Invariants of tensors 2147961 208076735 2008-04-25T10:05:49Z Tomeasy 5540175 /* Application for engineering */ renamed section {{Expert-subject|Mathematics}} In [[mathematics]], in the fields of [[multilinear algebra]] and [[representation theory]], '''invariants of tensors''' are coefficients of the [[characteristic polynomial]] of the [[tensor]] ''A'': :<math>\ \det (\mathbf{A}-\lambda \mathbf{E}) = 0 </math> The first invariant of an ''n''&times;''n'' tensor A (<math>I_A</math>) is the coefficient for <math>\lambda^{n-1}</math> (coefficient for <math> \lambda^n </math> is always 1), the second invariant (<math>II_A</math>) is the coefficient for <math>\lambda^{n-2}</math>, etc., the n-th invariant is the free term. The definition of the ''invariants of tensors'' and specific notations used through out the article were introduced into the field of [[Rheology]] by [[Ronald Rivlin]] and became extremely popular there. In fact even the [[Trace (linear algebra)|trace]] of a tensor <math>A</math> is usually denoted as <math>I_A</math> in the textbooks on rheology. ==Properties== The first invariant (trace) is always the sum of the diagonal components: :<math>\ I_A=A_{11}+A_{22}+ \dots + A_{nn}=\mathrm{tr}(\mathbf{A})</math> The n-th invariant is just <math>\pm \det \mathbf{A}</math>, the determinant of <math>\mathbf{A}</math> (up to sign). The invariants do not change with rotation of the coordinate system (they are objective). Obviously, any function of the invariants only is also objective. ==Calculation of the invariants of symmetric 3&times;3 tensors== Most [[Application of tensor theory in engineering science|tensors used in engineering]] are symmetric 3&times;3. For this case the invariants can be calculated as: :<math>\ I_A= tr(\mathbf{A}) = A_{11}+A_{22}+A_{33}=A_1+A_2+A_3</math> :<math>\ II_A= \frac{1}{2} \left( tr(\mathbf{A})^2 - \mathbf{A}^T \cdot \mathbf{A} \right) = A_{11}A_{22}+A_{22}A_{33}+A_{11}A_{33}-A_{12}^2-A_{23}^2-A_{13}^2 =A_1A_2+A_2A_3+A_1A_3</math> :<math>\ III_A=\det (\mathbf{A})= A_1 A_2 A_3</math> where <math>A_1</math>, <math>A_2</math>, <math>A_3</math> are the [[eigenvalues]] of tensor ''A''. Because of the [[Cayley-Hamilton theorem]] the following equation is always true: :<math>\ \mathbf{A}^3 - I_A \mathbf{A}^2 +II_A \mathbf{A} -III_A \mathbf{E}= 0</math> where E is the 2nd order Identity Tensor. A similar equation holds for tensors of higher order. ==Engineering application== A scalar valued tensor function ''f'' that depends merely on the three invariants of a symmetric 3&times;3 tensor <math>\mathbf{A}</math> is objective, i.e., independent from rotations of the coordinate system. Thus objectivity is fulfilled if :<math>\ f\ (A_{ij})=f(I_A,II_A,III_A)</math>. Moreover, the mere dependence on this tensor's invariants is a necessary condition for the objectivity of the function. Reason behind is that the tensor's six [[degrees of freedom (physics and chemistry)|degrees of freedom]] are reduced by three degrees of freedom for arbitrary rotation in three dimensional space. Therefore, the objective function depends at most on three degrees of freedom. A common application to this is the evaluation of the potential energy as function of the deformation tensor (stress and strain tensors). Exhausting the above theorem the energy potential reduces to a function of 3 scalar parameters rather than 6. Consequently, experimental fits and computational efforts may be eased significantly. ==See also== *[[Symmetric polynomial]] *[[Elementary symmetric polynomial]] *[[Newton's identities]] *[[Invariant theory]] ==References== {{unreferenced|date=October 2007}} <references/> [[Category:Tensors]] [[Category:Invariant theory]] [[Category:Linear algebra]] {{math-stub}}