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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**Extra info for Continuum Mechanics and Theory of Materials**

**Example text**

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 .