Stress-strain relations compliances

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... 

For single crystals with transverse dimensions large enough to permit a plane wave condition to be attained, the results are unambiguous and virtually free from theoretical assumptions. Five independent elastic constants (stiffnesses or compliances) are required to describe the linear elastic stress-strain relations for hexagonal materials. Only three independent constants are required for cubic (y-Ce, Eu, Yb) materials. Since there are no single crystal elastic constant data for the cubic rare earth metals, this discussion will concentrate on the relationships for hexagonal symmetry. 

From knowledge of the shear relaxation modulus, the memory function, or the creep compliance function of a particular material, its stress-strain relations for... 

The stress-strain relations (6.25) are of the familiar form employed in linear viscoelasticity, except that the retardation spectra incorporate now aging affects, and all instantaneous compliances age with time. Recalling (6.18) for a discrete spectrum of retardation times Xr, expressions (6.25) read... 

The matrices [5] and [Q] appearing in Equations 8.42 and 8.43 are called the reduced compliance and stiffness matrices, while [5 ] and Q in Equations 8.54 and 8.55 are known as the transformed reduced compliance and stillness matrices. In general, the lamina mechanical properties, from which compliance and stiffness can be calculated, are determined experimentally in principal material directions and provided to the designer in a material specification sheet by the manufacturer. Thus, a method is needed for transforming stress-strain relations from off-axis to the principal material coordinate systan. 

The purpose of introducing the concepts developed in Equation 8.56 through Equation 8.63 has been to provide the tools needed to present a straightforward derivation of the transformed reduced compliance and stiffness matrices [5] and [Q based on matrix algebra. The derivation makes use of the following sequence of operations to obtain stress-strain relations in the reference (x, y) coordinate system ... 

The stiffness and compliances in stress-strain and strain-stress relations are fourth-order tensors because they relate two second-order tensors ... 

There are many types of deformation and forces that can be applied to material. One of the foundations of viscoelastic theory is the Boltzmann Superposition Principle. This principle is based on the assumption that the effects of a series of applied stresses acting on a sample results in a strain which is related to the sum of the stresses. The same argument applies to the application of a strain. For example we could apply an instantaneous stress to a body and maintain that stress constant. For a viscoelastic material the strain will increase with time. The ratio of the strain to the stress defines the compliance of the body ... 

The proportionality constant in stress/strain measurements is known as the modulus of elasticity, the E modulus, or Young s modulus. The E modulus is always related to the cross section of the sample before stretching. The reciprocal of the modulus of elasticity is known as the tensile compliance. 

Related to shear stress around 3 dir. (strain for compliance, s)... 

In one dimension Hooke s law defines stiffness (or modulus) as stress divided by strain alternatively, compliance is strain divided by stress. In three dimensions the stress and strain tensors are related through stiffness [c] and compliance [5] tensors (note the confusing nomenclature). As stress and strain are each second-rank symmetric tensors, [c] and [5] are each fourth-rank symmetric tensors each component of strain is linearly related to all nine components of stress, and vice versa, so there are 81 components in the stiffness and compliance tensors, which when written out in full form a 9 X 9 array. 

A plot of shear stress versus shear strain will have a slope equal to the shear modulus. Hence, putting the results of Equation 8.50 into Equation 8.42 gives an expression relating compliance to the measured engineering shear modulus. 

Creep behavior is similar to viscous flow. The behavior in Equation 14.17 shows that compliance and strain are linearly related and inversely related to stress. This linear behavior is typical for most amorphous polymers for small strains over short periods of time. Further, the overall effect of a number of such imposed stresses is additive. Non-creep-related recovery... 

Equation (5.7) related the stress tensor to the strains throngh the modulus matrix, [Q, also known as the reduced stiffness matrix. Oftentimes, it is more convenient to relate the strains to the stresses through the inverse of the rednced stiffness matrix, [2] called the compliance matrix, [5]. Recall from Section 5.3.1.3 that we introduced a quantity called the compliance, which was proportional to the inverse of the modulus. The matrix is the more generalized version of that compliance. The following relationship then holds between the strain and stress tensors ... 

Consider two experiments carried out with identical samples of a viscoelastic material. In experiment (a) the sample is subjected to a stress CT for a time t. The resulting strain at f is ei, and the creep compliance measured at that time is D t) = e la. ln experiment (b) a sample is stressed to a level CT2 such that strain i is achieved immediately. The stress is then gradually decreased so that the strain remains at f for time t (i.e., the sample does not creep further). The stress on the material at time t will be a-i, and the corresponding relaxation modulus will be y 2(t) = CT3/C1. In measurements of this type, it can be expected that az> 0 > ct and Y t) (D(r)) , as indicated in Eq. (11-14). G(t) and Y t) are obtained directly only from stress relaxation measurements, while D(t) and J(t) require creep experiments for their direct observation. Tliese various parameters can be related in the linear viscoelastic region described in Section 11.5.2. 

Many other examples of stress or strain measurements through Raman spectroscopy are still primarily qualitative [18, 27]. Much of this stems from the fact that Raman spectroscopy provides only limited additional information (generally only in the form of frequency shifts) from potentially complicated strain distributions. Furthermore, care must be taken when extracting stresses from measured Raman shifts as key mechanical properties such as Young s modulus (which is related to the compliance or stiffness matrix elements) may be diameter dependent in NWs [61]. Still, Raman mapping with submicron spatial resolution and careful polarization analyses may help clarify the piezospectroscopic properties of semiconductor NWs in ongoing research. 

Creep-compliance studies conducted in the linear viscoelastic range also provide valuable information on the viscoelastic behavior of foods (Sherman, 1970 Rao, 1992). The existence of linear viscoelastic range may also be determined from torque-sweep dynamic rheological experiments. The creep-compliance curves obtained at all values of applied stresses in linear viscoelastic range should superimpose on each other. In a creep experiment, an undeformed sample is suddenly subjected to a constant shearing stress, Oc. As shown in Figure 3 1, the strain (y) will increase with time and approach a steady state where the strain rate is constant. The data are analyzed in terms of creep-compliance, defined by the relation ... 

Many of the earlier commercial viscometers have been of the constant shear rate design where the speed of rotation was the controlled variable and the resulting shear stress was measured. However, viscometers and rheometers in which the shear stress is controlled and resulting shear rate or strain can be measured are available now. Because they provide an opportunity to conduct studies related to yield stress, creep-compliance, stress relaxation, and rate of breakdown of weak structures, it seems... 

As mentioned above, it is very difficult, for experimental reasons, to measure the relaxation modulus or the creep compliance at times below 1 s. In this time scale region, dynamic mechanical viscoelastic functions are widely employed (5,6). However, in these methods the measured forces and displacements are not simply related to the stress and strain in the samples. Moreover, in the case of dynamic experiments, inertial effects are frequently important, and this fact must be taken into account in the theoretical methods developed to calculate complex viscoelastic functions from experimental results. 

Diffraction at high pressure also provides an opportunity to measure some combinations of elastic moduli directly, because the pressure is a stress which results in a strain that is expressed as a change in the unit cell parameters. The compressibility of any direction in the crystal is directly related to the components of the elastic compliance tensor by ... 

Although creep-compliance (Kawabata, 1977 Dahme, 1985) and stress-relaxation techniques (Comby et al., 1986) have been used to study the viscoelestic properties of pectin solutions and gels, the most common technique is small-deformation dynamic measurement, in which the sample is subjected to a low-amplitude, sinusoidal shear deformation. The resultant stress response may be resolved into an in-phase and 90° out-of-phase components the ratio of these stress components to applied strain gives the storage and loss moduli (G and G"), which can be related by the following expression ... 

In Figure 5.8d an intermediate behavior, called viscoelastic, is depicted such a relation is often called a creep curve, and the time-dependent value of the strain over the stress applied is called creep compliance. On application of the stress, the material at first deforms elastically, i.e., instantaneously, but then it starts to deform with time. After some time the material thus exhibits flow for some materials, the strain can even linearly increase with time (as depicted). When the stress is released, the material instantaneously loses some of it deformation (which is called elastic recovery), and then the deformation decreases ever slower (delayed elasticity), until a constant value is obtained. Part of the deformation is thus permanent and viscous. The material has some memory of its original shape but tends to forget more of it as time passes. 

The Boltzmann superposition principle is one of the simplest but most powerful principles of polymer physics.2 We have previously defined the shear creep compliance as relating the stress and strain in a creep experiment. Solving equation (2-6) for strain gives... 

These corrections to the elastic constants were calculated using the above equation and compared with those determined numerically. The numerical approach was to apply small stresses (Sj Voigt notation) of + 0.2 GPa and —0.2 GPa with the elastic compliances related to the resulting strains, e, to e6, by... 

In the case that the macroscopical uniform stress o, 0°, 0° is acting at far field of the crack homogenization model, the macroscopical uniform strain °, °, ° is generated. The relation between the strain and the stress is expressed by using compliance A,ju] as... 

The minus sign in Eqs. (5.10) and (5.11) is the consequence of the strain lagging behind the stress. It can be shown that, for a viscoelastic liquid, the dynamic creep compliances are related to the creep compliance J(t) by the one-sided Fourier transforms ... 

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