Materials Characterization and NDE Formula Sheet

This formula sheet collects screening equations used in materials characterization, mechanical testing, and non-destructive evaluation. It supports test-plan review, inspection qualification, material release, defect disposition, process validation, and evidence audits.

The formulas are useful only when the measurement method represents the material, product form, defect type, geometry, surface condition, orientation, and acceptance decision. They do not replace material standards, NDE procedures, inspector qualification, probability-of-detection studies, fracture assessment, or project-specific acceptance criteria.

Notation

Symbol	Meaning	Typical unit
x_m	measured value	varies
x_{ref}	reference or calibrated value	varies
\bar{x}	sample mean	varies
s	sample standard deviation	varies
u_c	combined standard uncertainty	varies
U	expanded uncertainty	varies
k	coverage factor	dimensionless
L_L, L_U	lower and upper acceptance limits	varies
n	sample count, diffraction order, or electron count depending on context	dimensionless
d	number of observed defects, lattice spacing, or diameter depending on context	varies
p	defect fraction or probability	dimensionless
C	confidence level	dimensionless
F	force	N
A_0	original cross-sectional area	mm2 or m2
\sigma_e	engineering stress	MPa or Pa
\epsilon_e	engineering strain	dimensionless
E	elastic modulus	GPa or Pa
v	ultrasonic wave speed or CT voxel size depending on context	m/s or length
t_e	ultrasonic echo time of flight	s
z	reflector depth from one inspection surface	m or mm
f	ultrasonic frequency	Hz
\lambda	wavelength	m or mm
A_d	defect area	mm2
d_{eq}	equivalent circular defect diameter	mm
a_c	critical defect size	mm
a_d	qualified detection size	mm
M_a	detection margin	mm

Measurement Error and Repeatability

Absolute measurement error against a reference:

Relative error:

Percent relative error:

Sample mean:

Sample standard deviation:

Coefficient of variation:

Use

These relationships are the first check on calibration and repeatability. They are weak if the reference is not traceable, if the sample does not represent the real material state, or if repeated measurements are made at different physical locations in a heterogeneous material.

Measurement Uncertainty and Guard Bands

For independent uncertainty components:

Expanded uncertainty:

Guarded lower value for a capacity, remaining thickness, property, or dimension that must exceed a lower limit:

Accept against a lower limit only if:

Guarded upper value for a defect size, contaminant, damage measure, or demand that must stay below an upper limit:

Accept against an upper limit only if:

Use

Guard bands prevent a marginal measurement from being treated as stronger evidence than it is. The direction of the guard band matters: subtract uncertainty from beneficial quantities and add uncertainty to harmful quantities. This is a decision rule, not a claim that the true value is known exactly.

Resolution, Tolerance, and Quantization

Simple resolution-to-tolerance screening:

where R is measurement resolution and T is the tolerance band. This is a rule of thumb, not a universal metrology requirement.

Quantization uncertainty for a uniformly rounded reading:

Minimum detectable change by a simple resolution screen:

Use

Resolution is not accuracy. A display with small increments can still be biased by calibration, fixture compliance, probe placement, surface preparation, thresholding, temperature, or operator method. Use resolution checks as a gate before deeper uncertainty analysis.

Sampling and Defect Occurrence

Observed defect fraction in a sampled population:

Estimated affected count in a population of size N:

Probability of seeing at least one defect in n independent sampled units when the true defect probability is p:

Sample count needed to have confidence C of seeing at least one defect if the true defect probability is at least p:

Observed pass fraction:

Use

Sampling formulas assume the sampled units represent the population. They can fail badly when defects cluster by heat, build plate position, laminate zone, supplier lot, weld shift, operator, coating batch, or geometry. Sampling evidence should be linked to process knowledge, not only arithmetic.

Mechanical Test Reductions

Engineering stress:

Engineering strain:

Elastic modulus from two points in the linear region:

Percent elongation:

Hardness range acceptance:

Minimum local hardness margin:

Use

Mechanical test reductions are valid only when the specimen geometry, orientation, product form, surface condition, rate, temperature, fixture, extensometry, and failure location are controlled. Hardness can support process control, but it is not proof of fatigue strength, fracture toughness, corrosion resistance, or full mechanical performance.

Defect Detection and Critical Size

Detection margin:

where a_c is the critical defect size and a_d is the qualified detection size.

Guarded detection condition:

Defect size utilization:

where:

Simple probability of detection from a validation set:

False call rate:

Use

Detection margin is an engineering link between inspection and failure analysis. A method is not adequate merely because it can produce an image. It must be qualified for the defect type, size, orientation, material attenuation or contrast, surface condition, access, geometry, operator procedure, and reporting threshold.

Ultrasonic Testing

Pulse-echo reflector depth from a single inspection surface:

where v is wave speed in the material and t_e is round-trip echo time.

Thickness from back-wall echo:

Wavelength:

Near-field length for a circular normal-beam probe:

where D is active probe diameter.

Amplitude ratio in decibels:

Signal-to-noise ratio in decibels:

Use

Ultrasonic formulas assume the sound path and wave mode are known. Velocity calibration, couplant, probe angle, frequency, attenuation, grain structure, curvature, surface roughness, defect orientation, and reference blocks can dominate the result. Do not infer defect severity from echo depth alone.

X-Ray CT and Imaging Resolution

Minimum reliably measurable feature from voxel size:

where v_x is voxel size and m is the required number of voxels across the feature. Typical screening values of m are 3 to 5, but validated procedures may require more.

Physical length represented by a voxel count:

Porosity or void fraction in a region of interest:

Area fraction from segmented image data:

Equivalent circular defect diameter:

Aspect ratio of a mapped indication:

Use

Voxel size is not validated detection capability. Density, thickness, beam hardening, scatter, reconstruction filters, segmentation threshold, noise, partial-volume effects, and calibration can make real detection worse than the voxel calculation.

XRD and XRF Screening

Bragg’s law:

Lattice spacing:

Composition guard band for a lower elemental limit:

Composition guard band for an upper elemental limit:

Composition deviation from target:

Use

XRD supports phase, structure, texture, and selected residual-stress evidence. XRF supports alloy sorting and elemental screening. Neither method proves mechanical properties, heat treatment quality, defect absence, fatigue life, corrosion performance, or biomedical compatibility by itself.

Inspection Coverage and Evidence Completion

Inspection coverage fraction:

Evidence completion fraction:

Risk priority number:

Expected missed-defect count in a screened population:

Use

Coverage and evidence completion are management metrics, not technical proof. A 100 percent scan with the wrong method is weak evidence. A complete checklist is only strong when the listed evidence items represent the actual defect, material, geometry, uncertainty, operator, acceptance criterion, and failure consequence.

Worked Example: Ultrasonic Delamination Screen with Sampling and Guard Bands

A carbon-fiber composite panel is inspected after impact. The team uses a pulse-echo ultrasonic method and a grid inspection plan. Use:

Quantity	Value
laminate thickness	18\ \text{mm}
calibrated ultrasonic velocity	v=2900\ \text{m/s}
indication echo time	t_e=8.6\ \mu\text{s}
probe frequency	f=5\ \text{MHz}
mapped indication area	A_d=310\ \text{mm}^2
zone equivalent-diameter limit	d_{limit}=18\ \text{mm}
sizing expanded uncertainty	U_d=1.5\ \text{mm}
inspection zones in panel	N=120
sampled zones	n=30
zones with reportable indications	d=4
confidence calculation defect probability	p=0.10
depth uncertainty components	0.25,\ 0.18,\ 0.30\ \text{mm}
coverage factor	k=2

Step 1: Estimate Indication Depth

Pulse-echo depth is:

Convert time:

Substitute:

Therefore:

Engineering comment: the result depends directly on velocity calibration. A composite laminate may have direction-dependent wave speed, so the calibration standard should represent the material, layup, temperature, and sound path.

Step 2: Check Wavelength Scale

Wavelength is:

Therefore:

Engineering comment: wavelength helps interpret the scale of the acoustic interaction, but it is not a validated defect-sizing limit. Attenuation, noise, ply interfaces, probe focusing, orientation, and calibration reflectors still control detectability.

Step 3: Convert Mapped Area to Equivalent Diameter

Equivalent circular diameter is:

Apply the conservative sizing uncertainty:

Compare with the zone limit:

The guarded indication size exceeds the acceptance limit.

Engineering comment: if damage size is harmful, add sizing uncertainty before comparison. Without the guard band, the indication already exceeds the limit; uncertainty strengthens the hold decision.

Step 4: Estimate Sampled Defect Fraction

Observed defect fraction:

Estimated affected zones in the panel:

Engineering comment: this is a sampling estimate, not proof that exactly 16 zones are affected. Impact damage often clusters, so the inspection plan should expand around detected indications rather than rely on random extrapolation alone.

Step 5: Check Probability of Seeing at Least One Defect

If the true zone defect probability were 10 percent, the probability of observing at least one defect in 30 sampled zones would be:

The sampling plan is likely to find at least one defect if defects are randomly distributed at that rate.

Engineering comment: this probability does not prove adequate coverage of the damage field. Because the method found reportable indications, the engineering response should move from sampling to targeted mapping and disposition.

Step 6: Guard the Depth Estimate

Combined depth uncertainty:

Expanded uncertainty:

Depth interval:

or approximately:

Engineering comment: the depth estimate is good enough for a preliminary ply-group decision only if the procedure can resolve the relevant interfaces. If acceptance depends on the exact ply interface, the inspection needs a calibrated reference laminate or a secondary method.

Decision

The panel should not be released as a routine pass. The guarded equivalent diameter exceeds the zone limit, the sampled defect fraction suggests the possibility of a wider damage field, and the indication depth lies in a range that may be structurally relevant.

The appropriate engineering action is to map the full affected region, compare the indication to the structural damage-tolerance basis, review compression-after-impact or bearing allowables, and document repair, replacement, or engineering concession with post-repair inspection evidence.

Common Mistakes

Common mistakes include treating resolution as accuracy, using a single coupon or scan as population evidence, reporting NDE images without a qualified detection size, applying guard bands in the wrong direction, and averaging away local defects that control failure.

Other errors include using ultrasonic depth without velocity calibration, claiming CT detection from voxel size alone, using XRF chemistry as proof of heat treatment, applying tensile results to fatigue or fracture without a damage model, and leaving inspector competency or procedure revision out of the release record.