Polymers & Composites Workbench

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Engineering reference

Polymers & Composites: theory, method, and sources

This materials science workspace publishes 11 governing equations, 6 stated assumptions, 1 documented boundary, and 2 sources so the numbers it returns can be checked rather than taken on trust.

Calculations run locallyCalculation & source methodology

How this tool works

Ten modules cover number and weight average molecular weight, degree of polymerization, the Fox equation for copolymer glass transition, rubber crosslink density, longitudinal and transverse composite modulus, composite density and fibre mass fraction, thermal expansion, the Halpin–Tsai model, and critical fibre length.

Longitudinal and transverse modulus are calculated from the same fibre volume fraction, which shows directly how large the anisotropy of a unidirectional laminate is.

Calculators and topics covered

polymers
composites
fiber reinforcement
glass transition
molecular weight
Fox equation
rule of mixtures
Halpin Tsai
critical fiber length

Core equations

\text{number average molecular weight:}\quad M_{n} = \frac{\sum N_{i} M_{i}}{\sum N_{i}}

\text{weight average:}\quad \begin{gathered}M_{w} = \frac{\sum N_{i} M_{i}^{2}}{\sum N_{i} M_{i}}\\\text{dispersity}\; \text{Đ} = \frac{M_{w}}{M_{n}}\end{gathered}

\text{degree of polymerization:}\quad DP = \frac{M}{M_{\mathrm{repeat}}}

\text{Fox equation:}\quad \frac{1}{T_{g}} = \sum \left(\frac{w_{i}}{T_{gi}}\right)

\text{rubber elasticity:}\quad n = \frac{E}{3 RT}

\text{longitudinal modulus (Voigt):}\quad E_{1} = V_{f}\; E_{f} + \left(1 - V_{f}\right) E_{m}

\text{transverse modulus (Reuss):}\quad \frac{1}{E_{2}} = \frac{V_{f}}{E_{f}} \frac{+ \left(1 - V_{f}\right)}{E_{m}}

\text{composite density:}\quad \rho _{c} = V_{f} \rho _{f} + \left(1 - V_{f}\right) \rho _{m}

\text{longitudinal CTE:}\quad \alpha _{1} = \frac{V_{f} E_{f} \alpha _{f} + V_{m} E_{m} \alpha _{m}}{V_{f} E_{f} + V_{m} E_{m}}

\text{Halpin–Tsai:}\quad E = \frac{E_{m} \left(1 + \xi \eta V_{f}\right)}{1 - \eta V_{f}}\; \text{with}\; \eta = \frac{\frac{E_{f}}{E_{m}} - 1}{\frac{E_{f}}{E_{m}} + \xi }

\text{critical fibre length:}\quad L_{c} = \frac{\sigma _{f}\; d}{2 \tau _{i}}

Method and assumptions

Assumptions

Composites are treated as unidirectional with perfectly aligned, uniformly distributed continuous fibres.
The fibre–matrix interface is assumed perfect, with no debonding or void content.
The rule of mixtures gives an upper bound longitudinally and the inverse rule a lower bound transversely; real laminates fall between them.
The Fox equation assumes fully miscible components with no phase separation.
Viscoelastic behaviour, moisture absorption, and time- or temperature-dependent creep are not modelled.
Halpin–Tsai results depend strongly on the reinforcement geometry factor ξ, which must be chosen to match the fibre form.

Limitations and design boundaries

Educational screening models only. Material certification, process qualification, applicable standards, statistical variability, environmental conditioning, and professional engineering judgment are required for real selection or design.

Sources and references

Primary sources are preferred for ratings, standards, manufacturer data, and externally defined constants.

Source policy

Callister and Rethwisch, Materials Science and Engineering: An Introduction
Askeland and Wright, The Science and Engineering of Materials