A constitutive equation is the governing equation that relates stress, o(t), to strain, &(t), where t...

 

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A constitutive equation is the governing equation that relates stress, o(t), to strain, &(t), where t is time. For instance, for a linear elastic material, the constitutive equation is the Hooke's law: o(t) = E-a(t), where E is the Young's modulus. (a) Derive the constitutive equation of the two-parameter Voigt model, in which a spring and a dashpot are placed in parallel with each other. Denote the spring constant as k and the dashpot viscosity as n. (b) In a constant-stress-rate (CSR) test, a tensile stress is applied and it is increased at a constant rate, Ro; i.e. o(t) = Rot. If a polymer can be characterized by the two- parameter Voigt model, derive its strain, a(t), for the CSR test. Hint: Ordinary differential equation (ODE) may be solved either directly or by using the Laplace Transform method. A constitutive equation is the governing equation that relates stress, o(t), to strain, &(t), where t is time. For instance, for a linear elastic material, the constitutive equation is the Hooke's law: o(t) = E-a(t), where E is the Young's modulus. (a) Derive the constitutive equation of the two-parameter Voigt model, in which a spring and a dashpot are placed in parallel with each other. Denote the spring constant as k and the dashpot viscosity as n. (b) In a constant-stress-rate (CSR) test, a tensile stress is applied and it is increased at a constant rate, Ro; i.e. o(t) = Rot. If a polymer can be characterized by the two- parameter Voigt model, derive its strain, a(t), for the CSR test. Hint: Ordinary differential equation (ODE) may be solved either directly or by using the Laplace Transform method.

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a Calculate the total stress a0D a pStress in dashpot Stress in spring Write the relation for st... View the full answer