Hi, let me try to Answer your questions as what i understand.

1. Fracture parameters are used for fracture prediction and toughness characterization. For mode I fracture of elastic media, the three most commonly used parameters are: the mode I energy release rate, GI, the mode I stress intensity factor, KI, and the path-independent J-integral (Broek, 1986). The critical value of these parameters, i.e., their values at the instant that crack growth initiates, is called the fracture toughness and is generally assumed to be an intrinsic material property.

now here you can calculate the fracture toughness value using strain energy release rate (Gc) . A number of means are available by which the material property Gc can be measured. One of these is known as compliance calibration, which employs the concept of compliance as a ratio of deformation to applied load : C=𝛿/𝑃)

total strain energy U=1/2 𝑃δ = 1/2 𝐶𝑃^2

G(strain energy release rate)=𝜕𝑈/𝜕𝐴 = 1/2 𝑃^2/𝐵 (𝜕∁ )/𝜕𝑎

here your sample size and structure depends, if you can assume you sample as a beam. we can calculate the deflection like this:

Beam theory gives the deflection as:𝛿/2= (𝑃𝑎^3)/3𝐸𝐼, where 𝐼= (𝐵ℎ^3)/12

then the compliance, 𝐶=𝛿/𝑃= (2𝑎^3)/3𝐸𝐼c

finally we can define 𝐺_𝑐= 1/2 𝑃^2/𝐵*(2𝑎^2)/𝐸𝐼=(12𝑃^2 𝑎^2)/(𝐵^2 ℎ^3 𝐸)

if once we have the Gc value we can obtained the fracture toughness ( the so called stress intensity factor), which are used as a measure that quantifies the severity of crack , using this equation we can relate them 𝐺=𝐾^2/𝐸^′.

the above equations will helps you to calculate fracture toughness form load-displacement curve.

2. https://en.wikipedia.org/wiki/Fracture_toughness

3. Knott et al. introduced a concept in 1973. A crack tip locates in the material and it seems like a line running from one location of the component to another location as a result the high stress is concentrated at the crack tip. That’s why a crack tip analysis is useful for getting the stress field and displacement. The fracture toughness values may also serve as a basis in material property characterization, performance evaluation, and quality assurance for typical engineering structures.

Therefore , notched beam (crack tip) is highly important to determine the fracture toughness.