C R Chen et al. Acta Materialia 55(2007)409-421 The calculation procedure to find the residual stress and aaz= aAZ -A=0.6x 10-K- as the substitute state in laminate specimens has been heavily investigated CTE in the AZ-material. The second 2D computation used (see e.g. [19, 20]. We use a three-dimensional (3D) model the corresponding plane stress model of the specimen, consisting of a slice of length 8 in the y- direction and covering the area0≤x≤M2,0≤z≤B/2.4.3. Beam bending Symmetry conditions are applied along 2=0 and x= h/ 2, and the y-displacements are fixed on the bottom of the As outlined above, four-point-bend tests are performed layer. Only in the end regions far from the crack plane will on rather slender beams. The classical beam-bending the the residual stress state depend on the y-coordinate. Since ory could be used to evaluate the stress state the specimen is very long in y-direction, the residual stres- uncracked composite beam; for details, see e.g ses can be assumed to be independent of the y-coordinate. Since we need to determine the stress state in the specimen Therefore, an unknown but spatially constant displace- with a crack of length a, finite element calculations are per ment u] is assumed along the upper boundary y=8. The formed. The beam is replaced by a two-dimensional plane finite element program system ABAQUS(htp!∥ strain model covering the area0≤x≤ h and y≥0.Note www.hks.com)isengagedforthecomputationusingthattheplanestrainmodelcanbetreatedasaplanestress eight-node 3D solid elements model by replacing Youngs modulus E by E= E/(1-v) It is well known that the behavior of a crack is deter- Only x-displacements ux are allowed at y=0. The speci mined not only by the stresses perpendicular to the crack men is fixed at the point P, in the x-direction; the load F plane y=0. The in-plane and out-of-plane constraints also is applied at point P2(see Fig. 1). The mesh consists of play a role, i.e. the stresses in x-and z-directions. The mate- eight-node plane strain elements rial can freely move in x-direction, so that no correspond To model a realistic stress state of the fracture mechan- g residual stresses will appear In the z-direction residual ics specimens, the following procedure is applied.First, the stresses o, Ar will appear which infiuence the strain energy uncracked and unloaded specimen is subjected to a thermal loading by a temperature difference AT=-1140C to cal The materials of the A- and AZ-layers are modeled as culate the thermal residual stresses. Then in the unloaded linear-elastic. The corresponding material properties are specimen a crack of length a is introduced by a node release taken from Table 1. The longitudinal residual stresses o, Ar technique. Subsequently, the specimen is loaded by pre in the symmetry plane z=0 and in the side-surface plane scribing the load Fat the load application point. The final z= B/2 are plotted between 0<x<h/2 in Fig. 2. Fig. 3 stress and strain distribution within the specimen is used shows the variation of o, Ar from the midsection to the for the evaluation of the crack driving force, which is side-surface for sections in the middle of the various described beloy laminae In addition to the 3D computation, two simple 2D com- 4. 4. Calculation of the crack driving force putations were also performed. The first one used a plane strain model covering the area0≤x≤h,0≤y≤ s with The study of stress intensity factors, as well as of their assumes no displacement in the 2-direction, u,=0. To mechanics research in composite materials (see e. g Simha, Kolednik et al. [13, 26-28]. References to the open formulations can be taken from these extensive papers Specifically, Section 3 of [28] provides the corresponding equations which are reshaped below in the specific form a composites with constant material properties within each lamina 。-atz=1.25of3 D model The concept of configurational forces considers a mate- rial inhomogeneity as an additional defect in the material (besides the crack) which induces an additional contribu- tion to the crack driving force. This contribution has been plane stress b called the material inhomogeneity term Cinh. The thermo- dynamic force at the crack tip, denominated as the local near-tip crack driving force Jtip, is the sum of the nominally x[ mm] applied far-field crack driving force Far and the material inh [28]: Fig. 2. Thermal residual stresses o,. AT along the x-direction at 2=0 and 25 mm for a plane stress, plane strain and three-dimensional (3D)model Jtip=Far CiThe calculation procedure to find the residual stress state in laminate specimens has been heavily investigated (see e.g. [19,20]). We use a three-dimensional (3D) model of the specimen, consisting of a slice of length d in the y￾direction and covering the area 0 6 x 6 h/2, 0 6 z 6 B/2. Symmetry conditions are applied along z = 0 and x = h/ 2, and the y-displacements are fixed on the bottom of the layer. Only in the end regions far from the crack plane will the residual stress state depend on the y-coordinate. Since the specimen is very long in y-direction, the residual stres￾ses can be assumed to be independent of the y-coordinate. Therefore, an unknown but spatially constant displace￾ment uy is assumed along the upper boundary y = d. The finite element program system ABAQUS (http:// www.hks.com) is engaged for the computation, using eight-node 3D solid elements. It is well known that the behavior of a crack is deter￾mined not only by the stresses perpendicular to the crack plane y = 0. The in-plane and out-of-plane constraints also play a role, i.e. the stresses in x- and z-directions. The mate￾rial can freely move in x-direction, so that no correspond￾ing residual stresses will appear. In the z-direction residual stresses rz,DT will appear which influence the strain energy density (see Appendix 1). The materials of the A- and AZ-layers are modeled as linear-elastic. The corresponding material properties are taken from Table 1. The longitudinal residual stresses ry,DT in the symmetry plane z = 0 and in the side-surface plane z = B/2 are plotted between 0 6 x 6 h/2 in Fig. 2. Fig. 3 shows the variation of ry,DT from the midsection to the side-surface for sections in the middle of the various laminae. In addition to the 3D computation, two simple 2D com￾putations were also performed. The first one used a plane strain model covering the area 0 6 x 6 h, 0 6 y 6 d with unit thickness in the z-direction. The plane strain model assumes no displacement in the z-direction, uz ” 0. To avoid any stresses due to the global shrinkage of the spec￾imen, we set a A ¼ 0 as the substitute CTE in the A-material and a AZ ¼ aAZ aA ¼ 0:6  106 K1 as the substitute CTE in the AZ-material. The second 2D computation used the corresponding plane stress model. 4.3. Beam bending As outlined above, four-point-bend tests are performed on rather slender beams. The classical beam-bending the￾ory could be used to evaluate the stress state in the uncracked composite beam; for details, see e.g. [21,22]. Since we need to determine the stress state in the specimen with a crack of length a, finite element calculations are per￾formed. The beam is replaced by a two-dimensional plane strain model covering the area 0 6 x 6 h and y P 0. Note that the plane strain model can be treated as a plane stress model by replacing Young’s modulus E by E* = E/(1 m 2 ). Only x-displacements ux are allowed at y = 0. The speci￾men is fixed at the point P1 in the x-direction; the load F is applied at point P2 (see Fig. 1). The mesh consists of eight-node plane strain elements. To model a realistic stress state of the fracture mechan￾ics specimens, the following procedure is applied. First, the uncracked and unloaded specimen is subjected to a thermal loading by a temperature difference DT = 1140 C to cal￾culate the thermal residual stresses. Then in the unloaded specimen a crack of length a is introduced by a node release technique. Subsequently, the specimen is loaded by pre￾scribing the load F at the load application point. The final stress and strain distribution within the specimen is used for the evaluation of the crack driving force, which is described below. 4.4. Calculation of the crack driving force The study of stress intensity factors, as well as of their relevance for crack growth, has been a topic of fracture mechanics research in composite materials (see e.g. [1,23–25]). In the current investigation the concept of con- figurational forces is used. Here we refer to the works by Simha, Kolednik et al. [13,26–28]. References to the open literature with respect to this concept and other related formulations can be taken from these extensive papers. Specifically, Section 3 of [28] provides the corresponding equations which are reshaped below in the specific form for composites with constant material properties within each lamina. The concept of configurational forces considers a mate￾rial inhomogeneity as an additional defect in the material (besides the crack) which induces an additional contribu￾tion to the crack driving force. This contribution has been called the material inhomogeneity term Cinh. The thermo￾dynamic force at the crack tip, denominated as the local, near-tip crack driving force Jtip, is the sum of the nominally applied far-field crack driving force Jfar and the material inhomogeneity term Cinh [28]: Jtip ¼ Jfar þ Cinh: ð3Þ Fig. 2. Thermal residual stresses ry,DT along the x-direction at z = 0 and 1.25 mm for a plane stress, plane strain and three-dimensional (3D) model. 412 C.R. Chen et al. / Acta Materialia 55 (2007) 409–421