D. Leguillon et al /Journal of the Mechanics and Physics of Solids 48(2000)2137-2161 2139 material 2 cracklE material 1 Fig 1. The notched bimaterial specimen under four-point bending continuous across the interface. The isotropy assumption is not essential and in-axis orthotropy can be considered as well. There is a transverse crack lying in one of the components, denoted material 1(Youngs modulus E1, Poissons ratio vi), and perpendicular to the interface. Material 2(E2, v2) is unbroken. The crack length is L and its tip is located at a distance from the interface (L+/=e/2). This distance is assumed to be small with respect to a characteristic dimension of the specimen(th thickness e for instance, k<<e) and to the crack length (<<L). We set ee, e is a small dimensionless parameter(the dimensionless ligament width). This specimen is submitted to a given loading geometry (a four-point bending for instance as illus- trated on Fig. 1) In the following the thickness e is set to unity. This is a way to define dimen sionless lengths; each physical length and displacement is divided by e. The corre- sponding domain is denoted n2 Numerical results are presented essentially for three different contrasts between the elastic properties of the components. A strong contrast characterized either by EE2=0. 1 or EE2=10, and the absence of contrast E/ E2=l(homogeneous specimen). In each case Poissons ratio is taken to be identical (V=V2=0. 3). Special situations are considered for comparison with the He and Hutchinson(HH)criterion to match with their data. In that comparison, Dundurs coefficients(a, B)are used to characterize the mismatch between two isotropic materials(Dundurs, 1969),a<0 corresponding to a material I stiffer than the other and reciprocally 3. Far and near fields- matched asymptotics Details of the matched asymptotics procedure are presented in Leguillon(1993): the main lines are recalled here. The displacement solution UE to the problem can be described with the help of two expansions U(x1x2)=U(x1-x2)+f(e)U(x1x2)+.where f(e)0 as e-0 The first term U(xI, x2) is the solution to the so-called"unperturbed problem""iD. Leguillon et al. / Journal of the Mechanics and Physics of Solids 48 (2000) 2137–2161 2139 Fig. 1. The notched bimaterial specimen under four-point bending. continuous across the interface. The isotropy assumption is not essential and in-axis orthotropy can be considered as well. There is a transverse crack lying in one of the components, denoted material 1 (Young’s modulus E1, Poisson’s ratio n1), and perpendicular to the interface. Material 2 (E2, n2) is unbroken. The crack length is L and its tip is located at a distance l from the interface (L+l=e/2). This distance is assumed to be small with respect to a characteristic dimension of the specimen (the thickness e for instance, l,,e) and to the crack length (l,,L). We set l=ee, e is a small dimensionless parameter (the dimensionless ligament width). This specimen is submitted to a given loading geometry (a four-point bending for instance as illus￾trated on Fig. 1). In the following the thickness e is set to unity. This is a way to define dimen￾sionless lengths; each physical length and displacement is divided by e. The corre￾sponding domain is denoted Ve . Numerical results are presented essentially for three different contrasts between the elastic properties of the components. A strong contrast characterized either by E1/E2=0.1 or E1/E2=10, and the absence of contrast E1/E2=1 (homogeneous specimen). In each case Poisson’s ratio is taken to be identical (n1=n2=0.3). Special situations are considered for comparison with the He and Hutchinson (HH) criterion to match with their data. In that comparison, Dundurs coefficients (a, b) are used to characterize the mismatch between two isotropic materials (Dundurs, 1969), a,0 corresponding to a material 1 stiffer than the other and reciprocally. 3. Far and near fields — matched asymptotics Details of the matched asymptotics procedure are presented in Leguillon (1993); the main lines are recalled here. The displacement solution Ue to the problem can be described with the help of two expansions Ue (x1,x2)5U0 (x1,x2)1f1(e)U1 (x1,x2)1… where f1(e)→0 as e→0. (1) The first term U0 (x1, x2) is the solution to the so-called “unperturbed problem” in