** Theme: Design of marine constructions. 1

2 Outline: Updated: 11. marts Introduction Notes 2. Geometrical nonlinearity - strain measures Cook 17.1, Geometrical nonlinearity - appl. in buckling analysis Cook Stress stiffness Cook Buckling Cook Material nonlinearity - introduction Cook Material nonlinearity - solution methods Cook 17.6, Contact nonlinearity Cook Nonlinear dynamic problems Cook Nonlinear dynamic problems Cook Literature: Noter A. Kristensen: Cook Cook, R. D. 2002: Concepts and applications of finite element analysis. John Wiley & Sons 2

3 7. Material nonlinearity - solution methods Outline: Last time 4 Incremental plasticity relations 5 Incremental plasticity relations - the yield function 7 Incremental plasticity relations - the flow rule 8 Incremental plasticity relations - the hardening rule 9 Incremental stress-strain relations 10 Plasticity: Formulation for von Mises theory 13 Examples in FEM Assignments 3

4 Last time Yield criterion or yield function, i.e. defines the state of stress at which material response changes from elastic to plastic. Flow rule, i.e. relates plastic strain increments to stress increments after the onset of initial yielding. Hardening rule, i.e. predicts the change in the yield surface due to plastic strains. Thus, basically two hardening rules exist: isotropic hardening F = σ σ 0 - Bauschinger effect is ignored but the elastic range expands kinematic hardening F = σ α σ Y - Bauschinger effect is included but elastic range remains constant 4

5 Incremental plasticity relations For strains larger than ε y a strain increment dε is given by: dε = dε e + dε p or {dε} = {dε e } + {dε p } where dε e is the elastic part and dε p is the plastic strain part excluding contributions from thermal and creep strain. When ε > ε y an increment dε e only exists if E t 0 and is associated with a stress increment dσ thus or dσ = Edε e dσ = E(dε dε p ) dσ = E t dε dσ = H p dε p {dσ} = [E]{dε e } {dσ} = [E]({dε} {dε p }) {dσ} = [E t ]{dε} {dσ} = [H p ]{dε p } where H p is termed the strain-hardening parameter or plastic modulus and E t the tangent modulus given as ( E t H p = E t = E 1 E ) 1 (E t /E) E + H p 5

6 Incremental plasticity relations If H p = 0 E t = 0 the material is termed elastic-perfectly plastic behaviour. The three elements in elastic-plastic analysis are: Yield criterion or yield function, i.e. defines the state of stress at which material response changes from elastic to plastic. Flow rule, i.e. relates plastic strain increments to stress increments after the onset of initial yielding. Hardening rule, i.e. predicts the change in the yield surface due to plastic strains. 6

7 Incremental plasticity relations - the yield function A yield function can be written in a general form as F = F ({σ},{α},w p ) where {α} and W p control the hardening, i.e. how a given "yield surface"in space is modified (location and size) due to a plastic strain response. For F < 0 elastic conditions F = 0 yielding is beginning or is in progress F > 0 not physical obtainable 7

8 Incremental plasticity relations - the flow rule The flow rule is given by i.e. {dε p } = {dε p x} = { } Q dλ { Q x } dλ The flow rule is called associated (ductile materials/metals) if Q = F and non-associated (soil, granualar) otherwise. 8

9 Incremental plasticity relations - the hardening rule Isotropic hardening can be represented the plastic work per unit volume W p thus W p = {dσ} T dε p Kinematic hardening is given by vector determining the translation of the yield surface in general stress state given as and {α} = C {dε p } C = 2 3 H p Plastic flow implies that dε p x + dε p y + dε p z = 0 thus {α} = α x + α y + α z =

10 Incremental stress-strain relations During an increment of plastic straining df = 0 thus { } { } F F {dσ} + {dα} + F dw p = 0 α W p By substitution {dσ} = [E] ( {dε e } { } ) Q dλ { } Q dw p = {dσ} T dλ { } Q {dα} = C dλ 10

11 Incremental stress-strain relations The resulting equation to solve for the plastic multiplier dλ is where P λ is the row matrix P λ = { } T F [E] dλ = P λ {dε} { } { Q F α { } T F [E] } T C { } Q F {σ} T α W p { } Q Both work hardening and strain hardening are included in this expression. 11

12 Incremental stress-strain relations Rearranging yield {dσ} = [E ep ]{dε} ( [E ep ] = [E] I { } ) Q P λ where I is a unit matrix and [E ep ] is the generalized tangent modulus. If Q = F [E ep ] is symmetric. For F < 0 (yield has not occurred) or F = 0 and df < 0 (unloading from plastic state) then [E ep ] = [E] The tangent stiffness matrix [k t ] is now given by [k t ] = [B] T [E ep ][B]dV 12

13 Plasticity: Formulation for von Mises theory Von Mises theory imply associative plasticity, i.e. F = Q where yielding is postulated to occur when the effective stress σ e = 1 2 (σ x σ y ) 2 + (σ y σ z ) 2 + (σ z σ x ) 2 + 6(τ 2 xy + τ 2 yz + τ 2 xz) have reached a limiting value (for most metals). The resulting expression for isotropic hardening is given as F = σ e σ 0 where σ 0 is the largest value of σ e in the previous plastic straining. F < 0 elastic condtions F = 0 yielding F > 0 not possible 13

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