Elasticity (Stress-Strain & Moduli) Imagine every solid object—a steel beam, a rubber band, or a bone—is actually made of billions of tiny atoms connected by invisible springs. These 'springs' are the electromagnetic bonds holding the atoms together. When you apply a force (stress), you aren't just moving the object; you are stretching or compressing these billions of tiny springs. Elasticity is simply the measurement of how stiff these springs are and how well they snap back to their original position when you let go. If the springs are very stiff (like in steel), it takes a huge force to stretch them even a tiny bit (High Young's Modulus). If they are loose (like in rubber), they stretch easily. However, if you stretch the springs too far, they get permanently bent out of shape or snap entirely; that's when you cross the 'Elastic Limit' into 'Plastic Deformation'. Elasticity (Stress-Strain & Moduli) Solids are not perfectly rigid. Push, pull, or twist them and they change length, volume, or shape by tiny amounts. Inside, interatomic bonds act like springs that resist deformation and try to restore the original arrangement when the external force is removed. This restoring tendency is what we call elasticity. The everyday feel of stiffness is measured by elastic moduli. Young’s modulus tells how hard it is to change length of a rod or wire. Bulk modulus tells how hard it is to squeeze the volume of any material, including fluids. Modulus of rigidity (shear modulus) tells how hard it is to change shape at constant volume. Another important idea is Poisson’s ratio, which captures how stretching in one direction is accompanied by shrinking sideways. We also read the full story of a material from its stress–strain curve: first a straight line region that obeys Hooke’s law, then yielding and plastic flow, finally necking and fracture. Elastic energy stored during deformation powers devices from bows to tendons. In medicine, bone strength and tissue compliance are direct applications of these ideas. Throughout, we will be careful about conditions: small deformations, uniform specimens, and clear sign conventions. Once these are fixed, the math becomes a clean language for reasoning about real structures. Crowded elevator analogy: people prefer an equilibrium spacing. Push in (compression) and everyone pushes back; pull apart (tension) and they reach to stay together. Moderate disturbance and they restore; too much and the arrangement fails (plasticity or fracture). remember Elasticity Property of a body to regain original shape and size after deforming forces are removed, within its elastic limit. Deforming force External force that changes length, volume, or shape of a body. Restoring force Internal force developed in a deformed body that opposes the deformation; equal and opposite to the deforming force within the elastic limit. Stress Internal restoring force per unit area. Tensile/compressive along normal, shear along tangent, and hydraulic when applied from all sides. Fractional change in dimension: = X/X . Longitudinal ( L/L ), volume ( V/V ), or shear (small angle in radians). Strain Elastic limit Maximum stress up to which the body returns completely to original configuration on unloading. For small deformations: stress is proportional to strain; the constant of proportionality is an elastic modulus. Hooke’s law Stress is the internal reaction distributed over area and is defined as force per unit area on a cross-section. For normal loading, tensile stress tries to elongate while compressive stress tries to shorten. Shear stress is due to a tangential force per unit area that tends to slide one layer over another. Strain measures how much deformation occurs relative to the original size and is dimensionless. For small shear, the strain is the angle in radians (since when is small). These two, stress and strain, are the pair that Hooke’s law links in the elastic region. Stress equals restoring force per unit cross-sectional area. 7.1 Definition of Stress Units of stress are pascal, 1 , Pa =1 , N/m 2 . Dimensionally [M 1 L -1 T -2 ] . In many contexts stress acts uniformly only if the specimen is long, thin, and the load is applied axially without bending. Real labs reduce bending by using a pointer and hanger aligned with the wire. Definition of Strain Generic definition valid for length, volume, or angle (as appropriate). 7.2 Longitudinal strain is L/L for a rod under tension or compression. Volume strain is V/V for isotropic squeezing by pressure. Shear strain is the angular distortion in radians: if the top face of a block slides by x over height h , then x/h for small angles. Young’s modulus ( Y ) Ratio of longitudinal stress to longitudinal strain for a rod or wire within elastic limit. Ratio of uniform pressure change (hydraulic stress) to volume strain; measures resistance to compression. Bulk modulus ( B ) Ratio of shear stress to shear strain; measures resistance to shape change at (approximately) constant volume. Modulus of rigidity ( ) Negative of lateral strain divided by longitudinal strain; often reported as a positive number in magnitude. Poisson’s ratio ( ) Young’s Modulus of a Wire 7.3 For a uniform wire of length L , area A under a small axial load F producing extension L . Young’s modulus depends only on the material and temperature, not on the wire’s length or area. Thicker wires stretch less for the same load because L 1/A , not because their Y is different. Typical values: steel 2 10 11 , Pa , glass 6 10 10 , Pa , rubber 10 6 , Pa . Bulk Modulus 7.4 Negative sign ensures B>0 since V<0 for an increase in pressure. Bulk modulus quantifies compressibility. Water is slightly compressible with B 2.2 10 9 , Pa . Gases have much smaller effective B at constant temperature. A larger B means a smaller volume change for the same pressure change. Compressibility is C=1/B . Shear Modulus 7.5 Shear stress divided by shear strain ( in radians). Use this to determine a material's resistance to angular deformation when subjected to twisting or transverse loads. Shear deformations occur in riveted joints, beams under transverse loads, or when you twist a jelly cube. For small distortions, is small and in radians. Units of are pascal, like Y and B . Poisson’s Ratio 7.6 Lateral strain divided by longitudinal strain with a negative sign. Quantifies the ratio of lateral contraction strain to the applied longitudinal strain during stretching. When a wire is stretched, its diameter decreases. Most solids have 0< <0.5 . The upper bound 0.5 corresponds to perfectly incompressible behavior under uniaxial loading. Some special foams can have negative (auxetic), meaning they get fatter when stretched. tip Small-angle rule: shear strain is an angle in radians. Use only when is small (typically less than 10 ). For larger angles, the linear shear formula underestimates the true strain. Do not drop the minus sign in B=-V , P/ V . Many answers come out with the wrong sign if you take V as a negative number but forget the explicit minus, or vice versa. neet-alert B = - ,V , dP dV Isotropic, homogeneous material Small, uniform volumetric strain Hydrostatic loading (same pressure from all sides) Bulk modulus B = -V , dP dV Elastic energy density u= 1 2 Linear elasticity (Hooke’s law holds) Quasi-static loading from zero to final stress u = 1 2 , , Elastic Energy of a Wire Total elastic potential energy stored when a wire is stretched from zero load to F . 7.7 This energy represents the potential energy stored in the spring, which corresponds to the area under the linear portion of the stress-strain graph. On a stress–strain graph, the energy density equals the area under the curve up to the working strain. In the linear region this area is a triangle of base and height , giving u= 1 2 . Beyond yield the curve is not linear, so the area must be found by actual shape, not by the triangle shortcut. Bulk modulus relates pressure change to fractional volume change: B=-V , P/ V . High B means low compressibility. Energy stored per unit volume in elastic deformation is u= 1 2 in the Hookean region. For isotropic solids, the four constants Y, B, , are not independent. Knowing any two determines the others. Two standard relations are Y=3B(1-2 ) and Y=2 (1+ ) . These are derived from 3D linear elasticity under symmetry. Real exam questions often give Y and and ask for or B . 7.8 Links linear stiffness to volumetric stiffness with lateral contraction captured by . Moduli Relation 1 This relation holds for isotropic, linear elastic materials under conditions where stress and strain are measured in three dimensions. 7.9 Connects Young’s modulus to shear modulus for isotropic materials. Moduli Relation 2 This relationship is valid for isotropic, linear elastic materials under uniaxial stress conditions. Eliminate Y 7.10 Express Poisson’s ratio in terms of B and . This relationship is valid for isotropic, linear elastic materials subjected to small strains, connecting Poisson's ratio to the Bulk and Sh Another useful inversion identity between the three moduli. 7.11 Alternative Combined Relation This relationship holds for isotropic, homogeneous materials that are subjected to small strains and are assumed to obey linear elasticity t Bounds and limits: stability requires -1< <0.5 . Most solids lie in 0.2 to 0.4 . In the limit 0.5 (near-incompressible), Y 3B(1-2 ) forces B for finite Y . For =0 , there is no lateral strain and Y=3B=2 . Materials with negative (auxetics) expand laterally when stretched; they are rare and engineered. Material Y (Pa) B (Pa) η (Pa) Poisson’s ratio Indicative room-temperature values (orders of magnitude; vary by composition and treatment). Steel 2.0× 10 11 1.6× 10 11 7.5× 10 10 0.30 Copper 1.1× 10 11 1.4× 10 11 4.2× 10 10 0.34 Glass 6.0× 10 10 3.5× 10 10 2.6× 10 10 0.22 Bone (cortical) 1.0× 10 10 ≈0.3 Rubber 1.0× 10 6 3.3× 10 5 ≈0.49 Water (fluid) 2.2× 10 9 Steel cable cross-section transitioning into a blue atomic lattice connected by springs with arrows showing tensile force. Steel cable cross-section with labeled outer wires; a glowing cubic lattice with spring-like bonds shows atomic structure under tensile force. Side-by-side diagram comparing stiffness of steel and rubber using internal spring analogies. Infographic: steel bar with tightly packed springs (high Y ) vs rubber band with loose long coils (low Y ); deformation arrows marked. Visualization of plastic deformation beginning along slip planes with an inset stress–strain graph. Atomic lattice with orange slip planes at yield; inset shows stress–strain curve with elastic region, yield point, and plastic region. Linear up to proportional limit A, slight nonlinearity to elastic limit B, yield plateau near C, rising to ultimate tensile strength D with necking, then drop to fracture E. Stress Pa Typical stress–strain curve for a ductile metal. custom Proportional limit A Hookean End of elastic region B Elastic limit Yield point C Yield UTS D Maximum Fracture E Drop Stress dependent Strain control Strain Reading the curve: in OA, stress is proportional to strain and unloading traces the same line back to zero. Between A and B, Hooke’s law still approximately holds. Beyond B, permanent set appears after unloading. At C, plastic flow starts at nearly constant or slowly rising stress. D marks ultimate tensile strength (maximum engineering stress). After D, necking reduces the local area and the measured stress falls until break at E. Brittle materials show a short elastic region and fracture soon after the linear part. “Steel is more elastic than rubber” means that for the same small strain, steel requires much larger stress. High Y means stronger restoring tendency per unit strain. remember Breaking stress is a material property; breaking force depends on cross-sectional area: F break = break ,A . Breaking force is a property of the material. Stress is the same as pressure; both are external forces per area. Pressure is external and isotropic in fluids; stress is internal in solids (can be tensile, compressive, shear). Same unit, different physical meaning. Y–relations at a glance: “1-2, 1+” • Y=3B(1-2 ) → think ‘bulk resists two sideways squeezes’. • Y=2 (1+ ) → think ‘shear plus Poisson add-on’. Say: Y-three-B one-minus-two-sigma; Y-two-eta one-plus-sigma. tip Sign convention: take tensile stress and elongation as positive. Poisson’s ratio often reported without the negative sign; when using the definition, keep = - ,( lateral strain )/( longitudinal strain ) . Wire extension using Young’s modulus L = 2.0 m r = 1.0 mm = 1.0 10 -3 , m F = 100 N Y = 2.0 10 11 , Pa A steel wire of length L=2.0 , m and radius r=1.0 , mm is stretched by a load F=100 , N . For steel, take Y=2.0 10 11 , Pa . Find the extension. Extension L Use Y=FL/(A , L) with A= r 2 . easy m (≈ 0.32 mm) Scaling: for the same material and load, L L/A . Doubling the length doubles extension; doubling the radius cuts extension by a factor of four. This proportional reasoning is often faster than full calculation. In parallel, the extension is the same for both wires. Hence L=F 1 L/(Y A 1)=F 2 L/(Y A 2) . So F 1/F 2=A 1/A 2= r 1 2 /r 2 2 =1 2 /2 2 =1/4 . Also F 1+F 2=300 , N . Forces F 1, F 2 and common extension L Two copper wires of equal length L are clamped in parallel to a rigid support and carry a total load of F=300 , N . Their radii are r 1=1.0 , mm and r 2=2.0 , mm . Find the force shared by each wire and the common extension. Take Y Cu =1.1 10 11 , Pa , L=1.5 , m . Total load F = 300 N r1 = 1.0 mm, r2 = 2.0 mm Y = 1.1 10 11 , Pa L = 1.5 m N, m medium hard Pa (≈ 120 MPa, compressive on heating) A steel rod fixed between two rigid walls is heated by T=50 C . Take Y=2.0 10 11 , Pa and coefficient of linear expansion =1.2 10 -5 , K -1 . Find the thermal stress developed. Ignore buckling. ΔT = 50 °C Y = 2.0 10 11 , Pa α = 1.2 10 -5 , K -1 Thermal stress in constrained rod If expansion is fully prevented, strain = , T , so S=Y =Y T . Stress S Fix geometry first: compute area A= r 2 or thickness–width product. Decide loading mode: axial (use Y ), hydrostatic (use B ), shear/twist (use ). Check linear regime: strains 10 -3 typically safe for metals. For series (same force) add extensions; for parallel (same extension) add forces. For energy, use U= 1 2 F L=u , Volume . Wire problems: a reliable checklist Units and dimensions Stress/moduli: pascal (Pa) = N m -2 ; dimension [M L -1 T -2 ] . Strain: dimensionless (radian for shear). Energy density: J m -3 (same dimensions as pressure). Compressibility: Pa -1 . Energy density u= 1 2 is per unit volume. To get total energy, multiply by volume AL . Many options differ only by this factor. neet-alert Biophysical links: Bones behave roughly elastically for small strains, with Y 10 10 , Pa . Tendons store and release elastic energy during running. Blood vessels show nonlinear elasticity; the initial low-stiffness regime allows pulse smoothing while high-stiffness at larger strain prevents overexpansion. Understanding where Hooke’s law holds is key for safe physiological ranges. Quantity Symbol Formula Applies to Notes Stress σ=F/A All solids Tensile (+), compressive (−) by convention Strain ε=ΔX/X All deformations Dimensionless Young’s modulus Y=FL/(AΔL) Rods/wires Axial small strain Bulk modulus B=−VΔP/ΔV Solids, liquids, gases Uniform pressure Shear modulus η=( F t /A)/θ Solids θ in radians Poisson’s ratio σ (nu) −(Δd/d)/(ΔL/L) Solids 0<σ<0.5 typically Energy density u=½ σε Hookean region Area under curve How to use the visual tool: increase Applied Load to watch stress rise linearly with strain initially. Changing Material Stiffness rotates the initial slope (higher Y → steeper line). Increasing Wire Diameter reduces stress for the same load and thus reduces extension, even though Y stays the same. Extending Original Length increases absolute extension but not strain. Hooke’s law is valid only in the small-strain elastic region. Past the elastic limit, stress is not proportional to strain. Hooke’s law applies for all deformations until fracture. Do not mix true stress with engineering stress in necking. Unless stated, NEET problems use engineering stress F/A 0 with original area A 0 . neet-alert Thermal stress revisited: heating a free rod causes no stress because it expands. Stress arises only when expansion or contraction is restrained. The sign depends on the sense: heating with restraint gives compressive stress; cooling with restraint gives tensile stress. Many quick questions hinge on recognizing whether ends are fixed or free. Dimensional checks are powerful. From Y=FL/(A L) , units are ( N )( m )/( m 2 m )= N/m 2 . From u= 1 2 , units are Pa 1 = J/m 3 , equal to pressure units, which makes sense as energy per volume. Worked relation example: Given Y=2.0 10 11 , Pa and =0.3 , find B . Using B= Y 3(1-2 ) , B= 2.0 10 11 3(1-0.6) = 2.0 10 11 1.2 =1.67 10 11 , Pa . The large value implies near-incompressibility compared with axial stretch response. Shear visualization: consider a cube of jelly of height h . A small tangential force F on the top face of area A shifts the top by x . Shear stress is F/A . Shear strain is x/h . If you release it and no permanent distortion remains, the jelly is behaving elastically in shear; determines how strongly it resists this sliding. Why cannot exceed 0.5 for stable isotropic solids: if =0.5 , volume does not change for uniaxial stretch (perfect incompressibility). If it were larger, the equations predict negative bulk modulus, which would be unstable. This is why rubbers approach 0.5 but do not cross it. Microscopic picture of yielding: once stress crosses the yield point, dislocations in the crystal move, allowing layers of atoms to slide along slip planes at almost constant stress. That is why the curve flattens near the yield region. Work hardening at higher strain increases the stress again before necking. Practical measurement of Y : The Searle’s apparatus and optic-lever techniques use long thin wires to magnify tiny extensions. Errors arise from kinks, temperature drift, and misalignment. Using long length L and small radius r gives measurable L while still keeping strain small. Safety factors: Structures are designed so the working stress is far below yield. The ratio of breaking stress to working stress is the factor of safety. For surgical implants or bridges, fatigue life also matters, but in elementary calculations we restrict to static loads and elastic response. Choosing the right model: If both length and shape change matter, pick the dominant mode. A pressurized submarine hull needs B for volumetric change plus bending stiffness for shell stability. A guitar string primarily uses Y . A gel pad under shear uses . Common unit conversions: 1 , GPa =10 9 , Pa , 1 , MPa =10 6 , Pa . Engineering data sheets often list Y in GPa and strength in MPa. Keep consistent units when substituting. Example of compressibility: An air bubble in water shrinks slightly as depth increases because external pressure rises. For small changes, V/V - P/B . In water, even a pressure change of 2.2 , MPa (~220 m depth) reduces volume by only about 0.1 % . Energy interpretation: Storing elastic energy is like charging a mechanical capacitor. Young’s modulus corresponds to stiffness per strain, while energy density is proportional to square of strain for a Hookean solid. This is why small strains store little energy but large, unsafe strains store disproportionately more. Dimensional design tip: For a target extension limit L under load F , choose area A F L Y , L . For example, if F=500 , N , L=2 , m , Y=2 10 11 , Pa , and L =0.2 , mm , then A 2.5 10 -6 , m 2 , giving radius r 0.89 , mm . Temperature effect: With rising temperature, moduli generally decrease because atomic bonds soften. An exception is Invar (iron–nickel alloy) whose expansion coefficient and elasticity change very little with temperature; it is used where dimensional stability is critical. Elastic hysteresis: Ideally, loading and unloading paths coincide. Real rubbers show a loop due to internal friction; the area of the loop is energy dissipated as heat. For NEET-level numericals we usually ignore hysteresis unless stated. Strength vs stiffness: High Y means stiff; high breaking stress means strong; high area under the total stress–strain curve means tough. A tough material may not be very stiff. Rubber is compliant (low Y ) but can absorb much energy before breaking (high toughness). Surface effects in thin wires: As radius decreases to sub-millimeter, surface imperfections dominate failure, lowering the observed breaking stress. That is why test specimens are polished and carefully prepared for reproducible results. When formulas fail: Non-uniform cross-sections, bending loads, large deflections, and temperature gradients violate the simple one-dimensional Hooke model. The neat relations among Y, B, , assume isotropy and linearity; anisotropic crystals or composites need tensor descriptions. Worked shear example idea (conceptual): If a square gel block of side 10 , cm is displaced by 1 , mm at the top under a tangential force, the shear strain is 0.001/0.10=0.01 rad. For =3 10 5 , Pa (rubber-like), the shear stress is 3 10 3 , Pa . Volume change from B : A pressure increase of 1.0 , MPa on a 0.5 , m 3 tank of water gives V=-V , P/B=-0.5 10 6 /(2.2 10 9 ) -2.3 10 -4 , m 3 (about 0.23 , L ). Cross-checking relations numerically: With Y=2.0 10 11 , Pa and =0.3 , =Y/[2(1+ )] 7.7 10 10 , Pa . Plug into =(3B-2 )/(6B+2 ) to verify B 1.67 10 11 , Pa gives back 0.3 . Stress Internal restoring force per unit area: =F/A . normal stress shear stress hydraulic stress Fractional deformation: = X/X ; dimensionless. longitudinal strain volume strain shear strain Strain Axial stiffness: Y=FL/(A , L) . modulus of elasticity (linear) Young’s modulus Volumetric stiffness: B=-V , P/ V . incompressibility modulus Bulk modulus Shear response: =(F t /A)/ . shear modulus Modulus of rigidity Poisson’s ratio Lateral contraction per unit axial extension: =-( d/d)/( L/L) . nu Energy per unit volume in the elastic region: u= 1 2 . Elastic energy density Maximum engineering stress on the stress–strain curve before necking leads to fracture. Ultimate tensile strength (UTS) Key terms recap