Universit à degli Studi di Bari Università MECHANICS 3 prof. ing. Livio Quagliarella The body The body is a system modelled as: point Kinetics study rigid body Static study deformable body Stress strain description Body mass 2/45 Introduction • Knowledge of a material’s properties, and how it behaves under various loading conditions is essential in design –Materials are selected for specific applications dependent on their properties and characteristics • The behaviour of a material under load is markedly different depending on whether the material response is elastic or plastic Rheology Different materials deform differently under the same state of stress. The material response to a stress is known as rheology. Ideal materials fall into one of the following categories: o Elasticity. o Viscosity. o Plasticity. Mechanical analogues Stress-Strain relations σ = Cε Elastic σs = η Viscous dεs dt slip = 0 if σ s < μσ n Plastic The behavior of real materials is better described by combining simple models in series or parallel. For example: A visco-elastic (or Maxwell) solid: dashpot A visco-plastic (or Bingham) material: An elasto-plastic (Prandtl) material: spring Elasticity The one-dimensional stress-strain relationship may be written as: σ = Cε n where C is an elastic constant. Note that: o The response is instantaneous. o Here the strain is the infinitesimal strain. The material is said to be linear elastic if n=1. Hooke’s law: σ = Cε In three dimensions, Hooke’s law is written as: σ ij = Cijkl ε kl where Cijkl is a matrix whose entries are the stiffness coefficients. It thus seems that one needs 81(!) constants in order to describe the stress strain relations. σ 11 = C1111ε11 + C1112ε 12 + C1113ε 13 + C1121ε 21 + C1122ε 22 + C1123ε 23 + C1131ε 31 + C1132ε 32 + C1133ε 33 σ 12 = C1211ε 11 + C1212ε 12 + C1213ε13 + C1221ε 21 + C1222ε 22 + C1223ε 23 + C1231ε 31 + C1232ε 32 + C1233ε 33 σ 13 = C1311ε 11 + C1312ε 12 + C1313ε13 + C1321ε 21 + C1322ε 22 + C1323ε 23 + C1331ε 31 + C1332ε 32 + C1333ε 33 σ 21 = C2111ε11 + C2112ε 12 + C2113ε13 + C2121ε 21 + C2122ε 22 + C2123ε 23 + C2131ε 31 + C2132ε 32 + C2133ε 33 σ 22 = C2211ε 11 + C2212ε 12 + C2213ε 13 + C2221ε 21 + C2222ε 22 + C2223ε 23 + C2231ε 31 + C2232ε 32 + C2233ε 33 σ 23 = C2311ε11 + C2312ε 12 + C2313ε 13 + C2321ε 21 + C2322ε 22 + C2323ε 23 + C2331ε 31 + C2332ε 32 + C2333ε 33 σ 31 = C3111ε 11 + C3112ε12 + C3113ε 13 + C3121ε 21 + C3122ε 22 + C3123ε 23 + C3131ε 31 + C3132ε 32 + C3133ε 33 σ 32 = C3211ε11 + C3212ε 12 + C3213ε13 + C3221ε 21 + C3222ε 22 + C3223ε 23 + C3231ε 31 + C3232ε 32 + C3233ε 33 σ 33 = C3311ε11 + C3312ε12 + C3313ε13 + C3321ε 21 + C3322ε 22 + C3323ε 23 + C3331ε 31 + C3332ε 32 + C3333ε 33 The case of isotropic materials o A material is said to be isotropic if its properties are independent of direction. o In that case, the number of non-zero stiffnesses (or compliances) is reduced to 12, all are a function of only 2 elastic constants. Young modulus: E= σy εy The case of isotropic materials Poisson’s ratio: ν =− εx εy Poisson’s ratio of incompressible isotropic materials equals 0.5; real materials are compressible and their Poisson ratio is less than 0.5. Elasticity • For the tensile bar, the external load has been assumed to be low enough that the bar will resume its initial shape once the external load is removed • This state of elastic σ deformation is possible only when the external load is within certain Elastic limit limits Loading • In the elastic range, the load-displacement or stress-strain curve is linear E Unloading – loading and unloading follow the same path ε Plasticity • If the loading is increased, it will reach a certain limit whereby elastic deformation would end and plastic deformation would start A yield strength or yield point is the material property defined as the • This limit is known as the stress at which a material begins to elastic limit and beyond this σ deform plastically. point the material is said to have yielded σy Elastic limit • The loading is thus beyond Loading the elastic limit Unloading –Permanent or irreversible deformation • Stress corresponding to yielding is called the yield strength, denoted by σy E Point of yielding ε Elastic vs. Plastic Behavior • If the strain disappears when the stress is removed, the material is said to behave elastically. • The largest stress for which this occurs is called the elastic limit. • When the strain does not return to zero after the stress is removed, the material is said to behave plastically. 2 - 14 Elastic-Plastic Stress-Strain Curves Ultimate tensile strength (UTS), often shortened to tensile strength (TS) or ultimate strength is the maximum stress that a material can withstand while being stretched or pulled before breaking. E = tan α = α σ ε True Stress-Strain Curve tensione di rottura curva nominale G= tensione di rottura tensione τ G= di snervamento γ Limite di snervamento G= E 2(1 + ν ) τ γ curva reale G= limite elastico τ G= γ limite di elasticità lineare Modulo di Young E = tan α = α E 2(1 + ν ) σ ε 16/62 STRESS-STRAIN TESTING • Typical tensile specimen • Typical tensile test machine load cell Adapted from Fig. 6.2, Callister 6e. specimen extensometer moving cross head gauge (portion of sample with = length reduced cross section) • Other types of tests: --compression: brittle materials (e.g., concrete) --torsion: cylindrical tubes, shafts. Stress-Strain Test 2 - 18 G= G= τ γ Stress-Strain Diagram: Ductile Materials E 2(1 + ν ) α E = tan α = 2 - 19 σ ε Stress-Strain Diagram: Brittle Materials 2 - 20 YOUNG’S MODULI: COMPARISON Metals Alloys 1200 1000 800 600 400 E(GPa) 200 100 80 60 40 109 Pa Graphite Composites Ceramics Polymers /fibers Semicond Diamond Si carbide Tungsten Al oxide Molybdenum Si nitride Steel, Ni <111> Tantalum Si crystal Platinum <100> Cu alloys Zinc, Ti Silver, Gold Glass-soda Aluminum Carbon fibers only CFRE(|| fibers)* Aramid fibers only AFRE(|| fibers)* Glass fibers only Magnesium, Tin GFRE(|| fibers)* Concrete GFRE* 20 10 8 6 4 2 1 0.8 0.6 0.4 0.2 Eceramics > Emetals >> Epolymers CFRE* GFRE( fibers)* Graphite Polyester PET PS PC CFRE( fibers)* AFRE( fibers)* Epoxy only PP HDPE PTFE LDPE Wood( grain) 13 TENSILE STRENGTH: COMPARISON Metals/ Alloys Tensile strength, TS (MPa) 5000 3000 2000 1000 300 200 100 40 30 20 Graphite/ Ceramics/ Polymers Semicond Composites/ fibers C fibers Aramid fib E-glass fib Steel (4140)qt Diamond W (pure) Ti (5Al-2.5Sn)a Steel (4140)a Si nitride Cu (71500)cw Cu (71500)hr Al oxide Steel (1020) Al (6061)ag Ti (pure)a Ta (pure) Al (6061)a Si crystal <100> Glass-soda Concrete Graphite AFRE(|| fiber) GFRE(|| fiber) CFRE(|| fiber) TS(ceram) ~TS(met) ~ TS(comp) >> TS(poly) Room T values Nylon 6,6 PC PET PVC PP HDPE Based on data in Table B4, Callister 6e. fiber) a = annealed fiber) hr = hot rolled fiber) ag = aged cd = cold drawn cw = cold worked qt = quenched & tempered AFRE, GFRE, & CFRE = aramid, glass, & carbon fiber-reinforced epoxy fiber) composites, with 60 vol% fibers. wood(|| fiber) GFRE( CFRE( AFRE( LDPE 10 wood( 19 1 Viscoelasticity Some features that are observed in polymeric materials that do not seem to be noticeable in metals or ceramics 1. Mechanical properties depend on Temperature 2. Mechanical properties depend on Strain Rate 3. Creep (noticed in metals at high temperatures) 4. Stress Relaxation 5. Hysteresis Comportamento viscoso La viscosità (μ) quantifica la resistenza dei fluidi allo scorrimento e dipende dal tipo di fluido e dalla temperatura; la viscosità si misura in Pascal per secondo, (Pa·s). Vi sono materiali solidi che sia pure in maniera assai minore rispetto ai fluidi mostrano fenomeni di scorrimento detto appunto scorrimento viscoso cui si associa il comportamento elastico, o plastico o elasto-plastico. Nei materiali a comportamento viscoso la deformazione dipende dal valore del carico, dalla sua velocità di applicazione, dalla temperatura e dal tempo di applicazione. Non esiste un valore di soglia; il materiale scorre fino a quando il carico è applicato. 24/62 viscous behavior Alcune proprietà dei materiali viscoelastici sono : • scorrimento viscoso" (o creep), in presenza di un carico applicato costante, la deformazione cresce con il tempo; • rilassamento se si mantiene costante la deformazione, lo sforzo decresce con il tempo; • la rigidità del materiale dipende dalla velocità di applicazione del carico; • in presenza di un carico ciclico, si verifica il fenomeno dell’isteresi (un ritardo periodico), con conseguente dissipazione (sotto forma di calore) di energia meccanica; Dipendono dalla temperatura ed iniziano a manifestarsi per: 1 T > Tf 3 T f è la temperatura di fusione dove Tf 25/62 Creep • Take a tension specimen made from a polymer and and put on a series of constant stresses on it. • We observe Creep: Progressive strain (deformation) over time at constant stress (load), usually at high temperatures General behavior: the creep curve • Three creep regimes are obtained: I. Primary or transient creep (decreasing strain rate) II. Secondary or steady-state creep (constant strain rate) III. Tertiary or accelerated creep (increase strain rate) Stress Relaxation • Think of a polymer specimen loaded with a constant strain. | Note that both linear elastic and viscous fluid behaviors are present. | Note that there seems to be some residual stress at the end, i.e. the material does release of tension not completely recover. There is both elasticity and plasticity. Stress Relaxation: Progressive loss of stress (load) over time under constant strain (deformation), usually at high temperatures Curva sforzo-deformazione solido viscoso Al variare della temperatura si ha un’altra famiglia di curve 29/62 Hysteresis • Polymers often don’t load and unload on the same line on the stress-strain curve. • The difference in areas under those curves represents energy loss (often to heat). • This means that polymers can have inherent energy damping. • This means plastic springs may not be as good an idea as plastic dampers. Hysteresis viscoelastoplastico viscoelastico • • • • Very complex behavior! Difficult to model. Great sensitivity to temperature. Great sensitivity to strain rate. 31/62 stiffness STRUCTURE PROPERTIES STIFFNESS DEPENDS ON : • MATERIAL • SHAPE AND DIMENTIONS • CONSTRAINTS TYPE AND DISTRIBUTION • LOAD SYSTEM 32/62 osso Bending stifness 33/62 osso torsion stifness 34/62 Stress concentration: Hole Discontinuities of cross section may result in high localized or concentrated stresses. K= σ max σ ave 2 - 35 Stress concentration: Hole σ0 ρt a x x’ 2a σ0 Per il materiali duttili, capaci di considerevole deformazione plastica, la presenza di fori, intagli ecc. non causa concentrazione degli sforzi tali da portare a rottura, perché ogni concentrazione degli sforzi viene limitata dalla deformazione plastica. 36/62 Stress concentration: Fillet chamfer 2 - 37 Fatigue Failure It has been recognized that a metal subjected to a repetitive or fluctuating stress will fail at a stress much lower than that required to cause failure on a single application of load. Failures occurring under conditions of dynamic loading are called fatigue failures. Fatigue failure is characterized by three stages Crack Initiation Crack Propagation Final Fracture Jack hammer component, shows no yielding before fracture. Crack initiation site Fracture zone Propagation zone, striation Crank shaft Gear tooth failure Fatica Hip stem prosthesis 41/62 Sollecitazioni dinamiche - Fatica 500x 42/62 1000x Fatigue Failure – Type of Fluctuating Stresses Alternating stress σa = most dangerous σmax σmin 2 Mean stress σm = σmax + σmin 2 Sollecitazioni dinamiche - Fatica breaking yield Endurance limit 2 1 La rottura per fatica avviene in modo fragile anche per materiali duttili. Le prove di fatica vengono condotte a vari valori di tensione e viene registrato il numero di cicli per causare la rottura. Per il dimensionamento degli organi meccanici si usano diagrammi ottenuti da tali prove in cui in ascissa è il numero di cicli e in ordinata la tensione massima. 44/62 Fatigue • Fatigue properties are shown on S-N diagrams. • A member may fail due to fatigue at stress levels significantly below the ultimate strength if subjected to many loading cycles. • When the stress is reduced below the endurance limit, fatigue failures do not occur for any number of cycles. 2 - 45 improve the fatigue strength Per migliorare la resistenza a fatica è possibile: • indurre uno stato di compressione sulla superficie; ad esempio eseguire lavorazioni di pallinatura; • effettuare trattamenti di indurimento superficiale; in qualche caso, i rivestimenti duri (ad es. l’anodizzazione dura), possono ridurre la vita a fatica, anche del 40-50%; • migliorare la finitura superficiale del pezzo; • ottimizzare la geometria: aumentando i raggi di raccordo, evitando intagli e spigoli interni; • utilizzare materiali appropriati. 46/62 Friction and Wear No relative movement Quando due corpi sono premuti l’uno contro l’altro nascono delle pressioni di contatto e uno stato di tensione quando i due corpi strisciano tra loro, a causa delle microasperità si ha l’usura relative movement 47/62 Tribology (dal greco tribos: strofinare o scorrere) Tribology comprises the science and technology of interacting surfaces in relative motion; that is, friction, lubrication and wear. Tribology is a vast and interdisciplinary subject, ranging from the fundamental physics of surface contact and adhesion to the application of advanced materials and lubricants to solve practical industrial friction and wear problems. What is Friction • Force tangential to the interface of two contacting bodies = Ff. –Dynamic and static –Dynamic produces heat Ff = μ N Friction Force Normal Force Assumptions: Ff independent of contact area, μ = constant Real Contact Area Schematic illustration of the interface of two bodies in contact showing real areas of contact at the asperities. In engineering surfaces, the ratio of the apparent-to-real areas of contact can be as high as 4 to 5 orders of magnitude. More Complicated Models Exist • Contact Mechanics In actuality, as N increases, contact area increases, thereby affecting μ. μ is a non-linear function of N. What else might μ vary with?? Definition of Surface Wear • Wear - Damage to a solid surface involving progressive loss of material due to contact and relative motion with another surface. 13 types of wear!! • Erosion – Damage to a solid surface involving progressive loss of material due to mechanical interaction between that surface and a fluid, impinging liquid or solid particles. 5 kinds of erosion Types of Friction • Kinetic friction occurs when force is applied to an object and the object moves. • Examples: Sliding Friction: pushing an object across a surface Rolling Friction: between wheels and a surface Fluid Friction: opposes the motion of objects traveling through a fluid (air or water) Affecting Friction • To reduce the amount of friction, apply a lubricant between two surfaces. • Motor oil, wax, and grease are examples. • Friction can also be reduced by rolling, rather than pushing, an object. Affecting Friction • Friction increases as surfaces are made rougher. • Friction increases when the force between two objects is increased. LUBRICATION Why do we need it? y y lower the friction prevent wear ↓ wear ↓ friction remove heat and contamination DUCTILITY, %EL • Plastic tensile strain at failure: Engineering tensile stress, σ Adapted from Fig. 6.13, L − Lo x100 %EL = f Lo smaller %EL (brittle if %EL<5%) larger %EL (ductile if %EL>5%) Lo Ao Af Lf Callister 6e. Engineering tensile strain, ε • Another ductility measure: %AR = Ao − A f x100 Ao • Note: %AR and %EL are often comparable. --Reason: crystal slip does not change material volume. --%AR > %EL possible if internal voids form in neck. 20 HARDNESS • Resistance to permanently indenting the surface. • Large hardness means: --resistance to plastic deformation or cracking in compression. --better wear properties. e.g., 10mm sphere apply known force (1 to 1000g) d D most plastics measure size of indent after removing load brasses easy to machine Al alloys steels file hard cutting tools Smaller indents mean larger hardness. nitrided steels diamond increasing hardness Adapted from Fig. 6.18, Callister 6e. (Fig. 6.18 is adapted from G.F. Kinney, Engineering Properties and Applications of Plastics, p. 202, John Wiley and Sons, 1957.) 28 La Resilienza la resilienza indica la capacità di un materiale di assorbire energia in caso di urto, ovvero di sopportare gli urti i materiali fragili assorbono poca energia i materiali duttili assorbono molta energia pendolo di Charpy 59/62 TOUGHNESS È l’energia necessaria per la frattura di un materiale sotto un carico statico. (UNDER STATIC LOAD) È rappresentata dall’area sottesa alla curva reale σ-ε della prova di trazione. 61/62
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