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3.1  Material models

• Classical lamination theory
• 3D anisotropy

3.1.1  Classical lamination theory

Both isotropic and orthotropic materials are considered. In these cases, the general form of the 3D elastic material law is

(3.1)

Plate formulation consists in assuming one dimension, the thickness along x₃, negligible compared with the surface dimensions. Thus, vertical stress σ₃₃=0 on the bottom and upper faces, and assumed to be neglected throughout the thickness,

(3.2)

and for isotropic material,

(3.3)

By eliminating σ₃₃, the plate constitutive law is written, with engineering notations,

(3.4)

The reduced stiffness coefficients Q_ij (i,j = 1,2,4,5,6) are related to the 3D stiffness coefficients C_ij by

(3.5)

The reduced elastic law for an isotropic plate becomes,

(3.6)

and

(3.7)

Under Reissner-Mindlin's kinematic assumption the linearized strain tensor is

(3.8)

So, the strain vector is written,

(3.9)

with є^m the membrane, κ the curvature or bending, and γ the shear strains,

(3.10)

Note that the engineering notation with γ₁₂=u_1,2+u_2,1 is used here rather than the tensor notation with є₁₂=(u_1,2+u_2,1)/2 . Similarly κ₁₂=β_1,2+β_2,1, where a factor 1/2 would be needed for the tensor.

The plate formulation links the stress resultants, membrane forces N_αβ, bending moments M_αβ and shear forces Q_α3, to the strains, membrane є^m, bending κ and shearing γ,

(3.11)

The stress resultants are obtained by integrating the stresses through the thickness of the plate,

(3.12)

with α, β = 1, 2.

Therefore, the matrix extensional stiffness matrix [A], extension/bending coupling matrix [B], and the bending stiffness matrix [D] are calculated by integration over the thickness interval [hb ht]

(3.13)

An improvement of Mindlin's plate theory with transverse shear consists in modifying the shear coefficients F_ij by

(3.14)

where k_ij are correction factors. Reddy's 3^rd order theory brings to k_ij=2/3. Very commonly, enriched 3^rd order theory are used, and k_ij are equal to 5/6 and give good results. For more details on the assessment of the correction factor, see [46].
For an isotropic symmetric plate (hb=−ht=h/2), the in-plane normal forces N₁₁, N₂₂ and shear force N₁₂ become

(3.15)

the 2 bending moments M₁₁, M₂₂ and twisting moment M₁₂

(3.16)

and the out-of-plane shearing forces Q₂₃ and Q₁₃,

(3.17)

One can notice that because the symmetry of plate, that means the reference plane is the mid-plane of the plate (x₃(0)=0) the extension/bending coupling matrix [B] is equal to zero.

Using expression (3.13) for a constant Q_ij, one sees that for a non-zero offset, one has

(3.18)

where is clearly appears that the constitutive matrix is a polynomial function of h, h³, x₃(0)²h and x₃(0)h. If the ply thickness is kept constant, the constitutive law is a polynomial function of 1,x₃(0),x₃(0)².

3.1.2  3D anisotropy

The constitutive equation in Voigt notation is of the form

(3.19)