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1、85-500-01 Stress-Strain Relationships and Behaviour,Dr. W. Altenhof, P. Eng.Department of Mechanical, Automotive and Materials EngineeringSunday, October 21, 2018,2,Introduction,We have discussed the following types of deformation: Elastic Plastic Creep Equations to describe stress-strain(-temperatu
2、re,-strain rate, etc.) behaviour are needed Developed to describe the behaviour of materials Basis of numerical (FEA) codes that calculate the behaviour of complex structures under complex loading conditions Necessary to adequately design parts and structures These equations are referred to as const
3、itutive equations which must account for: 3-dimensional behaviour anisotropy of materials,3,Rheological Models,Models which utilize springs, dashpots and/or friction to describe deformation behaviour are referred to as rheological models In the figure below stress, strain and strain rate are express
4、ed as a function geometry, deformation and applied load. Other considerations could also exist Temperature Others?,4,Rheological Models - continued,Elastic deformation is generally characterized by a proportional relationship between stress and strainGood approximation for metals and most polymers,P
5、lastic deformation is characterized by a threshold stress value, before which there is no plastic deformation: the yield stress (o). Permanent deformation results.,5,Rheological Models continued again,Steady state creep proceeds at a constant rate under a constant force. Velocity, , is proportional
6、to applied force. Dashpot constant c is proportionality constant. Permanent deformation results.,Transient creep involves a reduction in creep behaviour as time elapses. Spring results in recovery of deformation. Permanent deformation results.,6,Rheological Models elastic/plastic considerations,Diff
7、erent rheological models can describe elastic (spring) /plastic (frictional slider) behaviour Rigid (infinite E) perfectly plastic No deformation recovery during unloading Elastic perfectly plastic Recovery during unloading Elastic linear hardening Recovery during unloading,7,Rheological Models elas
8、tic/plastic linear hardening considerations,Consider the last model presentedTotal strain is combination of regions 1 and 2 (spring 1 and (spring 2 and frictional slider): Prior to yielding, slider prevents motion, so: (stress is carried through frictional slider) After yielding, slider exhibits con
9、stant stress. Stress within spring 2 is (-o), total strain becomes:Differentiation of above expression gives:Springs 1 and 2 in series, ET is lower than E1 and E2,8,Rheological Models creep behaviour,Creep deformation is described as a viscous behaviour Figure illustrates (a) steady-state creep with
10、 elastic strain (b) transient creep with elastic strain Elastic strain appears immediately with value /E1 and is recovered upon unloading Linear viscoelasticity assumed where:,For steady creep (a):Integrating 3rd equation yields:Creep strains are not recovered,9,Rheological Models creep behaviour,Fo
11、r transient creep (b):creep strain is analyzed by recognizing stress through dashpot and spring 2:separating variables and integrating gives (for a constant applied stress):total strain:,Removal of stress (recovery) results in strains asymptotically approaching zero Time necessary to recover c (esti
12、mated) can be determined from:,10,Rheological Models creep behaviour (relaxation),Previous cases dealt with situation where applied stress was constant Condition where constant applied strain may also exist For steady model:Resulting stress as a function of time:,11,Elastic Deformation,For materials
13、 loaded in one direction (assume x-direction) two mechanical properties are required to describe linear elastic behaviour Elastic modulus (E=x/x) and Poissons ratio (=-y /x= -z/ x) Stress application in x-direction will result in straining in y and z directions For elastic deformations is typically
14、0.3 and does not vary outside the range of 0 to 0.5 Linear elastic model is a good approximation of the actual elastic behaviour of many materials,12,Hookes Law in Three Dimensions,Applying principle of superposition:,Shear strains are not influenced by shear stresses on other planes, thus:,13,Volum
15、etric strain,Normal elastic strains (stretching atomic bonds) cause volume changes Elastic strains are small so L dL Shear strains cause distortion and no volume change Volumetric strain is defined by: Incorporating Hookes law gives: Physical limit on to be 0.5 If is equal to 0.5 no volume change fo
16、r any applied stress Values greater than 0.5 resultin volume reduction for tensilestress application - unrealistic,14,Hydrostatic (Mean) Stress,The hydrostatic stress is the average of the normal stresses Bulk modulus is ratio of hydrostatic stress to volumetric strain Compressibility is the inverse
17、 of bulk modulus Note both hydrostatic stress and volumetric strain are invariant (independent of coordinate system definition),15,Thermal Strains,Materials expand with increasing temperature and contract with decreasing temperatures The vibration amplitude of atoms increases with heat Attraction/re
18、pulsion curve is not symmetric causing the average distance between atoms to increase beyond the equilibrium position For isotropic materials thermal strains are consistent in all directions and related to change in temperature from reference temperature, the coefficient of thermal expansion is a function of temperature, if large temperature changes exist, analysis must consider variation in ,
