06 Energy Thoery.pdf

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 − 𝑭

Here, 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:

𝑆_𝑡 𝛿u^𝑇 T𝑑𝑆 + _𝑆

𝑡 𝛿u^𝑇 f𝑑𝑉= _𝑉 𝛿𝜀^𝑇 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 𝛿𝜀^𝑇 𝜎𝑑𝑉,𝛿F = _𝑆

𝑡 𝛿𝑢^𝑇𝑇𝑑𝑆 + _𝑉 𝛿𝑢^𝑇𝑓𝑑𝑉

∴ 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

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

1. Energy in Deforming Materials

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 𝐹_𝑐𝑜𝑛 in one-dimensional case,

Potential Energy:

the work done in moving a system from some standard configuration to the current

configuration

A conservative force 𝐹

_𝑐𝑜𝑛

in one-dimensional case,

7

1. Energy in Deforming Materials

1.1 Work and Energy in Particle Mechanics

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

1. Energy in Deforming Materials

1.1 Work and Energy in Particle Mechanics

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=-F

_spr 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

1. Energy in Deforming Materials

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

1.1 Work and Energy in Particle Mechanics

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

1. Energy in Deforming Materials

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

1. Energy in Deforming Materials

1.2 The Principle of Work and Kinetic Energy

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

1. Energy in Deforming Materials

a block attached to a spring and dragged along a rough surface

1.2 The Principle of Work and Kinetic Energy

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

1. Energy in Deforming Materials

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. Energy in Deforming Materials

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. Energy in Deforming Materials

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. Energy in Deforming Materials

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

2. Elastic Strain Energy



_x



_x



_y

 E 1 

( )



_y



_y



_x

 E 1 

( )



_z

  

_{x y}

  E (  )



_x



_x



_y



_z

 E 1  

( )



_y



_y

 

_{z x}

 E 1  

( )



_z

 1  

_z



_x

 

_y

( )

Strain Energy Density

 Hook’s las in Plane stress

21

2. Elastic Strain Energy

Strain Energy Density

 Strain energy of a volume under plane stress – Strain energy by normal strain 𝑈

₁

𝑈

₁

=

¹

2

𝑎

²

𝜎

_𝑥

𝑎𝜀

_𝑥

+

¹

2

𝑎

²

𝜎

_𝑦

𝑎𝜀

_𝑦

=

^𝑎3

2

𝜎

_𝑥

𝜀

_𝑥

+ 𝜎

_𝑦

𝜀

_𝑦

– Strain energy by shear strain 𝑈

₂

𝑈

₂

= 1

2 𝑉𝛿 = 1

2 𝑎

²

𝜏

_𝑥𝑦

𝑎𝛾

_𝑥𝑦

= 𝑎

³

2 𝜏

_𝑥𝑦

𝛾

_𝑥𝑦

– Total strain energy 𝑈

𝑈 = 𝑈

₁

+ 𝑈

₂

22

2. Elastic Strain Energy

Force on x-plane

Distance of x-plane

thickness 𝑎 𝑎

𝛾_𝑥𝑦

Strain Energy Density

 Strain energy density, 𝑢

– Strain energy in a unit volume

 Stain energy density of a volume under plane stress

𝑢 = 𝑢

₁

+ 𝑢

₂

= 1

2 𝜎

_𝑥

𝜀

_𝑥

+ 𝜎

_𝑦

𝜀

_𝑦

+ 𝜏

_𝑥𝑦

𝛾

_𝑥𝑦

23

2. Elastic Strain Energy

𝑈₁ = 1

2 𝑎²𝜎_𝑥 𝑎𝜀_𝑥 +1

2 𝑎²𝜎_𝑦 𝑎𝜀_𝑦 = 𝑎³

2 𝜎_𝑥𝜀_𝑥 + 𝜎_𝑦𝜀_𝑦 𝑈₂ = 1

2𝑉𝛿 = 1

2 𝑎²𝜏_𝑥𝑦 𝑎𝛾_𝑥𝑦 = 𝑎³

2 𝜏_𝑥𝑦𝛾_𝑥𝑦

Strain Energy Density

 Stain energy density of a volume under plane stress

24

2. Elastic Strain Energy

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 +F_apl= 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.

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  R_c=aF/L

26

3. Virtual Work

A loaded rigid bar

3.2 Principle of Virtual Work: deformable bodies

 External virtual work : W

_ext

= F

_apl

x

 The internal virtual work done by spring force :

W

_int

= - kx x  W

_int

= - 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

W

_ext

= U

4.1 The Principle of Minimum Potential Energy

 the total potential energy attains a stationary value (maximum or minimum) at the actual displacement (u

₁

);

𝜫 = 𝑼 − 𝑭

Here, 𝑼 : strain energy stored in the body

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

𝜹𝜫 = 𝜹𝑼 − 𝜹𝑭(= 𝜹𝑾

_𝒆𝒙𝒕

)

 Taking the total potential energy to

be a function of displacement u, one has

 True displacement = u

₁

, u

₂

, or u

₃



u

₂ ^:

stable equilibrium point

u

₁

, u

₃

: uns table 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= A

₀

(1+x/L )

 The Exact solution u=( FL / EA

₀

)ln(1+ x /L), u(0)=0

 The principle of minimum pontential energy

 First, substituting in the exact solution leads to

29

4. The Principle of Minimum Potential Energy

−𝐹 −𝑊

_𝑒𝑥𝑡

4.2 The Rayleigh-Ritz Method

 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

4. The Principle of Minimum Potential Energy

Exact and (Ritz) approximate solution for axial problem

=-0.347

4.2 The Rayleigh-Ritz Method

 The accuracy of the solution by using as the trial function a quadratic instead of a linear one.

 A trial function. u=  +  x+  x

²

. Since u(0)=0  u=  x+  x

²

 The two unknowns can be obtained from the two conditions

31

4. The Principle of Minimum Potential Energy