Solid Mechanics

Fracture Mechanics

Solid MechanicsPresented by

Calvin M. Stewart, PhD

MECH 5390-6390

Fall 2020

Outline

• Interatomic View of Fracture

• Linear Elasticity• Equilibrium of Stress• Compatibility Equations of Strain• Airy Stress Functions

• Stress Concentration Factors• Circular Hole• Elliptical Hole

• Limitations of SCF Approach

• Need for LEFM

Interatomic View of FracturePresented by

Calvin M. Stewart, PhD

MECH 5390-6390

Fall 2020

Interatomic View of Fracture

• A material fractures when sufficient stress and work are applied on the atomic level to break the bonds that hold atoms together. The bond strength is supplied by the attractive forces between atoms.

• Interatomic Forces➢Attractive Forces, Fa

➢Repulsive Forces, Fr

➢Applied Force, P

• Equilibrium Spacing, x0

P(x) = Applied Force

Interatomic View of Fracture

• Bond Energy, Eb

• where x0 is the equilibrium spacing and Pis the applied force. Idealize the applied force, P(x) function as

• For simplicity, small displacement,

( )o

b

x

E P x dx

=

sinc

xP P

=

c

xP P

=

Note: Distance, λ

Interatomic View of Fracture

• For small displacement, the Bond Stiffness, k is

• Multiple Both sides by number of bonds per unit area, Ab and the gage length, x0

• Simplify to find Cohesive Stress, σc

ck P

=

b o c b oA kx P A x

=

0

c

E

x

=

0

bEAk

x=Note:

2

b c b oA E P A x

=

0   ,c

Ex

=

Interatomic View of Fracture

• Surface Energy

• Surface energy is equal to one-half of the fracture energy because two surfaces are created when a material fractures.

• Replace λ to find Cohesive Stress, σc

0

1sin

2s c c

xdx

= =

0

sc

E

x

=

Linear ElasticityElasticity, Equilibrium Equations of Stress, Compatibility Equations of Strain, Airy Stress Functions

Elasticity

• Generalized Hooke’s Law1, ,ij ijkl kl ij ijkl klC S S C −= = =

ij

ij

ijklC

ijklS

Stress Tensor 2nd order 32=9 terms

Strain Tensor 2nd order 32=9 terms

Stiffness Tensor 4th order 34=81 terms

Compliance Tensor 4th order 34=81 terms

11 1111 11 1112 12 1133 33

12 1211 11 1212 12 1233 33

33 3311 11 3312 12 33333 33

.......

.......

...............................................................

.......

C C C

C C C

C C C

= + + +

= + + +

= + + +

Elasticity

• For isotropic, homogenous, elastic materials, the 3D form of Hooke’s laws can be written as

• σij is the stress tensor

• εij is the strain tensor

1 1 2

1

ij ij kk ij

ij ij kk ij

E

E E

= + + −

+= −

Young’s Modulus

Poisson’s Ratio

E

Elasticity

• Symmetry makes the tensors reduce to 6 terms which we will express with indices x, y, z

• σx , σy , σz Normal stress components

• τxy , τzx , τyz Shear stress components

• εx , εy , εz Normal strain components

• εxy , εzx , εyz Shear strain components

( )

( )

( )

1

1

1

x x y z

y y x z

z z x y

E

E

E

= − +

= − +

= − +

1

1

1

xy xy

zx zx

yz yz

E

E

E

+=

+=

+=

Elasticity

• Fracture mechanics mostly deals with 2-dimensional problems, in which case no quantity depends on the z coordinate. Two special cases are plane stress and plane strain conditions.

Plane Stress Plane Strain

Elasticity – Plane Stress

0z yz zx = = =

Plane Stress – Stress is zero across a particular plane. In this case, the component is Thin in the z direction.

( )

( )

1

1

1

x x y

y y x

xy xy

E

E

E

= −

= −

+=

( )

0z

z x yE

=

−= +

Elasticity – Plane Strain

xy

Plane Strain – Strain is zero across a particular plane. In this case, the component is Thick in the z direction.

0z yz zx = = =

Equilibrium Equations of Stress

• When we speak of Equilibrium in Solid Mechanics, we are referring to the Newton Laws of Motion given as

• Equilibrium only exists if the left hand side (LHS) and right hand side (RHS) of the equation are equal. In the case where a=0 and v≥0,

• Which is called “static equilibrium”

m =F a

=F 0

Equilibrium Equations of Stress

• Consider a finite element of infinitesimal volume, dV subject to static equilibrium with stresses acting in the x direction.

Equilibrium Equations of Stress

• The equilibrium of forces in the x direction is

• Repeating this process in the y and z direction and simplification furnishes the equilibrium equations of stress

2D case of plane stress and plane strain

Compatibility Equations of Strain

• There are six strain measurements εij that rise from three independent displacements, u, v, w. As such there exists six constraint equations, also called the Compatibility Conditions.

• If Compatibility is not satisfied than gaps, overlaps, or discontinuities would exist in the strain field.

• Let us assume the 2D case, where

• Differentiate by x and then y

x

du

dx = y

dv

dy =

1

2xy

du dv

dy dx

= +

x xy

ij

xy y

=

( )2

xy

d

dxdy

Compatibility Equations of Strain

• Start

• Double Derivative

• Simplify

2 3 31

2

xyd d u d v

dxdy dxdydy dxdydx

= +

2 2 2

2xyd d du d dv

dxdy dydy dx dxdx dy

= +

• Further Simplify to

2 22

2xy yx

d dd

dxdy dydy dxdx

= +

1

2xy

du dv

dy dx

= +

2D Compatibility Equation

Compatibility Equations of Strain

2 22

2 22

xy yxd dd

dxdy d y d x

= +

2 2 2

2 22

yz y zd d d

dydz d z d y

= +

2 22

2 22 zx xz

d dd

dzdx d x d z

= +

2

y xy yz zxd d d dd

dzdx dy dz dx dy

= + −

2yz xyzxz

d ddd d

dxdy dz dx dy dz

= + −

2xy yzx zx

d dd dd

dydz dx dy dz dx

= + −

3D Compatibility Equations

George Biddell Airy

• In 1862, Airy presented a new technique to determine the strain and stress field within a beam.[14] This technique, sometimes called the Airy stress function method, can be used to find solutions to many two-dimensional problems in solid mechanics.

• For example, it was used by H. M. Westergaard to determine the stress and strain field around a crack tip and thereby this method contributed to the development of fracture mechanics.

George Biddell Airy (1801-1892)

Airy Stress Function

• Any Stress field solution for an elastic problem must satisfy both equilibrium and compatibility, Airy (in 1863) introduced a function φ(x,y) to satisfy this requirement.

• Straightforward substation shows that this stress field• Always fulfils the equilibrium of stress

• Only fulfils the compatibility equations of strain if the stress function is a solution of the so-called biharmonic equation.

2 2 2

2 2, ,xx yy xy

d d d

dy dx dxdy

= = = −

Airy Stress Function

• This function automatically satisfies

• Equilibrium: & Compatibility:

• By Applying Hooke’s Law for a state of plane strain gives

0

0

xyxx

xy yy

dd

dx dy

d d

dx dy

+ =

+ =

2 22

2 22

xy yxd dd

dxdy d y d x

= +

( )2 2 22 2

2 2 2 21 2

yy yy xyxx xxd d dd d

dx dy dx dy dxdy

− + − + =

Airy Stress Function

• Taking,

• And Applying the Airy definition,

• We find,

( )2 2 22 2

2 2 2 21 2

yy yy xyxx xxd d dd d

dx dy dx dy dxdy

− + − + =

2 2 2

2 2, ,xx yy xy

d d d

dy dx dxdy

= = = −

4 4 4

4 2 2 4

2 2

4

2 0

0

d d d

dx dx dy dy

+ + =

=

=

The Biharmonic Equation !!! Sweet!!! ☺

Found in Continuum Mechanics of Linear Elastic Structures and Viscous Incompressible Fluids

Note: φ is a real valued and has units of Force and can be written in alternative coordinate systems such as polar or cylindrical.

Polar Coordinates

• Sometimes it is better to express our answer in polar coordinates, particularly when examining the stress field in the vicinity of a crack.

Strain-Displacement Stress-Strain

Plane strain

Plane stress

Polar Coordinates

• Similarly, equilibrium and compatibility can be expressed in polar coordinates as follows

Equilibrium Compatibility

Polar Coordinates

• Even the Airy Stress function can be expressed in polar coordinates

Airy Stress Function

Stress ConcentrationsCircular Hole

Stress Concentrations

• Stress Concentrations are geometric discontinuities that lead to local increase in the stress field. Examples: holes, grooves, sharp corners, fillets, welds, surface defects

• Stress Concentration Factor (SCF), KT characterizes the amplification of the stress

• Need: Stress field at the Notch!!!!

local

remote

SCF

=

Internal Force Lines

Circular Hole

• Elasticity problems can be solved by finding an Airy Stress function, φ(x,y) which satisfies the biharmonic equation and satisfies the boundary conditions of the posed problem.

• Direction solutions of the governing equations are, for the most part, not available.

• Consequently, an indirect approach called the semi-inverse method is often employed to solve a specific elasticity problem.

Derivation

See Tablet Derivation

A more detailed derivation with explanations is given in“Principles of Fracture Mechanics” by R.J. Sanford – Chapter 2

Circular Hole

• Using the semi-inverse method, we can find the stress concentration factor in an infinity wide plate with a circular hole. Assume uniaxial loading we can show that KT=3

local

remote

SCF

=

local

remote

Circular Hole

• We can also find the solutions for the biaxial case,

3 2 = − =

3 4 = + =

3 2 = − + = −

3 4 = − − = −

Elliptical Hole

• The first quantitative evidence for the stress concentration effect of flaws was provided by Inglis, who analyzed elliptical holes in flat plates

• The ratio σA/σ is defined as the stress concentration factor, KT. When a = b, the hole is circular and KT = 3.0, a well-known result.

Derivation

See Tablet Derivation

A more detailed derivation with explanations is given in“Principles of Fracture Mechanics” by R.J. Sanford – Chapter 2

Elliptical Hole

• As the major axis, a, increases relative to b, the elliptical hole begins to take on the appearance of a sharp crack. For this case, Inglis found it more convenient to express in terms of the radius of curvature, ρ:

• When a>>b,

Elliptical Hole

• The previous equation predicts an infinite stress at the tip of an infinitely sharp crack, where ρ = 0.

• This result caused concern when it was first discovered because no material is capable of withstanding infinite stress.

• A material that contains a sharp crack theoretically should fail upon the application of an infinitesimal load.

Limitations of SCF Approach

• It turns out that the strength-of-materials assumption, where fracture is controlled by the stress at the tip of the crack, is not a valid failure model in general.

• Any hairline flaw on a structure would result in instant failure.

• When there is a significant stress gradient in the structure, failure is generally not governed by a single high-stress point.

Need for LEFM

• The apparent paradox of a sharp crack motivated Griffith to develop the Fracture Energy theory rather than use local stress.

• It also motivated Irwin to develop the Stress Intensity Factor approach where a consistent and finite parameter is extracted from the stress field equations.

• Both approaches are a part of LEFM.

Summary➢ Fracture at the atomic level resolves around the energy required to break atomic

bounds.

➢ At the continuum level, Elasticity Equations describe the relationship between stress and strain.

➢ Equilibrium Equations of Stress ensure that Newton’s laws of motion are satisfied.

➢ Compatibility Equations of Strain ensure that the relationships between displacement and strain is satisfied.

➢ The Airy Stress field solution for an elastic problem must satisfy both equilibrium and compatibility.

Summary➢ The Stress Concentration Factor approach breakdown for sharp cracks where it

predicts Infinite Stress.

➢ Thus the LEFM approach is needed if we are to deal with cracked structures.

➢ The Fracture Energy and Stress Intensity Factor are LEFM approaches.

Homework 3

• Using the polar definition of the Airy Stress Function, derive the polar form of the biharmonic equation using the polar compatibility equation. Show all the steps to the solution.

References

• Janssen, M., Zuidema, J., and Wanhill, R., 2005, Fracture Mechanics, 2nd Edition, Spon Press

• Anderson, T. L., 2005, Fracture Mechanics: Fundamentals and Applications, CRC Press.

• Sanford, R.J., Principles of Fracture Mechanics, Prentice Hall

• Hertzberg, R. W., Vinci, R. P., and Hertzberg, J. L., Deformation and Fracture Mechanics of Engineering Materials, 5th Edition, Wiley.

• https://www.fracturemechanics.org/

CONTACT INFORMATION

Calvin M. Stewart

Associate Professor

Department of Mechanical Engineering

The University of Texas at El Paso

500 W. University Ave, Suite A126, El Paso, TX 79968-0521

Ph: 915-747-6179

cmstewart@utep.edu

me.utep.edu/cmstewart/