Orthotropic material

Most simple materia characterization tests are perfomned with a known load or stress. The resulting displacement or strain is then measured. The engineering constants are generally the slope of a stress-strain curve (e.g., E = o/e) or the slope of a strain-strain curve (e.g., v = -ey/ej5 for Ox = a and all other stresses are zero). Thus, the components of the compliance (Sy) matrix are determined more directly than those of the stiffness (Cy) matrix. For an orthotropic material, the compliance matrix components in terms of the engineering constants are... 

Note that an orthotropic material that is stressed in principal material coordinates (the 1, 2, and 3 coordinates) does not exhibit either shear-extension or shear-shear coupling. Recall that an orthotropic material has nine independent constants because... 

Thus, three reciprocal relations must be satisfied for an orthotropic material. Moreover, only 2, V13, and V23 need be further considered because V21, V31, and V32 can be expressed in terms of the first-mentioned group of Poisson s ratios and the Young s moduli. The latter group of Poisson s ratios should not be forgotten, however, because for some tests they are what is actually measured. 

Because the stiffness and compliance matrices are mutually inverse, it follows by matrix algebra that their components are related as follows for orthotropic materials ... 

The preceding restrictions on engineering constants for orthotropic materials are used to examine experimental data to see if they are physically consistent within the framework of the mathematical elasticity model. For boron-epoxy composite materials, Dickerson and DiMartino [2-3] measured Poisson s ratios as high as 1.97 for the negative of the strain in the 2-direction over the strain in the 1-direction due to loading in the 1-direction (v 2)- The reported values of the Young s moduli for the two directions are E = 11.86 x 10 psi (81.77 GPa) and E2 = 1.33x10 psi (9.17 GPa). Thus,... 

Show that the determinant inequality In Equation (2.48) tor orthotropic materials correctly reduces to v< 1/2 for isotropic materials. 

STRESS-STRAIN RELATIONS FOR PLANE STRESS IN AN ORTHOTROPIC MATERIAL... 

For orthotropic materials, imposing a state of plane stress results in implied out-of-plane strains of... 

However, as mentioned previously, orthotropic laminae are often constructed in such a manner that the principal material coordinates do not coincide with the natural coordinates of the body. This statement is not to be interpreted as meaning that the material itself is no longer orthotropic instead, we are just looking at an orthotropic material in an unnatural manner, i.e., in a coordinate system that is oriented at some angle to the principal material coordinate system. Then, the basic question is given the stress-strain relations In the principal material coordinates, what are the stress-strain relations in x-y coordinates ... 

In summary, the engineering constants for anisotropic materials and orthotropic materials loaded in non-principal material coordinates can be most effectively thought of In strictly functional terms ... 

Identity Equation (2.97) by interpreting Equation (2.88) using Equation (2.90) as well as Equations (2.91) and (2.62). Explain tbe key logical step that enables you to use both Equations (2.90) and (2.91) for anisotropic materials and Equations (2.62) and (2.88) for orthotropic materials in this problem. That Is, in what way can we interpret a material as satistying both definitions ot a material ... 

Show that the apparent extensional modulus of an orthotropic material as a function of 0 [the first of Equations (2.97)] can be written as... 

That Is, show that an orthotropic material can have an apparent Young s modulus that either exceeds or is less than the Young s moduli in both principal material directions. In doing so, derive the conditions for which each type of behavior exists, i.e., derive the inequalities. Plot E E, for some contrived materials that exemplify these relations. 

Qi are for anisotropic materials. Qy for orthotropic materials are obtained by deleting Ug and Uy from the definitions of Qy. 

The foregoing example is but one of the difficulties encountered in analysis of orthotropic materials with different properties in tension and compression. The example is included to illustrate how basic information in principal material coordinates can be transformed to other useful coordinate directions, depending on the stress field under consideration. Such transformations are simply indications that the basic information. 

For orthotropic materials, certain basic experiments can be performed to measure the properties in the principal material coordinates. The experiments, if conducted properly, generally reveal both the strength and stiffness characteristics of the material. Recall that the stiffness characteristics are... 

Hill [2-22] proposed a yield criterion for orthotropic materials ... 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

B. M. Lempriere, Poisson s Ratio in Orthotropic Materials, AIAA Journal, November 1968, pp. 2226-2227. 

Oscar Hoffman, The Brittle Strength of Orthotropic Materials, Journal of Composite Materials, April 1967, pp. 200-206. 

The mechanics of materials approach to the micromechanics of material stiffnesses is discussed in Section 3.2. There, simple approximations to the engineering constants E., E2, arid elasticity approach to the micromechanics of material stiffnesses is addressed. Bounding techniques, exact solutions, the concept of contiguity, and the Halpin-Tsai approximate equations are all examined. Next, the various approaches to prediction of stiffness are compared in Section 3.4 with experimental data for both particulate composite materials and fiber-reinforced composite materials. Parallel to the study of the micromechanics of material stiffnesses is the micromechanics of material strengths which is introduced in Section 3.5. There, mechanics of materials predictions of tensile and compressive strengths are described. 

The stress-strain reiations in principal material coordinates for a lamina of an orthotropic material under plane stress are... 

The special single-layered configurations treated in this section are isotropic, specially orthotropic, generally orthotropic, and anisotropic. The generally orthotropic configuration cannot, of course, be distinguished from an anisotropic layer from the analysis point of view, but does have only the four independent material properties of an orthotropic material. 

Because of the analytical complications involving the stiffnesses Ai6, A26, D g, and D26, a laminate is sometimes desired that does not have these stiffnesses. Laminates can be made with orthotropic layers that have principal material directions aligned with the laminate axes. If the thicknesses, locations, and material properties of the laminae are symmetric about the middle surface of the laminate, there is no coupling between bending and extension. A general example is shown in Table 4-2. Note that the material property symmetry requires equal [Q j], of the two layers that are placed at the same distance above and below the middle surface. Thus, both the orthotropic material properties, [Qjjlk. of the layers and the angle of the principal material directions to the laminate axes (i.e., the orientation of each layer) must be identical. 

Antisymmetry of a laminate requires (1) symmetry about the middle surface of geometry (i.e., consider a pair of equal-thickness laminae, one some distance above the middle surface and the other the same distance below the middle surface), but (2i some kind of a reversal or mirror image of the material properties [Qjjlk- In fact, the orthotropic material properties [Qjj], are symmetric, but the orientations of the laminae principal material directions are not symmetric about the middle surface. Those orientations are reversed from 0° to 90° (or vice versa) or from + a to - a (a mirror image about the laminate x-axis). Because the [Qjj]k are not symmetric, bending-extension coupling exists. 

Consider two laminae with principal material directions at -t- a and - a with respect to a reference axis. Prove that fr orthotropic materials... 

For cross-ply laminates, a knee in the load-deformation cun/e occurs after the mechanical and thermal interactions between layers uncouple because of failure (which might be only degradation, not necessarily fracture) of a lamina. The mechanical interactions are caused by Poisson effects and/or shear-extension coupling. The thermal interactions are caused by different coefficients of thermal expansion in different layers because of different angular orientations of the layers (even though the orthotropic materials in each lamina are the same). The interactions are disrupted if the layers in a laminate separate. 

Note that no assumptions involve fiber-reinforced composite materials explicitly. Instead, only the restriction to orthotropic materials at various orientations is significant because we treat the macroscopic behavior of an individual orthotropic (easily extended to anisotropic) lamina. Therefore, what follows is essentially a classical plate theory for laminated materials. Actually, interlaminar stresses cannot be entirely disregarded in laminated plates, but this refinement will not be treated in this book other than what was studied in Section 4.6. Transverse shear effects away from the edges will be addressed briefly in Section 6.6. 

A specially orthotropic laminate has either a single layer of a specially orthotropic material or multiple specially orthotropic layers that are symmetrically arranged about the laminate middle surface. In both cases, the laminate stiffnesses consist solely of A, A 2> 22> 66> 11> D 2, D22, and Dgg. That is, neither shear-extension or bend-twist coupling nor bending-extension coupling exists. Thus, for plate problems, the transverse deflections are described by only one differential equation of equilibrium ... 

The stiffnesses in Equation (5.39) are equivalent to the stiffnesses of an equivalent orthotropic material with principal material axes of orthotropy at 45° to the plate sides. The orthotropic bending stiffnesses of the equivalent material can be shown to be... 

A specially orthotropic laminate has either a single layer of a specially orthotropic material or multiple specially orthotropic layers that are symmetrically arranged about the laminate middle surface. In both... 

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