Thermo-elastic extended meshfree method for fracture without crack tip enrichment

Abstract

This is the first manuscript presenting an extended meshfree method for thermo- elastic fracture which does not exploit a crack tip enrichment. The crack is modeled by partition of unity enrichment of the displacement and temperature field. Only a step function is employed that facilitates the implementation. To ensure that crack tip is at the correct position, a Lagrange multiplier field ahead of the crack tip is introduced along a line. The Lagrange multiplier nodal parameters are discretised with the available meshfree functions. Two benchmark examples illustrate the efficiency of the method.

Keywords

meshfree method / thermo-elasticity

Introduction

Meshfree methods have allowed the solution of many problems where classical methods such as finite elements had difficulties. Phenomena such as fracture [ 1− 7], large deformations [ 8− 12] and fluid-structure interaction [ 13− 16] are classical applications where meshfree methods have advantages. Concerning fracture, partition of unity enrichment originally employed in the extended finite element method [ 17, 18] are effective methods to model fracture also in meshfree methods as in Ref. [ 19− 23]. Alternatives to XFEM or XIGA [ 24, 25] are efficient remeshing techniques such as proposed by Refs. [ 26− 30] or multiscale methods [ 31− 39]. Parition of unity methods are based on the decomposition of the displacement field into a continuous displacement field and a discontinuous displacement field. The discontinuous displacement field has been often achieved by adding additional degrees of freedom which are multiplied with enriched shape functions. Normally, one distinguishes between crack tip enrichment functions and step enrichment functions for fracture problems. Crack tip enrichment functions in the context of meshfree methods have been used for instance by [ 21, 40, 41]. While it is straightforward to omit the crack tip enrichment in finite element analysis, it causes difficulties in meshfree methods [ 42, 43]. One difficulty is the wrong positioning of the crack tip. Some simple criteria based on the adjustment of the domain of influence have been proposed by [ 43] but they seem to be inadequate for 3D and require also relocation of the nodes. Another alternative is based on Lagrange multipliers [ 42]. A good overview on modeling fracture with partition of unity methods in three dimensions is given in Ref. [ 44]. Alternative methods to partition of unity enriched meshfree methods employ the visibility method [ 2, 45, 46], transparency or diffraction method [ 47, 48] or simpler strategies of enrichment such as the cracking particles method [ 49] or slight variations of it [ 50, 51].

While most of the above mentioned approaches are focus on fracture for pure mechanical problems, there are comparatively few methods for thermo-elastic fracture. A few contributions exist in the context of the extended finite element method, see e.g., Refs. [ 52− 54]. However, there are only a few papers on thermo-elastic partition of unity enriched meshfree methods. All approaches employ a crack tip enrichment. In Ref. [ 55] for instance, the formulation presented is incorrect as it shows a symmetric final system of equations though the thermo-elasticity equations are uncoupled. In particular, there does not exist a partition of unity enriched meshfree method for thermo-elastic fracture. This formulation is proposed in this manuscript.

The manuscript is structured as follows: In the next section, the governing equations in weak form is stated. The discretization of the mechanical and thermal field is explained next before the final discrete system of equations are derived. Two benchmark problems conclude the manuscripts.

Weak form of the governing equations

The weak form of our governing equations are the equation of equilibrium and the heat equation. In the uncoupled thermo-elasticity problem, the thermal field influences the mechanical field but not vice versa. The weak form can be stated as variational form: Find the displacement field u∈ U and temperature field T∈ \mathit{\Gamma } for all test functions δu∈ U₀ and δT∈ {\mathit{\Gamma }}_{\mathrm{0}} such that

(1) \begin{array}{l}{\displaystyle \underset{\mathrm{\Omega }}{\int }\delta {\mathit{\in }}_{ij}^M{\sigma }_{ij}\mathrm{d}\mathrm{\Omega }}-{\displaystyle \underset{\mathrm{\Omega }}{\int }\delta u_i{b}_i\mathrm{d}\mathrm{\Omega }}-{\displaystyle \underset{{\mathrm{\Gamma }}_t}{\int }\delta u_i\overline{t_i}\mathrm{d}\mathrm{\Gamma }=\mathrm{0}},\hfill \\ {\displaystyle \underset{\mathrm{\Omega }}{\int }\delta q_i\mathit{k}{q}_i\mathrm{d}\mathrm{\Omega }}+{\displaystyle \underset{\mathrm{\Omega }}{\int }\delta TQ\mathrm{d}\mathrm{\Omega }}-{\displaystyle \underset{{\mathrm{\Gamma }}_q}{\int }\delta T\bar{q}\mathrm{d}\mathrm{\Gamma }=\mathrm{0}},\hfill \end{array}

where Ω denotes the domain of the continuum and the boundary \mathrm{\Gamma }={\mathrm{\Gamma }}_u\cup {\mathrm{\Gamma }}_t\cup {\mathrm{\Gamma }}_c={\mathrm{\Gamma }}_T\cup {\mathrm{\Gamma }}_q with {\mathrm{\Gamma }}_u\cap {\mathrm{\Gamma }}_t=\mathrm{0} , {\mathrm{\Gamma }}_c\cap {\mathrm{\Gamma }}_t=\mathrm{0} , {\mathrm{\Gamma }}_u\cap {\mathrm{\Gamma }}_c=\mathrm{0} ; Γ_T consists of boundaries where temperature and flux is imposed on Γ_q ; traction and displacements are imposed on Γ_t and Γ_u, respectively, and Γ_c denotes the crack boundary. The solution spaces of admissible test functions δu_i and δT and trial functions u_i ∈ U and T∈ \mathit{\Gamma } are given by

(2) \begin{array}{l}U=\{u_i|u_i\in H^{\mathrm{1}}, u_i=\overline{u_i} \mathrm{o}\mathrm{n} {\mathrm{\Gamma }}_u, u_i \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s} \mathrm{o}\mathrm{n} {\mathrm{\Gamma }}_c\},\hfill \\ U_{\mathrm{0}}=\{\delta u_i|\delta u_i\in H^{\mathrm{1}}, \delta u_i={\mathrm{0}}_i \mathrm{o}\mathrm{n} {\mathrm{\Gamma }}_u, \delta u_i \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s} \mathrm{o}\mathrm{n} {\mathrm{\Gamma }}_c\},\hfill \\ \mathrm{\Gamma }=\{T|T\in H^{\mathrm{1}}, T=\overline{T_i} \mathrm{o}\mathrm{n} {\mathrm{\Gamma }}_T, T \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s} \mathrm{o}\mathrm{n} {\mathrm{\Gamma }}_T\},\hfill \\ {\mathrm{\Gamma }}_{\mathrm{0}}=\{\delta T|\delta T\in H^{\mathrm{1}}, \delta T=\mathrm{0} \mathrm{o}\mathrm{n} {\mathrm{\Gamma }}_T, \delta T \mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s} \mathrm{o}\mathrm{n} {\mathrm{\Gamma }}_T\},\hfill \end{array}

where H is a Sobolev space.

Discetization of the displacement and temperature field

The thermo-elasticity problem requires the discretization of the displacement field and the temperature field. Accounting for the jump in both fields across the crack surface, the approximation is given by

(3) u^h\left(X\right)={\displaystyle \sum _{I\in N}{N}_{I }\left(X\right)u_I }+{\displaystyle \sum _{I\in N_c}{\tilde{N}}_I \left(X\right){\tilde{\mathbf{u}}}_I,}{T}^h\left(X\right)={\displaystyle \sum _{I\in N}{N}_{I }\left(X\right)T_I }+{\displaystyle \sum _{I\in N_c}{\tilde{N}}_I \left(X\right){\tilde{T}}_I},

where u_I and T_I are the nodal parameters of the continuous displacement and temperature field, respectively, while the nodal parameters {\tilde{u}}_I and {\tilde{T}}_I denote the nodal parameters of the discontinuous displacement and temperature field; N and N_c are the set of nodes in the entire discretization and close to the crack surface, respectively. The shape functions N_{I }(X) are constructed based on a moving least squares approximation as in the element-free Galerkin method [ 46] and are given by:

(4) N_I\left(\mathbf{X}\right)={\mathbf{p}}^{\mathrm{T}}\left(\mathbf{x}\right)A^{-\mathrm{1}}\left(\mathbf{x}\right){\mathbf{R}}_I\left(\mathbf{x}\right),

where

R_{I }\left(\mathbf{X}\right)=w\left(\mathrm{x}-{\mathrm{x}}_I,\mathit{h}\right)p^{\mathrm{T}}\left({\mathrm{X}}_I\right)

(5) {\mathbf{A}}_I\left(\mathbf{X}\right)=\sum _{I\in W}{\displaystyle w\left(\mathrm{x}-{\mathrm{x}}_I,\mathit{h}\right)p\left({\mathrm{X}}_I\right)p^{\mathrm{T}}\left({\mathrm{X}}_I\right)}.

w(X−X_I,, h) denoting the weighting function with support size h and with linear polynomial basis functions p_I (x) [1 x y]. The enriched (or discontinuous shape functions {\tilde{N}}_I(X) are constructed by multiplying the standard shape functions N_I (X) with a discontinuous enrichment function which ensures that the displacement field and the temperature field is discontinuous along the crack surface. We have used the step function:

(6)

where n is the normal vector to the crack surface and x_c is a point on the crack surface; n\cdot (x-x_c)=\mathrm{0} identifies a point on the crack surface. However, this will never occur in practice as commonly the distance of quadrature points in the interior of the domain (not on the crack surface) are determined. The crack surface is represented by a set of piecewise linear crack segments.

Discrete governing equations

To derive the discrete equations, we substitute the discretization of the displacement field and temperature field as well as their spatial derivative into the weak form, Eq. (1). After some algebra, the well-known discrete equation is obtained:

(7) KD=F,

where K denotes the system’s global stiffness matrix, F is the global force vector and the nodal parameters are stored in the vector D={\left[\mathbf{u},\tilde{\mathbf{u}},\mathbf{T},\tilde{\mathbf{T}}\right]}^{\mathrm{T}} where the nodal parameters of Eq. (3) are stored in the associated vectors, e.g., T_I is stored in T. This system of equation is smaller compared to the approach in Ref. [ 55]. However, in order to ensure that the crack tip ends at the correct position, we introduce a Lagrange multiplier field as suggested by Ref. [ 43] for a pure mechanical problem. We extend it here to thermo-mechanical problems. Therefore, we need to ensure that also the additional degrees of freedom of the thermal field vanish yielding. The discretization of the Lagrange multiplier field for the mechanical and thermal part is given by:

(8) \begin{array}{l}{\lambda }_M^h(X)={\displaystyle \sum _{I\in N}{N}_{I }(X)}{I}_{MI },\hfill \\ {\lambda }_T^h(X)={\displaystyle \sum _{I\in N}{N}_{I }(X)}{I}_{TI },\hfill \end{array}

the index M and T referring to the mechanical and the thermal field, respectively. Enforcing the crack to close along the extension line of the crack introduced into the formulation as constraint condition finally yields the following system of equations:

(9) \tilde{\mathbf{K}}\tilde{D}=\tilde{F},

or

(10) \left[\begin{array}{llllll}{\mathrm{K}}_{MM}^{uu}& {\mathrm{K}}_{MM}^{u\tilde{u}}& \mathrm{0}& {\mathrm{K}}_{MT}^{uT}& {\mathrm{K}}_{MT}^{u\tilde{T}}& \mathrm{0}\\ {\mathrm{K}}_{MM}^{\tilde{u}u}& {\mathrm{K}}_{MM}^{\tilde{u}\tilde{u}}& {\mathrm{G}}_{ML}^{\tilde{u}L}& {\mathrm{K}}_{MT}^{u\tilde{T}}& {\mathrm{K}}_{MT}^{\tilde{u}\tilde{T}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \left[{\mathrm{G}}_{ML}^{\tilde{u}L}\right]& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& {\mathrm{K}}_{TT}^{TT}& {\mathrm{K}}_{TT}^{T\tilde{T}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& {\mathrm{K}}_{TT}^{\tilde{T}T}& {\mathrm{K}}_{TT}^{\tilde{T}\tilde{T}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& {\left[{\mathrm{G}}_{TL}^{\tilde{T}L}\right]}^{\mathrm{T}}\end{array}\right]\left[\begin{array}{l}u\\ \tilde{u}\\ {\mathrm{I}}^M\\ \mathrm{T}\\ \tilde{\mathrm{T}}\\ {\mathrm{I}}^T\end{array}\right]\left[\begin{array}{l}{\mathrm{F}}_M^u\\ {\mathrm{F}}_M^{\tilde{u}}\\ \mathrm{0}\\ {\mathrm{F}}_T^T\\ {\mathrm{F}}_T^{\tilde{T}}\\ \mathrm{0}\end{array}\right].

In these equations, the subscripts M, T and L refer to the mechanical field, temperature field and the Lagrange-multiplier field similarly to the notation of the superscripts which refer to the degrees of freedom (DOF). The matrices are given by:

(11) \begin{array}{l}{\mathrm{K}}_{MM}^{uu}={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{B}}_{Mu}C{B}_{Mu}^{T}d\mathrm{\Omega }},\hfill \\ {\mathrm{K}}_{MM}^{u\tilde{u}}={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{B}}_{Mu}C{B}_{M\tilde{u}}^{T}d\mathrm{\Omega }},\hfill \\ {\mathrm{K}}_{MM}^{\tilde{u}\tilde{u}}={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{B}}_{M\tilde{u}}C{B}_{M\tilde{u}}^{T}d\mathrm{\Omega }},\hfill \\ {\mathrm{K}}_{MT}^{uT}={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{B}}_{Mu}{C}_{\alpha T}{N}_{Tu}^{T}d\mathrm{\Omega }},\hfill \\ {\mathrm{K}}_{MT}^{u\tilde{T}}={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{B}}_{Mu}{C}_{\alpha T}{N}_{T\tilde{T}}^{T}d\mathrm{\Omega }},\hfill \\ {\mathrm{K}}_{MT}^{u\tilde{T}}={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{B}}_{M\tilde{T}}{C}_{\alpha T}{N}_{Tu}^{T}d\mathrm{\Omega }},\hfill \\ {\mathrm{K}}_{MT}^{\tilde{u}\tilde{T}}={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{B}}_{M\tilde{u}}{C}_{\alpha T}{N}_{T\tilde{T}}^{T}d\mathrm{\Omega }},\hfill \\ {\mathrm{K}}_{TT}^{TT}={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{B}}_{TT}D{B}_{TT}^{T}d\mathrm{\Omega }},\hfill \\ {\mathrm{K}}_{TT}^{T\tilde{T}}={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{B}}_{TT}D{B}_{T\tilde{T}}^{T}d\mathrm{\Omega }},\hfill \\ {\mathrm{K}}_{TT}^{\tilde{T}\tilde{T}}={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{B}}_{T\tilde{T}}D{B}_{T\tilde{T}}^{T}d\mathrm{\Omega }}.\hfill \end{array}

where the derivatives of the EFG shape functions (the standard and the enriched ones) are stored in the vector B_IJI = M,T standard for the mechanical or temperature and J referring to the DOF; α_T is the thermal conductivity and D is a diagonal matrix containing the diffusity k on the main diagonal (assuming isotropy). The right hand side vectors are given by

(12) \begin{array}{l}{\mathrm{F}}_M^u={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{N}}_{Mu}\mathrm{bd}\mathrm{\Omega }}+{\displaystyle \underset{{\mathrm{\Gamma }}_t}{\int }{\mathrm{N}}_{Mu}\bar{t}d\mathrm{\Gamma }},\hfill \\ {\mathrm{F}}_M^a={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{N}}_{M\tilde{u}}\mathrm{bd}\mathrm{\Omega }}+{\displaystyle \underset{{\mathrm{\Gamma }}_t}{\int }{\mathrm{N}}_{M\tilde{T}}\bar{t}d\mathrm{\Gamma }},\hfill \\ {\mathrm{F}}_T^u={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{N}}_{TT}\mathrm{Qd}\mathrm{\Omega }}+{\displaystyle \underset{{\mathrm{\Gamma }}_q}{\int }{\mathrm{N}}_{TT}\bar{q}d\mathrm{\Gamma }},\hfill \\ {\mathrm{F}}_T^a={\displaystyle \underset{\mathrm{\Omega }}{\int }{\mathrm{N}}_{T\tilde{T}}\mathrm{Qd}\mathrm{\Omega }}+{\displaystyle \underset{{\mathrm{\Gamma }}_q}{\int }{\mathrm{N}}_{T\tilde{T}}\bar{q}d\mathrm{\Gamma }}.\hfill \end{array}

All integrals are evaluated by 4 × 4 Gauss quadrature based on a background mesh with bi-linear FE shape functions which is formed by the meshfree nodes. Background cells cut by the discontinuity are integrated by modifying the quadrature weights as proposed in Ref [ 10].

Results

Center crack problem

Consider a rectangular plate with a center crack as shown in Fig. 1(a). We set the length Lequal to the width W and assume temperature boundary conditions of 10 degree celsius. Though there is no analytical solution available, the results can be compared to results from other researchers [ 29, 35] who studied this problem as well. Table 1 illustrates that the results presented in this manuscript are in close agreement to the solutions in [ 56, 57]. The temperature distribution in the specimen at the end of the computation is depicted in Fig. 2(a).

Inclined crack problem

Next, we modify the first problem by inclining the crack as shown in Fig. 2 which also illustrates the temperature distribution. The stress intensity factors are again compared to a solution presented in Refs. [ 56, 57] and they show a good agreement, see the Tables 2 and 3. Table 2. In addition to the results for various angles α for a selected a/W value, also results for different values of a/W are shown for an angle of 30 degree.

Conclusiton and research perspectives

We presented a new partition of unity enriched meshfree method for thermo- elastic fracture. Therefore, the EFG method has been employed. The displacement field and temperature field have been enriched with step functions. The method has been tested for stationary cracks. The results obtained by this method was compared to two well-known benchmark problems available in the literature and the agreement of the presented method and other computational methods were good. Furthermore, the method requires less degrees of freedom due to the presence of an enrichment. In the future, the method will be used to model more complex problems including crack growths.

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