In Nonlinear Finite Elements for Continua and Structures by Belytschko and al. it is stated:

As we have already noted, the strong form, or generalized momentum balance, consists of the momentum equation, the traction boundary conditions and the traction continuity conditions, which are respectively: $$ \begin{gathered} \frac{\partial \sigma_{j i}}{\partial x_j}+\rho b_i=\rho \dot{v}_i \text { in } \Omega \\ n_j \sigma_{j i}=\bar{t}_i \text { on } \Gamma_{t_i} \\ [[n_j \sigma_{j i}]]=0 \text { on } \Gamma_{\mathrm{int}} \end{gathered} $$ where $\Gamma_{\text {int }}$ is the union of all surfaces (lines in two dimensions) on which the stresses are discontinuous in the body. These are usually material interfaces.

Is it not contradictory to admit the presence of stress discontinuities at the material interfaces and to write $[[n_j \sigma_{j i}]]=0 \text { on } \Gamma_{\mathrm{int}}$ ? Does anyone have a simple example to understand this?


1 Answer 1


I have checked the reference you provided and, in my opinion, there is a typo in the book. What it should say is "where $\Gamma_{int}$ is the union of all surfaces (lines in two dimensions) on which the strains are discontinuous in the body"

If you check references on jump conditions (e.g. page 2 of this article) you'll see that in order for a condition to have a jump on the stress (flux in the general problem) there needs to be an external source. If there were no external source equilibrium would not hold on an infinitesimal section at the interface

Let's work out a simple example. Assume you have the following set-up:

Problem set-up

Here the red line (call it $\Gamma_{int}$) represents a material interface, i.e. the zone to the left has stiffness $k_1$ and the part to the right has stiffness $k_2$, and $p$ is a well-behaved applied traction. The governing equation is, considering $k_i$ to be constants $$ -ku''=p(x)\qquad 0\leq x\leq L\\ u(0)=0\qquad\qquad\qquad\quad $$

Let's loot at the jump of stress at $x=a$. Assume you have an infinitesimal section centered around $a$, as in the following figure.

Section around a

Equilibrium for this section is $$ \int_{a-\frac{dx}{2}}^{a+\frac{dx}{2}} p(s) ds +T^+ - T^-=0 $$

(Here $s$ is just an integration variable, to avoid confusion with the notation for the interval length) Since $p(a)$ is finite, the integral goes to $0$ as $dx$ gets small, and you are left with $$ T^+ - T^-=k_2\,\,u'(a^+)-k_1\,\,u'(a^-)=[[k\,\,u'(a)]]=0. $$

As you can see, equilibrium requires the jump on $\Gamma_{int}$ to vanish. Strains on the other hand are discontinuous. From the previous equation you can solve for the difference at the interface, namely $$ u'(a^+)=\frac{k_1}{k_2}\,\,u'(a^-) $$ You can easily set up a similar example in 2D or 3D and arrive at the same conclusion (or a similar one involving equilibrium of the whole body).


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