fem solid mechanics

8/11/2019 FEM Solid Mechanics

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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

PlasticityViscoplasticity, creepFracturing materials

Geometrical nonlinearities

Contact problemsEquilibrium in deformed configurationFinite strains


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Linear and nonlinear problemsGeometrical nonlinearities:Equilibrium in deformed configuration

P

P


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No linealidad Geomtrica

Fuerza

aplicada

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

Tensi

na

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]

Sealcontactpressure[MPa]

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]

Sealcontactpressure[Mpa]

MU (Sphere)

100% Pc (Sphere)

197% Pc (Sphere)


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Linear and nonlinear problemsMaterial + Geometrical nonlinearities + ContactFinite 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 thecompatibility 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 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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Equilibrium in the deformed configuration


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The Cauchy Stress TensorEquilibrium in the deformed configuration

b is the force per unit massIn dynamic analyses include in f the inertia forces.


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Torques equilibrium

(symmetry)


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The Cauchy Stress Tensor and Physic

In nonlinear problems there are a numberof stress measures that are used during calculations:

Kirchhoff stress tensor

Second Piola-Kirchhoff stress tensorBiot stress tensoretc.

They are only mathematical tools.

The final results with significance for usshould be expressed in terms ofthe Cauchy stress tensor.


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Constitutive Relations

Phenomenological constitutive relations
http://www.mts.com/en/Material/Dynamic/index.asp


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P

Elastic material


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Constitutive RelationsHOOKEs LAWLinear elastic and isotropic materials


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Elasto-plastic materials

Ingredients:

Yield surface: in the 3D stress space describes the locus of the pointswhere the plastic behavior is initiated.

Flow rule: describes the evolution of the plastic deformations.

Hardening law: describes the evolution of the yield surface duringthe plastic deformation process.


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Elasto-plastic materials

: deviatoric stress tensor


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Elasto-plastic materialsVon Mises yield function (metals)


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Elasto-plastic materialsVon Mises yield function (metals)


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Elasto-plastic materials: The flow rule

fgmetalsplasticityassociatedFor

g

ij

ijP

ijP

ijE

ij

)(

g: plastic potential


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fgmetalsplasticityassociatedFor

g

ij

ijP

ijP

ijE

ij

)(

g: plastic potential


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For metals (von Mises + associated plasticity)

0P

zz

P

yy

P

xx

The plastic flow is incompressible


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Elasto-plastic materials:Hardening

t

t+ t

Plastic loading

ISOTROPIC HARDENING


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The isotropic hardening does not modelthe Bauschinger effect (Cyclic loading / unloading)


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t

t+ tKINEMATIC HARDENING


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Viscoplasticity

In plasticity:

We need viscoplasticity to model the experimental fact:


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A 1D problem

fB R

Adxdx

d)(

A: transversal areaE: Youngs modulus

A

dxf Bx

x=0 x=L

fB

: load per unit lengthR : concentrated load

u(x) : unknown


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A 1D problem

R

dx

duEA

u

Lx

0)0( Essential (rigid) boundary condition

Natural boundary condition


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A 1D problemEquilibrium:

0Bf

dx

dA

Constitutive equation:

E

Kinematic relation:

dx

du


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A 1D problem

02

2B

f

dx

udAE

At every point inside the bar we must fulfill:

u(x) is an arbitrary function

u(0)=0 (condition)

Hence, 00

2

2

dxufdx

udAE

L

B


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A 1D problem

Integrating by parts,

Lx

L L

BuRdxufdxA

0 0

Virtual work of internal forces = Virtual work of external forces


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Please notice that the PVW represents

Equilibriumand NOT

Energy Conservation


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General 3D case

b: Loads per unit volumet: Loads per unit surface

Please notice that the integral is calculated atthe deformed (unknown) configuration


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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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