Mechanical Properties of Engineered Materials 2008 Part 4 pps

The basic definitions of stress and strain are presented in this chapteralong with experimental methods for the measurement and application ofstrain and stress.. Simple experimental metho

The basic definitions of stress and strain are presented in this chapteralong with experimental methods for the measurement and application ofstrain and stress The chapter starts with the relationships between appliedloads/displacements and geometry that give rise to the basic definitions ofstrain and stress Simple experimental methods for the measurement ofstrain and stress are then presented before describing the test machinesthat are often used for the application of strain and stress in the laboratory.

The forces applied to the surface of a body may be resolved into nents that are perpendicular or parallel to the surface,Figs 3.1(a)–3.1(c) In

compo-FIGURE 3.1 Different types of stress: (a)] uniaxial tension; (b) uniaxial pression; (c) twisting moment (After Ashby and Jones, 1996 Courtesy ofButterworth Heinemann.)

com-cases where uniform forces are applied in a direction that is perpendicular tothe surface, i.e., along the direction normal to the surface, Figs 3.1(a) and3.1(b), we can define a uniaxial stress, , in terms of the normal axial load,

Pn, divided by the cross-sectional area, A This gives

¼Applied load (normal to surface)

cross-sectional area ¼Pn

It is also apparent from the above expression that stress has SI units ofnewtons per square meter (N/m2) or pascals (Pa)] Some older texts andmost engineering reports in the U.S.A may also use the old English units

of pounds per square inch (psi) to represent stress In any case, uniaxialstress may be positive or negative, depending on the direction of appliedload Figs 3.1(a) and 3.1(b) When the applied load is such that it tends tostretch the atoms within a solid element, the sign convention dictates thatthe stress is positive or tensile, Fig 3.1(a)] Conversely, when the appliedload is such that it tends to compress the atoms within a solid element, theuniaxial stress is negative or compressive, Fig 3.1(b) Hence, the uniaxialstress may be positive (tensile) or negative (compressive), depending on thedirection of the applied load with respect to the solid element that is beingdeformed

Similarly, the effects of twisting [Fig 3.1(c)] on a given area can becharacterized by shear stress, which is often denoted by the Greek letter,and is given by:

Applied load (parallel to surface)cross-sectional area ¼Ps

Shear stress also has units of newtons per square meter square (N/m2)

or pounds per square inch (psi) It is induced by torque or twisting momentsthat result in applied loads that are parallel to a deformed area of solid, Fig.3.1(c) The above definitions of tensile and shear stress apply only to caseswhere the cross-sectional areas are uniform

More rigorous definitions are needed to describe the stress and strainwhen the cross-sectional areas are not uniform Under such circumstances, it

is usual to define uniaxial and shear stresses with respect to infinitesimallysmall elements, as shown in Fig 3.1 The uniaxial stresses can then be defined

as the limits of the following expressions, as the sizes, dA, of the elementstend towards zero:

¼ lim

dA!0

PndA

 

ð3:3Þ

dA!0

PsdA

 

ð3:4Þ

where Pn and Ps are the respective normal and shear loads, and dA is thearea of the infinitesimally small element The above terms are illustratedschematically inFigs 3.1(a) and 3.1(c), with F being equivalent to P.Unlike force, stress is not a vector quantity that can be describedsimply by its magnitude and direction Instead, the general definition ofstress requires the specification of a direction normal to an area element,and a direction parallel to the applied force Stress is, therefore, a secondrank tensorquantity, which generally requires the specification of two direc-tion normals An introduction to tensor notation will be provided inChap 4.However, for now, the reader may think of the stress tensor as a matrix thatcontains all the possible components of stress on an element This conceptwill become clearer as we proceed in this chapter

The state of stress on a small element may be represented by gonal stress components within a Cartesian co-ordinate framework (Fig.3.2) Note that there are nine stress components on the orthogonal faces of

ortho-FIGURE3.2 (a) States of Stress on an Element, (b) positive shear stress and (c)Negative shear stress Courtesy of Dr Seyed M Allameh

the cube shown inFig 3.2 Hence, a 3
3 matrix may be used to describeall the possible uniaxial and shear stresses that can act on an element Thereader should note that a special sign convention is used to determine thesuffixes in Fig 3.2 The first suffix, i, in the ij or ij terms corresponds tothe direction of normal to the plane, while the second suffix, j, corresponds

to the direction of the force Furthermore, when both directions are tive or negative, the stress term is positive Similarly, when the direction ofthe load is opposite to the direction of the plane normal, the stress term isnegative

posi-We may now describe the complete stress tensor for a generalizedthree-dimensional stress state as

½  ¼ yxxx xyyy xzyz

24

3

Note that the above matrix, Eq (3.5), contains only six independent termssince ij ji for moment equilibrium The generalized state of stress at apoint can, therefore, be described by three uniaxial stress terms ( xx, yy,

i, can beestimated from expressions of the form:

i  E
T ¼ Eið
1
2ÞðT  T0Þ ð3:22Þwhere Ei is the Young’s modulus in the i direction,
is the thermal expan-sion coefficient along the direction i, subscripts 1 and 2 denote the twomaterials in contact, T is the actual/current temperature, and T0 is thereference stress-free temperature below which residual stresses can build

up Above this temperature, residual stresses are relaxed by flow processes.Interfacial residual stress considerations are particularly important inthe design of composite materials This is because of the large differencesthat are typically observed between the thermal expansion coefficients ofdifferent materials Composites must, therefore, be engineered to minimizethe thermal residual strains/stresses Failure to do so may result in cracking

if the residual stress levels are sufficiently large Interfacial residual stresslevels may be controlled in composites by the careful selection of compositeconstituents that have similar thermal expansion coefficients However, this

is often impossible in the real world It is, therefore, more common forscientists and engineers to control the interfacial properties of composites

by the careful engineering of interfacial dimensions and interfacial phases tominimize the levels of interfacial residual stress in different directions

Let us now consider the simple case of a two-dimensional stress state on anelement in a bar of uniform rectangular cross sectional area subjected touniaxial tension,[Fig 3.6] If we now take a slice across the element at anangle,, the normal and shear forces on the inclined plane can be resolvedusing standard force balance and basic trigonometry The dependence of thestress components on the plane angle,, was first recognized by Oligo Mohr

He showed that the stresses, x0x0, y0y0, x0 y0 along the inclined plane aregiven by the following expressions:

ð3:24Þand

C ¼ xxþ yy

Note the sign convention that is used to describe the plane angle, 2y,and the tensile and shear stress components in Fig 3.7 It is important toremember this sign convention when solving problems involving the use ofMohr’s circle Failure to do so may result in the wrong signs or magnitudes

of stresses The actual construction of the Mohr’s circle is a relatively simpleprocess once the magnitudes of the radius, R, and center position, C, havebeen computed using Eqs (3.24) and (3.25), respectively Note that the locus

of the circle describes all the possible states of stress on the element at thepoint, P, for various values of between 08 and 1808 It is also important tonote that several combinations of the stress components ( xx, yy, xy) mayresult in yielding, as the plane angle,, is varied These combinations will bediscussed inChap 5

When a generalized state of triaxial stress occurs, three Mohr’s circles[Fig 3.8(a)] may be drawn to describe all the possible states of stress Thesecircles can be constructed easily once the principal stresses, 1, 2, and 3,

FIGURE 3.6 Schematic of stresses on a plane inclined across an element.Courtesy of Dr Seyed M Allameh

of a solid For the simple case of a uniaxial displacement of a solid with auniform cross-sectional area[Fig 3.3(a)], the axial strain,&# 34; , is can be definedsimply as the ratio of the... thecalculation of stresses and strains in many practical problems Nevertheless,the reader should retain a picture of the generalized state of stress on anelement, as we develop the basic concepts of mechanical