which means that the principal stresses are obtained by solving the previous eigenvalue problem, the principal directions are the eigenvectors of the prob- lem. The eigenvalues A are obtained by noticing that the last identity can b satisfied for non-trivial n only if the factor is singular, i. e, if its determinant vanishes 01-入12013 22 23 which leads to the characteristic equation 3+12-I2入+I3=0 where 1=0i=011+022+a33 h2=5(10y-oy1)=0102+0203+0301-(v12+2+)(14) I 3= det[o]=loiill are called the stress invariants because they do not depend on the coordinate ystem of choic Linear and angular momentum balance We are going to derive the equations of momentum balance in integral form, since this is the formulation that is more aligned with our integral"approach in this course. We start from the definition of linear and angular momentum For an element of material at position x of volume dv, density p, mass pdv which remains constant, moving at a velocity v, the linear momentum is pvdv and the angular momentum xx(pvdv). The total momenta of the body are obtained by integration over the volume as pvdv and/x×pvd respectively. The principle of conservation of linear momentum states that the rate of change of linear momentum is equal to the sum of all the external forces acting on the body Dt/pvdv fdV+tds 16) 6� � � � � � � which means that the principal stresses are obtained by solving the previous eigenvalue problem, the principal directions are the eigenvectors of the prob￾lem. The eigenvalues λ are obtained by noticing that the last identity can be satisfied for non-trivial n only if the factor is singular, i.e., if its determinant vanishes: � � �σ11 − λ σ12 σ13 � � � σ21 σ22 − λ σ23 � � = 0 � σ31 σ32 σ33 − λ� which leads to the characteristic equation: −λ3 + I1λ2 − I2λ + I3 = 0 where: I1 = σii = σ11 + σ22 + σ33 (13) 1� � � � I2 = σiiσjj − σijσji = σ11σ22 + σ22σ33 + σ33σ11 − σ2 23 + σ2 12 + σ2 31 (14) 2 I3 = det[σ] = �σij� (15) are called the stress invariants because they do not depend on the coordinate system of choice. Linear and angular momentum balance We are going to derive the equations of momentum balance in integral form, since this is the formulation that is more aligned with our “integral” approach in this course. We start from the definition of linear and angular momentum. For an element of material at position x of volume dV , density ρ, mass ρdV which remains constant, moving at a velocity v, the linear momentum is ρvdV and the angular momentum x × (ρvdV ). The total momenta of the body are obtained by integration over the volume as: ρvdV and x × ρvdV V V respectively. The principle of conservation of linear momentum states that the rate of change of linear momentum is equal to the sum of all the external forces acting on the body: D ρvdV = fdV + tdS (16) Dt V V S 6