KIT Workshop - TEQIP · KIT Workshop Dr. Pritam Chakraborty Asst. Prof., AE, IIT Kanpur 18th January, 2020 Outreach Audi., IIT Kanpur

KIT Workshop

Dr. Pritam Chakraborty Asst. Prof., AE, IIT Kanpur

18th January, 2020 Outreach Audi., IIT Kanpur

Outline

Elasto-plasticity Overview

1D Elasto-plasticity and viscoplasticity ODEs, solution strategy, MATLAB code

Multidimensional Elasto-plasticity ODEs, solution strategy

Large Deformation, Incremental Form and Updated Lagrangian FEM

Elasto-Plasticity

Uniaxial behavior of a class of materials (e.g. metals and metallic alloys)

A typical UTM A typical tensile

specimen

Deformation of the gauge section Force-Displacement

From Extensometer

From

Loa

d Ce

ll

Specimen Deformation

Uniform Elongation

Localized Deformation or Necking

Fracture

Stress – Strain

Obtained from the force-displacement curve

Engineering stress and strain:

True stress and strain:

2: Proportional Limit 3: Elastic Limit 4.: Offset Yield strength

Elasto Perfectly Plastic

Linear Hardening

Nonlinear Hardening

Elasto-Plastic material is characterized by a permanent (plastic) deformation after unloading

Stress – Strain

Idealization

1D Elasto-Plasticity

For uniaxial tensile loading till onset of necking, strain and stress are homogenous in the gauge section

X

Y Z

Assuming incompressible deformation even in elastic regime

If the rate of pulling is slow, one can assume the problem to be 1D quasi-static, such that

where σ = σ(x, t), P=P(x,t) and A=A(x,t)

Constitutive Equation

εp εe

σ Both in the elastic and plastic regime

where εe is elastic strain and is due to lattice stretching. This strain is released on unloading. Plastic strain (εp) in metals happens due to motion of dislocations and is activated by the applied stress. Remains as a permanent deformation.

The total strain

State Variable and Rate Equation

ε1

ε2

State Variable: For an elastic material, the state of the material (stress, deformation, etc.) can be uniquely defined by the total strain (ε1) However, for elasto-plastic materials, an additional variable is required to define the state. Plastic strain can be considered as the other variable.

1

2

Rate Form: Proposing algebraic equations to describe evolution of variables (stress, strain, plastic strain) under general loading-unloading scenario is not convenient. Rate form for evolutions are thus proposed and can be integrated to get to the current state.

Constitutive Equation in Rate Form

Yield stress is the resistance of the material to deform plastically – intrinsic resistance Physically can be related to dislocation density With plastic deformation as more dislocations are generated

they resist the motion of others – increasing resistance Define an internal parameter, q, that captures the internal state

of the material The evolution of ‘q’ depends on plastic strain and can be

assumed as

To define a law for plastic strain rate, the condition of yielding is applied

Which means that both under tension or compression the quantity like DD increases along with the yield stress

Yield condition

Plastic flow (non zero plastic strain) can happen when applied stress = yield stress in rate independent materials Thus, f(σ, σY) = |σ| - σY(q) = 0 is the “yield condition” in both tension and

compression f(σ, σY) is the yield function If f(σ, σY) < 0, then there won’t be any plastic flow

If f(σ(t), σY(t)) = 0, and the material is yielding, then at any small time interval of time,

∆t -> 0, f(σ(t+ ∆t), σY(t+ ∆t)) = 0 Thus,

Thus, if and

Yield Condition Consistency Condition

Flow rule

The plastic strain rate can be defined as

where is the plastic multiplier and is >= 0, is the flow direction and is the flow potential The flow is defined as associative since

Simplified flow rule

The plastic multiplier is obtained from the consistency condition

Since,

Nonlinear PDE – 1D Plasticity

Assuming small strain, current configuration ~ reference configuration Thus, nonlinearity is only due to material behavior

f(σ, σY) = |σ| - σY(q) = 0 Governing Equation

Kinematics Constitutive

If f(σ, σY) = 0

Initial and Boundary Conditions – 1D Plasticity

Boundary Conditions: u(x = 0, t) = 0 u(x = L) = u0 Initial Conditions: u(x, t = 0) = 0 σ(x, t = 0) = 0 εp (x, t = 0) = 0 q(x, t=0) = 0

L

A(x)

u0

Assume a very slow change of area such that deformation is homogeneous

Solving Methodology

Assume a total time, T0, and a constant velocity, v0, such that v0 = u0/T0 For rate independent model, the choice of T0 is immaterial

Divide 0 to T0 into equal number of intervals

The size of the intervals affects convergence

Solve the weak form of force equilibrium equation at every time point tn+1, knowing

the state (u, σ, εp, q) of its previous time point at tn using NR

Derive weak form by separation of variables Show forward and backward Euler methods Derive the incremental strain Derive the numerical integration of elasto-plastic model (User material) Show MATLAB code of 1d elasto-plastic user material

1D Elasto-Viscoplasticity

In rate dependent plasticity, the plastic response depends on the rate of loading

The yield and consistency condition eliminates the strain rate dependence in the model

In rate dependent plasticity, there is no yield surface and hence no consistency condition

A model for plastic multiplier needs to prescribed independently

Overstress Model

Proposed by Perzyna (1971)

Power Law Model

Proposed by Peirce et. al. (1984)

Comparison

In overstress model plastic strain rate is non zero only when reference stress is exceeded

In power law, plastic strain also possible below reference strain, thus, the model can also be used to model creep

Derive the numerical integration of elasto-viscoplastic model (User material) Show MATLAB code of 1d elasto-viscoplastic user material

Show MATLAB FEM code for elasto-plastic 1D bar

Multi-axial Stress State

A component can be subjected to multi-axial and non-uniform loading (e.g. wings of aircraft, blades of turbine, axles, etc.) – multi-axial stress state. Non-uniform geometry can cause local

multi-axial stress state (e.g. cut-outs, holes, while necking, etc.) even under uniaxial loading. Can lead to local plasticity

Uniaxial

Multi-axial Uniaxial

Multi-axial

Uniaxial Observations

Uniaxial tensile, compression and torsion tests – elasto-plastic response under pure normal and shear stress Tests on samples machined out at different

orientations from a material block From the stress-strain response deduce

information on anisotropy Material Block

Samples

Plastic Strain

Similar to 1D, the strain tensor can be divided into an elastic and plastic part Under small strain assumption and in the indicial notation

In the rate form

Hypo-Elasticity

Stress at a point is due to the elastic-stretching of the lattice (metal plasticity) In the rate form

Einstein’s notation - Repeated index means summation For Isotropic elasticity 𝛿𝛿𝑖𝑖𝑖𝑖 = � 1 if i = j

0 otherwise

Kronecker Delta

Hydrostatic and Deviatoric Tensor

Tensors can be additively decomposed into hydrostatic and deviatoric part Deviatoric component is traceless (𝐴𝐴𝑖𝑖𝑖𝑖 = 0)

Hydrostatic stress – Pressure For strain rate: Hydrostatic – rate of volume change, Deviatoric – shape change For incompressible deformation, the total strain rate is only deviatoric

Yield Surface

From experiments on rate independent materials it is observed that plastic deformation happens only when some critical stress (σy) is exceeded For multi-axial, all the stress states that cause plastic

deformation can be considered to constitute a continuous surface – Yield Surface Divides the space into an elastic and plastic region

Yield Surface (Curve) in Principal Stress Space under

Plane Stress assumption

Elastic

Plastic

Von-Mises Yield Criterion

Applicable to rate independent isotropic elasto-plastic materials Experimental observation – hydrostatic stress has negligible influence on plastic

behaviour of most metals The model considers the shear stress on octahedral plane Pressure axis

Octahedral plane

Octahedral plane has normal equally inclined to the principal stress axis Thus, for a given state of stress

J2 – 2nd invariant of Deviatoric Stress Tensor

Von-Mises Yield Criterion

The yield criterion is defined as

J2 in terms of stress components

Under uniaxial tension along 1 – yielding happens when

From the yield criterion this gives

Final form where

Flow Rule

Plastic strain rate is non-zero when yield criterion is satisfied Thus, if the flow rule is defined as

Then

Furthermore, for the material to yield continually, over time dt, the stress state has to remain on the yield surface Thus,

Normality Rule

Plastic flow direction is chosen normal to the yield surface Maximizes plastic dissipation Thus,

Since plastic potential (φ) is same as yield function, the flow rule is defined as associative

Isotropic Strain Hardening

Plastic strain is a manifestation of dislocation motion (metals) The motion is obstructed by other dislocations, resulting in an intrinsic

resistance of the material (yield stress) For isotropic elasto-plastic behaviour, a scalar internal variable (q) can be

defined that is related to the microstructure (dislocation density, etc.) An empirical function can be proposed to capture the variation of internal

resistance to q or σY = σY(q) – strain hardening

Evolution of Internal Variable

During plastic deformation the dislocation multiply (Frank-Reed source), annihilate, etc. Thus, the microstructure evolves with plastic deformation

Thus, a rate form is proposed for the evolution of the internal variable, q, that

is relates the microstructure The plastic multiplier gives a scalar measure of extent of plasticity

Thus,

Von-Mises Elasto-Plastic Model

σY = σY(q)

Numerical Integration

While solving an initio-boundary value problem with the material response given by Vonmises plasticity, the rate equations need to be solved at every Gauss point at time, tn+1, for a prescribed

to obtain stress and tangent moduli and

Derive radial return mapping algorithm Derive tangent moduli

Large Deformation and Updated Lagrangian

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