Formula sheet
Polymer Composite and Ceramic Materials Formula Sheet
Materials engineering formula sheet for polymers, composites, and ceramics covering mixture rules, anisotropy, creep, moisture, voids, thermal stress, fracture, proof screening, and validation.
This formula sheet collects first-pass calculations for polymer, composite, and ceramic material systems. Use it for hand checks, design reviews, test planning, and interpretation of material data. It is not a substitute for qualified allowables, material standards, supplier data, coupon testing, environmental conditioning, or project-specific design rules.
Polymers, composites, and ceramics are strongly controlled by process history, defects, interfaces, temperature, moisture, loading rate, and inspection capability. A formula is useful only when the material system behind the formula matches the real part.
Conventions and Units
State units before comparing values:
- modulus, stress, and strength: Pa, MPa, or GPa;
- density: kg/m^3 or g/cm^3;
- thickness, flaw size, and bond length: m or mm;
- temperature change: K or deg C difference;
- moisture content and void content: percent or fraction, stated clearly;
- creep time: s, h, or years with temperature and stress level;
- probability quantities: dimensionless, with sample volume or area basis.
Do not mix mass fraction and volume fraction. Composite stiffness formulas usually use volume fraction. Purchasing and batch records often use mass fraction.
Volume Fraction from Mass and Density
For constituents with masses m_i and densities \rho_i, volume fraction is:
For a fiber and matrix composite:
and:
where V_f is fiber volume fraction and V_m is matrix volume fraction.
Worked Check
A coupon contains m_f=58\ \text{g} carbon fiber and m_m=42\ \text{g} epoxy. Use:
Then:
The fiber volume fraction is about 48.2\%.
Engineering Comment
This result is lower than the mass fraction because carbon fiber is denser than the matrix. If a stiffness calculation accidentally uses mass fraction as volume fraction, the predicted modulus can be badly biased.
Composite Density
For a void-free composite:
When mass fractions w_i are known:
These equations assume no voids and no significant density change during cure, sintering, crystallization, or chemical reaction.
Longitudinal Rule of Mixtures
For a unidirectional composite loaded along the fiber direction:
where:
- E_1 is longitudinal modulus;
- E_f is fiber modulus;
- E_m is matrix modulus.
Strength is sometimes screened similarly:
but this is much less reliable than the modulus relation because failure depends on fiber defects, matrix strain limit, interface quality, compression behavior, waviness, stress concentration, and test method.
Worked Check
Use:
Then:
The composite is stiff in the fiber direction, but this value says nothing about transverse modulus, interlaminar tension, bearing, impact damage, or delamination.
Transverse Modulus Screen
A simple inverse rule for transverse modulus is:
so:
This is a screening model. More accurate composite micromechanics depends on fiber packing, matrix constraint, interface quality, voids, temperature, moisture, and test direction.
Worked Check
Using the same material system:
Therefore:
The transverse modulus is about 5\% of the longitudinal modulus. This is why ply direction and load path matter.
Shear Modulus and Poisson Ratio
For an isotropic phase:
where G is shear modulus and \nu is Poisson ratio.
For a composite lamina, do not assume isotropy. A unidirectional ply needs direction-specific properties such as E_1, E_2, G_{12} and \nu_{12}. If only isotropic formulas are used, bearing, torsion, shear transfer, and free-edge behavior can be misread.
Laminate Modulus Screen
For a rough axial modulus screen of a laminate loaded in one direction:
where:
- E_{x,k} is the effective modulus contribution of ply k in the load direction;
- t_k is ply thickness;
- t=\sum_k t_k is total laminate thickness.
This formula is only a screening estimate. Real laminate analysis transforms orthotropic ply stiffness, assembles laminate stiffness matrices, and checks coupling, symmetry, balance, damage tolerance, open holes, compression after impact, and free-edge effects.
Specific Stiffness and Specific Strength
For weight-sensitive designs:
These ratios are useful for comparison, but they do not include joining, inspection, fire, damage tolerance, environmental knockdowns, cost, process yield, or repair.
Average Adhesive Shear Stress
For a simple bonded overlap carrying load P:
where b is bond width and L_b is effective bond length.
Worked Check
If:
then:
Engineering Comment
Average shear is not peak shear. Real joints have peel stress, edge stress, adherend bending, surface preparation sensitivity, cure history, temperature, moisture, fatigue, and inspection limits. A low average value can still fail at the edge.
Void Content from Density
If theoretical void-free density is \rho_{theory} and measured composite density is \rho_{meas}:
where V_v is void volume fraction.
Worked Check
For:
the void fraction is:
So:
Engineering Comment
Void content should be interpreted with process route and inspection evidence. A small void fraction may be acceptable in one laminate and unacceptable near a highly loaded bond, pressure boundary, fatigue detail, or compression-critical zone.
Moisture Uptake
Moisture content by mass is:
where m_t is conditioned mass and m_d is dry mass.
For early-time one-dimensional diffusion in a thin plate, a screening relation is:
where:
- M_{\infty} is saturation moisture content;
- h is plate thickness;
- D is diffusion coefficient;
- t is exposure time.
This approximation is valid only at early uptake and for the assumed geometry and boundary condition.
Polymer Creep Compliance
For linear viscoelastic screening under constant stress:
where:
- \epsilon(t) is strain at time t;
- J(t) is creep compliance;
- \sigma is applied stress.
If an initial elastic strain and creep strain are separated:
Worked Check
A polymer clip carries:
At service temperature, the creep compliance at one year is:
Convert stress to GPa:
Then:
So:
Engineering Comment
The value is meaningful only for the stated temperature, stress, material condition, and time. Polymers may show nonlinear creep near high stress, elevated temperature, chemical exposure, or long service time.
Creep Retention Factor
A simple retention factor can compare long-term property to short-term property:
or for allowable stress:
Use the factor only with the same test temperature, environment, strain rate, and failure criterion.
Thermal Strain and Thermal Stress
Free thermal strain is:
If expansion is fully restrained in a simple linear elastic member:
For a plane-strain-like restraint, a conservative screen may use:
where \alpha is coefficient of thermal expansion.
Worked Check
A ceramic layer with:
is fully restrained during:
Simple restrained stress:
Engineering Comment
This stress is a warning, not a full thermal-stress solution. Coating thickness, substrate stiffness, temperature gradient, creep, cracking, residual stress, edge geometry, and interface toughness can dominate the real failure mode.
Fracture Toughness and Critical Flaw Size
For a brittle material or ceramic under tensile stress:
Failure is screened when:
Critical flaw size is:
where:
- K_I is stress intensity factor;
- K_{IC} is fracture toughness;
- Y is geometry factor;
- a is flaw size;
- \sigma is tensile stress.
Worked Check
A ceramic has:
with:
and tensile stress:
Then:
Engineering Comment
If inspection cannot reliably detect flaws below this size in the critical region, the design needs lower stress, tougher material, proof screening, compressive residual stress, geometry change, or a different inspection basis.
Weibull Failure Probability
For brittle strength scatter, a common Weibull screening form is:
where:
- P_f is probability of failure;
- \sigma is applied tensile stress;
- \sigma_0 is scale stress for reference volume V_0;
- m is Weibull modulus;
- V is stressed volume.
This relation is sensitive to stress distribution, flaw population, volume basis, surface condition, proof testing, and statistical evidence.
Proof Stress Screening
If a brittle component survives proof stress \sigma_p, and service tensile stress is \sigma_s, a simple proof ratio is:
This is not a complete reliability proof. Proof testing can damage some materials, miss time-dependent degradation, or fail to represent multiaxial stress and contact damage.
Permeability Flux
For a thin polymer barrier with steady one-dimensional transport:
where:
- J is molar or mass flux, depending on the units of permeability P;
- \Delta p is pressure or concentration driving difference;
- h is barrier thickness.
Permeability depends on penetrant, polymer chemistry, crystallinity, temperature, humidity, aging, fillers, defects, and seams.
Rheology and Viscosity
For simple Newtonian shear:
where:
- \tau is shear stress;
- \mu is dynamic viscosity;
- \dot{\gamma} is shear rate.
Many polymer melts, filled resins, adhesives, and slurries are non-Newtonian. Their apparent viscosity can change with shear rate, temperature, filler loading, cure state, moisture, and residence time.
Damping and Loss Factor
Loss factor can be screened as:
where E' is storage modulus and E'' is loss modulus.
High damping may help vibration and acoustic response, but the same viscoelastic behavior may increase heat buildup, creep, hysteresis, and temperature sensitivity.
Measurement and Validation Checks
For a material property used in release:
or:
where MS is margin of safety.
When the property comes from test data, record:
- specimen orientation and geometry;
- process route, lot, cure, sinter, heat treatment, or conditioning;
- test temperature, humidity, strain rate, and environment;
- measurement uncertainty and calibration status;
- defect population and inspection method;
- statistical basis and outlier treatment;
- whether the coupon represents the real part.
Common Mistakes
Common errors include:
- using mass fraction where volume fraction is required;
- applying longitudinal composite modulus to transverse, shear, bearing, or interlaminar loading;
- using short-term polymer strength for long-term creep-controlled service;
- ignoring moisture, temperature, ultraviolet exposure, chemicals, or strain rate;
- treating void content as harmless without checking location and load path;
- using ceramic average strength without flaw size, surface finish, proof test, or volume effect;
- accepting a material change without repeating process, inspection, and validation evidence.
Validity Limits
These formulas are screening tools. They are most useful for early sizing, plausibility checks, test planning, and independent review. They become unsafe when used beyond their assumptions: nonlinear polymer behavior, damaged composites, thick laminates, impact damage, complex joints, multiaxial ceramic stress, thermal gradients, residual stress, manufacturing defects, aging, or unvalidated material substitutions.
For release decisions, connect every formula to material lot, process route, environmental conditioning, inspection evidence, uncertainty, and the failure mode that actually controls the design.