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This paper aims to present a nonlocal gradient plasticity damage model to demonstrate the crack pattern of a body, in an elastic and plastic state, in terms of damage law. The main objective of this paper is to reconsider the nonlocal theory by including the material in-homogeneity caused by damage and plasticity. The nonlocal nature of the strain field provides a regularization to overcome the analytical and computational problems induced by softening constitutive laws. Such an approach requires C1 continuous approximation. This is achieved by using an isogeometric approximation (IGA). Numerical examples in one and two dimensions are presented.

In this work, the authors propose a nonlocal elastic plastic damage model. The nonlocal nature of the strain field provides a regularization to overcome the analytical and computational problems induced by softening constitutive laws. An additive decomposition of strains in to elastic and inelastic or plastic part is considered. To obtain stable damage, a higher gradient order is considered for an integral equation, which is obtained by the Taylor series expansion of the local inelastic strain around the point under consideration. The higher-order continuity of nonuniform rational B-splines (NURBS) functions used in isogeometric analysis are adopted here to implement in a numerical scheme. To demonstrate the validity of the proposed model, numerical examples in one and two dimensions are presented.

The proposed nonlocal elastic plastic damage model is able to predict the damage in an accurate manner. The numerical results are mesh independent. The nonlocal terms add a regularization to the model especially for strain softening type of materials. The consideration of nonlocality in inelastic strains is more meaningful to the physics of damage. The use of IGA framework and NURBS basis functions add to the nonlocal nature in approximations of the field variables.

The method can be extended to 3D. The model does not consider the effect of temperature and the dissipation of energy due to temperature. The method needs to be implemented for more real practical problems and compare with experimental work. This is an ongoing work.

The nonlocal models are suitable for predicting damage in quasi brittle materials. The use of elastic plastic theories allows to capture the inelastic deformations more accurately.

The nonlocal models are suitable for predicting damage in quasi brittle materials. The use of elastic plastic theories allows to capture the inelastic deformations more accurately.

The present work includes the formulation and implementation of a nonlocal damage plasticity model using an isogeometric discretization, which is the novel contribution of this paper. An implicit gradient enhancement is considered to the inelastic strain. During inelastic deformations, the proposed strain tensor partitioning allows the use of a distinct potential surface and distinct failure criterion for both damage and plasticity models. The use of NURBS basis functions adds to more nonlocality in the approximation.

This study aims to develop a new analysis approach devised by incorporating a gradient-enhanced microplane damage model (GeMpDM) into isogeometric analysis (IGA), which shows computational stability and capability in accurately predicting crack propagations in structures with complex geometries.

For the non-local microplane damage modeling, the maximum modified von-Mises equivalent strain among all microplanes is regularized as a representative quantity. This characterization implies that only one additional governing equation is considered, which improves computational efficiency dramatically. By combined use of GeMpDM and IGA, quasi-static and dynamic numerical analyses are conducted to demonstrate the capability in predicting crack paths of complex geometries in comparison to FEM and experimental results.

The implicit scheme with the adopted damage model shows favorable numerical stability and the numerical results exhibit appropriate convergence characteristics concerning the mesh size. The damage evolution is successfully controlled by a tension-compression damage factor. Thanks to the advanced geometric design capability of IGA, the details of crack patterns can be predicted reliably, which are somewhat difficult to be acquired by FEM. Additionally, the damage distribution obtained in the dynamic analysis is in close agreement with experimental results.

The paper originally incorporates GeMpDM into IGA. Especially, only one non-local variable is considered besides the displacement field, which improves the computational efficiency and favorable convergence characteristics within the IGA framework. Also, enjoying the geometric design ability of IGA, the proposed analysis method is capable of accurately predicting crack paths reflecting the complex geometries of target structures.

As the dimensions of structures are scaled down to the micro‐ and nano‐domains, the mechanical behavior becomes size dependent and thus, the classical elasticity solutions cannot be expected to hold. In particular, recent experimental investigations of fatigue strength of metals show pronounced strengthening due to the influences of both grain size and small geometrical dimensions. This paper aims to provide a simple, yet rigorous analytical model in order to address the aforementioned size effects.

The present study employs a framework based on the type II, strain gradient elasticity theory by Mindlin, embedded into a thermodynamics‐based formulation which considers both mechanical behavior parameters and material lengths, as internal variables, in order to model metal fatigue.

A thermodynamics‐based, second gradient elasto‐plastic formulation with an explicit material length, which captures the size effects in fatigue of small‐scale metal components, has been established. From a physical viewpoint, the evolution of the internal length in the constitutive equations with the evolution of the intrinsic wavelength (e.g. persistent slip bands spacing) can be identified signifying the splitting of the grains into sub‐regions and consequently, the softening of the material.

The major novelty of the proposed modeling is that the internal characteristic length considered is not a fixed parameter, but evolves with the plastic effective strain amplitude obtained from cyclic loading.

The purpose of this paper is to propose an efficient and accurate numerical model that employs isogeometric analysis (IGA) for the geometrically nonlinear analysis of functionally graded plates (FGPs). This model is utilized to investigate the effects of boundary conditions, gradient index, and geometric shape on the nonlinear responses of FGPs.

A geometrically nonlinear analysis of thin and moderately thick functionally graded ceramic-metal plates based on IGA in conjunction with first-order shear deformation theory and von Kármán strains is presented. The displacement fields and geometric description are approximated with nonuniform rational B-splines (NURBS) basis functions. The Newton-Raphson iterative scheme is employed to solve the nonlinear equation system. Material properties are assumed to vary along the thickness direction with a power law distribution of the volume fraction of the constituents.

The present model for analysis of the geometrically nonlinear behavior of thin and moderately thick FGPs exhibited high accuracy. The shear locking phenomenon is avoided without extra numerical efforts when cubic or high-order NURBS basis functions are utilized.

This paper shows that IGA is particularly well suited for the geometrically nonlinear analysis of plates because of its exact geometrical modelling and high-order continuity.

This paper aims to propose a gradient-based shape optimization framework in which traditional time-consuming conversions between computer-aided design and computer-aided engineering and the mesh update procedure are avoided/eliminated. The scheme is general so that it can be used in all cases as a black box, no matter what the objective and/or design variables are, whilst the efficiency and accuracy are guaranteed.

The authors integrated CAD and CAE by using isogeometric analysis (IGA), enabling the present methodology to be robust and accurate. To overcome the difficulty in evaluating the sensitivities of objective and/or constraint functions by analytic method in some cases, the authors adopt the finite difference method to calculate these sensitivities, thereby providing a universal approach. Moreover, to further eliminate the inefficiency caused by the finite difference method, the authors advance the exact reanalysis method, the indirect factorization updating (IFU), to exactly and efficiently calculate functions and their sensitivities, which guarantees its generality and efficiency at the same time.

The proposed isogeometric gradient-based shape optimization using our IFU approach is reliable and accurate, as well as general and efficient.

The authors proposed a gradient-based shape optimization framework in which they first integrate IGA and the proposed exact reanalysis method for applicability to structural response and sensitivity analysis.

This paper aims to present a multi-scale stochastic damage model (SDM) for concrete and apply it to the stochastic response analysis of reinforced concrete shear wall structures.

The proposed SDM is constructed at two scales, i.e. the macro-scale and the micro-scale. The general framework of the SDM is established on the basis of the continuum damage mechanics (CDM) at the macro-scale, whereas the detailed damage evolution is determined through a parallel fiber buddle model at the micro-scale. The parallel buddle model is made up of micro-elements with stochastic fracture strains, and a one-dimensional random field is assumed for the fracture strain distribution. To represent the random field, a random functional method is adopted to quantify the stochastic damage evolution process with only two variables; thus, the numerical efficiency is greatly enhanced. Meanwhile, the probability density evolution method (PDEM) is introduced for the structural stochastic response analysis.

By combing the SDM and PDEM, the probabilistic analysis of a shear wall structure is performed. The mean value, standard deviation and the probability density function of the shear wall responses, e.g., shear capacity, accumulated energy consumption and damage evolution, are obtained.

It is noted that the proposed method can reflect the influences of randomness from material level to structural level, and is efficient for stochastic response determination of shear wall structures.

The stress state near the notch affects fatigue damage directly, but quantifying the stress field is difficult. The purpose of this study is to provide a mathematical description method of the stress field near the notch to achieve a reliable assessment of the fatigue life of notched specimens.

Firstly, the stress distribution of notched specimens of different materials and shapes under different stress levels is investigated, and a method for calculating the stress gradient impact factor is presented. Then, the newly defined stress gradient impact factor is used to describe the stress field near the notch, and an expression for the stress at any point along a specified path is developed. Furthermore, by combining the mathematical expressions for the stress field near the notch, a multiaxial fatigue life prediction model for notched shaft specimens is established based on the damage mechanics theory and closed solution method.

The stress gradient factor for notched specimens with higher stress concentration factors (V60-notch, V90-notch) varies to a certain extent when the external load and material change, but for notched specimens with relatively lower stress concentration factors (C-notch, U-notch, stepped shaft), the stress gradient factor hardly varies with the change in load and material, indicating that the shape of the notch has a greater influence on the stress gradient. It is also found that the effect of size on the stress gradient factor is not obvious for notched specimens with different shapes, there is an obvious positive correlation between the normal stress gradient factor and the normal stress concentration factor compared with the relationship between the shear stress gradient factor and the stress concentration factor. Moreover, the predicted results of the proposed model are in better agreement with the experimental results of five kinds of materials compared with the FS model, the SWT model, and the Manson–Coffin equation.

In this paper, a new stress gradient factor is defined based on the stress distribution of a smooth specimen. Then, a mathematical description of the stress field near the notch is provided, which contains the nominal stress, notch size, and stress concentration factor which is calculated by the finite element method (FEM). In addition, a multiaxial fatigue life prediction model for shaft specimens with different notch shapes is established with the newly established expressions based on the theory of damage mechanics and the closed solution method.

In reinforced concrete (RC) structures, an evidence of damage is the presence of cracking. In order to evaluate the effect of damage on cracking pattern and natural frequency in RC slabs, two of such structures with different dimensions and reinforcement ratios were tested, in which cracks were induced through application of static load, followed by modal tests using impact excitation. The paper aims to discuss this issue.

The gradient of the fundamental natural frequency along the decay, the crack opening rate and also a global damage index based on changes of the fundamental natural frequency were evaluated.

The behaviour of the aforementioned gradient was distinct for both slabs, increasing monotonically with the cracking level for the slab with lowest reinforcement ratio, and increasing until 33 per cent of the collapse load and then decreasing afterwards for the slab with the highest ratio. Changes of the gradient were consistent with changes of the crack opening rate. Both results of gradient changes and cracking pattern brought evidence that the balance between open (old) and breathing (new) cracks differed between the slabs, and may be responsible for such differences.

Damage assessment in RC structures using vibration tests is mostly concentrated on beams. In this work, an advance is made by investigating slabs. The lack of a unique pattern of changes of the gradient implies that its absolute value is not generally suitable for the association with the damage level. However, the impact tests can be effectively used to detect early damage on slabs using this proposed parameter.

This paper presents a comparison of various models for strain‐softening due to damage such as cracking or void growth, as proposed recently in the literature. Continuum‐based models expressed in terms of softening stress—strain relations, and fracture‐type models expressed in terms of softening stress—displacement relations are distinguished. From one‐dimensional wave propagation calculations, it is shown that strain‐localization into regions of finite size cannot be achieved. The previously well‐documented spurious convergence is obtained with continuum models, while stress—displacement relations cannot model well smeared‐crack situations. Continuum models may, however, be used in general if a localization limiter is implemented. Gradient‐type localization limiters appear to be rather complicated; they require solving higher‐order differential equations of equilibrium with additional bourdary conditions. Non‐local localization limiters, especially the non‐local continuum with local strain, in which only the energy dissipating variables are non‐local, is found to be very effective, and also seems to be physically realistic. This formulation can correctly model the transition between homogeneous damage states and situations in which damage localizes into small regions that can be viewed as cracks. The size effect observed in the experimental and numerical response of specimens in tension or compression is shown to be a consequence of this progressive transition from continuum‐type to fracture‐type formulations.