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Fig. 3. Geometric interpretation of the the polynomial invariants: (a) K, and (b) K3. J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-445 with y, > 0. It should be noted that K; /E! 9 and K; /E! ? are coupled volumetric—isochoric terms. In Proof (4) we have seen that powers of tr[CM] are polyconvex; these functions represent powers of the stretch in the oreferred direction. As such it seems to be elemental that stretches in the isotropy plane also make sense as specific ansatz functions. For the construction of such further mixed terms we use the redundant structural ensor (3.25). We obtain the polynomial invariant

Figure 3 Geometric interpretation of the the polynomial invariants: (a) K, and (b) K3. J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-445 with y, > 0. It should be noted that K; /E! 9 and K; /E! ? are coupled volumetric—isochoric terms. In Proof (4) we have seen that powers of tr[CM] are polyconvex; these functions represent powers of the stretch in the oreferred direction. As such it seems to be elemental that stretches in the isotropy plane also make sense as specific ansatz functions. For the construction of such further mixed terms we use the redundant structural ensor (3.25). We obtain the polynomial invariant

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Fig. 1. Preferred directions ‘a and ‘a in the neighborhoods Pz C Zand Py C ¥ of the points X € Zand x € Y, respectively J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-445
Fig. 1. Preferred directions ‘a and ‘a in the neighborhoods Pz C Zand Py C ¥ of the points X € Zand x € Y, respectively J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-445
Fig. 2. Physical interpretation of the polyconvex invariant function /, /B! - J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-443 a ee ee a ree For the physical interpretation of the isotropic invariant function J; /E! > we consider a cylinder of unit length and unit diameter (see Fig. 2a). A deformation described by the deformation gradient F = 1+ (A- l)a®a, with llal| = = I, leads to a final configuration as outlined by the outer cylinder in Fig. 2b. The isochoric part Gr F, ie. the term F := (det F) ‘°F, and thus the quadratic function in the remainder h/B? = trC = ||F||’, controls only the isochoric part of the deformation. In the considered example only the shaded volume in Fig. 2b is affected by this invariant. For the anisotropic case we will discuss this from a different point of view. ee pe as ee, eee ae ee - ge Tap ae ee eR. we
Fig. 2. Physical interpretation of the polyconvex invariant function /, /B! - J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-443 a ee ee a ree For the physical interpretation of the isotropic invariant function J; /E! > we consider a cylinder of unit length and unit diameter (see Fig. 2a). A deformation described by the deformation gradient F = 1+ (A- l)a®a, with llal| = = I, leads to a final configuration as outlined by the outer cylinder in Fig. 2b. The isochoric part Gr F, ie. the term F := (det F) ‘°F, and thus the quadratic function in the remainder h/B? = trC = ||F||’, controls only the isochoric part of the deformation. In the considered example only the shaded volume in Fig. 2b is affected by this invariant. For the anisotropic case we will discuss this from a different point of view. ee pe as ee, eee ae ee - ge Tap ae ee eR. we
For the proof of this statement see the Appendix, Lemma C.5 and Corollary C.6. The treatment of the isotropic case has been taken from Hartmann and Neff (2002). For the explicit derivations of the stress functions and the moduli we choose from (3.34) the terms On the natural domain of definition det F > 0 the given functions are convex in the variable det F. The terms in (3.33) are each polyconvex and lead to a stress free reference configuration. Furthermore, the following isochoric terms are polyconvex and stress free in the natural state:
For the proof of this statement see the Appendix, Lemma C.5 and Corollary C.6. The treatment of the isotropic case has been taken from Hartmann and Neff (2002). For the explicit derivations of the stress functions and the moduli we choose from (3.34) the terms On the natural domain of definition det F > 0 the given functions are convex in the variable det F. The terms in (3.33) are each polyconvex and lead to a stress free reference configuration. Furthermore, the following isochoric terms are polyconvex and stress free in the natural state:
Up to now we can conclude that we cannot use the invariant Js; and the polynomial invariant [Js as single terms for the construction of a free energy term. To take into account quadratic expressions of these terms within the ansatz functions we remember that Cof[C] is a quadratic function in the right Cauchy— Green tensor. Furthermore, it seems reasonable from a physical point of view to construct a polynomial mixed invariant, which reflects the deformation of a preferred area element of an infinitesimal volume of the considered body. With this geometric motivation we start with the characteristic polynomial of the matrix C (see the Cayley-Hamilton Theorem A.8). Multiplication of the characteristic polynomial with C~'M yields with CofC = AdjC
Up to now we can conclude that we cannot use the invariant Js; and the polynomial invariant [Js as single terms for the construction of a free energy term. To take into account quadratic expressions of these terms within the ansatz functions we remember that Cof[C] is a quadratic function in the right Cauchy— Green tensor. Furthermore, it seems reasonable from a physical point of view to construct a polynomial mixed invariant, which reflects the deformation of a preferred area element of an infinitesimal volume of the considered body. With this geometric motivation we start with the characteristic polynomial of the matrix C (see the Cayley-Hamilton Theorem A.8). Multiplication of the characteristic polynomial with C~'M yields with CofC = AdjC
Fig. 4. Interpretation of the polynomial invariants Ky /Is ® and K; if Ty ® for an assumed isochoric deformation process. Picture (a) shows the isochoric deformation of the uniaxial stretched cylinder of Fig. 2 and Picture (b) represents an energetic weighting of the parts controlled by K,/J,/? and K3/J;’°. J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-445
Fig. 4. Interpretation of the polynomial invariants Ky /Is ® and K; if Ty ® for an assumed isochoric deformation process. Picture (a) shows the isochoric deformation of the uniaxial stretched cylinder of Fig. 2 and Picture (b) represents an energetic weighting of the parts controlled by K,/J,/? and K3/J;’°. J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-445
J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-445 3.3. Spatial formulation For isotropic material response the Kirchhoff stresses t can be derived directly by the derivative of the free energy function with respect to the Finger tensor b := FF' (see e.g. Miehe, 1994). For the anisotropic case the Kirchhoff stresses and associated spatial moduli c can be computed via a push-forward operation of the second Piola—Kirchhoff stresses S and the associated moduli C, i.e.
J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-445 3.3. Spatial formulation For isotropic material response the Kirchhoff stresses t can be derived directly by the derivative of the free energy function with respect to the Finger tensor b := FF' (see e.g. Miehe, 1994). For the anisotropic case the Kirchhoff stresses and associated spatial moduli c can be computed via a push-forward operation of the second Piola—Kirchhoff stresses S and the associated moduli C, i.e.
which is the index representation of the Finger tensor 5. In an analogous way we get the derivatives of the other invariants. In direct notation we obtain the expressions
which is the index representation of the Finger tensor 5. In an analogous way we get the derivatives of the other invariants. In direct notation we obtain the expressions
With the X3-axis as the axis of symmetry we have a = (0,0, 1)", M = diag(0,0,1) and with the same index arrangement as in (4.68) we obtain from (4.69) the linearized moduli J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-445
With the X3-axis as the axis of symmetry we have a = (0,0, 1)", M = diag(0,0,1) and with the same index arrangement as in (4.68) we obtain from (4.69) the linearized moduli J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-445
Fig. 5. Tapered Cantilever. (a) System and boundary conditions; (b) discretization with 20 x 20 x 5 = 2000 eight-noded brick ele ments. In this example we consider a tapered cantilever clamped on the left hand side and subjected to a shearing deformation AF in vertical direction with ||F|| = 1 on the right hand side. Here 2 denotes the load parameter. The system and the boundary conditions are depicted in Fig. 5a and b shows the reference configuration, discretized with 20 x 20 x 5 eight-noded standard displacement elements in horizontal, vertical and thickness direction. The thickness of the specimen is set to 1.
Fig. 5. Tapered Cantilever. (a) System and boundary conditions; (b) discretization with 20 x 20 x 5 = 2000 eight-noded brick ele ments. In this example we consider a tapered cantilever clamped on the left hand side and subjected to a shearing deformation AF in vertical direction with ||F|| = 1 on the right hand side. Here 2 denotes the load parameter. The system and the boundary conditions are depicted in Fig. 5a and b shows the reference configuration, discretized with 20 x 20 x 5 eight-noded standard displacement elements in horizontal, vertical and thickness direction. The thickness of the specimen is set to 1.
The chosen material parameters of the polyconvex free energy function in the invariant setting are for the vanishing stresses in the reference configuration. The coefficients of the linearized moduli (4.70) are given by
The chosen material parameters of the polyconvex free energy function in the invariant setting are for the vanishing stresses in the reference configuration. The coefficients of the linearized moduli (4.70) are given by
Fig. 6. Tapered Cantilever. Deformed configurations for selected load levels: (a) at A = 130, (b) at A = 270 and (c) at 2 = 400. J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-44: the load parameter is increased by increments AA = 1 until a final value of 2 = 400 is reached. Fig. 6a—c depicts the deformed structures of the beam at 2 = 130, 2 = 270 and A = 400, respectively. In contrast to an isotropic material the anisotropic one leads to a salient out-of-plane bending deformation. This enormous out-of-plane bending is of course initiated by the orientation of the preferred direction.
Fig. 6. Tapered Cantilever. Deformed configurations for selected load levels: (a) at A = 130, (b) at A = 270 and (c) at 2 = 400. J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-44: the load parameter is increased by increments AA = 1 until a final value of 2 = 400 is reached. Fig. 6a—c depicts the deformed structures of the beam at 2 = 130, 2 = 270 and A = 400, respectively. In contrast to an isotropic material the anisotropic one leads to a salient out-of-plane bending deformation. This enormous out-of-plane bending is of course initiated by the orientation of the preferred direction.
Fig. 7. Geometric properties of the rectangular plate with centered hole. of the plate. The dimensions of the specimen are given in Fig. 7 and the thickness 1s set to ¢ = I. The system is subjected to tension in horizontal direction. The left and right boundaries are pulled up to a final length of the specimen of 42 units. Furthermore, we fixed the vertical degrees of freedom for the outer boundary of the plate. The analysis is performed under plane strain conditions and the specimen is discretized with 1800 four-noded standard displacement elements. The three different orientations of the preferred direction are
Fig. 7. Geometric properties of the rectangular plate with centered hole. of the plate. The dimensions of the specimen are given in Fig. 7 and the thickness 1s set to ¢ = I. The system is subjected to tension in horizontal direction. The left and right boundaries are pulled up to a final length of the specimen of 42 units. Furthermore, we fixed the vertical degrees of freedom for the outer boundary of the plate. The analysis is performed under plane strain conditions and the specimen is discretized with 1800 four-noded standard displacement elements. The three different orientations of the preferred direction are
Fig. 8. Tension of a rectangular plate with centered hole. Deformed configurations for material parameter set MS 1: (a) a, (b) a) and (c) a3, and material parameter set MS 2: (d) a), (e) a and (f) a3, respectively. For the simulations we choose for the material parameters in (7.100) the values
Fig. 8. Tension of a rectangular plate with centered hole. Deformed configurations for material parameter set MS 1: (a) a, (b) a) and (c) a3, and material parameter set MS 2: (d) a), (e) a and (f) a3, respectively. For the simulations we choose for the material parameters in (7.100) the values
and we may use the same ideas as in the proof to Lemma C.1 to conclude that the term is polyconvex.
and we may use the same ideas as in the proof to Lemma C.1 to conclude that the term is polyconvex.
One might be tempted to use some other ansatz terms in order to construct polyconvex strain energies. However, we have e.g. Since this last expression coincides with
One might be tempted to use some other ansatz terms in order to construct polyconvex strain energies. However, we have e.g. Since this last expression coincides with
If we choose 1/ di = 3 it turns out that A is not convex in t, hence (Theorem 3.5) W is not elliptic. We remark that the non-ellipticity is mainly due to the fact that J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-445
If we choose 1/ di = 3 it turns out that A is not convex in t, hence (Theorem 3.5) W is not elliptic. We remark that the non-ellipticity is mainly due to the fact that J. Schroder, P. Neff / International Journal of Solids and Structures 40 (2003) 401-445
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EngineeringLinear ElasticityInvariant TheoryFinite StrainConstitutive Equation
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