Maximum elastic strain

Noting that the maximum elastic strain energy ... 

The energy dissipated can be compared with the maximum elastic strain energy, W, which is stored in the material during the stress cycle. Because the elastic strain is proportional to the applied stress, W is equal to just half of the product of the maximum stress and strain (i.e., W = a0ei/2), and therefore... 

A shortening in relaxation time in the critically strained region makes some materials tough. The shift of relaxation time is attributed to strain-induced dilatation and can reach as much as five decades. Thermal history, on the other hand, dictates the initial state from which this dilatation starts and may be expressed in terms of excess entropy and enthalpy. The excess enthalpy at Tg is measurable by differential scanning calorimetry. Brittle to ductile transition behavior is determined by the strain-induced reduction in relaxation time, the initial amount of excess entropy, and the maximum elastic strain that the material can undergo without fracturing or crazing. 

The maximum strain energy corresponding to a maximum elastic strain that <5,53Since represents a reference level of the Lagrangian corresponding to an optimum state of strains, it can be assumed to be zero for convenience. is a constant indepen-... 

The maximum elastic strain theory (St. Venant s theory) states that inception of failure is due if the largest local strain, 3, within the material exceeds somewhere a critical value e. The failure criterion, therefore, is derived as... 

Problem Determine material specifications for a design of a shock absorber for a design that requires 63% of the elastic strain to be delayed (retarded) 2.0 s. Assume a Kelvin solid-type material behavior and assume the maximum elastic strain is limited to 0.01 when a constant stress of 1000 psi is applied. 

We imagine a finite-duration shock pulse arriving at some point in the material. The strain as a function of time is shown as the upper diagram in Fig. 7.11 for elastic-perfectly-plastic response (solid line) and quasi-elastic response generally observed (dash-dot line). The maximum volume strain = 1 - PoIp is designated... 

The elastic stress curve in figure perfectly follows elastic strain [2]. This constant is the elastic modulus of the material. In this idealized example, this would be equal to Young s modulus. Here at this point of maximum stretch, the viscous stress is not a maximum, it is zero. This state is called Newton s law of viscosity, which states that, viscous stress is proportional to strain rate. Rubber has some properties of a liquid. At the point when the elastic band is fully stretched and is about to return, its velocity or strain rate is zero, and therefore its viscous stress is also zero. 

At the instant of contact between a sphere and a flat specimen there is no strain in the specimen, but the sphere then becomes flattened by the surface tractions which creates forces of reaction which produce strain in the specimen as well as the sphere. The strain consists of both hydrostatic compression and shear. The maximum shear strain is at a point along the axis of contact, lying a distance equal to about half of the radius of the area of contact (both solids having the same elastic properties with Poisson s ratio = 1/3). When this maximum shear strain reaches a critical value, plastic flow begins, or twinning occurs, or a phase transformation begins. Note that the critical value may be very small (e.g., in pure simple metals it is zero) or it may be quite large (e.g., in diamond). 

Clarity requires that a distinction be made between elastic strain and plastic deformation. They both have units of length/length, but they are physically different entities. Elastic strain is recoverable (conservative) plastic deformation is not (non-conservative). At a dislocation core, where atoms exchange places via shear, the plastic displacement gradient is a maximum as it passes from zero some distance ahead of the core, up to the maximum, and then back to zero some distance back of the core. In crystals with distinct bonds, the gradient becomes indefinite (infinite) at the core center. 

The maximum shear strain criterion is now applied for the calculation of the creep curve up to fracture for increasing creep stress. The total creep strain of the fibre, q(f), is the sum of the elastic strain, cf, and the viscoelastic plus plastic strain, cj(f),... 

Let us consider also the regularities common for different types of extension. Dependencies of extension a upon elastic strain a are given in Fig. 5. Continuous lines in Fig. 5 indicate dependencies a(a) in extension at different constant strain velocities x. The higher x, the higher passes the dependency a(cx). The points with maximum a... 

In a typical indentation experiment the indenter is pressed onto the surface under investigation and the load is successively increased up to a certain maximum load. In the so-called compliance approach both load and indenter displacement are recorded and plotted as a load-displacement curve, the so-called compliance curve. If the experiment is exclusively run in the compressive load regime, the curve is also referred to as the load-penetration curve. Upon loading, elastic deformations occur succeeded by plastic ones. Upon releasing the imposed stress, elastic strain recovers immediately. 

See Curve A in Figure 1.) The maximum possible shift in relaxation time is determined by the limiting elastic strain, beyond which dilatation no longer continues. 

Classical theories of failure are based on concepts of maximum stress, strain, or strain energy and assume that the material is homogeneous and free from defects. Stresses, strains, and strain energies are typically obtained through elastic analyses. 

Elastic strain gages have a linearity within 1% for 10% of the maximal extension. For extensions of up to 30% of the maximum, the non-linearity increases to 4%. Another problem is the dead band or initial non-linearity due... 

These moments can be combined in a variety of ways depending on the criterion of operation wide variations exist between answers derived from the different formulae. The authors of the FMP Shaft Design Guide recommend that the maximum elastic shear strain energy theory is used. This results in an equivalent bending moment Af, c given by ... 

In another reference Sachs also suggests the use of the maximum elastic shear strain energy but uses a formula which generally gives different values of Mbe than equation (13.8), i.e ... 

Figure 4.30P 1 shows the compressive stress-strain curves for latex-modified mortars. Generally, the maximum compressive strain at failure increases with rising polymer-cement ratio, even though there is no pronounced change in the modulus of elasticity in compression. The maximum compressive strain at a polymer-cement ratio of 20% increases to 2 to 3 times that of unmodified mortar. 

Figures 4.31 and 4.321 1 represent the tensile stress-strain curves and the polymer-cement ratio-elon tion (i.e., maximum tensile strain at failure) relationships of SBR-modified concretes respectively. As seen in these figures, as the polymer-cement ratio is raised, die modulus of elasticity in tension decreases, and the elongation increases and is 2 to 3 times greater than that of unmodified concrete. This is explained by considering that the polymer films formed in the concrete may effectively halt propagating microcracks through their high tensile strength and elongation.

Where Young s modulus is needed in the above stress anafysis, the tensile creep modulus is inserted, using the value for the appropriate temperature and duration of loading, and for the maximum tensile strain in the component (as determined by the linear elastic anafysis). 

The maximum permissible deformation of the plastic hub in an annular snap-fit joint is limited by the maximum permissible strain or proportional hmit of the material. This limit is typically 50 percent of the strain at break for most reinforced plastics. It can be upwards of 60 to 70 percent of strain at break for more elastic polymers. 

Analogous to principal stresses, there are principal strains acting on the principal planes of the strains. Also as in the principal stresses, the shear strains on the principal planes of those strains equal zero, i.e., the normal strains on these planes are actually the principal strains. Following convention, the maximum principal strain of the three is called the major principal strain , while the smallest strain is known as the minor principal strain . In an isotropic elastic material, the principal planes of strain coincide with the principal planes of stress. In a manner similar to that in Eqs. (1.22e) and (1.23), it is possible to write the following for the strain ... 

---