Mechanical Properties of Materials

the aggregate of indexes that characterize the resistance of a material to a load acting on it, the degree to which it will deform under the load, and its behavior in the process of failure. Accordingly, the properties are measured in stresses (usually in kilograms-force per sq mm [kgf/mm²] or meganewtons per sq m [MN/m²]), strains (in percent), specific strain and failure energy (usually in kgf . m/cm² or megajoules per sq m [MJ/m²]), and rate of development of the process of failure under a static or repeated load (most often in millimeters per second or per thousand cycles of the load). The mechanical properties of materials are determined in mechanical tests of specimens of various shapes.

Materials in structures are subjected to the most varied loads (Figure 1), working under tension, compression, bending, twisting, shear, and other factors, or they may be subjected to the joint action of several types of loads, such as extension and bending. The conditions of use of materials are also diverse in terms of the temperature, environment, rate of application of the load, and the nature of its change in time. Accordingly, there are many indexes of the mechanical properties of materials and many methods of mechanical tests. Tests of tension, hardness, and impact bending are most widely used for metals and structural plastics; brittle structural materials, such as ceramics and cermets, are often tested for compression and static buckling. It is also important to evaluate the mechanical properties of composition materials in shearing tests.

Figure 1. Diagrams of strain under various modes of stress: (a) tension, (b) compression, (c) bending, (d) twisting (the broken line indicates the original shape of the specimens)

Stress-strain diagram. A load applied to a specimen causes its deformation. The correlations between load and deformation are described by the stress-strain diagram (Figure 2). The deformation of the specimen (upon tension, an incremental length Δl) initially is proportional to the growing load P, and then at the point n the proportionality is broken. However, for the deformation to increase, a further increase in the load P is required. When Δ l > Δ li the deformation develops without application of an external force, under a gradually decreasing load. The appearance of the stress-strain diagram does not change if the stress σ = P/F₀ is plotted on the ordinate and the relative extension δ = Δ l/l₀ is plotted on the abscissa (where F₀ and l₀ are the initial cross section and calculated length of the specimen, respectively).

Figure 2. Typical stress-strain diagram upon tension of structural metals

The strength of materials is measured by the stresses that characterize the load per unit cross section of the specimen σ = P/F₀ (kgf/mm²). The stress σ_n = P_n/F₀, at which the proportionality of the increase in deformation to the load is disrupted, is called the limit of proportionality. For a load P < P_n, relief of the load on the specimen leads to disappearance of the deformation that arose under the applied force. Such deformation is said to be elastic. The character of deformation may not change if the load slightly exceeds P_n —as before, it will retain its elastic character. The maximum load that the specimen endures without the appearance of residual plastic deformation upon unloading defines the elastic limit of the material:σ_e = P_e/F₀.

In nonmetallic structural materials (plastics or rubber) an applied load may cause elastic, highly elastic, or permanent deformation. In contrast to elastic deformation, highly elastic deformation disappears with time rather than immediately after relief of the load. Highly elastic reinforced polymers (such as Fiberglas and coal-based plastics) fail upon extension of 1–3 percent. Highly elastic deformation appears in some reinforced polymers in the final stages of loading. The modulus of high elasticity is lower than the elastic modulus; therefore, the stress-strain diagram in this case tends to be deflected toward the abscissa.

Elastic properties. In the elastic domain stress and strain are interrelated by a proportionality constant. Upon extension σ = Eδ, where E is the modulus of normal elasticity, which is numerically equal to the tangent of the angle of inclination of the rectilinear segment of the curve σ = σ(δ) to the strain axis (Figure 2). During tensile testing of a cylindrical or flat specimen, a triaxial deformed state (increment of length in the direction of action of the applied forces and a decrease in linear dimensions in the two other mutually perpendicular directions; δ₁ > 0; δ₂ = δ₃ < 0) corresponds to the uniaxial stressed state (σ₁ > 0; σ₂ = 0 = σ₃ = 0). The relation between transverse deformation and stretch (Poisson’s ratio), Jμ = δ₂/δ₁, varies in a very narrow range within the limits of elasticity for the main structural materials (0.27–0.30 for steels, 0.30–0.33 for aluminum alloys). Poisson’s ratio is one of the main calculated characteristics. If μ and E are known, then the shear modulus G = E/[2(1 + μ)], and the bulk modulus K = E/[3(l − 2μ)] can be determined by calculation. Strain gauges are used to determine E, G, and μ.

Resistance to plastic deformation. A noticeable irreversible plastic deformation, which does not disappear upon relief of the load, appears in addition to the steadily increasing elastic deformation under loads P > P_e. The stress under which relative plastic deformation (or, upon extension, elongation) reaches a prescribed value (0.2 percent, according to the All-Union State Standard), is called the arbitrary yield point and is designated as σo.2 = P^y/F₀. In practice, the accuracy of modern testing methods is such that σ_n and σ_e are determined with assigned tolerances in terms of deviation from proportionality (an increase of 25–50 percent in cot [90 — α]) and the value of permanent deformation (0.003–0.05 percent), respectively, and we speak of the arbitrary limits of proportionality and the elasticity. The load-elongation curve of structural metals may have a maximum (point v in Figure 2), or it may be discontinuous when the maximum load P is reached. The relation P_v/F₀ = σ_v characterizes the tensile (ultimate) strength of a material. When a maximum is present on the load-elongation curve in the region of loads that lie to the left of v on the curve, the specimen deforms uniformly over the entire length l₀, gradually decreasing in diameter but retaining the original cylindrical or prismatic shape. Metals are strengthened during plastic deformation; therefore, despite the decrease in the cross section of the specimen, an increasing load must be applied for further deformation. The quantity σ₁ like the arbitrary σ_0.2, σ_n, and σ_e, characterizes the resistance of metals to plastic deformation. On the segment of the stress-strain diagram to the right of v, the shape of the stretched specimen changes: a period of concentrated deformation sets in and is expressed in the appearance of a “neck.” The decrease in cross section at the neck “overtakes” the strengthening of the metals. This accounts for the decrease in external load over the segment P_v−P_f.

In many structural materials resistance to plastic deformation in the elastic-plastic domain is virtually identical upon extension and compression. Appreciable differences in this characteristic during extension and compression are characteristic of certain metals and alloys (for example, magnesium alloys and high-tensile steels). Resistance to plastic deformation is estimated particularly often (for example, in product quality control or the monitoring of heat treatment conditions) on the basis of results of hardness tests conducted by impressing a hard tip in the form of a sphere (Brinell or Rockwell hardness), cone (Rockwell hardness), or pyramid (Vickers hardness). Hardness tests do not require disruption of the integrity of the part and therefore are the most widely used means of monitoring mechanical properties. Brinell hardness (HB) upon indentation of a ball with a diameter D under a load P characterizes the mean compressive stress, arbitrarily calculated per unit surface of a spherical indentation with a diameter d:

Plasticity characteristics. Plasticity of structural materials upon extension is evaluated in terms of the elongation δ = (l_f − l₀)/l₀ X l00 percent, or the constriction ϕ = (F₀ − F_f)/F₀ X 100 percent; upon compression, in terms of the shortening Δ = (h₀ − h_f)/h₀ X 100 percent (where h₀ and h_f are the initial and final heights of the specimen) ; and upon twisting, by the maximum angle of twist θ of the working part of the specimen, in radians, or by the relative shear γ = θr (where r is the radius of the specimen). The final ordinate of the stress-strain diagram (point f in Figure 2) characterizes the resistance to breaking S_f of the metal, which is defined as S_f = P_f/F_f (where F_f is the actual area at the point of the break).

Failure characteristics. Failure does not take place instantaneously (at point f), but rather develops with time. The onset of failure may correspond to some intermediate point on the segment vf, and the entire process ends as the load gradually decreases to zero. The position of the point f on the stress-strain diagram is determined largely by the rigidity of the testing machine and the inertia of the measuring system. This makes the quantity S_f largely arbitrary.

Upon extension, many structural metals (steels, including high-tensile steel, and also heat-resistant chromium-nickel alloys and soft aluminum alloys) fail after significant plastic deformation, resulting in the formation of a neck. The fracture surface often lies at an angle of approximately 45° to the direction of the tensile force (for example, in high-tensile aluminum alloys). Under certain conditions (for example, in the testing of cold-short steels in liquid nitrogen or hydrogen and upon exposure to tensile stresses and a corrosive environment for metals that are inclined to corrode under stress) failure takes place in sections perpendicular to the tensile force (even fracture), without macroplastic deformation.

The strength of materials achieved in structural elements depends not only on the mechanical properties of the metal itself but also on the shape and size of the part (the shape and scale effects), the elastic energy accumulated in the loaded structure, the character of the effective load (static, dynamic, or of periodically varying magnitude), the diagram of applied external forces (uniaxial and biaxial elongation, elongation with applied bending, and so on), operating temperature, and environment. The dependence of the strength and plasticity of metals on shape is characterized by notch sensitivity, which is usually estimated from the ratio of the tensile strength of notched and smooth samples (in cylindrical samples the notch is usually cut in the form of a circular recess; in strips it is cut in the form of a central aperture or lateral recesses). For many structural materials this ratio is greater than 1 under static load because of the significant local plastic deformation at the apex of the notch. The sharper the notch, the lower the local plastic deformation and the greater the share of even fracture in the fractured cross section. A well-developed even fracture can be produced in most structural materials at room temperature under laboratory conditions if samples of large cross section (the more plastic the material, the thicker the cross section) are subjected to stretching or bending after a narrow special slit with an artificial crack (Figure 3) was made in the sample. When a broad, flat sample

Figure 3. Specimen with a fatigue crack especially generated at the peak of the notch to determine K,c. Tests for eccentric tension (a) and axial tension (b).

is stretched, plastic deformation is impeded and is limited to a small region measuring 2r_y (crosshatched area in Figure 3,b) that is adjacent to the end of the crack. Even fracture is usually characteristic of operational failures of structural elements.

Such indexes as the critical coefficient of stress intensity under plane deformation K_1c and the failure viscosity G_1c = K²_1c/E, which were proposed by the American scientist G. R. Irwin as constants for the conditions of brittle fracture, have come into wide usage. Here the failure process is considered over time and the indexes K_1c(G_1c) refer to the critical moment when the stable development of the crack is disrupted; the crack becomes unstable and propagates spontaneously when the energy necessary to increase its length is less than the energy of elastic deformation reaching the apex of the crack from neighboring elastically stressed zones of the metal.

Designating the thickness of the sample t and the size of the crack 2l_cr, we proceed from the following requirement: l_cr ≥ 2.5 (K_1c/σ0.2)² ≤ t. The coefficient of stress intensity K takes into account not only the value of the load, but also the length of the increasing crack: K = . ; (; takes into account the geometry of the crack and sample) is expressed in kgf/mm^3/2 or MN/m^3/2. The inclination of structural materials to brittle fracture under operating conditions can be assessed from K_1c or G_1c.

Impact tests of prismatic samples that have a notch on one side are widely used to evaluate the quality of metal. Here an estimate is made of the impact strength (in kg .m/cm² or MJ/ m²)—the work of deformation and fracture of the sample, arbitrarily referred to the cross section formed by the notch. Impact tests of samples with a fatigue crack artificially produced at the apex of the notch are widely used. On the whole, the work of fracture in such samples α_fr is in satisfactory agreement with such a fracture characteristic as K_1c and still better with the ratio (K_1c/σ0.2)²

Time dependence of strength. Plastic deformation strength and resistance to breaking decrease with increasing time of action of the load. In metals at room temperature this becomes especially noticeable upon exposure to a corrosive agent (corrosion under stress) or other active agent (the Rebinder effect). The phenomenon of creep—that is, the increase in plastic deformation with time under constant stress (Figure 4,a)—is observed at high temperatures. The creep strength of metals is evaluated by the arbitrary creep limit—most often by the stress under which plastic deformation reaches 0.2 percent in 100 hours—and is designated as σ0.2/100. The higher the temperature t, the more pronounced the creep and the greater the decrease in the fracture strength of the metal over time (Figure 4,b). The last property is characterized by the long-term strength, which is the stress that induces rupture of the material at a given temperature over a given time (for example, σ^t100 and σ^t1,000). The temperature-time dependence of strength and deformation is more marked for polymers than for metals. When plastics are heated, highly elastic reversible deformation is observed. Beginning at a certain higher temperature, irreversible deformation associated with the transition of the material to the state of viscous flow develops.

Figure 4. Change in the mechanical properties of structural materials as a function of time (or the number of cycles)

Another important mechanical property of materials—the tendency toward stress relaxation, that is, toward a gradual decrease in stress under conditions in which the total (elastic and plastic) deformation maintains a constant, prescribed value (for example, in tightened bolts)—is also associated with creep. Stress relaxation is due to an increase in the share of the plastic component of the total deformation and to a decrease in its elastic component.

If a metal is acted upon by a load that varies periodically according to some law (for example, a sine law), then as the number N of cycles of the load increases, the metal’s strength decreases (Figure 4,c)—the metal “fatigues.” For structural steel such a decrease in strength is observed before N = (2 − 5) X 10⁶ cycles. Accordingly, we speak of the fatigue limit of structural steel, by which is usually meant the stress amplitude (σ_max − σ_min)/2, below which the steel does not fracture under a repeated variable load. When ǀσ_minǀ = ǀσ_maxǀ the fatigu limit is designated by the symbol σ₋₁. The fatigue curves of aluminum, titanium, and magnesium alloys usually do not have a horizontal segment; therefore, the fatigue strength of these alloys is characterized by restricted fatigue limits (corresponding to a given N). The fatigue strength also depends on the frequency of load application. The resistance of materials at low frequencies and at high values of repeated loading (slow, or short-time, fatigue) is not unambiguously related to fatigue limits. Under repeated variable loads, in contrast to a static load, notch sensitivity is always manifested—that is, the fatigue limit is lower in the presence of a notch than for a smooth sample. For convenience the notch sensitivity during fatigue is expressed by the relation

K_σ = σR/σRⁿ

(where R = σ_min/σ_max characterizes the asymmetry of the cycle). In the fatigue process it is possible to single out the period that precedes the formation of the center of fatigue fracture, and also the period that follows it, which is sometimes very long (the period of development of the fatigue crack). The more slowly the crack develops, the more reliably the material functions in a structure. The rate of development of the fatigue crack dl/dN is related to the coefficient of stress intensity by the exponential function dl/dN = cKⁿ.

Thermal fatigue resistance is also distinguished. In this case the stresses that appear in the material are caused by the fact that, for various reasons (for example, because of the shape of the part or the nature of its attachment), the thermal displacements that occur upon a cyclic change in temperature cannot be realized. The thermal fatigue resistance also depends on many other properties of a material, such as the coefficient of linear expansion, thermal diffusivity, elastic modulus, and the elastic limit.