Linear elastic limit

Reiterer et al. (1999) observed that their 200 p,m thick specimens stretched considerably once the elastic limit had been exceeded, but only where the MFA is quite large (> 15-20°). Then it stretches even further for each increment of strain - in this region, beyond the linear elastic limit, the material deforms irrecoverably by viscoelastic or plastic flow. Finally the sample breaks in tension. The strength of the material, i.e. the failure stress, is read from the y-axis. The stiffness of all woods ranges from 0.5-20 GPa and strength ranges from 1-40 MPa, from the corewood of low density species to the outerwood of very dense species. 

One reason that the loop curves are bent downward so that the slope increases as the strain increases is obviously the strain-stiflFening eflFect. As the specimen is stretched beyond the linear elastic limit, the modulus of the material and hence the slope of the curve increases with increasing strain. 

The preceding equations provided a reasonable foundation for predicting DE behavior. Indeed the assumption that DEs behave electronically as variable parallel plate capacitors still holds however, the assumptions of small strains and linear elasticity limit the accuracy of this simple model. More advanced non-linear models have since been developed employing hyperelasticity models such as the Ogden model [144—147], Yeoh model [147, 148], Mooney-Rivlin model [145-146, 149, 150] and others (Fig. 1.11) [147, 151, 152]. Models taking into account the time-dependent viscoelastic nature of the elastomer films [148, 150, 151], the leakage current through the film [151], as well as mechanical hysteresis [153] have also been developed. 

The importance of inherent flaws as sites of weakness for the nucleation of internal fracture seems almost intuitive. There is no need to dwell on theories of the strength of solids to recognize that material tensile strengths are orders of magnitude below theoretical limits. The Griffith theory of fracture in brittle material (Griflfith, 1920) is now a well-accepted part of linear-elastic fracture mechanics, and these concepts are readily extended to other material response laws. 

Linear-elasticity, of course, is limited to small strains (5% or less). Elastomeric foams can be compressed far more than this. The deformation is still recoverable (and thus elastic) but is non-linear, giving the plateau on Fig. 25.9. It is caused by the elastic... 

The basic assumptions of fracture mechanics are (1) that the material behaves as a linear elastic isotropic continuum and (2) the crack tip inelastic zone size is small with respect to all other dimensions. Here we will consider the limitations of using the term K = YOpos Ttato describe the mechanical driving force for crack extension of small cracks at values of stress that are high with respect to the elastic limit. 

The simplified failure envelopes differ little from the concept of yield surfaces in the theory of plasticity. Both the failure envelopes (or surfaces) and the yield surfaces (or envelopes) represent the end of linear elastic behavior under a multiaxial stress state. The limits of linear elastic... 

When required, combined with the use of computers, the finite element analysis (FEA) method can greatly enhanced the capability of the structural analyst to calculate displacement and stress-strain values in complicated structures subjected to arbitrary loading conditions. In its fundamental form, the FEA technique is limited to static, linear elastic analysis. However, there are advanced FEA computer programs that can treat highly nonlinear dynamic problems efficiently. 

When an engineering plastic is used with the structural foam process, the material produced exhibits behavior that is easily predictable over a large range of temperatures. Its stress-strain curve shows a significantly linearly elastic region like other Hookean materials, up to its proportional limit. However, since thermoplastics are viscoelastic in nature, their properties are dependent on time, temperature, and the strain rate. The ratio of stress and strain is linear at low strain levels of 1 to 2%, and standard elastic design... 

The above results are derived from linear elastic fracture mechanics and are strictly valid for ideally brittle materials with the limit of the process zone size going to zero. In order to apply this simple framework of results, Irwin (1957) proposed that the process zone, r be treated as an effective increase in crack length, Sc. With this modification, the fracture toughness becomes... 

The resulting stress was measured, and a discrete Fourier transform was performed to obtain the elastic and viscous moduli. The experimental variables in FTMS are the fundamental frequency, f, and the strain amplitudes, Yi, at each frequency, i. Each of the other frequencies are harmonics (integer multiples) of the fundamental frequency. The fundamental frequency was set at 1 rad/s, while the harmonics were chosen to be 2, 5, 10, 25, 40, 50, and 60 rad/s. The summation of the strain amplitudes at each frequency was below the linear viscoelastic limit of the NOA 61 sample. 

These results indicate that in the present linear elastic model, the limiting velocity for the screw dislocation will be the speed of sound as propagated by a shear wave. Even though the linear model will break down as the speed of sound is approached, it is customary to consider c as the limiting velocity and to take the relativistic behavior as a useful indication of the behavior of the dislocation as v — c. It is noted that according to Eq. 11.20, relativistic effects become important only when v approaches c rather closely. 

The point at which the stress-strain curve ceases to be linear and at which plastic deformation begins to occur. Also known as the elastic limit. 

The mathematical treatment of a three-dimensional generalization of this linear elastic model is more complex. In this case the angular deformation is not limited to the on-plane bending, but also off-plane twisting takes place. This model is depicted schematically in Fig. 38. 

It must be noted that the fracture mechanics framework described above only applies when plastic deformation of the material is limited. Substantial plastic deformation may accompany propagation of existing defects in structures fabricated from relatively low-strength materials, e.g., carbon steels. In these cases, the linear elastic stress intensity factor, K, does not accurately apply in structural design. Alternately, elastic-plastic fracture mechanics methods may apply. ... 

We are all familiar with a typical stress-strain curve as depicted in Figure 14.1. As we know, almost all materials behave linearly. That is, the strain (amount of distortion) produced in a material is directly proportional to the stress placed on the material. This linear behavior holds until a point at which the material, if released, will not return to its exact original shape or dimensions. This point is called the yield point or elastic limit. When we strain a material beyond its elastic limit, we cause plastic deformation. 

This means that in the elastic region, pressure and density are linearly related. Beyond the elastic region, the wave velocity increases with pressure or density and Pip is not linearly proportional. Wave velocity continues to increase with stress or pressure throughout the region of interest. Therefore, up to the elastic limit, the sound velocity in a material is constant. Beyond the elastic limit, the velocity increases with increasing pressure. Let us look at a major implication of this fact. Consider the pressure wave shown in Figure 14.3. 

There have been a number of studies that demonstrate that crystallized AMF and butter exhibit linear (ideal) viscoelastic behavior at low levels of stress or strain (4), where the strain is directly proportional to the applied stress. For most materials, this region occurs when the critical strain (strain where structure breaks down) is less than 1.0%, but for fat networks, the strains typically exceed 0.1% (4, 66). Ideally, within the LVR, mUkfat crystal networks will behave like a Hookean solid where the stress is directly proportional to the strain (i.e., a oc y), as shown in Figure 15 (66, 68). Within the elastic region, stress will increase linearly with strain up to a critical strain. Beyond that critical strain (strain at the limit of linearity), deformation of the network will occur at a point known as the yield point. The elastic limit quickly follows, beyond which permanent deformation and sample fracture occurs. Beyond these points, the structural integrity of the network is compromised and the sample breaks down. 

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