With the judgment a complete and eternal separation takes place between the ungodly and the godly. The former are delivered over to eternal damnation, a condition which in Scripture is also called eternal death (“eternal death, eternal damnation, is a condition most miserable through the aggregation of multitudinous evils, and to last forever.” HOLL., (978).)  The Holy Scriptures say of them that they are in hell (ᾅδης, שְׁאֹל, a place of torment,  in which they suffer, according to the degree of their ungodliness,  in bodily and spiritual pains,  for their sins, eternally.*
On these four points, Schmid assembles the following citations from the leading Lutheran Theologians-
 HOLLAZIUS. (979): “Death eternal is the separation of the unbelieving soul from the beatific sight of God and eternal enjoyment.” QUEN. (I, 565): “Death eternal or damnation is that most unhappy state in which, from the just judgment of God, men who remain unbelieving to the end, being excluded from the beatific sight of God, and associated in the infernal prison with devils, will be tortured eternally (in soul, immediately after its departure from the body, and in both parts of their composite being, at length when sentence has been passed at the final judgment) with the most severe and ineffable torments, to the praise of the divine truth, and the glory and exultation of the godly.” HOLL. (979): “Death eternal is named likewise the second death, Rev. 2:11; 20:6, because it occasions the forfeiture of that other life which man was able to attain when the present life had been completed; besides it is called corruption, Jude 12; Matt. 7:13; everlasting destruction, 2 Thess. 1:9; not as though eternal death were an annihilation of substance, but because it is the forfeiture or the want of happiness, and shame, and everlasting contempt, Dan. 12:2, since there is nothing more contemptible, in the eyes of God, the angels, and the blessed, than the damned, for they will be an abhorring unto all flesh, Is. 66:24; everlasting punishment, Matt. 25:46; tribulation and anguish, Rom. 2:9.” QUEN. (I, 551) presents scriptural proofs from Ps. 49:15, 20; Is. 66:24; Dan. 12:2; Zach. 9:11; Matt. 3:10; Luke 3:17: Matt. 5:22; 25:46; 8:12; Luke 13:27, 28; Matt. 10:28; 13:40, 42; 22:13; 25:41, 46; Luke 16:23. (GRH. (XX, 169) adds: “Reasons and arguments sought (1) from the condition of divine justice, 2 Thess. 1:6; (2) from the curse of the divine law, Deut. 27:26; Matt. 19:16; Rom. 4:15; 1 Thess. 1:10; (3) from the deformity and confusion of sin, Rom. 6:23; (4) from the witness of one’s own conscience; (5) from the tasting of the pains of hell, 2 Sam. 22:6; Ps. 18:5; 30:3; 49:15; 86:13; 88:4; 116:3; (6) from the article concerning the descent of Christ ad inferos; (7) from the resurrection of the wicked; (8) from the administration of the final judgment.”)
 GERHARD. (XX, 175); “The name hell can be received in a twofold manner: (1) for eternal death; (2) for that place (ποῦ), in which they suffer, and to eternity will sustain that most miserable condition and those ineffable tortures. By reason of the former signification, the devils are said to carry about with them their own hell wherever they wander. By reason of the latter, it is said that on the day of judgment they will be cast into hell, and be confined there. In the former signification, the name hell is received internally and formally; in the latter, externally, objectively, and locally, the term used in the article being received in a general sense, according to the language of Scripture, Luke 16:28. What hell is, in the former signification, cannot be understood, more correctly than by collecting and distributing, into certain classes, the descriptions by which, in the Holy Scriptures, the extreme misery of the damned is prefigured. But what hell is, in the latter signification, pertains to a question that is extremely difficult and obscure. Some altogether reject the latter signification, and think that hell should not be defined except by the sense of divine wrath, and of the eternal curse and horror of conscience. But there is no apparent reason why a certain place (ποῦ), in which the damned suffer their punishments, should be denied.” HOLL. (984): “It is certain that the infernal prison is in a real locality (Luke 16:28; 1 Pet. 3:19), separate from the abode of the blessed (Rev. 22:15; Luke 16:23). It is also probable that the same is outside of this habitable world (2 Pet. 3:10: John 12:31; Matt. 8:12); but where this place definitely is, is unknown to men during the present life.”
 HOLLAZIUS. (990): “The punishments of hell differ in degree, according to the quality and measure of sins, Matt. 11:24. Luke 12:47; Matt. 23:15.”
 HAFFENREFFER. (691): “They are the most exquisite pains of soul and body (for both had sinned), arising from the fear and sense of the most just wrath and vengeance of God against sins, the most sad consciousness of which they carry about with them, the baseness of which is manifest, and of which, likewise, no remission afterwards, and, therefore, no mitigation or end can be hoped for. Whence, in misery, they will execrate, with horrible lamentation and wailing, their former impiety, by which they carelessly neglected the admonitions of their brethren and all the means of attaining salvation; but in vain. For in perpetual anguish, with dreadful trembling, in shame, confusion, and ignominy, in inextinguishable fire, in weeping and gnashing of teeth, amidst that which is eternal and terrible, torn away from the grace and favor of God, they must quake among devils, and will be tortured without end to eternity. These future torments of the damned far surpass all the penetration of the human mind, so that we are not sufficient to ever comprehend in thought their greatness; therefore, what they will be, or of what nature, cannot be at all expressed in words. Scripture, nevertheless, in order to show that these tortures are the greatest and most exquisite, likens them to those things by which, in this life, pain both of soul and body is accustomed to be excited. For this reason they are compared now to the gnashing of teeth, now to worms, now to the most sorrowful darkness, and whatever other matters of sadness and of the most complete pain can be mentioned, Is. 66:24; Matt. 5:22; 8:12; Rev. 19:20.” QUEN. (I, 562): “The form (of eternal death) is the entire mass of evils intended for the damned. These are partly privative, and partly positive. The privative are: (1) forfeiture of the beatific sight of God; Matt. 25:41; 22:13; 8:12; (2) separation from the society of all the good, Matt. 8:11, 12; 22:13; Luke 16:23, 26; (3) exclusion from heavenly light, rest, and happiness, Matt. 8:12; 22:13; 25:30; 2 Thess. 1:6, 8, 9; (4) entire denial of pity, divine as well as human, Prov. 1:26; Ps. 52:6, 7; 58:10; Luke 16:24, 25; (5) despair of every kind, Rev. 6:16, 17. Of the positive, some are internal, and others external. The internal are those which the damned experience within themselves, viz., the inexplicable pains and tortures of soul, Ps. 18:4, 5; Is. 66:24; Mark 9:44, 46, 48.” (HOLL. (982): “Their intellect will recognize God as the most just judge and the most severe avenger of sins, Ps. 139:7; 2 Thess. 1:9; Wis. 5:3. Their will will be tortured by hatred to God, the greatest sorrow, and raging impatience.”) “The external are those most sorrowful evils, outside of themselves, that they deeply feel, namely, association with devils, Matt. 25:41; a most foul dwelling-place, Matt. 25:30; and most painful burning without being consumed, Luke 16:23, 24; Rev. 14:10, 11; 20:10.” HOLL. (983) answers the question concerning the nature of the fire: “The bodies of the damned will be tortured in infernal fire, properly so called, and, therefore, material. For the sentence of the judge announces a fire, Matt. 25:41, from which smoke ascends, Rev. 14:10, whose flames burn, Luke 16:24. That, therefore, to which the Holy Ghost has ascribed the name, the properties, and the effects of true fire, is not metaphorical, but fire properly so called. But to the infernal fire, etc. Therefore.—But this will not be the element of fire, but that which is altogether peculiar. Ordinary fire burns only bodies; the infernal fire will act also upon souls. The former ceases when fuel fails; the latter does not stand in need of nourishment properly so called. But to desire to explain the nature of infernal fire more explicitly, is a matter of curiosity rather than of profit.”
*The Doctrinal Theology of the Evangelical Lutheran Church, Verified from the Original Sources. (C. A. Hay & H. E. Jacobs, Trans.) (Second English Edition, Revised according to the Sixth German Edition., pp. 657–660).