Element 3d v2 deform direction free#
76, 2261–2266 (1994)Īlipour, M.M., Shariyat, M.: Analytical layerwise free vibration analysis of circular/annular composite sandwich plates with auxetic cores.
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Furthermore, it was found that the cell aspect ratio has an important role in the mechanical properties of WAW auxetic structures.Īlderson, K.L., Pickles, A.P., Neale, P.J., Evans, K.E.: Auxetic polyethylene: The effect of a negative Poisson’s ratio on hardness. The thickness of cell walls does not alter NPR, yet relatively increases the cell stiffness. The obtained results revealed that with the increase in the IRE angle, NPR increases in the vertical direction and decreases in two other horizontal directions it also increases the cell stiffness in three perpendicular directions. Subsequently, the effects of parameters, such as the initial re-entrant (IRE) angle, the cell thickness, and the auxetic cell aspect ratio on the mechanical properties were investigated. Fairly good correlations were found between the present analytical results and the results of FE simulations. For validating the proposed analytical model, the 3D finite element (FE) simulations were used. To this end, the principle of virtual work along with the concept of lumped plasticity (plastic hinges) were utilized. Due to the periodic pattern of the auxetic structure, a unit cell was chosen as a representative volume element (RVE) and its mechanical properties were assessed by developing analytical formulations in the elastic region. = v1.rotate(angle=a, axis=vector(x,y,z)).The present paper was conducted to calculate Young's modulus and Negative Poisson's ratio (NPR) of a warp and woof (WAW) 3D auxetic structure in three perpendicular directions using an analytical method under uniaxial loading and considering the elastic–plastic behavior of materials. (0,0,1), for a rotation in the xy plane around the z axis. V2 = rotate(v1, angle=a, axis=vector(x,y,z)) There is a function for rotating a vector: Two vectors are normalized, the dot product gives the cosine of the angleīetween the vectors, which is often useful. Which is an ordinary number equal to mag(A)*mag(B)*cos(diff_angle(A,B)). The magnitude of this vector is equal mag(A)*mag(B)*sin(diff_angle(A,B)).ĭot(A,B) or A.dot(B) gives the dot product of two vectors, Hand bend from A toward B, the thumb points in the direction In a direction defined by the right-hand rule: if the fingers of the right Ĭross(A,B) or A.cross(B) gives the cross product of two vectors, a vector perpendicular to the plane defined by A and B, Magnitude, the difference of the angles is calculated to be zero. For convenience, if either of the vectors has zero To calculate the angle between two vectors (the "difference" V2.hat = v1 # changes the direction of v2 to that of v1 You can change the direction of a vector without changing its magnitude:
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Norm(A) # A/|A|, normalized magnitude of 1 You can reset the magnitude to 1 with norm(): V2.mag2 = 2.7 # sets squared magnitude of v2 to V2.mag = 5 # sets magnitude to 5 no change in direction
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It is possible to reset the magnitude or the Vector.random() produces a vector each of whose components is a random number in the range -1 to +1 Proj(A,B) = A.proj(B) = dot(A,norm(B))*norm(B), the vector projection of A along BĬomp(A,B) = A.comp(B) = dot(A,norm(B)), the scalar projection of A along BĪ.equals(B) is True if A and B have the same components (which means that they have the same magnitude and the same direction). Norm(A) = A.norm() = A/|A|, a unit vector in the direction of the vectorĪ/|A|, a unit vector in the direction of the vector an alternative to A.norm(), based on the fact that unit vectors are customarily written in the form ĉ, with a "hat" over the vectorįor convenience, norm(vec(0,0,0)) or vec(0,0,0).hat is calculated to be vec(0,0,0).ĭot(A,B) = A.dot(B) = A dot B, the scalar dot product between two vectorsĬross(A,B) = A.cross(B), the vector cross product between two vectorsĭiff_angle(A,B) = A.diff_angle(B), the angle between two vectors, in radians Mag2(A) = A.mag2 = |A|*|A|, the vector's magnitude squared Mag(A) = A.mag = |A|, the magnitude of a vector The following functions are available for working with vectors: This is a convenient way to make a separate copy of a vector.
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It is okay to make a vector from a vector: vector(v2) is still vector(10,20,30). You can refer to individual components of a vector: Vectors can be added or subtracted from each other, or multiplied by an This creates a 3D vector object with the given components x, y, and z. The vector object is not a displayable object but isĪ powerful aid to 3D computations.