ModuleModule #13a#13aModule Module #13a#13a

Rheological Models for Deformation and TheRheological Models for Deformation and The Schematic Representation of Real

Materials Behavior

SUGGESTED READING*• J. Lemaitre and J-L. Chaboche, Mechanics of Solid Materials (Cambridge University

Press, Cambridge, 1990) pp. 99-108.• S.M. Edelglass, Engineering Materials Science (The Ronald Press, New York, 1966) pp.

437-445.• N.E. Dowling: Ch. 5, pp. 171-182

Prof. M.L. Weaver

*This list does not mean that you need to read all of these chapters. It has been assembled toprovide you with suggested reading from that you may be using OR referring to in your course.Most of these chapters cover similar material. Any “required” reading will be noted separately.

IntroductionIntroduction• Qualitatively, we can describe deformation by

several adjectives:– elastic.

– plastic.p

– viscous,

– or combinations of the aboveor combinations of the above.

It b i t ti t d fi d f ti i t• It can be instructive to define deformation in terms of mechanical elements that respond to loads in the same manner as engineering materials

Prof. M.L. Weaver

the same manner as engineering materials.

Models for Deformation BehaviorModels for Deformation Behavior• Mechanical elements used to aid understanding of

various forms of deformation.– Linear springs

– Frictional sliders These things are

– Viscous dashpots

– Spring clips

ganalogues for deformation behavior.

Spring clips

• Characteristics of all can be related to modes of deformation (and the adjectives mentioned on the previous viewgraph)

Prof. M.L. Weaver

previous viewgraph).

Rheological models for Rheological models for deformationdeformation

Rheological models for Rheological models for deformationdeformation

When load removed, spring contracts

M t f i ti

Load extension

Must overcome friction.Critical force for motion Po=μmg. (corresponds to a critical yield stress). Above Po block accelerates

When load removed, block stays puto

Occurs at constant rate under constant force.

Slows down over time.(a) + (c)

Prof. M.L. Weaver

From Norman E. Dowling; Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, Third Edition; (Pearson Education, Inc., Upper Saddle River, NJ, 2007) p. 173.

Models for deformation Models for deformation beyond yieldingbeyond yielding

Models for deformation Models for deformation beyond yieldingbeyond yielding

Pure plastic

Elastic/plastic

Elastic + linear hardeningElastic + linear hardening

Prof. M.L. Weaver

From Norman E. Dowling; Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, Third Edition; (Pearson Education, Inc., Upper Saddle River, NJ, 2007) p. 176.

Rheological models can be used to represent the stress-strain behavior of a material under loadof a material under load.

Let’s assume that this bar deforms elastically up to the yield point and then deforms plastically with no work hardening. We can model this

Prof. M.L. Weaver

From Norman E. Dowling; Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, Third Edition; (Pearson Education, Inc., Upper Saddle River, NJ, 2007) p. 174.

We can model this.

What will the stress strain curve look like?

Elastic deformation occurs up to the proportional limit Then the materialElastic deformation occurs up to the proportional limit. Then the material yields at σo. This can be modeled as follows.

Material “stretches” w/ increasing load until yield point (σo) is reached. This is reversible.Beyond σ plastic deformation occurs at constant stress Plastic deformation is irreversible

Prof. M.L. Weaver

Figures from Norman E. Dowling; Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, Third Edition; (Pearson Education, Inc., Upper Saddle River, NJ, 2007) p. 176.

Beyond σo, plastic deformation occurs at constant stress. Plastic deformation is irreversible.

Models for creepModels for creepModels for creepModels for creep

Steady-state creep + elastic strain

T i l i iTransient creep + elastic strain

Prof. M.L. Weaver

From Norman E. Dowling; Mechanical Behavior of Materials: Engineering Methods for Deformation, Fracture, and Fatigue, Third Edition; (Pearson Education, Inc., Upper Saddle River, NJ, 2007) p. 178.