We call nonlinear dilatational strain gradient elasticity the theory in which the specific class of dilatational second gradient continua is considered: those whose deformation energy depends, in an objective way, on the gradient of placement and on the gradient of the determinant of the gradient of placement. It is an interesting particular case of complete Toupin–Mindlin nonlinear strain gradient elasticity: indeed, in it, the only second gradient effects are due to the inhomogeneous dilatation state of the considered deformable body. The dilatational second gradient continua are strictly related to other generalized models with scalar (one-dimensional) microstructure as those considered in poroelasticity. They could be also regarded to be the result of a kind of “solidification” of the strain gradient fluids known as Korteweg or Cahn–Hilliard fluids. Using the variational approach we derive, for dilatational second gradient continua the Euler–Lagrange equilibrium conditions in both Lagrangian and Eulerian descriptions. In particular, we show that the considered continua can support contact forces concentrated on edges but also on surface curves in the faces of piecewise orientable contact surfaces. The conditions characterizing the possible externally applicable double forces and curve forces are found and examined in detail. As a result of linearization the case of small deformations is also presented. The peculiarities of the model is illustrated through axial deformations of a thick-walled elastic tube and the propagation of dilatational waves.

我们将非线性膨胀应变梯度弹性称为理论，其中考虑了特定的膨胀第二梯度连续体类别：那些变形能量以客观的方式取决于放置梯度和放置梯度决定因素的梯度的理论。这是完整的Toupin-Mindlin非线性应变梯度弹性的一个有趣的特殊情况：实际上，其中唯一的第二个梯度效应是由于所考虑的可变形体的不均匀膨胀状态引起的。膨胀第二梯度连续体与具有孔隙弹性的标量（一维）微观结构的其他广义模型严格相关。它们也可以被认为是应变梯度流体（称为Korteweg或Cahn-Hilliard流体）的一种“固化”结果。使用变分方法，我们得出了拉格朗日和欧拉描述中的扩张第二梯度连续的欧拉-拉格朗日平衡条件。特别是，我们表明，所考虑的连续体可以支持集中在边缘上的接触力，也可以支持分段可定向接触面的表面上的曲面曲线上的接触力。找到并详细检查了表征可能在外部施加的双力和弯曲力的条件。作为线性化的结果，还出现了小变形的情况。通过厚壁弹性管的轴向变形和膨胀波的传播来说明模型的特殊性。