Titolo presentazione

sottotitolo

Milano, XX mese 20XX

Introduction and abaqus installation procedure

Equilibrium and biharmonic equations in polar coordinates

Practical lessons in Fracture MechanicsMaster of Science in Civil Engineering

Nicola Cefis

Nome Cognome, assoc.prof. ABC Dept.

General information abut the teacher of practical lessons

Name: Nicola Cefis

Affiliation: Dipartimento di Ingegneria Civile e Ambientale del Politecnico di Milano

Office: Building 5 – Leonardo Campus

Mail: nicola.cefis@polimi.it

Skype: nicola.cefis_1

Nome Cognome, assoc.prof. ABC Dept.

Table of contents

1. How to install Abaqus CAE

2. Equilibrium equations in cartesian cordinates

3. Equilibrium equations in polar cordinates

4. Lamè problem of the thick cylinder with internal and external pressure

Nome Cognome, assoc.prof. ABC Dept.

HOW TO INSTALL ABAQUS CAE

Nome Cognome, assoc.prof. ABC Dept.

How to install Abaqus CAE

https://www.software.polimi.it/en/software-download/students/

Nome Cognome, assoc.prof. ABC Dept.

How to install Abaqus CAE

Nome Cognome, assoc.prof. ABC Dept.

How to install Abaqus CAE

Nome Cognome, assoc.prof. ABC Dept.

How to install Abaqus CAE

Create new accountor

log in (if you are already registered on the 3DS Academy website)

Nome Cognome, assoc.prof. ABC Dept.

How to install Abaqus CAE

After log in or registration…

Nome Cognome, assoc.prof. ABC Dept.

How to install Abaqus CAE

Accept the terms of use…

Download and install Abaqus v.19

Nome Cognome, assoc.prof. ABC Dept.

EQUILIBRIUM EQUATIONS IN CARTESIAN (X, Y) COORDINATES

Equilibrium equations in

cartesiancoordinates

Equilibrium equations in

polarcoordinates

Airy’s stress function in

polar coordinates

Example of application:

Lamè’s problem

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in cartesian (x, y) coordinates

Stresses at the incremental faces (A’ and B’)

A’

B’

Stresses at the base faces (A and B)

A

B

Compact form Extended form

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in cartesian (x, y) coordinates

Resultant forces at the incremental faces (A’ and B’)

A’

B’

Resultant forces at the base faces (A and B)

A

B

Resultant of internal forces

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in cartesian (x, y) coordinates

Equilibrium equation in x direction

Compact form

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in cartesian (x, y) coordinates

Equilibrium equation in x direction

Compact form

Compact form

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in cartesian (x, y) coordinates

Equilibrium equation in x direction

Compact form

Compact form

Extended form

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in cartesian (x, y) coordinates

Equilibrium equation in x direction

Compact form

Compact form

Extended form

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in cartesian (x, y) coordinates

Equilibrium equation in y direction

Compact form

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in cartesian (x, y) coordinates

Equilibrium equation in y direction

Compact form

Compact form

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in cartesian (x, y) coordinates

Equilibrium equation in y direction

Compact form

Compact form

Extended form

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in cartesian (x, y) coordinates

Equilibrium equation in y direction

Compact form

Compact form

Extended form

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in cartesian (x, y) coordinates

Equilibrium equations in x,y direction

Nome Cognome, assoc.prof. ABC Dept.

EQUILIBRIUM EQUATIONS IN POLAR (r, q) COORDINATES

Equilibrium equations in

cartesiancoordinates

Equilibrium equations in

polarcoordinates

Airy’s stress function in

polar coordinates

Example of application:

Lamè’s problem

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in polar (r, q) coordinates

Stresses at the incremental faces (A’ and B’)

A’

B’

Stresses at the base faces (A and B)

A

B

Compact form Extended form

hp: zero volume forces

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in polar (r, q) coordinates

Resultant forces at the incremental faces (A’ and B’)

A’

B’

Resultant forces at the base faces (A and B)

A

B

hp: zero volume forces

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in polar (r, q) coordinates

hp: zero volume forces

Equilibrium equation in radial direction (r)

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in polar (r, q) coordinates

hp: zero volume forces

Equilibrium equation in radial direction (r)

high order term

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in polar (r, q) coordinates

hp: zero volume forces

Equilibrium equation in radial direction (r)

high order term

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in polar (r, q) coordinates

hp: zero volume forces

Equilibrium equation in radial direction (r)

high order term

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in polar (r, q) coordinates

hp: zero volume forces

Equilibrium equation in hoop direction (q)

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in polar (r, q) coordinates

hp: zero volume forces

Equilibrium equation in hoop direction (q)

high order term

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in polar (r, q) coordinates

hp: zero volume forces

Equilibrium equation in hoop direction (q)

high order term

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in polar (r, q) coordinates

hp: zero volume forces

Equilibrium equation in hoop direction (q)

high order term

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in polar (r, q) coordinates

hp: zero volume forces

Equilibrium equations in r, q direction

Nome Cognome, assoc.prof. ABC Dept.

Equilibrium equations in polar (r, q) coordinates

hp: zero volume forces

Equilibrium equations in r, q direction

We can solve the system through the

Airy’s stress function φ

Nome Cognome, assoc.prof. ABC Dept.

AIRY’S STRESS FUNCTION φ IN POLAR CORDINATES

Equilibrium equations in

cartesiancoordinates

Equilibrium equations in

polarcoordinates

Airy’s stress function in

polar coordinates

Example of application:

Lamè’s problem

Nome Cognome, assoc.prof. ABC Dept.

Definition of the Airy’s stress function φ

Let's go back to the x,y coordinate system. The state of stress can be expressed through the Airy’sstress function φ:

Nome Cognome, assoc.prof. ABC Dept.

Definition of the Airy’s stress function φ

Let's go back to the x,y coordinate system. The state of stress can be expressed through the Airy’sstress function φ:

The Laplacian of the Airy’s stress function can be expressed as:

Nome Cognome, assoc.prof. ABC Dept.

Definition of the Airy’s stress function φ

Let's go back to the x,y coordinate system. The state of stress can be expressed through the Airy’sstress function φ:

Let us consider the first invariant of the stress tensor to define the Airy’s stress function in polar coordinates

The Laplacian of the Airy’s stress function can be expressed as:

Nome Cognome, assoc.prof. ABC Dept.

Definition of the Airy’s stress function φ

Let's go back to the x,y coordinate system. The state of stress can be expressed through the Airy’sstress function φ:

Let us consider the first invariant of the stress tensor to define the Airy’s stress function in polar coordinates

The Laplacian of the Airy’s stress function can be expressed as:

Nome Cognome, assoc.prof. ABC Dept.

Definition of the Airy’s stress function φ

To use the previous relationship we formulate the derivatives of the Airy’s stress function φ in polar coordinate :

Nome Cognome, assoc.prof. ABC Dept.

Definition of the Airy’s stress function φ

To use the previous relationship we formulate the derivatives of the Airy’s stress function φ in polar coordinate :

First derivatives

Nome Cognome, assoc.prof. ABC Dept.

Definition of the Airy’s stress function φ

To use the previous relationship we formulate the derivatives of the Airy’s stress function φ in polar coordinate :

First derivatives

Nome Cognome, assoc.prof. ABC Dept.

Definition of the Airy’s stress function φ

To use the previous relationship we formulate the derivatives of the Airy’s stress function φ in polar coordinate :

First derivatives

Nome Cognome, assoc.prof. ABC Dept.

Definition of the Airy’s stress function φ

To use the previous relationship we formulate the derivatives of the Airy’s stress function φ in polar coordinate :

First derivatives

Nome Cognome, assoc.prof. ABC Dept.

Definition of the Airy’s stress function φ

To use the previous relationship we formulate the derivatives of the Airy’s stress function φ in polar coordinate :

First derivatives

Nome Cognome, assoc.prof. ABC Dept.

Definition of the Airy’s stress function φ

To use the previous relationship we formulate the derivatives of the Airy’s stress function φ in polar coordinate :

Second derivatives

Nome Cognome, assoc.prof. ABC Dept.

Definition of the Airy’s stress function φ

Second derivatives

Nome Cognome, assoc.prof. ABC Dept.

Definition of the Airy’s stress function φ

Second derivatives

Nome Cognome, assoc.prof. ABC Dept.

Definition of the Airy’s stress function φ

Final results

Equilibrium equations:

Compatibility equations:

Nome Cognome, assoc.prof. ABC Dept.

APPLICATION EXAMPLE: LAME’S PROBLEM OF THE THICK CYLINDER WITH INTERNAL AND EXTERNAL

PRESSURE (1852)

Equilibrium equations in

cartesiancoordinates

Equilibrium equations in

polarcoordinates

Airy’s stress function in

polar coordinates

Example of application:

Lamè’s problem

Nome Cognome, assoc.prof. ABC Dept.

Lamè’s problem of the thick cylinder with internal and external

pressure (1852)

Field equations

Nome Cognome, assoc.prof. ABC Dept.

Lamè’s problem of the thick cylinder with internal and external

pressure (1852)

Field equations

Boundary conditions

Nome Cognome, assoc.prof. ABC Dept.

Lamè’s problem of the thick cylinder with internal and external

pressure (1852)

Nome Cognome, assoc.prof. ABC Dept.

Lamè’s problem of the thick cylinder with internal and external

pressure (1852)

Nome Cognome, assoc.prof. ABC Dept.

Lamè’s problem of the thick cylinder with internal and external

pressure (1852)

General integral of the fourth order equation and its derivatives

Nome Cognome, assoc.prof. ABC Dept.

Lamè’s problem of the thick cylinder with internal and external

pressure (1852)

General integral of the fourth order equation and its derivatives

Stress inside thick cylinder (3 unknows):

Nome Cognome, assoc.prof. ABC Dept.

Lamè’s problem of the thick cylinder with internal and external

pressure (1852)

I must calculate the integration constants

A, B, C

❖ Static boundary conditions: 2 equations

❖ Consistency based considerations: 1 equation

The circumferential displacement must be equal in phi = 0 and phi = 2pi from which it is obtained B=0

Nome Cognome, assoc.prof. ABC Dept.

Lamè’s problem of the thick cylinder with internal and external

pressure (1852)

Final results:

Firma convenzione

Politecnico di Milano e Veneranda Fabbrica

del Duomo di Milano

Aula Magna – Rettorato

Mercoledì 27 maggio 2015

Introduction and abaqus installation procedure

Equilibrium and biharmonic equations in polar coordinates

Practical lessons in Fracture MechanicsMaster of Science in Civil Engineering

Nicola Cefis