I am trying to understand the dyadic operation for a while since I am interested in Elasticity problems. I believe an intuitive understanding (rather than assuming) will give me good problem solving abilities.
- The following equation is shown in almost all lecture notes on elasticity. However, I don't understand how this result is arrived. Can you actually derive this result? Or is this result rather intuitive? $$ \bf{A}-\rm{\dfrac{1}{3}}\bf(I:A)I=(\mathbb I-\rm\dfrac{1}{3}\bf(I \otimes I)):A $$
- Also how does a fourth order Identity tensor look like? I see the following notation everywhere. $$ \bf \mathbb I= \rm \delta_{ij}\delta_{kl}\bf e_i \otimes e_j \otimes e_k \otimes e_l $$ Not being able to understand this could be attributed to my lack of understanding of the dyadic operator. Every lecture note elaborates on scalars. But not on tensors. Please show the actual matrix. It could help in understanding the operation better.
- Also how does the dyad works between vectors and tensors? How do you numerically compute $\bf I \otimes I$ in the following $$ \bf \mathbb{E}=\rm \lambda \bf I \otimes I + \rm 2 \mu \bf\mathbb{I}^{\rm sym} $$ Ive taken the above equations from Stanford Notes
I believe I have detailed my issue. Also let me know if there are any notes or books that could detail on these trivia.