Linear Viscoelasticity

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Viscoelastic Material Analysis

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Presentation transcript:

Linear Viscoelasticity

Elastic Response

Viscous Response

Maxwell Model

Creep

Stress Relaxation due to Maxwell

Voigt Model

Creep due to Voigt Relaxation

Combination of Maxwell and Voigt

Burgers Model

Generalized Models

Continues Distribution: Maxwell

Continues Distribution: Voigt

Superposition Principle

Dynamic Response Output: Input: Viscoelastic body For a dashpot: Stress: Moduli:

Complex Representation:

Time Scales

TTT (if it would be right….) It is because

Master curve for Polymers

Composition of Relaxations: phase shift

Comparison of E(T) and E(t)

General Constitutive Law We can re-write this in the form: than we generalize the Elastic law: If we define and, for example: It might be shown that

Laplace Transform

Properties of Laplace Transform

Linear Viscoelasticity (no time, so far)

Laplace… Laplace transform of this function leads to Similarly for m Finally:

Examples of Operators Boltzmann kernel Boltzmann without singularity No infinite rate of deformation

Homework Pick a Linear viscoelastic moduli Solve the Lame problem