Continuum Mechanics and Theory of Materials by Peter Haupt

By Peter Haupt

This treatise makes an attempt to painting the information and normal ideas of the idea of fabrics in the framework of phenomenological continuum mechanics. it's a well-written mathematical advent to classical continuum mechanics and offers with suggestions resembling elasticity, plasticity, viscoelasticity and viscoplasticity in nonlinear fabrics. the purpose of a common thought of fabric behaviour is to supply a categorised diversity of chances from which a consumer can decide upon the constitutive version that applies most sensible. The e-book might be useful to graduate scholars of fabrics technological know-how in engineering and in physics. the recent variation contains extra analytical tools within the classical concept of viscoelasticity. This ends up in a brand new conception of finite linear viscoelasticity of incompressible isotropic fabrics. Anisotropic viscoplasticity is totally reformulated and prolonged to a common constitutive conception that covers crystal plasticity as a different case.

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1. 3 x3 - surface Figure 1. 4: Curvilinear coordinates In connection with the identifications ~ +-t x +-t P, the spatial coordinates (xl, x 2, x 3 ) can be interpreted as curvilinear coordinates in the Euclidean space 1E3 . 11) (Fig. 1. 4) with the property gi. 9 k = 8~. ) 3 Questions arising in the context of a reference frame are to be investigated in Chap. 4. , = -=-=- , 8y. 8y. 13) g. =g .. g =~-8. l = 8 ~i 8~k 8~i 8~k 1 1 1 1 which are reciprocal to each other in the sense of gil gZk = 8~.

X o is a reference point (for example, the centre of mass) and X o (t) a vector-valued function of time, which represents the motion of the material reference point. <> The definition ascertains the fact that a rigid body motion consists of a rotation Q(t) about a point X o and a translation xo(t). The rotation Q(t) depends merely upon time and is also - as can be proved - independent of the choice of point X o . 71) F(X, t) = Q(t) , whereas the corresponding Cauchy-Green tensors are unit tensors, As far as continuum mechanics is concerned, rigid body motions are lacking in interest: attention is drawn to the deformations that deviate from rigid body motions.

1 Kinematics 36 Differentiating the identity 4>(X) = CP(xR(X, t), t) with respect to X and observing the chain rule, it foHows that Grad4>(X) = FTgradcp(x, t) or gradcp(x, t) = F T - 1 Grad4>(X) . 81) Both gradient vectors are normal on the material surfaces 4>(X) = C and cp(x, t) = C. Their absolute values are the normal derivatives of the family parameter C. If = Grad4>(X) No IGrad4>(x)1 is the normal unit vector on the material surface 4>(X) = C, then the derivative in the direction of No is given by dC dN = IGrad4>(X)1 .

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