Elasticity of Materials PDF

Lesson 4 ELASTICITY OF MATERIALS CONTENTS STRESS AND DEFORMATION all units) is test Cof Theory TRACTION AND CONTRACTION. YOUNG'S MODULUS BENDING OTHER ELASTIC PROCESSES: TORSION, SHEAR AND COMPRESSION BIOLOGICAL APPLICATIONS: ELASTIC OF BONES AND MUSCLES STRESS AND DEFORMATION CAUSE → EFFECT...

Lesson 4 ELASTICITY OF MATERIALS CONTENTS STRESS AND DEFORMATION all units) is test Cof Theory TRACTION AND CONTRACTION. YOUNG'S MODULUS BENDING OTHER ELASTIC PROCESSES: TORSION, SHEAR AND COMPRESSION BIOLOGICAL APPLICATIONS: ELASTIC OF BONES AND MUSCLES STRESS AND DEFORMATION CAUSE → EFFECT CAUSE: Stress (deforming agent) EFFECT: Deformation (elongation, contraction, shear, compression, torsion…) • Permanent: Non elastic material • Non permanente: elastic material STRESS AND DEFORMATION Flexion → Buckling Traction → Elongation Torsion → Twisting Contraction → Shortening Shear → Cutting TRACTION AND CONTRACTION Elastic Limit Tension s ∆𝐿 𝐷𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 = 𝐿 𝐹 𝑇𝑒𝑛𝑠𝑖𝑜𝑛 = 𝑆 Young’s Modulus (𝚼) Constant measuring how deformable a material is (linear region) 𝐹 𝑡𝑒𝑛𝑠𝑖𝑜𝑛 𝑆 Υ= = 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 Δ𝐿 𝐿 O Breaking point Linear limit Deformation REGIONS OB: Elastic region (OA linear & AB non linear) BC: Plastic Zone OA: Linear dependence. Hooke’s Law. B: Elastic Limit. ↑ Tension (B) Permanent deformation C: Breaking Point ↑ Tensión (C) the bar breaks. TRACTION AND CONTRACTION • Cause: Force (F) perpendicular to S • Effect - Deformation parallel to F: change in length (Δ𝐿). Length L and section S. Υ: Young’s Modulus =Υ • Effect - Deformacion perpendicular to F: change in radius (Δ𝑟). Radius r and length L 𝜎: Poisson’s coefficient = −𝜎 • The stress at which failure occurs is called the tensile strength or, in the case of compression, the compressive strength. For bone, the tensile and compressive strength is equivalent to 200 and 270 MN/m2 • The representative value of Young's modulus for a bone is different for tensile and compressive stresses, which are 16 and 9 GN/m2, respectively. Biological explanation: The main purpose of bones is to resist compressive loads exerted by contractile muscles. The biceps of the right arm of a certain person has a maximum cross-sectional area of 12 cm2. What is the tension in the muscle if it exerts a force of 300 N? Solution: 230769.23 N/m2 A wire 1.5 m long has a cross section of area 2.4 mm2. It hangs vertically and stretches 0.32 mm when a 10-kg block is tied to its lower end. Find (a) the stress, (b) the strain, and (c) the Young's modulus for this wire. BENDING • Cause: Bending Moment, perpendicular to the axis (M) • Effect: Bending (𝜒) 1 𝜒= 𝑀 𝛶𝛤 Γ = ∫ 𝑧 𝑑𝑆, S being the section where M is applied Γ : depends on how far the deforming fibers are from the neutral fiber. Farther fibers means greater Γ. SHEAR • Cause: Force (𝐹 ) parallel (tangential) to 𝑆 • Effect: variation of the angles of the edges (𝜃) s 𝑆ℎ𝑒𝑎𝑟 𝑡𝑒𝑛𝑠𝑖𝑜𝑛 = 𝑆ℎ𝑒𝑎𝑟 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 = ∆ = tan 𝜃 , 𝑆ℎ𝑒𝑎𝑟 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 = 𝐹 𝑆 𝜃 is the shear angle 𝑆ℎ𝑒𝑎𝑟 𝑇𝑒𝑛𝑠𝑖𝑜𝑛 𝐹 = 𝑆ℎ𝑒𝑎𝑟 𝐷𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑆 𝑡𝑔𝜃 TORSION AND COMPRESSION TORSION COMPRESSION • Cause: Torque momentum parallel to the axis (M) • Effect: Torsion Angle (𝛼) • Cause: increase in pressure (Δ𝑝) • Effect: decrease in volume (Δ𝑉) Is the only elastic deformation for both fluids and solids. z p dz p p p p dy p dx y x 𝑀 =constant 𝐾 𝛼 angle betweenthe fore and es 𝑇𝑜𝑟𝑠𝑖𝑜𝑛 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡: 𝐾 = 𝑓(Υ, 𝜎) Δ𝑝 = −𝜒 Δ𝑉 𝑉 Υ 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑏𝑖𝑙𝑖𝑡𝑦 𝑚𝑜𝑑𝑢𝑙𝑢𝑠: 𝜒 = 3 1 − 2𝜎 𝜒 > 0, 𝑡ℎ𝑒𝑛 𝜎 < 0,5 BIOLOGICAL APPLICATIONS BONES • Very slightly deformable material. • Support and protection function:  High Young's modulus value (low deformability)  High modulus of rupture value (avoid fractures)  Lowest possible weight. • Long Bones (e.g.: femur)  Supports large compressions and prevents buckling: epiphysis.  High resistance to deformation: structure of the diaphysis (resistant bone material tube surrounding a soft part (bone marrow) that achieves maximum resistance to bending.  Existence of trabeculae (small partitions) that give consistency to the bone: different for each bone and for each person. BIOLOGICAL APPLICATIONS BONES • Composite structure of bones:  Composite: Material whose mechanical properties exceed those of their respective components  Formed by fibers of a hard and resistant element with high Young's modulus, embedded in a soft and deformable medium with a very low modulus of elasticity  Structural part of bone is made up of mineral bone lamellae (calcium and magnesium phosphates and carbonates with a chemical structure similar to apatite) surrounded by collagen.  Structure of the body has very little deformability (mineral matter) and is capable of withstanding high stresses and impacts without causing cracks that would propagate rapidly if collagen did not exist. BIOLOGICAL APPLICATIONS MUSCLES: Muscular contraction mechanism • Passive elastic behavior:  Muscles lengthen when subjected to external forces and shrink when external forces disappear. • Active contractile behavior:  The voluntary or spontaneous shortening of the muscles is not an elastic phenomenon but some parts are inserted inside others having, when extended, a much greater length than when it is furled.  Striated muscles are composed of: fibers (separated by a cell membrane) → fiber bundle → fibril → filaments (thick and thin).  The arrangement of the filaments in a periodic structure: SARCOMERE, functional unit of the fibril.  During muscle contraction, thin filaments are shortened by sliding over thick ones. BIOLOGICAL APPLICATIONS MUSCLES as a Linear Motor • In the contraction mechanism, the muscle acts as a motor of linear displacement (only in one direction): it exerts forces when contracting but not when extending.. • Two-way movements are produced by antagonistic muscles. • The muscle can operate in two ways during contraction.  ISOMETRIC: the muscle exerts force without contracting (example: trying to move an object without success).  ISOTONIC: the muscle contracts by exerting a constant force. • When the muscle has to perform a force, nerve impulses are generated, which causes the sarcomeres to contract. • If the external force is greater than that carried out by the muscle, its elastic elements are stretched, but there is no movement (isometric contraction).. • If there is a possibility of movement, the contraction of the active elements of the muscle is accompanied by a deformation of the passive elastic elements, first lengthening to generate force, and then contracting to generate movement.