Dear Mehrab,

Thank you for your thoughtful question regarding the computation of Normalized Total Fracture Energy (GF) in concrete specimens, particularly in the context of varying geometric dimensions.

You are correct that the total fracture energy (GF) is typically determined as the area under the load–displacement (or load–CMOD) curve divided by the fractured ligament area, which for a notched beam is usually expressed as:

GF=WfAlig=∫0δuP(δ) dδb⋅(d−a)G_F = \frac{W_f}{A_{\text{lig}}} = \frac{\int_0^{\delta_u} P(\delta) \, d\delta}{b \cdot (d - a)}GF​=Alig​Wf​​=b⋅(d−a)∫0δu​​P(δ)dδ​

Where:

• WfW_fWf​ is the work of fracture (energy absorbed until failure)
• P(δ)P(\delta)P(δ) is the load–displacement function
• bbb is the specimen width
• ddd is the specimen depth
• aaa is the notch depth
• Alig=b⋅(d−a)A_{\text{lig}} = b \cdot (d - a)Alig​=b⋅(d−a) is the ligament area
• δu\delta_uδu​ is the ultimate displacement

Normalized Total Fracture Energy (GFnorm):

To normalize GFG_FGF​, researchers often aim to account for geometric effects (e.g., size, shape, or volume dependence) in order to make energy values comparable across specimens of different dimensions. A common method is to normalize fracture energy by a characteristic length, a nominal strength, or by using dimensionless formulations.

One widely accepted approach (as recommended in RILEM TC 50-FMC and by Hillerborg) is to normalize GFG_FGF​ by the tensile strength ftf_tft​ and elastic modulus EEE to obtain a characteristic length lchl_{ch}lch​:

lch=E⋅GFft2l_{ch} = \frac{E \cdot G_F}{f_t^2}lch​=ft2​E⋅GF​​

However, if you're strictly referring to normalized GF with respect to geometry, an alternative is to use:

GF,norm=GF(d−a)G_{F,\text{norm}} = \frac{G_F}{(d - a)}GF,norm​=(d−a)GF​​

Or more generally:

GF,norm=Wfb⋅(d−a)2G_{F,\text{norm}} = \frac{W_f}{b \cdot (d - a)^2}GF,norm​=b⋅(d−a)2Wf​​

This accounts for ligament length scaling, often used when comparing fracture energies across specimens with varying ligament heights.

Key References:

Final Note:

The choice of normalization method should align with the goal of your analysis—whether for size effect compensation, comparison across studies, or scaling analysis. Always document the rationale behind the selected normalization approach in your methodology.

Should you require a deeper discussion or example calculation templates in MATLAB or Excel, I’d be happy to assist further.

Warm regards, Dr. Gitonga Muriithi