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Constitutive

Constitutive - EN224 Linear Elasticity Division of...

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EN224: Linear Elasticity Division of Engineering 1.3 Constitutive Law for Linear Elastic Solids Objective: find relationship between , assuming infinitesimal motion As before, we will begin by reviewing constitutive models for large deformations. We will define an elastic solid from thermodynamic considerations. We make two key assumptions regarding the behavior of the solid: ASSUMPTION 1: Local Action Stress at a point X depends only on the deformation of the immediate neighborhood of X. Implication: we need only specify the response of the solid to homogeneous deformation (recall that all smooth deformations are locally homogeneous). ASSUMPTION 2: Equation of State State is completely characterized by Lagrangean strain E and temperature θ Hence, there exists a specific internal energy u ( E, s), where s is the specific entropy. Measuring u( E ,s) and s( E , θ ) would completely characterize the material. Given u( E ,s) and s( E , θ ), we may define the Helmholz free energy Given u or F , we can find the stress - strain - temperature response of the solid: I: Isothermal deformations Consider quasi - static state change Then Now, recall the first law of thermodynamics Where dw is the work done on the solid per unit reference volume, and dq is the heat flow into the solid per unit reference volume. Generated by www.PDFonFly.com at 1/10/2010 9:33:48 PM URL: http://www.engin.brown.edu/courses/en224/Constitutive.html
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From the power identity and the second law: Where is the second Piola - Kirchhoff stress.
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