Abstract: Using Fourier transform and based on the Mindlin's 2nd gradient model that involves two length scale parameters, the Green's function, the Eshelby tensor, and the Eshelby-like tensor for a spherical inclusion are derived. It is proved that the Eshelby tensor consists of two parts; the classical Eshelby tensor and a gradient part including the length scale parameters which enable the interpretation of the size effect. When the strain gradient is not taken into account, the obtained Green's function and Eshelby tensor reduce to its analogue based on the classical elasticity. The Eshelby tensor in and outside the inclusion, the volume average of the gradient part and the Eshelby-like tensor are explicitly obtained. Unlike the classical Eshelby tensor, the results show that the components of the new Eshelby tensor vary with the position and the inclusion dimensions. It is demonstrated that the contribution of the gradient part should not be neglected.
Abstract: A size-dependent Euler–Bernoulli beam model, which
accounts for nonlocal stress field, strain gradient field and higher
order inertia force field, is derived based on the nonlocal strain
gradient theory considering velocity gradient effect. The governing
equations and boundary conditions are derived both in dimensional
and dimensionless form by employed the Hamilton principle. The
analytical solutions based on different continuum theories are
compared. The effect of higher order inertia terms is extremely
significant in high frequency range. It is found that there exists
an asymptotic frequency for the proposed beam model, while for
the nonlocal strain gradient theory the solutions diverge. The effect
of strain gradient field in thickness direction is significant in low
frequencies domain and it cannot be neglected when the material
strain length scale parameter is considerable with beam thickness.
The influence of each of three size effect parameters on the natural
frequencies are investigated. The natural frequencies increase with
the increasing material strain gradient length scale parameter or
decreasing velocity gradient length scale parameter and nonlocal
parameter.
Abstract: The stress-strain relationship of concrete under flexure is one of the essential parameters in assessing ultimate flexural strength capacity of RC beams. Currently, the concrete stress-strain curve in flexure is obtained by incorporating a constant scale-down factor of 0.85 in the uniaxial stress-strain curve. However, it was revealed that strain gradient would improve the maximum concrete stress under flexure and concrete stress-strain curve is strain gradient dependent. Based on the strain-gradient-dependent concrete stress-strain curve, the investigation of the combined effects of strain gradient and concrete strength on flexural strength of RC beams was extended to high strength concrete up to 100 MPa by theoretical analysis. As an extension and application of the authors’ previous study, a new flexural strength design method incorporating the combined effects of strain gradient and concrete strength is developed. A set of equivalent rectangular concrete stress block parameters is proposed and applied to produce a series of design charts showing that the flexural strength of RC beams are improved with strain gradient effect considered.
Abstract: In order to calculate the flexural strength of
normal-strength concrete (NSC) beams, the nonlinear actual concrete
stress distribution within the compression zone is normally replaced
by an equivalent rectangular stress block, with two coefficients of α
and β to regulate the intensity and depth of the equivalent stress
respectively. For NSC beams design, α and β are usually assumed
constant as 0.85 and 0.80 in reinforced concrete (RC) codes. From an
earlier investigation of the authors, α is not a constant but significantly
affected by flexural strain gradient, and increases with the increasing
of strain gradient till a maximum value. It indicates that larger
concrete stress can be developed in flexure than that stipulated by
design codes. As an extension and application of the authors- previous
study, the modified equivalent concrete stress block is used here to
produce a series of design charts showing the maximum design limits
of flexural strength and ductility of singly- and doubly- NSC beams,
through which both strength and ductility design limits are improved
by taking into account strain gradient effect.