A thin flat plate lies in the plane and carries a uniform state of plane stress

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids

Invariance, Ch Index notation and Cartesian tensors

Problem

A thin flat plate lies in the xyxy plane and carries a uniform state of plane stress. In axes x,y,zx,y,z (with zz normal to the plate) the stress tensor has components σ11=80 MPa\sigma_{11}=80\,\mathrm{MPa}, σ22=20 MPa\sigma_{22}=20\,\mathrm{MPa}, σ12=σ21=40 MPa\sigma_{12}=\sigma_{21}=40\,\mathrm{MPa}, and every component carrying a 33 index is zero. ((a)) Write σij\sigma_{ij} as the sum of an isotropic part 13σkkδij\tfrac13\sigma_{kk}\delta_{ij}, an antisymmetric part, and a symmetric traceless part. Give the numerical value of the scalar in the isotropic part, and say why the antisymmetric part is zero here. ((b)) Axes are turned by an angle θ\theta about the zz-axis, with rotation matrix R11=R22=cos⁡θ\mathsf R_{11}=\mathsf R_{22}=\cos\theta, R12=−R21=sin⁡θ\mathsf R_{12}=-\mathsf R_{21}=\sin\theta, R33=1\mathsf R_{33}=1, all other components zero. Starting from σij′=RikRjlσkl\sigma'_{ij}=\mathsf R_{ik}\mathsf R_{jl}\sigma_{kl}, derive the formula for the shear component on the rotated axes, σ12′(θ)\sigma'_{12}(\theta), in terms of σ11,σ22,σ12\sigma_{11},\sigma_{22},\sigma_{12} and θ\theta. ((c)) Find the angle θ\theta, between 0∘0^\circ and 90∘90^\circ, at which the shear stress on the rotated plane is greatest in magnitude, and give that maximum shear stress.

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