Kinematics and constitutive relations in the stress-gradient theory: interpretation by homogenization

Abstract

The stress-gradient theory has a third order tensor as kinematic degree of freedom, which is workconjugate to the stress gradient. This tensor was called micro-displacements just for dimensional reasons. Consequently, this theory requires a constitutive relation between stress gradient and micro-displacements, in addition to the conventional stress-strain relation. The formulation of such a constitutive relation and identification of the parameters therein is difficult without an interpretation of the micro-displacement tensor.The present contribution presents an homogenization concept from a Cauchy continuum at the micro-scale towards a stress-gradient continuum at the macro-scale. Conventional static boundary conditions at the volume element are interpreted as a Taylor series whose next term involves the stress gradient. A generalized Hill-Mandel lemma shows that the micro-displacements can be identified with the deviatoric part of the first moment of the microscopic strain field. Kinematic and periodic boundary conditions are provided as alternative to the static ones. The homogenization approach is used to compute the stress-gradient properties of an elastic porous material. The predicted negative size effect under uni-axial loading is compared with respective experimental results for foams and direct numerical simulations from literature.

Morphology-dependent Hashin–Shtrikman bounds on the effective properties of stress-gradient materials

“…The vector field u has dimension of a classical displacement. For heterogeneous Cauchy materials that are homogenized as stress-gradient materials, Hütter et al (2020) have recently shown that φ can be interpreted as the trace-free part of the first moment of the microscopic strain-field. The above field equations must be complemented with appropriate boundary conditions…”

“…Polizzotto (2018) and references therein. Such models are relevant for nanocomposites exhibiting "negative" size-effect (Tran et al, 2018) and foams (Hütter et al, 2020).…”

“…Homogenization of heterogeneous, stressgradient materials as homogeneous, Cauchy materials was then considered in the work by Tran et al (2018), where a simplified stress-gradient model, akin to the simplified strain-gradient model of Aifantis (1992, 1997), was also proposed. The converse homogenization problem (Cauchy material at the microscale, stress-gradient material at the macroscale) has recently been addressed by Hütter et al (2020), who offer a microscopic interpretation of the generalized strain that is workconjugate to the gradient of the stresses.…”

A finite element implementation of the stress gradient theory

“…These new degrees of freedom, called microdisplacements in [14], are conjugate to the stress gradient tensor in the generalised work of internal forces. The microstructural origin of the microdisplacements was recently highlighted based on an original homogenisation approach in [18]. In contrast, the strain gradient model is solely based on the classic displacement degrees of freedom.…”

A finite deformation isogeometric finite element approach to fibre-reinforced composites with fibre bending stiffness

“…This is characteristic for strain gradient theories and in contrast to stress gradient theories in which negative size effects in the sense "smaller is softer" are observed, cf. [10][11][12]. One specific approach to generalised continua is the couple-stress theory, as discussed in detail in [13,14].…”