Chapter 7: Mechanical Properties

Fundamentals of Mechanical Properties and Structural Loading

• Mechanical properties quantify a material's behavioral response, deformation, or structural failure when subjected to applied external mechanical forces or loads.

• Key mechanical properties in materials science and structural engineering include:

  • Strength: The ability of a material to withstand applied forces without undergoing permanent deformation or mechanical failure.

  • Stiffness: The resistance of a material to undergoing elastic (recoverable) deformation when subjected to mechanical loading.

  • Hardness: The localized surface resistance of a material to scratching, abrasion, or permanent surface indentation.

  • Ductility: The capacity of a material to sustain significant plastic deformation prior to fracture.

  • Toughness: The total capacity of a material to absorb energy and deform plastically up to the point of fracture.

• Primary mechanical loading configurations encountered in engineering service:

  • Tension: Pulling forces applied along the longitudinal axis of a member (e.g., steel cables supporting ski lift assemblies or suspension bridge suspenders).

  • Torsion: Twisting moments (torques) applied about the longitudinal axis of a cylindrical shaft (e.g., ski lift drive shafts or vehicle transmission shafts).

  • Biaxial Tension: Tensile forces acting simultaneously across mutually perpendicular directions on a curved surface (e.g., cylindrical pressurized gas containment tanks).

  • Compression: Pushing forces that shorten a structural element along its axis (e.g., vertical bridge piers and structural truss members supporting deck loads).

  • Hydrostatic Compression: Multiaxial equal compressive stress exerted inward normal to all surfaces of a submerged body (e.g., aquatic organisms or submersibles under water).

Engineering Stress and Strain

• Engineering Stress:

  • Defined as the internal resisting force per unit original cross-sectional area, having SI units of Pascals ($\text{Pa}$ or $\text{N/m}^2$), typically expressed in megapascals ($1\,\text{MPa} = 10^6\,\text{N/m}^2$).

  • Tensile (Normal) Stress ($\sigma$): Force component acting perpendicularly (normal) to the original cross-sectional area:     $\sigma = \frac{F_t}{A_0}$     where $F_t$ represents the applied normal tensile load in Newtons ($\text{N}$) and $A_0$ is the original cross-sectional area before load application in square meters ($\text{m}^2$).

  • Shear Stress ($\tau$): Force component acting parallel (tangential) to the plane of the cross-sectional area:     $\tau = \frac{F_s}{A_0}$     where $F_s$ is the applied shear load in Newtons ($\text{N}$) and $A_0$ is the shear plane area in square meters ($\text{m}^2$).

• Engineering Strain:

  • Represents the geometric deformation of a material normalized against its original dimensions; it is an inherently dimensionless quantity.

  • Tensile (Longitudinal) Strain ($\epsilon$): The change in gauge length divided by the initial gauge length along the tensile axis:     $\epsilon = \frac{\Delta L}{L_0} = \frac{L - L_0}{L_0}$     where $L_0$ is the initial gauge length, $L$ is the instantaneous gauge length, and $\Delta L$ is the elongation. Tensile strain is taken as positive ($\epsilon > 0$).

  • Lateral (Transverse) Strain ($\epsilon_L$): The dimensional variation perpendicular to the tensile loading axis:     $\epsilon_L = -\frac{\Delta w}{w_0}$     where $w_0$ is the original width or diameter and $\Delta w$ is the change in lateral width. Under tensile loading, lateral strain is negative ($\epsilon_L < 0$) due to lateral contraction.

  • Shear Strain ($\gamma$): The angular deformation resulting from shear stress, defined as the tangent of the angular displacement angle $\theta$ from an initially perpendicular coordinate frame:     $\gamma = \frac{\Delta x}{y} = \tan(\theta)$     where $\Delta x$ is the transverse linear displacement, $y$ is the orthogonal distance from the fixed base, and $\theta$ is the shear angle in radians.

• Torsional Shear Stress and Strain in Cylindrical Members:

  • Torsion produces a state of pure shear across concentric circular planes perpendicular to the member's longitudinal axis.

  • The shear stress $\tau$ at any radial distance $r$ from the center axis of a circular specimen subjected to torque $T$ is governed by:     $\tau = \frac{T r}{J}$     where $T$ is the applied torque, $r$ is the radius, and $J$ is the polar moment of inertia.

  • For a solid cylindrical specimen of diameter $d$ (where radius $r = \frac{d}{2}$):     $J = \frac{\pi d^4}{32}$     $\tau = \frac{T \left(\frac{d}{2}\right)}{\frac{\pi d^4}{32}} = \frac{16 T}{\pi d^3}$

  • The torsional shear strain $\gamma$ on the outer cylindrical surface is given by:     $\gamma = \tan(\phi) = \frac{r \theta}{L}$     where $\theta$ is the angle of twist (degree of rotation) in radians along the length $L$, and $r$ is the outer radius of the cylindrical specimen.

Elastic Deformation and Linear Elastic Properties

• Mechanism of Elastic Deformation:

  • Elastic deformation is strictly reversible: upon removal of an applied load, all accumulated strain completely recovers, and the component returns to its exact initial dimensions.

  • At the atomic level, applied elastic loads cause interatomic bonds to stretch and atoms to displace slightly from their equilibrium positions without rupturing neighbor bonds.

  • Elastic behavior is most commonly linear (where stress remains strictly proportional to strain), but can be non-linear (as observed in certain elastomers and polymers).

• Hooke's Law and Modulus of Elasticity:

  • In the linear-elastic regime, tensile stress and tensile strain are directly proportional according to Hooke's Law:     $\sigma = E \epsilon$

  • The constant of proportionality $E$ is the Modulus of Elasticity, or Young's Modulus, typically quantified in gigapascals ($1\,\text{GPa} = 10^3\,\text{MPa} = 10^9\,\text{N/m}^2$).

  • Young's modulus corresponds to the slope of the linear-elastic region of the engineering stress-strain curve.

• Interatomic Basis of Elastic Modulus:

  • The macroscopic modulus of elasticity is directly proportional to the slope of the interatomic force-versus-separation curve ($F$ vs. $r$) evaluated at the equilibrium interatomic separation distance $r_0$:     $E \propto \left(\frac{dF}{dr}\right)_{r_0}$

  • Strongly bonded materials (such as refractory metals and ceramics with deep interatomic potential energy wells) exhibit steep force slopes at $r_0$, resulting in high values of $E$.

  • Weakly bonded materials (such as polymers held primarily by secondary van der Waals interactions) exhibit shallow slopes at $r_0$, producing low values of $E$.

• Elastic Shear Modulus ($G$):

  • Relates shear stress and shear strain in the linear-elastic regime:     $\tau = G \gamma$     where $G$ is the shear modulus (typically in $\text{GPa}$).

  • For isotropic materials (materials whose physical properties are directionally independent), the shear modulus $G$ is related to Young's modulus $E$ and Poisson's ratio $\nu$ by:     $G = \frac{E}{2(1 + \nu)}$

• Poisson's Ratio ($\nu$):

• When an isotropic material is stretched under uniaxial tension, it elongates in the axial direction and simultaneously contracts in the transverse (lateral) directions.

• Poisson's ratio is defined as the negative ratio of the lateral strain $\epsilon_L$ to the axial tensile strain $\epsilon$:   $\nu = -\frac{\epsilon_L}{\epsilon}$

• Poisson's ratio is a dimensionless parameter.

• Typical values across distinct material classes:

  • Metals: $\nu \approx 0.33$

• Ceramics: $\nu \approx 0.25$

• Polymers: $\nu \approx 0.40$

Plastic Deformation, Yielding, and Tensile Testing

• Mechanism of Plastic Deformation:

  • Plastic deformation is permanent and non-reversible upon the removal of the applied mechanical load.

  • At the atomic level, plastic deformation involves the breaking of atomic bonds and the collective sliding (shearing) of adjacent atomic planes past one another via dislocation movement.

  • When an elastically and plastically deformed sample is unloaded, the interatomic bonds relax to their original spacing (recovering the elastic portion of strain), while the atomic planes remain permanently displaced (retaining the plastic strain component):     $\delta_{\text{total}} = \delta_{\text{elastic}} + \delta_{\text{plastic}}$

  • Unloading from the plastic region proceeds along a linear path that has a slope identical to the initial elastic modulus $E$, leaving a permanent plastic strain $\epsilon_p$.

• Yield Strength ($\sigma_y$):

  • Yield strength designates the stress level at which noticeable, macroscopic plastic deformation commences, measured in $\text{MPa}$.

  • $0.2\%$ Offset Method: Because the transition from elastic to plastic behavior is typically gradual without a distinct discontinuity, standard practice establishes $\sigma_y$ by constructing a parallel line offset by a plastic strain of $\epsilon_p = 0.002$ ($0.2\%$) from the origin. The intersection of this offset line with the stress-strain curve defines $\sigma_y$.

  • For a standard $2\text{ in}$ gauge length test specimen, an offset strain of $\epsilon_p = 0.002$ corresponds to an absolute plastic elongation of:     $\Delta L = \epsilon_p \times L_0 = 0.002 \times 2\text{ in} = 0.004\text{ in}$

• Tensile Strength ($TS$):

  • Tensile strength (also known as ultimate tensile strength) is the maximum engineering stress recorded on the engineering stress-strain curve, measured in $\text{MPa}$.

  • Phenomenological behavior:

  • In metals, tensile strength corresponds to the onset of localized necking (point $M$). Before this point, plastic deformation is uniformly distributed throughout the gauge length. Beyond $TS$, deformation concentrates into a localized neck, which acts as a stress concentrator until fracture occurs at point $F$.

  • In polymers, tensile strength occurs when polymer backbone molecular chains become aligned along the tensile axis and are immediately adjacent to chain rupture.

• Uniaxial Tensile Testing Apparatus and Specimen Standards:

  • Tensile testing machines employ a stationary load cell to measure instantaneous load, a motorized moving crosshead to apply controlled displacement, and an extensometer attached to the specimen to measure axial strain continuously.

  • Standard cylindrical tensile specimen geometry (ASTM standard):

  • Gauge length: $2.000\text{ in}$ ($2''$)

  • Reduced section diameter: $0.505\text{ in}$ (yielding an initial cross-sectional area $A_0 \approx 0.200\text{ in}^2$)

  • Reduced section length: $2\frac{1}{4}\text{ in}$ ($2.25''$)

  • Grip section diameter: $\frac{3}{4}\text{ in}$ ($0.75''$)

  • Transition fillet radius: $\frac{3}{8}\text{ in}$ ($0.375''$)

True Stress and True Strain

• Limitations of Engineering Quantities:

  • Engineering stress and strain fail to account for continuous changes in the cross-sectional area and gauge length that occur during deformation, particularly after localized necking initiates.

• True Stress and True Strain Definitions:

  • True Stress ($\sigma_T$): The instantaneous applied force $F$ divided by the instantaneous cross-sectional area $A_i$:     $\sigma_T = \frac{F}{A_i}$

  • True Strain ($\epsilon_T$): The natural logarithm of the ratio of the instantaneous length $L_i$ to the original length $L_0$:     $\epsilon_T = \ln\left(\frac{L_i}{L_0}\right)$

  • Conversion relationships up to the onset of necking (assuming constant volume during plastic deformation, $A_0 L_0 = A_i L_i$):     $\sigma_T = \sigma (1 + \epsilon)$     $\epsilon_T = \ln(1 + \epsilon)$

• Comparison Between Engineering and True Curves:

  • Prior to necking, engineering and true stress-strain curves track closely with small divergence.

  • Beyond the onset of necking (point $M$ on the engineering curve, corresponding to point $M'$ on the true curve), the engineering curve declines due to using the constant initial area $A_0$.

  • The true stress-strain curve continues to increase monotonically up to fracture, reflecting continued strain hardening within the localized neck.

Ductility, Toughness, and Resilience

• Ductility:

  • Quantifies the degree of plastic deformation sustained by a material prior to fracture.

  • Brittle materials display little to no plastic deformation prior to fracture.

  • Ductile materials display extensive plastic deformation prior to fracture.

  • Quantification measures:

  • Percent Elongation ($\%EL$):       $\%EL = \left(\frac{L_f - L_0}{L_0}\right) \times 100$       where $L_f$ is the fracture gauge length and $L_0$ is the original gauge length.

  • Percent Reduction in Area ($\%RA$):       $\%RA = \left(\frac{A_0 - A_f}{A_0}\right) \times 100$       where $A_0$ is the initial cross-sectional area and $A_f$ is the cross-sectional area at the fracture surface.

• Toughness:

  • Defined as the mechanical energy absorbed per unit volume of material up to fracture, with SI units of Joules per cubic meter ($\text{J/m}^3$).

  • Graphically represented by the total area under the engineering stress-strain curve integrated from zero strain to the fracture strain.

  • Characteristic toughness across material classes:

  • Metals: Large toughness, resulting from an optimal combination of high strength and high ductility (absorbing both elastic and significant plastic deformation energy).

  • Ceramics: Small toughness, exhibiting high strength but negligible ductility (absorbing only elastic strain energy prior to brittle fracture).

  • Unreinforced Polymers: Very small toughness, exhibiting substantial ductility but extremely low strength.

• Interplay of Strength and Ductility in Steels:

  • Maximum toughness requires an optimal balance between strength and ductility; neither extreme alone maximizes absorbed energy.

  • High Carbon Steel: Highest yield and tensile strength, but minimal ductility; exhibits moderate toughness.

  • Medium Carbon Steel: Combines high strength with substantial ductility, achieving the greatest area under the curve and the highest toughness.

  • Low Carbon Steel: Greatest ductility, but comparatively low yield and tensile strength; exhibits lower toughness than medium carbon steel.

• Modulus of Resilience ($U_r$):

  • The capacity of a material to absorb energy when deformed elastically and release that energy upon unloading, expressed in $\text{J/m}^3$.

  • Defined as the area under the engineering stress-strain curve up to the yield strain ($\epsilon_y$):     $U_r = \int_0^{\epsilon_y} \sigma\,d\epsilon$

  • Assuming linear-elastic behavior up to yielding, this simplifies to:     $U_r \cong \frac{1}{2} \sigma_y \epsilon_y = \frac{1}{2} \sigma_y \left(\frac{\sigma_y}{E}\right) = \frac{\sigma_y^2}{2E}$     where $\sigma_y$ is the yield strength and $E$ is Young's modulus.

  • High resilience requires a high yield strength paired with a low modulus of elasticity (a requirement typical for mechanical spring applications).

Hardness and Indentation Testing Techniques

• Definition and Engineering Relevance:

  • Hardness represents a material's resistance to localized surface plastic indentation, penetration, scratching, or abrasive wear.

  • Materials exhibiting high hardness demonstrate high resistance to compression-induced plastic deformation and superior wear characteristics.

  • Hierarchy of increasing hardness across materials:     $\text{Most Plastics} < \text{Brasses \& Al Alloys} < \text{Easy to Machine Steels} < \text{File Hard Cutting Tools} < \text{Nitrided Steels} < \text{Diamond}$

• Rockwell Hardness Testing:

  • Non-destructive method measuring the net increase in indentation depth following the application and removal of a major load.

  • Testing sequence: A minor load of $10\,\text{kg}$ is initially applied to seat the indenter and establish a zero reference datum, followed by the application and subsequent release of a specified major load.

  • Standard scales run up to 130, but test results are practically useful and reliable in the range between 20 and 100 on any specific scale.

  • Common Rockwell scales:

  • Scale A: Diamond cone indenter ($120^\circ$ angle), $60\,\text{kg}$ major load.

  • Scale B: $\frac{1}{16}\text{ in}$ diameter hardened steel ball indenter, $100\,\text{kg}$ major load.

  • Scale C: Diamond cone indenter ($120^\circ$ angle), $150\,\text{kg}$ major load.

  • Designation: $80\text{ HRB}$ designates a hardness of 80 on the Rockwell B scale.

  • Superficial Rockwell Testing: Used for thin specimens or thin hardened cases; operates with a $3\,\text{kg}$ minor load and major loads of $15\,\text{kg}$, $30\,\text{kg}$, or $45\,\text{kg}$.

• Brinell Hardness Testing ($HB$):

  • Uses a $10\text{-mm}$ diameter hardened steel or tungsten carbide sphere indenter.

  • Applied load $P$ ranges between $500\,\text{kg}$ and $3000\,\text{kg}$ (commonly $3000\,\text{kg}$ for steels).

  • The Brinell Hardness number is calculated using:     $HB = \frac{2P}{\pi D \left[D - \sqrt{D^2 - d^2}\right]}$     where $P$ is the applied load in $\text{kg}$, $D$ is the indenter diameter in $\text{mm}$ ($10\text{ mm}$), and $d$ is the diameter of the surface impression in $\text{mm}$.

  • Empirical correlation between Brinell Hardness and Tensile Strength ($TS$) for plain carbon and alloy steels:     $TS\,(\text{psi}) = 500 \times HB$     $TS\,(\text{MPa}) = 3.45 \times HB$

• Microhardness Testing Techniques (Vickers and Knoop):

  • Vickers Microhardness ($HV$):

  • Employs a square-base diamond pyramid indenter with a face angle of $136^\circ$.

  • Formula:       $HV = \frac{1.854 P}{d_1^2}$       where $P$ is the load in $\text{kg}$ and $d_1$ is the mean length of the two indentation diagonals in $\text{mm}$.

  • Knoop Microhardness ($HK$):

  • Employs an elongated diamond pyramid indenter producing an indent with a length-to-width ratio of $l/b = 7.11$ and width-to-depth ratio of $b/t = 4.00$.

  • Formula:       $HK = \frac{14.2 P}{l^2}$       where $P$ is the load in $\text{kg}$ and $l$ is the length of the long diagonal in $\text{mm}$.

Design and Safety Factors

• Engineering Margin of Safety:

  • Structural designs account for uncertainties in service loading conditions, environmental degradation, manufacturing flaws, and statistical variances in material properties.

  • To guarantee that mechanical components operate strictly within the elastic regime without undergoing unexpected yielding, allowable working stresses are defined relative to the yield strength.

• Working Stress and Factor of Safety Formulas:

  • The design working stress ($\sigma_{\text{working}}$) is defined as:     $\sigma_{\text{working}} = \frac{\sigma_y}{N}$     where $\sigma_y$ is the yield strength and $N$ is the design factor of safety.

  • Typical values of $N$ range between $1.2$ and $4.0$.

• Sizing Problem: Cylindrical Rod Under Tensile Loading:

  • Objective: Calculate the minimum required diameter ($d$) to prevent yielding in a cylindrical rod made of 1045 plain carbon steel subjected to an axial tensile force $F = 220{,}000\,\text{N}$, using a factor of safety $N = 5$.

  • Material properties for 1045 plain carbon steel:

  • Yield strength: $\sigma_y = 310\,\text{MPa} = 310 \times 10^6\,\text{N/m}^2$

  • Tensile strength: $TS = 565\,\text{MPa} = 565 \times 10^6\,\text{N/m}^2$

  • Step 1: Compute allowable working stress:     $\sigma_{\text{working}} = \frac{\sigma_y}{N} = \frac{310 \times 10^6\,\text{N/m}^2}{5} = 62 \times 10^6\,\text{N/m}^2 = 62\,\text{MPa}$

  • Step 2: Formulate the cross-sectional area in terms of diameter $d$:     $A_0 = \frac{\pi d^2}{4}$

  • Step 3: Relate working stress to force and cross-sectional area:     $\sigma_{\text{working}} = \frac{F}{A_0} = \frac{F}{\frac{\pi d^2}{4}} = \frac{4 F}{\pi d^2}$

  • Step 4: Solve algebraically for diameter $d$:     $d = \sqrt{\frac{4 F}{\pi \sigma_{\text{working}}}}$

  • Step 5: Substitute numerical parameters:     $d = \sqrt{\frac{4 \times 220{,}000\,\text{N}}{\pi \times (62 \times 10^6\,\text{N/m}^2)}}$     $d = \sqrt{\frac{880{,}000}{1.9478 \times 10^8}} = \sqrt{4.5179 \times 10^{-3}} \approx 0.0672\,\text{m}$

  • Minimum required rod diameter:     $d = 0.067\,\text{m} = 6.7\,\text{cm}$