Materials Selection and Mechanical Properties Formula Sheet

This formula sheet collects first-pass calculations used to compare material-process systems. Use it to screen specific stiffness, specific strength, allowable stress, utilization, elastic constants, anisotropy, thermal strain, corrosion allowance, property scatter, cost normalization, inspection evidence, and lifecycle tradeoffs.

The formulas do not select a material by themselves. A valid selection must specify material grade, product form, processing route, heat treatment, surface condition, inspection method, environment, temperature, load spectrum, repair approach, and failure consequence. A property table value is only useful when it represents the material state that will actually be manufactured and validated.

Symbols and Units

Symbol	Meaning	Common unit
E	Young’s modulus	Pa or GPa
G	shear modulus	Pa or GPa
\nu	Poisson’s ratio	dimensionless
\rho	density	kg/m^3
\sigma_y	yield strength	Pa or MPa
\sigma_u	ultimate tensile strength	Pa or MPa
\sigma_{allow}	allowable stress	Pa or MPa
\sigma_{demand}	applied or demanded stress	Pa or MPa
N	factor of safety or margin factor	dimensionless
u	utilization	dimensionless
\alpha	coefficient of thermal expansion	1/K
\Delta T	temperature change	K
CR	corrosion rate	mm/year
C_m	material cost per mass	currency/kg
m	mass	kg

Use consistent units before calculating a ratio. For example, compare E/\rho only after all moduli are in Pa and all densities are in kg/m^3.

Specific Stiffness and Specific Strength

Specific stiffness:

Specific strength based on yield:

Specific strength based on ultimate tensile strength:

These ratios are useful for weight-sensitive screening, but they do not include geometry, buckling, joining, fatigue, corrosion, manufacturing route, cost, or inspection evidence.

Worked Example: Stiffness-Limited Bracket Screening

Candidate A has:

Candidate B has:

Specific stiffness:

The two candidates are nearly equivalent on this simple stiffness-per-density metric. The decision should move to strength, corrosion, process route, joining, inspection, cost, and availability instead of declaring a winner from E alone.

Allowable Stress, Factor of Safety, and Utilization

Allowable stress from a factor of safety:

Stress utilization:

Margin:

Strength check:

The limiting stress may be yield strength, ultimate strength, compressive strength, bearing strength, creep strength, fatigue strength, or an environment-specific reduced value. State the failure mode before applying the check.

Worked Example: Yield-Based Screening

A machined part sees nominal stress:

The material yield strength in the specified heat treatment is:

Use:

Allowable stress:

Utilization:

The static yield screen passes. It does not clear fatigue, fracture, corrosion, weld quality, stress concentration, temperature, or process-route scatter.

Elastic Constants

For isotropic linear elastic materials:

Bulk modulus:

These relations do not apply to strongly anisotropic composites, wood, textured metals, additive-manufactured parts with directional properties, cracked materials, foams, or nonlinear polymers without additional assumptions.

Common Material Indices

Material indices are useful only for a defined loading mode, geometry family, and objective. Common first-pass indices are:

Design objective	Simplified index to maximize	Typical assumption
light tension member, stiffness limited	E/\rho	axial stiffness, variable area
light tension member, strength limited	\sigma_{allow}/\rho	axial strength, variable area
light beam, bending stiffness limited	E^{1/2}/\rho	similar beam shape, variable scale
light plate, bending stiffness limited	E^{1/3}/\rho	similar plate form, variable thickness
low cost stiffness member	E/(\rho C_m)	material cost dominates
low cost strength member	\sigma_{allow}/(\rho C_m)	material cost dominates

These are screening indices, not universal laws. Manufacturing minimum thickness, joining, buckling, fatigue, corrosion, fire, temperature, and inspection can overturn an index ranking.

Worked Example: Strength per Cost

Two candidate material-process systems have:

Candidate	Allowable stress	Density	Material cost
A	180\ \text{MPa}	7800\ \text{kg/m}^3	2.5\ \text{currency/kg}
B	130\ \text{MPa}	2700\ \text{kg/m}^3	5.0\ \text{currency/kg}

Use:

Candidate A:

Candidate B:

Candidate B is slightly better on this narrow strength-per-material-cost index. The difference is small enough that manufacturing yield, machining time, joining, corrosion protection, availability, and inspection may dominate the real lifecycle decision.

Anisotropy and Directional Knockdown

Directional stiffness ratio:

Directional strength ratio:

If a load can rotate relative to material direction, a conservative directional knockdown may be used:

where 0<k_{dir}\le1 is justified by test data, laminate analysis, product-form data, or design standard.

Anisotropy is not limited to composites. Rolled plate, extrusions, forgings, welds, additive-manufactured parts, wood, and oriented polymers can all have direction-dependent properties.

Thermal Strain and Thermal Stress

Free thermal strain:

If axial thermal expansion is fully restrained in a linear elastic member:

Thermal stress utilization:

Worked Example: Constrained Thermal Expansion

A bar has:

It is fully restrained while temperature rises by:

Thermal strain:

Thermal stress:

If allowable stress is 160\ \text{MPa}:

The screen passes for this simplified axial case. Real assemblies may have partial restraint, nonlinear contact, yielding, creep, buckling, fatigue from thermal cycles, or stress concentration at welds and attachments.

Corrosion Allowance and Section Loss

Thickness loss from uniform corrosion:

Remaining thickness:

Relative area loss for a plate strip with unchanged width is:

Uniform corrosion allowance does not cover pitting, crevice corrosion, galvanic attack, erosion-corrosion, hydrogen damage, coating defects, or inaccessible inspection zones.

Worked Example: Corrosion Allowance

A carbon-steel plate has initial thickness:

Expected uniform corrosion rate is:

The intended service interval is:

Thickness loss:

Remaining thickness:

Area loss for a constant-width strip:

The result supports a uniform-corrosion screen only. If inspection history shows pitting or under-deposit corrosion, minimum measured thickness and local stress concentration become more important than average loss.

Property Scatter and Characteristic Values

For test results with mean \bar{x} and standard deviation s_x, a simple lower characteristic value may be screened as:

where k is selected from the design basis, sample size, confidence requirement, and property distribution.

Property knockdown:

Use characteristic values for release decisions when property scatter, process variability, and failure consequence matter. Do not mix a lower-bound strength with an average modulus unless the decision basis allows it.

Validation and Evidence

Selection quantity	Evidence to request
elastic modulus	tensile test, flexural test, supplier data tied to product form
yield or ultimate strength	certificate, heat treatment record, representative tensile test
density	material standard, measured mass and volume, porosity evidence
anisotropy	directional test coupons, laminate data, rolling direction or build orientation
hardness	hardness map, conversion basis, correlation to heat treatment
fracture toughness	test standard, thickness validity, temperature and environment
fatigue property	S-N data matching surface, mean stress, environment and survival basis
corrosion allowance	exposure data, coupon test, inspection history, coating condition
NDE credit	method qualification, access, probability of detection, acceptance criteria
substitution approval	equivalence matrix, risk review, validation plan, configuration control

Material validation should connect property, product form, process route, inspection method, and load case. A certificate is not enough if the purchased form, heat treatment, weld region, print orientation, or surface condition differs from the data behind the calculation.

Common Mistakes

Common mistakes include selecting by a single maximum property, comparing density without checking stiffness or strength, using yield strength when fatigue or fracture governs, treating isotropic formulas as valid for anisotropic materials, ignoring product form, and assuming a supplier data-sheet value represents the worst case.

Other mistakes are lifecycle-related: ignoring corrosion allowance, coating maintenance, repair welding, inspection access, property scatter, joining compatibility, and end-of-life replacement. A material choice is defensible only when the selected material-process-surface system remains valid under the actual load, environment, manufacturing route, inspection plan, and failure consequence.