A simplified version of the strain gradient elasticity theory, in which all gradient effects are related to the first scalar invariant of the infinitesimal strain tensor, i.e., to the dilatation, is developed. Two variants of the theory with different forms of boundary conditions are derived using the variational approach. The first variant is derived taking into account independence of the dilatation variation on the body surface and it has simplified traction boundary conditions formulated only with respect to the total stress tensor. The second variant is derived following a general procedure exploiting the surface divergence theorem which results in a more complex form of boundary conditions on the body surfaces and edges. Correctness of the presented formulations of the theory is discussed. Examples of analytical solutions for the problems of pure bending, pressurized sphere and radial vibrations of sphere are obtained and compared for both variants of the theory.

应变梯度弹性理论的简化版本，其中所有梯度效应都与第一个标量不变量相关产生了极小应变张量的“张量”，即达到张量。使用变分方法可以得出理论的两个变体，它们具有不同形式的边界条件。在考虑到体表上扩张变化的独立性的情况下得出第一变体，并且简化了仅针对总应力张量制定的牵引边界条件。第二个变体是根据利用表面发散定理的一般过程得出的，该过程导致在体表和边缘上的边界条件的形式更加复杂。讨论了该理论提出的公式的正确性。获得了用于纯弯曲，加压球体和球体径向振动问题的解析解的示例，并针对该理论的两种变体进行了比较。