M-theta Method on Mode I and II Failure of Two Cameroonian Hardwoods in Bending

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INTRODUCTION
The wood species lovoa trichilioides and triplochiton scleroxylon are subject of an increasingly high exploitation and exportation each year in Cameroon and belong to the first five types of hardwoods exploited in the country and used in the various ways [1], [2]. They are very little used for construction, but important for socioeconomic function of the country.
Often used in an empirical way, Cameroonian woods are the source of many security and dimension problems, for instance when they are used for the construction of bridges and heavy roof frames. In this context, wood material has a special interest to revive construction sector. Furthermore, due to the economic cost, the design of structure is required not for oversized dimension, but to respect the limit range of the security. Therefore, fracture behavior of wood is important to be more investigated to better understand the crack process in the structure [3].
In this regard, many work have been done to evaluate the characteristics of fracture and their consequences on wood structure facing crack problems. Reference [4] investigated in 2002 the influence of the physical and mechanical characteristics on the behavior of wood during machining. The determination of fracture energy and fracture process zone length in Mode I fracture of European spruce wood were reported by [5]. Reference [6] work on fracture energy of wood and root burl wood of Thuya. The mode I fracture of tropical woods using grid method has been developed by [7]. The Influence of loading rate on the mode II fracture toughness of vinyl ester GRP where investigated by [8] in 2017. In this work bending simulation used the Mtheta method introduced by [9]- [11] in the plane constrain for linear elastic fracture mechanics to model crack growth on two Cameroonian hardwoods.  In the absence of the body forces, the equilibrium equations are considering in the form of ( . . (2 ) .
By setting a change of variables [12], [13] as follow below,   The combination of equation (5), (6) and (7) The elements 11 12 22 33 , , , S S S S are the elastic compliance tensor components for an orthotropic symmetry. Considering the orthotropic directions (L, R, T), the elastic compliance tensor components are directly related to the elastic orthotropic characteristics for an opening mode solicitation in (R, L) plane with the crack oriented in the L direction of fiber. In the plane stress their expression is stated in the form ; Where ϕ' and χ' denote the derivative.
By combining (10) and (4), we obtain the deformation and by integration we deduced the displacement as follow:

DISPLACEMENTS AND STRESSES
From Airy's function, let set (12) in order to deduce the relation between the stress intensity factor and stress.
Taking the surface area of the crack free in tension (Fig. 2), for ϴ = ± π permit to deduce that 0 (13) y xy

 
Therefore (10) and (13) imply that, , For each values of ϴ, (13) and (14) can be presented as follow: (15)  Is obtained by imposing that deformation energy should be finite at the vicinity of the crack, therefore n = 1 [12], [13]. By comparing to the isotropic medium, the stress intensity factor is related to the field of stresses by (16). (16) proves that the singular field of stresses is dependent on the characteristic of materials in contrary to the isotropic case. Also, the stress intensity factor is related to the field of displacement by: p and q are defined as stated in (11).
From [13], the crack opening intensity factor and the stress intensity factor are related. Their expressions are given by the relation (18) (19) are the reduced elastic compliances for orthotropic behavior develop by [13] and cited by [16] and [17], which derive from the crack opening displacement vectors x u and y u for each mode of failure considered (Mode I and Mode II). 2 [14], [15], is one of the widely applied in rate-independent quasi-static fracture analysis, as a parameter characterizing crack tip field. It could be used for linear-elastic as well as elastic-plastic material behavior and the definition is given by (20): (20) x xy yx y

. Contour of integration.
Г is an arbitrary curvilinear contour oriented by its normal vector n surrounding the crack type in the external failure zone, W is the strain energy density, σ ij is the component stress and u i is the displacement vector. The integral form supposes that the crack is oriented in the x direction Fig. 3. However, energetic criterions necessitate a mixed-mode separation in order to isolate open and shear cinematic effects. In this regard, the generalization of the M-integral to orthotropic material has been developed by [10]. M-integral takes the following notation: Finite element analysis (FEA) is employed to transform this form in terms of a surface integral by introducing a θ scalar field which is continuous and differentiable for θ =1 and θ = 0 inside and outside the ring as it is shown in Fig. 4. The vector field  r must be continuous and differentiable on the considered S domain. Then, the general modeling form of integral is given by [16], [17]. .
where the virtual field v is given by the Sih's singular form presented in equation (17).

IV. GEOMETRY OF SENB SIMULATION SPECIMEN
The geometry of the specimen is firstly generated according to the SENB specimen. This specimen is adapted to model easily in three point bending the evolution of crack. The obtained specimen dimensions are presented in Fig. 5 with the following values base on [18] work.  The geometry is calculated using a plane stress assumption, using a unitary loading force corresponding to the determination of stress intensity factor and the energy release rate values. This finite element mesh in which a circular discretization around the crack tip ( Fig. 6 (v)), allows to easily define the θ vector field and the integration crown using parametric elements. Mθ-integral performed the virtual finite element displacement fields ( Fig. 6 (iii) and (iv)) and deformation ( Fig. 6 (i)  V.

RESULTS AND DISCUSSIONS
The simplified numerical routine (flow diagram) used to compute the fracture parameters based on Mθ-integral is sketched in Fig. 7. The Fig. 8 below evaluates the path independence of the methods on the specimen according to crown. By considering different crack orientations, it is observed the unvarying results versus different crowns for energy release rate evolution. Across singularity in the crack tip or near zero the mechanical fields cause a disturbance through the integration process for the crown 0, which is the origin of the perturbation. Nevertheless, there is a great stability with the increase of crown sizes. This satisfied the theoretical assumption of path independence in agreement with [19] work.
From the result, in what follows crown six, will be used to performed calculations of the stress intensity factors and energy release rate. obtained by [20], [21]. Hence the total energy of the two wood is reduced to the released strain energy consequences of the decrement and induced their brittle fracture. This type of result would occur in a "fixed-grips" apparatus with the applied load and the apparatus clamped into position. The same arguments can be exactly being applied for a "dead-weight" loading, where the fracture surface energy corresponds to a decrease in potential energy of the loading system [21]. The apparent drop in mode II toughness of stress intensity factor and energy release rate ( Fig.   9 (iii), (iv)) and (Fig. 10 (c), (d)) observed as crack length increases for the two hardwoods were obtained by [8] on four-point bending end notched flexure (4ENF) tests. The causes of this behavior is not yet known but, [8] observed that the crack tends to open under large displacements and tended to migrate away from the mid-thickness of the specimen and towards face of tension. They remark that mode I fracture in addition induce to the intended mode II fracture for the opening of the crack and the measured of fracture toughness is modify by the mixed mode. In this respect, this study reveals the decreasing of the energies for the both two modes of fracture; which imply future investigation of the effect of mixed mode to overcome some conclusion on these behaviors.

VI. CONCLUSION AND PERSPECTIVE
In this study, the evolution of energy release rate and stress intensity factors of two woods species have been investigated on bending. The finite element mesh of the two hardwoods in different modes were presented. The independently computation results of the stress intensity factor and the energy release rate for mode I and mode II fracture are proved. The pathindependence of Mθ-integral result have been showed. It has been observed the quasi decrement of energy release rate and stress intensity factors of the both two hardwoods species. The future investigation will be consecrated to study mixes mode fracture on the two hardwoods to evaluate his effect and finally it will be important to make an experimental setup result to compare with the numerical results.