Finite Element Analysis for Engineers by Frank Rieg, Reinhard Hackenschmidt, Bettina Alber-Laukant

By Frank Rieg, Reinhard Hackenschmidt, Bettina Alber-Laukant

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2-1: Stress-strain diagram Between σ and ε there is a linear connection to the limit of proportionality. This is valid for most metallic materials. e. 2 Stress-Strain Relations 35 E = tan α = σ dσ or = ε dε σ = E · ε resp. F = K · U This corresponds exactly to the Hooke’s law of springs. These kinds of materials are also called Hookean materials. E is, so to speak, a spring constant of the material, and is determined through tensile tests. By experiments we know that a tensile bar contracts crosswise to the direction of tension – you can execute this experiment at home with a sealing ring.

2-3: Stresses in space with “usual” naming The shear stresses are labelled as follows: The first index states the plane in which the shear stress appears, the second index states the direction of the stress. The surface is defined by the surface-normal, which stands vertically on the surface. 2-4: Stresses in space in FEA naming Here we can easily recognize the theorem of the equality of the shear stresses (cf. e. in general σij = σji. Why? 2 Stress-Strain Relations 39 Thereby we have to consider that the normal stresses do not contribute; they cancel each other.

2-3: Stresses in space with “usual” naming The shear stresses are labelled as follows: The first index states the plane in which the shear stress appears, the second index states the direction of the stress. The surface is defined by the surface-normal, which stands vertically on the surface. 2-4: Stresses in space in FEA naming Here we can easily recognize the theorem of the equality of the shear stresses (cf. e. in general σij = σji. Why? 2 Stress-Strain Relations 39 Thereby we have to consider that the normal stresses do not contribute; they cancel each other.

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