Hygrothermal stresses in laminates

•Changing environment conditions (temperature and moisture) have an important effect on the properties which are matrix dominated.

Ch i t t d i t t t i d lli f th l t i•Change in temperature and moisture content induces swelling of the polymer matrix.

Temperature effect on polymers

Variousepoxyresins

Glass transition temperature

The maximum usage temperatureis slightly smaller than Tg

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•Thermal expansion of unidirectional composites is strongly anisotropic•Longitudinal CTE of Kevlar has a negative value•Longitudinal CTE of Graphite composite close to zeroLongitudinal CTE of Graphite composite close to zero

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Thermal expansion: Thermal strains for an orthotropic material in (L,T) frame:(no thermal shear in L‐T frame)

Hygroscopic expansion:Changes in moisture concentration are responsible for  swelling of the matrix materialMoisture‐induced strains in orthotropic material in (L,T) frame:p ( , )(no moisture‐induced shear in L‐T frame)

Moisture diffusion is governed by the diffusion equation

Fick’s second lawDz = mass diffusivity along zz y g

Very similar to the heat conduction equation:

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Change of coordinates of hygrothermal expansion coefficients

Strain tensor:

Th lThermal shear

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Thermal strains Hygroscopic strain

•Do  not produce a resultant force or moment when the body iscompletely free to expand, bend and twist. •An individual lamina is restrained by the other laminae and is not•An individual lamina is restrained by the other laminae and is not free to expand. This induces thermal stresses.•The thermal stresses are internal stresses: they are self‐equilibrated.

Total strains = mechanical strains + hygrothermal strains

M h i lMechanicalstrains

Total lamina strainsFollow the kinematicsof Kirchhoff plates:

Linear over the thhickness7

Mechanical strains(associated with stresses)

If there is no external loading the resultant forces {N} and moments {M} are such thatIf there is no external loading, the resultant forces {N} and moments {M} are such that{N}=0  {M}=0  (hygrothermal loads are self‐equilibrated)

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Th l b l d f ti l ti fThe global deformations are solution of:

Thermal loads:

F t i l i tFor a symmetric laminateB=0

{MT}={MH}=0

Thermal stresses are unavoidable in the fabrication of composites.The residual stresses due to curing have a significant effect on failureThe residual stresses due to curing have a significant effect on failureand should not be neglected in the design.Non‐symmetric laminates will experience warping during cooling. 9

Example 6.10: Non‐symmetric two‐ply laminate (glass‐epoxy)(5mm at 0° and 3mm at 45°) Fabricated at 125°C and cooled at room temperature 25°C

Stiffness matrix of one plyStiffness matrix of one plyin principal material axes:

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Thermal loads:

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Thermal loads:

Thermal momentsGl b l d f ti f th l i tGlobal deformations of the laminate:

Non‐symmetric laminate

Warping !!12

F h lFor each ply,

Same for all plies

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Stresses in the 0° ply

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The residual stresses are self‐equilibrated

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Logitudinal CTE (micromechanical model)

1. The fibers and the matrix experience the same strain

2. The load is shared between the fibers and the matrix

Transverse CTE (Schapery): 

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Approximation for  Vf>0.25:

Example: Glass‐epoxy system

The CTE in longitudinal directiongis dominated by the fiber.

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Moisture expansion coefficient

Extension:

1. Isotropic material:

Relative change of volume:

Moisture content: 

Relative volume change: 

2 C it2. Composites:•Polymer matrices absorb moisture; inorganic fibers do not.•The expansion in the longitudinal direction is negligiblebecause of the high stiffness of the fibers

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Thermal conductivity

Longitudinal direction (rule of mixture)

Transverse coefficient may be computed according to Halpin‐Tsai equation:

kf and k refer to the fibers and thekf and km refer to the fibers and thematrix in the appropriate direction

(fiber anisitropy)

ab

Direction of measurement

Fiber crosssection

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Direction of measurement

For circular fibers, a=b

Example: Find kL and kT for glass‐epoxy and carbon‐epoxy for Vf=60% (circular fibers) 

Isotropic

Anisotropic

Glass‐epoxy Carbon‐epoxy

Anisotropic !

•Thermal conduction is very anisotropic. •For carbon epoxy composites, the thermal 

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p y p ,conductivity of the matrix is negligible.