II.1 Theoretical background

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Chapter II.1
Theoretical background

The purpose of this Chapter is to summarize the classical laminate theory, and to provide the information needed by the user of composite classes to understand a few conventions that have been assumed for the programming of Classical Laminate Analysis in FeResPost (axes, angles, numbering of layers,...).

The programmer will find a presentation of the classical laminate theory that follows closely what is programmed in C++ language in FeResPost. However those who are interested in studying the theory, or who are not familiar with it are referred to more extensive presentations of the classical laminate theory [Gay97, Pal99]. Only for the out-of-plane shear behavior of the laminate, is the presentation original, even though inspired by information found in [Sof04a].

The Chapter is organized as follows:

Section II.1.1 presents the conventions used for the numbering of plies, orientations of plies, laminate axes...
Section II.1.2 summarizes the calculation rules for the rotation of tensors and vectors. Some of the notions and notations used in the rest of the Chapter are introduced in the section.
The constitutive equations describing ply materials behavior are given in section II.1.3. One also introduces several notations that are used in the rest of the Chapter and one presents the calculation rules that must be used to perform material property rotations.
In section II.1.5, the calculation of laminate in-plane and flexural properties is described. Note that the influences of temperature and moisture are not considered in the section.
Section II.1.6 is devoted to the out-of-plane shear behavior of laminates. The explanation is more detailed than what is given in section II.1.5. Here again, temperature and moisture effects are not considered.
Section II.1.7 is devoted to the influence of temperature and moisture. One considers the influence of these loading contributions on the in-plane, flexural and out-of-plane shear laminate behavior and load response.
Section II.1.8 presents the calculation of load response of laminates submitted to a loading. Among other-things, one explains how the ply stresses and strains are calculated.
Section II.1.10 is devoted to the calculation criteria available in FeResPost.
Section II.1.13 is devoted to a presentation of the units of many of the quantities introduced in this Chapter.

II.1.1 Conventions
II.1.2 Rotation in XY plane and algebraic notations
II.1.3 Materials and plies
  II.1.3.1 Plies
  II.1.3.2 Materials and constitutive equations
  II.1.3.3 In-plane properties
  II.1.3.4 Out-of-plane shear properties
II.1.4 Thickness and mass of laminate
II.1.5 In-plane and flexural laminate behavior
II.1.6 Out-of-plane shear of laminate
  II.1.6.1 Out-of-plane shear equilibrium equations
  II.1.6.2 Triangular distribution of in-plane stresses
  II.1.6.3 Out-of-plane shear stress partial derivative equations
  II.1.6.4 Integration of out-of-plane shear stress equation
  II.1.6.5 Approximations with out-of-plane shear forces
  II.1.6.6 Out-of-plane laminate shear stiffness
  II.1.6.7 Calculation algorithm for shear stiffness
  II.1.6.8 Ply out-of-plane shear stresses
II.1.7 CTE and CME calculations
  II.1.7.1 In-plane and flexural thermo-elastic behavior
  II.1.7.2 Out-of-plane shear thermo-elastic behavior
  II.1.7.3 Hygrometric behavior of laminates
  II.1.7.4 Full sets of equations
II.1.8 Calculation of load response
  II.1.8.1 In-plane and flexural response
  II.1.8.2 Out-of-plane shear response
  II.1.8.3 Out-of-plane T/C deformation
II.1.9 Accelerating the calculation of load response
  II.1.9.1 Calculation of laminate loads and strains
  II.1.9.2 Calculation of plies stresses and strains
II.1.10 Failure theories
  II.1.10.1 Tresca criterion (2D)
  II.1.10.2 Von Mises criterion (2D)
  II.1.10.3 Von Mises criterion (3D)
  II.1.10.4 Maximum stress criterion
  II.1.10.5 Maximum stress criterion (3D)
  II.1.10.6 Maximum strain criteria (2D)
  II.1.10.7 Maximum strain criterion (3D)
  II.1.10.8 Combined strain criterion (2D)
  II.1.10.9 Tsai-Hill criterion
  II.1.10.10 Tsai-Hill criterion (version b)
  II.1.10.11 Tsai-Hill criterion (version c)
  II.1.10.12 Tsai-Hill criterion (3D)
  II.1.10.13 Tsai-Hill criterion (3D version b)
  II.1.10.14 Tsai-Wu criterion
  II.1.10.15 Tsai-Wu criterion (3D)
  II.1.10.16 Hoffman criterion
  II.1.10.17 Puck criterion
  II.1.10.18 Puck “b” criterion
  II.1.10.19 Puck “c” criterion
  II.1.10.20 Hashin criteria
  II.1.10.21 Hashin criteria (3D)
  II.1.10.22 Yamada-Sun criterion
  II.1.10.23 Yamada-Sun criterion (version b)
  II.1.10.24 3D honeycomb criterion
  II.1.10.25 Honeycomb shear criterion
  II.1.10.26 Honeycomb simplified shear criterion
  II.1.10.27 Inter-laminar shear criterion
II.1.11 Temperature diffusion in laminates
  II.1.11.1 Material thermal parameters
  II.1.11.2 In-plane and out-of-plane components
  II.1.11.3 In-plane rotations of vectorial and tensorial properties
  II.1.11.4 Integration along the laminate thickness
II.1.12 Moisture diffusion in laminates
II.1.13 Units

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