Three-Dimensional Non linear Laminate Media Analysis

Lam3DNL is an advanced software tool designed to predict the nonlinear stress/strain response and failure behavior of composite laminates

Nonlinear lamina constitutive relations for the composites are represented using the Ramberg-Osgood equation [3]. Piece-wise linear increments in laminate stress and strain are calculated and superimposed to formulate the overall effective nonlinear response. Individual ply stresses and strains are monitored to calculate instantaneous ply stiffnesses for the incremental solution and to establish ply failure levels. The progressive-ply failure approach allows for stress unloading in a ply and discrimination of the various potential modes of failure.

Defining Nonlinear Lamina Constitutive Relations

Material nonlinearity in our laminate analysis is accounted for on the lamina or ply level. The nonlinear lamina constitutive relations (i.e., stress-vs.-strain relations) for each of the principal lamina directions are defined with the Ramberg-Osgood equation [3]. For the treatment of unidirectional lamina in our three-dimensional analysis, this would include the fiber direction (1), in-plane transverse direction (2), transverse normal direction (3), interlaminar shear directions (23 and 13), and the in-plane shear direction (12).

The Ramberg-Osgood equation provides an expression for stress written explicitly in terms of strain and three unique parameters,

. (13)

Here E o is the initial modulus, is the asymptotic stress level, and n is a shape parameter for the stress versus strain curve. Figure 2 graphically illustrates the significance of these parameters with a typical nonlinear stress-vs.-strain relationship.

Figure 2. Ramberg-Osgood parameters definitions

For computational considerations, it is desired to define the instantaneous or tangent lamina stiffness as a continuous function of strain. Taking the derivative of equation (13) with respect to strain, the following expression is obtained:

, (14)

where E t is the instantaneous or tangent lamina stiffness modulus expressed explicitly in terms of strain and the three Ramberg-Osgood parameters.

A unique set of Ramberg-Osgood parameters for each of the principal directions in the lamina is required. A fitting routine was implemented to find the Ramberg-Osgood parameters which realistically represent the stress/strain response for each of the four materials used in the study. As an example, the data fit to equation (13) is illustrated in Figure 3 for the nonlinear 12-shear direction stress/strain response of the E-glass/MY750 material. A full account of all the Ramberg-Osgood parameters used in our analysis is provided in the Results section of this paper.

2.3 Incremental Approach (Solution Strategy)

The nonlinear response of the laminate is generated through the summation of piece-wise linear increments in stress over a pre-established load schedule. An incremental form of equation (1) is used to determine the linear increments in laminate stress-and-strain. The laminate stiffness matrix is updated at the end of each stress increment (based on all current ply strain levels) during the incremental loading strategy. The schematic presented in Figure 4 provides a mathematical representation of the incremental loading strategy for an arbitrary laminate.

Assume that at point (a), corresponding to the end of the nth stress increment, the strain and stress state of the laminate is known ( ). From this point, the objective is to determine the strain and stress state at point (b) or ( ). The effective laminate stiffness matrix at the end of stress increment n, , is computed from an incremental form of the laminated media model constitutive relation, equation (1). With the increment in load defined, , the corresponding increment in laminate strain, , is calculated from an inverse form of equation (1):

(i, j = 1, 2, 3, 4, 5, 6). (15)

Individual ply stress and strain increments are calculated according to the equations presented previously. A cumulative summation is maintained to track the total stress-and-strain levels in each ply of the laminate. The tangent modulus values for each ply and material direction are calculated according to equation (14) and used in the determination of the laminate stiffness matrix for the next laminate stress increment calculation.

The entire nonlinear response for the laminate is obtained by the cumulative sum of all stress and strain increments throughout the entire stress loading history. The implementation of a progressive ply failure methodology into this incremental loading strategy is described in the next section.

2.4 Lamina Failure Methodology

Failure of individual plies and their effect on the overall laminate response during incremental loading are accounted for in our analysis. Our ply failure predictions are based on the well-established Maximum Strain Failure Criterion [8, 30]. The Maximum Strain Failure Criterion predicts that a material will fail when the strain in any direction exceeds its corresponding allowable level. The principal ply strains in the six directions ( , , , , , and ) are compared to their corresponding maximum strain allowables:

if > 0 and if > Y1T, then the failure mode is fiber tension, (16a)

if < 0 and if | | > Y1C, then the failure mode is fiber compression, (16b)

if > 0 and if > Y2T, then the failure mode is transverse tension, (16c)

if < 0 and if | | > Y2C, then the failure mode is transverse compression, (16d)

if > 0 and if > Y3T, then the failure mode is transverse tension, (16e)

if < 0 and if | | > Y3C, then the failure mode is transverse compression, (16f)

if | | > Y23, then the failure mode is interlaminar shear, (16g)

if | | > Y13, then the failure mode is interlaminar shear, (16h)

and if | | > Y12, then the failure mode is in-plane shear. (16i)

In equations (16a) through (16i), Y1T is the maximum tensile strain in the 1-direction (longitudinal), Y1C is the maximum compressive strain in the 1-direction, Y2T is the maximum tensile strain in the 2-direction (transverse), Y2C is the maximum compressive strain in the 2-direction, Y3T is the maximum tensile strain in the 3-direction (out-of-plane), Y3C is the maximum compressive strain in the 3-direction, Y23 is the maximum shear strain in the 23-plane, Y13 is the maximum shear strain in the 13-plane, and Y12 is the maximum shear strain in the 12-plane.

As the laminate is loaded and laminate strains develop, the individual ply strains are monitored. When ply failure is predicted in any ply, according to the maximum strain failure criteria, the incremental loading to that point is stopped and the entire laminate stress vs. strain response is recorded. The modulus associated with the particular mode of failure in the failed ply is then reduced to an insignificant value (as well as the associated Poisson’s ratio), and the incremental loading strategy is repeated from the beginning (all stresses and strains are set to zero). The loading procedure is continued until the next failure in a ply is detected. The corresponding modulus value is again discounted, the laminate response is recorded, and the procedure is repeated. This progressive ply failure response is repeated until final failure is determined, which is assumed when the laminate looses sufficient stiffness such that it cannot carry any load without undergoing an arbitrarily excessive amount of deformation (say greater than 5% strain).

The entire laminate response is determined by the stress vs. strain response up to the point of failure, and then the load is assumed to drop to the level of the subsequent stress vs. strain curve response. The load path then continues until failure and drops again. This methodology essentially corresponds to progressive ply failure where the load in failure plies is redistributed to adjacent plies under a displacement controlled load path history.

2.5 Thermal Residual Stresses

Thermal residual stresses resulting from thermal expansion mismatch in adjacent plies in the laminates during cool down from the stress-free state at the cure temperature were not accounted for in the predictions. Their actual calculation follows straightforwardly from the analysis derivation described in the previous section. For completeness, however, a full description of their determination is given elsewhere [23]. It is acknowledged that the inclusion of thermal residual stresses will have some effect on the ultimate laminate strength predictions. The exact effect, however, will depend on the specific laminate architecture and loading considered.

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