A generalized theory of nonlocal elasticity is elaborated. The proposed integral type nonlocal formulation is based on attenuation functions being assumed as the convolution product of n first order (Eringen type) kernels. The theory stems from a generalized higher-order constitutive relation between the nonlocal stress and the local strain. Inspired by the Eringen two-phase local/nonlocal integral model, this theory can also be thought of as the constitutive relation for an (n + 1)-phase material, in which one phase has local elastic behavior, and the remaining n phases comply with nonlocal elasticity of higher order. The theory is supported by a suitable thermodynamic framework. In the spirit of Eringen’s 1983 paper, the particular family of attenuation functions adopted are the Green functions associated with generalized Helmholtz type differential operators of order n —which suggests denoting this model as a generalized nonlocal elasticity theory of n-Helmholtz type. Besides the integral type nonlocal formulation, elegant and compact expressions for the differential and integro-differential counterpart are derived. For n = 1 this formulation straightforwardly leads to the Aifantis 2003 implicit gradient elasticity theory with simultaneous stress gradients and strain gradients, which was postulated to eliminate stress and strain singularities from crack tips and dislocation lines. For n = 2 an implicit gradient elasticity formulation with bi-Helmholtz type stress and strain gradients is obtained. The paper is complemented by a companion Part II on the particularization of the generalized theory of nonlocal elasticity for the one-dimensional case, along with some applications in statics and dynamics.

阐述了非局部弹性的广义理论。所提出的整数类型非局部公式是基于将衰减函数假定为n个一阶（Eringen类型）内核的卷积的。该理论源于非局部应力与局部应变之间的广义高阶本构关系。受Eringen两相局部/非局部积分模型的启发，该理论也可以被视为（n  + 1）相材料的本构关系，其中一相具有局部弹性行为，其余n相符合较高阶的非局部弹性。该理论得到合适的热力学框架的支持。根据Eringen 1983年论文的精神，采用的特定衰减函数族是与n阶广义Helmholtz型微分算子相关的Green函数-这建议将该模型表示为n -Helmholtz型广义非局部弹性理论。除了整数类型的非局部公式外，还导出了微分和整数微分对应项的简洁明了的表达式。对于n = 1此公式直接导致Aifantis 2003隐式梯度弹性理论，同时具有应力梯度和应变梯度，被假定为消除裂纹尖端和位错线的应力和应变奇异性。对于n  = 2，获得了具有双亥姆霍兹型应力和应变梯度的隐式梯度弹性公式。本文的第二部分是对一维情况下非局部弹性广义理论的特殊性的补充，以及在静力学和动力学中的一些应用。