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By Daniel I. Rosenthal

Deformations (Mechanics) lines and stresses

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Sample text

2-20) yields (2-22) In practice it is m o r e expedient to r e p r e s e n t the experimental d a t a b y the conventional stress-strain diagram, ac= fc(e). Relation (2-22) t h e n p r o vides a convenient m e a n s for transforming the ac = fc(s) diagram into the true stress-strain diagram, a = /(e). T h e conversion is illustrated graphically in Fig. 2-9. 2-10 WORK OF DEFORMATION T h e conventional crc= fc(e) stress-strain diagram is particularly useful in the c o m p u t a t i o n of t h e w o r k of deformation.

O n e c o n c e r n s the forces acting on t h e solid medium, the other the deformations occurring in various p a r t s of the medium. Unlike t h e stress-strain (or strain-rate) relationships they c a n n o t b e derived from t h e behavior of e l e m e n t a r y particles, although they can b e c h e c k e d experimentally. T h e y are axiomatic s t a t e m e n t s , or principles. 4-1 PRINCIPLES OF EQUILIBRIUM OF FORCES AND MOMENTS T h e s e principles form the foundation of t h e c o u r s e of statics.

2-9 Conventional cr and true cr stress-strain diagram. c 36 Plasticity application, thus dW = Fdx (2-23) If U is t h e initial length of t h e b a r a n d A 0 its initial cross-sectional area, then dx = hde (2-24) and F = Aoac (2-25) Substitution of E q s . (2-24) and (2-25) into E q . (2-23) yields dW = A0lo(Tcde (2-26) W e replace A0lo b y V0 and n o t e that t h e volumetric strain ev is solely d u e t o t h e elastic c o m p o n e n t ee of t h e total strain e. T h e value of e„, therefore, is, (see E q .

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