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FEM in Solid Mechanics

Part I

Eduardo Dvorkin

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Contents

Part 1

Linear and nonlinear problems in solid mechanics

Fundamental equations:

Kinematics

The stress tensor

Equilibrium

Constitutive relations

The principle of virtual work

Part 2

FEM in solid mechanics

Elasto-plasticity

Structural elements

Nonlinear problems- Collapse

Dynamic problems

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Linear and nonlinear problemsMaterial nonlinearities

Plasticity

Viscoplasticity, creep

Fracturing materials

Geometrical nonlinearities

Contact problems

Equilibrium in deformed configuration

Finite strains

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Linear and nonlinear problemsGeometrical nonlinearities:

Equilibrium in deformed configuration

P

P

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No linealidad Geométrica

Fue

rza

aplic

ada

Desplazamiento vertical

MATERIAL ELASTIC NAME = 1,

E = 21000 NU = 0.3

EGROUP TRUSS NAME = 1,

DISPLACEMENTS = LARGE,

MATERIAL = 1,

INT = 2,

area = 10.

Desplazamiento vertical

Ten

sió

n a

xial

P

20

10

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Linear and nonlinear problemsMaterial + Geometrical nonlinearities + Contact: Collapse

Buckle arrestors

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Linear and nonlinear problemsMaterial + Geometrical nonlinearities + Contact: Collapse

Contact forces

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Linear and nonlinear problems

Material + Geometrical nonlinearities + Contact

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Linear and nonlinear problemsMaterial + Geometrical nonlinearities + Contact

OCTG connections

Structural Analysis

Detail of

the seal area

0

500

1,000

1,500

2,000

2,500

3,000

0123456

Distance along seal area [mm]

Sea

l co

nta

ct p

ress

ure

[M

Pa]

MU (Cone)

100% Pc (Cone)

197% Pc (Cone)

0

500

1,000

1,500

2,000

2,500

3,000

0123456

Distance along seal area [mm]

Sea

l co

nta

ct p

ress

ure

[M

pa

]

MU (Sphere)

100% Pc (Sphere)

197% Pc (Sphere)

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Linear and nonlinear problemsMaterial + Geometrical nonlinearities + Contact

Finite elasto – plastic strains

OCTG connections: Failure analysis

disp = 10.60 mm

disp = 16.60 mm

disp = 20.06 mm

disp = 31.91 mm

disp = 46.72 mm

disp = 5.77

mm

disp = 4.10 mm

Make Up

disp = 52.65 mm

Equivalent

PlasticStrain

6.72%4.29%2.73

%1.11

%0.71%0.45%

1.74

%

10.5%

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Cartesian Coordinate System

x (z1)

y (z2)

z (z3)

k

k: point or material particle

Continuos body

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Kinematics of the Continuous Media

Lagrangian description

Eulerian description

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Pop Quiz # 2

Eulerian formulation

t

v

t

va

Is “a” the acceleration?

NO

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Kinematics of the Continuous Media

),,(

),,(

),,(

22

22

22

22

22

22

),,(

),,(

),,(

zyxw

zyxv

zyxu

zyxw

zyxv

zyxu

zzzyzx

yzyyyx

xzxyxx

zzzyzx

yzyyyx

xzxyxx

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Kinematics of the Continuous MediaThe second transformation is only possible if the

compatibility equations are fulfilled

0

0

0

02

02

02

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

zyxxyx

zyxyxz

zyxxzy

xzzx

zyyz

yxxy

xyzxyzzz

xyzxyzyy

xyzxyzxx

zxxxzz

yzzzyy

xyyyxx

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The Cauchy Stress Tensor

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The Cauchy Stress Tensor

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The Cauchy Stress TensorDefinition:

(i=1,2,3)

(Above we use the summation convention)

P

t

n

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Pop Quiz # 3Water tank

H

A

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The Cauchy Stress Tensor

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The Cauchy Stress Tensor

Equilibrium in the deformed configuration

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The Cauchy Stress TensorEquilibrium in the deformed configuration

b is the force per unit mass

In dynamic analyses include in f the inertia forces.

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The Cauchy Stress TensorTorques equilibrium

(symmetry)

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The Cauchy Stress Tensor and Physic

In nonlinear problems there are a number

of stress measures that are used during calculations:

Kirchhoff stress tensor

Second Piola-Kirchhoff stress tensor

Biot stress tensor

etc.

They are only mathematical tools.

The final results with significance for us

should be expressed in terms of

the Cauchy stress tensor.

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Constitutive Relations

Phenomenological constitutive relations

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Constitutive Relations

P

Elastic material

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Constitutive RelationsHOOKE’s LAW

Linear – elastic and isotropic materials

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Constitutive RelationsElasto-plastic materials

Ingredients:

Yield surface: in the 3D stress space describes the locus of the points

where the plastic behavior is initiated.

Flow rule: describes the evolution of the plastic deformations.

Hardening law: describes the evolution of the yield surface during

the plastic deformation process.

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Constitutive RelationsElasto-plastic materials

: deviatoric stress tensor

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Constitutive RelationsElasto-plastic materials

Von Mises yield function (metals)

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Constitutive RelationsElasto-plastic materials

Von Mises yield function (metals)

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Constitutive RelationsElasto-plastic materials: The flow rule

fg

metalsplasticityassociatedFor

g

ij

ijP

ijP

ijE

ij

)(

g: plastic potential

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Constitutive RelationsElasto-plastic materials: The flow rule

fg

metalsplasticityassociatedFor

g

ij

ijP

ijP

ijE

ij

)(

g: plastic potential

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Constitutive RelationsElasto-plastic materials: The flow rule

For metals (von Mises + associated plasticity)

0P

zz

P

yy

P

xx

The plastic flow is incompressible

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Constitutive RelationsElasto-plastic materials:Hardening

t

t+ t

Plastic loading

ISOTROPIC HARDENING

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Constitutive RelationsElasto-plastic materials:Hardening

The isotropic hardening does not model

the Bauschinger effect (Cyclic loading / unloading)

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Constitutive RelationsElasto-plastic materials:Hardening

t

t+ tKINEMATIC HARDENING

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Constitutive RelationsViscoplasticity

In plasticity:

We need viscoplasticity to model the experimental fact:

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The principle of virtual workA 1D problem

fB R

Adxdx

d)(

A: transversal area

E: Young’s modulus

A

dxf Bx

x=0 x=L

fB : load per unit length

R : concentrated load

u(x) : unknown

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The principle of virtual workA 1D problem

Rdx

duEA

u

Lx

0)0( Essential (rigid) boundary condition

Natural boundary condition

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The principle of virtual workA 1D problem

Equilibrium:

0Bfdx

dA

Constitutive equation:

E

Kinematic relation:

dx

du

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The principle of virtual workA 1D problem

02

2Bf

dx

udAE

At every point inside the bar we must fulfill:

δu(x) is an arbitrary function

δu(0)=0 (condition)

Hence, 00

2

2

dxufdx

udAE

L

B

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The principle of virtual workA 1D problem

Integrating by parts,

Lx

L L

B uRdxufdxA0 0

Virtual work of internal forces = Virtual work of external forces

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The principle of virtual work

Please notice that the PVW represents Equilibriumand NOT

Energy Conservation

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The principle of virtual workGeneral 3D case

b: Loads per unit volume

t: Loads per unit surface

Please notice that the integral is calculated at

the deformed (unknown) configuration

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The principle of virtual work

No material restriction (applies to any material)

No kinematics restriction (large or small strains)

No loads restriction (conservative or non-conservative)

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References