Stress-strain relations anisotropic

The three-dimensional thermoelastic anisotropic strain-stress relations are... 

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. 

The mechanical properties of an anisotropic elastic solid where the stresses are linearly related to the strains are defined by the generalized Hooke s law where each component of stress can relate to all six independent components of strain, and equivalently each component of strain can relate to all six independent components of stress. In the former case we have, for example... 

Equations (2.9) and (2.10) are representative of all isotropic, homogeneous solids, regardless of the stress-strain relations of a solid. What is strongly materials specific and uncertain is the appropriate value for shear stress, particularly if materials are in an inelastic condition or anisotropic, inhomogeneous properties are involved. The limiting shear stress controlled by strength is termed r. ... 

Figure 2-5 Physical Significance of the Anisotropic Stress-Strain Relations...

Note that the transformed reduced stiffness matrix Qy has terms in all nine positions in contrast to the presence of zeros in the reduced stiffness matrix Qy. However, there are still only four independent material constants because the lamina is orthotropic. In the general case with body coordinates x and y, there is coupling between shear strain and normal stresses and between shear stress and normal strains, i.e., shear-extension coupling exists. Thus, in body coordinates, even an orthotropic lamina appears to be anisotropic. However, because such a lamina does have orthotropic characteristics in principal material coordinates, it is called a generally orthotropic lamina because it can be represented by the stress-strain relations in Equation (2.84). That is, a generally orthotropic lamina is an orthotropic lamina whose principai material axes are not aligned with the natural body axes. 

Other anisotropic elasticity relations are used to define Chentsov coefficients that are to shearing stresses and shearing strains what Poisson s ratios are to normal stresses and normal strains. However, the Chentsov coefficients do not affect the in-plane behavior of laminaeS under plane stress because the coefficients are related to S45, S46, Equation (2.18). The Chentsov coefficients are defined as... 

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... 

Derive the thermoelastic stress-strain relations for an orthotropic lamina under plane stress, Equation (4.102), from the anisotropic thermoelastic stress-strain relations in three dimensions. Equation (4.101) [or from Equation (4.100)]. 

Using relations (2.5) and (2.6) we can determine the elasticity tensor which describes the linear relation between components of the stress and strain tensors. 2 slr.ss = CEstta n is therefore an expression of Hooke s law for anisotropic crystals... 

The fundamental quantities in elasticity are second-order tensors, or dyadicx the deformation is represented by the strain thudte. and the internal forces are represented by Ihe stress dyadic. The physical constitution of the defurmuble body determines ihe relation between the strain dyadic and the stress dyadic, which relation is. in the infinitesimal theory, assumed lo be linear and homogeneous. While for anisotropic bodies this relation may involve as much as 21 independent constants, in the euse of isotropic bodies, the number of elastic constants is reduced lo two. 

The complete stress-strain relation requires the six as to be written in terms of the six y components. The result is a 6 x 6 matrix with 36 coefficients in place of the single constant, Twenty-one of these coefficients (the diagonal elements and half of the cross elements) are needed to express the deformation of a completely anisotropic material. Only three are necessary for a cubic crystal, and two for an amorphous isotropic body. Similar considerations prevail for viscous flow, in which the kinematic variable is y. 

The first Lame constant (A) has no physical interpretation. However, both Lame constants are related to other elastic moduli. To see this, recall that the Young s modulus, E, is defined as the ratio of normal stress to normal strain. Hence, for an elastically isotropic body, E is given by (cn-Ci2)(cn-b 2ci2)/(cn-I-C12), or /r(3A-b 2/r)/(A-b/r). It should be emphasized that the Young s modulus is anisotropic for all crystal classes, including the cubic class, so this relation would never apply to any monocrystal. 

Employing the concepts of stress and conjugate strain, and their proper mathematical formulation as. second-rank tensors, now enables us to deal with mechanical work in a general anisotropic piece of matter. One realization of sudi a system are fluids in confinement to whidi this book is devoted. However, at the core of our subsequent treatment are thermal properties of confined fluids. In other words, we need to understand the relation between medianical work represented by stress-strain relationships and other forms of energy such as heat or chemical work. This relation will be formally... 

The complicated morphology of crystalline polymer solids and the coexistence of crystalline and amorphous phases make the stress and strain fields extremely nonhomogeneous and anisotropic. The actual local strain in the amorphous component is usually greater and that in the crystalline component is smaller than the macroscopic strain. In the composite structure, the crystal lamellae and taut tie molecules act as force transmitters, and the amorphous layers are the main contributors to the strain. Hence in a very rough approximation, the Lennard-Jones or Morse type force field between adjacent macro-molecular chain sections (6, 7) describes fairly well the initial reversible stress-strain relation of a spherulitic polymer solid almost up to the yield point, i.e. up to a true strain of about 10%. 

Deformation is measured by a quantity known as strain (strain is a relative extension or contraction of dimension). Strain is similarly a tensor of the second rank having nine components (3x3 matrix). The relation between stress and strain in the elastic regime is given by the classical Hooke s law. It is therefore obvious that the Hooke s proportionality constant, known as the elastic modulus, is a tensor of 4 rank and is represented by a (9 x 9) matrix. Before further discussion we note the following. The stress tensor consists of 9 elements of which stability conditions require cjxy=(jyx, stress components in the symmetric stress matrix are only six. Similarly there are only six independent strain components. Therefore there can only be six stress and six strain components for an elastic body which has unequal elastic responses in x, y and z directions as in a completely anisotropic solid. The representation of elastic properties become simple by following the well known Einstein convention. The subscript xx, yy, zz, yz, zx and xy are respectively represented by 1, 2, 3, 4, 5 and 6. Therefore Hooke s law may now be written in a generalized form as. 

Biot, M. A., 1954.r/ieory of Stress-strain Relations in Anisotropic Viscoelasticity and Relaxation Phenomena, J. Appl. Phys., 25(11) ppl385-1391. 

When written in matrix form these equations relate the properties to the crystallographic directions. For ceramics and other crystals the piezoelectric constants are anisotropic. For this reason, they are expressed in tensor form. The directional properties are defined by the use of subscripts. For example, d i is the piezoelectric strain coefficient where the stress or strain direction is along the 1 axis and the dielectric displacement or electric field direction is along the 3 axis (i.e., the electrodes are perpendicular to the 3 axis). The notation can be understood by looking at Figure 31.19. 

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