2024 Fall CIVL1113 Engineering Mechanics & Materials - Material 2 Fundamentals PDF

2024 Fall- CIVL1113 Engineering Mechanics & Materials Material 2 Fundamentals Prof. Jing YU HW627, Department of Civil Engineering The University of Hong Kong Email: [email protected] （Consultation: Tue 10:30-...

2024 Fall- CIVL1113 Engineering Mechanics & Materials Material 2 Fundamentals Prof. Jing YU HW627, Department of Civil Engineering The University of Hong Kong Email: [email protected] （Consultation: Tue 10:30-11:30） Content Elasticity Plasticity Fracture Fatigue Creep & Relaxation 2 Elasticity: Linear Elastic Behavior All materials behave elastically at low stress levels Stress σ On unloading, the deformation is fully recovered Most materials also exhibit a linear relationship between stress and strain Loading/Unloading follows the same line Linear elasticity is the basic assumption for engineering analysis in many cases Strain ε 3 Elasticity: Physical Basis Materials consist of atoms held together by bond force Bond force varies with distance between atoms in a linear manner when displacement is small The bonding between adjacent atoms can be modelled as linear springs When force is applied, the atoms have to move apart or closer to Bond behaves like spring each other to maintain equilibrium In reality, bond force extends Displacement of atom as bond stretch under load beyond the first layer of atoms 4 Elasticity: The Poisson’s Effect There are interactions between atoms in all directions Interactions of atoms along directions inclined to the loading direction lead to lateral deformation, which is known as the Poisson’s Effect Vertical component of spring force pulls atoms away from one another Horizontal component of spring force moves atoms closer to one another 5 Elasticity: Definitions of E and ν Under loading along the x-direction Young’s modulus E = stress/strain = σx/εx σx εy σx Poisson’s ratio ν = - (εy/εx) εx with the y-direction perpendicular to the loading The negative sign results in a positive Poisson’s ratio for most common materials Post-Class Exploration: Material with negative Poisson’s ratio In addition, shear modulus (G) is defined as the ratio between shear stress and total shear strain γ For an isotropic material (with the τ G = τ/γ same properties in all directions) G = E/[2(1+ ν)] Derivation: Any book on Theory of Elasticity 6 Modulus and Nature of Bonding Magnitude of E governed by the strength of bond between atoms Materials with primary bond (covalent, ionic or metallic bond) exhibit high stiffness Ceramics, Metals Materials with secondary bond (hydrogen bond, van der Waal’s force) exhibit low stiffness Polymers, Timber Consists of covalently bonded chains held together by weak secondary bonds or occasional cross-links Relative sliding of chains lead to low modulus 7 E Value for Common Materials Diamond 1000 Wood (// grain) 9-16 Steel 190-207 Ice (H2O) 9.1 All values in GPa Aluminum 69 Epoxies 3-6 (no need to remember these values) Glass 69 Polyesters 1-5 Concrete 20 - 40 Wood ( | grain) 0.6-1 E governs the stiffness of a material but does not reflect its strength Aluminum and glass have similar E value but very different strength and failure mode Post-Class Exploration: What causes higher strength? 8 Engineering Significance of Young’s Modulus Together with member size, E governs deflection of a structure Important for tall buildings and long-span bridges where design is governed by deflection When material with lower E is used (e.g., aluminum replaces steel to reduce weight and prevent environmental corrosion), deflection should be checked Members made of material with lower E are easier to fail by buckling Moment of inertia can be increased to increase bucking load Material damage results in reduction in E and wave speed in materials is proportional to the square root of E Measurement of wave speed provides a means for non-destructive damage detection of structural members 9 Modulus of Composite Materials Composites can be made in different ways Layers of different materials Particles inside a matrix Fibers inside a matrix It is of interest to see how the modulus of the composite relates to those of its components Here, only two limiting cases, with the components working in parallel and in series, are considered CASE 1 CASE 2 Loading perpendicular Loading along aligned direction to aligned direction The parallel case (Case 1) applies to reinforced concrete members 10 Loading along the Fiber Direction σ Phase B: area= VbA Phase A: area= VaA Both phases are under the SAME STRAIN ε εa = εb = ε The stresses in the two phases are then given by: σa = Eaεa = Eaε σb = Ebεb = Ebε The force carried by each phase is equal to the stress in each phase multiplied by the area: Fa = VaA σa = VaA Eaε Fb = VbA σb = VbA Ebε Force equilibrium requires F = Fa + Fb, which gives: Aσ = (VaA Ea + VbA Eb) ε or, E = σ/ε = (VaEa + VbEb) (Rule of Mixture) 11 Loading perpendicular to the Fiber Direction (Optional) σ H Phase B: total length = VbH Phase A: total length = VaH Both phases are under the SAME STRESS σ σa = σb = σ The strains in each of the two phases are: εa = σ / Ea εb = σ / Eb The total extension in phases A and B are given by: ea = εa Va H eb = εb Vb H For the composite, the total extension is given by e = εH, where ε is the average strain over the thickness. Since e = ea + eb, εH = (σ/Ea)VaH + (σ/Eb)VbH or E = σ/ε = ( Va/Ea + Vb/Eb )-1 12 Upper and Lower Bounds of Modulus (Optional) E Parallel Model Ea Eb Series Model Va 1.0 The parallel and series models give upper and lower bounds of the modulus Experimental data for composite with any geometry and arrangement of the phases should lie within these bounds If not, either there is error in the measurement OR there are additional phases (e.g., phase formed at the interface of materials, voids) that have not been considered 13 Reinforced Concrete Column under Compression 200 Applied loading = 1000 kN Es = 200 GPa, Ec = 26.7 GPa The volume fraction of steel is given by: 6π(12.5)2/(200x400) = 0.0368 400 Since steel and concrete are loaded in parallel, the effective modulus is given by: E = (0.0368) (200) + (1-0.0368) (26.7) = 33.08 GPa φ25 rebar Shortening of the column [1000 x 103 (N) / (200 x 400 (mm2)x 33.08 x 103 (N/mm2) )] x 3000mm = 3.78 x 10-4 x 3000 (mm) = 1.134 mm F/A L E Stress in steel = 3.78 x 10-4 x 200 x 103 (N/mm2) = 75.6 N/mm2 (or 75.6 MPa) Stress in concrete = 3.78 x 10-4 x 26.7 x 103 = 10.1 MPa 14 Plasticity: Physical Basis Under low stress levels, elastic behavior arising from bond force between atoms Two puzzles about yielding If yielding is related to the breakage of bonds, the yield stress calculated from bond forces between atoms should be around one order of magnitude below the Young’s modulus However, yield stress of metals is two to three orders of magnitude below E The variation of bond force (which is arising from electrostatic attraction/repulsion) with atomic distance DOES NOT have the same shape as the stress vs strain curve of metals Explanation: theory of dislocations 15 Plasticity: Dislocation Movement under Shear Stress Dislocation A C A C A C E A C E G A C E G B D B D B D F B D F H B D F H b Under shear stress, bonds can break and reform one by one During this process, an extra plane of atoms will form and this is called a dislocation Movement of the dislocation from one side to the other side of a block of atoms results in relative movement equal to the atomic distance Continuous application of shear stress enables this process to continue, leading to the increase in strain associated with yielding 16 Plasticity: Dislocations and Yielding Behavior In reality, dislocations are pre-existing in materials Under shear stress, the material can be considered as making up of many elastic blocks sliding relative to one another due to movement of the dislocations The sliding due to dislocation movement is not recoverable but the elastic deformation between the blocks can be fully recovered There is hence residual strain after unloading Unloading slope is equal to initial elastic slope ∆ due to sliding Before Elastic After Full Loading Stage Yielding Unloading 17 Implications of the Dislocation Theory Yielding is resulted from shear stresses Under direct tension, yielding would occur along inclined planes Any criterion for yielding to occur must be based on shear Yielding along inclined planes Yielding is accompanied by sliding of atomic blocks in different directions – Blocks get into the way of one another, making movement more difficult – Increased stress is required for continued movement Hardening By increasing the resistance to dislocation movement (e.g., through alloying), the yield strength can be increased, but ductility may be reduced – High strength steel often less ductile than low strength steel 18 Modeling of Plastic Behavior of Materials Stress Rigid-Perfectly Plastic Model Elastic-Perfectly Plastic Model Strain The elastic-perfectly plastic model (linear elastic behavior followed by constant stress after yielding) is often adopted in structural analysis It is a good approximation for mild steel It simplifies analysis for hardening material and gives conservative results If one is only interested in the prediction of collapse load, a rigid perfectly plastic model neglecting the elastic part can be used 19 Yielding under General Loading Conditions σy τxy Prof. Wang will teach you: σx 1) Tresca yield Criterion 2) Von Mises yield Criterion Question: What is the combination of axial and shear stresses for yielding to occur ? Answer is provided by the yield criterion A physically correct yield criterion must be related to shearing of the material 20 Effect of Multiaxial Tension on Plastic Behavior According to both criteria, tensile yield σy strength along a certain direction is increased if tension is also applied in the lateral directions τxy This is consistent with test results σx However, increase in yield strength is accompanied by reduction in ductility This may be a problem in some cases Welding of steel members may result in multiaxial stress states leading to brittle failure with limited plastic yielding 21 Plastic Behavior of Parallel System (Optional) A parallel system is defined as one in which loading is carried by a number of members working together Many real structures are parallel systems Building with many columns plus an internal core to carry the vertical load and wind load Bridge with multiple spans, each contributing to carrying the traffic load of other spans In a parallel system, members usually do not fail at the same time The behavior of parallel system with ductile members is illustrated by a simple example 22 Plastic Behavior of Parallel System: Key Observations (Optional) σ F u - uy 1 3 2 σy AE/L 2L L Rigid Fy Post-Class Derivation: Bar (if you are interested) εy ε 2AE/L u : applied displacement F: total Force uy u After yielding, the stiffness of the system (i.e., change of Force vs change in displacement) is reduced Unloading after yielding follows the initial slope Residual displacement exists There will be residual stresses in the members With sufficient ductility, ultimate load capacity is the sum of the values for individual members Not the case for brittle materials 23 Fracture (Crack) Mode I – Opening mode (a tensile stress normal to the plane of the crack) Mode II – Sliding mode (a shear stress acting parallel to the plane of the crack and perpendicular to the crack front) Mode III – Tearing mode (a shear stress acting parallel to the plane of the crack and parallel to the crack front) 24 Fast Fracture Fast fracture is the sudden failure of materials due to crack propagation Unlike plastic failure accompanied by large deformation, brittle fracture gives little warning Cracks exist in materials due to Imperfect densification during material forming Damage/scratches during transportation and installation Repeated loading which may cause cracks to form Welding resulting in residual stresses and phase change Materials exhibiting brittle failure include glass, rock and plain concrete Concrete considered quasi-brittle due to crack bridging effect of aggregates Under certain circumstances, materials that are normally ductile can also undergo brittle fracture 25 Physical Basis of Fast Fracture When loading is applied to a member with a crack, the stress in front of the crack is very high reaches infinity at the crack tip if the material stays elastic As no real material can carry infinite stress, yielding (or inelastic behavior such as microcracking) will occur within a small region in front of the crack tip Loading Yielding within a small region Direction may not lead to crack Stress propagation Critical The crack propagates if the Stress energy released during the Distance from Inelastic propagation is sufficient to Zone Crack Tip overcome the energy required 26 Physical Basis: Energy Required for Crack Propagation Atoms on the surface of materials are under unbalanced force so their energy level is higher It takes energy to create new surfaces When the crack propagates, the inelastic (or yielding) zone in front of the crack will also increase in size Additional energy is required Surface atom: Bond force is unbalanced Extension of Inelastic Zone Internal atom: Bond force balanced Additional Crack Area formed on all sides due to crack propagation 27 Physical Basis: Where is Energy Coming From (Optional) Any member will become easier to deform if a crack in it increases in size If displacement is fixed, loading will decrease energy is released If load is fixed, it will move with increased displacement so there is work done Part of the work done stored as additional energy in the system The remaining part is available for crack propagation For small da, the available energy is the same for both cases P,∆ Load Before Crack Load d∆ Propagation Work Done = Pd∆ P da Before a Energy Additional Released Energy Stored After Crack = Pd∆/2 Propagation After Displacement Displacement Crack extended by da Fixed Displacement Fixed Load 28 Criterion for Fracture to Take Place (Optional) Physically, fracture will take place when the energy released (G) from the system reaches the critical energy release rate (Gc) of the material Under small scale yielding (i.e., when the yield zone or inelastic zone in front of the crack tip is very small) G = K2/E, where K is the stress intensity factor which governs the stress field in front of the crack tip The fracture criterion is then given by: K = Kc = (EGc)1/2 Kc is also called the fracture toughness which can be obtained from testing of a cracked member with known K Stress When the crack tip is approached, K the variation of stress with σ= distance from crack tip is 2πr independent of loading configuration Distance from Crack Tip, r 29 Transition between Gross Yielding and Brittle Fracture (Optional) Consider the simple case with a small through crack of size 2a within a wide plate with width >>a Crack size = 2a Under uniaxial stress σ, K is given by σ(πa)1/2 Stress for brittle fracture is: σF = Kc/(πa) 1/2 Stress for gross yielding (the whole section to yield) is: σ = σY Transition between yielding and fracture occurs at crack size aT given by: aT = (1/π) (Kc/σY)2 30 Transition between Gross Yielding and Brittle Fracture (Optional) Gross Brittle Yielding Fracture σ σ = σy Gross Yielding if a < aT Brittle Fracture if a > aT σ = K c / πa aT Crack Size, a With decreasing temperature or increasing strain rate, σY increases as less energy or time is available for dislocation movement With higher yield strength, the yield zone in front of the crack tip is smaller, so Kc is reduced aT becomes smaller so it is easier for brittle failure to occur With cyclic loading, a crack can grow in size May lead to reduced failure stress and change to a brittle failure mode 31 Fatigue Fatigue is the reduction in material strength under repeated cyclic loading Assuming full stress reversal (cycling between tension and compression of the same magnitude), strength will decrease with the number of cycles The plot of strength or stress amplitude (i.e., the variation of stress) vs number of cycles is called the S-N diagram There is usually a threshold which represents the lower bound of strength when the number of cycles is very high In experiments, the stress is fixed to find the number of cycles to failure In practice, the allowable number of cycles is the design criterion and the corresponding strength (or stress amplitude) is determined σ Stress Fatigue Amplitude Strength Threshold time Number of Cycles, N 32 Physical Basis of Fatigue of Metals (Optional) When there are locations of stress concentration (bends, corners), yielding under cyclic loading leads to the formation of surface extrusions and intrusions The intrusion acts like a notch to initiate crack propagation With continued cyclic loading, new crack surfaces are formed and folded forward during unloading to extend the crack Amount of crack growth depends on the difference in opening at maximum and minimum stress which is governed by ∆σ or ∆K Extrusions K min Preferred Sliding Direction Crack Crack K max New Surface Formed Initiation Intrusions produces Propagation a Sharp Notch New Surface Folds Forward K min Crack Initiated from Notch Tip 33 Fracture-Based Fatigue Modeling (Optional) Based on test data, the crack propagation rate was found to be related to the variation of K during cyclic loading as follows: Paris’ Law (Paris–Erdogan equation): da/dN = C(∆K)n a is the crack length. da/dN is the fatigue crack ∆K = K(σmax) – K(σmin) growth for a load cycle. Material coefficients C and n are obtained experimentally and also Number of Cycles to Failure is then: depend on environment, 𝑎𝑎𝑓𝑓 frequency, temperature 𝑑𝑑𝑑𝑑 𝑁𝑁𝑓𝑓 = and stress ratio 𝑎𝑎𝑖𝑖 𝐶𝐶[Δ𝐾𝐾]𝑛𝑛 ai : Initial Crack Size obtained from Inspection ( if no crack is found, take ai as the detection threshold, i.e. the smallest size that can be detected) af : Final Crack Size obtained from K(σmax, af) = Kc 34 Phenomenon of Time Dependent Behavior Strain Constant Stress Stress Constant Strain Creep Strain Instantaneous Elastic Strain Time Time Creep: increase of strain with time Relaxation: decrease in stress with time If creep/relaxation behavior is linear (i.e., the ratio of stress and strain at a given time is constant), one can define: Creep Compliance J(t) = ε(t)/σ (under constant stress) Relaxation Modulus Er(t) = σ(t)/ε (under constant strain) J(t) and Er(t) can be determined from testing 35 Physical Basis of Time-Dependent Behavior Atoms or molecules can re-arrange under loading and the process takes time Illustration with polymers Stiff polymer chains interact with each other through weak bonding (van der Waal’s forces) Weak bonds can break and reform, enabling sliding between polymers Polymer chains Chains stretched by applied strain Chains stretched under loading Chains shorten and relax due Sliding leads to strain increase with time to relative sliding 36 Stress-Strain Behavior of Time-Dependent Material (Optional) High Loading rate σ Intermediate Loading Rate Low Loading Rate ε Time dependent behavior due to internal movements of molecules High loading rate – Insufficient time for movement, stiff response but similar loading/unloading behavior Low loading rate – A significant portion of the movement can occur, flexible response but similar loading/unloading behavior Intermediate loading rate – Most significant difference between loading and unloading behavior – Highest energy absorbed in each cycle 37 Implications to Structural Design Long term deformation may be much higher than that at the short term Should provide enough allowance between attachments (such as wall panels) to the primary structure As members in a building may not be under the same stress, they deform to different levels. The long term difference due to creeping has to be checked Damping of structures under vibration due to energy absorbed during each loading cycle Damping should be frequency dependent (as loading/unloading rate is proportional to frequency) but this is seldom considered in practice because damping of structures is hard to quantify Creeping and relaxation affects prestressed concrete members Creeping can lead to stress reversal 38 Implications to Structural Design: Prestressed Concrete Cable anchored on one side and Cable under tension anchored pulled on the other side on the other side as well Cable force released so it tends to shorten Beam creeps and shortens Eccentric compression induces upward bending Steel cable shortens with the beam Stress in cable is reduced Prestress effect is partially lost Pre-stress is often applied to concrete beams to reduce deflection by inducing displacement in opposite direction Loss of prestress occurs due to Creeping of concrete as illustrated above Relaxation behavior of the steel cable itself Re-stressing the cable may be necessary 39 When is time-dependent behavior important Movement of molecules requires energy More significant when temperature is high Homologous temperature is defined as: T/Tm, where Tm is the melting point of the material with T and Tm in absolute temperature (deg C + 273) Creeping becomes important for metals when T/Tm > 0.3-0.4, and for ceramics when T/Tm >0.4-0.5 For polymers, creeping becomes significant when the glass transition temperature is approached as van der Waal’s forces between chains start to become ineffective At room temperature, creeping is significant for concrete, wood, most polymers as well as lead, tin and glass Under working stress, concrete, wood and polymers exhibit linear creep behavior 40 Recap Linear Elasticity and Physical Basis Definitions of E and ν, as well as G Modulus of Composite Materials: Rule of Mixture Plasticity Behavior and Theory of Dislocations Modeling of Plastic Behavior of Materials Three Modes of Fracture Physical Basis of Fast Fracture Fatigue (reduction in material strength under repeated cyclic loading) Time Dependent Behavior & Physical Basis Creep (increase of strain with time) Relaxation (decrease in stress with time) 41 Final remarks Linear elasticity is the basic assumption for engineering analysis in many cases Failure of Materials: brittle failure (fracture) & ductile failure (yielding) Depending on the conditions (such as temperature, state of stress, loading rate) most materials can fail in a brittle or ductile manner or both. However, for most practical situations, a material may be classified as either brittle or ductile. Think-Pair-Share: What is the difference between Material Failure & Structural Failure 42