non-linear finite element methods in solid mechanics

Politecnico di Milano, February 17, 2017, Lesson 5

Non-Linear Finite Element Methods in Solid MechanicsAttilio Frangi, [email protected]

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Politecnico di Milano, February 17, 2017, Lesson 5 2

Outline

Lesson 1-2: introduction, linear problems in staticsLesson 3: dynamicsLesson 4: locking problemsLesson 5: geometrical non-linearitiesLesson 6-7: small strain plasticity


Lesson 5: Introduction to non linear analysis

1. Sources of non-linearitiesUnilateral contactFracture propagationNon-linear constitutive lawsGeometrical non-linearities

2. Methods of numerical solutionNon linear equations: Newton-like iterative algorithmsExample of iterative algorithm: large transformations

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Example of analytical solution: spherical rigid indenter against a deformable surface (Hertz solution, 1882)

Radius of the contact region a and indentation depth depend non-linearly on force P

Example: Hertz contact

The behaviour of a system of two solids in contact is a nonlinear functionof external loading even if the deformable solid is linear elastic


Unilateral contact (infinitesimal transformations)

Unilateral contact without friction

slave surface

master surface

Select a master surface and a slave surface.

The nodes of the slave surface cannot penetrate the master surface

: gap along normal direction

no tangential force

compression!

no compenetration

complementarity

Sc area of potential contact





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Fracture propagation





Plasticity; lessons 6-7

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Geometrical non-linearities

shear bucklingof a membrane

more than one possible solution.Transition betweendifferent solutionsvia buckling


Geometrical non-linearities parachute instability


Tetra Pak

Example of highly non-linear problem

contact with friction fracture with unknown path non-linear constitutive law (damage like) large displacements


Geometrical non-linearities, “very large” transformations

lagrangian (typical of solid mechanics) vs eulerian (typical of fluid mechanics) approaches

Tetra Pak


Geometrical non-linearities, “very large” transformations

casting applications





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Numerical solution of a non-linear scalar equation

Find u such that r(u)=0

Iterative procedure: create a sequence u(k) → u such that r(u)=0

Newton-Raphson method: truncated first order series expansion of r(u(k+1)) around u(k)

and solution of the associated linear equation

u(1) u(2)u(0)

u

r(u)

EXERCISE

apply to r(u)=-3+(u+1)2

1) compute r’2) linearization - expansion


Quadratic convergence of the Newton-Raphson method

Newton-Raphson method: quadratic convergence speed in the vicinity of the solution

Setting e(k) = u(k) - u (error w.r.t. solution) for any k, one has:

Taylor expansion with remainder for r(u(k)) and r’(u(k)) around the exact solution u


Divergence examples of the Newton-Raphson method


Convergence of the Newton-Raphson method

u(0)=0

u

r(u)

u2(0)

u2(1) u2

(3)

r(u) = 0

sequence of N sub-problems

the solution of i-th sub-problemis employed as initial guess for(i+1)-th sub-problem

r(u) = (N-i)/N r(u(0)) i=1 … N r(u(0))


Modified Newton-Raphson method - 1

u

u(0) u(1) u(2)

r(u)r’(u) replaced with a constant K, which gives

u(3)

The convergence is only linear near the solution

K might be the tangentat the initial estimate


Modified Newton-Raphson method - 2

Approximate tangent with segment passing through the two previous estimates

u

u(0) u(1) u(2)

r(u)


Non linear system: Newton-like iterative algorithms

The analysis of structures often leads to the solution of a system of non-linear equations:

(typically: weak enforcement of equilibrium with PPV + constitutive laws)see e.g. the examples in the sequel

In the linear elastic case (lessons 1-3) one would have:

Newton-like algorithms: iterative approaches for the numerical solution ofa system of non-linear equations

total value of displacement or increment in an interative procedure


Solution of a system of a non-linear system with NR technique





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Geometrical non-linearities: buckling of a beam in compression


Buckling of a beam in compression

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deformation gradientGreen Lagrange tensor

(second) Piola (Kirchhoff)

first Piola Kirchhoff

Saint Venant Kirchhoff model for large displacements butsmall strain case (typical of buckling analysis)

Summary of background



here assumed givenhere assumed 0


Geometrical non-linearities, finite transformations

velocity strain tensor


Newton iterative procedure

given the current iterate

find the new iterate

through a linearization of the residuum:



Linearization of the residuum:


………

Element procedures for a T6


loading sequence

initialisation of each step

non-linear finite element methods in solid mechanics

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