Smoothed reduction of fracture mechanics solutions to 1D cracked models

Abstract In the framework of fracture mechanics 2 or 3 dimensional modelling approaches are mostly employed in order to simulate cracks. However, in many applications 1D models that represent cracked structural members can be proved to be very useful. A crack contributes to abrupt stiffness reduction, causing “jump” in the displacement field close to the crack, in the present work the observation that the presence of cracks causes also stress and strain redistribution at some distance from the crack tips as resulted from fracture mechanics solutions is underlined. An appropriate “smoothed” stiffness reduction is introduced as a function from the distance from the crack consistent energetically with the Linear Elastic Fracture Mechanics (LEFM) solutions. These smoothed 1D solutions capture the strain variations in some distance from the crack(s) achieving very good accuracy. Some modelling examples in static and dynamic (eigenfrequencies, mode shapes) cases are examined and compared to existing 1D models, 3D FEM simulation and experimental data.


INTRODUCTION
The main task when modelling cracks into a fracture mechanics' framework, is to evaluate the stress field near crack tip, in order to assess the possible unstable crack growth. In the context of structural damage identification where the main objective is to localize and quantify possible damages, modelling cracks mainly implies stiffness degradation, since such an approach produces measurable changes in structural response. Depending on the type of the examined structural system, different models (usually based on the finite element method) are utilized; cracks are usually simulated either as a springs (e.g. [1][2][3]), either by introducing local stiffness degradations expressed as modulus of elasticity reductions (see for example in [4]), or related to overall finite element stiffness e.g. [5][6][7][8] and many others. Damage identification is formulated as an inverse problem in the model parameter domain, in the usual formalism it is decomposed in various stages [9]. Stage one deals with the determination of the existence of damage. The rest of the stages deal with the identification of the 1 Corresponding author 2 geometric location, type and quantification of the potential damages (cracks). The most common practice, is the adaptation of finite element model updating schemes. Cracks and damages are represented as an overall stiffness reduction corresponding to the damaged structural member/ finite element.
The main motivation of the presented work is to develop a method that models cracks that is not dependent on the size of the FEM discretisation while at the same time exploit the fact that the effect of the presence of the crack in the stain field may not always be absolutely local. The work presented here is composed of three parts. In part one, the equivalence of the spring models and the models representing damage as a stiffness modulus reduction are examined by introducing an appropriate stiffness function. In part two, based on the strain energy as a function of the distance from the crack tip built on the celebrated solution of Inglis for stresses on an elliptical hole [10] and the concept of energy release of fracture mechanics, the appropriate form of the introduced stiffness function is derived. In the third part, numerical validation of the proposed method for modelling cracks and its possible use in damage identification is demonstrated.

CRACK MODELS -ANALYSIS OF DAMAGES PRESENTED AS STIFFNESS MODULI REDUCTION
In the case of one dimensional structural members, two methods are usually invoked in order to describe crack(s). The first one is based on the idea to use an equivalent spring at the location of the crack and its stiffness is defined by fracture mechanics principles. This modelling method has the advantage to achieve the necessary "jump" in the structural member's displacement field around the crack and to introduce localized damages. The second method is based on the idea to describe damage (or crack) as stiffness reduction of the appropriate finite element [11]. In order to analyse the two methods and develop an alternative one, an appropriate stiffness function needs to be introduced.
Such a function for the case of an element of length 2δ is depicted in Figure 1. EA or EI is the bar or beam element stiffness, where E is the modulus of elasticity, A is the area of the structural element cross section, I is the cross section moment of inertia, s is the percentage of the remaining stiffness in the damaged element and y0 is the element's middle point (in the longitudinal direction).
In the following analysis, which in the context of the present work without loss of generality is focused on truss structures, the expression of displacement field of a bar element as a function of s will be derived. The case of localized damages as implemented with the spring model will be extracted as a special case of the distributed case (i.e. the stiffness reduction model) and both modelling methods will be directly related with s. The bar element stiffness as a function of the abscissa value y along the element is defined as follows: 3 and Η(y) is the Heaviside step function. The bar element under equilibrium is governed by the following differential equation: where u(y) is the bar element displacement field, F is the axial force developed at the bar element and ε is the standard engineering strain. Integrating both sides of Eq. (4) results into the following expression: where δ(y) is the Dirac delta function. Three cases are defined with respect to the abscissa value y along the element: Although the displacement field u(y) is described by a continuous function, its derivatives are not: As δ tends to 0 the following observations need to be addressed: (i) for a finite value of s the derivative is continuous everywhere except at y0 where it takes a finite value. However this does not influence u(y) as it is observed from Eq. (6), therefore no change in the displacement field is noticed and consequently no noticeable stiffness reduction of the larger component related to the undamaged case.
(ii) Modelling cracks using springs with stiffness derived from fracture mechanics, actually produces a "jump" in the displacement field around the crack. If where w is a constant of finite value. Therefore: At this part of our analysis, it is possible correlate the first and the second modelling methods, since it is observed that: where Cf is the stiffness of the spring adopted in the first modeling method.
Closing the analysis of the two methods an observation regarding the first crack model can be inferred, as the jump in the displacement field Δu is of finite value the stiffness function value (s) needs to be equal to zero at the location of the crack. Furthermore, the first model does not predict any variation in the strain field near the crack. The second modelling scheme is unsatisfactory since the length of the crack influence depends on the FE discretization. Therefore, both modeling schemes yield unrealistic behavior of the cracked structural members.

STIFFNESS RELAXATION CRACK MODEL BASED ON FRACTURE ENERGY AT A DISTANCE FORM THE CRACK TIP
Significant work has been carried out aiming to develop finite element models for simulating cracked structural members. The equivalent model is often developed by invoking stress intensity factor (SIF).
For cracks of mode I the equation for SIF is given by the following expression [12]: where R is the uniform stress, αc is the crack length (or half-length for a central crack), Fc is the correction function considering the finite dimensions and b is the width (or half width in case of a central crack) of the bar structural member. The correction function for the case of bar structural members with prismatic cross-section and a central crack is given by: while the strain energy due to a central crack is given by [11]: where E is the modulus of elasticity and t is the structural member thickness.

Crack Energy Function
As indicated by fracture mechanics principles, stress (and strain) concentration is observed near cracks. In this part of the study, a new stiffness relaxation model will be derived using the redistribution of stresses (based on an energetic approach) in a finite distance from the crack. The mathematical tools invoked are the solution of a plate with a central elliptic hole which limits to a crack as described in [10] along with a crack energy expression similar to that derived by Griffith in [13] but different in the sense that in the current study the expression of the crack energy is formulated as a function of the distance from the crack tip. Similar to the prodromal studies [10,13] curvilinear coordinates are employed. In particular, α and β define a family of confocal ellipses for the case that (cosh 2 cos 2 ) sinh 2 (cosh 2 cosh 2 2 cos 2 ) (cosh 2 cos 2 ) sin 2 (cosh 2 cosh 2 ) (cosh 2 cos 2 ) ( 1) cosh 2 ( 1) cos 2 2 cosh 2 8 6 where c is the half length of the focal line, E is the modulus of elasticity, μ is the shear modulus

Stiffness Relaxation Model
Since the range of interest for the distribution of the crack energy is defined for m yy  and aiming to develop a simple representation of the stiffness function, the following expression is considered as a very good approximation of the energy distribution: The total excess energy due to the crack is defined according to Eq. (14) while its spatial distribution is determined with the aid of Eqs. (20) to (23). Therefore, Θ is computed by the following relation:  Figure 3 shows the corresponding stiffness function of Eq. (29); both for a steel bar structural member with E=200Gpa, σ=0.265, R=100Mpa, 2b= 0.4m, t=0.4m and half crack length αc=0.13m, (the results are achieved assuming plane stress conditions).

NUMERICAL STUDY
Aiming to present the efficiency of the proposed model of the cracks two levels of comparison are employed herein: (i) in the first one, the new model is compared against two usually applied approaches for modelling cracks in the case of 1D simulations, (ii) in the second one, the new method is integrated into a damage identification approach and is compared against a 3D implementation.

Cracked Bar Finite Element Implementing the Proposed Stiffness Function
In order to derive the stiffness coefficients kij of the stiffness matrix for the proposed cracked bar finite

Comparison with Existing Models for 1D Simulations of Cracked Structural Elements
In the first level of comparison employed in the current study, the proposed modelling approach is compared against two highly used modelling practices for the case of 1D simulations of cracked 9 structural members. The three models are compared with reference to the normal strain and displacement profiles along the length of the structural element achieved. For this case an indicative test case is used corresponding to a steel structural element of length L=2m, of rectangular crosssection (2b×t) 2b= 0.4m, t=0.4m, subjected to a tensile force F=10MN (applied uniformly to one edge of the bar, while the other edge is fixed). The material properties considered are modulus of elasticity E=200GPa, poison ration σ=0.3 while a crack located at the half-length of the structural element is considered with half-length equal to αc=0.15m. Figures 4 to 6 depict the profiles of the normal strains obtained for the steel beam. In particular, Figure 4 illustrates the normal strains εyy (defined at the cross section) derived when implementing the first modelling method where a linear spring is used at the location of the crack. The FE discretization used for the implementation of the first method is composed by 20 1D beam elements and one spring. Figure 5 illustrates the normal strains derived when implementing the second modelling method applying the appropriate cracked finite element at the location of the crack. The FE discretization used for the implementation of the second method is composed by 21 1D beam elements. Finally, Figure 6 illustrates the normal strains profile obtained when implementing the proposed modelling method. The FE discretization used for the implementation of the third method is composed by 20 1D beam elements whose stiffness matrices were derived using Eq. (31).
Comparing the three methods it can be seen that first and second ones result into unrealistic profiles of the normal strain, exhibiting no variation.
Additionally, in order to further examine the differences between the three modelling approaches, the displacement profiles along the bar structural element were also obtained using the three crack representation methods. The displacements' profiles obtained are depicted in Figures 7, 8 and 9 for the three modelling methods, respectively. Similar to the normal strain profiles, those obtained for the first and second method develop displacement profiles with sharp discontinuities, while the proposed one displace a smoother one.

Application to Damage Identification
In this part of the numerical study, it is presented how the performance of damage identification procedures based on measured static strains can be improved with the aid of the proposed 1D crack model. For this purpose, the same steel structural member is used (i.e. length L=2m, rectangular crosssection, 2b=0.4m, t=0.4m, also subjected to the same tensile force F=10MN, applied uniformly to one edge of the bar, while the other edge is fixed, with E=200GPa and σ=0.3). A structural damage is also considered at the half-length of the structural element, having though smaller value; the crack implemented has half-length equal to αc=0.10m.
In this part of the comparative study, the proposed 1D modelling approach of cracked structural elements is compared against a more detailed numerical model. Thus, in order to define the basis of 10 comparison, the structural element was numerically simulated by means of the FE method using a fine 3D discretization. In particular, a numerical model composed by 15,116 3D solid finite elements is used and the numerical analysis was performed with ANSYS R17.0 software. The type solid element used is the SOLID186, which is a higher order 3-D 20-node solid element that exhibits quadratic displacement behavior. In order to identify possible damages in the 3D numerical simulation strain gauges are implemented along the longitudinal direction of the structural element.
In particular, strain gauges were placed every 0.3m at perpendicular faces and they were simulated using appropriate zero stiffness elements (see Figures 10 and 11). Strain gauges were simulated with the use of SHELL181 shell elements with zero stiffness.
The proposed crack representation refers to 1-dimensional modeling of a cracked structural element; thus, its strain and displacement fields refer to "averaged" values. Therefore, an appropriate definition of the "averaged" values needs to be adopted able to translate 2D fields obtained from the measurements into 1D ones, which is not a trivial task. For the test case examined herein, without loss of generality, only central cracks are assumed to be present. Furthermore, strain gauges are assumed to be in proximity to the crack in order to capture the "averaged 1D strain" labeled as yy,i  by applying the following relation:  are the maximum and minimum strains measured for the i th cross section (see Table 1). The "averaged 1D strains" are derived assuming a linear approximation as described in   Figure 13; as it can be observed the strain field obtained by the FE analysis compared to that obtained using the proposed modeling method are also in good agreement. It should be noted that it is not possible to perform a similar identification procedure implementing any other 1D crack model, since they are not able to reproduce the strain field variation due to the presence of a crack.

CONCLUSIONS
A new model for representation of cracked bars is presented in the current study. The new model was developed by introducing an appropriate stiffness function derived from the analytical solution of a cracked plate. As it was observed through the numerical tests performed, the new model captures with satisfactory accuracy effects that are revealed only in 2 or 3 dimensional numerical simulation of the strain field variation near the crack, while preserving simplicity and computational efficiency.
For comparative reasons the proposed modelling approach was compared with existing crack models for static analysis. In addition, it was also implemented in the framework of damage identification using measured static strains; worth mentioning that such a task was not able to be performed with the existing 1D crack models. In future work, it is intended to study the efficiency of the proposed new model in case of dynamic problems like the calculation of eigenfrequencies and eigenmodes while its efficiency will also be examined when integrated in case of damage identification problems using vibration data. Integrals I1 and I2 present a singularity at π/2 and 3π/2 therefore they are appropriately computed:  T  T  T  T  T  T  T  T  T  I