1. ENERGY APPROACH
When the available energy for crack growth exceeds the material resistance, the crack expansion occurs, in other words fracture occurs. For fracture, Griffith is the first person who suggests an energy approach.
Irwin, has come up with the concept of strain energy (strain energy release rate) . In linear elastic materials, the energy release rate represents the elastic (potential) energy per unit crack area required for a very small crack extension.
The speed of energy release rate while fracture occurs is equal to the critical energy dissipation rate, which is the measure of fracture toughness
the strain energy release rate for an infinite plate including 2a crack lengths and subjected to tensile stress in the direction perpendicular to the crack axis is:
a = Half of a crack length, = applied stress at infinity, E = elastic modulus

2. During Fracture = c
= material toughness or material resistance to fracture. c
= energy release rate(crack driving force) c

3. GRIFFITH APPROACH
Through thickness crack in plate

4. GRIFFITH BRITTLE FRACTURE THEORY
If the elastic energy because of the stresses that will form around the crack is equal to the surface energy of crack then crack propagates.
Note: In an elastic, brittle, and axially loaded plate as a result of crack, decreasing potential energy and the increase in surface energy is in equilibrium
Equation.1
Potential Energy: depends on released stored elastic energy and the resulting work due to external forces Potential energy is reduced because the crack growth releases the elastic strain energy. Potential energy reduction in the plate containing the crack:
Equation 2 Ua
: change in elastic strain energy of a plate which includes crack initiation a
: half of a crack length t
: thickness
: Elastic modulus E

5. Total surface area of the crack :
Surface Energy: The growth rate of the crack is proportional to the surface energy on the plate or material. The increase in surface energy increases the crack growth rate. Surface energy according to Griffith is:
energy / unit area
Total surface area of the crack :
Specific surface energy :
Potential energy change caused by crack initiation :
Equation 3
U0 : potential energy of plate without crack
Equation 4
By taking the derivative of the potential energy according to the crack length and equating to zero :

6. For crack propagation, we take the derivative of the potential energy according to the crack length and equate to zero :
Equation5
Equation 6
From the above equation, the quatiGriffith eon is obtained:
According to the sign of the second derivative, the stability of the fracture is decided.   The negative sign represents instability. The crack grows constantly.
Equation 7
If Equation 6 is to be written in the new form, the uniaxial stress expression is:
Equation 8
Stress for plane stress (three axis stress) :
Equation 9

7. , strain energy release rate
The Griffith theory applies to elastic, brittle materials containing sharp cracks. In apparently, brittle crystal-structured materials, there is usually some local plastic deformation around crack tip. In such materials, besides the energy required to form a new crack surface, the plastic strain energy around the crack must also be included in the calculation.
, strain energy release rate
critical strain energy release rate. c
values can be calculated by the Charpy-notched experiment in the laboratory. c

8. solid model which has an embedded crack and subjected to tensile stress

9. Kalınlığı boyunca çatlak içeren çekmeye maruz plakada enerji metodu ile elde edilen kırılma gerilmesi The fracture stress, which is obtained with energy method of the plate including through thickness crack and subjected to tensile stress, is :

10. Stress concentration, fracture and Griffith Approach
Thin bar
Volume
Strain Enegy
thin
Plate which is Uniformly loaded

11. Griffith Energy Approach
In order to get energy release rate, there must be a break between the atomic bonds so that surface energy is formed .Surface energy is material property.
The energy balance in crack propagation:
and

12. The relationship between K and G
Plane stress case
When examined in terms of fracture criterion, the relationship between KIC and GC
Griffith Equation :
Griffith-Irwin Relationship :
For ductile materials
: can be neglected and equation can be written as:

13. If there is growth in the crack length
The elastic energy (strain energy) that accumulates in the material due to the applied P force in the event of the absence or growth of a crack,
Note: ΔL = c P , c: compliance.
If there is growth in the crack length
ΔL = c P
d(ΔL) = P dc + c dP
1) Sample dimension will increase( dL > 0 )
2) Force will drop( dP < 0)
If da>0 (crack growth)=>
Thermodynamic Balance of the System:
dW + dQ = dUel + dUk +dUs |
here w = is the work done by the external force(P)
Q = system heat

14. here w = is the work done by the external force(P) Q = system heat
Thermodynamic Balance of the System :
dW + dQ = dUel + dUk +dUs
here w = is the work done by the external force(P)
Q = system heat
Uel = strain energy in the system
Uk = kinetic energy
Us = surface energy |