3. Energy release rate and Griffith theory

Linear elastic fracture mechanics (LEFM) rests on a single energetic idea: a crack advances only when the mechanical energy released by its extension is sufficient to supply the energy consumed in creating new surfaces. This chapter develops that idea quantitatively, defines the energy release rate G, states the Griffith criterion, and connects G to the stress intensity factor K that governs the near-tip stress field.

3.1 Energy balance during crack extension

Consider a cracked body of thickness B loaded by prescribed forces or displacements. Let U be the elastic strain energy stored in the body, F the work done by external forces, and Γ the energy dissipated in forming new crack surfaces. For a virtual crack advance da, the first law of thermodynamics requires

dF − dU = dΓ ≥ 0.

The left-hand side is the net energy available for crack growth. Griffith (1921) defined the energy release rate (also called the crack driving force) as the energy available per unit of new crack area:

G = −dΠ/dA,    A = B·a,

where Π = U − F is the total potential energy of the system. The minus sign reflects that Π decreases as the crack grows under fixed loading. G has units of energy per unit area (J/m²), equivalently force per unit length (N/m).

For a linear elastic body, the potential energy can be written in terms of the compliance C(a) = δ/P:

Π = −½ P δ = −½ P² C(a).

Differentiating with respect to crack area at constant load gives the compliance form of the energy release rate:

G = ½ P² / B · dC/da.

This expression is the basis of the experimental compliance method: measure C(a) for several crack lengths, fit a curve, and evaluate its slope. It also shows that G is a global quantity — it depends on specimen geometry and loading, not on the local stress field.

3.2 Griffith's criterion and the critical energy release rate

Griffith balanced the released elastic energy against the surface energy γs of the two new crack faces. For a through-thickness crack of length 2a in an infinite plate under remote tension σ, the Inglis solution gives

Π(a) = Π0 − π σ² a² B / E′ ,

so that

G = π σ² a / E′ ,

where E′ = E for plane stress and E′ = E/(1 − ν²) for plane strain. Crack growth occurs when G reaches a material threshold, the critical energy release rate Gc:

G ≥ Gc   (crack propagation).

For ideally brittle materials Gc = 2γs. In engineering alloys, Gc is dominated by plastic dissipation near the tip and is orders of magnitude larger than 2γs; it is then identified with the fracture toughness measured in terms of G. The Griffith stress for a central crack follows directly:

σf = √( E′ Gc / (π a) ).

Note the characteristic √(1/a) scaling: doubling the crack length reduces the failure stress by about 30 %. This size effect is the central engineering message of Griffith theory.

3.3 Relationship between G and the stress intensity factor

Irwin (1957) showed that the energy release rate and the stress intensity factor are two descriptions of the same crack-tip field. For mode I,

G = KI² / E′ .

For mixed-mode loading the contributions add:

G = ( KI² + KII² ) / E′ + KIII² / (2μ),

with μ the shear modulus. Because G is a scalar, it cannot distinguish mode mixity; the individual K components must be retained when the crack path or the mode-dependent toughness matters. The equivalence also provides the practical conversion between the two toughness measures:

Gc = KIc² / E′ .

For a typical structural steel with KIc = 50 MPa·√m, E = 210 GPa and ν = 0.3, plane strain gives E′ ≈ 231 GPa and Gc ≈ 10.8 kJ/m². The same steel in plane stress would give Gc ≈ 11.9 kJ/m², illustrating why thickness must be reported alongside toughness data.

Sign and reference conventions used throughout this course. Crack length is measured from the notch root; a increases in the direction of growth. G is defined as −dΠ/dA and is therefore positive for a growing crack under load control. Plane strain uses E′ = E/(1 − ν²); plane stress uses E′ = E. All stress intensity factors follow the standard √(πa) normalization, K = Y σ√(πa).

3.4 Worked example: edge-cracked plate

Take a single-edge-notched tension (SENT) specimen: width W = 50 mm, thickness B = 10 mm, crack length a = 10 mm, applied load P = 40 kN, E = 70 GPa (aluminium), ν = 0.33. The geometry factor for this configuration is approximately

Y(a/W) = 1.12 - 0.231(a/W) + 10.55(a/W)^2
       - 21.72(a/W)^3 + 30.39(a/W)^4

a/W = 0.20  ->  Y = 1.12 - 0.0462 + 0.422
                  - 0.1738 + 0.0486
              = 1.371

The remote stress is σ = P/(BW) = 40 000/(0.010 × 0.050) = 80 MPa. Hence

K_I = Y * sigma * sqrt(pi * a)
    = 1.371 * 80e6 * sqrt(pi * 0.010)
    = 1.371 * 80e6 * 0.1772
    = 19.4 MPa*sqrt(m)

E'  = E / (1 - nu^2) = 70e9 / (1 - 0.33^2)
    = 78.5 GPa

G   = K_I^2 / E'
    = (19.4e6)^2 / 78.5e9
    = 4.79 kJ/m^2

If the measured toughness of this alloy is Gc = 15 kJ/m², the applied G is only about one third of the critical value, so the crack is stable. Increasing the load to 140 kN would raise KI to 34 MPa·√m and G to 14.7 kJ/m², essentially at the threshold — a useful check when planning a test or assessing an inspection finding.

3.5 From energy release rate to simulation

In finite element practice, G is rarely obtained from a closed-form compliance. Two standard routes are used:

  • Virtual crack closure technique (VCCT): extract nodal forces Fy ahead of the tip and relative displacements Δv behind it, then compute GI = Fy Δv / (2B Δa). Requires a pre-existing crack path and a refined tip mesh.
  • J-integral: evaluate the contour integral J = ∫Γ (W dy − t·∂u/∂x ds) around the tip. For linear elasticity J = G, and it is path-independent, which makes it robust to mesh details.

Both methods return G in J/m²; dividing by Gc gives the normalized driving force used in crack-growth laws such as da/dN = C(ΔG)m.

Griffith Energy Balance: G(a) vs. G_c crack length a energy rate G_c G(a) = πσ²a / E′ a* Stable: G < G_c no crack growth Unstable: G ≥ G_c crack propagates slope ∝ σ²/E′ driving force G(a) material resistance G_c critical crack length a*

The figure summarizes the Griffith instability condition. The driving force rises linearly with crack length for fixed remote stress, while the material resistance Gc is a horizontal threshold. Their intersection defines the critical crack length a* = E′Gc/(πσ²). Below a* the crack is dormant; at a* it becomes unstable and, under load control, propagates without further increase in applied load.

Common pitfalls. (1) Using E instead of E′ in plane strain underestimates G by up to 10 % for ν = 0.3. (2) Applying the Griffith formula to a short crack in a small specimen ignores the finite-width correction Y(a/W). (3) Treating Gc as a single material constant when it is strongly thickness- and constraint-dependent. (4) Confusing G (energy per unit area) with K (stress × √length) — they are related but not interchangeable when mode mixity is present.

3.6 Summary

The energy release rate G = −dΠ/dA quantifies the driving force for crack growth in terms of global energy changes. Griffith's criterion G ≥ Gc predicts the onset of propagation and yields the characteristic σf ∝ 1/√a size effect. Irwin's relation G = K²/E′ bridges the energetic and stress-intensity viewpoints, allowing toughness data to be converted between Gc and Kc and enabling finite element post-processing via VCCT or the J-integral. These three quantities — G, Gc, and K — form the quantitative foundation on which the remainder of this course builds.

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