Elastic behavior orthotropic

Thus far four composites listed in Table I have been studied. NbTi/Cu is discussed briefly here. From its microstructure and manufacture, a rectangular cross-section bar, it was assumed that this composite has orthorhombic (orthotropic) symmetry in its physical properties. Materials with this symmetry have nine independent elastic constants. While deviations from elastic behavior were small, nine independent elastic constants were verified. Four specimens were prepared (Fig. 16) and 18 ultrasonic wave velocities were determined by propagating differently polarized waves in six directions, (100) and (110). An example cooling run is shown in Fig. 17 for E33, Young s modulus along the filament axis. These data typify the composites studies a wavy, irregular modulus/temperature curve. 

Under quasi-static loading, or in conditions under which short-term loading responses are expected to occur [27], both meniscal and discal tissues may be modeled as linear elastic and orthotropic. Under a constant load rate, the non-linear behavior may be described by an exponential stress-strain relationship given by... 

Erdogan and Ratwani( ) presented an analytical solution based on a one-dimensional model for calculating stresses in a stepped lap joint. One adherend was treated as isotropic and the second as orthotropic, and linear elastic behavior was assumed. The thickness variation of the stresses in both the adherends and the adhesive was neglected. 

Dias et al. [71] presented a computational model for plain woven fabrics which can represent known elastic behavior in deformation such as planar extension, shearing and out-of-plane bending, drape, and buckling. They assumed the fabric to be an orthotropic linear elastic continuum and discretized by a mesh of triangles. Then each triangle links three particles which are capable to measure the stress and strain of the imderlying medium. For the planar deformation, they assume the hypothesis of the plate imder plane stress Irom the classical theory of elasticity. For the out-of-plane deformation, they allow linear elasticity and... 

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... 

The macromechanical behavior of a lamina was quantitatively described in Chapter 2. The basic three-dimensional stress-strain relations for elastic anisotropic and orthotropic materials were examined. Subsequently, those relations were specialized for the plane-stress state normally found in a lamina. The plane-stress relations were then transformed in the plane of the lamina to enable treatment of composite laminates with different laminae at various angles. The various fundamental strengths of a lamina were identified, discussed, and subsequently used in biaxial strength criteria to predict the off-axis strength of a lamina. 

Elastic constants and strength properties needed for the characterization of directional behavior can be used. The number of constants needed depends primarily on the complexity of the construction, that is, whether it is isotropic, planar isotropic, or orthotropic. With these stiffness constants known, the directional stiffness properties can be calculated using textbook equations. 

As wood is orthotropic, each behavior law involves nine independent terms. In fact, it is more common to define the inverse of that, for the case of linear elasticity, leads to the generalized Hooke s law ... 

For materials with orthorhombic symmetry (or orthotropic symmetry), there are three mutually orthogonal axes, each of two-fold rotational symmetry. Translation of the material in the direction of any of these axes leaves the material behavior unaltered, but the translation distances required to recover the same lattice in crystalline materials are distinct, in general. Nine independent constants (cn, C22, C33, C44, C55, cge, C z, C23, C12) are required to specify the elastic response of a general orthorhombic material. For a material in which one of the three symmetry axes has fourfold rotational symmetry (or tw o of the translational invariance distances are equal), the number of independent constants is reduced to six this is achieved by requiring that C22 = cn, C55 = C44 and C23 = C12, for example. 

An element will have hoth geometric and material properties. Spatially, an element is defined by its nodes however, additional geometric input is usually required for line and surface elements. For structural analysis the minimum material property is the modulus of elasticity. In most cases, Poisson s ratio or shear modulus must also be specified. If an orthotropic material is used then the orientation of the material must be specified as well as the elastic constants relative to each principal axis. If post-yield behavior is to be modelled then an elasto-plastic material model must be applied and the yield and hardening behavior defined. Constitutive adhesive and sealant models are discussed in more detail in O Chap. 23. Additional material properties will also be required for dynamic or thermal analysis. 

It is apparent from Equations 8.42 and 8.43 that four material elastic properties (compliance or stiffness) are needed to characterize the in-plane behavior of a linear elastic orthotropic lamina. It is convenient to define these material properties in terms of measured engineering constants (Young s moduli, El > d Ej, shear modulus Glt, and Poisson s ratios u,lt and (Xtl). The longitudinal Young s... 

Hence, the four measured engineering constants E, E, Glt, and Plt are sufficient to describe the linear elastic macromechanical behavior of a thin orthotropic composite lamina subjected to plane stress. 

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