An approach to limit states in An approach to limit states in - - PowerPoint PPT Presentation

Mechanical Engineering COPPE-UFRJ

An approach to limit states in An approach to limit states in advanced materials advanced materials

Lavinia Borges Lavinia Borges and Fernando and Fernando Duda Duda

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Solid Mechanics and Materials Group Solid Mechanics and Materials Group Professors Professors

Fernando Alves Rochinha Fernando Pereira Duda Lavinia Alves Borges Nestor Zouain Pereira Students 16 DSc 4 MSc 8 Scientific Initiation Program

• Structural Integrity

Plasticity and Viscoplasticity Fatigue Damage Limit States

• Structural Dynamic
• Advanced Materials

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Limit States:

– Limit analysis : Model and adaptive remesh

– Shakedown for FGM beam

Outline

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Limit Load

Plastic Collapse Structural Analysis Plastic Flow Metal Forming Geomechanics; - Bearing capacity Failure Criterion Composites- Failure Prediction.

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Limit Analysis of a continuum

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Limit Analysis: Discrete Model - Mixed Elements

Nodes for velocities interpolation (QUADRATIC) Nodes for stress interpolation (linear)

(T) 1 v v T

q n T

v

≤ = ⋅ ⋅ =

ℜ ∈ ℜ ∈

f F B max min α

Algorithm

Newton-like formula associated with the set

• f

all equalities included in the

• ptimality

conditions, followed by a step relaxation and stress scaling in

• rder

to preserve the plastic admissibility constraint.

m j f f F F

• B

f B

j j j j T

, , 1 ) ( ) ( 1 ) ( …

• =

≥ ≤ = = ⋅ = = ∇ − λ λ α λ T T 1 v T T v

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Estimators based on derivatives recovery

The interpolation error as an indicator of the approximate solution Hr(uh) is the recovered Hessian matrix

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ηT= Global error

Hr - eigenvectors matrix

error direction absolute value of the

Hr eigenvalues

value of error

Error estimator

Local error η=

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Derivative Recuperation: Weighted Average or Patch Recovering

In order to approximate representing strong variations in the derivatives the adapted mesh becomes oriented by means of stretching its elements in the direction of maximum curvature of function graph

S(N) N N M M

^

^

Stretched Mesh and the transformed domain defined by advancing frontal technique

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For each node N of the mesh: 1 . Define the patch associated to N.

• 2. For known s(N), h(N), e1 and e2, built the metric tensor S(N).
• 3. Transform the elements of the patch.
• 4. In each element compute the grad(uh).
• 5. Compute recovered gradient gradr(uh).
• 6. Transform gradr(uh) to the original domain by

∇R uh(N) = ST gradRuh(N) Taking the first derivative as a new field we reapply the algorithm to recover the Second derivative First Derivative Recovery Algorithm

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Adaptive Procedure

1.For each element compute the local error ηT and then the global error η 2.Given Nel in the new mesh, compute the expected local error indicator, equally distributed on all elements, by The decreasing or increasing rate of element size is estimated by This parameter at nodal level β(N) is computed by the same approach adopted for the recovering the derivatives. 4.Compute the size of the new element h k+1 and the stretching s(Nk) node N, by

where λ 1 and λ2 are the absolute eigenvalue of Hessian matrix.

• 5. h k+1 _scaling

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Plate with imperfections (Diez, 1999)

Initial mesh 1908 dof

Plane Stress Plane Strain pc= 1.0350 σY pc= 0.9123 σY

Final mesh:7064 Final mesh: 8194 dof

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Slip-line

2 4 6 8 10 12 14

DOF(Thousands)

1.4 1.42 1.44 1.46 1.48 1.5 Collapse load Numerical Slip-line

700 7000 DOF 0.01 0.1 1 Global Error Global error

a a a a P

Initial - 773 DOF

plastic multiplier field

After five Steps - 12247 DOF

plastic multiplier field

Frictionless extrusion through a square die – Reduction 1/2

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Slip-line

plastic multiplier field plastic multiplier field

Initial - 707 DOF After four Steps - 10255 DOF

Collapse load DOF(Thousands)

2 4 6 8 10 12 1.9 1.92 1.94 1.96 1.98 2 Numerical Slip-line

700 7000 DOF 0.01 0.1 1

Global Error

Global error

2a a 2a a P

Frictionless extrusion through a square die – Reduction 2/3

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Equal channel angular extrusion (ECAE)

Method for deforming materials to very high plastic strains, with no net change in the billet’s shape. By grain refinement:

• control materials structure,
• texture and
• physico-mechanical properties.

F = 1.0 y x 20 20 5 5 E = 2100000

ν = 0.3

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Plastic Zone –Experimental Segal (1999) Plastic Zone and velocity field - Numerical

Nanostructured metals by severe plastic deformation

Winther and Huang, 2003

Grain subdivision Dislocations accumulate in walls

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Limit States:

Basic direct methods for safety assessment of engineering structures subjected to variable loading. Inelastic structures submitted to variable loads may undergo one of the following types of failure modes:

• alternating plasticity (plastic shakedown),
• incremental collapse(ratcheting)
• instantaneous collapse (plastic collapse) - Limit analysis

The objective in this analysis is the computation of the load amplification factor, α, for the domain of variable loadings, that ensures elastic shakedown.

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Shakedown limits (Bree diagram) for an Al2O3 / FGM / Ni 3-layer beam ρd = 0.6 (1.2 mm FGM in 2 mm height of 3 layer beam)

∆T

p p

∆T ∆T ∆T

p

∆T

p

∆T

p