Stiffness classical lamination theory

Laminate plate and shell stiffness classical lamination theory (CLT)... 

Ciassicai lamination theory is derived in Section 4.2. Then, special stiffnesses of practical interest are classified and examined in Section 4.3. Next, the theoretical stiffnesses obtained by classical lamination theory are compared with experimental results in Section 4.4. In Section 4.5, the strengths of various laminates are predicted. Finally, the stresses between the laminae of a laminate are examined in Section 4.6 and found to be a proba lei causg of delamination of some laminates. 

In preceding sections, laminate stiffnesses were predicted on the basis of combination of lamina stiffnesses in accordance with classical lamination theory. However, the actual, practical realization of those laminate stiffnesses remains to be demonstrated. The purpose of this section is to compare predicted laminate stiffnesses with measured laminate stiffnesses to determine the validity of classical lamination theory. Results for two types of laminates, cross-ply and angle-ply laminates, are presented. 

The predicted strengths in Figure 4-44 are generally somewhat above the measured values. The predicted and observed stiffnesses, both initial (below the knee) and final, are in very good agreement. Thus, the stiffness aspects of classical lamination theory, as well as the present strength-analysis procedure, are verified. 

The significance of interlaminar stresses relative to laminate stiffness, strength, and life is determined by Classical Lamination Theory, i.e., CLT stresses are accurate over most of the laminate except in a very narrow boundary layer near the free edges. Thus, laminate stiffnesses are affected by global, not local, stresses, so laminate stiffnesses are essentially unaffected by interlaminar stresses. On the other hand, the details of locally high stresses dominate the failure process whereas lower global stresses are unimportant. Thus, laminate strength and life are dominated by interlaminar stresses. 

A collection of the basic building block, a lamina, was bonded together to form a laminate in Chapter 4. The behavior restrictions were covered in the section on classical lamination theory. Special cases of laminates were discussed to learn about laminate characteristics and behavior. Predicted and measured laminate stiffnesses were favorably compared to give credence to classical lamination theory. Then, the strength of laminates was discussed and found to be reasonably predictable. Finally, interlaminar stresses were analyzed because of their apparent strong influence on laminate strength (and life). 

Laminate stiffness analysis predicts the constitutive behaviour of a laminate, based on classical lamination theory (CLT). The result is often given in the form of stiffness and compliance matrices. Engineering constants, i.e. the in-plane and flexural moduli, Poisson s ratios and coefficients of mutual influence, are further derived from the elements of the compliance matrix. Analyses are continuously needed in structural design since it is essential to know the constitutive behaviour of laminates forming the structure. The results are also the necessary input data for all other macromechanical analyses. A computer code for the stiffness analysis is a valuable tool on account of the extensive calculations related to the analysis. 

This chapter is devoted to the analysis of the elastic properties and their characterization for laminated advanced composites. It starts with a general overview of composite stiffness and then moves to lamina analysis focused on unidirectional reinforced composites. The analysis of laminated composites is addressed through the classical lamination theory (CLT). The last section describes full-field techniques coupled with inverse identification methods that can be employed to measure the elastic constants. 

The Equivalent Constraint Model (ECM) was introduced by Fan and Zhang (1993) with the aim to analyse the in-situ constraint effects on damage evolution in a particular lamina within a multidirectional laminate. In this model, all the laminae below and above the chosen lamina are replaced with homogeneous layers having the equivalent constraining effect. It is assumed that the in-plane stiffness properties of the equivalent constraint layers can be calculated from the classical lamination theory, provided stresses and strains in them are known. 

The reduced stiffness properties of the damaged /i, layer can be determined by applying the classical lamination theory to the ECM/r laminate after replacing the explicitly damaged layer with an equivalent homogeneous one. Constitutive equations of the homogeneous layer, equivalent to the explicitly damaged one, have the form... 

It may be seen from Fig. 3 that, as transverse crack density increases, all stiffness properties of the laminate are significantly reduced. Longitudinal and transverse moduli of the undamaged laminate, calculated from the classical lamination theory, are 166.5 GPa, shear modulus 44 GPa, Poisson s ratio 0.19. When transverse cracking in the 90° layer reaches saturation, the laminate longitudinal and shear moduli are predicted to lose more than 45% of their value. Inclusion of tensile residual stresses into the analysis would lead to even more significant reduction in the longitudinal modulus and Poisson s ratio, but reduction in shear modulus would remain the same. Predictions for a [O/OOi], SiC/CAS laminate are shown in Fig. 4. 

CMC laminates with macrocracks both in the 90° and 0° plies. To capture simultaneous accumulation of damage both in the 90° and the 0° plies, two ECM laminates were analysed simultaneously, as a coupled problem, instead of the original one. Following analysis of stresses in the explicitly damaged layer(s) of ECM laminates, closed form expressions for the reduced stiffness properties of the damaged laminate were derived representing them as functions of crack densities in the 90° and 0° plies. Residual thermal stresses were neglected in the stress analysis, but their value was estimated from the classical lamination theory. 

The mechanical stiffness coefficients, which are unchanged in comparison to the classical lamination theory, have been arranged just as they appear in the latter. They may be determined in accordance with Eqs. (6.4b). As the remaining coefficients are involved with the negated electric field strengths E and appertaining resultants G3 of the electric flux density, they have to be established in consideration of the group association represented by the... 

Classical laminated plate tlieoiy is used to determine the stiffness of laminated composites. Details of the Kitchoff-Love hypothesis on which the theory is based can be found in standard texts (1,7,51). Essentially, the strains in each ply of the laminate ate represented as middle surface strains plus... 

Alternatively, tests can be used to obtain the basic stiffness properties of the material form and their corresponding range measured by some statistical property such as the standard deviation. In two-dimensional cases where there are no significant loads in the out-of-plane direction, the basic orthotropic stiffness properties in Eqn (6.1) can be measured experimentally. Then, the classical laminated plate theory described in previous sections for determination of stiffness can be used effectively to model these sttuctures. Alternatively, the four basic stiffiiesses for 3-D woven composites can be... 

However, especially for three-dimensional stmcture, or, even, in two-dimensional stmcture with significant out-of-plane loads, the stiffness averaging of the classical laminated plate theory is not sufficient. In such cases, a specialized finite element model such as the binary model by Cox, Carter, and Fleck [25] can be used. In this model, Monte Carlo simulation allows the user to randomly orient tows and to randomly assign strength properties to the different components. 

Once the new matrix properties accounting for porosity are known, the ply-level stiffnesses can be determined using micromechanics, and then the classical laminated plate theory outlined in previous sections can be used. 

---