Polymers

Materials Science · FE Reference Handbook section

Materials Science

2 formulas

10 exam-style examples

~49 min

All Materials Science lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Polymers within Materials Science. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what polymers describes physically and when it applies.
State every one of the 2 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: percent versus fraction in composition and strain.

Lecture

Why this section exists. Polymers is the part of Materials Science that lets you connect a steel, concrete or polymer specimen under test to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as a definition, a phase-diagram read, or a one-line property calculation. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. percent versus fraction in composition and strain. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Concrete cylinder under axial load in a compression testing machine. — Photo 1. Where this shows up in practice: polymers.
Wikimedia Commons, public domain

Materials Science — Polymers: reference schematic for orienting the symbols used in this section.

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes a steel, concrete or polymer specimen under test. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 2 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Notation used in this section

LOG E or LOG σ	Quantity produced by "LOG E or LOG σ" — read its definition and unit from the handbook line directly above the equation.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

Polymers are classified as thermoplastics that can be melted and reformed. Thermosets cannot be melted and reformed.
Tg Tm
TEMPERATURE

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Composite modulus by the rule of mixtures — Polymers

A fiber-reinforced polymer composite contains 60% fiber by volume. The fiber modulus is 76 GPa and the polymer matrix modulus is 4.5 GPa. Compute the longitudinal and transverse moduli of the composite material.

Given

V_f = 0.60
E_f = 76 GPa
E_m = 4.5 GPa

Find

E_longitudinal and E_transverse

Start with the thinking

Loaded along the fibers the strains are equal, giving the direct rule of mixtures.
Loaded across the fibers the stresses are equal, giving the inverse (harmonic) rule.

Step-by-step solution

Formula
$E_{∥} = V_{f} E_{f} + (1 - V_{f}) E_{m}$
Substituting
$E_{∥} = 0.60 (76) + 0.40 (4.5) = 47.40 GP a$
Formula
$1/ E_{⊥} = V_{f} / E_{f} + (1 - V_{f}) / E_{m}$
Substituting
$1/ E_{⊥} = 0.60/76 + 0.40/4.5 = 0.09678$
Evaluate
$E_{⊥} = 10.33 GP a$

Answer: E_∥ = 47.4 GPa, E_⊥ = 10.33 GPa

Why the other options are there

40.3 GPa (simple average)
E_⊥ = 47.4 GPa (same as longitudinal)

Reference: FE Reference Handbook — Materials Science → Polymers

Example 2

Composite modulus by the rule of mixtures — Polymers (2)

A fiber-reinforced polymer composite contains 40% fiber by volume. The fiber modulus is 40 GPa and the polymer matrix modulus is 5.0 GPa. Compute the longitudinal and transverse moduli of the composite material.

Given

V_f = 0.40
E_f = 40 GPa
E_m = 5.0 GPa

Find

E_longitudinal and E_transverse

Start with the thinking

Loaded along the fibers the strains are equal, giving the direct rule of mixtures.
Loaded across the fibers the stresses are equal, giving the inverse (harmonic) rule.

Step-by-step solution

Formula
$E_{∥} = V_{f} E_{f} + (1 - V_{f}) E_{m}$
Substituting
$E_{∥} = 0.40 (40) + 0.60 (5.0) = 19.00 GP a$
Formula
$1/ E_{⊥} = V_{f} / E_{f} + (1 - V_{f}) / E_{m}$
Substituting
$1/ E_{⊥} = 0.40/40 + 0.60/5.0 = 0.13000$
Evaluate
$E_{⊥} = 7.69 GP a$

Answer: E_∥ = 19.0 GPa, E_⊥ = 7.69 GPa

Why the other options are there

22.5 GPa (simple average)
E_⊥ = 19.0 GPa (same as longitudinal)

Reference: FE Reference Handbook — Materials Science → Polymers

Example 3

Composite modulus by the rule of mixtures — Polymers (3)

A fiber-reinforced polymer composite contains 35% fiber by volume. The fiber modulus is 76 GPa and the polymer matrix modulus is 3.5 GPa. Compute the longitudinal and transverse moduli of the composite material.

Given

V_f = 0.35
E_f = 76 GPa
E_m = 3.5 GPa

Find

E_longitudinal and E_transverse

Start with the thinking

Loaded along the fibers the strains are equal, giving the direct rule of mixtures.
Loaded across the fibers the stresses are equal, giving the inverse (harmonic) rule.

Step-by-step solution

Formula
$E_{∥} = V_{f} E_{f} + (1 - V_{f}) E_{m}$
Substituting
$E_{∥} = 0.35 (76) + 0.65 (3.5) = 28.87 GP a$
Formula
$1/ E_{⊥} = V_{f} / E_{f} + (1 - V_{f}) / E_{m}$
Substituting
$1/ E_{⊥} = 0.35/76 + 0.65/3.5 = 0.19032$
Evaluate
$E_{⊥} = 5.25 GP a$

Answer: E_∥ = 28.9 GPa, E_⊥ = 5.25 GPa

Why the other options are there

39.8 GPa (simple average)
E_⊥ = 28.9 GPa (same as longitudinal)

Reference: FE Reference Handbook — Materials Science → Polymers

Example 4

Composite modulus by the rule of mixtures — Polymers (4)

A fiber-reinforced polymer composite contains 50% fiber by volume. The fiber modulus is 51 GPa and the polymer matrix modulus is 2.5 GPa. Compute the longitudinal and transverse moduli of the composite material.

Given

V_f = 0.50
E_f = 51 GPa
E_m = 2.5 GPa

Find

E_longitudinal and E_transverse

Start with the thinking

Loaded along the fibers the strains are equal, giving the direct rule of mixtures.
Loaded across the fibers the stresses are equal, giving the inverse (harmonic) rule.

Step-by-step solution

Formula
$E_{∥} = V_{f} E_{f} + (1 - V_{f}) E_{m}$
Substituting
$E_{∥} = 0.50 (51) + 0.50 (2.5) = 26.75 GP a$
Formula
$1/ E_{⊥} = V_{f} / E_{f} + (1 - V_{f}) / E_{m}$
Substituting
$1/ E_{⊥} = 0.50/51 + 0.50/2.5 = 0.20980$
Evaluate
$E_{⊥} = 4.77 GP a$

Answer: E_∥ = 26.8 GPa, E_⊥ = 4.77 GPa

Why the other options are there

26.8 GPa (simple average)
E_⊥ = 26.8 GPa (same as longitudinal)

Reference: FE Reference Handbook — Materials Science → Polymers

Example 5

Composite modulus by the rule of mixtures — Polymers (5)

A fiber-reinforced polymer composite contains 45% fiber by volume. The fiber modulus is 65 GPa and the polymer matrix modulus is 3.5 GPa. Compute the longitudinal and transverse moduli of the composite material.

Given

V_f = 0.45
E_f = 65 GPa
E_m = 3.5 GPa

Find

E_longitudinal and E_transverse

Start with the thinking

Loaded along the fibers the strains are equal, giving the direct rule of mixtures.
Loaded across the fibers the stresses are equal, giving the inverse (harmonic) rule.

Step-by-step solution

Formula
$E_{∥} = V_{f} E_{f} + (1 - V_{f}) E_{m}$
Substituting
$E_{∥} = 0.45 (65) + 0.55 (3.5) = 31.18 GP a$
Formula
$1/ E_{⊥} = V_{f} / E_{f} + (1 - V_{f}) / E_{m}$
Substituting
$1/ E_{⊥} = 0.45/65 + 0.55/3.5 = 0.16407$
Evaluate
$E_{⊥} = 6.10 GP a$

Answer: E_∥ = 31.2 GPa, E_⊥ = 6.10 GPa

Why the other options are there

34.3 GPa (simple average)
E_⊥ = 31.2 GPa (same as longitudinal)

Reference: FE Reference Handbook — Materials Science → Polymers

Example 6

Composite modulus by the rule of mixtures — Polymers (6)

A fiber-reinforced polymer composite contains 30% fiber by volume. The fiber modulus is 76 GPa and the polymer matrix modulus is 4.0 GPa. Compute the longitudinal and transverse moduli of the composite material.

Given

V_f = 0.30
E_f = 76 GPa
E_m = 4.0 GPa

Find

E_longitudinal and E_transverse

Start with the thinking

Loaded along the fibers the strains are equal, giving the direct rule of mixtures.
Loaded across the fibers the stresses are equal, giving the inverse (harmonic) rule.

Step-by-step solution

Formula
$E_{∥} = V_{f} E_{f} + (1 - V_{f}) E_{m}$
Substituting
$E_{∥} = 0.30 (76) + 0.70 (4.0) = 25.60 GP a$
Formula
$1/ E_{⊥} = V_{f} / E_{f} + (1 - V_{f}) / E_{m}$
Substituting
$1/ E_{⊥} = 0.30/76 + 0.70/4.0 = 0.17895$
Evaluate
$E_{⊥} = 5.59 GP a$

Answer: E_∥ = 25.6 GPa, E_⊥ = 5.59 GPa

Why the other options are there

40.0 GPa (simple average)
E_⊥ = 25.6 GPa (same as longitudinal)

Reference: FE Reference Handbook — Materials Science → Polymers

Example 7

Composite modulus by the rule of mixtures — Polymers (7)

A fiber-reinforced polymer composite contains 60% fiber by volume. The fiber modulus is 71 GPa and the polymer matrix modulus is 2.5 GPa. Compute the longitudinal and transverse moduli of the composite material.

Given

V_f = 0.60
E_f = 71 GPa
E_m = 2.5 GPa

Find

E_longitudinal and E_transverse

Start with the thinking

Loaded along the fibers the strains are equal, giving the direct rule of mixtures.
Loaded across the fibers the stresses are equal, giving the inverse (harmonic) rule.

Step-by-step solution

Formula
$E_{∥} = V_{f} E_{f} + (1 - V_{f}) E_{m}$
Substituting
$E_{∥} = 0.60 (71) + 0.40 (2.5) = 43.60 GP a$
Formula
$1/ E_{⊥} = V_{f} / E_{f} + (1 - V_{f}) / E_{m}$
Substituting
$1/ E_{⊥} = 0.60/71 + 0.40/2.5 = 0.16845$
Evaluate
$E_{⊥} = 5.94 GP a$

Answer: E_∥ = 43.6 GPa, E_⊥ = 5.94 GPa

Why the other options are there

36.8 GPa (simple average)
E_⊥ = 43.6 GPa (same as longitudinal)

Reference: FE Reference Handbook — Materials Science → Polymers

Example 8

Composite modulus by the rule of mixtures — Polymers (8)

A fiber-reinforced polymer composite contains 30% fiber by volume. The fiber modulus is 52 GPa and the polymer matrix modulus is 2.5 GPa. Compute the longitudinal and transverse moduli of the composite material.

Given

V_f = 0.30
E_f = 52 GPa
E_m = 2.5 GPa

Find

E_longitudinal and E_transverse

Start with the thinking

Loaded along the fibers the strains are equal, giving the direct rule of mixtures.
Loaded across the fibers the stresses are equal, giving the inverse (harmonic) rule.

Step-by-step solution

Formula
$E_{∥} = V_{f} E_{f} + (1 - V_{f}) E_{m}$
Substituting
$E_{∥} = 0.30 (52) + 0.70 (2.5) = 17.35 GP a$
Formula
$1/ E_{⊥} = V_{f} / E_{f} + (1 - V_{f}) / E_{m}$
Substituting
$1/ E_{⊥} = 0.30/52 + 0.70/2.5 = 0.28577$
Evaluate
$E_{⊥} = 3.50 GP a$

Answer: E_∥ = 17.4 GPa, E_⊥ = 3.50 GPa

Why the other options are there

27.3 GPa (simple average)
E_⊥ = 17.4 GPa (same as longitudinal)

Reference: FE Reference Handbook — Materials Science → Polymers

Example 9

Composite modulus by the rule of mixtures — Polymers (9)

A fiber-reinforced polymer composite contains 60% fiber by volume. The fiber modulus is 64 GPa and the polymer matrix modulus is 2.5 GPa. Compute the longitudinal and transverse moduli of the composite material.

Given

V_f = 0.60
E_f = 64 GPa
E_m = 2.5 GPa

Find

E_longitudinal and E_transverse

Start with the thinking

Loaded along the fibers the strains are equal, giving the direct rule of mixtures.
Loaded across the fibers the stresses are equal, giving the inverse (harmonic) rule.

Step-by-step solution

Formula
$E_{∥} = V_{f} E_{f} + (1 - V_{f}) E_{m}$
Substituting
$E_{∥} = 0.60 (64) + 0.40 (2.5) = 39.40 GP a$
Formula
$1/ E_{⊥} = V_{f} / E_{f} + (1 - V_{f}) / E_{m}$
Substituting
$1/ E_{⊥} = 0.60/64 + 0.40/2.5 = 0.16938$
Evaluate
$E_{⊥} = 5.90 GP a$

Answer: E_∥ = 39.4 GPa, E_⊥ = 5.90 GPa

Why the other options are there

33.3 GPa (simple average)
E_⊥ = 39.4 GPa (same as longitudinal)

Reference: FE Reference Handbook — Materials Science → Polymers

Example 10

Composite modulus by the rule of mixtures — Polymers (10)

A fiber-reinforced polymer composite contains 30% fiber by volume. The fiber modulus is 76 GPa and the polymer matrix modulus is 2.5 GPa. Compute the longitudinal and transverse moduli of the composite material.

Given

V_f = 0.30
E_f = 76 GPa
E_m = 2.5 GPa

Find

E_longitudinal and E_transverse

Start with the thinking

Loaded along the fibers the strains are equal, giving the direct rule of mixtures.
Loaded across the fibers the stresses are equal, giving the inverse (harmonic) rule.

Step-by-step solution

Formula
$E_{∥} = V_{f} E_{f} + (1 - V_{f}) E_{m}$
Substituting
$E_{∥} = 0.30 (76) + 0.70 (2.5) = 24.55 GP a$
Formula
$1/ E_{⊥} = V_{f} / E_{f} + (1 - V_{f}) / E_{m}$
Substituting
$1/ E_{⊥} = 0.30/76 + 0.70/2.5 = 0.28395$
Evaluate
$E_{⊥} = 3.52 GP a$

Answer: E_∥ = 24.6 GPa, E_⊥ = 3.52 GPa

Why the other options are there

39.3 GPa (simple average)
E_⊥ = 24.6 GPa (same as longitudinal)

Reference: FE Reference Handbook — Materials Science → Polymers

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given a steel, concrete or polymer specimen under test, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

Polymers contains 2 relations; you must be able to find this page in under 15 seconds.
Exam style: a definition, a phase-diagram read, or a one-line property calculation.
Unit rule: percent versus fraction in composition and strain.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

percent versus fraction in composition and strain
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.

Polymers

Learning objectives

Lecture

Theory, developed

Notation used in this section

Handbook notes for this section

Core formulas for this FE topic

LOGEorLOGσ

Theabovecurveshowsthetemperaturedependentstrength(σ)ormodulus(E)forathermoplasticpolymer.

Worked exam-style examples

Self-check

Chapter summary

Common traps in this section