Part-through cracks computation in an Euler–Bernoulli beam model

Finding how and when cracks develop in a 3D continuum has immediate relevance in applications. The seminal paper of Griffith [1] rooted the problem in an energetic framework where crack development is driven by the competition between the elastic energy stored within the deforming body and the crack energy. Formulation in the modern variational framework was only achieved recently [2] and pinpointed the troubles of Griffith’s theory to handle initiation [3‐5]. The need for numerical implementation lead to the development of phase-field models [6, 7]: a damage field modeling the phase transition of the material from pristine to damaged is introduced and regularized by the addition of higher gradients. Refinements allow to mimic various material behavior [8‐10], and the framework proves successful for describing the crack initiation and propagation in many situations [11] - see for example [12] for a review.

A beam theory requires describing the behavior of each section associated with a position on a centre-line, a geometric abstraction grasping the shape of a slender body. For some geometries, the loads result in a tensile stress all over the sections, legitimating the use of the brittle assumption [13, 14]. When bending is included, the non-zero yet limited load-bearing capacity in tension together with the different capacities of a brittle material in tension and compression result in the preferential development of cracks on the tensiled side of the body [15, 16]. This asymmetry results in significant variation of the sections characteristic geometrical quantities [17, 18] and requires special care when handling the description of the beam with developed cracks [19‐21]. Recent attempts at developing a phase-field theory of beams or shells [22, 23] grasping the initiation of the fracture stemmed from the idea that a damage field can be used to represent the crack penetration depth in the beam [24]. In [25], the shape of the damage field is postulated whereas [26] attempts at precisely describing the strain field alteration due to the existence of the cracks.

We propose here to investigate crack initiation by describing each cross section using two geometric fields measuring the crack penetration on each side of a slender body thus parametrizing the geometry of partially cracked sections. Characterizing the deformation of the sections using the Euler–Bernoulli hypothesis, we get the dependence of the bending moment in both the transversal displacement field and the geometric fields by direct integration. We formulate the problem of the equilibrium of a slender body allowed to crack in this framework, and regularize the energy functional in the spirit of the variational brittle fracture by the addition of higher gradients of the geometric fields. The solution fields are obtained as global minimizers along a time-stepping procedure. The compatibility of the minimizers with the geometric description of the body, with the irreversibility of the cracking process, and the crack localization on the tensiled side are ensured with additional constraints. From our smeared modeling, we retrieve the envelope of the crack zones in our dimension-reduced model.

Consider a homogeneous connected body \(\Omega \subset \mathbb {R}^3\) with Lipschitz boundary \(\partial \Omega \). The body is in equilibrium with internal forces of density \(\underline{f}_b\) and surfacic forces of density \(\underline{f_s}\) whose work we will denote \(\mathcal {P}=\int _\Omega \underline{f}_b\cdot \underline{u} \text {d}\Omega + \int _{\partial \Omega } \underline{f_s}\cdot \underline{u}\text {d}S\). Under the effect of the forces the body deforms following a displacement field \(\underline{u}\) and stores energy as deformation energy of density \(W(\nabla \underline{u})\). The eventual cracks of the body form a set \(\Gamma \subset \mathbb {R}^2\) - supposed smooth - endowed with a surface energy of density G according to the Griffith postulate. Altogether, the total energy of the solid contains the three terms from above:The strain energy density W is the double contraction of the strain tensor \(\underline{\underline{\varepsilon }}\) with the stress tensor \(\underline{\underline{\sigma }}\). Assuming small strains, the strain tensor is the symmetric part of the displacement gradient \(\underline{\underline{\varepsilon }}=\frac{1}{2}(\nabla \underline{u}+\nabla \underline{u}^t)\). The main difficulty of the formulation - besides the eventual complexity of the strain energy density W - is the dependence of the functional on the unknown cracked set \(\Gamma \). The elastic energy density is then given as The now classical method of regularization [6, 7, 10] consists in introducing a smooth scalar field \(d:\Omega \rightarrow [0,1]\) measuring the local degradation of the material. This field is deemed to be 0 if the material is intact and 1 wherever the material is completely destroyed e.g there is no longer pressure transmission. The addition of a term depending in the scalar field gradients [7, 27] ensures well-posedness for the new problem consisting of finding simultaneously the unknown displacement and damage fields [28]. A characteristic length \(\ell \) controlling the transition size of the scalar field appears in the regularized form of the energy and can be traced with the modeling of nonlocal materials [15]. In addition, the strain energy density has to be modulated by the local value of the damage field , in order to account for the effect of the damage on the local stiffness. Physical considerations [10] and need to faithfully represent various material behavior [9] lead to different modulations of the bulk energy density. In case the load is purely tensile and the material homogeneous isotropic, a choice suggested by the reference above is \(\tilde{W}(d,\nabla \underline{u}):=(1-d)^2W(\nabla \underline{u})\). In turn, the energy to minimize is:The load dependence in time (or in some time-like control parameter) is contained in the potential energy \(\mathcal {P}\). Using a gamma-convergence argument [6], one shows that the regularized energy (3) converges effectively to the original energy (1) when the characteristic length \(\ell \) goes to zero.

In the following section, we develop an Euler–Bernoulli beam model, where the damage is characterized by the crack penetration depth. For this purpose, we follow the beam slicing into sections and specify the crack topology, on every section. In effect we retrieve a damage field whose geometric meaning is clear: it corresponds to the crack penetration depth on each side of the beam. The stress resultants and their modulations by the crack depth are obtained by direct integration. Finally, the model is regularized by the addition of higher gradients.

In this subsection we derive from the 3D model (3) a 1D version of the penalized elastic energy density. Consider a beam, described by translating a section \([-b/2,b/2]\times [-h/2,h/2]\) along a straight line [0, L], b, h and L being, respectively, the breadth, height and length of the beam. The domain occupied by the beam is \(\Omega = [-b/2,b/2]\times [-h/2,h/2]\times [0,L]\). We use the frame where \(\hat{z}\) points in the direction of the centre-line, and \(\hat{y},\hat{x}\) are in the principal directions of inertia of the section, y being the height coordinate of any point of the section. Let us associate a displacement field \(w(z)\hat{y}\), deforming the centre line in its plane, and make the Euler–Bernoulli hypothesis that sections perpendicular to the centre line remain perpendicular to the centre line and deform rigidly. For a point at arclength z along the centre-line at a height y away from the neutral axis and under the small displacement assumption, the transversal displacement u isWith our hypothesis on the displacement field and neglecting Poisson effect, only the tensile stress \(\sigma _{zz}\) is nonvanishing:Consider now that the beam sections are allowed to evolve. From the Euler–Bernoulli hypotheses, the beam sections remain plane, thus we prescribe cracks to follow the beam slicing and reduce the section at particular positions. That is, we assign to each position z along the centre-line of the curve scalar fields \(0\le h^-(z),h^+(z) \le h\) characterizing the damage, on each side, to each section. Effectively, we selected the damage field \(d:\Omega \rightarrow [0,1]\) appearing in (3) to be 0 in the part of the beam delimited by \(h^-,h^+\) and 1 outside it:where we denoted \(\mathbb {I}_\Gamma (y)\) the indicator function of the set \(\Gamma \), taking value 1 if \(y\in \Gamma \) and 0 otherwise. To get the contribution of the damage to the bulk energy density, we start from the 3D formulation (3). The density of elastic energy per unit length is obtained by integrating the energy density over the beam section at position z. Taking into account our kinematic assumption (5) and our prior on the damage field (6), we get the lineic density of elastic energy for the beam:where the moment of inertia of the section depends on the crack penetration depthsIn the second line we introduced dimensionless quantities \(h^+/h\) and \(h^-/h\) corresponding to the damaged fraction of the height of the beam section. These quantities characterize the damage to the section as damage fields: they have minimal value 0 without damage and maximal value 1 when the damage spans a whole section. They are the 1D analogous of the damage field d appearing in the first term of Eq. (3).

We now turn to the second term appearing in (3) accounting for the energy cost of creating new cracks surfaces. According to Griffith theory and in the case of a perfectly brittle material, the energy expenditure is the product of the created crack surface with the fracture toughness G. Using the fields notation we introduced above, we get that the crack energy for a crack localized at \(z_0\) is:The fundamental issue of the previous expression is that the crack is assumed to be perfectly localized at a single position \(z_0\): the energy above does not have a density. We show in the next subsection that this results in an infinite required stress for crack propagation from a pristine material, precluding the calculation of cracks without regularization. In the next section, we include a regularization depending on the higher gradients of \(h^+,h^-\) that allow the problem to be well posed (Fig. 1).

Currently we did not introduce asymmetry in the crack development. To keep track of the side on which the crack initiation takes place, one can complexify the expression of the strain energy [29], but we find it more convenient to postpone the resolution of this issue further. As the energy minimization has to follow a time-stepping procedure, we will enforce at this latter stage the evolution of \(h^+,h^-\) depending on the sign of the curvature \(w_{,zz}\). Further, note that the modulations of the sections surface and second moment of area are different for the bending part and for the tensile part, similarly to [26].

In this subsection we show that the initiation of cracks depends on the size of the mesh discretization.

Let us, just for this subsection, consider that the a crack is localized at arclength \(z_0\) of the beam, and that this crack occupies a small portion of the beam between the arclengths \(z_0-\varepsilon /2\) and \(z_0+\varepsilon /2\) - the length \(\varepsilon \) can be taken to be a typical mesh size. Without loss of generality, due to assuming that the crack initiates on the tensiled side of the bent beam, we set \(h^-\equiv 0\) and thus concern ourselves only with \(h^+\) that we assume constant on the interval under consideration. In this setup, \(h^+\) is a scalar parameter characterizing the crack penetration depth in the beam element, and \(0\le h^+/h \le 1\) is a dimensionless parameter characterizing the crack penetration depth. Considering a Griffith-like dissipation potential, together with the results of Sect. 2.1 allows to write the energy stored in the deteriorating part of the beamFrom the Griffith criterion, the crack propagates whenever the elastic energy release rate is larger than the material toughness. Using that here the damage value \(h^+\) is a simple scalar, the derivative of \(\mathcal {E}_\varepsilon \) against \(h^+/h\) is readily computed:Light calculations using expression (11) for \(I(h^+,0)\) yields the cracking criterion at length \(h^+\). Letting \(h^+\) go to zero, one retrieves a crack initiation criterion, determined by the geometry of the beam and the material constants:Crack nucleation is possible with this model only if the crack is spatially extended. That means, only smeared cracks can be computed, and the result will be more akin to a crack envelope than a localized crack. As the relapse of the bending energy is conditioned by the reduction of the section, a localized crack cannot induce the energy spending necessary for crack opening. We keep in mind this feature of the formulation, and introduce in the next section such a characteristic length.

From the general damage field formulation (3) and the previous section, we can readily formulate the problem of the beam equilibrium in the variational framework. Considering a distributed load \(p_t(z)\) whose potential energy is \(\mathcal {P}=\int p_tw\), we get the regularized energy of the solid from the previous subsection. The singular term is approximated by a convex function of the damage fields and their gradient similar to [7, 10]: The independence of the evolution of the two crack fronts on both sides of the beam is reflected in each term being independently squared. Note that this formulation of the energy is consistent with our convention for the value of \(h^+,h^-\): the first part of the second line term is minimal when \(h^+=h^-=0\). A dependence on a scale parameter \(\ell \) is introduced to control the length over which the fields vary. Decreasing \(\ell \) effectively favors sharp variations of the section.

Despite constraining the evolution of cracks to follow the beam slicing into sections as per the Euler–Bernoulli hypothesis, the whole geometry of the beam and the cracks path is specified through \(h^+,h^-\). In practice, given a load \(p_t(z)\) we look for a displacement field w and fields \(h^+,h^-\) minimizing the energy (18). From now on and for all the remaining of this paper we will assume the beam to have a straight initial shape and an initial rectangular cross-section as per the calculations of Sect. 2.1. Further we restrict ourselves to proportional loads \(p_t(z)=tp(z)\).

This section is devoted to the method of computation of the unknown fields \(h^+,h^-,w\) following a quasistatic increase of the load \(p_t\) and with initial fields \(h^+_0,h^-_0,w_0\) provided. The computation of the whole evolution roots in a time-stepping procedure. The load is discretized following instants \(t_n=n \delta t\), and the fields are computed from the previous solutions. We use the staggered scheme introduced in [29] ensuring a robust and reliable resolution of the equations. The displacement field and the damage fields are computed at two different steps. We will denote \(w_n,h^+_n,h^-_n\) the unknown fields at time step \(t_n\).

The solution unknown fields have to satisfy the stationarity equations for the functional defined (18). Following the operator split, we look at the stationarity equations for the displacement field \(w_n\) and the geometric fields \(h^+_n,h^-_n\) separately.

At time \(t_n,n\ge 1\) the first step of the staggered scheme is to compute the solution displacement \(w_n\) given the value of fields \(h^+_{n-1},h^-_{n-1}\). The solution transversal displacement field \(w_n\) is searched as a minimizer of the restricted functional \(\mathcal {E}(\cdot ,h^+_{n-1},h^-_{n-1})\). Satisfying the Euler equation for the functional on the interior of the interval where the displacement is defined (0, L):Providing boundary conditions suited to any problem at hand ensures well-posedness for the two point linear boundary value problem for the displacement field \(w_n\). Boundary conditions will be specified within the examples in Sect. 4.

The second step of the staggered scheme is to solve for the geometric fields \(h^+_n,h^-_n\). The resolution is much more delicate than for the damage fields, as the problems are highly nonlinear and constraints have to be taken into account. The system of boundary value problems is deduced from setting the first variation of the functional (18) to be zero and is supplemented with the natural boundary conditions for the fields \(h^+_n,h^-_n\), namely:Note that the subordination of the geometric fields evolution to the curvature sign is enforced here: depending on the geometric field considered and the sign of the beam curvature \(w_{n,zz}\), the stationarity equation has to be adapted. Precisely, if the curvature sign \(w_{n,zz}(z)\) is positive on some given interval \((z_0,z_1)\), then the field \(h_n^+(z)\) has to satisfy the Euler equation for the functional \(\mathcal {E}(w_n,\cdot ,h^-_n)\) on that interval \((z_0,z_1)\), whereas \(h_n^-\) has to remain constant. If the curvature is negative, then it is \(h_n^-\) that has to satisfy the stationarity equations whereas \(h_n^+\) is prescribed to be constant.

In addition, constraints on the fields value have to be satisfied as given in (??) to ensure that the section reduction due to the damage remains coherent with the initial geometry of the beam. Finally, the Unilateral Stationarity constraint [2] is enforced by requiring that for all n, all \(0 \le z \le L\):The damage is irreversible, and cracks can only open further as the load increases.

The first load case we investigate is that of a beam whose two ends are clamped, loaded in the middle of its span with a single point force F. The mechanical model can be seen in Fig. 2. The boundary condition for the associated model are the geometrical conditions on the beam centre-line:Namely there is no transversal displacement for the beam centre line at the clamped points, and the centre-line of the beam there is horizontal. The results of the resolution following the algorithm of Sect. 3.2 are shown in Fig. 3.

The moment diagram for the mechanical model of Fig. 2 when the beam section is kept constant has three extrema at the extremal and middle points of the beam \(z=0,z=L/2,z=L\). On the left of Fig. 3, the damage field obtained after a gradual load increase is showed. The zones of large damage are figured in red. As expected, fracture initiate at the 3 points of maximal strain: on the tensiled, top parts of the beam at the supports \(z=0, z=L\), and bottom part at the midpoint \(z=L/2\). On the right hand side of the figure, the original shape of the beam is figured in black and the damaged surface of the beam in red. The essential features of the cracking pattern are grasped through the dimension reduction: the shape of the damage zone obtained through the two-dimensional damage field model matches that of the depth crack penetration on the right. On the other hand, our model predicts a strong smearing of the damage despite our small characteristic length. Neglecting the effect of the crack on the stress distribution induces a diffusion of the damage farther away from the point maximal moment.

The second load case we investigate is that of a simple cantilevered beam loaded in the middle and the end of its span with single point forces \(F_1,F_2\). The mechanical model is given in Fig. 4. The boundary conditions to associate are the followingThat is, the transversal displacement and the angle of the centre line with the horizontal are prescribed to be zero at the attach point \(z=L\), whereas the centre-line curvature and its derivative are zero at the free end.

As we investigate the case where \(F1=F2\), the moment diagram for the mechanical model of Fig. 4, computed with the Euler–Bernoulli hypothesis has two maxima at \(z=L/2, z=L\). The solution obtained by the damage field model is given in Fig. 5 left, with the most damaged zones in red. The damage concentrates on the tensiled side of the beam at the attach point, initiating at the point of maximal moment. The damaged zone expands far away from the support with our model, probably for the same reason as in Sect. 4.1. Nonetheless the appearance point of the crack at the support matches in both simulation.