Abstract
“Strength” of a material can be interpreted multifariously. Strength as related to “ease” or “difficulty” of mechanical deformation is the focus of this chapter. Along the way a whole series of corresponding strength parameters is presented and discussed. The basic elastic, i.e., reversible, and plastic, i.e., permanent, deformation modes are presented. The treatment of elasticity incorporates the phenomenon of rubber elasticity (elastomeric behavior) as well as viscoelasticity/anelasticity (mechanical hysteresis). The treatment of plasticity involves an extensive discussion of the tensile stress–strain curve and the role of dislocations in plastic deformation. The yielding criteria of Tresca and von Mises, for cases of more than uniaxial loading, are presented and discussed. The plastic deformation of a single crystal, encompassing the definition of critical resolved shear stress, and its relation to the plastic deformation of polycrystals are dealt. Next to the discussion of classical hardness parameters, an emphasis is put on nanoindentation and hardness profiling on nanoscale. Ways to strengthen materials by microstructural manipulation are presented: strain hardening, reduction of grain size (the role of the Hall–Petch relation (and its (in)validity for nanosized materials) is dealt with), solid solution hardening and precipitation/dispersion hardening. The chapter ends with the consideration of material failure by crack propagation, creep and fatigue and the role of residual, internal stress. Some practical examples illustrate the theoretical considerations; for example, the combined effect of creep and residual stress gradient to explain the whiskering phenomenon (e.g., of thin films in the microelectronic industry) and the role of surface hardening to increase dramatically the fatigue resistance (e.g., of components in the automotive industry) are highlighted.
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Notes
- 1.
Erroneously, it is sometimes thought that this experience derives from the conservation of volume. However, there is no conservation of volume upon elastic deformation. Only if the Poisson ratio equals ½, volume is conserved during deformation (cf. discussion of Eq. (12.17)).
- 2.
Apart from the fundamental, scientific interest in materials with negative Poisson constants, their potential applications, e.g. as shock-absorbing material (sound deadening layers) and fasteners raise practical interest.
- 3.
Cork is a cellular solid that exhibits a Poisson ratio close to zero. This is of obvious importance for its application as stopper of a wine bottle: the stopper must be inserted and removed easily.
- 4.
This result is a direct consequence of the recognition that in the principal system the x, y and z components of the total stress acting on the plane considered are \(\sigma^{p}_{x} \cos \left( {n,x} \right)\), \(\sigma^{p}_{y} \cos \left( {n,y} \right)\) and \(\sigma^{p}_{z} \cos \left( {n,z} \right)\).
- 5.
These results can be obtained as follows (e.g. see Timoshenko and Goodier 1982). The total stress, σtot, acting on a plane of arbitrary orientation (not a principal plane), in the frame of reference given by the principal axes, is given by Eq. (12.13). Decompose this total stress into the normal component of the total stress acting on this plane, σn (this normal stress is given by the first three terms at the right-hand side of Eq. (12.12), since the shearing stress components in this equation are zero (principal system)) and the shearing stress acting on this plane (cf. Fig. 12.7a). Evidently, this shearing stress is given by \(\left( {\sigma^{2}_{\text{tot}} {-}\sigma^{2}_{n} } \right)^{1/2}\). Two of the direction cosines defining the orientation of the plane (in the principal system) are independent, because a relation of the type cos2(n, x) + cos2(n, y) + cos2(n, z) = 1 holds. Eliminate one of the direction cosines in the expression for the shearing stress by using the relation mentioned. Differentiate the resulting expression for the shearing stress with respect to the two remaining, independent direction cosines. Equating the two resulting differentials to zero leads to values for the maximal shearing stress and the orientation of the corresponding shearing plane.
- 6.
One may ask into which forms of energy the work done is transformed. Obviously, considering the straining of the body concerned upon loading, it appears that the predominant part of the work done is transformed into elastic strain energy. However, consider a gas that is compressed adiabatically (i.e. there is no heat exchange between the system considered and its surroundings). The adiabatic compression induces an increase of temperature of the gas. Similarly, compression of a solid leads to an increase of temperature, albeit a very small one. This minute increase of temperature of a solid upon loading of, say, a couple of tenths Kelvin, corresponds to a very small thermal strain (cf. Sect. 3.1) to be distinguished from the mechanical, elastic strain. This thermal strain, as compared to the elastic strain, is negligible. Would not this not be the case, one should have to distinguish between adiabatic and isothermal elastic constants.
- 7.
If F = constant ⋅ l, with F and l as force and extension, it follows for the work done = energy stored U:
U = ∫F dl = constant ⋅ ∫ l dl = constant ⋅ ½(lend)2 = ½Fend ⋅ lend, with Fend and lend as the final values of F and l. In the text above, the roles of Fend and lend are taken by F and εxdx.
- 8.
If this would be the case, a specific cycle of loading and unloading the six forces could be devised that would lead to net production of energy, which would violate the first law of thermodynamics (conservation of energy).
- 9.
The backbone of the polymeric chain is a string of covalently bonded carbon atoms.
- 10.
An amorphous polymer beneath the glass transition temperature, Tg, behaves as a “glass”, showing linearly elastic deformation and brittle fracture. Above Tg the amorphous polymer behaves as a rubbery solid, until at still higher temperatures a viscous liquid results (see Sect. 12.7).
- 11.
The notion “viscosity” of a material is used to indicate the resistance of the material against flow invoked by shear forces.
- 12.
Ductility is the ability of a material to undergo plastic deformation. The term toughness is used to indicate the (plastic deformation) energy which can be absorbed until fracture occurs (i.e. the area under the stress–strain curve until fracture): ductile materials are normally tougher than brittle materials. Evidently, a high toughness requires ductility, but also considerable strength. Usually ductility and toughness both increase or both decrease upon manipulation of the microstructure. Ideally, a material for structural application is both strong and tough. However, usually a strong material has modest toughness and vice versa. Consequently, material treatments and material developments aim at achieving an optimal combination of strength and toughness. For example, see the discussion on tempering of steels in the “Intermezzo: Tempering of iron-based interstitial martensitic specimens” at the end of Chap. 9. Yet, there are ways to establish the simultaneous occurrence of high strength and large toughness. Such a route implies the involvement of different mechanisms to establish plastic behaviour which operate at different length scales (cf. Sect. 1.4), for example (i) by applying a hard phase with a microstructure that allows local relieve of high loads by a small amount of plastic deformation, such as provided by sliding along fibres contained in the hard-phase matrix (of course, dislocations in (metallic) materials in fact have a similar role in plastic deformation of the body concerned (cf. Sect. 5.2)), and (ii) by subcritical (micro)crack growth (cf. Sect. 12.15) (Ritchie 2011).
- 13.
In ferrite (α-Fe) these tetragonal stress fields around the interstitial atoms are not aligned and, as a result, the average crystal structure of ferrite remains (body centred) cubic, whereas in martensite, because of the higher concentration of interstitials, the interaction, as discussed in the above sense, of the tetragonal stress fields around the individual interstitials, causes an alignment of the individual stress fields such that the EF-axes for the interstitials become aligned (in other words: only one of the three types of octahedral interstices becomes occupied) and, as a result, the average crystal structure of martensite is (body centred) tetragonal (see Sect. 9.5.2.1). The octahedral interstice in face-centred cubic iron (γ-Fe) is regular and thereby the misfit-stress field introduced upon introduction of an interstitial is spherical (isotropic; here, in first-order approximation, the host matrix (Fe) is assumed to possess elastically isotropic properties), and thus, the (average) crystal structure remains cubic for also large amounts of dissolved interstitial solute (see also the discussion on interstitial diffusion in Sect. 8.5.3 and footnotes 6 and 7 in Sect. 8.6; further note that, although ferrite shows anisotropic elastic behaviour, an overall hydrostatic stress field will lead to an isotropic distortion due to the cubic crystal symmetry of ferrite).
- 14.
The elastic theory of misfitting inclusions in a matrix has been developed by Eshelby (1956). His analysis not only has provided fundamental insight into the elastic deformations due to the inclusion of a misfitting point defect in a matrix, but also has been the basis for understanding the stress fields around misfitting precipitates in a matrix (cf. Sect. 12.18).
- 15.
Any state of stress can be subdivided into a hydrostatic state of stress plus a so-called deviatoric state of stress. The total strain energy can be written, for this special case, as a sum of the strain energies of the hydrostatic state of stress component and the deviatoric state of stress component (in general the strain energies of two superimposed states of stress are not additive). The first strain energy contribution is called “strain energy of dilatation”; the latter strain energy contribution is called “strain energy of distortion”. For the von Mises criterion it then is assumed that the hydrostatic state of stress component \(\left( {\sigma^{P}_{x} = \sigma^{P}_{y} = \sigma^{P}_{z} } \right)\) makes a negligible contribution to the deformation (incompressible body), so that the “strain energy of distortion” is decisive for the occurrence of yielding.
- 16.
In rock-salt type crystals six equivalent, from a crystallographic point of view (slip plane: {110}; slip direction: <110>), slip systems can be indicated, but only two of these are independent. Consequently, according to the discussion in the main text, a polycrystal of rock-salt type is brittle. Only at higher temperatures, if another slip system becomes operative as well (slip plane: {001}; slip direction <110>), significant plastic deformation of a polycrystal of rock-salt type becomes feasible. This contrasts strongly with many polycrystalline metals, which can experience extensive plastic deformation already at room temperature.
- 17.
Although this approach to hardness testing is no longer used in materials science, it is interesting to note that a “scratch test” is (still) often used to investigate the adherence of a thin layer on a substrate.
- 18.
Four-sided pyramids are use as tips in Vickers and Knoop hardness testers (cf. Fig. 12.34a, b). Three-sided pyramid tips are common in nanoindentation (Berkovich and cube-corner tips), because these are easier to produce with sharp tips than four-sided pyramids for such applications (cf. Figs. 12.34c and 12.36b).
- 19.
The projected contact area, Ac, can be written as the product constant . h 2c , with hc as the depth of contact between indenter and specimen at maximum load, i.e. the distance along the indenter axis that the specimen is in contact with the indenter (cf. Fig. 12.39). The constant in this expression depends on the shape of the tip of the indenter; for example, for the three-sided pyramid-type Berkovich indenter (cf. footnote 18), often used in nanoindentation experiments, it holds that the constant in case of ideal tip shape equals 24.5; for the ideal cube-corner tip the constant equals 2.6.
- 20.
The geometrically necessary dislocations (GNDs) are complemented by the “statistically stored dislocations (SSDs)” such that the total dislocation density is given by the sum of the densities of GNDs and SSDs. Upon plastic deformation the SSDs can be conceived as those dislocations which are introduced in the absence of macroscopic/mesoscopic plastic strain gradients as referred to in the above text. In a strict sense any dislocation is introduced to comply with a geometrical incompatibility….
- 21.
Similarly as discussed above for dislocations: the role of geometrically necessary dislocations (GNDs), to accommodate macroscopic/mesoscopic plastic strain gradients, e.g. as occurring in the vicinity of grain boundaries upon plastic deformation of a massive polycrystalline material, can be taken by twins: “geometrically necessary twins (GNTs)” (Sevillano 2008).
- 22.
- 23.
- 24.
This needs not hold for surface-hardened (case hardened) metallic components; see the Epilogue after Sect. 12.18, as the conclusion of this book.
- 25.
For metallic components subjected to fatigue loading at high temperatures not only the fatigue–creep interaction has to be recognized: the simultaneously occurring oxidation, as a striking example of profound interaction with the environment that occurs as well, can drastically influence service life too.
- 26.
The sum of the forces acting on a cross section through the body must be nil: force balance. Similarly, a balance of moments is required.
- 27.
- 28.
Another classical case-hardening method is carburizing. Carburizing, as nitriding, is a thermochemical surface treatment to improve the mechanical properties (wear, fatigue) which depend on the quality of the surface adjacent material of the component (see also the “Intermezzo: Thermochemical surface engineering; nitriding and carburizing of iron and steels” in Sect. 4.4.2). Carbon and nitrogen are offered by an outward, e.g. gaseous, atmosphere and diffuse into the surface region of the component at elevated temperature. In case of carburizing the treatment is carried out at higher temperatures such that the matrix is austenitic. Upon quenching a hard, martensitic microstructure is induced in the carburized case. The (tendency to) volume expansion associated with the martensite formation contributes to the development of a compressive macrostress parallel to the surface in the surface region. In case of nitriding the treatment is carried out at a lower temperature such that the matrix is ferritic. The precipitation of nitrides in the nitrided surface region leads to the high hardness of the nitrided case. The (tendency to) volume expansion by the precipitation of the nitrides contributes to the development of a compressive macrostress parallel to the surface in the surface region.
- 29.
According to the idealized sketch in Fig. 12.59a, the value of the local fatigue strength of the case-hardened component at the case/core transition, Sf(zf), equals the fatigue strength of the component before case hardening (case (i) above: the fatigue strength of the material before case hardening does not depend on depth), and thus, Sf(zf). = S 0f . In reality, the fatigue limit for failure initiating at the surface can be smaller than the intrinsic fatigue limit of the material, e.g. due to atmospheric influences. This is one reason why Sf(zf) can be larger than the value measured for S 0f for the not case-hardened component.
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Mittemeijer, E.J. (2021). Mechanical Strength of Materials . In: Fundamentals of Materials Science. Springer, Cham. https://doi.org/10.1007/978-3-030-60056-3_12
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