Search for Thermally Stable Laminates 9/9-3... · CDW-15, Kanazawa, Oct. 18-19, 2010 3 Introduction...
Transcript of Search for Thermally Stable Laminates 9/9-3... · CDW-15, Kanazawa, Oct. 18-19, 2010 3 Introduction...
CDW-15, Kanazawa, Oct. 18-19, 2010 1
Search for Thermally Stable Laminates
Georges Verchery,ISMANS, France.
CDW-15, Kanazawa, Oct. 18-19, 2010 2
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.
CDW-15, Kanazawa, Oct. 18-19, 2010 3
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).
CDW-15, Kanazawa, Oct. 18-19, 2010 5
CLPT
● Thermoelastic:
where:
{ N = A 0 B − R T
M = B 0
D − S T
T=T current−T 0 uniform in the thickness
CDW-15, Kanazawa, Oct. 18-19, 2010 7
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
CDW-15, Kanazawa, Oct. 18-19, 2010 8
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
CDW-15, Kanazawa, Oct. 18-19, 2010 9
Free Deformation / Thermal Stability
● Free Deformation:
conditions on the loads.
● Thermal Stability:
conditions on the material properties.
CDW-15, Kanazawa, Oct. 18-19, 2010 10
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
CDW-15, Kanazawa, Oct. 18-19, 2010 11
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)
CDW-15, Kanazawa, Oct. 18-19, 2010 12
Thermal Stability
● In the literature:
● In fact, for isolaminar laminates,these conditions imply:
R isotropic and S=0
A and B square symmetric
CDW-15, Kanazawa, Oct. 18-19, 2010 13
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.
CDW-15, Kanazawa, Oct. 18-19, 2010 14
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 %
CDW-15, Kanazawa, Oct. 18-19, 2010 15
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.
CDW-15, Kanazawa, Oct. 18-19, 2010 16
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.
CDW-15, Kanazawa, Oct. 18-19, 2010 17
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.
CDW-15, Kanazawa, Oct. 18-19, 2010 18
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.
CDW-15, Kanazawa, Oct. 18-19, 2010 19
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.
CDW-15, Kanazawa, Oct. 18-19, 2010 20
Appendix 1: Review of some continuum mechanics results
● Strains from various origins are additive:
● Specially:
=∑
linear elastic: mec = Q−1
linear thermal: th = T
CDW-15, Kanazawa, Oct. 18-19, 2010 21
Appendix 1: Some continuum mechanics results
● Linear thermoelasticity:
or (more conveniently) for applying CLPT:
= Q−1 T
=Q − T
CDW-15, Kanazawa, Oct. 18-19, 2010 22
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
CDW-15, Kanazawa, Oct. 18-19, 2010 23
Appendix 1: Some continuum mechanics results
● Uniform change of temperature:
- expansion and shear,
- homothetic only if isotropic:
* isotropic material,
* cubic symmetry.
CDW-15, Kanazawa, Oct. 18-19, 2010 25
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
CDW-15, Kanazawa, Oct. 18-19, 2010 26
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
CDW-15, Kanazawa, Oct. 18-19, 2010 27
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.