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plane_cst-lst - Plane Problems Constitutive Equations...

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1 Plane Problems: Constitutive Equations Constitutive equations for a linearly elastic and isotropic material in plane stress (i.e., σ z = τ xz = τ yz =0): where the last column has the initial (thermal) strains which are 0 , xy0 0 0 = = = γ α ε ε T y x • Rewriting in a compact form and solving for the stress vector, where
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2 Plane Problems: Approximate Strain-Displacement Relations • From the above, by definition , , x v y u y v x u xy y x + γ ε ε
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3 Plane Problems: Strain-Displacement Relations As the size of the rectangle goes to zero, in the limit,
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4 Plane Problems: Displacement Field Interpolated Interpolating the displacement field, u(x,y) and v(x,y), in the plane finite element from nodal displacements, where entries of matrix N are the shape (interpolation) functions N i . From the previous two equations, where B is the strain-displacement matrix .
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5 Stiffness Matrix and strain energy Strain energy density of an elastic material (energy/volume) ε ε
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