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Search for Thermally Stable Laminates

Georges Verchery,ISMANS, France.

georges.verchery@m4x.org

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Introduction (1)

● Growing interest in various mechanical couplings:● Application to adaptive, multifunctional structures,

specially in aerodynamics (blades, wings, ...),● More comprehensive understanding of laminate

behaviour.

● Effects of non mechanical loadings, due to temperature, moisture, chemical reactions, ..., specially due to curing during manufacture.

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Introduction (2)

« Thermal Stability » was introduced in works by Winckler, Chen, Cross et al., Haynes and Armanios, and others.

They derived stacking sequences of coupled laminates which should remain flat during the post-cure cooling and more generally during uniform changes of temperature.

They considered UD linear thermohygroelastic plies and used the Classical laminated plate theory (CLPT).

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CLPT

● Elastic behaviour:

{ N = A 0

B

M = B 0

D

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CLPT

● Thermoelastic:

where:

{ N = A 0 B − R T

M = B 0

D − S T

T=T current−T 0 uniform in the thickness

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CLPT

● General:

{ N = A 0

B N0

M = B 0 D M0

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Free Deformation

● Is it possible to have zero stresses?

● Compatibility of both generalized strains,

which is obtained when:

● Example: curing strains.

{ 0 = A 0 B − R T

0 = B 0

D − S T

T uniform in x and y

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Thermal Stability

● Is it possible for a plate to remain flat?

● 6 equations for 3 unknown:

generally impossible,

mathematical compatibility of eqs.

{ A 0 = R T

B 0 = S T

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Free Deformation / Thermal Stability

● Free Deformation:

conditions on the loads.

● Thermal Stability:

conditions on the material properties.

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Thermal Stability

● Mathematical compatibility condition of

is:

general, highly non linear, scarcely useful.

{ A 0 = R T

B 0 = S T

S = B A−1 R

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Thermal Stability

● Restricting to isolaminar laminates, compatibility condition of

is:

{ A 0 = R T

B 0 = S T

A , B , R square symmetric and S=0(so R isotropic)

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Thermal Stability

● In the literature:

● In fact, for isolaminar laminates,these conditions imply:

R isotropic and S=0

A and B square symmetric

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Thermal Stability

● Consequences 1:

● General condition satisfied.● Anisotropy is limited for A, B, R, even D.● Stability extends to T linear in x and y.

● Consequences 2:

● Balanced fabrics starting with 1 ply.

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Thermal Stability - Examples

● Winckler UD and balanced fabrics solutions (1):

● W is UD

● BF or BF equivalent to W

● same A, B, R, small difference for D:

– relative deviation for D11 or D22 :

DD

∣sin 2∣

8 10 %

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Thermal Stability - Examples

● Winckler UD and balanced fabrics solutions (2):● same principal axes for A, D, in which B is full,

● principal axes for B at 22°30', whatever is ● B maximum for 22°30', with A isotropic,

● for BF B divided by 2.

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Thermal Stability - Examples

For other published solutions with UD

plies (from 5 plies and up), it can be

checked that A and B are square

symmetric.

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Some Conclusions

● Thermal stability limits anisotropy to square

symmetry, so does not take advantage of the

high anisotropy of UD,

● BF can be used instead of UD, with less plies,

and easier control of anisotropy,

● Thermal stability extends to linear variation of

temperature.

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Pending questions

● Warp and weft might behave slightly differently, so

balance might be imperfect in « balanced fabrics »,

● Extension to other swelling phenomena (hygral,

chemical, etc.) might be questionned,

● Hybrid laminates ? No results up to now.

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Some References

● S.I. Winckler (1985). Hygrothermally curvature stable laminates with tension-torsion coupling, Journal of the American Helicopter Society, 31(7): 56-58.

● H.P. Chen (2003). Study of hygrothermal isotropic layup and hygrothermal curvature stable coupling composite laminates, Proceedings of the 44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference, AIAA 2003-1506, April 7-10, 2003, Norfolk, VA, USA.

● R.J. Cross, R.A. Haynes and E.A. Armanios (2008). Families of hygrothermally stable asymmetric laminated composites, Journal of Composite Materials, 42(7): 697-716.

● R. Haynes and E. Armanios (2009). Overview of hygrothermally stable laminates with improved extension-twist coupling, Proceedings of the 17th International Conference on Composite Materials, July 27-31, 2009, Edinburgh, Scotland.

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Appendix 1: Review of some continuum mechanics results

● Strains from various origins are additive:

● Specially:

=∑

linear elastic: mec = Q−1

linear thermal: th = T

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Appendix 1: Some continuum mechanics results

● Linear thermoelasticity:

or (more conveniently) for applying CLPT:

= Q−1 T

=Q − T

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Appendix 1: Some continuum mechanics results

● Free deformation ?

- no geometrical constraints

- no external and internal forces

- compatibility of strains● For homogeneous material, free deformation is

possible when T is linear in the coordinates:

T linear : = 0 , = T

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Appendix 1: Some continuum mechanics results

● Uniform change of temperature:

- expansion and shear,

- homothetic only if isotropic:

* isotropic material,

* cubic symmetry.

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Expanded and sheared

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Appendix 2: Square symmetry

● Example: ply reinforced by a balanced fabric

● Two (orthogonal) principal directions X, Y with identical properties:

T= T XX T XY 0T XY T XX 0

0 0 T SS

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Appendix 2: Square symmetry

● In arbitrary axes:

● Invariant condition:

in which R1 is the (orthotropy) quadratic invariant

defined as:

T= T 11 T 12 T16

T 12 T 11 −T16

T 16 −T16 T 66

R1=0

64 R12= T 11−T 22

2 4 T16T 26

2

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Appendix 2: Square symmetry

● Expansion and shear are uncoupled:

- for C square symmetric

and pure (isotropic) expansion, C is also pure (isotropic) expansion.

- for C square symmetric

- for C square symmetric and pure shear, C is also pure shear.

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Appendix 2: Square symmetry

● For laminates (according to CLPT):

● In particular, for isolaminar laminate:

R1k=0 ⇒ R1=0 , R1=0 , R1=0

all plies membrane coupling bending

B = R0 c −c s−c c −ss −s −c