Milano NSM 9

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9. Plasticity

Marino Arroyo & Anna Pandolfi

An Introduction to Nonlinear Solid Mechanics

Doctoral School Politecnico di Milano

November 2014


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NLSM Nov 2014 Marino Arroyo & Anna Pandolfi

Uniaxial Cyclic Behaviors

Elastic material: the path followed during

loading is also followed during unloading.

Typical of a reversible behavior.

Dissipative material: on the way back tothe origin, an isteretic cycle is drawn,

testifying disspation of energy.

Plasticity: a residual irreversible

deformation characterizes the cycle.

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Ideal Plasticity Types 263

Plasticity manifests itself when a stress

threshold is reached (yield stress). Beyond

such limit, irreversible deformations

appear, otherwise the behavior is fullyelastic. The limit is called yield condition:

Total strains (compatible) are the sum ofan elastic part (related to the stress), and a

plastic part, dependent on the load history:

Perfect plasticity: the plastic behavior isfully described by plastic strains.

Hardening: the threshold evolves with the

loading history and new variables are

needed to describe the behavior.


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Basic Elements of Plasticity

The plastic strains evolve according to evolution equations (flow rule):

The evolution of the yield function needs the definition of a hardening function.

A consistency condition guarantees that stress never violates the yield function.

In 3D a plasticity model needs:

An elastic law;

A yield condition;

A flow rule;

A hardening rule.

At the attainment of the plasticity condition the

material may behave elastically (upon unloading)or develop plastic deformations.

The effective choice is dictated by the external

loading and is incremental by nature.

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Incremental Linearized 3D Plasticity

Generalize the previous concepts to 3D, use Cauchy stresses, small strains and

Hooke law.

The solution at the beginning of the increment is known. Want to compute the

solution at the end of the increment

Additive decomposition of incremental strains:

Linearized elasticity:

Yield function:

Flow-rule, where is the plastic multiplier or magnitude, and gy defines the

direction:

A hardening rule will introduce additional internal variables q.

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J2 (or Von Mises) Plasticity

Assumptions:

Mises yield function;

Zero volumetric plastic strains;

Sole internal variable: yield strain y.

Plastic strains follow the gradient of the yield function gy = fy (normality).

The constitutive law splits into volumetric and deviatoric parts:

The total strain is known from boundary conditions. The unknowns are the

plastic part of the deviatoric strain ep, and the deviatoric stress s.

Start from the solution at the end of the previous step:

Express the deviatoric part of the stress as a function of the effective strain:

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J2 Plasticity Equations

State the consistency of the scalar Mises yield function at the end of the step:

Set up the hardening rule:

Flow-rule, ep and s are parallel:

The deviatoric constitutive law shows that e is also parallel to s:

write the Mises law in scalar terms to obtain the plastic multiplier :

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Effective Stress Scalar Function

The scalar equation is often called effective stress function and for more

complicated behaviors is also written as:

The detection of the effective stress is reduced to the search of the zero of a

nonlinear scalar function.

Any plasticity problem may be reduced

to a nonlinear scalar function where

the effective stress is a function

of the effective plastic strain:

which accounts for the flow rule.

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Linear Hardening and Perfect Plasticity

For a material with isotropic hardening,

the effective stress function reduces to a

linear relationship between effective

plastic strain and effective stress:

By adding the flow-rule, obtain:

For ideal plasticity, ET= 0, and it results:

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Elastic Predictor

An elastic-plastic step is performed only if a

preliminary elastic step detects a violation of

the yield condition.

By imposing an increment of the

displacement, a purely elastic predictor step

is computed:

If the resulting effective stress is less then

the current yield stress (as in the unloading

case), the response is elastic.

Otherwise, the response is elastic-plastic:

and in order to recover the solution the

previous equations must be used.

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Radial Return

Interpretation on the deviatoric plane.

A indicates the stress at the end of the

previous step.

B indicates the predictor step.

Vector CB indicates the corrector step:

The final stress is represented by the point

C and lies on the yield surface:

Points B and C are aligned in the direction of the normal to the plasticity

condition, along the radius of Mises cylinder (circle), that justifies the name of

radial return for the method.

The integration of the elastic-plastic law estimates the plastic deformations by

means of backward differences. The algorithm is exact if the direction of the

deviatoric stress does not change during the plastic corrector.

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Consistent Stiffness Matrix

The integration algorithms for nonlinear problems use the Newton-Raphson

iterative procedure. To guarantee a good convergence rate, it is important that

the stiffness matrix is consistent with the integration algorithm. In the case of

total stress and total strains, this implies:

In particular, the computation of the stiffness matrix must be performed in a

correct way, in order to include all the variables which define the materialbehavior. It must be done by a correct application of the chain rule. For

example, the bilinear elastic-plastic model is described through the effective

stress, and the correct derivation of the stiffness matrix requires:

The procedure can be easily extended to other plasticity conditions (as Tresca

or Drucker-Prager).

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Finite Deformation Elastoplasticity

Several formulations and proposals:

Green e Naghdi [1965, 1971], Lee [1969, 1981], Sim [1985, 1992], Ortiz

[1985, 1992, 1999, 2001], Lubliner [1998],

For a correct formulation, it is important the choice of: stress and strain

measures, elastic constitutive law, flow-rule.

In recent times, plenty of finite formulations have been developed: the

equilibrium is enforced at the final time of the step in terms of total stresses. Thenumeric integration is used only to compute the inelastic strain increment.

In a material description, it is customary to decompose the deformation gradient

into the product of an elastic and a plastic part.

In rate formulations, stress, strains and rotations are expressed in rates. All the

physical quantities are numerically integrated. This introduces numerical andconceptual errors, and produces responses in some cases physically not

consistent.

Rates may be considered only in the parts of the law where they are really

necessary, i.e. the flow rule.

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Multiplicative Decomposition

The deformation

gradient F is

decomposed as:

The inelastic partFp

corresponds to the attainment (at the ideal time) of anintermediate configuration where all the inelastic deformation have taken place

and the body is relaxed and unstressed. Here incompressibility is assumed.

The elastic part Fe is related to the attainment of a final configuration, through an

additional elastic deformation which is totally responsible of the stress.

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Deformation and Velocity Gradients

The elastic part of the deformation gradient is expressed as:

The volume change is related to the elastic deformation only:

The rate of the deformation gradient is expressed as:

The velocity gradient formally decomposes in the sum of two terms:

Note the both le and lp are defined in the current (final) configuration (they are

spatial quantities). While this is correct for le, for lp a definition in the

intermediate configuration would be more appropriate.

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Work Conjugate Measures of Stress and Strain

The elastic part of the deformation gradient obeys the polar decomposition:

In the material description, we use the Hencky strain tensor referred to the

intermediate configuration (logarithmic mapping):

The stress corresponds to the Cauchy tensor in the current configuration but it isreferred to the intermediate configuration (barred), thus it includes Re and J:

Stress and strain rates are conjugate in the power expression:

The effective plastic velocity gradient (barred) in the intermediate configuration

is defined as:

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Governing Equations

Elastic constitutive law: linear elasticity extended to the finite deformations,

decomposed into volumetric and deviatoric parts:

Mises yield function expressed in terms of deviatoric stress in the intermediate

configuration:

Flow-rule (enforcing normality) for the evolution of the plastic strain:

The normal n, plastic velocity gradient and its symmetric and skew-symmetric

parts are all defined in the intermediate configuration:

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Solution Procedure

Given F(t), evaluate a predictor elastic stress (at ideal time ):

Elastic deformation gradient

Elastic polar decomposition

Elastic logarithmic deformation tensor (logarithmic mapping):

Cauchy stress in the intermediate configuration:

Effective stress in the intermediate configuration:

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Solution Procedure (cont)

Yield condition check:

If : elastic solution, compute Cauchy stress at final time:

Start a new step.

Otherwise, plastic corrector:

The effective stress function is solved numerically, computing the value ofthe effective stress in the intermediate configuration and the value of the

plastic deformations:

Compute the plastic multiplier:

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Solution Procedure (end)

Compute the deviatoric stress in the intermediate configuration:

Compute the total final stress (add the volumetric part), referred to the

intermediate configuration:

Compute the final Cauchy stress, referred to the final configuration:

The plastic deformation gradient in the final configuration is computed by

integration of the incremental expression (exponential mapping):

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