Isotropic material stress-strain relations

For plane stress on isotropic materials, the strain-stress relations are... 

Unified equations that couple rate-independent plasticity and creep [114] are not readily available for SOFC materials. The data in the hterature allows a simple description that arbitrarily separates the two contributions. In the case of isotropic hardening FEM tools for structural analysis conveniently accept data in the form of tabular data that describes the plastic strain-stress relation for uniaxial loading. This approach suffers limitations, in terms of maximum allowed strain, typically 10 %, predictions in the behaviour during cycling and validity for stress states characterised by large rotations of the principal axes. 

A key element in the experimental determination of the stiffness and strength characteristics of a lamina is the imposition of a uniform stress state in the specimen. Such loading is relatively easy for isotropic materials. However, for composite materials, the orthotropy introduces coupling between normal stresses and shear strains and between shear stresses and normal and shear strains when loaded in non-principal material coordinates for which the stress-strain relations are given in Equation (2.88). Thus, special care must be taken to ensure obtaining... 

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... 

Viscoelasticity deals with the dynamic or time-dependent mechanical properties of materials such as polymer solutions. The viscoelasticity of a material in general is described by stresses corresponding to all possible time-dependent strains. Stress and strain are tensorial quantities the problem is of a three dimensional nature (8), but we shall be concerned only with deformations in simple shear. Then the relation between the shear strain y and the stress a is simple for isotropic materials if y is very small so that a may be expressed as a linear function of y,... 

The fracture toughness, a term defined by Irwin (1956, 1960) to characterize brittleness, provides a measure of the conditions required for catastrophic crack propagation in a material (see Section 1.6). One fracture toughness parameter is the surface fracture energy y, defined as one-half G, the critical strain energy release rate above which catastrophic failure occurs. In turn G is related to another convenient toughness parameter, the critical stress intensity factor a measure of the stress field at the crack tip. For fracture of an isotropic material in a plane strain modet (Baer, 1964, p. 946) ... 

Elastic constant n. Any of the several constants of a constitutive relationship between stress (of any mode) and strain in a material. For an isotropic material stressed in its elastic range, there are (at any temperature) four interrelated constants tensile modulus , shear modulus G, bulk modulus By and Poisson s ratio ji. Two expressions of the relations are... 

It is convenient for users of these elastic constants to get them from tables summarizing all these relations, such as Table 1.4. For isotropic materials, two independent elastic constants are sufficient (as indicated in the Table 1.4) for describing a stress-strain relation. There are different stress-strain constants for various other deformation conditions. 

X10. The next three rows present the viscosity rj, the surface tension, and its tenqterature dependence, in the liquid state. The next properties are the coefficient of linear thermal expansion a and the sound velocity, both in the solid and in the liquid state. A number of quantities are tabulated for the presentation of the elastic properties. For isotropic materials, we list the volume compressihility k = —(l/V)(dV/dP), and in some cases also its reciprocal value, the bulk modulus (or compression modulus) the elastic modulus (or Young s modulus) E the shear modulus G and the Poisson number (or Poisson s ratio) fj,. Hooke s law, which expresses the linear relation between the strain s and the stress a in terms of Young s modulus, reads a = Ee. For monocrystalline materials, the components of the elastic compliance tensor s and the components of the elastic stiffness tensor c are given. The elastic compliance tensor s and the elastic stiffness tensor c are both defined by the generalized forms of Hooke s law, a = ce and e = sa. At the end of the list, the tensile strength, the Vickers hardness, and the Mohs hardness are given for some elements. 

For example, the stress-strain relation for shear strain in an isotropic material is <7 = Ge. For a viscoelastic material this relation is modified as follows ... 

It was observed empirically by Hooke that, for many materials under low strain, stress is proportional to strain. Young s modulus may then be defined as the ratio of stress to strain for a material under uniaxial tension or compression, but it should be noted that not all materials (and this includes polymers) obey Hooke s law rigorously. This is particularly so at high values of strain but this section only considers the linear portion of the stress-strain curve. Clearly, reality is more complicated than described previously because the application of stress in one direction on a body results in a strain, not only in that direction, but in the two orthogonal directions also. Thus, a sample subjected to uniaxial tension increases in length, but it also becomes narrower and thinner. This quickly leads the student into tensors and is beyond the scope of this chapter. The subject is discussed elsewhere [21-23]. There are four elastic constants usually used to describe a macroscopically isotropic material. These are Young s modulus, E, shear modulus, G, bulk modulus, K, and Poisson s ratio, v. They are defined in Figure 9.2 and they are related by Equations 9.1-9.3. 

It is important to appreciate that plasticity is different in kind from elasticity, where there is a unique relationship between stress and strain defined by a modulus or stiffness constant. Once we achieve the combination of stresses required to produce yield in an idealized rigid plastic material, deformation can proceed without altering stresses and is determined by the movements of the external constraints, e.g. the displacement of the jaws of the tensometer in a tensile test. This means that there is no unique relationship between the stresses and the total plastic deformation. Instead, the relationships that do exist relate the stresses and the incremental plastic deformation, as was first recognized by St Venant, who proposed that for an isotropic material the principal axes of the strain increment are parallel to the principal axes of stress. 

Let us consider in more detail the elasticity of nematic rubbers, which is at the heart of understanding their specific properties. Consider a weakly crosslinked network with junction points sufficiently well spaced to ensure that the conformational freedom of each chain section is not restricted. We recall that for a conventional isotropic network the stress-strain relation for simple stretching (compression) of a unit cube of material can be derived as [62, 63] ... 

As discussed in the previous section, strains and stresses are closely linked together through material characteristics and deformation mechanisms. In solid mechanic, it is useful to mathematically express this relation through constitutive laws. Also, as aforementioned, numerical modelling now requires three-dimensional constitutive models. One of the simplest expressions of a constitutive law is the Hooke s law (4) in Voigt notation (Mase and Mase, 1999), for isotropic materials. In this case, assuming the small strain theory, only two... 

The elastic increment of stress and strain is related to an isotropic material property matrix [D] by... 

The simple shear is an isochoric deformation that is possible in every compressible, homogeneous, and isotropic hyperelastic material. The constimtive relation (38) shows that the shear stress related to the shear strain y is given by ... 

To this point the relations between stress and strain (constitutive equations) for viscoelastic materials have been limited to one-dimension. To appreciate the procedure for the extension to three-dimensions recall the generalized Hooke s law for homogeneous and isotropic materials given by Eqs. 2.28,... 

One property of second-order tensors such as stress and strain is that one can identify certain principal planes on which extreme values of the magnitudes occur. In general, complex, three-dimensional stress states such as shown in Fig. Ic can be resolved into principal stresses as shown in the same figure. Maximum and minimum normal stresses occur on these principal planes where shear stresses vanish. These principal stresses (or strains) are of significant importance in several failure criteria for homogeneous, isotropic materials [ 16], and may be important in the failure of adhesives as well [17]. Because of the natural planes associated with the bond plane, however, normal and shear stresses acting on these bond planes are often examined and reported in adhesion-related literature. 

Stresses and strains are related through constitutive equations for linear elastic, homogeneous, isotropic materials, these relations include... 

The integral that defines the stresses and strains aronnd the crack tip in a material undergoing linear and nonlinear deformations is called the /-integral, which is path-independent. For perfectly linear elastic, brittle and isotropic materials, the /-integral can be directly related to the fracture toughness [6], thus ... 

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. ... 

If at every point of a material there is one plane in which the mechanical properties are equal in all directions, then the material is called transversely isotropic. If, for example, the 1-2 plane is the plane of isotropy, then the 1 and 2 subscripts on the stiffnesses are interchangeable. The stress-strain relations have only five independent constants ... 

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. 

Stress-Strain Relations as Equations of State. Simple theory of elasticity assumes that the material is isotropic and that induced stresses and strains are linearly related to each other as long as they are small. The theory further assumes that the stress and the strain tensors always have the same axes. Poisson s ratio and... 

Considering the liner as an isotropic elastic plastic material, the incremental stress-strain relations can be rewritten as ... 

Stress-strain tests were mentioned on page 24 and in Fig. 11-12. In such a tensile lest a parallel-sided strip is held in two clamps that are separated at aconstant speed, and the force needed to effect this is recorded as a function of clamp separation. The test specimens are usually dogbone shaped to promote deformation between the clamps and deter flow in the clamped portions of the material. The load-elongation data are converted to a stress-strain curve using the relations mentioned on p. 24. These are probably the most widely used of all mechanical tests on polymers. They provide useful information on the behavior of isotropic specimens, but their... 

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