522 Mechanics of Materials 2 §12.5 Stress-Stroin curves at different strain rates Stress o Stress o Fracture stress for g Frocture stress for Es eld" stress for 3 d,>e2> Stron e Stroin e (a)For "brittle"piastic (b)For "tough"plastic Fig.12.9.Stress-strain curves at different strain rates & with A and B linear differential operators with respect to time,or as: do d2o de d2e Aoo+dr+Aa=Boe (12.21) In most cases this equation can be simplified to two terms on either side of the expression, the first relating to stress (or strain)the second to its first differential.This will be shown below to be equivalent to describing viscoelastic behaviour by mechanical models composed of various configurations of springs and dashpots.The simplest of these models contain one spring and one dashpot only and are due to Voigt/Kelvin and Maxwell. (a)Voigt-Kelvin Model The behaviour of Hookean solids can be simply represented by a spring in which stress is directly and linearly related to strain, i.e. Os=EEs The Newtonian liquid,however,needs to be represented by a dashpot arrangement in which a piston is moved through the Newtonian fluid.The constant of proportionality relating stress to strain rate is then the coefficient of viscosity n of the fluid. i.e. OD nED (12.22) In order to represent a viscoelastic material,therefore,it is necessary to consider a suitable combination of spring and dashpot.One such arrangement,known as the Voigt-Kelvin model, combines the spring and dashpot in parallel as shown in Fig.12.10. The response of this model,i.e.the relationship between stress o,strain and strain rate gis given by: 0=0十0D and since the strain is common to both parts of the parallel model ss p =s o=E8+n (12.23)5 22 Mechanics of Materials 2 912.5 Stress - Strain curves at different strain rates 6 Stress u Stress u - Strum c (a 1 Fa "brittle" plastic Strain c ( b 1 For ''toqh" pbstlc Fig. 12.9. Stress-strain curves at different strain rates i. with A and B linear differential operators with respect to time, or as: d2E dt2 da d2a dE dt dt Ao~ + AI - + A2 - + . . . = Bo& +B1- +B2- +. . . (12.2 1 ). dt2 In most cases this equation can be simplified to two terms on either side of the expression, the first relating to stress (or strain) the second to its first differential. This will be shown below to be equivalent to describing viscoelastic behaviour by mechanical models composed of various configurations of springs and dashpots. The simplest of these models contain one spring and one dashpot only and are due to VoigdKelvin and Maxwell. (a) Voigt-Kelvin Model is directly and linearly related to strain, i.e. a, = EE, The Newtonian liquid, however, needs to be represented by a dashpot arrangement in which a piston is moved through the Newtonian fluid. The constant of proportionality relating stress to strain rate is then the coefficient of viscosity q of the fluid. The behaviour of Hookean solids can be simply represented by a spring in which stress i.e. CD = V&D (12.22) In order to represent a viscoelastic material, therefore, it is necessary to consider a suitable combination of spring and dashpot. One such arrangement, known as the Voigt-Kelvin model, combines the spring and dashpot in parallel as shown in Fig. 12.10. The response of this model, i.e. the relationship between stress a, strain E and strain rate & is given by: (T = a, + a0 and since the strain is common to both parts of the parallel model ES = ED = E .. ~=E~+qtf (12.23)