Stress and Strain Calculations

Table 3. Critical Cracking Stresses and Strains Calculated from the Approach of Hu and Evans ...

In order to achieve better agreement between the stresses and strains calculated using Odemark s concept and those from the theory of elasticity, a correction factor f is applied to the above equation. Thus,... 

The proposed blood-myocardium composite model enables the regional analysis to be realistically implemented by application of the finite element method at reasonable computer cost. It permits not only regional stress and strain calculations but more importantly the in vivo quantification of myocardial fiber contraction of the heart during cardiac cycles. The diastolic and systolic fiber strains can be easily computed at any instant of a cardiac cycle by use of Eqs. (8) and (9) if the myocardial layer thickness t and the blood volume fraction/in that layer are known. Presently, the layer thickness t can be calculated using the three-dimensional displacement data of the implanted markers which can be monitored by the biplane or computer-aided tomographic technique, and the blood volume... 

The cyclic stress-strain curve can be used, for example, to perform finite element simulations of cyclic loadings. To simulate the complete experiment in the computer, it would be necessary to obtain information on the hardening of the material (isotropic and kinematic hardening) and to determine a material model that correctly describes it. This would be an extremely comphcated procedure. Furthermore, the entire number of cycles would have to be calculated, which would require an immense amount of computing time. Instead, the flow curve, taken by the finite element software to be monotonous, can be replaced by the cyclic stress-strain curve. A single, monotonous loading of the component is then simulated. Stresses and strains calculated in this way correspond well with those in the cyclically loaded component. 

One of the simplest ways to model polymers is as a continuum with various properties. These types of calculations are usually done by engineers for determining the stress and strain on an object made of that material. This is usually a numerical finite element or finite difference calculation, a subject that will not be discussed further in this book. 

Numerical simulation of a complex dynamic fracture application can be illustrated by calculations of impact induced damage in a ceramic cylinder. The computer model used was originally developed for oil shale explosive fragmentation (Grady and Kipp, 1980), with various extended applications considered by Boade et al. (1981) and Chen et al. (1983). In this model, stress and strain are related through... 

This linear relationship between stress and strain is a very handy one when calculating the response of a solid to stress, but it must be remembered that most solids are elastic only to very small strains up to about 0.001. Beyond that some break and some become plastic - and this we will discuss in later chapters. A few solids like rubber are elastic up to very much larger strains of order 4 or 5, but they cease to be linearly elastic (that is the stress is no longer proportional to the strain) after a strain of about 0.01. 

Example 3.10 If a moment of Af, = 100 Nm/m is applied to the unidirectional composite described in the previous Example, calculate the curvatures which will occur. Determine also the stress and strain distributions in the global and local (1-2) directions. 

Example 3.12 For the laminate [0/352/ - determine the elastic constants in the global directions using the Plate Constitutive Equation. When stresses of = 10 MN/m, o-y = —14 MN/m and = —5 MN/m are applied, calculate the stresses and strains in each ply in the local and global directions. If a moment of 10(X) N m/m is added, determine the new stresses, strains and curvatures in the laminate. The plies are each 1 mm thick. 

For the next interface, z — —4 mm, the new values of , Sy and yxy can be calculated and hence the stresses in the global and local co-ordinates. / = 1 and f = 2 need to be analysed for this interface but there will be continuity across the interface because the orientation of the plies is the same in both cases. However, at z = —3 mm there will be a discontinuity of stresses in the global direction and discontinuity of stresses and strains in the local directions due to the difference in fibre orientation in plies 2 and 3. 

The tensile modulus can be determined from the slope of the linear portion of this stress-strain curve. If the relationship between stress and strain is linear to the yield point, where deformation continues without an increased load, the modulus of elasticity can be calculated by dividing the yield strength (pascals) by the elongation to yield ... 

Texture has a number of component attributes, and some of them can be assessed by mechanical means. The texture or firmness of cooked potatoes is evaluated by subjecting each sample to a compression test using a universal testing machine equipped with a load cell. Cooked potato cylinders are compressed in a single-cycle compression-decompression test. Uniaxial compression is measured with an Instron machine with a lOON load cell. Measurements are performed on hot potato cylinders (depth 12 mm, height 10 mm) from 15 potatoes immediately after cooking, at a deformation rate of 20 mm/min. Stress and strain at fracture are calculated by the Instron series IX version 7.40 software and means of 15 repetitions are calculated. 

The temperature, fiber tension, stresses, and strains vary only in the radial directions. An elasticity solution is employed to calculate the six components of the stresses and strains. The solution procedure follows the established techniques of elasticity solutions. A displacement field is assumed that satisfies the equilibrium equations and the compatibility conditions. The latter requires that at each interface the displacements and the normal stresses in adjacent... 

It should be noted that the stresses usually used are engineering stresses calculated from the ratio of force and original cross section area whereas true stress is the ratio of the force and the actual cross sectional area at that deformation. Clearly, the relationship between stress and strain depends on the definition of stress used and taking the case of tensile strain, for example, the true stress is equal to the engineering stress multiplied by the extension ratio. 

With dumb-bells, it is assumed that stress and strain are uniform throughout the gauge length and, hence, the calculation of stress presents no difficulties. Modulus as such is not normally measured but the stress quoted for a given elongation. It is sometimes debated whether the mean or the minimum cross sectional area should be used for ultimate stress but whatever the arguments in favour of the minimum, it is rather difficult to measure this and the mean is normally used. 

As has been mentioned previously, rings present more of a problem because of non-uniformity of stress and strain. ISO 37 calculates strength at break from force divided by twice the cross-sectional area but this is not the true strength (see Section 5.1). The elongation at break is calculated on the increase in internal circumference on the assumption that failure starts at the internal, most highly stressed, surface. To be precise, a small correction... 

A compression stress/strain test is in many ways easier to carry out than a tensile test, and in view of the large number of applications of rubber in compression, should be more often used. Frequently, it would be logical for the test piece to be the complete product and a compressive force applied as it would be in service. Usually a constant rate of deformation would be appropriate and the force and corresponding deformation recorded without attempts at calculating the resultant stresses and strains. 

For many types of flow, calculations are complex, Where they can be made at all. Ihey require the methods of tensor analysis of the stresses and strains involved. One important relationship that is widely useful in die systematic study of fluid flow in the equation of continuity. 

What is involved in the calculation of modulus First, we mean by modulus in materials science a relation between stress and strain in a bulk sample under practical conditions. This means in effect a testing rate or frequency usually less than a few kilohertz or at the most in the ultrasonic region of, say, 10 MHz. We also suppose the sample to be a representative volume element of size suitable for the test method and we assume its elastic properties to be uniform over this RVE. 

Typically, in compression tests a cylindrical piece of the test sample is compressed between smooth plates using a Material Tester. Assuming constant volume, the stress and strain (Hencky strain) are calculated from the force, displacement data. However,... 

Extensive theoretical investigations devoted to calculation of residual stresses have been carried out for metals. The principal theme of this work is assumption that residual stresses and strains are the result of differences between pure elastic and elastic-plastic deformations under fixed loading.127 128 The same mechanism, i.e., the appearance of plastic deformed zones, is responsible for the residual stresses arising during crystallization of metals, which occurs on quenching from the melt or cooling after welding. 

The four variables in dynamic oscillatory tests are strain amplitude (or stress amplitude in the case of controlled stress dynamic rheometers), frequency, temperature and time (Gunasekaran and Ak, 2002). Dynamic oscillatory tests can thus take the form of a strain (or stress) amplitude sweep (frequency and temperature held constant), a frequency sweep (strain or stress amplitude and temperature held constant), a temperature sweep (strain or stress amplitude and frequency held constant), or a time sweep (strain or stress amplitude, temperature and frequency held constant). A strain or stress amplitude sweep is normally carried out first to determine the limit of linear viscoelastic behavior. In processing data from both static and dynamic tests it is always necessary to check that measurements were made in the linear region. This is done by calculating viscoelastic properties from the experimental data and determining whether or not they are independent of the magnitude of applied stresses and strains. 

For this purpose the "reduced" stress and strain are calculated from the following equations ... 

Fabrication of seam-welded aluminum tube for hydroforming Calculation of stress and strain... 

The first quantitative study of deformation mechanisms in ABS polymers was made by Bucknall and Drinkwater, who used accurate exten-someters to make simultaneous measurements of longitudinal and lateral strains during tensile creep tests (4). Volume strains calculated from these data were used to determine the extent of craze formation, and lateral strains were used to follow shear processes. Thus the tensile deformation was analyzed in terms of the two mechanisms, and the kinetics of each mechanism were studied separately. Bucknall and Drinkwater showed that both crazing and shear processes contribute significantly to the creep of Cycolac T—an ABS emulsion polymer—at room temperature and at relatively low stresses and strain rates. 

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