A small block of a photoelastic material is tested by applying strain along two…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Linear algebra and vector spaces

Mathematical Methods for Physics, Ch Matrices, determinants and systems of linear equations

Problem

A small block of a photoelastic material is tested by applying strain along two perpendicular directions in its plane,

x

and

y

, and measuring the resulting stress. Strain is dimensionless; stress is measured in

\mathrm{MPa}

. A unit strain applied purely along

x

produces a stress response

(3.0, 1.0)\,\mathrm{MPa}

(the components being the stress along

x

and along

y

). A unit strain applied purely along

y

produces a stress response

(1.0, 3.0)\,\mathrm{MPa}

. (a) Write the response as a matrix equation

\boldsymbol{\sigma} = \mathsf{A}\boldsymbol{\varepsilon}

, stating what each row and column of

\mathsf{A}

means. (b) Evaluate

\det\mathsf{A}

and say what it tells you about how a small patch of strain space is rescaled when mapped to stress space. (c) State whether

\mathsf{A}

is symmetric, orthogonal or neither, and say in words what

A_{12} = A_{21}

means for the two measurements.