Classical Laminate  theory has been extensively used to describe the behavior of composite materials under mechanical, thermal, and hygothermal loading conditions. Many references are available where classical lamination theory is utilized to describe composite material behavior. In Classical Lamination Theory, the plate is assumed to have infinite dimensions and the whole panel undergoes the same thermal gradients. In this process however the heat is applied locally, and the rest of the panel effectively acts as a barrier to curvature growth. At the completion of every layer or cycle, every point on the panels surface experiences the same thermal history at different times. As a result the thermal signature of this process is experienced everywhere but at an offset time corresponding to the time difference between each point on the surface. Since Classical lamination Theory is time independent, this model can be used as a method for determining the stress fields through the thickness as long as the panel is sufficiently large. One important difference however is that since the heated region is local, the solid boundary acts as a mechanism for constraint. This unique boundary is included in the model as a “partial constraint” and the method by which it is applied to the model is explained later. The effect of partial constraint is not as significant in the simplified residual stress model as the heat input is restricted to the surface ply and the remaining preconsolidated material acts as a similar partial constraint. In the second-generation model, the effect of partial constraint becomes a more critical mechanism for limiting the curvature growth as the actual heat input is applied through the thickness of the laminate. This increases the difference in absolute temperatures between the cooler solid boundary and the temperatures within the heat-affected zone.

Basic Assumptions

Classical Lamination Theory predicts the behavior of the laminate within the framework of the following assumptions. [Pister 1959, Reissner 1961] Firstly, each layer of the laminate is assumed to be both quasihomogeneous and orthotropic. The laminate is thin with its lateral dimensions is much larger than its thickness and that the laminate is only loaded in the in-plane directions. This assumption is especially important for the condition of clamping the laminate. All out of plane displacements are assumed to be small relative to the thickness of the laminate, and in-plane displacements vary linearly through the thickness of the laminate. All displacements are small compared with the thickness of the laminate, and normal distances from the middle surface remain constant. Provided that these conditions are met, it is possible to determine the reference plane strains and curvatures as a linear summation of the thermal strains and curvatures for each ply.

A number of important assumptions are made in this simplified residual stress model in order to avoid the complexities associated with a moving localized heat source on a composite laminate. The first assumption in the model is that the panel undergoes the same thermal cycle across the panel surface. Since classical lamination theory is time independent (not including the effects of viscoelastic relaxation) this assumption is valid for a constant velocity manufacturing process, since every point on the surface observes the same thermal processing cycle. So although this analysis cannot provide information about localized residual stresses during panel manufacture it can predict the overall response of the laminate based on this assumption.

Composite  Strains
The incremental strains are a function of temperature alone as follows:

               

where aL and at are the coefficients of thermal expansion of the composite lamina and are based on the micromechanics model and the instantaneous resin properties, fiber properties and the volume fraction of fibers. DT is the temperature gradient applied to the surface ply. These strains are then transformed to the global coordinate system, through a second order tensoral transformation of the principle strain increments as follows:

               

These strains are applied to each layer within the composite to determine the effective plate loads and stresses. The increments in stress and deformation development are computed and superimposed to provide a complete profile of the stress field. To carry out this procedure the laminate is discretized into a number of nodes or layers corresponding to the centerline dimensions of each ply.

Each layer is assigned a set of thermoelastic properties that are a function of time, temperature and position in the z, or thickness direction. The thermoelastic properties of each lamina in its principle coordinate system are defined in terms of the plane stress stiffness coefficients, Qij. These coefficients are related to the effective constitutive mechanical properties of each layer as follows:

       

The principle coordinate system of each layer or lamina may be orientated at some angle, q to the principle coordinate system.

The lamina properties, , in the global coordinate system are determined from the following tensoral transformation equations:

               

The effective in-plane force and moment resultants, when transformed in the global laminate system through a second-order tensoral transformation are given by:

               

where represents the transformed process-induced incremental stress-free strains in each ply in the global coordinate system, n is the number of plies, and represents the transformed plane stress stiffness matrix coefficients obtained from the instantaneous composite properties of each ply.

The resulting plate loads are then used to compute the effective laminate response in terms of the in-plane strains, and curvatures, , through the [abcd] matrix as follows:

               

The total incremental ply strains are then computed through the classical strain-displacement equations:

               

where z is the distance from the laminate mid plane to the ply center. The incremental ply stresses are based on the difference between the laminate response strain increments and the stress-free process induced strain increments through the following expression:

       

The laminate stiffness is obtained by appropriate averaging through the thickness.

The laminate force and moment resultants are found through integration of stresses across the thickness of each lamina.

Laminate strains are assumed to vary linearly through the thickness as a function of the laminate mid-plane strains, eo and curvatures, k .

Substituting the assumed strain field into the transformed ply stiffness relations and performing integration yields the well known laminate stiffness relationship

A general outline of the CLT model described above is shown below. The primary inputs to this model are the composite material constituent properties; ply thickness, stacking sequence, through thickness temperature gradients, degree of crystallinity and temperature dependent resin stiffness (if material models are included).

Flow chart summary for Thermoelastic analysis of laminated composites using Classical Lamination Theory