Strain increment

But the two curves still do not exactly match, as Fig. 8.7 shows. The reason is a displacement of (for example) u = l f2 in tension and compression gives different strains) it represents a drawing out of the tensile specimen from 1q to 1.5 1q, but a squashing down of the compressive specimen from /q to 0.5/q. The material of the compressive specimen has thus undergone much more plastic deformation than the material in the tensile specimen, and can hardly be expected to be in the same state, or to show the same resistance to plastic deformation. The two conditions can be compared properly by taking small strain increments... 

The increase in stress needed to produce further strain in the plastic region. Each strain increment strengthens or hardens the material so that a larger stress is needed for further strain. 

Stress relaxation. In a stress-relaxation test a plastic is deformed by a fixed amount and the stress required to maintain this deformation is measured over a period of time (Fig. 2-33) where (a) recovery after creep, (b) strain increment caused by a stress step function, and (c) strain with stress applied (1) continuously and (2) intermittently. The maximum stress occurs as soon as the deformation takes place and decreases gradually with time from this value. From a practical standpoint, creep measurements are generally considered more important than stress-relaxation tests and are also easier to conduct. 

The initial strain eu is changed at time (, to c, and the stress is the sum of that induced by the separate strain increments. 

Figure 7.3. Determination of elastic and viscous components. Incremental stress-strain curve constructed by stretching a specimen in strain increments of 2 to 5% and allowing the specimen to relax to an equilibrium stress before an additional strain increment is added. The elastic fraction is defined as the equilibrium stress divided by the initial stress. (Adapted from Silver, 1987.)...

The orientation of the two components of a compatible (1), amorphous 50% NC blend is shown in Figure 4. The films have been strained to 25, 50, and 75% in successive cycles, with each strain increment followed by a relaxation to zero stress. The orientation of the PCL and NC have been followed by using the carbonyl (1,728 cm 1) and N02 (1,660 cm 1) stretching peaks, respectively. 

Volume fraction of CNT (%) % increment of Young s modulus % increment of strain % increment of toughness... 

A cup could be designed that would apply equal strain increments to all elements of the material without gross movement of the materials within the cups, or... 

Neither of these extreme situations is feasible. For case 1 no practically conceivable cup shape could produce equal strain increments and in case 2 a material with the necessary degree of plasticity would normally be incompatible with a potential need to develop adequate pressing load because the material could be extruded from between the pockets at relatively low pressure. Alternatively, the product specification may exclude modification of the material or it is impossible to remove the plasticizing constituents after briquetting if... 

Division of the strain increment by the stress increment yields ... 

This idea can be used to formulate an integral representation of linear viscoelasticity. The strategy is to perform a thought experiment in which a step function in strain is applied, e t) = Cq H t), where H t) is the Heaviside step function, and the stress response a t) is measured. Then a stress relaxation modulus can be defined by E t) = <7(t)/ o Note that does not have to be infinitesimal due to the assumed superposition principle. To develop a model capable of predicting the stress response from an arbitrary strain history, start by decomposing the strain history into a sum of infinitesimal strain increments ... 

In the limit as the number of strain increments goes to infinity, the stress response becomes... 

Heats of formation are calculated as a sum of the bond energies and other stabilizing and destabilizing (e.g., strain) increments for the structure. MM4 calculations include terms for contributions of higher-energy conformations. For a set of hydrocarbons ranging from methane and ethane to adamantane and bicyclo[2.2.2]octane, the heats of formation are calculated with a standard deviation of 0.353 kcal/mol. The MM4 system has also been applied to alkenes, ° aldehydes, and ketones. ... 

If the viscous strain of Kelvin model is f at to t = to + Ar, and the stress remains constant during the Al time increment, the viscous strain increment of the generalized Kelvin model during At can be derived from equation (4) as... 

The initial strain increment includes thermal strain increment and creep increment, that is ... 

After the creep strain increment is calculated, the corresponding creep stress increment can be solved using finite element method. Song (2002). 

Apart from yield criterion, one is interested in the constitutive relations. In the elastic constitutive relation, the stress is related to strain however in the plastic constitutive relation stress can be related to strain-rate or strain-increment. In 1872, M. Levy used an incremental constimtive equation, which was later proposed by von Mises. Levy s paper was not known outside France. Levy-Mises relation considers that the increments of plastic strain increments are in proportion to deviatoric components, i.e.. 

A strain ei is applied to a Maxwell element at time ti and a second strain increment 62 is applied at time t2- Without assuming the applicability of the BSP, calculate the stress in the element immediately after the application of 62 and hence calculate the stress at any time t > ti- Show that this stress is the same as would have been calculated by assuming applicability of the BSP. 

Figure 6. Contours of volumetric strain increment of surrounding rocks when working face advances 40 m.

As mentioned above, the calculatimi for a given strain increment is the key to calculate the backstress-tensor a for a defined strain path because... 

Plastic flow in simple tension, (b) a strain increment, and (c) successive increments of displacement... 

It is useful and instructive to consider conditions of uniform finite straining where integration of strain increments can be performed analytically. Uniform straining is taken to mean that the principal axes of strain do not rotate. Plastic flow in a simple tensile test (Fig. 3a) prior to the onset of necking fulfils this condition. 

Initially, the original gauge length 1q increases by an infinitesimal increment 81. This increment is represented graphically in Fig. 3b, whence strain increment can be formed as... 

For subsequent equal displacements 81 (Fig. 3c), corresponding strain increments can be formed ... 

An invariant function of the strain increments similar to a can be derived which when applied to simple tension reduces to the longitudinal strain ... 

In order for a theory of this type to be useful it is necessary that it is not only descriptive but also predictive. The Hill criterion offers such a possibility in that it allows a prediction of the plastic strain increments at yield, provided sufficient yield stresses have been determined to evaluate F, G, H, L, M and N explicitly. By making the conventional assumption of normality (see Ref. 5, Chapters 2 and 10), it follows that the components... 

We have now discussed in turn, the stresses required to produce yield, the relationship between stress and plastic strain increment, the structural reorientation occurring as a result of yield, and the relationship between constant strain-rate yield and features of non-linear recoverable creep deformation. Theoretical models to describe the behaviour have ranged from single crystal plasticity through to the oriented continuum ideas of plasticity and viscoelasticity. On many points both the experimental data and the interpretations appear almost contradictory and it is therefore helpful to see if any common ground can be established. 

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