Topics in Ship Structuresocw.snu.ac.kr/sites/default/files/NOTE/06 Energy Thoery.pdf · Topics in Ship Structures 06 Energy Theory. OPen INteractive Structural Lab Energy approach

OPen INteractive Structural Lab

Reference : Solid Mechanics Books in

The University of AUCKLAND

2017. 9

by Jang, Beom Seon

Topics in Ship Structures

06 Energy Theory


Energy approach in Fracture Mechanics

Griffith theory for brittle material (1920’s)

“A crack in a component will propagate if the total energy of the

system is lowered with crack propagation.”

“ if the change in elastic strain energy due to crack extension > the

energy required to create new crack surfaces, crack propagation will

occur”

Irwin (1940’s) extended the theory for ductile materials.

“ the energy due to plastic deformation must be added to the surface

energy associated with creation of new crack surfaces”

“ local stresses near the crack tip are of the general form”

2

0. INTRODUCTION


Minimum total potential energy principle

Fundamental concept used in physics, chemistry, biology, and engineering.

It dictates that a structure or body shall deform or displace to a position

that (locally) minimizes the total potential energy, with the lost potential

energy being converted into kinetic energy (specifically heat).

The total potential energy of an elastic body, Π, is defined as follows:

𝜫 = 𝑼− 𝑭Here, 𝑼 : strain energy stored in the body

𝑭 : the work done by external loads

Examples

A rolling ball will end up stationary at the bottom of a hill, the point of

minimum potential energy. It rolls downward under the influence of gravity,

friction produced by its motion transfers energy in the form of heat of the

surroundings with an attendant increase in entropy.

Deformation of spring under gravity stretches and vibrates and finally stops

due to the structural damping.

3

0. Introduction


Minimum total potential energy principle

This energy is at a stationary position when an infinitesimal variation from

such position involves no change in energy.

𝛿𝛱 = 𝛿(𝑈 − 𝐹)

The principle of minimum total potential energy may be derived as a special

case of the virtual work principle for elastic systems subject to conservative

forces.

The equality between external and internal virtual work (due to virtual

displacements) is:

𝑆𝑡𝛿u𝑇 T𝑑𝑆 + 𝑆𝑡

𝛿u𝑇 f𝑑𝑉= 𝑉 𝛿휀𝑇 𝜎𝑑𝑉

𝑢 = vector of displacements

𝑇= vector of distributed forces acting on the part of the surface

𝑓= vector of body forces

In the special case of elastic bodies

𝛿U = 𝑉 𝛿휀𝑇 𝜎𝑑𝑉,𝛿F = 𝑆𝑡𝛿𝑢𝑇 𝑇𝑑𝑆 + 𝑉 𝛿𝑢𝑇 𝑓𝑑𝑉

∴ 𝛿𝑈 = 𝛿𝐹

The basis for developing the finite element method.

4

0. Introduction


0. Introduction

Energy

Kinetic energies (which are due to movement)

Potential energies (which are stored energies – energy that a piece of

matter has because of its position or because of the arrangement of its

parts)

Elastic strain energy, gravitational potential energy

A rubber ball

A rubber ball held at some height above the ground has (gravitational)

potential energy.

When dropped, this energy is progressively converted into kinetic

energy. When the ball.

hits the ground it begins to deform elastically and, in so doing, the

kinetic energy is progressively converted into elastic strain energy.

In any real material undergoing deformation, at least some of the

supplied energy will be converted into heat.

5

1. Energy in Deforming Materials

Reference : http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_I/index.html


Work and Energy in Particle Mechanics

The work W done by F : 𝐹𝑠𝑐𝑜𝑠𝜃

Positive for 0 ≤ 𝜃 ≤ 90, Negative for 90 ≤ 𝜃 ≤ 180

Work done is

1.1 Work and Energy in Particle Mechanics 6


Reference : http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_I/index.html

Force acting on a particle, which

moves through a displacement

A varying force moving a particle along a path


1.1 Work and Energy in Particle Mechanics

Conservative Forces

The work done in moving a particle between two points is

independent of the taken path and dependent only on the position of

the object.

A conservative force 𝐹𝑐𝑜𝑛 in one-dimensional case,

Potential Energy:

the work done in moving a system from some standard

configuration to the current

configuration

A conservative force 𝐹𝑐𝑜𝑛 in one-dimensional case,

7




Potential Energy:

the work done in moving a system from some standard configuration

to the current

Potential energy has the following characteristics:

a. The existence of a force field.

b. To move something in the force field, work must be done.

c. The force field is conservative.

d. There is some reference configuration.

e. The force field itself does negative work when another force is

moving something against it.

f. It is recoverable energy.

Ex) Gravity potential energy, Spring pentation energy in spring system

8




Spring system example

a body attached to the coil of a spring is extended slowly by a force F ,

overcoming the spring (restoring) force F spr (so that there are no

accelerations and F=-Fspr at all times).

Work done by F in extending the spring to a distance x is

The corresponding work done by the conservative spring force F spr is

9


a force extending an elastic spring force-extension curve for a spring



The definition for the potential energy U

the negative of the work done by a conservative force in moving the

system from some standard configuration to the current configuration.

In general then, the work done by a conservative force is related to the

potential energy through

Dissipative (Non-Conservative) Forces

the work done by the pulling force F keeps increasing, and the work

done is not simply determined by the final position of the block, but

by its complete path history.

The energy used up in moving the block is dissipated as heat (the

energy is irrecoverable).

10


Dragging a block over a frictional surface


1.2 The Principle of Work and Kinetic Energy

In general, a mechanics problem can be solved using either

Newton’s second law or the principle of work and energy.

The rate of change of kinetic energy is, using Newton’s second law

F =ma ,

The change in kinetic energy over a time interval (t 0, t 1 ) is then

The work done over this time interval is

Work Energy Principle :

11




The principle of work and kinetic energy: .

the total work done by the external forces acting on a particle equals

the change in kinetic energy of the particle

a standard non-homogeneous second order linear ordinary

differential equation.

The solution for initial position 𝑥0 and the initial velocity 𝑥0

where, 𝜔 = 𝑘/𝑚

12


a block attached to a spring and

dragged along a rough surface



The principle of work and kinetic energy: .

The change in kinetic energy of the block is

the work done by the spring force =the negative of the potential

energy change.

this energy loss by Friction :

13


a block attached to a spring and

dragged along a rough surface


1.3 The Principle of Conservation of Mechanical Energy

The work-energy principle

it is assumed that there is no energy loss,

1. The total work done by the external forces acting on a body

equals the change in kinetic energy of the body:

2. The total work done by the external forces acting on a body,

exclusive of the conservative forces, equals the change in the total

mechanical energy of the body

The special case where there are no external forces,

14



1.3 The Principle of Conservation of Mechanical Energy

The principle of conservation of energy

the total energy of a system remains constant – energy cannot be

created or destroyed, it can only be changed from one form.

The principle of conservation of mechanical energy

It is assumed there is no energy dissipation.

if a system is subject only to conservative forces, its mechanical energy

remains constant.

Where i : initial, f : final

15



1.4 Deforming Materials

There will be a complex system of internal forces acting between the

molecules, even when the material is in a natural (undeformed)

equilibrium state .

If external forces are applied, the material will deform and the molecules

will move, and hence not only will work be done by the external forces, but

work will be done by the internal forces.

In the special case where no external forces act on the system : Free

vibration.

The case where the kinetic energy is unchanging : quasi-static

16



1.4 Deforming Materials

Conservative Internal Forces

Suppose one could apply an external force to pull two of these

molecules apart.

The work done by the external forces equals the change in potential

energy plus the change in kinetic energy,

U : elastic strain energy, the energy due to the molecular arrangement

relative to some equilibrium position.

17

1. Energy in Deforming Materia

external force pulling two molecules/particles apart


1.5 Non-Conservative Internal Forces

Internal friction

The molecules would slide over each other.

Frictional forces would act between the molecules, very much like the

frictional force between the block and rough surface.

H is the energy dissipated during the deformation and will depend on

the precise deformation process.

18

1. Energy in Deforming Materia

external force pulling two molecules/particles apart


Strain Energy Density 19

2. Elastic Strain Energy


Strain Energy Density

Strain and stress of a volume element under stress

20


x x yE

1

( )

y y xE

1

( )

z x yE

( )

x x y zE

1

( )

y y z xE

1

( )

z z x yE

1

( )



Hook’s las in Plane stress

21




Strain energy of a volume under plane stress

– Strain energy by normal strain 𝑈1

𝑈1 =1

2𝑎2𝜎𝑥 𝑎휀𝑥 +

1

2𝑎2𝜎𝑦 𝑎휀𝑦 =

𝑎3

2𝜎𝑥휀𝑥 + 𝜎𝑦휀𝑦

– Strain energy by shear strain 𝑈2

𝑈2 =1

2𝑉𝛿 =

1

2𝑎2𝜏𝑥𝑦 𝑎𝛾𝑥𝑦 =

𝑎3

2𝜏𝑥𝑦𝛾𝑥𝑦

– Total strain energy 𝑈

𝑈 = 𝑈1 + 𝑈2

22


Force on x-plane

Distance of x-plane

thickness 𝑎𝑎

𝛾𝑥𝑦



Strain energy density, 𝑢– Strain energy in a unit volume

Stain energy density of a volume under plane stress

𝑢 = 𝑢1 + 𝑢2 =1

2𝜎𝑥휀𝑥 + 𝜎𝑦휀𝑦 + 𝜏𝑥𝑦𝛾𝑥𝑦

23


𝑈1 =1

2𝑎2𝜎𝑥 𝑎휀𝑥 +

1

2𝑎2𝜎𝑦 𝑎휀𝑦 =

𝑎3

2𝜎𝑥휀𝑥 + 𝜎𝑦휀𝑦

𝑈2 =1

2𝑉𝛿 =

1

2𝑎2𝜏𝑥𝑦 𝑎𝛾𝑥𝑦 =

𝑎3

2𝜏𝑥𝑦𝛾𝑥𝑦



Stain energy density of a volume under plane stress

24



3.1 Principle of Virtual Work

When the mass is in equilibrium

the mass is not in fact at its equilibrium position but at an (incorrect)

non-equilibrium position x +x, x= virtual displacement.

The virtual work W done by a force to be the equilibrium force times

this small imaginary displacement x.

The total virtual work is

25

3. Virtual Work

If the system is in equilibrium(-kx +Fapl = 0),

total virtual work is zero, W=0.

Alternatively, if the virtual work is zero then,

since x is arbitrary, the system must be in

equilibrium.

A force extending an elastic spring


3.1 Principle of Virtual Work

Example

Let point C undergo a virtual displacement u.

From similar triangles, the displacement at B = (a/L)u

Total virtual work

The beam is in equilibrium when W=0 Rc=aF/L

26

3. Virtual Work

A loaded rigid bar


3.2 Principle of Virtual Work: deformable bodies

External virtual work : Wext= Fapl x

The internal virtual work done by spring force :

Wint = - kx x Wint = - U

U : the virtual potential energy change which occurs when the

spring is moved a distance.

The internal virtual work of an elastic body is the (negative of the)

virtual strain energy.

The principle of virtual work can be expressed as

27

3. Virtual Work

Wext= U


4.1 The Principle of Minimum Potential Energy

the total potential energy attains a stationary value (maximum or

minimum) at the actual displacement (u1 );

𝜫 = 𝑼− 𝑭Here, 𝑼 : strain energy stored in the body

𝑭 : the work done by external loads (=“potential energy” of the applied loads

𝜹𝜫 = 𝜹𝑼− 𝜹𝑭(= 𝜹𝑾𝒆𝒙𝒕)

Taking the total potential energy to

be a function of displacement u, one has

True displacement = u1, u2, or u3

u2 : stable equilibrium point

u1, u3 : unstable equilibrium point

28

4. The Principle of Minimum Potential Energy

u1 u2 u3

u1 u2


4.2 The Rayleigh-Ritz Method

The principle of minimum potential energy is used to obtain

approximate solutions to problems which are otherwise difficult or,

more usually, impossible to solve exactly.

Ex) A uniaxial bar of length L, young’s modulus E and varying cross-

section A= A0(1+x/L )

The Exact solution u=(FL/EA0)ln(1+x/L), u(0)=0

The principle of minimum pontential energy

First, substituting in the exact solution leads to

29


−𝐹−𝑊𝑒𝑥𝑡

=-0.347



Suppose now that the solution was unknown

Rayleigh Ritz method An estimate of the solution can be made in

terms of some unknown parameter(s), 𝜫 is minimized to find the

parameters.

A trial function. u=+ x. Since u(0)=0 u= x

30


Exact and (Ritz) approximate

solution for axial problem=-0.347

From true solution



The accuracy of the solution by using as the trial function a

quadratic instead of a linear one.

A trial function. u=+ x+x2 . Since u(0)=0 u= x+x2

The two unknowns can be obtained from the two conditions

31


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Topics in Ship Structuresocw.snu.ac.kr/sites/default/files/NOTE/06 Energy Thoery.pdf · Topics in Ship Structures 06 Energy Theory. OPen INteractive Structural Lab Energy approach