Lam3D is an advanced software tool designed to predict laminate response and failure behavior of various laminates under a broad range of loading conditions.

In this work, the analytic model developed by Chou et al is used to predict the effective laminate stress/strain response. It is also used to calculate ply-level stresses and strains during incremental loading for failure and strength prediction. The following section outlines the laminated media model upon which our analysis is based.

Chou et al. [2] use a control volume approach to yield a closed-form solution to the problem of effective homogeneous property determination for a laminated media composed of individual layers. Unlike the works of White and Angona [24], Postma [25], Rytov [26], Behrens [27], and Salamon [28], which required the individual layers to be isotropic, Chou et al. [2] permitted general anisotropy of the layers. The analysis is based on the assumptions that all interlaminar stresses are continuous across ply interfaces and that all in-plane strains are continuous through the thickness dimension of a representative volume element (i.e., a repeating sublaminate configuration).

The following expression is used to represent the effective (i.e., homogeneous) stress/strain constitutive relationship for an N-layered laminate (see Figure 1):

Figure 1. Laminate configuration

for (i, j = 1, 2, 3, 4, 5, 6). (1)

The barred notation is used to denote that the relationship applies in the global x-y-z coordinate system of the laminate. The asterisk superscript is used here to denote the "average" or effective laminate stress and strain quantities. In-plane strains are assumed uniform (i.e., constant within each ply) and equal to the effective strains of the laminate. Mathematically, this is expressed as

for (i = 1, 2, 6; k = 1, 2, ..., N), (2)

where represents the strain in the k th ply of the laminate (see ply numbering convention in Figure 1). To ensure stress continuity across ply interfaces, all ply stress components associated with the out-of-plane direction (i.e., z-direction) are assumed uniform and equal to the corresponding effective stresses in the laminate. Mathematically, this is expressed as

for (i = 3, 4, 5; k = 1, 2, ..., N), (3)

where represents the stress in the k th ply of the laminate.

All remaining effective laminate strains and stresses are assumed to be the volume average of all their corresponding ply strain and stress components, respectively. Mathematically, these assumptions are expressed as

for (i = 3, 4, 5) (4)

and for (i = 1, 2, 6), (5)

where V k is the ratio of the original (i.e., undeformed) volume of the k th ply over the original volume of the entire laminate. The constitutive equation for each ply in the laminate is written below (equation (6)) using the superscript notation.

for ( i, j = 1, 2, 3, 4, 5, 6; k=1,2, ..., N). (6)

(For completeness, the ply stiffness matrix coefficients ( ) are defined in terms of the lamina engineering constants and layer orientations in the Appendix.)

Equations (1) through (6) represent 12N+6 linear algebraic equations with 12N+12 unknowns. Solution to equations (1) through (6) yields the following effective three-dimensional stress/strain constitutive relation, which can be used as an equivalent (i.e., homogeneous) representation for the laminated media where the coefficients in the laminate stiffness matrix, , are given by

for (i, j = 1, 2, 3, 6), (7)

for (i=1, 2, 3, 6; j = 4, 5), (8)

and

for (i, j = 4,5), (9)

where

. (10)

The effective stress/strain constitutive relation for the laminated media is therefore given by equations (1) and (7) through (10).

In determining the individual ply-level stresses and strains, the assumption is made that the applied mechanical loading on the laminated media ( ) is known, uniform, and represents the "average" or "effective" stress acting on the sublaminate configuration. The associated "effective" or "smeared" laminate strains ( ) can be obtained directly from the inversion of equation (1). From the assumption made in equation (2), all in-plane strain values (defined in the global x-y-z coordinate system) for plies 1 through N are therefore known. Similarly, from the assumption made in equation (2), all out-of-plane stresses for plies 1 through N are known (also defined in the global x-y-z coordinate system). The out-of-plane ply strains and in-plane ply stresses remain to be determined.

Sun and Liao [29] derived the following expression for determination of the remaining out-of plane ply strains

. (11)

Once all of the ply strains are known, the remaining in-plane ply stresses can be calculated straightforwardly through the following relation

. (12)

 

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