Pdf stress and strain

The stress is said to be direct stress when the area being stressed is at right angles to the line of action of the external forces, as when the material is in tension or compression.

Example A bar of material with a cross-sectional area of 50 mm2is subject to tensile forces of N. What is the tensile stress? Stress and strain 33 Example A pipe has an outside diameter of 50 mm and an inside diameter of 45 mm and is acted on by a tensile force of 50 kN.

What is the stress acting on the pipe? The cross-sectional area of the pipe is! An - d2 , where D is the external diameter and d the internal diameter. Thus we might, for example, have a strain of 0. This would indicate that the change in length is 0. Figure 3. When it is subject to tensile forces it increases in length by 0. What is the strain? This increases by force applied A 0. This is termed shear. Shear stresses are not direct stresses since the forces Figure 3.

With shear, the area over which forces act is in the same plane as the line of action of the forces. With tensile and compressive stresses, changes in length are produced; with shear stress there is an angular change 4. Example Figure 3. The area of the adhesive in contact with the component is mm2. The weight of the component results in a force of 30 N being applied to the adhesive-component interface.

The material is said to be elastic. If measurements are made of the extension at different forces and a graph plotted, then the extension is found to be proportional to the force and the material is said to obey Hooke's law.

Such a graph applies to only one particular length and cross-sectional area of a particular material. We can make the graph more general so that it can be applied to other lengths and cross-sectional areas of the same material by dividing the extension by the original length to give the strain and the force by the cross-sectional area to give the stress Figure 3. Then we have, for a material that obeys Hooke's law, the stress proportional to the strain: stress strain Figure 3.

Initially the graph is a straight line and the material obeys Hooke's law. The point at which the straight line behaviour is not followed is called the limit of proportionality. With low stresses the material springs back completely to its original shape when the stresses are removed, the material being said to be elastic. At higher forces this does not occur and the material is then said to show some plastic behaviour.

The term plastic is used for that part of the behaviour Figure 3. This point often coincides with the point on a stress-strain graph at which the graph stops being a straight line, i. Limit of Sample breaks proportionality g!

The term tensile strength is used for the maximum value of the tensile stress that a material can withstand without breaking, i. Strengths are often millions of pascals and so MPa is often used, 1 h4Pa being Pa or 1 Pa. Typically, carbon and low alloy steels have tensile strengths of to MPa, copper alloys 80 to l MPa and aluminium alloys to MPa. With some materials, e. Then, it has an almost constant value up to 20 mm and increases for l higher than 20 mm.

The radius 2 mm corresponds to the reference model. These two errors have a sharp drop in the beginning of the test and they stabilize thereafter.

The sharp drop corresponds to the dynamic homogenisation process. Contrarily to results of the previous section, the increase of the radius q tends to rise the errors on strain measurements. This is for the same reasons Fig. That the time needed by the wave to travel a characteristics distance. On the cal heterogeneity remains.

Both parameters rise with the radius q see also Fig. Indeed, when the radius in- creases, the loading wave has to do longer distance and the stress homogenisation process takes more time. This distance is equal to 9. We should notice that the increase in nr and ar with the radius q is smooth. The average value of nr increases from 1. On the other hand, the average value of ar rises to 1. Six values are considered: 0. The case of 4-mm-width geometry corresponds to the reference model.

We plot in Fig. It is equal to For the highest values heterogeneous in the whole central and transition zones. Therefore, the cross-sectional areas of the tran- see Fig. The constant value equals The average value [nr] is also plotted in Fig.

It is almost constant with the width w. It varies between 1. This gives a quasi-constant characteristic distance d Fig. Nevertheless, the geometrical induced heterogeneity is slightly different. This gives an average value [ar] which vary slightly with width w Fig. No clear tendency can be observed. Then it goes down to reach 0. Subsequently, it increases to reach 2. Then, this value equals 1. That is why we compute the tion of the transition zones, rises when increasing the width w.

The drop is rather norm. The same conclusions can be made for eters are highly correlated. The time average values, [nr] and [ar], are plotted in Fig. However, there is sharp drop between 1 and 5 ls. The and geometrical parameters. It is worth to notice that sh depends on sr. The time needed to achieve a constant loading velocity is called here rise time.

The time sh 1 is approximated We plot in Fig. Thus shown. We should notice here the second kind of behaviour. The error on the stress measure is shown in Fig. Two kinds of behaviour are observed. The second ing velocity. We considered six values: 0.

Indeed, the stress homog- enisation process occurs almost at the same time see the variation of d in Fig. However, at high loading velocity high strain rates larger strains are achieved before the time sh Fig. That is why, the error nr increases with the loading velocity. Discussion In this section we will discuss separately the accuracy of strain and stress measurement. It assumes that defor- mation only occurs at the central zone. Hence, only the length of this zone is considered in Eq.

The second measure is newly pro- posed in this paper and is given in Eq. Subsequently, they take an almost con- This corresponds to a nominal strain rate ranging from to stant value. Hence, we plot in Fig. A more pronounced lin- ear behaviour can be observed. Regarding this study, we suggest that only the second measure should ne used. Stress measurement In this study, we considered the conventional stress measure, assuming that the applied load is instantly transmitted to any Fig.

The error on the stress measure generally decreases, with time, while oscillating. Indeed, the second and third error on- of the stress measure. The favor- see Sections 3. Assuming this case, we study, in Fig. There- iation of the error on the stress measure in terms of the dimension- fore, we plot in Fig. An almost linear tendency is The wave velocity C0 and test duration s0 are introduced to take into observed.

In this case, the tively. Dynamic behaviour of high-strength sheet steel in dynamic tension: experimental and numerical analyses.

J Strain Anal Eng Des ;— For this purpose, we hydraulic machine for testing at intermediate strain rates. Int J Impact Eng ; The dynamic tensile behavior of tough, ultrahigh- ters, on the strain and stress errors, is investigated. In this study, strength steels at strain-rates from 0.

Int J Impact Eng we considered two strain measures. Quasi static and dynamic shearing of and the second is new. Torsional split Hopkinson bar tests at strain rates above always give better results than the conventional one.

Exp Mech ;—9. Int J Plast ;— Secondly, shear testing of adhesively bonded assemblies. Int J Modern Phys B ;—6. In this frame- for various steel sheets. Int J Mater Form —6. Biaxial testing of sheet materials at high strain rates using viscoelastic bars.

Exp Mech ; The dynamic compression testing of solids by the method of the split Hopkinson pressure bar. J Mech Phys Solids ;— References [24] Gorham DA. Specimen Inertia in high strain rate compression. J Phys D Appl Phys ;— The effect of specimen dimensions on high strain rate [1] Zhao H. Material behaviour characterisation using SHPB techniques, tests and compression measurements of copper.

Comput Struct ;— Kinematic analysis consequently interprets geometric observations and appeals to geometric assumptions. For example, 2. Both and an ellipsoidal ooid is assumed to have been spheri- are most generally expressed in analytical terms as cal before deformation. Kinematic problems structures. However, analogous is prerequisite to dynamic analysis, but dynamic analy- dynamic questions must be posed as inverse problems sis of geometric features made without an intervening e.

As a kinematic interpretations of the geometric features. Indeed, graphical methods appeals to rheological and environmental assumptions. Wojtal, ; Marrett and For example, brittle fault striae are interpreted kine- Allmendinger, Additionally, kinematic solutions matically as the slip direction, which in turn is to problems are typically unique e. To the extent that kinematic interpretations in uniqueness makes kinematic analysis more amenable and rheological assumptions are valid, dynamic analy- to intuitive insight and understanding than is com- sis genetically explains structures.

In summary, strain and all other kinematic quan- tities descriptively interpret what movements produced the structures; stress and all related dynamic quantities 3. Terminology genetically interpret why the structures formed. These conclusions conform to the thinking of Sander , Confusion in the terminology, and possibly the phil- p. Ramsay, ible foundation for the genetic dynamic consideration , p. Without knowledge appear to be a meaningless subtlety or a careless error.

Stress terms connote stresses do not superpose in time, only in space. Below we address the use and misuse of The distinction between the descriptive interpret- a variety of structural terms and highlight their ations of kinematics and the genetic interpretations of descriptive and genetic implications. The genetic understanding of structure provided by dynamic analy- 3. Extension and contraction, tension and compression sis is deeper than the descriptive understanding that results from kinematic analysis, but this deeper under- Extension and contraction signify the strains that standing comes at a price.

The pression, respectively. Tectonic events should not, stresses with negative contractional elongations''. Furthermore, the sional' being used as opposites include: procedure used for the analysis i. Stress from petrofabrics and Mercier, ; 2.

A good overview of these Jeyakumaran and Keer, ; piezometric methods is given by Passchier and Trouw 5. To take one example, calcite twin lamellae can and fault motion Carey-Gailhardis and Mercier, be used to directly determine stress magnitudes, the ; assumptions being made that stress was homogeneous 6.

Passchier and Trouw indicate, however, that 7. This view was echoed by Burkhard , Robinson et al. The assumptions of paleostress analysis al.

In addition, the piezometric been applied to faults and fault bends by Wolfe et al. Similarly, the name pressure solution cleavage is a stress term 3. Special problems with tension commonly used to describe a structure. Kasahara, , p. Lacombe et al.

McKenzie, ; pression in all but the uppermost few hundred meters Marrett and Allmendinger, Paleostress of the crust Rubin, Current dynamic interpret- dynamic analyses of fault-slip data e. Gross and Engelder, Are the structures tive tension fracture would be a more appropriate but or the inferred stresses being characterized?

It is prefer- unwieldy genetic term than tension fracture. Strain is a change in length per unit fractures.

It seems unnecessary, and potentially mis- original length, while stress is a force per unit area. Strain terms are closely related to observations and McGrath and Davison, when the structure is consequently are most appropriate to describe the simply a vein.

Fracture propagation about the genesis of natural structures. Strain kin- ematic analyses have the following advantages over The terminology of fracture propagation modes stress dynamic analyses: a they are more directly re- modes I, II and III concerns the propagating tip of a lated to observed structures; b they are less computa- fracture Pollard and Segall, , and does not refer tionally and analytically intensive; and c they are to the geometry of the resultant fracture.

For example, more intuitive but shallower. Fracture mode dikes and normal faults should be particularly avoided terminology is only appropriate when fracture propa- because these structures almost always form in a com- gation and rock mechanics are addressed.

Transtension and transpression mixes description with genetic interpretation. The terms are often because they refer to measurable aspects of movement. Strain terms that could be used to describe structures Funding for Peacock was provided by a Natural formed under transpressional stress include oblique Environment Research Council ROPA award to Rob contraction, convergent transcurrence, or prolate trans- Knipe.

Marrett was supported by BDM-Oklahoma currence. Strain terms that could be used to describe subcontract G4S U. As with any description of strain, it is David Pollard, Dave Sanderson and Sue Treagus are important to state the orientations of the strain: e. This does not imply that they agree with all of the deformation described?

Tectonic analysis of fault slip data sets. Mesozoic and Cenozoic Knopf, E. Structural Petrology. Geological extension recorded by metamorphic rocks in the Funeral Society of America Memoir 6. Mountains, California. Biddle, K. In: Biddle, K. Society of Economic tal transform zone eastern France. Paleontologists and Mineralogists, pp. Laurent, P. Determining deviato- Billings, M.

Structural Geology. Prentice-Hall, New York. Dynamic basis of volcanic spreading. Journal of synthetic and natural polycrystals. Marrett, R. Kinematic analysis of fault- Burkhard, M.