School of Engineering · Playlist

Composite Material Behavior

Fiber direction, stacking sequence, and the elastic response of a ply or laminate.

  1. Fiber Direction and Elastic Stiffness
Materials and Manufacturing Processes/Composites

Fiber Direction and Elastic Stiffness

Predict how one unidirectional composite layer stretches and skews when its fibers turn relative to the same applied stress.

Fiber direction changes elastic response

Fiber direction changes elastic response

A fiber-reinforced composite combines a reinforcement with a surrounding matrix. In a unidirectional layer, the reinforcement is continuous and parallel. The layer is much stiffer along the fibers than across them, so its elastic response depends on direction.

Stress is force divided by loaded area. This lesson compares patches under the same uniform axial stress, so each square millimeter carries the same applied force. Strain is deformation divided by original dimension. Elastic stiffness connects stress to strain in the reversible small-strain model.

The illustrative material uses E1 = 40 GPa along the fibers, E2 = 10 GPa across them, G12 = 5 GPa, and ν12 = 0.30. These round teaching values do not describe a named composite and are not design allowables.

Review Plastics as Materials for reinforcement, matrix, thermosets, and thermoplastics. Review Plastic Forming Processes for hand layup and other forming routes.

Predict the least stretch, then turn the fibers; Interactive

Predict the least stretch, then turn the fibers

Read the directions: the fiber angle is the starting, unloaded direction measured from the horizontal load axis. In this drawing, +45° slopes up to the right and −45° slopes down to the right. Axial means along the load; transverse means across it. The drawn fibers follow the exaggerated deformation, so their visible slope can differ from the starting angle on the control.

Predict first: identical free material patches carry uniform axial stress σx = 10 MPa, with no applied transverse or shear stress. Which orientation stretches least: 0°, 45°, or 90°?

Same illustrative lamina and same 10 MPa stress
Fiber angleEffective axial stiffnessAxial strainTransverse strainEngineering shear strain
0°40.00 GPa0.0250%−0.0075%0 µε
+45°12.90 GPa0.0775%−0.0225%−375 µε
−45°12.90 GPa0.0775%−0.0225%+375 µε
90°10.00 GPa0.1000%−0.0075%0 µε

Result: 0° stretches least because the stiff fiber direction is aligned with the load. Opposite 45° angles have the same axial stiffness and stretch, but their shear strain has opposite sign.

This is a freely deforming material patch loaded by uniform edge traction. It is not a tensile-test coupon held in rigid grips: grips would restrain the off-axis skew and create another boundary-value problem.

Off-axis extension can also create shear strain

Off-axis extension can also create shear strain

At 0° and 90°, the loading direction matches a principal material direction. The patch stretches and narrows without extension–shear coupling. At an off-axis angle, an axial stress can also make the free patch skew. Turning +45° to −45° mirrors that skew while leaving the axial stiffness unchanged.

The interactive drawing exaggerates every strain by the same factor of 120. A larger drawn change therefore means a larger calculated strain; the display does not secretly rescale each angle to fill the frame.

Supporting detail: Addendum 1 — How the axial stiffness is calculated.

Stiffness does not predict strength; Practice

Stiffness does not predict strength

A smaller strain under the same stress means greater elastic stiffness. It does not say when the material fails, which direction is safest, or which laminate should be built. Strength needs failure data and criteria; a laminate also needs its complete stacking sequence, ply thicknesses, bonding, environment, loading, and boundary conditions.

This lesson models one thin, continuous-fiber lamina in small-strain linear elasticity and plane stress. It omits failure, damage, buckling, bending, temperature, moisture, manufacturing defects, grips, and interaction between multiple plies.

Supporting detail: Addendum 2 — Primary mechanics source and model boundary.

Practice axial strain Practice angle symmetry Practice model limits

Stack plies into a laminate and find one equally stiff in every direction; Interactive

Stack plies into a laminate and find one equally stiff in every direction

A real part is rarely one ply. A laminate is a stack of bonded plies, each with its own fiber angle. The stacking sequence is written from the top face down: [0/90] is a 0° ply on top of a 90° ply, and a subscript s means the list is mirrored about the midplane, so [0/90]s is [0/90/90/0]. Every ply here is the same illustrative ply as the card above, 0.125 mm thick.

Challenge: using only 0°, +45°, −45° and 90° plies, build a laminate that is equally stiff in every in-plane direction. Then find out why an unsymmetric stack curls when it is pulled.

4 0°3 90°2 90°1 0°midplanex 0°y 90°TOP FACEply angle10203040 GPa0°90°45°135°180°
8 0°7 +45°6 −45°5 90°4 90°3 −45°2 +45°1 0°midplanex 0°y 90°TOP FACEply angle10203040 GPa0°90°45°135°180°
Same ply, two balanced symmetric stacks. Left, cross-ply [0/90]s: 25.21 GPa along 0° and 90° but only 14.82 GPa at 45°. Right, [0/+45/−45/90]s: 20.74 GPa in every in-plane direction, so the shaded rose is a circle. The dotted outline is one 0° ply on its own (40 GPa along the fibers, 10 GPa across), the same ply as the fiber-orientation lesson above. Radius = modulus; outer ring 40 GPa.
Same illustrative plies, 0.125 mm each. Ex is flat-held (curvature held at zero).
Stack (top first)SymmetricBalancedQuasi-isotropicEx at 0° / 45° / 90°, GPaLargest |Bij|
[0]yesyesno40.00 / 12.90 / 10.000
[0/90]noyesno25.21 / 14.82 / 25.21239.8 N
[0/90]syesyesno25.21 / 14.82 / 25.210
[+45/−45]noyesno14.82 / 25.21 / 14.82119.9 N
[±45]syesyesno14.82 / 25.21 / 14.820
[0/+45/−45/90]syesyesyes20.74 / 20.74 / 20.740

A single 0° ply gives 40.00 GPa along x and 10.00 GPa along y, the fiber-orientation result for the same ply.

UNSYMMETRICheight ×30[0/90], 0° ply on topNx 0.50 N/mm, M = 0
SYMMETRICheight ×30[0/90]sNx 0.50 N/mm, M = 0
Same pull, Nx = 0.5 N/mm, no applied moment. [0/90] curls: κx −7.87×10⁻⁴ mm⁻¹ · κy 0 mm⁻¹ · κxy 0 mm⁻¹ (principal radius about 1271 mm). [0/90]s stays flat: all curvatures are zero because B = 0. Height exaggerated ×30 in both drawings of a 50 mm square patch; the dashed outline is the unloaded patch and in-plane stretch is not drawn. With no load, both are flat.

The polar plot is a stiffness rose. Its radius in each direction is the laminate's effective in-plane modulus Ex(θ) for a pull in that direction, computed with the laminate held flat. A circle means the same in-plane stiffness every way.

Symmetric, balanced and quasi-isotropic are three different properties

Symmetric, balanced and quasi-isotropic are three different properties

  • Symmetric: the plies above the midplane are a mirror image of the plies below it. Then the coupling matrix B is zero, so in-plane loads and bending stay separate. [0/90]s is symmetric; [0/90] is not.
  • Balanced: every +θ ply has a −θ ply of the same thickness somewhere in the stack (0° and 90° plies need no partner). Then A16 = A26 = 0: there is no extension–shear coupling in the flat-held in-plane response, so a pull along x or y held flat does not shear the laminate in its plane. A balanced and symmetric stack also has B = 0, so it does not shear when free either; a balanced unsymmetric stack such as [0/+45/−45/90] can still shear and twist when free to curve.
  • Quasi-isotropic: the in-plane stiffness matrix A is the same in every direction, so the rose is a circle. Equal numbers of 0°, +45°, −45° and 90° plies do this, and because A ignores the order of the plies, [0/+45/−45/90]s and [+45/0/−45/90]s both qualify. It describes in-plane stiffness only; bending stiffness D still depends on order and direction.

An unsymmetric laminate does not bend just because B is not zero. With no applied force and no applied moment, this model predicts no deformation at all. B only links a load to a response: pull an unsymmetric panel in its plane and it also curls or twists, even though no moment is applied. A symmetric panel under the same pull stays flat: it stretches (and, if unbalanced, also shears in its plane) but does not curl or twist.

Flat-held and free stiffness differ for unsymmetric laminates

Flat-held and free stiffness differ for unsymmetric laminates

The two stiffness numbers for an unsymmetric laminate answer different questions. Flat-held (curvature-constrained) stiffness uses A alone and assumes something stops the panel from curving. Free stiffness lets it curve with zero applied moment and solves the full relation including B and D; it is never higher, and in most directions lower, because the panel can curl as well as stretch. For a symmetric laminate the two are identical.

Real unsymmetric panels can also warp as they cool after curing, because each ply shrinks differently in different directions. That temperature effect is not modeled here, and the curled panel in the interactive is not a picture of it: every curl drawn above is the response to the stated in-plane load.

Supporting detail: Addendum 3 — How the laminate stiffness and the free-panel response are calculated.

This is linear, small-deflection classical laminate theory with perfectly bonded plies. It does not predict strength, first-ply failure, delamination, buckling, temperature or moisture effects, or how a real panel behaves once its deflection is comparable to its thickness.

Supporting detail: Addendum 4 — Primary mechanics source and model boundary.

Supporting detail

Addendum

Additional evidence, standards context, and background referenced from the main text.

  1. Materials and Manufacturing Processes/Composites/Fiber Direction and Elastic Stiffness

    Addendum 1. How the axial stiffness is calculated

    Let m = cos θ and n = sin θ. For this plane-stress orthotropic layer, the compliance in the load direction is S̄11 = m⁴/E1 + n⁴/E2 + m²n²(1/G12 − 2ν12/E1). Then Ex = 1/S̄11 and εx = σx/Ex. The full stress and strain transformation supplies transverse and shear strain.

    Back to reference 1

  2. Materials and Manufacturing Processes/Composites/Fiber Direction and Elastic Stiffness

    Addendum 2. Primary mechanics source and model boundary

    NASA Reference Publication 1351, Basic Mechanics of Laminated Composite Plates, sections I and II, defines isotropic and orthotropic behavior, develops plane-stress compliance and off-axis transformation, and identifies extension–shear coupling. Its stated scope separates elastic response from strength. The constants and examples on this page are original illustrative teaching inputs, not values from that report.

    Back to reference 2

  3. Materials and Manufacturing Processes/Composites/Fiber Direction and Elastic Stiffness

    Addendum 3. How the laminate stiffness and the free-panel response are calculated

    Each ply's reduced stiffness Q (MPa) comes from E1, E2, G12 and ν12, and is rotated to the ply angle to give Q̄, using engineering shear strain γxy. With z measured up from the midplane and each ply running from zbottom to ztop (mm):

    A = Σ Q̄ (ztop − zbottom) in N/mm; B = Σ Q̄ (ztop² − zbottom²)/2 in N; D = Σ Q̄ (ztop³ − zbottom³)/3 in N·mm.

    The force resultants N (N/mm) and moment resultants M (N·mm per mm) relate to midplane strain ε⁰ and curvature κ (1/mm) by N = A ε⁰ + B κ and M = B ε⁰ + D κ. The interactive applies N with M = 0 and solves all six equations together. The drawn surface is w = −(κxx² + κyy² + κxyxy)/2, exaggerated by the stated factor.

    For the rose, a pull Nθ along direction θ has components (cos²θ, sin²θ, sinθ cosθ) Nθ. Flat-held: Ex(θ) = 1/(h · vᵀA⁻¹v) with v = (cos²θ, sin²θ, sinθ cosθ) and laminate thickness h. Free: the same, with the in-plane block of the inverted six-by-six A-B-D matrix in place of A⁻¹.

    Reversing the ply order (the bottom ply listed first) leaves A and D unchanged and reverses the sign of every B term, so the same pull makes the opposite curl. A single ply is symmetric about its own midplane, so it reproduces the one-ply result above: 40 GPa along the fibers and 10 GPa across.

    Back to reference 3

  4. Materials and Manufacturing Processes/Composites/Fiber Direction and Elastic Stiffness

    Addendum 4. Primary mechanics source and model boundary

    NASA Reference Publication 1351, Basic Mechanics of Laminated Composite Plates (A. T. Nettles, 1994) defines balanced and symmetric laminates, derives the A, B and D matrices and the laminate constitutive equations, shows that a symmetric laminate has B = 0, and works an unsymmetric laminate example in which a tensile load alone bends the plate. It also notes that unsymmetric laminates can warp from processing temperature changes, which this lesson excludes. The ply constants, ply thickness, loads and panel size here are original illustrative teaching inputs, not values from that report and not design data.

    Back to reference 4