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We present a general form of the gradient-enhanced damage theory and its numerical implementation using finite element method (FEM) in modeling quasi-static brittle crack growth in one- (1D), two- (2D) and three-dimensional (3D) bodies. Coupled equations of the equilibrium and a new implicit gradient damage formulation are introduced to govern the deformation of the solid and evolution of the damage. The resulting nonlocal damage evolution equation featuring the growth of diffusive crack is integrated with a characteristic length scale to eliminate the common mesh-bias issue in FEM implementation. In contrast to the traditional gradient-enhanced damage approaches, the nonlocal damage field here is defined as the primary variable of the damage evolution equation without interpolation mismatch between the displacement and nonlocal damage fields. For derivation of the material constitutive law and local damage parameter, a novel strain energy density (SED) function based on the energy limiter theory for brittle crack growth problems under small strain regime is introduced. To further improve the performance of the developed model, an initial SED threshold, which is used for determining the critical point when damage starts to initiate in the material, is integrated into the novel energy limiter theory. For preventing nonphysical failure in compression domains, the spectral decomposition technique for the strain tensor is adopted to split the reference SED. With integrating the energy limiter into the developed theory, unlike the conventional nonlocal damage theories where the interpretation of the length scale is still ambiguous, the developed nonlocal damage model defines the length scale parameter as the problem-dependent factor. The performance and ability of the proposed model are demonstrated via a set of representative numerical examples in 1D, 2D and 3D fracture problems.

Contact-type ultrasonic test is commonly used in construction industry where gel-couplant is applied to the material being tested and the transducers to assure that wave propagation will travel through without any air gaps. However, this method has disadvantages, since surface preparation is considered prior to testing. Another method of testing without the worry of air gaps that causes scattering of waves before it reaches the medium is the use of the noncontact ultrasonic test. In particular, the air-coupled ultrasonic test is done in this paper for reinforced concrete beams. Sixteen plain concrete cube specimens under the compression test and six reinforced concrete beam specimens under the four-point bending test are made with water-cement ratio of 40% and 60%. The plain concrete cubes are investigated to establish the relationship of the contact ultrasonic test and load. Added parameters are considered to investigate the sensitivity of the contact and noncontact ultrasonic test in reinforced concrete beams. These are ultrasonic wave path and the neutral axis index. It shows that the higher water-cement ratio produces good sensitivity in the noncontact ultrasonic test, since it produces more cracks on the tension face. Lower water-cement ratio gives good sensitivity with load for the contact ultrasonic test, since it has its ultrasonic wave path passing through the concrete experiencing compression. In addition, the neutral axis index for a member subjected to bending is an important factor in assessing the sensitivity of both contact and noncontact ultrasonic test.

This paper aims at enhancing the capability of the smoothing gradient-enhanced damage method in modeling localization failure, especially mixed-mode fracture in quasi-brittle materials, e.g., double edge notch — DEN by Nooru-Mohamed test. As a novelty, we integrate an alternative technique for estimation of the equivalent strain based on Ottosen’s failure criterion, the four-parameter strain equivalent method that characterizes the sensitivity of deformation stages and loading path conditions, into our damage theory to form a new approach for better capturing the mixed-mode fracture behavior. Our approach allows to use the same low-order finite elements for approximation of both displacements and equivalent strain fields without any stress oscillation, highly suitable for practice. The new approach owns several desirable features of an advanced method including mesh insensitivity, and numerical examples in two-dimension indicate the applicability of the developed model in capturing the crack paths observed in experiments. Interpreting principal stresses in the failure envelopes further sheds light on insight into the damage behavior of the developed model.

This paper presents a novel smoothing gradient damage model, which goes beyond certain limitations of conventional methods, for accurate prediction of localized failure in quasi-brittle materials. The proposed method is particularly tailored to low-order finite elements such as 4-node quadrilateral or 3-node triangular elements. The low-order elements are preferable in practice as they can automatically be generated for problems even with complex geometries at low computational cost. In order to eliminate spurious damage growth and correct wrong prediction of shear band induced by using the constant gradient parameter in terms of conventional models, we thus introduce a novel modified evolving anisotropic nonlocal gradient parameter, which aims to control the behavior of nonlocal interactions of damage microprocess during the entire loading history in a more appropriate manner. Unlike conventional approaches, the novel modified evolving gradient parameter heavily depends on the principal stress and equivalent strain states, which serves to reduce the impact of localized deformation. The stress fields thus play a crucial role in the present formulation as they greatly affect the orientation and intensity of nonlocal interactions. The quality of raw stresses becomes critical (e.g., stress oscillation) once two unknowns (i.e., displacements and nonlocal equivalent strain) are approximated simultaneously using the same orders of interpolation functions (e.g., linear–linear). A smoothing technique is thus adopted to smooth out the raw stresses. The stresses after smoothing are shown adequately in the estimation of the new gradient parameter, providing much better solutions. In addition, to precisely capture the softening in quasi-brittle materials, the original energy norm is decomposed into tensile and compressive parts to form a new bi-energy norm. The scalar equivalent strain is thus estimated through this new bi-energy norm, which is obviously able to distinguish tensile and compressive conditions. To further enhance the capability of the present damage model, the material softening process is also determined through fracture energy in terms of fracture mechanics. Comparison of the present results with reference solutions derived from experimental data and other numerical methods for benchmark applications, regarding the nonlocal equivalent strain, damage profile, structural force–displacement curves, etc., confirms the accuracy and superior performance of the proposed approach for characterizing localized failure in quasi-brittle materials.

The recently developed local enriched consecutive-interpolation 4-node quadrilateral element (XCQ4) is extended to study transient dynamic stress intensity factors (DSIFs) for isotropic solids and anisotropic composite materials containing stationary cracks. The XCQ4 involves both nodal values and averaged nodal gradients as interpolation conditions to smooth the distribution of unknown variables. The possible physical properties of enriched nodes in consecutive-interpolation procedure (CIP) would be thought as non-locality feature, which could improve the accuracy of results and eliminate the non-smooth stresses among inter-elements. In XCQ4, the crack is determined by level set function and enriched by Heaviside and crack-tip enrichment functions with special anisotropic enriched crack-tip functions. Time-dependent discrete equations for dynamic cracks are solved by Newmark time integration scheme at each time step without considering the effects of velocity-based global damping matrix. The proposed method is verified through a series of numerical examples of transient fracture in both isotropic and anisotropic materials. Numerical DSIFs are compared with reference solutions available in literature. The behavior of dynamic response is explored in specimens with complex configuration under step and sine loads.

Failure modeling of quasi-brittle materials in terms of continuum damage mechanics model and finite element approach is presented in this paper. A stress-based implicit gradient-enhanced damage model using low-order finite elements is developed in which the characteristic length is a stress level dependent parameter. A simple formula for the calculation of the characteristic length of failure localization in the spatial discretization of materials is also taken, which is to additionally suppress the sensitivity of geometrical meshes. The displacements and nonlocal equivalent strain fields are approximated in the framework of low-order finite elements, i.e., the same interpolation functions are used for both primary variables. The spurious damaged zones and stress oscillation induced by the standard strain-based gradient damage can be overcome by using the proposed damage approach. Quasi-brittle materials such as concrete and limestone are considered and some numerical examples with different configuration are examined. The obtained numerical results are analyzed and are validated against reference solutions including experimental data and other existing numerical approaches