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Analysis of edge stresses in composite laminates under combined thermo-mechanical loading, using a complementary energy approach

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Abstract

An approximate method is presented to investigate the interlaminar stresses near the free edges of composite laminate plates that are subjected to a combined thermo-mechanical loading. The method is based upon admissible function representations of stresses which account for the effects of both the global mismatches and the local mismatches in two of the elastic properties, the Poisson's ratio and the coefficient of mutual influence. For this purpose, new thermo-mechanical mismatch terms are defined to reflect an effective deformation under the combined thermo-mechanical loading. Closed form solutions of all the stress components are sought by minimizing the complementary energy with respect to the unknown functions, in the stress representations, of the width coordinate. These unknown functions are determined by solving five ordinary differential equations along with a set of free edge boundary conditions, which allow complex as well as real roots for their exponential decaying rates. The resulting solutions satisfy the stress equilibrium and all of the boundary conditions exactly, but compatibility is met in a weak form. Numerical examples are given for several typical laminates, and are compared with previous results obtained by finite element and other approximate methods. It is found that the present approximate method yields interlaminar stress results in an efficient, fast and yet reliable way. It is also concluded that unlike some previous approximate methods, the current method is numerically robust and stable.

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Abbreviations

A, B, D :

Laminate stiffness matrices as defined by (27)

b :

Half laminate width

ΔT (k) :

Temperature difference at the top of the k-th ply

ΔT (k)(x 3):

Temperature difference in the k-th ply given as a function of through-thickness coordinate x 3

e (k)0,1,2,3,12 :

Thermal strains of the k-th ply in the laminate axes

h :

Laminate thickness

M 11 :

Applied uniform bending load (moment/unit length)

M Tij :

Applied thermal moments (moment/unit length)

N :

Total number of plies

N 11 :

Applied thermal loads (force/unit length)

N Tij :

Applied thermal loads (force/unit length)

Q (k)ij :

Ply stiffness for the k-th ply in the laminate axes

S (k)ij :

Ply compliance for the k-th ply in the laminate axes

t (k) :

Thickness of the k-th ply

(x 1, x 2, x 3):

Local coordinate system with the out-of-plane coordinate x 3 defined at the bottom of each ply

(x 1, x 2, z):

Global coordinate system with the out-of-plane coordinate z defined at the mid-plane of the laminate

z (k) :

Global out-of-plane coordinate for the top of the k-th ply

ε mech11 :

Applied mechanical axial strain

ε tot11 :

Applied total (including thermal) axial strain

εtot :

Total in-plane strain vector at arbitrary plane as defined by (32)

ε0 :

Total in-plane strain vector at the midplane

η12,1 :

Laminate coefficient of mutual influence

η (k)12,1 :

Coefficient of mutual influence for the k-th ply

η (k)12, 1E :

Equivalent coefficient of mutual influence for the k-th ply as defined by (23)

K :

Total curvature vector

μα,β,z :

Coefficients of thermal expansion for a uni-ply in its material principal axes

μ (k)11,22,12,33 :

Coefficients of thermal expansion for the k-th ply in the laminate axes

V12 :

Laminate Poisson's ratio

ν (k)12 :

Poisson's ratio for the k-th ply

ν (k)12E :

Equivalent Poisson's ratio for the k-th ply as defined by (22)

σ i j :

Total stress components in tensor notation

\(\widetilde\sigma _{ij} \) :

Far-field stress components

σ c ij :

Companion stress components

References

  • Altus, E.; Rotem, A.; Shmueli, M. 1980: Free Edge Effect in Angle Ply Laminates — A New Three-Dimensional Finite Difference Solution. J. Compos. Mater. 14: 21–30

    Google Scholar 

  • Herakovich, C. T. 1976: On Thermal Edge Effects in Composite Laminates. Int. J. Mech. Sci. 18: 129–134

    Google Scholar 

  • Jones, R. M. 1975: Mechanics of Composite Materials. New York, McGraw Hill

    Google Scholar 

  • Kassapoglou, C. 1990: Determination of Interlaminar Stresses in Composite Laminates Under Combined Loads. J. Reinf. Plast. and Compos. 9: 33–58

    Google Scholar 

  • Kim, T.; Atluri, S. N. 1994: Interlaminar Stresses in Composite Laminates under Out-of-plane Shear/bending. AIAA J. 32(8): 1700–1708

    Google Scholar 

  • Kim, T.; Atluri, S. N. 1995: Optimal Through-Thickness Temperature Gradients for Control of Interlaminar Stresses in Composites. AIAA J. 33(4): 730–738

    Google Scholar 

  • Murthy, P. L. N.; Chamis, C. C. 1989: Free-Edge Delamination: Laminate Width and Loading Conditions Effects. J. Compos. Tech. & Res. 11(1): 15–22

    Google Scholar 

  • Nishioka, T.; Atluri S. N. 1982: Stress Analysis of Holes in Angle-Ply Laminates: An Efficient Assumed Stress “Special-Hole-Element” Approach and a Simple Estimation Method. Computers & Struct. 15(2): 135–147

    Google Scholar 

  • Pagano, N. J. 1978: Stress Fields in Composite Laminates. Int. J. Solids and Struct. 14(5): 385–400

    Google Scholar 

  • Pipes, R. B.; Pagano, N. J. 1970: Interlaminar Stresses in Composite Laminates under Uniform Axial Extension. J. Compos. Mater. 4: 538–548

    Google Scholar 

  • Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. 1986: Numerical Recipes, The Art of Scientific Computing. Cambridge University Press

  • Rose, C. A. 1992: An Approximate Solution for Interlaminar Stresses in Laminated Composite. Ph.D. Thesis, Applied Mech. Program, University of Virginia, Charlotteville, VA 22901

  • Rose, C. A.; Herakovich, C. T. 1993: An Approximate Solution for Interlaminar Stresses in Composite Laminates. Compos. Eng. 3(3): 271–285

    Google Scholar 

  • Rybicki, E. F. 1971: Approximate Three-Dimensional Solutions for Symmetric Laminates under In-Plane Loading. J. Compos. Mater. 5: 354–360

    Google Scholar 

  • Salamon, N. J. 1978: Interlaminar Stresses in a Layered Composite Laminate in Bending. Fibre Sci. and Tech. 11: 305–317

    Google Scholar 

  • Wang, A. S. D.; Crossman, F. W. 1976: Calculation of Edge Stresses in Multi-Layered Laminates by Sub-Structuring. J. Compos. Mater. 12: 76–83

    Google Scholar 

  • Wang, A. S. D.; Crossman, F. W. 1977a: Some New Results on Edge Effect in Symmetric Composite Laminates. J. Compos. Mater. 11: 92–106

    Google Scholar 

  • Wang, A. S. D.; Crossman, F. W. 1977b: Edge Effects on Thermally Induced Stresses in Composite Laminates. J. Compos. Mater. 11: 300–312

    Google Scholar 

  • Wang, S. S.; Choi, I. 1982: Boundary-Layer Effects in Composite Laminates Part-1 Free Edge Stress Singularities. ASME. J. App. Mech. 49: 541–548

    Google Scholar 

  • Wang, S. S.; Choi, I. 1982: Boundary-Layer Effects in Composite Laminates Part-2 Free Edge Stress Solutions and Basic Characteristics. ASME J. App. Mech. 49: 549–560

    Google Scholar 

  • Webber, J. P. H.; Morton, S. K. 1993: An Analytical Solution for the Thermal Stresses at the Free Edges of Laminated Plates. Compos. Sci. and Tech. 46: 175–185

    Google Scholar 

  • Yin, W. L. 1994: Analysis of the Temperature Gradient Effect on the Free-edge Interlaminar Stresses in Layered Structures. Proc. 35th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conf. 5: 2528–2536

    Google Scholar 

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Authors and Affiliations

  1. Computational Mechanics Center Georgia Institute of Technology, 30332-0356, Atlanta, Georgia, USA

    T. Kim & S. N. Atluri

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  1. T. Kim
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  2. S. N. Atluri
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Communicated by S. N. Atluri, 30 August 1994

This work was supported by a grant from the Federal Aviation Administration, to the Center of Excellence for Computational Modeling of Aircraft Structures at Georgia Institute of Technology.

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Kim, T., Atluri, S.N. Analysis of edge stresses in composite laminates under combined thermo-mechanical loading, using a complementary energy approach. Computational Mechanics 16, 83–97 (1995). https://doi.org/10.1007/BF00365862

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