cauchy stress tensor math boldsymbol p j boldsymbol sigma cdot boldsymbol f text j det boldsymbol f math alternatively define nominal stress tensor math boldsymbol n math transpose first piola kirchhoff stress tensor math boldsymbol n boldsymbol p j boldsymbol f cdot boldsymbol sigma math balance laws become math begin align rho det boldsymbol f rho qquad text balance mass rho ddot mathbf x boldsymbol nabla circ cdot boldsymbol n rho mathbf b qquad text balance linear momentum boldsymbol f cdot boldsymbol n boldsymbol n cdot boldsymbol f qquad text balance angular momentum rho dot e boldsymbol n dot boldsymbol f boldsymbol nabla circ cdot mathbf q rho
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