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1、Lesson content:Classical Linear Viscoelasticity Appendix 3: Linear Viscoelasticity Theory30 minutesClassical Linear Viscoelasticity (1/9)The stress at time t is characterized by:where e (- ) = 0.E ( t ) is the relaxation modulus.Classical Linear Viscoelasticity (2/9)The relaxation modulus can be obt

2、ained from a standard stress relaxation test. The specimen has prescribed constant displacement (strain). The measured response is the force (stress) over time.Classical Linear Viscoelasticity (3/9)In this case e ( t ) = e0 H ( t ) andorwhere H is the Heaviside (step) function and d is the Dirac del

3、ta function.Classical Linear Viscoelasticity (4/9)The stress-strain relation can be inverted and strain at time t is characterized bywhere s (- ) = 0.J ( t ) is the creep function, or creep compliance.Classical Linear Viscoelasticity (5/9)The creep compliance can be obtained form a standard creep te

4、st.The specimen is loaded with a prescribed constant force (stress) and the measured response is the changing displacement (strain) over time.In this case s ( t ) = s0 H ( t ) andorClassical Linear Viscoelasticity (6/9) E ( t ) and J ( t ) are related through Abaqus uses this relation to convert use

5、r-supplied creep test data into relaxation data. This is valid only for linear viscoelasticity.Classical Linear Viscoelasticity (7/9)For a finite-strain viscoelasticity formulation it is important that the stress relaxation equation be written purely in terms of stress.Classical Linear Viscoelastici

6、ty (8/9)Use integration by parts to obtainorwhere s0 ( t ) E0 e ( t ) is the stress that would exist at the current strain state if the specimen were purely elastic.Classical Linear Viscoelasticity (9/9)A similar expression holds for the shear stress S in terms of the shear strain g (t) for a specimen in a time history of pure shear:Here S0(t) G0 g (t) is the shear stress that would exist at the current shear strain state if the specimen were purely elastic. Th